AI App Use In Hindi

AI App Use In Hindi — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Situated approach (artificial intelligence)

  In artificial intelligence research, the situated approach builds agents that are designed to behave effectively successfully in their environment. This requires designing AI "from the bottom-up" by focussing on the basic perceptual and motor skills required to survive. The situated approach gives a much lower priority to abstract reasoning or problem-solving skills. The approach was originally proposed as an alternative to traditional approaches (that is, approaches popular before 1985 or so). After several decades, classical AI technologies started to face intractable issues (e.g. combinatorial explosion) when confronted with real-world modeling problems. All approaches to address these issues focus on modeling intelligences situated in an environment. They have become known as the situated approach to AI. == Emergence of a concept == === From traditional AI to Nouvelle AI === During the late 1980s, the approach now known as Nouvelle AI (Nouvelle means new in French) was pioneered at the MIT Artificial Intelligence Laboratory by Rodney Brooks. As opposed to classical or traditional artificial intelligence, Nouvelle AI purposely avoided the traditional goal of modeling human-level performance, but rather tries to create systems with intelligence at the level of insects, closer to real-world robots. But eventually, at least at MIT new AI did lead to an attempt for humanoid AI in the Cog Project. === From Nouvelle AI to behavior-based and situated AI === The conceptual shift introduced by nouvelle AI flourished in the robotics area, given way to behavior-based robotics (BBR), a methodology for developing AI based on a modular decomposition of intelligence. It was made famous by Rodney Brooks: his subsumption architecture was one of the earliest attempts to describe a mechanism for developing BBAI. It is extremely popular in robotics and to a lesser extent to implement intelligent virtual agents because it allows the successful creation of real-time dynamic systems that can run in complex environments. For example, it underlies the intelligence of the Sony Aibo and many RoboCup robot teams. Realizing that in fact all these approaches were aiming at building not an abstract intelligence, but rather an intelligence situated in a given environment, they have come to be known as the situated approach. In fact, this approach stems out from early insights of Alan Turing, describing the need to build machines equipped with sense organs to learn directly from the real-world instead of focusing on abstract activities, such as playing chess. == Definitions == Classically, a software entity is defined as a simulated element, able to act on itself and on its environment, and which has an internal representation of itself and of the outside world. An entity can communicate with other entities, and its behavior is the consequence of its perceptions, its representations, and its interactions with the other entities. === AI loop === Simulating entities in a virtual environment requires simulating the entire process that goes from a perception of the environment, or more generally from a stimulus, to an action on the environment. This process is called the AI loop and technology used to simulate it can be subdivided in two categories. Sensorimotor or low-level AI deals with either the perception problem (what is perceived?) or the animation problem (how are actions executed?). Decisional or high-level AI deals with the action selection problem (what is the most appropriate action in response to a given perception, i.e. what is the most appropriate behavior?). === Traditional or symbolic AI === There are two main approaches in decisional AI. The vast majority of the technologies available on the market, such as planning algorithms, finite-state machines (FSA), or expert systems, are based on the traditional or symbolic AI approach. Its main characteristics are: It is top-down: it subdivides, in a recursive manner, a given problem into a series of sub-problems that are supposedly easier to solve. It is knowledge-based: it relies on a symbolic description of the world, such as a set of rules. However, the limits of traditional AI, which goal is to build systems that mimic human intelligence, are well-known: inevitably, a combinatorial explosion of the number of rules occurs due to the complexity of the environment. In fact, it is impossible to predict all the situations that will be encountered by an autonomous entity. === Situated or behavioral AI === In order to address these issues, another approach to decisional AI, also known as situated or behavioral AI, has been proposed. It does not attempt to model systems that produce deductive reasoning processes, but rather systems that behave realistically in their environment. The main characteristics of this approach are the following: It is bottom-up: it relies on elementary behaviors, which can be combined to implement more complex behaviors. It is behavior-based: it does not rely on a symbolic description of the environment, but rather on a model of the interactions of the entities with their environment. The goal of situated AI is to model entities that are autonomous in their environment. This is achieved thanks to both the intrinsic robustness of the control architecture, and its adaptation capabilities to unforeseen situations. === Situated agents === In artificial intelligence and cognitive science, the term situated refers to an agent which is embedded in an environment. The term situated is commonly used to refer to robots, but some researchers argue that software agents can also be situated if: they exist in a dynamic (rapidly changing) environment, which they can manipulate or change through their actions, and which they can sense or perceive. Examples might include web-based agents, which can alter data or trigger processes (such as purchases) over the Internet, or virtual-reality bots which inhabit and change virtual worlds, such as Second Life. Being situated is generally considered to be part of being embodied, but it is useful to consider each perspective individually. The situated perspective emphasizes that intelligent behavior derives from the environment and the agent's interactions with it. The nature of these interactions are defined by an agent's embodiment. == Implementation principles == === Modular decomposition === The most important attribute of a system driven by situated AI is that the intelligence is controlled by a set of independent semi-autonomous modules. In the original systems, each module was actually a separate device or was at least conceived of as running on its own processing thread. Generally, though, the modules are just abstractions. In this respect, situated AI may be seen as a software engineering approach to AI, perhaps akin to object oriented design. Situated AI is often associated with reactive planning, but the two are not synonymous. Brooks advocated an extreme version of cognitive minimalism which required initially that the behavior modules were finite-state machines and thus contained no conventional memory or learning. This is associated with reactive AI because reactive AI requires reacting to the current state of the world, not to an agent's memory or preconception of that world. However, learning is obviously key to realistic strong AI, so this constraint has been relaxed, though not entirely abandoned. === Action selection mechanism === The situated AI community has presented several solutions to modeling decision-making processes, also known as action selection mechanisms. The first attempt to solve this problem goes back to subsumption architectures, which were in fact more an implementation technique than an algorithm. However, this attempt paved the way to several others, in particular the free-flow hierarchies and activation networks. A comparison of the structure and performances of these two mechanisms demonstrated the advantage of using free-flow hierarchies in solving the action selection problem. However, motor schemas and process description languages are two other approaches that have been used with success for autonomous robots. == Notes and references == Arsenio, Artur M. (2004) Towards an embodied and situated AI, In: Proceedings of the International FLAIRS conference, 2004. (online) The Artificial Life Route To Artificial Intelligence: Building Embodied, Situated Agents, Luc Steels and Rodney Brooks Eds., Lawrence Erlbaum Publishing, 1995. (ISBN 978-0805815184) Rodney A. Brooks Cambrian Intelligence (MIT Press, 1999) ISBN 0-262-52263-2; collection of early papers including "Intelligence without representation" and "Intelligence without reason", from 1986 & 1991 respectively. Ronald C. Arkin Behavior-Based Robotics (MIT Press, 1998) ISBN 0-262-01165-4 Hendriks-Jansen, Horst (1996) Catching Ourselves in the Act: Situated Activity, Interactive Emergence, Evolution, and Human Thought. Cambridge, Mass.: MIT Press.

  Read more →
• 

  Grammatical evolution

  Grammatical evolution (GE) is a genetic programming (GP) technique (or approach) from evolutionary computation pioneered by Conor Ryan, JJ Collins and Michael O'Neill in 1998 at the BDS Group in the University of Limerick. As in any other GP approach, the objective is to find an executable program, program fragment, or function, which will achieve a good fitness value for a given objective function. In most published work on GP, a LISP-style tree-structured expression is directly manipulated, whereas GE applies genetic operators to an integer string, subsequently mapped to a program (or similar) through the use of a grammar, which is typically expressed in Backus–Naur form. One of the benefits of GE is that this mapping simplifies the application of search to different programming languages and other structures. == Problem addressed == In type-free, conventional Koza-style GP, the function set must meet the requirement of closure: all functions must be capable of accepting as their arguments the output of all other functions in the function set. Usually, this is implemented by dealing with a single data-type such as double-precision floating point. While modern Genetic Programming frameworks support typing, such type-systems have limitations that Grammatical Evolution does not suffer from. == GE's solution == GE offers a solution to the single-type limitation by evolving solutions according to a user-specified grammar (usually a grammar in Backus-Naur form). Therefore, the search space can be restricted, and domain knowledge of the problem can be incorporated. The inspiration for this approach comes from a desire to separate the "genotype" from the "phenotype": in GP, the objects the search algorithm operates on and what the fitness evaluation function interprets are one and the same. In contrast, GE's "genotypes" are ordered lists of integers which code for selecting rules from the provided context-free grammar. The phenotype, however, is the same as in Koza-style GP: a tree-like structure that is evaluated recursively. This model is more in line with how genetics work in nature, where there is a separation between an organism's genotype and the final expression of phenotype in proteins, etc. Separating genotype and phenotype allows a modular approach. In particular, the search portion of the GE paradigm needn't be carried out by any one particular algorithm or method. Observe that the objects GE performs search on are the same as those used in genetic algorithms. This means, in principle, that any existing genetic algorithm package, such as the popular GAlib, can be used to carry out the search, and a developer implementing a GE system need only worry about carrying out the mapping from list of integers to program tree. It is also in principle possible to perform the search using some other method, such as particle swarm optimization (see the remark below); the modular nature of GE creates many opportunities for hybrids as the problem of interest to be solved dictates. Brabazon and O'Neill have successfully applied GE to predicting corporate bankruptcy, forecasting stock indices, bond credit ratings, and other financial applications. GE has also been used with a classic predator-prey model to explore the impact of parameters such as predator efficiency, niche number, and random mutations on ecological stability. It is possible to structure a GE grammar that for a given function/terminal set is equivalent to genetic programming. == Criticism == Despite its successes, GE has been the subject of some criticism. One issue is that as a result of its mapping operation, GE's genetic operators do not achieve high locality which is a highly regarded property of genetic operators in evolutionary algorithms. == Variants == Although GE was originally described in terms of using an Evolutionary Algorithm, specifically, a Genetic Algorithm, other variants exist. For example, GE researchers have experimented with using particle swarm optimization to carry out the searching instead of genetic algorithms with results comparable to that of normal GE; this is referred to as a "grammatical swarm"; using only the basic PSO model it has been found that PSO is probably equally capable of carrying out the search process in GE as simple genetic algorithms are. (Although PSO is normally a floating-point search paradigm, it can be discretized, e.g., by simply rounding each vector to the nearest integer, for use with GE.) Yet another possible variation that has been experimented with in the literature is attempting to encode semantic information in the grammar in order to further bias the search process. Other work showed that, with biased grammars that leverage domain knowledge, even random search can be used to drive GE. == Related work == GE was originally a combination of the linear representation as used by the Genetic Algorithm for Developing Software (GADS) and Backus Naur Form grammars, which were originally used in tree-based GP by Wong and Leung in 1995 and Whigham in 1996. Other related work noted in the original GE paper was that of Frederic Gruau, who used a conceptually similar "embryonic" approach, as well as that of Keller and Banzhaf, which similarly used linear genomes. == Implementations == There are several implementations of GE. These include the following.

  Read more →
• 

  Structured kNN

  Structured k-nearest neighbours (SkNN) is a machine learning algorithm that generalizes k-nearest neighbors (k-NN). k-NN supports binary classification, multiclass classification, and regression, whereas SkNN allows training of a classifier for general structured output. For instance, a data sample might be a natural language sentence, and the output could be an annotated parse tree. Training a classifier consists of showing many instances of ground truth sample-output pairs. After training, the SkNN model is able to predict the corresponding output for new, unseen sample instances; that is, given a natural language sentence, the classifier can produce the most likely parse tree. == Training == As a training set, SkNN accepts sequences of elements with class labels. The type of element does not matter; the only requirement is a defined metric function that gives a distance between each pair of elements of a set. SkNN is based on idea of creating a graph, with each node representing a class label. There is an edge between a pair of nodes if there is a sequence of two elements in the training set with corresponding classes. The first step of SkNN training is the construction of such a graph from training sequences. There are two special nodes in the graph corresponding to sentence beginnings and ends: if a sequence starts with class C, the edge between node START and node C should be created. Like regular k-NN, the second part of SkNN training consists of storing the elements of a training sequence in a certain way. Each element of the training sequences is stored in the node related to the class of the previous element in the sequence. Every first element is stored in the START node. == Inference == Labelling input sequences by SkNN consists of finding the sequence of transitions in the graph, starting from node START. Each transition corresponds to a single element of the input sequence. As a result, the label of each element is determined as the target node label of the transition. The cost of the path is defined as the sum of all transitions, with the cost of transition from node A to node B being the distance from the current input sequence element to the nearest element of class B, stored in node A. Determining an optimal path may be performed using a modified Viterbi algorithm (where the sum of the distances is minimized, unlike the original algorithm which maximizes the product of probabilities).

  Read more →
• 

  Bayesian hierarchical modeling

  Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis

  Read more →
• 

  Cloud Native Computing Foundation

  The Cloud Native Computing Foundation (CNCF) is a subsidiary of the Linux Foundation founded in 2015 to support cloud-native computing. == History == It was announced alongside Kubernetes 1.0, an open source container cluster manager, which was contributed to the Linux Foundation by Google as a seed technology. Founding members include Google, CoreOS, Mesosphere, Red Hat, Twitter, Huawei, Intel, RX-M, Cisco, IBM, Docker, Univa, and VMware. Today, CNCF is supported by over 450 members. In August 2018 Google announced that it was handing over operational control of Kubernetes to the community. == Projects == Argo is a collection of tools for getting work done with Kubernetes. Among its main features are Workflows and Events. It was accepted to CNCF on March 26, 2020 at the Incubating maturity level and then moved to the Graduated maturity level on December 6, 2022. cert-manager provisions and manages TLS certificates in Kubernetes. It was accepted to CNCF on November 10, 2020, moved to the Incubating maturity level on September 19, 2022, and then moved to the Graduated maturity level on September 29, 2024. Cilium provides networking, security, and observability for Kubernetes deployments using eBPF technology. It joined the CNCF at incubation level in October 2021 and the CNCF announced its graduation in October 2023. containerd is an industry-standard core container runtime. It is currently available as a daemon for Linux and Windows, which can manage the complete container lifecycle of its host system. In 2015, Docker donated the OCI Specification to The Linux Foundation with a reference implementation called runc. Since February 28, 2019 it is an official CNCF project. Its general availability and intention to donate the project to CNCF was announced by Docker in 2017. CoreDNS is a DNS server that chains plugins. Its graduation was announced in 2019. Dapr, the distributed application runtime, provides APIs for building secure and reliable microservices and agentic AI systems. Dapr was donated to the CNCF in November 2021 and joined at incubation level. The CNCF announced its graduation in November 2024. Envoy: Originally built at Lyft to move their architecture away from a monolith, Envoy is a high-performance open source edge and service proxy that makes the network transparent to applications. Lyft contributed Envoy to Cloud Native Computing Foundation in September 2017. etcd is a distributed key value store, providing a method of storing data across a cluster of machines. It became a CNCF incubating project in 2018 at KubeCon+CloudNativeCon North America in Seattle that year. Falco is an open source and cloud native runtime security initiative. It is the "de facto Kubernetes threat detection engine". It became an incubating project in January 2020 and graduated in February 2024. Flux is an open source project for powering GitOps in Kubernetes clusters. It provides the GitOps Toolkit, a set of Kubernetes APIs that allow you to define how configuration source code is securely pulled into your cluster and deployed by popular Kubernetes manifests rendering engines like Kustomize and Helm. The most recommended source mechanism is the OCIRepository API, which provides enhanced security and benefits from container image tooling out there. Flux has also notification integrations with popular services like Prometheus Alertmanager, PagerDuty, Slack and so on. Flux has graduated in CNCF in 2022. Harbor is an "open source trusted cloud native registry project that stores, signs, and scans content." It became an incubating project in September 2019 and graduated in June 2020. Helm is a package manager that helps developers "easily manage and deploy applications onto the Kubernetes cluster." It joined the incubating level in June 2018 and graduated in April 2020. Istio is a service mesh technology. It was accepted by CNCF in September 2022 and graduated on July 12, 2023. Jaeger, Created by Uber Engineering, Jaeger is an open source distributed tracing system inspired by Google Dapper paper and OpenZipkin community. It can be used for tracing microservice-based architectures, including distributed context propagation, distributed transaction monitoring, root cause analysis, service dependency analysis, and performance/latency optimization. The Cloud Native Computing Foundation Technical Oversight Committee voted to accept Jaeger as the 12th hosted project in September 2017 and became a graduated project in 2019. In 2020 it became an approved and fully integrated part of the CNCF ecosystem. Kubernetes is an open source framework for automating deployment and managing applications in a containerized and clustered environment. "It aims to provide better ways of managing related, distributed components across the varied infrastructure." It was originally designed by Google and donated to The Linux Foundation to form the Cloud Native Computing Foundation with Kubernetes as the seed technology. The "large and diverse" community supporting the project has made its staying power more robust than other, older technologies of the same ilk. In January 2020, the CNCF annual report showed significant growth in interest, training, event attendance and investment related to Kubernetes. Linkerd is CNCF's fifth member project, and the project that coined the term "service mesh". Linkerd adds observability, security, and reliability features to applications by adding them to the platform rather than the application layer, and features a "micro-proxy" to maximize speed and security of its data plane. Linkerd graduated from CNCF in July 2021. Open Policy Agent (OPA) is "an open source general-purpose policy engine and language for cloud infrastructure." It became a CNCF incubating project in April 2019. OPA graduated from CNCF in February 2021. Prometheus is a cloud monitoring tool sponsored by SoundCloud in early iterations. In August 2018, the tool was designated a graduated project by the Cloud Native Computing Foundation. It is now a Cloud Native Computing Foundation member project. Rook is CNCF's first cloud native storage project. It became an incubation level project in 2018 and graduated in October 2020. SPIFFE is an open standard and framework for workload identity, much the same way that OAuth is an open standard and framework for human identity. It is built from the ground up to accommodate modern computing environments, which operate with systems scale and velocity (as opposed to human scale and velocity), while still maintaining interoperability with existing technologies like OAuth and X.509 Public key infrastructure. Unlike other identity standards, SPIFFE supports multiple credential types for a single identity, ensuring that the highly varied needs of production environments are consistently met without compromise. SPIFFE joined the CNCF as a sandbox project in 2018, was accepted to incubation in 2020, and graduated in 2022. SPIRE is an open source identity provider for workloads based on the SPIFFE framework. It is highly pluggable, and fills the attestation and issuance needs required by any workload identity solution. The plugin interfaces it exposes allows users to write integrations with in-house systems, build internal self-service portals, and more. It is a very powerful building block for issuing short-lived identity credentials to dynamic cloud workloads. SPIRE became a CNCF Graduated project in 2022. The Update Framework (TUF) helps developers to secure new or existing software update systems, which are often found to be vulnerable to many known attacks. TUF addresses this widespread problem by providing a comprehensive, flexible security framework that developers can integrate with any software update system. TUF was CNCF's first security-focused project and the ninth project overall to graduate from the foundation's hosting program. TiKV provides a distributed key–value database. Vitess is a database clustering system for horizontal scaling of MySQL, first created for internal use by YouTube. It became a CNCF project in 2018 and graduated in November 2019. Contour is a management server for Envoy that can direct the management of Kubernetes' traffic. Contour also provides routing features that are more advanced than Kubernetes' out-of-the-box Ingress specification. VMWare contributed the project to CNCF in July 2020. Cortex offers horizontally scalable, multi-tenant, long-term storage for Prometheus and works alongside Amazon DynamoDB, Google Bigtable, Cassandra, S3, GCS, and Microsoft Azure. It was introduced into the ecosystem incubator alongside Thanos in August 2020. CRI-O is an Open Container Initiative (OCI) based "implementation of Kubernetes Container Runtime Interface". CRI-O allows Kubernetes to be container runtime-agnostic. It became an incubating project in 2019. gRPC is a "modern open source high performance RPC framework that can run in any environment." The project was formed in 2015 when Google decided to open sou

  Read more →
• 

  Q-learning

  Q-learning is a reinforcement learning algorithm that trains an agent to assign values to its possible actions based on its current state, without requiring a model of the environment (model-free). It can handle problems with stochastic transitions and rewards without requiring adaptations. For example, in a grid maze, an agent learns to reach an exit worth 10 points. At a junction, Q-learning might assign a higher value to moving right than left if right gets to the exit faster, improving this choice by trying both directions over time. For any finite Markov decision process, Q-learning finds an optimal policy in the sense of maximizing the expected value of the total reward over any and all successive steps, starting from the current state. Q-learning can identify an optimal action-selection policy for any given finite Markov decision process, given infinite exploration time and a partly random policy. "Q" refers to the function that the algorithm computes: the expected reward—that is, the quality—of an action taken in a given state. == Reinforcement learning == Reinforcement learning involves an agent, a set of states S {\displaystyle {\mathcal {S}}} , and a set A {\displaystyle {\mathcal {A}}} of actions per state. By performing an action a ∈ A {\displaystyle a\in {\mathcal {A}}} , the agent transitions from state to state. Executing an action in a specific state provides the agent with a reward (a numerical score). The goal of the agent is to maximize its total reward. It does this by adding the maximum reward attainable from future states to the reward for achieving its current state, effectively influencing the current action by the potential future reward. This potential reward is a weighted sum of expected values of the rewards of all future steps starting from the current state. As an example, consider the process of boarding a train, in which the reward is measured by the negative of the total time spent boarding (alternatively, the cost of boarding the train is equal to the boarding time). One strategy is to enter the train door as soon as they open, minimizing the initial wait time for yourself. If the train is crowded, however, then you will have a slow entry after the initial action of entering the door as people are fighting you to depart the train as you attempt to board. The total boarding time, or cost, is then: 0 seconds wait time + 15 seconds fight time On the next day, by random chance (exploration), you decide to wait and let other people depart first. This initially results in a longer wait time. However, less time is spent fighting the departing passengers. Overall, this path has a higher reward than that of the previous day, since the total boarding time is now: 5 second wait time + 0 second fight time Through exploration, despite the initial (patient) action resulting in a larger cost (or negative reward) than in the forceful strategy, the overall cost is lower, thus revealing a more rewarding strategy. == Algorithm == After Δ t {\displaystyle \Delta t} steps into the future the agent will decide some next step. The weight for this step is calculated as γ Δ t {\displaystyle \gamma ^{\Delta t}} , where γ {\displaystyle \gamma } (the discount factor) is a number between 0 and 1 ( 0 ≤ γ ≤ 1 {\displaystyle 0\leq \gamma \leq 1} ). Assuming γ < 1 {\displaystyle \gamma <1} , it has the effect of valuing rewards received earlier higher than those received later (reflecting the value of a "good start"). γ {\displaystyle \gamma } may also be interpreted as the probability to succeed (or survive) at every step Δ t {\displaystyle \Delta t} . The algorithm, therefore, has a function that calculates the quality of a state–action combination: Q : S × A → R {\displaystyle Q:{\mathcal {S}}\times {\mathcal {A}}\to \mathbb {R} } . Before learning begins, ⁠ Q {\displaystyle Q} ⁠ is initialized to a possibly arbitrary fixed value (chosen by the programmer). Then, at each time t {\displaystyle t} the agent selects an action A t {\displaystyle A_{t}} , observes a reward R t + 1 {\displaystyle R_{t+1}} , enters a new state S t + 1 {\displaystyle S_{t+1}} (that may depend on both the previous state S t {\displaystyle S_{t}} and the selected action), and Q {\displaystyle Q} is updated. The core of the algorithm is a Bellman equation as a simple value iteration update, using the weighted average of the current value and the new information: Q n e w ( S t , A t ) ← ( 1 − α ⏟ learning rate ) ⋅ Q ( S t , A t ) ⏟ current value + α ⏟ learning rate ⋅ ( R t + 1 ⏟ reward + γ ⏟ discount factor ⋅ max a Q ( S t + 1 , a ) ⏟ estimate of optimal future value ⏟ new value (temporal difference target) ) {\displaystyle Q^{new}(S_{t},A_{t})\leftarrow (1-\underbrace {\alpha } _{\text{learning rate}})\cdot \underbrace {Q(S_{t},A_{t})} _{\text{current value}}+\underbrace {\alpha } _{\text{learning rate}}\cdot {\bigg (}\underbrace {\underbrace {R_{t+1}} _{\text{reward}}+\underbrace {\gamma } _{\text{discount factor}}\cdot \underbrace {\max _{a}Q(S_{t+1},a)} _{\text{estimate of optimal future value}}} _{\text{new value (temporal difference target)}}{\bigg )}} where R t + 1 {\displaystyle R_{t+1}} is the reward received when moving from the state S t {\displaystyle S_{t}} to the state S t + 1 {\displaystyle S_{t+1}} , and α {\displaystyle \alpha } is the learning rate ( 0 < α ≤ 1 ) {\displaystyle (0<\alpha \leq 1)} . Note that Q n e w ( S t , A t ) {\displaystyle Q^{new}(S_{t},A_{t})} is the sum of three terms: ( 1 − α ) Q ( S t , A t ) {\displaystyle (1-\alpha )Q(S_{t},A_{t})} : the current value (weighted by one minus the learning rate) α R t + 1 {\displaystyle \alpha \,R_{t+1}} : the reward R t + 1 {\displaystyle R_{t+1}} to obtain if action A t {\displaystyle A_{t}} is taken when in state S t {\displaystyle S_{t}} (weighted by learning rate) α γ max a Q ( S t + 1 , a ) {\displaystyle \alpha \gamma \max _{a}Q(S_{t+1},a)} : the maximum reward that can be obtained from state S t + 1 {\displaystyle S_{t+1}} (weighted by learning rate and discount factor) An episode of the algorithm ends when state S t + 1 {\displaystyle S_{t+1}} is a final or terminal state. However, Q-learning can also learn in non-episodic tasks (as a result of the property of convergent infinite series). If the discount factor is lower than 1, the action values are finite even if the problem can contain infinite loops or paths. For all final states s f {\displaystyle s_{f}} , Q ( s f , a ) {\displaystyle Q(s_{f},a)} is never updated, but is set to the reward value r {\displaystyle r} observed for state s f {\displaystyle s_{f}} . In most cases, Q ( s f , a ) {\displaystyle Q(s_{f},a)} can be taken to equal zero. == Influence of variables == === Learning rate === The learning rate or step size determines to what extent newly acquired information overrides old information. A factor of 0 makes the agent learn nothing (exclusively exploiting prior knowledge), while a factor of 1 makes the agent consider only the most recent information (ignoring prior knowledge to explore possibilities). In fully deterministic environments, a learning rate of α t = 1 {\displaystyle \alpha _{t}=1} is optimal. When the problem is stochastic, the algorithm converges under some technical conditions on the learning rate that require it to decrease to zero. In practice, often a constant learning rate is used, such as α t = 0.1 {\displaystyle \alpha _{t}=0.1} for all t {\displaystyle t} . === Discount factor === The discount factor ⁠ γ {\displaystyle \gamma } ⁠ determines the importance of future rewards. A factor of 0 will make the agent "myopic" (or short-sighted) by only considering current rewards, i.e. r t {\displaystyle r_{t}} (in the update rule above), while a factor approaching 1 will make it strive for a long-term high reward. If the discount factor meets or exceeds 1, the action values may diverge. For ⁠ γ = 1 {\displaystyle \gamma =1} ⁠, without a terminal state, or if the agent never reaches one, all environment histories become infinitely long, and utilities with additive, undiscounted rewards generally become infinite. Even with a discount factor only slightly lower than 1, Q-function learning leads to propagation of errors and instabilities when the value function is approximated with an artificial neural network. In that case, starting with a lower discount factor and increasing it towards its final value accelerates learning. === Initial conditions (Q0) === Since Q-learning is an iterative algorithm, it implicitly assumes an initial condition before the first update occurs. High initial values, also known as "optimistic initial conditions", can encourage exploration: no matter what action is selected, the update rule will cause it to have lower values than the other alternative, thus increasing their choice probability. The first reward r {\displaystyle r} can be used to reset the initial conditions. According to this idea, the first time an action is taken the reward is used to set the value

  Read more →
• 

  Facial recognition system

  A facial recognition system is a technology potentially capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and works by pinpointing and measuring facial features from a given image. Development on similar systems began in the 1960s as a form of computer application. Since their inception, facial recognition systems have seen wider uses in recent times on smartphones and in other forms of technology, such as robotics. Because computerized facial recognition involves the measurement of a human's physiological characteristics, facial recognition systems are categorized as biometrics. Although the accuracy of facial recognition systems as a biometric technology is lower than iris recognition, fingerprint image acquisition, palm recognition or voice recognition, it is widely adopted due to its contactless process. Facial recognition systems have been deployed in advanced human–computer interaction, video surveillance, law enforcement, passenger screening, decisions on employment and housing, and automatic indexing of images. Facial recognition systems are employed throughout the world today by governments and private companies. Their effectiveness varies, and some systems have previously been scrapped because of their ineffectiveness. The use of facial recognition systems has also raised controversy, with claims that the systems violate citizens' privacy, commonly make incorrect identifications, encourage gender norms and racial profiling, and do not protect important biometric data. The appearance of synthetic media such as deepfakes has also raised concerns about its security. These claims have led to the ban of facial recognition systems in several cities in the United States. Growing societal concerns led social networking company Meta Platforms to shut down its Facebook facial recognition system in 2021, deleting the face-scan data of more than one billion users. The change represented one of the largest shifts in facial recognition usage in the technology's history. IBM also stopped offering facial recognition technology due to similar concerns. == History of facial recognition technology == Automated facial recognition was pioneered in the 1960s by Woody Bledsoe, Helen Chan Wolf, and Charles Bisson, whose work focused on teaching computers to recognize human faces. Their early facial recognition project was dubbed "man-machine" because a human first needed to establish the coordinates of facial features in a photograph before they could be used by a computer for recognition. Using a graphics tablet, a human would pinpoint facial features coordinates, such as the pupil centers, the inside and outside corners of eyes, and the widows peak in the hairline. The coordinates were used to calculate 20 individual distances, including the width of the mouth and of the eyes. A human could process about 40 pictures an hour, building a database of these computed distances. A computer would then automatically compare the distances for each photograph, calculate the difference between the distances, and return the closed records as a possible match. In 1970, Takeo Kanade publicly demonstrated a face-matching system that located anatomical features such as the chin and calculated the distance ratio between facial features without human intervention. Later tests revealed that the system could not always reliably identify facial features. Nonetheless, interest in the subject grew and in 1977 Kanade published the first detailed book on facial recognition technology. In 1993, the Defense Advanced Research Project Agency (DARPA) and the Army Research Laboratory (ARL) established the face recognition technology program FERET to develop "automatic face recognition capabilities" that could be employed in a productive real life environment "to assist security, intelligence, and law enforcement personnel in the performance of their duties." Face recognition systems that had been trialled in research labs were evaluated. The FERET tests found that while the performance of existing automated facial recognition systems varied, a handful of existing methods could viably be used to recognize faces in still images taken in a controlled environment. The FERET tests spawned three US companies that sold automated facial recognition systems. Vision Corporation and Miros Inc were founded in 1994, by researchers who used the results of the FERET tests as a selling point. Viisage Technology was established by an identification card defense contractor in 1996 to commercially exploit the rights to the facial recognition algorithm developed by Alex Pentland at MIT. Following the 1993 FERET face-recognition vendor test, the Department of Motor Vehicles (DMV) offices in West Virginia and New Mexico became the first DMV offices to use automated facial recognition systems to prevent people from obtaining multiple driving licenses using different names. Driver's licenses in the United States were at that point a commonly accepted form of photo identification. DMV offices across the United States were undergoing a technological upgrade and were in the process of establishing databases of digital ID photographs. This enabled DMV offices to deploy the facial recognition systems on the market to search photographs for new driving licenses against the existing DMV database. DMV offices became one of the first major markets for automated facial recognition technology and introduced US citizens to facial recognition as a standard method of identification. The increase of the US prison population in the 1990s prompted U.S. states to established connected and automated identification systems that incorporated digital biometric databases, in some instances this included facial recognition. In 1999, Minnesota incorporated the facial recognition system FaceIT by Visionics into a mug shot booking system that allowed police, judges and court officers to track criminals across the state. Until the 1990s, facial recognition systems were developed primarily by using photographic portraits of human faces. Research on face recognition to reliably locate a face in an image that contains other objects gained traction in the early 1990s with the principal component analysis (PCA). The PCA method of face detection is also known as Eigenface and was developed by Matthew Turk and Alex Pentland. Turk and Pentland combined the conceptual approach of the Karhunen–Loève theorem and factor analysis, to develop a linear model. Eigenfaces are determined based on global and orthogonal features in human faces. A human face is calculated as a weighted combination of a number of Eigenfaces. Because few Eigenfaces were used to encode human faces of a given population, Turk and Pentland's PCA face detection method greatly reduced the amount of data that had to be processed to detect a face. Pentland in 1994 defined Eigenface features, including eigen eyes, eigen mouths and eigen noses, to advance the use of PCA in facial recognition. In 1997, the PCA Eigenface method of face recognition was improved upon using linear discriminant analysis (LDA) to produce Fisherfaces. LDA Fisherfaces became dominantly used in PCA feature based face recognition. While Eigenfaces were also used for face reconstruction. In these approaches no global structure of the face is calculated which links the facial features or parts. Purely feature based approaches to facial recognition were overtaken in the late 1990s by the Bochum system, which used Gabor filter to record the face features and computed a grid of the face structure to link the features. Christoph von der Malsburg and his research team at the University of Bochum developed Elastic Bunch Graph Matching in the mid-1990s to extract a face out of an image using skin segmentation. By 1997, the face detection method developed by Malsburg outperformed most other facial detection systems on the market. The so-called "Bochum system" of face detection was sold commercially on the market as ZN-Face to operators of airports and other busy locations. The software was "robust enough to make identifications from less-than-perfect face views. It can also often see through such impediments to identification as mustaches, beards, changed hairstyles and glasses—even sunglasses". Real-time face detection in video footage became possible in 2001 with the Viola–Jones object detection framework for faces. Paul Viola and Michael Jones combined their face detection method with the Haar-like feature approach to object recognition in digital images to launch AdaBoost, the first real-time frontal-view face detector. By 2015, the Viola–Jones algorithm had been implemented using small low power detectors on handheld devices and embedded systems. Therefore, the Viola–Jones algorithm has not only broadened the practical application of face recognition systems but

  Read more →
• 

  Frequent pattern discovery

  Frequent pattern discovery (or FP discovery, FP mining, or Frequent itemset mining) is part of knowledge discovery in databases, Massive Online Analysis, and data mining; it describes the task of finding the most frequent and relevant patterns in large datasets. The concept was first introduced for mining transaction databases. Frequent patterns are defined as subsets (itemsets, subsequences, or substructures) that appear in a data set with frequency no less than a user-specified or auto-determined threshold. == Techniques == Techniques for FP mining include: market basket analysis cross-marketing catalog design clustering classification recommendation systems For the most part, FP discovery can be done using association rule learning with particular algorithms Eclat, FP-growth and the Apriori algorithm. Other strategies include: Frequent subtree mining Structure mining Sequential pattern mining and respective specific techniques. Implementations exist for various machine learning systems or modules like MLlib for Apache Spark.

  Read more →
• 

  Social software engineering

  Social software engineering (SSE) is a branch of software engineering that is concerned with the social aspects of software development and the developed software. SSE focuses on the socialness of both software engineering and developed software. On the one hand, the consideration of social factors in software engineering activities, processes and CASE tools is deemed to be useful to improve the quality of both development process and produced software. Examples include the role of situational awareness and multi-cultural factors in collaborative software development. On the other hand, the dynamicity of the social contexts in which software could operate (e.g., in a cloud environment) calls for engineering social adaptability as a runtime iterative activity. Examples include approaches which enable software to gather users' quality feedback and use it to adapt autonomously or semi-autonomously. SSE studies and builds socially-oriented tools to support collaboration and knowledge sharing in software engineering. SSE also investigates the adaptability of software to the dynamic social contexts in which it could operate and the involvement of clients and end-users in shaping software adaptation decisions at runtime. Social context includes norms, culture, roles and responsibilities, stakeholder's goals and interdependencies, end-users perception of the quality and appropriateness of each software behaviour, etc. The participants of the 1st International Workshop on Social Software Engineering and Applications (SoSEA 2008) proposed the following characterization: Community-centered: Software is produced and consumed by and/or for a community rather than focusing on individuals Collaboration/collectiveness: Exploiting the collaborative and collective capacity of human beings Companionship/relationship: Making explicit the various associations among people Human/social activities: Software is designed consciously to support human activities and to address social problems Social inclusion: Software should enable social inclusion enforcing links and trust in communities Thus, SSE can be defined as "the application of processes, methods, and tools to enable community-driven creation, management, deployment, and use of software in online environments". One of the main observations in the field of SSE is that the concepts, principles, and technologies made for social software applications are applicable to software development itself as software engineering is inherently a social activity. SSE is not limited to specific activities of software development. Accordingly, tools have been proposed supporting different parts of SSE, for instance, social system design or social requirements engineering. Consequently vertical market software, such as software development tools, engineering tools, marketing tools or software that helps users in a decision-making process can profit from social components. Such vertical social software differentiates strongly in its user-base from traditional social software such as Yammer.

  Read more →
• 

  Teacher forcing

  Teacher forcing is an algorithm for training the weights of recurrent neural networks (RNNs). It involves feeding observed sequence values (i.e. ground-truth samples) back into the RNN after each step, thus forcing the RNN to stay close to the ground-truth sequence. The term "teacher forcing" can be motivated by comparing the RNN to a human student taking a multi-part exam where the answer to each part (for example a mathematical calculation) depends on the answer to the preceding part. In this analogy, rather than grading every answer in the end, with the risk that the student fails every single part even though they only made a mistake in the first one, a teacher records the score for each individual part and then tells the student the correct answer, to be used in the next part. The use of an external teacher signal is in contrast to real-time recurrent learning (RTRL). Teacher signals are known from oscillator networks. The promise is, that teacher forcing helps to reduce the training time. The term "teacher forcing" was introduced in 1989 by Ronald J. Williams and David Zipser, who reported that the technique was already being "frequently used in dynamical supervised learning tasks" around that time. A NeurIPS 2016 paper introduced the related method of "professor forcing".

  Read more →
• 

  Principal component analysis

  Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector

  Read more →
• 

  Vladimir Batagelj

  Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks (blockmodeling). == Education and career == Vladimir Batagelj completed his Ph.D. at the University of Ljubljana in 1986 under the direction of Tomaž Pisanski. He stayed at the University of Ljubljana as a professor until his retirement, where he was a professor of sociology and statistics, while also being a chair of the Department of Sociology of the Faculty of Social Sciences. As visiting professor, he was taught at the University of Pittsburgh (1990-91) and at the University of Konstanz (2002). He was also a member of editorial boards of two journals: Informatica and Journal of Social Structure. His work has been cited over 11000 times. His book Exploratory Social Network Analysis with Pajek on blockmodeling, coauthored with Wouter de Nooy and Andrej Mrvar, is Batagelj's most cited work and has over 3300 citations. The book was translated into Chinese and Japanese. The revised and expanded third edition has been published by Cambridge University Press. In 1975, 11 years before completing his PhD, Batagelj published a solo paper in Communications of the ACM. Batagelj authored more than 20 textbooks in Slovenian, covering topics like TeX, combinatorics and discrete mathematics. He has also written extensively in the Slovenian popular science journal Presek. Batagelj has advised 9 Ph.D. students. == Pajek == Batagelj is particularly known for his work on Pajek, a freely available software for analysis and visualization of large networks. He began work on Pajek in 1996 with Andrej Mrvar, who was then his PhD student. == Awards and honors == First prizes for contributions (with Andrej Mrvar) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. In 2007 the book Generalized blockmodeling was awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association In 2007 he was awarded (together with Anuška Ferligoj) the Simmel Award by INSNA. In 2013, Vladimir Batagelj and Andrej Mrvar received the INSNA's William D. Richards Software award for their work on Pajek. == Selected bibliography == Vladimir Batagelj, Social Network Analysis, Large-Scale [1]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 8245–8265. Vladimir Batagelj, Complex Networks, Visualization of [2]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 1253–1268. Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press 2005 (ISBN 0-521-60262-9). ESNA in Japanese, TDU, 2010. Patrick Doreian, Vladimir Batagelj, Anuška Ferligoj, Mark Granovetter (Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)

  Read more →
• 

  Agentive logic

  Agentive logic (also called the logic of action or logic of agency) is the field of philosophical logic and logic in computer science that studies formal representations of agents, their actions, and their abilities. An agentive logic in the narrower sense is a formal system whose primitive operators express that an agent does something, can do something, or sees to it that something is the case. Agentive logics generalise modal logic by adding modalities indexed to agents and to actions. Typical examples include: STIT logics (from sees to it that) with operators of the form [ i s t i t : φ ] {\displaystyle [i\ {\mathsf {stit}}:\varphi ]} meaning that agent i {\displaystyle i} sees to it that φ {\displaystyle \varphi } holds; dynamic logics of action with program-like modalities [ α ] φ {\displaystyle [\alpha ]\varphi } and ⟨ α ⟩ φ {\displaystyle \langle \alpha \rangle \varphi } meaning, roughly, that after every (respectively, some) execution(s) of action α {\displaystyle \alpha } , φ {\displaystyle \varphi } holds; logics with explicit agentive operators such as "can do", "brings about", or "is able to ensure". Agentive logics are used in action theory in philosophy, in the semantics of natural language, in the theory of program verification, and in artificial intelligence, where they underpin formalisms for reasoning about actions, planning, and intelligent agents. == Terminology and scope == The adjective agentive derives from the Latin agens ("one who acts") and originally referred to the grammatical agent of a verb. In logical contexts it designates operators or predicates whose primary argument position is an agent rather than a proposition alone, for example A i φ {\displaystyle A_{i}\varphi } ("agent i {\displaystyle i} does φ {\displaystyle \varphi } ") or C i φ {\displaystyle C_{i}\varphi } ("agent i {\displaystyle i} can bring about φ {\displaystyle \varphi } "). In contemporary literature, agentive logic is sometimes used narrowly for formal reconstructions of St. Anselm's modal account of facere ("to do"). More broadly, the term is used interchangeably with logic of action or logic of agency to cover a family of modal and dynamic logics designed to capture the structure of action and choice. == Historical background == === Medieval and early modern roots === Medieval logicians already explored analogies between modalities of action and alethic modalities such as possibility and necessity, for instance, in discussions of obligation and power. An influential early agentive analysis is due to St. Anselm (11th century), who treated "doing φ {\displaystyle \varphi } " as a kind of modal operator on propositions, anticipating later modal logics of agency. Modern reconstructions of Anselm's theory show that the resulting "agentive logic" can be modelled with neighbourhood semantics and satisfies a recognisable square of opposition. === Modern logic of action === Modern study of the logic of action began in the mid-20th century, parallel to developments in deontic logic and tense logic. Early systems were proposed by Georg Henrik von Wright, Stig Kanger, and others, often motivated by questions about norms and responsibility. From the 1960s onward, two largely independent but eventually converging traditions emerged: a branching-time tradition, culminating in STIT logics, emphasising agents' choices among possible futures; and dynamic logics of programs and actions, developed within computer science to reason about program execution. In the 1990s and 2000s, action logics were further developed in connection with knowledge representation, planning, and multi-agent systems in AI, and with dynamic and update semantics in linguistics. == Core ideas == Despite their diversity, most agentive logics share some general themes: Agents are treated as explicit indices of modal operators, as in [ i d o e s ] φ {\displaystyle [i\ {\mathsf {does}}]\varphi } or C i φ {\displaystyle C_{i}\varphi } . Actions are represented either implicitly, via changes between possible worlds along an accessibility relation, or explicitly, as terms denoting primitive and composite actions. Choice and ability are captured by modalities describing what an agent can ensure, usually relative to assumptions about the environment and other agents. Formal properties such as closure under composition, interaction between different agents, and connections to obligation (what an agent ought to do) and knowledge (what an agent knows how to do) are investigated. == STIT logics == STIT ("sees to it that") logics, originating in work by Nuel Belnap and collaborators, treat agency in a branching-time framework. A STIT model consists of a partially ordered set of moments with a tree-like structure, sets of histories (maximal branches through the tree), and for each agent at each moment, a partition of the histories through that moment representing the choices available to the agent. Intuitively, an agent's action at a moment determines which equivalence class (choice cell) of histories becomes actual; a formula [ i s t i t : φ ] {\displaystyle [i\ {\mathsf {stit}}:\varphi ]} is true at a history–moment pair if φ {\displaystyle \varphi } holds on all histories in the choice cell corresponding to the agent's current action. Different STIT operators have been distinguished, notably: the Chellas STIT operator, often written [ i c s t i t : φ ] {\displaystyle [i\ {\mathsf {cstit}}:\varphi ]} , which requires only that the agent's choice guarantees φ {\displaystyle \varphi } ; and the deliberative STIT operator, [ i d s t i t : φ ] {\displaystyle [i\ {\mathsf {dstit}}:\varphi ]} , which additionally requires that φ {\displaystyle \varphi } is not already historically necessary. STIT frameworks have been extended with group agency operators, temporal modalities, epistemic operators, and deontic operators to study responsibility, collective action, and obligations under indeterminism. == Dynamic logics of action == Dynamic logic was originally developed to reason about the behaviour of computer programs, treating program execution as a kind of action. In propositional dynamic logic (PDL), action terms α , β , … {\displaystyle \alpha ,\beta ,\dots } denote abstract programs or actions, and formulas of the form [ α ] φ {\displaystyle [\alpha ]\varphi } and ⟨ α ⟩ φ {\displaystyle \langle \alpha \rangle \varphi } express that all, respectively some, terminating executions of α {\displaystyle \alpha } lead to states where φ {\displaystyle \varphi } holds. From the standpoint of agentive logic, dynamic logic provides: a language for building complex actions from primitives via sequencing, choice, and iteration (e.g., α ; β {\displaystyle \alpha ;\beta } , α ∪ β {\displaystyle \alpha \cup \beta } , α ∗ {\displaystyle \alpha ^{}} ); a Kripke semantics in which actions correspond to labelled accessibility relations; and proof systems (such as Hoare logic and weakest precondition calculi) for reasoning about the correctness of action sequences. Extensions such as concurrent dynamic logic add operators for parallel composition, allowing reasoning about interacting processes and concurrent actions. John-Jules Ch. Meyer and others have argued that dynamic logic is a natural base for logics of agents, by adding modalities for knowledge, belief, and ability on top of the action modalities. Dynamic logics have also been applied to normative reasoning, yielding dynamic deontic logics where actions are related to obligations and permissions, and to dynamic epistemic logics in which information-changing actions such as announcements are modelled as programs. == Situation calculus and other action formalisms == In artificial intelligence, reasoning about action and change is often based on first-order languages that explicitly represent situations, events, and fluents (time-varying properties). The best known is situation calculus, introduced by John McCarthy and developed extensively by Raymond Reiter. In such formalisms: action terms name primitive actions; a function symbol (often d o {\displaystyle {\mathsf {do}}} ) maps an action and a situation to a successor situation; and axioms describe which fluents hold in which situations and how actions change them. Reiter's successor state axioms give compact specifications of how each fluent changes under all actions, and precondition axioms specify when actions are possible. Related formalisms include the event calculus and fluent calculus, which provide alternative ways of representing events and their effects. While these systems are often first-order rather than modal, they are closely related to agentive logics: their action terms and transition structures can be seen as providing models for dynamic or STIT-style modalities, and conversely, dynamic logics can be used as abstract specification languages for such AI formalisms. == Ability, agency, and related modalities == Many agentive logics introduce explicit operators for ability or "can-do"

  Read more →
• 

  Quadratic unconstrained binary optimization

  Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing. == Definition == Let B = { 0 , 1 } {\displaystyle \mathbb {B} =\lbrace 0,1\rbrace } the set of binary digits (or bits), then B n {\displaystyle \mathbb {B} ^{n}} is the set of binary vectors of fixed length n ∈ N {\displaystyle n\in \mathbb {N} } . Given a symmetric or upper triangular matrix Q ∈ R n × n {\displaystyle {\boldsymbol {Q}}\in \mathbb {R} ^{n\times n}} , whose entries Q i j {\displaystyle Q_{ij}} define a weight for each pair of indices i , j ∈ { 1 , … , n } {\displaystyle i,j\in \lbrace 1,\dots ,n\rbrace } , we can define the function f Q : B n → R {\displaystyle f_{\boldsymbol {Q}}:\mathbb {B} ^{n}\rightarrow \mathbb {R} } that assigns a value to each binary vector x {\displaystyle {\boldsymbol {x}}} through f Q ( x ) = x ⊺ Q x = ∑ i = 1 n ∑ j = 1 n Q i j x i x j . {\displaystyle f_{\boldsymbol {Q}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}}=\sum _{i=1}^{n}\sum _{j=1}^{n}Q_{ij}x_{i}x_{j}.} Alternatively, the linear and quadratic parts can be separated as f Q ′ , q ( x ) = x ⊺ Q ′ x + q ⊺ x , {\displaystyle f_{{\boldsymbol {Q}}',{\boldsymbol {q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Q}}'{\boldsymbol {x}}+{\boldsymbol {q}}^{\intercal }{\boldsymbol {x}},} where Q ′ ∈ R n × n {\displaystyle {\boldsymbol {Q}}'\in \mathbb {R} ^{n\times n}} and q ∈ R n {\displaystyle {\boldsymbol {q}}\in \mathbb {R} ^{n}} . This is equivalent to the previous definition through Q = Q ′ + diag ⁡ [ q ] {\displaystyle {\boldsymbol {Q}}={\boldsymbol {Q}}'+\operatorname {diag} [{\boldsymbol {q}}]} using the diag operator, exploiting that x = x ⋅ x {\displaystyle x=x\cdot x} for all binary values x {\displaystyle x} . Intuitively, the weight Q i j {\displaystyle Q_{ij}} is added if both x i = 1 {\displaystyle x_{i}=1} and x j = 1 {\displaystyle x_{j}=1} . The QUBO problem consists of finding a binary vector x ∗ {\displaystyle {\boldsymbol {x}}^{}} that minimizes f Q {\displaystyle f_{\boldsymbol {Q}}} , i.e., ∀ x ∈ B n : f Q ( x ∗ ) ≤ f Q ( x ) {\displaystyle \forall {\boldsymbol {x}}\in \mathbb {B} ^{n}:~f_{\boldsymbol {Q}}({\boldsymbol {x}}^{})\leq f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In general, x ∗ {\displaystyle {\boldsymbol {x}}^{}} is not unique, meaning there may be a set of minimizing vectors with equal value w.r.t. f Q {\displaystyle f_{\boldsymbol {Q}}} . The complexity of QUBO arises from the number of candidate binary vectors to be evaluated, as | B n | = 2 n {\displaystyle \left|\mathbb {B} ^{n}\right|=2^{n}} grows exponentially in n {\displaystyle n} . Sometimes, QUBO is defined as the problem of maximizing f Q {\displaystyle f_{\boldsymbol {Q}}} , which is equivalent to minimizing f − Q = − f Q {\displaystyle f_{-{\boldsymbol {Q}}}=-f_{\boldsymbol {Q}}} . == Properties == QUBO is scale invariant for positive factors α > 0 {\displaystyle \alpha >0} , which leave the optimum x ∗ {\displaystyle {\boldsymbol {x}}^{}} unchanged: f α Q ( x ) = x ⊺ ( α Q ) x = α ( x ⊺ Q x ) = α f Q ( x ) {\displaystyle f_{\alpha {\boldsymbol {Q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }(\alpha {\boldsymbol {Q}}){\boldsymbol {x}}=\alpha ({\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}})=\alpha f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In its general form, QUBO is NP-hard and cannot be solved efficiently by any known polynomial-time algorithm. However, there are polynomially-solvable special cases, where Q {\displaystyle {\boldsymbol {Q}}} has certain properties, for example: If all coefficients are positive, the optimum is trivially x ∗ = ( 0 , … , 0 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(0,\dots ,0)^{\intercal }} . Similarly, if all coefficients are negative, the optimum is x ∗ = ( 1 , … , 1 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(1,\dots ,1)^{\intercal }} . If Q {\displaystyle {\boldsymbol {Q}}} is diagonal, the bits can be optimized independently, and the problem is solvable in O ( n ) {\displaystyle {\mathcal {O}}(n)} . The optimal variable assignments are simply x i ∗ = 1 {\displaystyle x_{i}^{}=1} if Q i i < 0 {\displaystyle Q_{ii}<0} , and x i ∗ = 0 {\displaystyle x_{i}^{}=0} otherwise. If all off-diagonal elements of Q {\displaystyle {\boldsymbol {Q}}} are non-positive, the corresponding QUBO problem is solvable in polynomial time. QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer. This is possible since QUBO can be reformulated as a linear constrained binary optimization problem. To achieve this, substitute the product x i x j {\displaystyle x_{i}x_{j}} by an additional binary variable z i j ∈ B {\displaystyle z_{ij}\in \mathbb {B} } and add the constraints x i ≥ z i j {\displaystyle x_{i}\geq z_{ij}} , x j ≥ z i j {\displaystyle x_{j}\geq z_{ij}} and x i + x j − 1 ≤ z i j {\displaystyle x_{i}+x_{j}-1\leq z_{ij}} . Note that z i j {\displaystyle z_{ij}} can also be relaxed to continuous variables within the bounds zero and one. == Applications == QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas. === Maximum Cut === Given a graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V = { 1 , … , n } {\displaystyle V=\lbrace 1,\dots ,n\rbrace } and edges E ⊆ V × V {\displaystyle E\subseteq V\times V} , the maximum cut (max-cut) problem consists of finding two subsets S , T ⊆ V {\displaystyle S,T\subseteq V} with T = V ∖ S {\displaystyle T=V\setminus S} , such that the number of edges between S {\displaystyle S} and T {\displaystyle T} is maximized. The more general weighted max-cut problem assumes edge weights w i j ≥ 0 ∀ i , j ∈ V {\displaystyle w_{ij}\geq 0~\forall i,j\in V} , with ( i , j ) ∉ E ⇒ w i j = 0 {\displaystyle (i,j)\notin E\Rightarrow w_{ij}=0} , and asks for a partition S , T ⊆ V {\displaystyle S,T\subseteq V} that maximizes the sum of edge weights between S {\displaystyle S} and T {\displaystyle T} , i.e., max S ⊆ V ∑ i ∈ S , j ∉ S w i j . {\displaystyle \max _{S\subseteq V}\sum _{i\in S,j\notin S}w_{ij}.} By setting w i j = 1 {\displaystyle w_{ij}=1} for all ( i , j ) ∈ E {\displaystyle (i,j)\in E} this becomes equivalent to the original max-cut problem above, which is why we focus on this more general form in the following. For every vertex in i ∈ V {\displaystyle i\in V} we introduce a binary variable x i {\displaystyle x_{i}} with the interpretation x i = 0 {\displaystyle x_{i}=0} if i ∈ S {\displaystyle i\in S} and x i = 1 {\displaystyle x_{i}=1} if i ∈ T {\displaystyle i\in T} . As T = V ∖ S {\displaystyle T=V\setminus S} , every i {\displaystyle i} is in exactly one set, meaning there is a 1:1 correspondence between binary vectors x ∈ B n {\displaystyle {\boldsymbol {x}}\in \mathbb {B} ^{n}} and partitions of V {\displaystyle V} into two subsets. We observe that, for any i , j ∈ V {\displaystyle i,j\in V} , the expression x i ( 1 − x j ) + ( 1 − x i ) x j {\displaystyle x_{i}(1-x_{j})+(1-x_{i})x_{j}} evaluates to 1 if and only if i {\displaystyle i} and j {\displaystyle j} are in different subsets, equivalent to logical XOR. Let W ∈ R + n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} _{+}^{n\times n}} with W i j = w i j ∀ i , j ∈ V {\displaystyle W_{ij}=w_{ij}~\forall i,j\in V} . By extending above expression to matrix-vector form we find that x ⊺ W ( 1 − x ) + ( 1 − x ) ⊺ W x = − 2 x ⊺ W x + ( W 1 + W ⊺ 1 ) ⊺ x {\displaystyle {\boldsymbol {x}}^{\intercal }{\boldsymbol {W}}({\boldsymbol {1}}-{\boldsymbol {x}})+({\boldsymbol {1}}-{\boldsymbol {x}})^{\intercal }{\boldsymbol {Wx}}=-2{\boldsymbol {x}}^{\intercal }{\boldsymbol {Wx}}+({\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\boldsymbol {1}})^{\intercal }{\boldsymbol {x}}} is the sum of weights of all edges between S {\displaystyle S} and T {\displaystyle T} , where 1 = ( 1 , 1 , … , 1 ) ⊺ ∈ R n {\displaystyle {\boldsymbol {1}}=(1,1,\dots ,1)^{\intercal }\in \mathbb {R} ^{n}} . As this is a quadratic function over x {\displaystyle {\boldsymbol {x}}} , it is a QUBO problem whose parameter matrix we can read from above expression as Q = 2 W − diag ⁡ [ W 1 + W ⊺ 1 ] , {\displaystyle {\boldsymbol {Q}}=2{\boldsymbol {W}}-\operatorname {diag} [{\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\bol

  Read more →
• 

  Self-organizing map

  A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. For example, a data set with p {\displaystyle p} variables measured in n {\displaystyle n} observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a two-dimensional "map" such that observations in proximal clusters have more similar values than observations in distal clusters. This can make high-dimensional data easier to visualize and analyze. A SOM is a type of artificial neural network but is trained using competitive learning rather than the error-correction learning (e.g., backpropagation with gradient descent) used by other artificial neural networks. The SOM was introduced by the Finnish professor Teuvo Kohonen in the 1980s and therefore is sometimes called a Kohonen map or Kohonen network. The Kohonen map or network is a computationally convenient abstraction building on biological models of neural systems from the 1970s and morphogenesis models dating back to Alan Turing in the 1950s. SOMs create internal representations reminiscent of the cortical homunculus, a distorted representation of the human body, based on a neurological "map" of the areas and proportions of the human brain dedicated to processing sensory functions, for different parts of the body. == Overview == Self-organizing maps, like most artificial neural networks, operate in two modes: training and mapping. First, training uses an input data set (the "input space") to generate a lower-dimensional representation of the input data (the "map space"). Second, mapping classifies additional input data using the generated map. The goal of training is to represent an input space with p dimensions as a map space with n dimensions, where p > n. Specifically, an input space with p variables is said to have p dimensions. A map space consists of components called "nodes" or "neurons", which are arranged as a hexagonal or rectangular grid with two dimensions. The number of nodes and their arrangement are specified beforehand based on the larger goals of the analysis and exploration of the data. Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After training, the map can be used to classify additional observations for the input space by finding the node with the closest weight vector (smallest distance metric) to the input space vector. == Learning algorithm == The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain. The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations. The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron whose weight vector is most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM grid are adjusted towards the input vector. The magnitude of the change decreases with time and with the grid-distance from the BMU. The update formula for a neuron v with weight vector Wv(s) is W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} , where s is the step index, t is an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s. Depending on the implementations, t can scan the training data set systematically (t is 0, 1, 2...T-1, then repeat, T being the training sample's size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function θ(u, v, s) (also called function of lateral interaction) depends on the grid-distance between the BMU (neuron u) and neuron v. In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and Mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates. In some implementations, the learning coefficient α and the neighborhood function θ decrease steadily with increasing s, in others (in particular those where t scans the training data set) they decrease in step-wise fashion, once every T steps. This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net. During mapping, there will be one single winning neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector. While representing input data as vectors has been emphasized in this article, any kind of object which can be represented digitally, which has an appropriate distance measure associated with it, and in which the necessary operations for training are possible can be used to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps. === Algorithm === Randomize the node weight vectors in a map For s = 0 , 1 , 2 , . . . , λ {\displaystyle s=0,1,2,...,\lambda } Randomly pick an input vector D ( t ) {\displaystyle {D}(t)} Find the node in the map closest to the input vector. This node is the best matching unit (BMU). Denote it by u {\displaystyle u} For each node v {\displaystyle v} , update its vector by pulling it closer to the input vector: W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} The variable names mean the following, with vectors in bold, s {\displaystyle s} is the current iteration λ {\displaystyle \lambda } is the iteration limit t {\displaystyle t} is the index of the target input data vector in the input data set D {\displaystyle \mathbf {D} } D ( t ) {\displaystyle {D}(t)} is a target input data vector v {\displaystyle v} is the index of the node in the map W v {\displaystyle \mathbf {W} _{v}} is the current weight vector of node v {\displaystyle v} u {\displaystyle u} is the index of the best matching unit (BMU) in the map θ ( u , v , s ) {\displaystyle \theta (u,v,s)} is the neighbourhood function, α ( s ) {\displaystyle \alpha (s)} is the learning rate schedule. The key design choices are the shape of the SOM, the neighbourhood function, and the learning rate schedule. The idea of the neighborhood function is to make it such that the BMU is updated the most, its immediate neighbors are updated a little less, and so on. The idea of the learning rate schedule is to make it so that the map updates are large at the start, and gradually stop updating. For example, if we want to learn a SOM using a square grid, we can index it using ( i , j ) {\displaystyle (i,j)} where both i , j ∈ 1 : N {\displaystyle i,j\in 1:N} . The neighborhood function can make it so that the BMU updates in full, the nearest neighbors update in half, and their neighbors update in half again, etc. θ ( ( i , j ) , ( i ′ , j ′ ) , s ) = 1 2 | i − i ′ | + | j − j ′ | = { 1 if i = i ′ , j = j ′ 1 / 2 if | i − i ′ | + | j − j ′ | = 1 1 / 4 if | i − i ′ | + | j − j ′ | = 2 ⋯ ⋯ {\displaystyle \theta ((i,j),(i',j'),s)={\frac {1}{2^{|i-i'|+|j-j'|}}}={\begin{cases}1&{\text{if }}i=i',j=j'\\1/2&{\text{if

  Read more →