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Visual descriptor

In computer vision, visual descriptors or image descriptors are descriptions of the visual features of the contents in images, videos, or algorithms or applications that produce such descriptions. They describe elementary characteristics such as the shape, the color, the texture or the motion, among others. == Introduction == As a result of the new communication technologies and the massive use of Internet in our society, the amount of audio-visual information available in digital format is increasing considerably. Therefore, it has been necessary to design some systems that allow us to describe the content of several types of multimedia information in order to search and classify them. The audio-visual descriptors are in charge of the contents description. These descriptors have a good knowledge of the objects and events found in a video, image or audio and they allow the quick and efficient searches of the audio-visual content. This system can be compared to the search engines for textual contents. Although it is relatively easy to find text with a computer, it is much more difficult to find concrete audio and video parts. For instance, imagine somebody searching a scene of a happy person. The happiness is a feeling and it is not evident its shape, color and texture description in images. The description of the audio-visual content is not a superficial task and it is essential for the effective use of this type of archives. The standardization system that deals with audio-visual descriptors is the MPEG-7 (Motion Picture Expert Group - 7). == Types == Descriptors are the first step to find out the connection between pixels contained in a digital image and what humans recall after having observed an image or a group of images after some minutes. Visual descriptors are divided in two main groups: General information descriptors: contain low level descriptors which give a description about color, shape, regions, textures and motion. Specific domain information descriptors: give information about objects and events in the scene. A concrete example would be face recognition. === General information descriptors === General information descriptors consist of a set of descriptors that covers different basic and elementary features like: color, texture, shape, motion, location and others. This description is automatically generated by means of signal processing. ==== Color ==== It's the most basic quality of visual content. Five tools are defined to describe color. The three first tools represent the color distribution and the last ones describe the color relation between sequences or group of images: Dominant color descriptor (DCD) Scalable color descriptor (SCD) Color structure descriptor (CSD) Color layout descriptor (CLD) Group of frame (GoF) or group-of-pictures (GoP) ==== Texture ==== It's an important quality in order to describe an image. The texture descriptors characterize image textures or regions. They observe the region homogeneity and the histograms of these region borders. The set of descriptors is formed by: Homogeneous texture descriptor (HTD) Texture browsing descriptor (TBD) Edge histogram descriptor (EHD) ==== Shape ==== It contains important semantic information due to human's ability to recognize objects through their shape. However, this information can only be extracted by means of a segmentation similar to the one that the human visual system implements. Nowadays, such a segmentation system is not available yet, however there exists a serial of algorithms which are considered to be a good approximation. These descriptors describe regions, contours and shapes for 2D images and for 3D volumes. The shape descriptors are the following ones: Region-based shape descriptor (RSD) Contour-based shape descriptor (CSD) 3-D shape descriptor (3-D SD) ==== Motion ==== It's defined by four different descriptors which describe motion in video sequence. Motion is related to the objects motion in the sequence and to the camera motion. This last information is provided by the capture device, whereas the rest is implemented by means of image processing. The descriptor set is the following one: Motion activity descriptor (MAD) Camera motion descriptor (CMD) Motion trajectory descriptor (MTD) Warping and parametric motion descriptor (WMD and PMD) ==== Location ==== Elements location in the image is used to describe elements in the spatial domain. In addition, elements can also be located in the temporal domain: Region locator descriptor (RLD) Spatio temporal locator descriptor (STLD) === Specific domain information descriptors === These descriptors, which give information about objects and events in the scene, are not easily extractable, even more when the extraction is to be automatically done. Nevertheless, they can be manually processed. As mentioned before, face recognition is a concrete example of an application that tries to automatically obtain this information. == Descriptors applications == Among all applications, the most important ones are: Multimedia documents search engines and classifiers. Digital library: visual descriptors allow a very detailed and concrete search of any video or image by means of different search parameters. For instance, the search of films where a known actor appears, the search of videos containing the Everest mountain, etc. Personalized electronic news service. Possibility of an automatic connection to a TV channel broadcasting a soccer match, for example, whenever a player approaches the goal area. Control and filtering of concrete audiovisual content, like violent or pornographic material. Also, authorization for some multimedia content.
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Minimum Population Search

In evolutionary computation, Minimum Population Search (MPS) is a computational method that optimizes a problem by iteratively trying to improve a set of candidate solutions with regard to a given measure of quality. It solves a problem by evolving a small population of candidate solutions by means of relatively simple arithmetical operations. MPS is a metaheuristic as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. For problems where finding the precise global optimum is less important than finding an acceptable local optimum in a fixed amount of time, using a metaheuristic such as MPS may be preferable to alternatives such as brute-force search or gradient descent. MPS is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means MPS does not require for the optimization problem to be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods. MPS can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. == Background == In a similar way to Differential evolution, MPS uses difference vectors between the members of the population in order to generate new solutions. It attempts to provide an efficient use of function evaluations by maintaining a small population size. If the population size is smaller than the dimensionality of the search space, then the solutions generated through difference vectors will be constrained to the n − 1 {\displaystyle n-1} dimensional hyperplane. A smaller population size will lead to a more restricted subspace. With a population size equal to the dimensionality of the problem ( n = d ) {\displaystyle (n=d)} , the “line/hyperplane points” in MPS will be generated within a d − 1 {\displaystyle d-1} dimensional hyperplane. Taking a step orthogonal to this hyperplane will allow the search process to cover all the dimensions of the search space. Population size is a fundamental parameter in the performance of population-based heuristics. Larger populations promote exploration, but they also allow fewer generations, and this can reduce the chance of convergence. Searching with a small population can increase the chances of convergence and the efficient use of function evaluations, but it can also induce the risk of premature convergence. If the risk of premature convergence can be avoided, then a population-based heuristic could benefit from the efficiency and faster convergence rate of a smaller population. To avoid premature convergence, it is important to have a diversified population. By including techniques for explicitly increasing diversity and exploration, it is possible to have smaller populations with less risk of premature convergence. === Thresheld Convergence === Thresheld Convergence (TC) is a diversification technique which attempts to separate the processes of exploration and exploitation. TC uses a “threshold” function to establish a minimum search step, and managing this step makes it possible to influence the transition from exploration to exploitation, convergence is thus “held” back until the last stages of the search process. The goal of a controlled transition is to avoid an early concentration of the population around a few search regions and avoid the loss of diversity which can cause premature convergence. Thresheld Convergence has been successfully applied to several population-based metaheuristics such as Particle Swarm Optimization, Differential evolution, Evolution strategies, Simulated annealing and Estimation of Distribution Algorithms. The ideal case for Thresheld Convergence is to have one sample solution from each attraction basin, and for each sample solution to have the same relative fitness with respect to its local optimum. Enforcing a minimum step aims to achieve this ideal case. In MPS Thresheld Convergence is specifically used to preserve diversity and avoid premature convergence by establishing a minimum search step. By disallowing new solutions which are too close to members of the current population, TC forces a strong exploration during the early stages of the search while preserving the diversity of the (small) population. == Algorithm == A basic variant of the MPS algorithm works by having a population of size equal to the dimension of the problem. New solutions are generated by exploring the hyperplane defined by the current solutions (by means of difference vectors) and performing an additional orthogonal step in order to avoid getting caught in this hyperplane. The step sizes are controlled by the Thresheld Convergence technique, which gradually reduces step sizes as the search process advances. An outline for the algorithm is given below: Generate the first initial population. Allowing these solutions to lie near the bounds of the search space generally gives good results: s k = ( r s 1 ∗ b o u n d 1 / 2 , r s 2 ∗ b o u n d 2 / 2 , . . . , r s n ∗ b o u n d n / 2 ) {\displaystyle s_{k}=(rs_{1}bound_{1}/2,rs_{2}bound_{2}/2,...,rs_{n}bound_{n}/2)} where s k {\displaystyle s_{k}} is the k {\displaystyle k} -th population member, r s i {\displaystyle rs_{i}} are random numbers which can be −1 or 1, and the b o u n d i {\displaystyle bound_{i}} are the lower and upper bounds on each dimension. While a stop condition is not reached: Update threshold convergence values ( m i n _ s t e p {\displaystyle min\_step} and m a x _ s t e p {\displaystyle max\_step} ) Calculate the centroid of the current population ( x c {\displaystyle x_{c}} ) For each member of the population ( x i {\displaystyle x_{i}} ), generate a new offspring as follows: Uniformly generate a scaling factor ( F i {\displaystyle F_{i}} ) between − m a x _ s t e p {\displaystyle -max\_step} and m a x _ s t e p {\displaystyle max\_step} Generate a vector ( x o {\displaystyle x_{o}} ) orthogonal to the difference vector between x i {\displaystyle x_{i}} and x c {\displaystyle x_{c}} Calculate a scaling factor for the orthogonal vector: m i n _ o r t h = s q r t ( m a x ( m i n _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle min\_orth=sqrt(max(min\_step^{2}-F_{i}^{2},0))} m a x _ o r t h = s q r t ( m a x ( m a x _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle max\_orth=sqrt(max(max\_step^{2}-F_{i}^{2},0))} o r t h _ s t e p = u n i f o r m ( m i n _ o r t h , m a x _ o r t h ) {\displaystyle orth\_step=uniform(min\_orth,max\_orth)} Generate the new solution by adding the difference and the orthogonal vectors to the original solution n e w _ s o l u t i o n = x i + F i ∗ ( x i − x c ) ∗ o r t h _ s t e p ∗ x o {\displaystyle new\_solution=x_{i}+F_{i}(x_{i}-x_{c})orth\_stepx_{o}} Pick the best members between the old population and the new one by discarding the least fit members. Return the single best solution or the best population found as the final result.
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Characteristic samples

Characteristic samples is a concept in the field of grammatical inference, related to passive learning. In passive learning, an inference algorithm I {\displaystyle I} is given a set of pairs of strings and labels S {\displaystyle S} , and returns a representation R {\displaystyle R} that is consistent with S {\displaystyle S} . Characteristic samples consider the scenario when the goal is not only finding a representation consistent with S {\displaystyle S} , but finding a representation that recognizes a specific target language. A characteristic sample of language L {\displaystyle L} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} where: l ( s ) = 1 {\displaystyle l(s)=1} if and only if s ∈ L {\displaystyle s\in L} l ( s ) = − 1 {\displaystyle l(s)=-1} if and only if s ∉ L {\displaystyle s\notin L} Given the characteristic sample S {\displaystyle S} , I {\displaystyle I} 's output on it is a representation R {\displaystyle R} , e.g. an automaton, that recognizes L {\displaystyle L} . == Formal Definition == === The Learning Paradigm associated with Characteristic Samples === There are three entities in the learning paradigm connected to characteristic samples, the adversary, the teacher and the inference algorithm. Given a class of languages C {\displaystyle \mathbb {C} } and a class of representations for the languages R {\displaystyle \mathbb {R} } , the paradigm goes as follows: The adversary A {\displaystyle A} selects a language L ∈ C {\displaystyle L\in \mathbb {C} } and reports it to the teacher The teacher T {\displaystyle T} then computes a set of strings and label them correctly according to L {\displaystyle L} , trying to make sure that the inference algorithm will compute L {\displaystyle L} The adversary can add correctly labeled words to the set in order to confuse the inference algorithm The inference algorithm I {\displaystyle I} gets the sample and computes a representation R ∈ R {\displaystyle R\in \mathbb {R} } consistent with the sample. The goal is that when the inference algorithm receives a characteristic sample for a language L {\displaystyle L} , or a sample that subsumes a characteristic sample for L {\displaystyle L} , it will return a representation that recognizes exactly the language L {\displaystyle L} . === Sample === Sample S {\displaystyle S} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} such that l ( s ) ∈ { − 1 , 1 } {\displaystyle l(s)\in \{-1,1\}} ==== Sample consistent with a language ==== We say that a sample S {\displaystyle S} is consistent with language L {\displaystyle L} if for every pair ( s , l ( s ) ) {\displaystyle (s,l(s))} in S {\displaystyle S} : l ( s ) = 1 if and only if s ∈ L {\displaystyle l(s)=1{\text{ if and only if }}s\in L} l ( s ) = − 1 if and only if s ∉ L {\displaystyle l(s)=-1{\text{ if and only if }}s\notin L} === Characteristic sample === Given an inference algorithm I {\displaystyle I} and a language L {\displaystyle L} , a sample S {\displaystyle S} that is consistent with L {\displaystyle L} is called a characteristic sample of L {\displaystyle L} for I {\displaystyle I} if: I {\displaystyle I} 's output on S {\displaystyle S} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . For every sample D {\displaystyle D} that is consistent with L {\displaystyle L} and also fulfils S ⊆ D {\displaystyle S\subseteq D} , I {\displaystyle I} 's output on D {\displaystyle D} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . A Class of languages C {\displaystyle \mathbb {C} } is said to have charistaristic samples if every L ∈ C {\displaystyle L\in \mathbb {C} } has a characteristic sample. == Related Theorems == === Theorem === If equivalence is undecidable for a class C {\textstyle \mathbb {C} } over Σ {\textstyle \Sigma } of cardinality bigger than 1, then C {\textstyle \mathbb {C} } doesn't have characteristic samples. ==== Proof ==== Given a class of representations C {\textstyle \mathbb {C} } such that equivalence is undecidable, for every polynomial p ( x ) {\displaystyle p(x)} and every n ∈ N {\displaystyle n\in \mathbb {N} } , there exist two representations r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} of sizes bounded by n {\displaystyle n} , that recognize different languages but are inseparable by any string of size bounded by p ( n ) {\displaystyle p(n)} . Assuming this is not the case, we can decide if r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} are equivalent by simulating their run on all strings of size smaller than p ( n ) {\displaystyle p(n)} , contradicting the assumption that equivalence is undecidable. === Theorem === If S 1 {\displaystyle S_{1}} is a characteristic sample for L 1 {\displaystyle L_{1}} and is also consistent with L 2 {\displaystyle L_{2}} , then every characteristic sample of L 2 {\displaystyle L_{2}} , is inconsistent with L 1 {\displaystyle L_{1}} . ==== Proof ==== Given a class C {\textstyle \mathbb {C} } that has characteristic samples, let R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} be representations that recognize L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} respectively. Under the assumption that there is a characteristic sample for L 1 {\displaystyle L_{1}} , S 1 {\displaystyle S_{1}} that is also consistent with L 2 {\displaystyle L_{2}} , we'll assume falsely that there exist a characteristic sample for L 2 {\displaystyle L_{2}} , S 2 {\displaystyle S_{2}} that is consistent with L 1 {\displaystyle L_{1}} . By the definition of characteristic sample, the inference algorithm I {\displaystyle I} must return a representation which recognizes the language if given a sample that subsumes the characteristic sample itself. But for the sample S 1 ∪ S 2 {\displaystyle S_{1}\cup S_{2}} , the answer of the inferring algorithm needs to recognize both L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , in contradiction. === Theorem === If a class is polynomially learnable by example based queries, it is learnable with characteristic samples. == Polynomialy characterizable classes == === Regular languages === The proof that DFA's are learnable using characteristic samples, relies on the fact that every regular language has a finite number of equivalence classes with respect to the right congruence relation, ∼ L {\displaystyle \sim _{L}} (where x ∼ L y {\displaystyle x\sim _{L}y} for x , y ∈ Σ ∗ {\displaystyle x,y\in \Sigma ^{}} if and only if ∀ z ∈ Σ ∗ : x z ∈ L ↔ y z ∈ L {\displaystyle \forall z\in \Sigma ^{}:xz\in L\leftrightarrow yz\in L} ). Note that if x {\displaystyle x} , y {\displaystyle y} are not congruent with respect to ∼ L {\displaystyle \sim _{L}} , there exists a string z {\displaystyle z} such that x z ∈ L {\displaystyle xz\in L} but y z ∉ L {\displaystyle yz\notin L} or vice versa, this string is called a separating suffix. ==== Constructing a characteristic sample ==== The construction of a characteristic sample for a language L {\displaystyle L} by the teacher goes as follows. Firstly, by running a depth first search on a deterministic automaton A {\displaystyle A} recognizing L {\displaystyle L} , starting from its initial state, we get a suffix closed set of words, W {\displaystyle W} , ordered in shortlex order. From the fact above, we know that for every two states in the automaton, there exists a separating suffix that separates between every two strings that the run of A {\displaystyle A} on them ends in the respective states. We refer to the set of separating suffixes as S {\displaystyle S} . The labeled set (sample) of words the teacher gives the adversary is { ( w , l ( w ) ) | w ∈ W ⋅ S ∪ W ⋅ Σ ⋅ S } {\displaystyle \{(w,l(w))|w\in W\cdot S\cup W\cdot \Sigma \cdot S\}} where l ( w ) {\displaystyle l(w)} is the correct label of w {\displaystyle w} (whether it is in L {\displaystyle L} or not). We may assume that ϵ ∈ S {\displaystyle \epsilon \in S} . ==== Constructing a deterministic automata ==== Given the sample from the adversary W {\displaystyle W} , the construction of the automaton by the inference algorithm I {\displaystyle I} starts with defining P = prefix ( W ) {\displaystyle P={\text{prefix}}(W)} and S = suffix ( W ) {\displaystyle S={\text{suffix}}(W)} , which are the set of prefixes and suffixes of W {\displaystyle W} respectively. Now the algorithm constructs a matrix M {\displaystyle M} where the elements of P {\displaystyle P} function as the rows, ordered by the shortlex order, and the elements of S {\displaystyle S} function as the columns, ordered by the shortlex order. Next, the cells in the matrix are filled in the following manner for prefix p i {\displaystyle p_{i}} and suffix s j {\displaystyle s_{j}} : If p i s j ∈ W → M i j = l ( p i s j ) {\displaystyle p_{i}s_{j}\in W\rightarrow M_{ij}=l(p_{i}s_{j})} else, M i j = 0 {\displaystyle M_{ij}=0} Now, we say row i {\displaystyle i} and t {\displaystyle t} are distinguishable if there exi
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Soft independent modelling of class analogies

Soft independent modelling by class analogy (SIMCA) is a statistical method for supervised classification of data. The method requires a training data set consisting of samples (or objects) with a set of attributes and their class membership. The term soft refers to the fact the classifier can identify samples as belonging to multiple classes and not necessarily producing a classification of samples into non-overlapping classes. == Method == In order to build the classification models, the samples belonging to each class need to be analysed using principal component analysis (PCA); only the significant components are retained. For a given class, the resulting model then describes either a line (for one Principal Component or PC), plane (for two PCs) or hyper-plane (for more than two PCs). For each modelled class, the mean orthogonal distance of training data samples from the line, plane, or hyper-plane (calculated as the residual standard deviation) is used to determine a critical distance for classification. This critical distance is based on the F-distribution and is usually calculated using 95% or 99% confidence intervals. New observations are projected into each PC model and the residual distances calculated. An observation is assigned to the model class when its residual distance from the model is below the statistical limit for the class. The observation may be found to belong to multiple classes and a measure of goodness of the model can be found from the number of cases where the observations are classified into multiple classes. The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant. == Application == SIMCA as a method of classification has gained widespread use especially in applied statistical fields such as chemometrics and spectroscopic data analysis.
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Process map

Process map is a global-system process model that is used to outline the processes that make up the business system and how they interact with each other. Process map shows the processes as objects, which means it is a static and non-algorithmic view of the processes. It should be differentiated from a detailed process model, which shows a dynamic and algorithmic view of the processes, usually known as a process flow diagram. There are different notation standards that can be used for modelling process maps, but the most notable ones are TOGAF Event Diagram, Eriksson-Penker notation, and ARIS Value Added Chain. == Global process models == Global characteristics of the business system are captured by global or system models. Global process models are presented using different methodologies and sometimes under different names. Most notably, they are named process map in Visual Paradigm and MMABP, value-added chain in ARIS, and process diagram in Eriksson-Penker notation – which can easily lead to the confusion with process flow (detailed process model). Global models are mainly object-oriented and present a static view of the business system; they do not describe dynamic aspects of processes. A process map shows the presence of processes and their mutual relationships. The requirement for the global perspective of the system as a supplementary to the internal process logic description results from the necessity of taking into consideration not only the internal process logic but also its significant surroundings. The algorithmic process model cannot take the place of this perspective since it represents the system model of the process. The detailed process model and the global process model represent different perspectives on the same business system, so these models must be mutually consistent. A macro process map represents the major processes required to deliver a product or service to the customer. These macro process maps can be further detailed in sub-diagrams. It is often the case that process maps cross different functional areas of the organization. Process maps are used by many companies to have a holistic view of all processes and the connections between them. Maps help in navigating the sub-processes and make understanding of the organization's operations easier. The process map shows relationships and dependencies between processes and its focus should be on core business processes of the organization. A process map can be seen as the most abstract level of the process architecture, and it acts as the introduction to the more detailed levels. A process map that is correctly designed is able to provide a general understanding of a company's operations. Designing the process map is an important and strategic step for the organization, and it is followed by further business process modelling implementation. == Context == Methodology for Modelling and Analysis of Business Process (MMABP) is a business process modelling methodology developed at the Department of Information Technology, Faculty of Informatics and Statistics of the Prague University of Economics and Business. The methodology is defined as a “general methodology for modelling business systems using informatics methods and approaches”. Methodology is used to analyse business processes and to develop a comprehensive model of the system. The goal of developing a model is to be used for process optimization. The model should be created following the characteristics and specifics of the organization in question and following external influences that can affect the organization. The model should be optimal from an economic perspective, but it should also be optimal from a factual perspective, meaning that it should be as simple as possible while maintaining complete functionality. Business system modelling is based on a two-dimensional approach: Real World structure (substance) – set of objects and their relationships Real World behaviour – set of mutually connected business processes Additionally, there are also two views of the systems: Global view of the system Detailed view of the system's parts This results in the need to model the system from four different perspectives in order to achieve the complete and comprehensive view of the business system. MMABP also proposes which notation languages can be used for modelling each perspective, and it also suggests some improvements to the notation languages in order to fit the purpose. Global view of the objects – Conceptual model (Class diagram) Detailed view of the objects – Object life cycle (State Chart) Global view of the processes – Process map (Eriksson-Penker Diagram/TOGAF Event Diagram/ARIS VAC) Detailed view of the processes – Model of the process flow (BPMN Diagram) Data Flow Diagram (DFD) is additional diagram used for describing the required functionalities of the information system. == Notation standards == === Eriksson-Penker Diagram === Eriksson-Penker diagram is a tool used in business model analysis and design. It is named after Hans-Erik Eriksson and Magnus Penker, who developed the concept in their book "Business modelling with UML: Business Patterns at Work”. Eriksson-Penker diagrams are used to map out the key components of a business model and how they interact with one another. The diagrams typically consist of a series of boxes and lines that represent the different elements of the business model, such as the value proposition, customer segments, channels, revenue streams, and key resources. The lines between the boxes represent the relationships and dependencies between the different elements of the business model. These diagrams are useful for visualizing and understanding the various components of a business model, and can help organizations identify potential areas for improvement or areas of risk. They can also be used as a communication tool to help stakeholders understand the business model and its underlying assumptions. These diagrams are useful for visualizing and understanding the various components of a business model, and can help organizations identify potential areas for improvement or areas of risk. They can also be used as a communication tool to help stakeholders understand the business model and its underlying assumptions. It is possible to use Eriksson-Penker diagrams to create a global process view of a business. In this case, a diagram would be used to map out the key processes and activities that are involved in the business, as well as the relationships and dependencies between these processes. For example, an Eriksson-Penker diagram could be used to depict the various steps involved in the product development process, from concept development to market launch. It could also be used to show how different functions within the organization, such as marketing, sales, and production, interact and depend on one another to support the overall business. Eriksson-Penker diagram is one of the most popular de facto standards that can be used for an object-oriented global view of business processes. It is developed as an extension of the UML, and it is often used together with the BPMN to compensate for the lack of possibility to model the global view with this widely accepted standard. === TOGAF Event Diagram === TOGAF (The Open Group Architecture Framework) is a framework for enterprise architecture that provides a common language and set of standards for designing, planning, implementing, and governing an enterprise's IT architecture. TOGAF event diagrams are diagrams used in the TOGAF framework to represent the flow of events within a system or process. The TOGAF Event Diagram is a visual representation of the events within an organization or system. It can be used to show the sequence of events that occur in a particular process, as well as the relationships between the events and the stakeholders involved. TOGAF Event Diagrams can be useful in creating a global process view because they provide a visual representation of the events, which can be helpful in understanding how the process fits into the larger context of the organization. TOGAF Event Diagram is the most perspective standard for the system view of processes today. It is used to represent the system of processes as well as their connections to the functional organizational structure. === ARIS Value Added Chain === ARIS (Architecture of Integrated Information Systems) is a methodology and a set of tools for designing and managing business processes. It is based on the idea that business processes are the core of an organization and that they can be modelled and optimized to improve efficiency and effectiveness. The ARIS methodology provides a framework for understanding and analysing business processes, as well as for designing and implementing improvements to those processes. It includes a set of graphical modelling languages and tools for creating process models, as well as a database for storing and managing pr
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Sharpness aware minimization

Sharpness Aware Minimization (SAM) is an optimization algorithm used in machine learning that aims to improve model generalization. The method seeks to find model parameters that are located in regions of the loss landscape with uniformly low loss values, rather than parameters that only achieve a minimal loss value at a single point. This approach is described as finding "flat" minima instead of "sharp" ones. The rationale is that models trained this way are less sensitive to variations between training and test data, which can lead to better performance on unseen data. The algorithm was introduced in a 2020 paper by a team of researchers including Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. == Underlying Principle == SAM modifies the standard training objective by minimizing a "sharpness-aware" loss. This is formulated as a minimax problem where the inner objective seeks to find the highest loss value in the immediate neighborhood of the current model weights, and the outer objective minimizes this value: min w max ‖ ϵ ‖ p ≤ ρ L train ( w + ϵ ) + λ ‖ w ‖ 2 2 {\displaystyle \min _{w}\max _{\|\epsilon \|_{p}\leq \rho }L_{\text{train}}(w+\epsilon )+\lambda \|w\|_{2}^{2}} In this formulation: w {\displaystyle w} represents the model's parameters (weights). L train {\displaystyle L_{\text{train}}} is the loss calculated on the training data. ϵ {\displaystyle \epsilon } is a perturbation applied to the weights. ρ {\displaystyle \rho } is a hyperparameter that defines the radius of the neighborhood (an L p {\displaystyle L_{p}} ball) to search for the highest loss. An optional L2 regularization term, scaled by λ {\displaystyle \lambda } , can be included. A direct solution to the inner maximization problem is computationally expensive. SAM approximates it by taking a single gradient ascent step to find the perturbation ϵ {\displaystyle \epsilon } . This is calculated as: ϵ ( w ) = ρ ∇ L train ( w ) ‖ ∇ L train ( w ) ‖ 2 {\displaystyle \epsilon (w)=\rho {\frac {\nabla L_{\text{train}}(w)}{\|\nabla L_{\text{train}}(w)\|_{2}}}} The optimization process for each training step involves two stages. First, an "ascent step" computes a perturbed set of weights, w adv = w + ϵ ( w ) {\displaystyle w_{\text{adv}}=w+\epsilon (w)} , by moving towards the direction of the highest local loss. Second, a "descent step" updates the original weights w {\displaystyle w} using the gradient calculated at these perturbed weights, ∇ L train ( w adv ) {\displaystyle \nabla L_{\text{train}}(w_{\text{adv}})} . This update is typically performed using a standard optimizer like SGD or Adam. == Application and Performance == SAM has been applied in various machine learning contexts, primarily in computer vision. Research has shown it can improve generalization performance in models such as Convolutional Neural Networks (CNNs) and Vision Transformers (ViTs) on image datasets including ImageNet, CIFAR-10, and CIFAR-100. The algorithm has also been found to be effective in training models with noisy labels, where it performs comparably to methods designed specifically for this problem. Some studies indicate that SAM and its variants can improve out-of-distribution (OOD) generalization, which is a model's ability to perform well on data from distributions not seen during training. Other areas where it has been applied include gradual domain adaptation and mitigating overfitting in scenarios with repeated exposure to training examples. == Limitations == A primary limitation of SAM is its computational cost. By requiring two gradient computations (one for the ascent and one for the descent) per optimization step, it approximately doubles the training time compared to standard optimizers. The theoretical convergence properties of SAM are still under investigation. Some research suggests that with a constant step size, SAM may not converge to a stationary point. The accuracy of the single gradient step approximation for finding the worst-case perturbation may also decrease during the training process. The effectiveness of SAM can also be domain-dependent. While it has shown benefits for computer vision tasks, its impact on other areas, such as GPT-style language models where each training example is seen only once, has been reported as limited in some studies. Furthermore, while SAM seeks flat minima, some research suggests that not all flat minima necessarily lead to good generalization. The algorithm also introduces the neighborhood size ρ {\displaystyle \rho } as a new hyperparameter, which requires tuning. == Research, Variants, and Enhancements == Active research on SAM focuses on reducing its computational overhead and improving its performance. Several variants have been proposed to make the algorithm more efficient. These include methods that attempt to parallelize the two gradient computations, apply the perturbation to only a subset of parameters, or reduce the number of computation steps required. Other approaches use historical gradient information or apply SAM steps intermittently to lower the computational burden. To improve performance and robustness, variants have been developed that adapt the neighborhood size based on model parameter scales (Adaptive SAM or ASAM) or incorporate information about the curvature of the loss landscape (Curvature Regularized SAM or CR-SAM). Other research explores refining the perturbation step by focusing on specific components of the gradient or combining SAM with techniques like random smoothing. Theoretical work continues to analyze the algorithm's behavior, including its implicit bias towards flatter minima and the development of broader frameworks for sharpness-aware optimization that use different measures of sharpness.
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Cartesian genetic programming

Cartesian genetic programming is a form of genetic programming that uses a graph representation to encode computer programs. It grew from a method of evolving digital circuits developed by Julian F. Miller and Peter Thomson in 1997. The term ‘Cartesian genetic programming’ first appeared in 1999 and was proposed as a general form of genetic programming in 2000. It is called ‘Cartesian’ because it represents a program using a two-dimensional grid of nodes. Miller's keynote explains how CGP works. He edited a book entitled Cartesian Genetic Programming, published in 2011 by Springer. The open source project dCGP implements a differentiable version of CGP developed at the European Space Agency by Dario Izzo, Francesco Biscani and Alessio Mereta able to approach symbolic regression tasks, to find solution to differential equations, find prime integrals of dynamical systems, represent variable topology artificial neural networks and more.
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Neural Networks (journal)

Neural Networks is a monthly peer-reviewed scientific journal and an official journal of the International Neural Network Society, European Neural Network Society, and Japanese Neural Network Society. == History == The journal was established in 1988 and is published by Elsevier. It covers all aspects of research on artificial neural networks. The founding editor-in-chief was Stephen Grossberg (Boston University). The current editors-in-chief are DeLiang Wang (Ohio State University) and Taro Toyoizumi (RIKEN Center for Brain Science). == Abstracting and indexing == The journal is abstracted and indexed in Scopus and the Science Citation Index Expanded. According to the Journal Citation Reports, the journal has a 2022 impact factor of 7.8.
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Contrastive Language-Image Pre-training

Contrastive Language-Image Pre-training (CLIP) is a technique for training a pair of neural network models, one for image understanding and one for text understanding, using a contrastive objective. This method has enabled broad applications across multiple domains, including cross-modal retrieval, text-to-image generation, and aesthetic ranking. == Algorithm == The CLIP method trains a pair of models contrastively. One model takes in a piece of text as input and outputs a single vector representing its semantic content. The other model takes in an image and similarly outputs a single vector representing its visual content. The models are trained so that the vectors corresponding to semantically similar text-image pairs are close together in the shared vector space, while those corresponding to dissimilar pairs are far apart. To train a pair of CLIP models, one would start by preparing a large dataset of image-caption pairs. During training, the models are presented with batches of N {\displaystyle N} image-caption pairs. Let the outputs from the text and image models be respectively v 1 , . . . , v N , w 1 , . . . , w N {\displaystyle v_{1},...,v_{N},w_{1},...,w_{N}} . Two vectors are considered "similar" if their dot product is large. The loss incurred on this batch is the multi-class N-pair loss, which is a symmetric cross-entropy loss over similarity scores: − 1 N ∑ i ln ⁡ e v i ⋅ w i / T ∑ j e v i ⋅ w j / T − 1 N ∑ j ln ⁡ e v j ⋅ w j / T ∑ i e v i ⋅ w j / T {\displaystyle -{\frac {1}{N}}\sum _{i}\ln {\frac {e^{v_{i}\cdot w_{i}/T}}{\sum _{j}e^{v_{i}\cdot w_{j}/T}}}-{\frac {1}{N}}\sum _{j}\ln {\frac {e^{v_{j}\cdot w_{j}/T}}{\sum _{i}e^{v_{i}\cdot w_{j}/T}}}} In essence, this loss function encourages the dot product between matching image and text vectors ( v i ⋅ w i {\displaystyle v_{i}\cdot w_{i}} ) to be high, while discouraging high dot products between non-matching pairs. The parameter T > 0 {\displaystyle T>0} is the temperature, which is parameterized in the original CLIP model as T = e − τ {\displaystyle T=e^{-\tau }} where τ ∈ R {\displaystyle \tau \in \mathbb {R} } is a learned parameter. Other loss functions are possible. For example, Sigmoid CLIP (SigLIP) proposes the following loss function: L = 1 N ∑ i , j ∈ 1 : N f ( ( 2 δ i , j − 1 ) ( e τ w i ⋅ v j + b ) ) {\displaystyle L={\frac {1}{N}}\sum _{i,j\in 1:N}f((2\delta _{i,j}-1)(e^{\tau }w_{i}\cdot v_{j}+b))} where f ( x ) = ln ⁡ ( 1 + e − x ) {\displaystyle f(x)=\ln(1+e^{-x})} is the negative log sigmoid loss, and the Dirac delta symbol δ i , j {\displaystyle \delta _{i,j}} is 1 if i = j {\displaystyle i=j} else 0. == CLIP models == While the original model was developed by OpenAI, subsequent models have been trained by other organizations as well. === Image model === The image encoding models used in CLIP are typically vision transformers (ViT). The naming convention for these models often reflects the specific ViT architecture used. For instance, "ViT-L/14" means a "vision transformer large" (compared to other models in the same series) with a patch size of 14, meaning that the image is divided into 14-by-14 pixel patches before being processed by the transformer. The size indicator ranges from B, L, H, G (base, large, huge, giant), in that order. Other than ViT, the image model is typically a convolutional neural network, such as ResNet (in the original series by OpenAI), or ConvNeXt (in the OpenCLIP model series by LAION). Since the output vectors of the image model and the text model must have exactly the same length, both the image model and the text model have fixed-length vector outputs, which in the original report is called "embedding dimension". For example, in the original OpenAI model, the ResNet models have embedding dimensions ranging from 512 to 1024, and for the ViTs, from 512 to 768. Its implementation of ViT was the same as the original one, with one modification: after position embeddings are added to the initial patch embeddings, there is a LayerNorm. Its implementation of ResNet was the same as the original one, with 3 modifications: In the start of the CNN (the "stem"), they used three stacked 3x3 convolutions instead of a single 7x7 convolution, as suggested by. There is an average pooling of stride 2 at the start of each downsampling convolutional layer (they called it rect-2 blur pooling according to the terminology of ). This has the effect of blurring images before downsampling, for antialiasing. The final convolutional layer is followed by a multiheaded attention pooling. ALIGN a model with similar capabilities, trained by researchers from Google used EfficientNet, a kind of convolutional neural network. === Text model === The text encoding models used in CLIP are typically Transformers. In the original OpenAI report, they reported using a Transformer (63M-parameter, 12-layer, 512-wide, 8 attention heads) with lower-cased byte pair encoding (BPE) with 49152 vocabulary size. Context length was capped at 76 for efficiency. Like GPT, it was decoder-only, with only causally-masked self-attention. Its architecture is the same as GPT-2. Like BERT, the text sequence is bracketed by two special tokens [SOS] and [EOS] ("start of sequence" and "end of sequence"). Take the activations of the highest layer of the transformer on the [EOS], apply LayerNorm, then a final linear map. This is the text encoding of the input sequence. The final linear map has output dimension equal to the embedding dimension of whatever image encoder it is paired with. These models all had context length 77 and vocabulary size 49408. ALIGN used BERT of various sizes. == Dataset == === WebImageText === The CLIP models released by OpenAI were trained on a dataset called "WebImageText" (WIT) containing 400 million pairs of images and their corresponding captions scraped from the internet. The total number of words in this dataset is similar in scale to the WebText dataset used for training GPT-2, which contains about 40 gigabytes of text data. The dataset contains 500,000 text-queries, with up to 20,000 (image, text) pairs per query. The text-queries were generated by starting with all words occurring at least 100 times in English Wikipedia, then extended by bigrams with high mutual information, names of all Wikipedia articles above a certain search volume, and WordNet synsets. The dataset is private and has not been released to the public, and there is no further information on it. ==== Data preprocessing ==== For the CLIP image models, the input images are preprocessed by first dividing each of the R, G, B values of an image by the maximum possible value, so that these values fall between 0 and 1, then subtracting by [0.48145466, 0.4578275, 0.40821073], and dividing by [0.26862954, 0.26130258, 0.27577711]. The rationale was that these are the mean and standard deviations of the images in the WebImageText dataset, so this preprocessing step roughly whitens the image tensor. These numbers slightly differ from the standard preprocessing for ImageNet, which uses [0.485, 0.456, 0.406] and [0.229, 0.224, 0.225]. If the input image does not have the same resolution as the native resolution (224×224 for all except ViT-L/14@336px, which has 336×336 resolution), then the input image is first scaled by bicubic interpolation, so that its shorter side is the same as the native resolution, then the central square of the image is cropped out. === Others === ALIGN used over one billion image-text pairs, obtained by extracting images and their alt-tags from online crawling. The method was described as similar to how the Conceptual Captions dataset was constructed, but instead of complex filtering, they only applied a frequency-based filtering. Later models trained by other organizations had published datasets. For example, LAION trained OpenCLIP with published datasets LAION-400M, LAION-2B, and DataComp-1B. == Training == In the original OpenAI CLIP report, they reported training 5 ResNet and 3 ViT (ViT-B/32, ViT-B/16, ViT-L/14). Each was trained for 32 epochs. The largest ResNet model took 18 days to train on 592 V100 GPUs. The largest ViT model took 12 days on 256 V100 GPUs. All ViT models were trained on 224×224 image resolution. The ViT-L/14 was then boosted to 336×336 resolution by FixRes, resulting in a model. They found this was the best-performing model. In the OpenCLIP series, the ViT-L/14 model was trained on 384 A100 GPUs on the LAION-2B dataset, for 160 epochs for a total of 32B samples seen. == Applications == === Cross-modal retrieval === CLIP's cross-modal retrieval enables the alignment of visual and textual data in a shared latent space, allowing users to retrieve images based on text descriptions and vice versa, without the need for explicit image annotations. In text-to-image retrieval, users input descriptive text, and CLIP retrieves images with matching embeddings. In image-to-text retrieval, images are used to find related text content. CLIP’s ability to connect vis
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Out-of-bag error

Out-of-bag (OOB) error, also called out-of-bag estimate, is a method of measuring the prediction error of random forests, boosted decision trees, and other machine learning models utilizing bootstrap aggregating (bagging). Bagging uses subsampling with replacement to create training samples for the model to learn from. OOB error is the mean prediction error on each training sample xi, using only the trees that did not have xi in their bootstrap sample. Bootstrap aggregating allows one to define an out-of-bag estimate of the prediction performance improvement by evaluating predictions on those observations that were not used in the building of the next base learner. == Out-of-bag dataset == When bootstrap aggregating is performed, two independent sets are created. One set, the bootstrap sample, is the data chosen to be "in-the-bag" by sampling with replacement. The out-of-bag set is all data not chosen in the sampling process. When this process is repeated, such as when building a random forest, many bootstrap samples and OOB sets are created. The OOB sets can be aggregated into one dataset, but each sample is only considered out-of-bag for the trees that do not include it in their bootstrap sample. The picture below shows that for each bag sampled, the data is separated into two groups. This example shows how bagging could be used in the context of diagnosing disease. A set of patients are the original dataset, but each model is trained only by the patients in its bag. The patients in each out-of-bag set can be used to test their respective models. The test would consider whether the model can accurately determine if the patient has the disease. == Calculating out-of-bag error == Since each out-of-bag set is not used to train the model, it is a good test for the performance of the model. The specific calculation of OOB error depends on the implementation of the model, but a general calculation is as follows. Find all models (or trees, in the case of a random forest) that are not trained by the OOB instance. Take the majority vote of these models' result for the OOB instance, compared to the true value of the OOB instance. Compile the OOB error for all instances in the OOB dataset. The bagging process can be customized to fit the needs of a model. To ensure an accurate model, the bootstrap training sample size should be close to that of the original set. Also, the number of iterations (trees) of the model (forest) should be considered to find the true OOB error. The OOB error will stabilize over many iterations so starting with a high number of iterations is a good idea. Shown in the example to the right, the OOB error can be found using the method above once the forest is set up. == Comparison to cross-validation == Out-of-bag error and cross-validation (CV) are different methods of measuring the error estimate of a machine learning model. Over many iterations, the two methods should produce a very similar error estimate. That is, once the OOB error stabilizes, it will converge to the cross-validation (specifically leave-one-out cross-validation) error. The advantage of the OOB method is that it requires less computation and allows one to test the model as it is being trained. == Accuracy and Consistency == Out-of-bag error is used frequently for error estimation within random forests but with the conclusion of a study done by Silke Janitza and Roman Hornung, out-of-bag error has shown to overestimate in settings that include an equal number of observations from all response classes (balanced samples), small sample sizes, a large number of predictor variables, small correlation between predictors, and weak effects.
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Vanishing gradient problem

In machine learning, the vanishing gradient problem is the problem of greatly diverging gradient magnitudes between earlier and later layers encountered when training neural networks with backpropagation. In such methods, neural network weights are updated proportional to their partial derivative of the loss function. As the number of forward propagation steps in a network increases, for instance due to greater network depth, the gradients of earlier weights are calculated with increasingly many multiplications. These multiplications shrink the gradient magnitude. Consequently, the gradients of earlier weights will be exponentially smaller than the gradients of later weights. This difference in gradient magnitude might introduce instability in the training process, slow it, or halt it entirely. For instance, consider the hyperbolic tangent activation function. The gradients of this function are in range [0,1]. The product of repeated multiplication with such gradients decreases exponentially. The inverse problem, when weight gradients at earlier layers get exponentially larger, is called the exploding gradient problem. Backpropagation allowed researchers to train supervised deep artificial neural networks from scratch, initially with little success. Hochreiter's diplom thesis of 1991 formally identified the reason for this failure in the "vanishing gradient problem", which not only affects many-layered feedforward networks, but also recurrent networks. The latter are trained by unfolding them into very deep feedforward networks, where a new layer is created for each time-step of an input sequence processed by the network (the combination of unfolding and backpropagation is termed backpropagation through time). == Prototypical models == This section is based on the paper On the difficulty of training Recurrent Neural Networks by Pascanu, Mikolov, and Bengio. === Recurrent network model === A generic recurrent network has hidden states h 1 , h 2 , … {\displaystyle h_{1},h_{2},\dots } , inputs u 1 , u 2 , … {\displaystyle u_{1},u_{2},\dots } , and outputs x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } . Let it be parameterized by θ {\displaystyle \theta } , so that the system evolves as ( h t , x t ) = F ( h t − 1 , u t , θ ) {\displaystyle (h_{t},x_{t})=F(h_{t-1},u_{t},\theta )} Often, the output x t {\displaystyle x_{t}} is a function of h t {\displaystyle h_{t}} , as some x t = G ( h t ) {\displaystyle x_{t}=G(h_{t})} . The vanishing gradient problem already presents itself clearly when x t = h t {\displaystyle x_{t}=h_{t}} , so we simplify our notation to the special case with: x t = F ( x t − 1 , u t , θ ) {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )} Now, take its differential: d x t = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) d x t − 1 = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) [ ∇ θ F ( x t − 2 , u t − 1 , θ ) d θ + ∇ x F ( x t − 2 , u t − 1 , θ ) d x t − 2 ] ⋮ = [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] d θ {\displaystyle {\begin{aligned}dx_{t}&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )dx_{t-1}\\&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )\left[\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )d\theta +\nabla _{x}F(x_{t-2},u_{t-1},\theta )dx_{t-2}\right]\\&\;\;\vdots \\&=\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]d\theta \end{aligned}}} Training the network requires us to define a loss function to be minimized. Let it be L ( x T , u 1 , … , u T ) {\displaystyle L(x_{T},u_{1},\dots ,u_{T})} , then minimizing it by gradient descent gives Δ θ = − η ⋅ [ ∇ x L ( x T ) ( ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ) ] T {\displaystyle \Delta \theta =-\eta \cdot \left[\nabla _{x}L(x_{T})\left(\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right)\right]^{T}} where η {\displaystyle \eta } is the learning rate. The vanishing/exploding gradient problem appears because there are repeated multiplications, of the form ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ∇ x F ( x t − 3 , u t − 2 , θ ) ⋯ {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\nabla _{x}F(x_{t-3},u_{t-2},\theta )\cdots } ==== Example: recurrent network with sigmoid activation ==== For a concrete example, consider a typical recurrent network defined by x t = F ( x t − 1 , u t , θ ) = W rec σ ( x t − 1 ) + W in u t + b {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\sigma (x_{t-1})+W_{\text{in}}u_{t}+b} where θ = ( W rec , W in ) {\displaystyle \theta =(W_{\text{rec}},W_{\text{in}})} is the network parameter, σ {\displaystyle \sigma } is the sigmoid activation function, applied to each vector coordinate separately, and b {\displaystyle b} is the bias vector. Then, ∇ x F ( x t − 1 , u t , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))} , and so ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ⋯ ∇ x F ( x t − k , u t − k + 1 , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) W rec diag ⁡ ( σ ′ ( x t − 2 ) ) ⋯ W rec diag ⁡ ( σ ′ ( x t − k ) ) {\displaystyle {\begin{aligned}&\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\cdots \nabla _{x}F(x_{t-k},u_{t-k+1},\theta )\\&=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-2}))\cdots W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-k}))\end{aligned}}} Since | σ ′ | ≤ 1 {\displaystyle \left|\sigma '\right|\leq 1} , the operator norm of the above multiplication is bounded above by ‖ W rec ‖ k {\displaystyle \left\|W_{\text{rec}}\right\|^{k}} . So if the spectral radius of W rec {\displaystyle W_{\text{rec}}} is γ < 1 {\displaystyle \gamma <1} , then at large k {\displaystyle k} , the above multiplication has operator norm bounded above by γ k → 0 {\displaystyle \gamma ^{k}\to 0} . This is the prototypical vanishing gradient problem. The effect of a vanishing gradient is that the network cannot learn long-range effects. Recall Equation (loss differential): ∇ θ L = ∇ x L ( x T , u 1 , … , u T ) [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] {\displaystyle \nabla _{\theta }L=\nabla _{x}L(x_{T},u_{1},\dots ,u_{T})\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]} The components of ∇ θ F ( x , u , θ ) {\displaystyle \nabla _{\theta }F(x,u,\theta )} are just components of σ ( x ) {\displaystyle \sigma (x)} and u {\displaystyle u} , so if u t , u t − 1 , … {\displaystyle u_{t},u_{t-1},\dots } are bounded, then ‖ ∇ θ F ( x t − k − 1 , u t − k , θ ) ‖ {\displaystyle \left\|\nabla _{\theta }F(x_{t-k-1},u_{t-k},\theta )\right\|} is also bounded by some M > 0 {\displaystyle M>0} , and so the terms in ∇ θ L {\displaystyle \nabla _{\theta }L} decay as M γ k {\displaystyle M\gamma ^{k}} . This means that, effectively, ∇ θ L {\displaystyle \nabla _{\theta }L} is affected only by the first O ( γ − 1 ) {\displaystyle O(\gamma ^{-1})} terms in the sum. If γ ≥ 1 {\displaystyle \gamma \geq 1} , the above analysis does not quite work. For the prototypical exploding gradient problem, the next model is clearer. === Dynamical systems model === Following (Doya, 1993), consider this one-neuron recurrent network with sigmoid activation: x t + 1 = ( 1 − ε ) x t + ε σ ( w x t + b ) + ε w ′ u t {\displaystyle x_{t+1}=(1-\varepsilon )x_{t}+\varepsilon \sigma (wx_{t}+b)+\varepsilon w'u_{t}} At the small ε {\displaystyle \varepsilon } limit, the dynamics of the network becomes d x d t = − x ( t ) + σ ( w x ( t ) + b ) + w ′ u ( t ) {\displaystyle {\frac {dx}{dt}}=-x(t)+\sigma (wx(t)+b)+w'u(t)} Consider first the autonomous case, with u = 0 {\displaystyle u=0} . Set w = 5.0 {\displaystyle w=5.0} , and vary b {\displaystyle b} in [ − 3 , − 2 ] {\displaystyle [-3,-2]} . As b {\displaystyle b} decreases, the system has 1 stable point, then has 2 stable points and 1 unstable point, and finally has 1 stable point again. Explicitly, the stable points are ( x , b ) = ( x , ln ⁡ ( x 1 − x ) − 5 x ) {\displaystyle (x,b)=\left(x,\ln \left({\frac {x}{1-x}}\right)-5x\right)} . Now consider Δ x ( T ) Δ x ( 0 ) {\displaystyle {\frac {\Delta x(T)}{\Delta x(0)}}} and Δ x ( T ) Δ b {\displaystyle {\frac {\Delta x(T)}{\Delta b}}} , where T {\displaystyle T} is large enough that the system has settled into one of the stable points. If ( x ( 0 ) , b ) {\displaystyle (x(0),b)} puts the system very close to an unstable point, then a tiny variation in x ( 0 ) {\displaystyle x(0)} or b {\displaystyle b} wo
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Common Voice

Common Voice is a crowdsourcing project started by Mozilla to create a free and open speech corpus. The project is supported by volunteers who record sample sentences with a microphone and review recordings of other users. The transcribed sentences are collected in a voice database available under the public domain license CC0. This license ensures that developers can use the database for voice-to-text and text-to-voice applications without restrictions or costs. == Aims == Common Voice aims to provide diverse voice samples. According to Mozilla's Katharina Borchert, many existing projects took datasets from public radio or otherwise had datasets that underrepresented both women and people with pronounced accents. == Voice database == The first dataset was released in November 2017. More than 20,000 users worldwide had recorded 500 hours of English sentences. In February 2019, the first batch of languages was released for use. This included 18 languages such as English, French, German and Mandarin Chinese, but also less prevalent languages like Welsh and Kabyle. In total, this included almost 1,400 hours of recorded voice data from more than 42,000 contributors. By July 2020 the database had amassed 7,226 hours of voice recordings in 54 languages, 5,591 hours of which had been verified by volunteers. In May 2021, following the work to add Kinyarwanda, the project received a grant to add Kiswahili. At the beginning of 2022, Bengali.AI partnered with Common Voice to launch the "Bangla Speech Recognition" project that aims to make machines understand the Bangla language. 2000 hours of voice was collected. In September 2022, it was announced that the Twi language of Ghana was the 100th language to be added to the database. As of December 2025, Mozilla Common Voice collects voice data for over 250 languages, with the most hours having been collected in English, Catalan, Kinyarwanda, Belarusian and Esperanto.
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IOS SDK

The iOS SDK (iOS Software Development Kit), formerly the iPhone SDK, is a software development kit (SDK) developed by Apple Inc. The kit allows for the development of mobile apps on Apple's iOS 17 and iPadOS operating systems. The iOS SDK is a free download for users of Macintosh (or Mac) personal computers. It is not available for Microsoft Windows PCs. The SDK contains sets giving developers access to various functions and services of iOS devices, such as hardware and software attributes. It also contains an iPhone simulator to mimic the look and feel of the device on the computer while developing. New versions of the SDK accompany new versions of iOS. In order to test applications, get technical support, and distribute apps through App Store, developers are required to subscribe to the Apple Developer Program. Combined with Xcode, the iOS SDK helps developers write iOS apps using officially supported programming languages, including Swift and Objective-C. Other companies have also created tools that allow for the development of native iOS apps using their respective programming languages. == History == While originally developing iPhone prior to its unveiling in 2007, Apple's then-CEO Steve Jobs did not intend to let third-party developers build native apps for the iOS operating system, instead directing them to make web applications for the Safari web browser. However, backlash from developers prompted the company to reconsider, with Jobs announcing on October 17, 2007, that Apple would have a software development kit (SDK) available for developers by February 2008. The SDK was released on March 6, 2008. == Features == The iOS SDK is a free download for Mac users. It is not available for Microsoft Windows. To test the application, get technical support, and distribute applications through App Store, developers are required to subscribe to the Apple Developer Program. The SDK contents are separated into the following sets: UIKit Multi-touch events and controls Accelerometer support View hierarchy Localization (i18n) Camera support Media OpenAL audio mixing and recording Video playback Image file formats Quartz Core Animation OpenGL ES Core Services Networking Embedded SQLite database Core Location Threads CoreMotion Mac OS X Kernel TCP/IP Sockets Power management File system Security The SDK also contains an iPhone simulator, a program used to simulate the look and feel of iPhone on the developer's computer. New SDK versions accompany new iOS versions. == Programming languages == The iOS SDK, combined with Xcode, helps developers write iOS applications using officially supported programming languages, including Swift and Objective-C. An .ipa (iOS App Store Package) file is an iOS application archive file which stores an iOS app. === Java === In 2008, Sun Microsystems announced plans to release a Java Virtual Machine (JVM) for iOS, based on the Java Platform, Micro Edition version of Java. This would enable Java applications to run on iPhone and iPod Touch. Soon after the announcement, developers familiar with the SDK's terms of agreement believed that by not allowing third-party applications to run in the background (answer a phone call and still run the application, for example), and not allowing an application to download code from another source, nor allowing an application to interact with a third-party application, Sun's development efforts could be hindered without Apple's cooperation. Sun also worked with a third-party company called Innaworks in attempts to get Java on iPhone. Despite the apparent lack of interest from Apple, a firmware leak of the 2007 iPhone release revealed an ARM chip with a processor with Jazelle support for embedded Java execution. === .NET === Novell announced in September 2009 that they had successfully developed MonoTouch, a software framework that let developers write native iPhone applications in the C# and .NET programming languages, while still maintaining compatibility with Apple's requirements. === Flash === iOS does not support Adobe Flash, and although Adobe has two versions of its software: Flash and Flash Lite, Apple views neither as suitable for the iPhone, claiming that full Flash is "too slow to be useful", and Flash Lite to be "not capable of being used with the Web". In October 2009, Adobe announced that an upcoming update to its Creative Suite would feature a component to let developers build native iPhone apps using the company's Flash development tools. The software was officially released as part of the company's Creative Suite 5 collection of professional applications. === 2010 policy on development tools === In April 2010, Apple made controversial changes to its iPhone Developer Agreement, requiring developers to use only "approved" programming languages in order to publish apps on App Store, and banning applications that used third-party development tools; the ban affected Adobe's Packager tool, which converted Flash apps into iOS apps. After developer backlash and news of a potential anti-trust investigation, Apple again revised its agreement in September, allowing the use of third-party development tools. === Mac Catalyst === Originally called "Project Marzipan", Mac Catalyst helps developers bring iPadOS app experiences to macOS, and make it easier to take apps developed for iPadOS devices to Macs by avoiding the need to write the underlying software code twice.
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Unique negative dimension

Unique negative dimension (UND) is a complexity measure for the model of learning from positive examples. The unique negative dimension of a class C {\displaystyle C} of concepts is the size of the maximum subclass D ⊆ C {\displaystyle D\subseteq C} such that for every concept c ∈ D {\displaystyle c\in D} , we have ∩ ( D ∖ { c } ) ∖ c {\displaystyle \cap (D\setminus \{c\})\setminus c} is nonempty. This concept was originally proposed by M. Gereb-Graus in "Complexity of learning from one-side examples", Technical Report TR-20-89, Harvard University Division of Engineering and Applied Science, 1989.
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Sum of absolute transformed differences

The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.
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