In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct
Fatsecret
Fatsecret, commonly styled as fatsecret, is a mobile application, website and API that helps people achieve their weight loss goals and find accurate nutrition information. It also offers a weight loss clinic with coaching and medically supported programs. The platform powers global health apps. == History == Fatsecret was founded in 2006 in Melbourne, Australia by Lenny Moses and Rodney Moses. As of 2019, Lenny serves as the company's CEO. The company is known for its calorie counting and meal tracking app, and by April 2016, the company claimed to have 45 million users of its services. In August 2018, a premium version of its app was released. Since August 2009, the company has operated the Fatsecret Platform API, which allows access to its global food and nutrition database. Fatsecret reportedly had 900,000 downloads of its app in January 2020. In an analysis of several Health & Fitness app subcategories for the United States in January 2021, Fatsecret was reported to have the highest 30 day user retention rate of top Calorie Counter + Meal Planner for Weight Loss apps.
Conceptions of Library and Information Science
Conceptions of Library and Information Science (CoLIS) is a series of conferences about historical, empirical and theoretical perspectives in Library and Information Science. == CoLIS conferences == CoLIS 1 1991 in Tampere, Finland CoLIS 2 1996 in Copenhagen, Denmark CoLIS 3 1999 in Dubrovnik, Croatia CoLIS 4 2002 in Seattle, US CoLIS 5 2005 in Glasgow, Scotland CoLIS 6 2007 in Borås, Sweden CoLIS 7 June 2010 in London, at City University London. CoLIS 8 August 19–22, 2013, in Copenhagen, Denmark, at The Royal School of Library and Information Science. CoLIS 9 June 27–29, 2016, in Uppsala, Sweden, at Uppsala University. CoLIS 10 June 16–19, 2019, in Ljubljana, Slovenia, Faculty of Arts CoLIS 11 May 29–June 1, 2022, in Oslo, Norway, Oslo Metropolitan University.
Kleene's algorithm
In theoretical computer science, in particular in formal language theory, Kleene's algorithm transforms a given nondeterministic finite automaton (NFA) into a regular expression. Together with other conversion algorithms, it establishes the equivalence of several description formats for regular languages. Alternative presentations of the same method include the "elimination method" attributed to Brzozowski and McCluskey, the algorithm of McNaughton and Yamada, and the use of Arden's lemma. == Algorithm description == According to Gross and Yellen (2004), the algorithm can be traced back to Kleene (1956). A presentation of the algorithm in the case of deterministic finite automata (DFAs) is given in Hopcroft and Ullman (1979). The presentation of the algorithm for NFAs below follows Gross and Yellen (2004). Given a nondeterministic finite automaton M = (Q, Σ, δ, q0, F), with Q = { q0,...,qn } its set of states, the algorithm computes the sets Rkij of all strings that take M from state qi to qj without going through any state numbered higher than k. Here, "going through a state" means entering and leaving it, so both i and j may be higher than k, but no intermediate state may. Each set Rkij is represented by a regular expression; the algorithm computes them step by step for k = -1, 0, ..., n. Since there is no state numbered higher than n, the regular expression Rn0j represents the set of all strings that take M from its start state q0 to qj. If F = { q1,...,qf } is the set of accept states, the regular expression Rn01 | ... | Rn0f represents the language accepted by M. The initial regular expressions, for k = -1, are computed as follows for i≠j: R−1ij = a1 | ... | am where qj ∈ δ(qi,a1), ..., qj ∈ δ(qi,am) and as follows for i=j: R−1ii = a1 | ... | am | ε where qi ∈ δ(qi,a1), ..., qi ∈ δ(qi,am) In other words, R−1ij mentions all letters that label a transition from i to j, and we also include ε in the case where i=j. After that, in each step the expressions Rkij are computed from the previous ones by Rkij = Rk-1ik (Rk-1kk) Rk-1kj | Rk-1ij Another way to understand the operation of the algorithm is as an "elimination method", where the states from 0 to n are successively removed: when state k is removed, the regular expression Rk-1ij, which describes the words that label a path from state i>k to state j>k, is rewritten into Rkij so as to take into account the possibility of going via the "eliminated" state k. By induction on k, it can be shown that the length of each expression Rkij is at most 1/3(4k+1(6s+7) - 4) symbols, where s denotes the number of characters in Σ. Therefore, the length of the regular expression representing the language accepted by M is at most 1/3(4n+1(6s+7)f - f - 3) symbols, where f denotes the number of final states. This exponential blowup is inevitable, because there exist families of DFAs for which any equivalent regular expression must be of exponential size. In practice, the size of the regular expression obtained by running the algorithm can be very different depending on the order in which the states are considered by the procedure, i.e., the order in which they are numbered from 0 to n. == Example == The automaton shown in the picture can be described as M = (Q, Σ, δ, q0, F) with the set of states Q = { q0, q1, q2 }, the input alphabet Σ = { a, b }, the transition function δ with δ(q0,a)=q0, δ(q0,b)=q1, δ(q1,a)=q2, δ(q1,b)=q1, δ(q2,a)=q1, and δ(q2,b)=q1, the start state q0, and set of accept states F = { q1 }. Kleene's algorithm computes the initial regular expressions as After that, the Rkij are computed from the Rk-1ij step by step for k = 0, 1, 2. Kleene algebra equalities are used to simplify the regular expressions as much as possible. Step 0 Step 1 Step 2 Since q0 is the start state and q1 is the only accept state, the regular expression R201 denotes the set of all strings accepted by the automaton.
SQL programming tool
In the field of software, SQL programming tools provide platforms for database administrators (DBAs) and application developers to perform daily tasks efficiently and accurately. Database administrators and application developers often face constantly changing environments which they rarely completely control. Many changes result from new development projects or from modifications to existing code, which, when deployed to production, do not always produce the expected result. For organizations to better manage development projects and the teams that develop code, suppliers of SQL programming tools normally provide more than facility to the database administrator or application developer to aid in database management and in quality code-deployment practices. == Features == SQL programming tools may include the following features: === SQL editing === SQL editors allow users to edit and execute SQL statements. They may support the following features: cut, copy, paste, undo, redo, find (and replace), bookmarks block indent, print, save file, uppercase/lowercase keyword highlighting auto-completion access to frequently used files output of query result editing query-results committing and rolling-back transactions inside cut paper === Object browsing === Tools may display information about database objects relevant to developers or to database administrators. Users may: view object descriptions view object definitions (DDL) create database objects enable and disable triggers and constraints recompile valid or invalid objects query or edit tables and views Some tools also provide features to display dependencies among objects, and allow users to expand these dependent objects recursively (for example: packages may reference views, views generally reference tables, super/subtypes, and so on). === Session browsing === Database administrators and application developers can use session browsing tools to view the current activities of each user in the database. They can check the resource-usage of individual users, statistics information, locked objects and the current running SQL of each individual session. === User-security management === DBAs can create, edit, delete, disable or enable user-accounts in the database using security-management tools. DBAs can also assign roles, system privileges, object privileges, and storage-quotas to users. === Debugging === Some tools offer features for the debugging of stored procedures: step in, step over, step out, run until exception, breakpoints, view & set variables, view call stack, and so on. Users can debug any program-unit without making any modification to it, including triggers and object types. === Performance monitoring === Monitoring tools may show the database resources — usage summary, service time summary, recent activities, top sessions, session history or top SQL — in easy-to-read graphs. Database administrators can easily monitor the health of various components in the monitoring instance. Application developers may also make use of such tools to diagnose and correct application-performance problems as well as improve SQL server performance. === Test data === Test data generation tools can populate the database by realistic test data for server or client side testing purposes. Also, this kind of software can upload sample blob files to database.
Doubao
Doubao (Chinese: 豆包) is an artificial intelligence assistant developed by ByteDance. == History == The chatbot was launched in August 2023. By November 2024, it had become China's most popular AI chatbot, with approximately 60 million monthly active users according to industry analytics. == Design == Doubao is powered by Volcano Engine (Volcengine), 120 trillion tokens consumed per day. == Variants == === Dola === The international version of Doubao is Dola which was launched in August 2023 as Cici. Dola is powered by OpenAI's GPT series of large language models and by Google's Gemini.
Single source of truth
In information science and information technology, single source of truth (SSOT) architecture, or single point of truth (SPOT) architecture, for information systems is the practice of structuring information models and associated data schemas such that every data element is mastered (or edited) in only one place, providing data normalization to a canonical form (for example, in database normalization or content transclusion). There are several scenarios with respect to copies and updates: The master data is never copied and instead only references to it are made; this means that all reads and updates go directly to the SSOT. The master data is copied but the copies are only read and only the master data is updated; if requests to read data are only made on copies, this is an instance of CQRS. The master data is copied and the copies are updated; this needs a reconciliation mechanism when there are concurrent updates. Updates on copies can be thrown out whenever a concurrent update is made on the master, so they are not considered fully committed until propagated to the master. (many blockchains work that way.) Concurrent updates are merged. (if an automatic merge fails, it could fall back on another strategy, which could be the previous strategy or something else like manual intervention, which most source version control systems do.) The advantages of SSOT architectures include easier prevention of mistaken inconsistencies (such as a duplicate value/copy somewhere being forgotten), and greatly simplified version control. Without a SSOT, dealing with inconsistencies implies either complex and error-prone consensus algorithms, or using a simpler architecture that's liable to lose data in the face of inconsistency (the latter may seem unacceptable but it is sometimes a very good choice; it is how most blockchains operate: a transaction is actually final only if it was included in the next block that is mined). Ideally, SSOT systems provide data that are authentic (and authenticatable), relevant, and referable. Deployment of an SSOT architecture is becoming increasingly important in enterprise settings where incorrectly linked duplicate or de-normalized data elements (a direct consequence of intentional or unintentional denormalization of any explicit data model) pose a risk for retrieval of outdated, and therefore incorrect, information. Common examples (i.e., example classes of implementation) are as follows: In electronic health records (EHRs), it is imperative to accurately validate patient identity against a single referential repository, which serves as the SSOT. Duplicate representations of data within the enterprise would be implemented by the use of pointers rather than duplicate database tables, rows, or cells. This ensures that data updates to elements in the authoritative location are comprehensively distributed to all federated database constituencies in the larger overall enterprise architecture. EHRs are an excellent class for exemplifying how SSOT architecture is both poignantly necessary and challenging to achieve: it is challenging because inter-organization health information exchange is inherently a cybersecurity competence hurdle, and nonetheless it is necessary, to prevent medical errors, to prevent the wasted costs of inefficiency (such as duplicated work or rework), and to make the primary care and medical home concepts feasible (to achieve competent care transitions). Single-source publishing as a general principle or ideal in content management relies on having SSOTs, via transclusion or (otherwise, at least) substitution. Substitution happens via libraries of objects that can be propagated as static copies which are later refreshed when necessary (that is, when refreshing of the copy-paste or import is triggered by a larger updating event). Component content management systems are a class of content management systems that aim to provide competence on this level. == Implementation == === Ontologic interactions === An acknowledged prerequisite (of the notion that any given single source of truth can exist) is that it depends on the ontologic condition that no more than a single truth (about any particular fact or idea) exists, an assertion that is ontologic in both the IT sense and the general sense of that word. In many instances, this presents no problem (for example, within particular namespaces, or even across them, as long as naming collisions or broader name conflicts are adequately handled). The broadest contexts (and thus thorniest, regarding ontologic discrepancies) require adequate epistemic regime comparison and reconciliation (or at least negotiation or transactional exchanges). An archetypal example of this class of reconciliation is that two theological seminary libraries, from two different religions (X and Y), could exchange information with an SSOT architecture, but the unification of truth would reside on the level of the statement that "religion X asserts that God is purple whereas religion Y asserts that God is green", rather than on the level of "God is purple" or "God is green". === Architectures or architectural features === An ideal implementation of SSOT is rarely possible in most enterprises. This is because many organisations have multiple information systems, each of which needs access to data relating to the same entities (e.g., customer). Often these systems are purchased as commercial off-the-shelf products from vendors and cannot be modified in trivial ways. Each of these various systems therefore needs to store its own version of common data or entities, and therefore each system must retain its own copy of a record (hence immediately violating the SSOT approach defined above). For example, an enterprise resource planning (ERP) system (such as SAP or Oracle e-Business Suite) may store a customer record; the customer relationship management (CRM) system also needs a copy of the customer record (or part of it) and the warehouse dispatch system might also need a copy of some or all of the customer data (e.g., shipping address). In cases where vendors do not support such modifications, it is not always possible to replace these records with pointers to the SSOT. For organisations (with more than one information system) wishing to implement a Single Source of Truth (without modifying all but one master system to store pointers to other systems for all entities), some supporting architectures are: Master data management (MDM) Event store and event sourcing (ES) ==== Master data management (MDM) ==== A master data management system typically serves as the source of truth for an organization's metadata, helping to ensure accuracy and consistency throughout that organizations multiple data sources. Typically the MDM acts as a hub for multiple systems, many of which could allow (be the source of truth for) updates to different aspects of information on a given entity. For example, the CRM system may be the "source of truth" for most aspects of the customer, and is updated by a call centre operator. However, a customer may (for example) also update their address via a customer service web site, with a different back-end database from the CRM system. The MDM application receives updates from multiple sources, acts as a broker to determine which updates are to be regarded as authoritative (the golden record) and then syndicates this updated data to all subscribing systems. The MDM application normally requires an ESB to syndicate its data to multiple subscribing systems. ==== Event store and event sourcing (ES) ==== In event oriented architectures, it has become increasingly common to find an implementation of the Event Sourcing pattern which stores the system state as an ordered sequence of state changes. To do this, you need an Event Store, a particular type of database designed to hold all the events that change the state of the system. The event store in an Event Sourcing + Command Query Responsibility Separation + Domain Driven Design + Messaging architecture is in fact a "single source of truth", with the additional advantage that it can also act as an Enterprise Service Bus as it can listen directly to the event store for status changes as everything passes by. In addition, by saving all the events, it also plays the role of Data Warehouse. One last advantage is that through this system the Shared Database pattern can be implemented, another technique not mentioned to obtain a single source of truth. ==== Data warehouse (DW) ==== While the primary purpose of a data warehouse is to support reporting and analysis of data that has been combined from multiple sources, the fact that such data has been combined (according to business logic embedded in the data transformation and integration processes) means that the data warehouse is often used as a de facto SSOT. Generally, however, the data available from the data warehouse are not used to update other systems; rather the DW becomes