AI Code For You

AI Code For You — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Scripped

  Scripped was an online screenplay services company offering three services: script writing, script registration, and script coverage. Scripped did not facilitate collaboration among screenwriters. It combined with Zhura in 2010. According to Techcrunch, Scripped had more than 60,000 writers as of March 2010. Scripped was administered by Sunil Rajaraman, Ryan Buckley and Zak Freer. Actor, writer, and director Edward Burns and screenwriter Steven E. de Souza joined Scripped's Board of Advisers in May 2008. In 2008, the company formed a partnership with Write Brothers, makers of Movie Magic Screenwriter software. On March 29, 2010, Scripped announced that it closed $250,000 in private investment and merged with competitor Zhura. Scripped's CEO, Sunil Rajaraman, remains the merged company's Chief Executive Officer. On April 1, 2015, citing a serious technical failure, Scripped shuttered its service. As part of the announcement, it was disclosed that their backup servers had failed as well, losing all of its users' stored scripts. The website URL currently redirects to WriterDuet's website, another online scriptwriting service; Scripped had advertised WriterDuet in Scripped's shutdown open letter. == Features == The Scripped Writer provided a built-in screenplay template which formatted the document to a standard for scripts as recommended by the AMPAS. The screenplay document was composed of seven elements: scene, action, character, dialog, parenthetical, transition and general. Each element had a specific style to which the Scripped Writer conformed as text was entered. Like other client-side screenplay software, Scripped offered Tab-Enter toggling between screenplay elements, making the writing process much faster. Text files could be imported into the Scripped Writer and automatically conformed to the screenplay template. Completed scripts could be exported as PDF files. In May 2011 the administrators of Scripped launched Scripted.com - a sister site focused on freelance writing jobs. Subsequent to the service's launch, the company was renamed to Scripted, Inc.

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  Anyword

  Anyword is a technology company that offers an artificial intelligence platform, using natural language processing to generate and optimize marketing text for websites, social media, email, and ads. The company also offers a complete managed service to publishers and brands to help them increase their revenue through social ads. It is used by National Geographic, Red Bull, The New York Times, BBC, Ted Baker, etc. The company has an office in New York, and Tel Aviv. == History == It was founded in 2013 — its original name was Keywee Inc. In March 2015, Anyword received $9.1 million in the Series A funding round led by a notable group of investors. In July 2016, the company was selected as an official Facebook Marketing Partner. In August 2019, Anyword was named Best Content Marketing Platform in the Digiday Technology Award winners. In November 2021, it raised $21 million in its Series B funding round.

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  Least-squares spectral analysis

  Least-squares spectral analysis (LSSA) is a class of methods for estimating a frequency spectrum by fitting sinusoids to data using a least-squares fit. Unlike Fourier analysis, the most widely used spectral method in science, data need not be equally spaced to use LSSA. Furthermore, while Fourier analysis generally amplifies long-period noise in long or gapped records, LSSA mitigates such problems. The first strictly least-squares LSSA method was developed in 1969 and 1971, and is known as the Vaníček method or the Gauss–Vaniček method, after its inventor Petr Vaníček and Carl Friedrich Gauss, the inventor of the least-squares method for error minimization. A widely known LSSA variant is the Lomb method or the Lomb–Scargle periodogram, based on dated computational simplifications of the Vaníček method introduced in the 1970s and 1980s, first by Nicholas R. Lomb and later by Jeffrey D. Scargle. Other LSSA variants have been subsequently developed. == Historical background == The close connections between Fourier analysis, the periodogram, and the least-squares fitting of sinusoids have been known for a long time. However, most developments are restricted to complete data sets of equally spaced samples. In 1963, Freek J. M. Barning of Mathematisch Centrum, Amsterdam, handled unequally spaced data by similar techniques, including both a periodogram analysis equivalent to what nowadays is called the Lomb method and least-squares fitting of selected frequencies of sinusoids determined from such periodograms — and connected by a procedure known today as the matching pursuit with post-back fitting or the orthogonal matching pursuit. Petr Vaníček, a Canadian geophysicist and geodesist of the University of New Brunswick, proposed in 1969 also the matching-pursuit approach for equally and unequally spaced data, which he called "successive spectral analysis" and the result a "least-squares periodogram". He generalized this method to account for any systematic components beyond a simple mean, such as a "predicted linear (quadratic, exponential, ...) secular trend of unknown magnitude", and applied it to a variety of samples, in 1971. Vaníček's strictly least-squares method was then simplified in 1976 by Nicholas R. Lomb of the University of Sydney, who pointed out its close connection to periodogram analysis. Subsequently, the definition of a periodogram of unequally spaced data was modified and analyzed by Jeffrey D. Scargle of NASA Ames Research Center, who showed that, with minor changes, it becomes identical to Lomb's least-squares formula for fitting individual sinusoid frequencies. Scargle states that his paper "does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced," and further points out regarding least-squares fitting of sinusoids compared to periodogram analysis, that his paper "establishes, apparently for the first time, that (with the proposed modifications) these two methods are exactly equivalent." Press summarizes the development this way: A completely different method of spectral analysis for unevenly sampled data, one that mitigates these difficulties and has some other very desirable properties, was developed by Lomb, based in part on earlier work by Barning and Vanicek, and additionally elaborated by Scargle. In 1989, Michael J. Korenberg of Queen's University in Kingston, Ontario, developed the "fast orthogonal search" method of more quickly finding a near-optimal decomposition of spectra or other problems, similar to the technique that later became known as the orthogonal matching pursuit. == Development of LSSA and variants == === The Vaníček method === In the Vaníček method, a discrete data set is approximated by a weighted sum of sinusoids of progressively determined frequencies using a standard linear regression or least-squares fit. The frequencies are chosen using a method similar to Barning's, but going further in optimizing the choice of each successive new frequency by picking the frequency that minimizes the residual after least-squares fitting (equivalent to the fitting technique now known as matching pursuit with pre-backfitting). The number of sinusoids must be less than or equal to the number of data samples (counting sines and cosines of the same frequency as separate sinusoids). The relationship between the DFT and the approximation of trigonometric functions using the least-squares method is well explained in (Strutz, 2017). A data vector Φ is represented as a weighted sum of sinusoidal basis functions, tabulated in a matrix A by evaluating each function at the sample times, with weight vector x: ϕ ≈ A x , {\displaystyle \phi \approx {\textbf {A}}x,} where the weights vector x is chosen to minimize the sum of squared errors in approximating Φ. The solution for x is closed-form, using standard linear regression: x = ( A T A ) − 1 A T ϕ . {\displaystyle x=({\textbf {A}}^{\mathrm {T} }{\textbf {A}})^{-1}{\textbf {A}}^{\mathrm {T} }\phi .} Here the matrix A can be based on any set of functions mutually independent (not necessarily orthogonal) when evaluated at the sample times; functions used for spectral analysis are typically sines and cosines evenly distributed over the frequency range of interest. If we choose too many frequencies in a too-narrow frequency range, the functions will be insufficiently independent, the matrix ill-conditioned, and the resulting spectrum meaningless. When the basis functions in A are orthogonal (that is, not correlated, meaning the columns have zero pair-wise dot products), the matrix ATA is diagonal; when the columns all have the same power (sum of squares of elements), then that matrix is an identity matrix times a constant, so the inversion is trivial. The latter is the case when the sample times are equally spaced and sinusoids chosen as sines and cosines equally spaced in pairs on the frequency interval 0 to a half cycle per sample (spaced by 1/N cycles per sample, omitting the sine phases at 0 and maximum frequency where they are identically zero). This case is known as the discrete Fourier transform, slightly rewritten in terms of measurements and coefficients. x = A T ϕ {\displaystyle x={\textbf {A}}^{\mathrm {T} }\phi } — DFT case for N equally spaced samples and frequencies, within a scalar factor. === The Lomb method === Trying to lower the computational burden of the Vaníček method in 1976 (no longer an issue), Lomb proposed using the above simplification in general, except for pair-wise correlations between sine and cosine bases of the same frequency, since the correlations between pairs of sinusoids are often small, at least when they are not tightly spaced. This formulation is essentially that of the traditional periodogram but adapted for use with unevenly spaced samples. The vector x is a reasonably good estimate of an underlying spectrum, but since we ignore any correlations, Ax is no longer a good approximation to the signal, and the method is no longer a least-squares method — yet in the literature continues to be referred to as such. Rather than just taking dot products of the data with sine and cosine waveforms directly, Scargle modified the standard periodogram formula so to find a time delay τ {\displaystyle \tau } first, such that this pair of sinusoids would be mutually orthogonal at sample times t j {\displaystyle t_{j}} and also adjusted for the potentially unequal powers of these two basis functions, to obtain a better estimate of the power at a frequency. This procedure made his modified periodogram method exactly equivalent to Lomb's method. Time delay τ {\displaystyle \tau } by definition equals to tan ⁡ 2 ω τ = ∑ j sin ⁡ 2 ω t j ∑ j cos ⁡ 2 ω t j . {\displaystyle \tan {2\omega \tau }={\frac {\sum _{j}\sin 2\omega t_{j}}{\sum _{j}\cos 2\omega t_{j}}}.} Then the periodogram at frequency ω {\displaystyle \omega } is estimated as: P x ( ω ) = 1 2 [ [ ∑ j X j cos ⁡ ω ( t j − τ ) ] 2 ∑ j cos 2 ⁡ ω ( t j − τ ) + [ ∑ j X j sin ⁡ ω ( t j − τ ) ] 2 ∑ j sin 2 ⁡ ω ( t j − τ ) ] , {\displaystyle P_{x}(\omega )={\frac {1}{2}}\left[{\frac {\left[\sum _{j}X_{j}\cos \omega (t_{j}-\tau )\right]^{2}}{\sum _{j}\cos ^{2}\omega (t_{j}-\tau )}}+{\frac {\left[\sum _{j}X_{j}\sin \omega (t_{j}-\tau )\right]^{2}}{\sum _{j}\sin ^{2}\omega (t_{j}-\tau )}}\right],} which, as Scargle reports, has the same statistical distribution as the periodogram in the evenly sampled case. At any individual frequency ω {\displaystyle \omega } , this method gives the same power as does a least-squares fit to sinusoids of that frequency and of the form: ϕ ( t ) = A sin ⁡ ω t + B cos ⁡ ω t . {\displaystyle \phi (t)=A\sin \omega t+B\cos \omega t.} In practice, it is always difficult to judge if a given Lomb peak is significant or not, especially when the nature of the noise is unknown, so for example a false-alarm spectr

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  Enterprise Objects Framework

  The Enterprise Objects Framework, or simply EOF, was introduced by NeXT in 1994 as a pioneering object-relational mapping product for its NeXTSTEP and OpenStep development platforms. EOF abstracts the process of interacting with a relational database by mapping database rows to Java or Objective-C objects. This largely relieves developers from writing low-level SQL code. EOF enjoyed some niche success in the mid-1990s among financial institutions who were attracted to the rapid application development advantages of NeXT's object-oriented platform. Since Apple Inc's merger with NeXT in 1996, EOF has evolved into a fully integrated part of WebObjects, an application server also originally from NeXT. Many of the core concepts of EOF re-emerged as part of Core Data, which further abstracts the underlying data formats to allow it to be based on non-SQL stores. == History == In the early 1990s NeXT Computer recognized that connecting to databases was essential to most businesses and yet also potentially complex. Every data source has a different data-access language (or API), driving up the costs to learn and use each vendor's product. The NeXT engineers wanted to apply the advantages of object-oriented programming, by getting objects to "talk" to relational databases. As the two technologies are very different, the solution was to create an abstraction layer, insulating developers from writing the low-level procedural code (SQL) specific to each data source. The first attempt came in 1992 with the release of Database Kit (DBKit), which wrapped an object-oriented framework around any database. Unfortunately, NEXTSTEP at the time was not powerful enough and DBKit had serious design flaws. NeXT's second attempt came in 1994 with the Enterprise Objects Framework (EOF) version 1, a complete rewrite that was far more modular and OpenStep compatible. EOF 1.0 was the first product released by NeXT using the Foundation Kit and introduced autoreleased objects to the developer community. The development team at the time was only four people: Jack Greenfield, Rich Williamson, Linus Upson and Dan Willhite. EOF 2.0, released in late 1995, further refined the architecture, introducing the editing context. At that point, the development team consisted of Dan Willhite, Craig Federighi, Eric Noyau and Charly Kleissner. EOF achieved a modest level of popularity in the financial programming community in the mid-1990s, but it would come into its own with the emergence of the World Wide Web and the concept of web applications. It was clear that EOF could help companies plug their legacy databases into the Web without any rewriting of that data. With the addition of frameworks to do state management, load balancing and dynamic HTML generation, NeXT was able to launch the first object-oriented Web application server, WebObjects, in 1996, with EOF at its core. In 2000, Apple Inc. (which had merged with NeXT) officially dropped EOF as a standalone product, meaning that developers would be unable to use it to create desktop applications for the forthcoming Mac OS X. It would, however, continue to be an integral part of a major new release of WebObjects. WebObjects 5, released in 2001, was significant for the fact that its frameworks had been ported from their native Objective-C programming language to the Java language. Critics of this change argue that most of the power of EOF was a side effect of its Objective-C roots, and that EOF lost the beauty or simplicity it once had. Third-party tools, such as EOGenerator, help fill the deficiencies introduced by Java (mainly due to the loss of categories). The Objective-C code base was re-introduced with some modifications to desktop application developers as Core Data, part of Apple's Cocoa API, with the release of Mac OS X Tiger in April 2005. == How EOF works == Enterprise Objects provides tools and frameworks for object-relational mapping. The technology specializes in providing mechanisms to retrieve data from various data sources, such as relational databases via JDBC and JNDI directories, and mechanisms to commit data back to those data sources. These mechanisms are designed in a layered, abstract approach that allows developers to think about data retrieval and commitment at a higher level than a specific data source or data source vendor. Central to this mapping is a model file (an "EOModel") that you build with a visual tool — either EOModeler, or the EOModeler plug-in to Xcode. The mapping works as follows: Database tables are mapped to classes. Database columns are mapped to class attributes. Database rows are mapped to objects (or class instances). You can build data models based on existing data sources or you can build data models from scratch, which you then use to create data structures (tables, columns, joins) in a data source. The result is that database records can be transposed into Java objects. The advantage of using data models is that applications are isolated from the idiosyncrasies of the data sources they access. This separation of an application's business logic from database logic allows developers to change the database an application accesses without needing to change the application. EOF provides a level of database transparency not seen in other tools and allows the same model to be used to access different vendor databases and even allows relationships across different vendor databases without changing source code. Its power comes from exposing the underlying data sources as managed graphs of persistent objects. In simple terms, this means that it organizes the application's model layer into a set of defined in-memory data objects. It then tracks changes to these objects and can reverse those changes on demand, such as when a user performs an undo command. Then, when it is time to save changes to the application's data, it archives the objects to the underlying data sources. === Using Inheritance === In designing Enterprise Objects developers can leverage the object-oriented feature known as inheritance. A Customer object and an Employee object, for example, might both inherit certain characteristics from a more generic Person object, such as name, address, and phone number. While this kind of thinking is inherent in object-oriented design, relational databases have no explicit support for inheritance. However, using Enterprise Objects, you can build data models that reflect object hierarchies. That is, you can design database tables to support inheritance by also designing enterprise objects that map to multiple tables or particular views of a database table. == Enterprise Objects (EOs) == An Enterprise Object is analogous to what is often known in object-oriented programming as a business object — a class which models a physical or conceptual object in the business domain (e.g. a customer, an order, an item, etc.). What makes an EO different from other objects is that its instance data maps to a data store. Typically, an enterprise object contains key-value pairs that represent a row in a relational database. The key is basically the column name, and the value is what was in that row in the database. So it can be said that an EO's properties persist beyond the life of any particular running application. More precisely, an Enterprise Object is an instance of a class that implements the com.webobjects.eocontrol.EOEnterpriseObject interface. An Enterprise Object has a corresponding model (called an EOModel) that defines the mapping between the class's object model and the database schema. However, an enterprise object doesn't explicitly know about its model. This level of abstraction means that database vendors can be switched without it affecting the developer's code. This gives Enterprise Objects a high degree of reusability. == EOF and Core Data == Despite their common origins, the two technologies diverged, with each technology retaining a subset of the features of the original Objective-C code base, while adding some new features. === Features Supported Only by EOF === EOF supports custom SQL; shared editing contexts; nested editing contexts; and pre-fetching and batch faulting of relationships, all features of the original Objective-C implementation not supported by Core Data. Core Data also does not provide the equivalent of an EOModelGroup—the NSManagedObjectModel class provides methods for merging models from existing models, and for retrieving merged models from bundles. === Features Supported Only by Core Data === Core Data supports fetched properties; multiple configurations within a managed object model; local stores; and store aggregation (the data for a given entity may be spread across multiple stores); customization and localization of property names and validation warnings; and the use of predicates for property validation. These features of the original Objective-C implementation are not supported by the Java implementation.

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  ConEmu

  ConEmu (short for Console emulator) is a free and open-source tabbed terminal emulator for Windows. ConEmu presents multiple consoles and simple GUI applications as one customizable GUI window with tabs and a status bar. It also provides emulation for ANSI escape codes for color, bypassing the capabilities of the standard Windows Console Host to provide 256 and 24-bit color in Windows. The program has a large range of customization, including custom color palettes for the standard 16 colors, hotkeys, transparency, an auto-hideable mode (similar to the way Quake originally displayed its developer console). Initially, the program was created as a companion to Far Manager, bringing some features common for graphical file managers to this console application (thumbnails and tiles, drag and drop with other windows, true color interface, and others). As of 2012, ConEmu could be used with any other Win32 console application or simple GUI tool (such as Notepad, PuTTY or DOSBox). ConEmu doesn't provide any shell itself, but rather allows using any other shell. It does provide a limited macro language, to control the hosted applications startup.

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  AlphaTensor

  AlphaTensor is an artificial intelligence system developed by DeepMind for discovering efficient matrix multiplication algorithms using reinforcement learning. Introduced in 2022, the system was based on AlphaZero and formulated the search for matrix multiplication algorithms as a single-player game called TensorGame. AlphaTensor was designed to search for new ways to multiply matrices with fewer scalar multiplication operations. Matrix multiplication is a fundamental operation in linear algebra, numerical analysis, scientific computing, computer graphics, and machine learning. The system discovered thousands of matrix multiplication algorithms, including algorithms that rediscovered known human-designed methods and others that improved on previously known results for particular matrix sizes and mathematical settings. == Background == Matrix multiplication is one of the basic operations in numerical computing. The standard algorithm for multiplying two square matrices has cubic time complexity, while faster algorithms such as the Strassen algorithm reduce the number of multiplication operations by using more complex algebraic decompositions. Finding optimal matrix multiplication algorithms can be difficult because it involves searching through a large space of possible tensor decompositions. AlphaTensor approached this problem by representing algorithm discovery as TensorGame, in which each move corresponds to an operation that reduces a tensor representing matrix multiplication. The goal of the game is to find a low-rank decomposition of the matrix multiplication tensor, corresponding to an efficient multiplication algorithm. == Development == AlphaTensor was developed by DeepMind and described in a paper published in Nature in October 2022. The system built on the reinforcement-learning approach used in AlphaZero, which had previously been applied to games such as Go, chess, and shogi. Unlike those games, TensorGame involved a very large search space, requiring changes to the AlphaZero-style search method and neural network architecture. DeepMind released source code and discovered algorithms associated with the publication through a public GitHub repository. == Results == AlphaTensor discovered matrix multiplication algorithms over both standard arithmetic and finite fields. One widely reported result was a method for multiplying 4 × 4 matrices over the field with two elements using 47 multiplication operations, improving on the 49 operations required by applying Strassen's algorithm recursively in that setting. The system also found algorithms optimized for particular computer hardware, including algorithms designed for graphics processing units and Tensor Processing Units. DeepMind stated that some of the hardware-specific algorithms improved practical execution time compared with commonly used algorithms on the tested hardware. == Significance == AlphaTensor was described as an example of using machine learning not only to apply existing algorithms, but to assist in discovering new ones. The work was connected to broader research in algorithm discovery, automated machine learning, program synthesis, and computational complexity theory, especially the open problem of determining the optimal complexity of matrix multiplication. AlphaTensor later became part of a broader group of Google DeepMind systems for algorithm and mathematical discovery, alongside systems such as AlphaDev and AlphaEvolve.

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  Artificial intelligence in Indonesia

  Artificial intelligence in Indonesia refers to development, use and governance of artificial intelligence in Indonesia. Indonesia has treated AI as a national policy area through the Strategi Nasional Kecerdasan Artifisial or National Artificial Intelligence Strategy for 2020–2045. Public discussion has focused on the role of AI in sectors such as health, agriculture, education, mobile technology and e-commerce. Recent developments include AI ethics guidance issued by the communications ministry. Proposals for a national AI roadmap and sovereign AI fund, investment in cloud and AI infrastructure, and local-language AI initiatives for Bahasa Indonesia and regional Indonesian languages. == National strategy == Indonesia's National Artificial Intelligence Strategy is known in Indonesian as Strategi Nasional Kecerdasan Artifisial or Stranas KA. The strategy was published as a long-term framework for the development and use of AI between 2020 and 2045. It is intended to guide ministries, government agencies, regional governments and other stakeholders. The strategy identifies five priority sectors: health services, bureaucratic reform, education and research, food security, and mobility and smart cities. OECD lists the Ministry of Research and Technology and the National Research and Innovation Agency as organisations associated with the strategy. The strategy was developed through consultation with public and private stakeholders. == Institutions == The Indonesian Artificial Intelligence Industry Research and Innovation Collaboration, known as KORIKA is the nodal agency for the national AI strategy. KORIKA describes its vision as creating a collaborative ecosystem to accelerate implementation of the national AI strategy towards Vision Indonesia 2045. The Ministry of Communication and Digital Affairs has also been involved in AI governance, digital policy and public communication. In 2025, Reuters reported that the ministry was preparing a national AI roadmap to give investors and developers a clearer view of Indonesia's market, infrastructure and computing capacity. == AI Governance == Indonesia has introduced policy guidance on the ethical use of artificial intelligence. The policy sets out ethical values for the development and use of AI. These include humanity, security, transparency, credibility and accountability, personal data protection, sustainable development and intellectual property protection. A UNESCO country profile on Indonesia noted that Indonesia had adopted a national AI strategy and had policy frameworks. It also identified gaps in internet access, gender inclusion, language datasets, digital talent and cybersecurity. UNESCO recommended that Indonesia update its AI standards, invest in ethical AI, strengthen research coordination and consider establishing a national agency for artificial intelligence. In May 2026, Antara News reported comments by Deputy Minister of Communication and Digital Affairs Nezar Patria. Who said that AI safety requires partnerships, shared standards and continuing dialogue. == Sectors == AI policy discussions in Indonesia have identified health, agriculture, education, government services, mobility and smart cities as areas where AI could be applied. Mobile technology and e-commerce have been discussed as important areas of AI adoption in Indonesia. Research on AI adoption in Indonesia by Siddhartha Paul Tiwari and Adi Fahrudin has also examined mobile and e-commerce sectors. UNESCO has also noted that Indonesia's large digital economy and startup ecosystem have supported AI adoption, while also pointing to challenges in talent, research capacity and cybersecurity. Indonesia is one of the developing-country markets attracting AI infrastructure investment, including data centres. == Challenges == Indonesia faces several challenges in developing and governing AI. These include gaps in computing infrastructure, uneven connectivity outside major cities, shortages of skilled workers, limited research funding, cybersecurity risks, misinformation, data leaks and the underrepresentation of Indonesian and indigenous languages in AI datasets. UNESCO noted that Bahasa is spoken by around 200 million people but remains underrepresented in AI. It also noted that Indonesia has more than 700 indigenous languages, many of which face the risk of extinction. UNESCO recommended stronger coordination in AI research and a more unified strategy for using AI in language preservation.

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  Ontology merging

  Ontology merging defines the act of bringing together two conceptually divergent ontologies or the instance data associated to two ontologies. This is similar to work in database merging (schema matching). This merging process can be performed in a number of ways, manually, semi automatically, or automatically. Manual ontology merging although ideal is extremely labour-intensive and current research attempts to find semi or entirely automated techniques to merge ontologies. These techniques are statistically driven often taking into account similarity of concepts and raw similarity of instances through textual string metrics and semantic knowledge. These techniques are similar to those used in information integration employing string metrics from open source similarity libraries.

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  Hierarchical Risk Parity

  Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .

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  Ontology (information science)

  In information science, an ontology encompasses a representation, formal naming, and definitions of the categories, properties, and relations between the concepts, data, or entities that pertain to one, many, or all domains of discourse. More simply, an ontology is a way of showing the properties of a subject area and how they are related, by defining a set of terms and relational expressions that represent the entities in that subject area. The field which studies ontologies so conceived is sometimes referred to as applied ontology. Every academic discipline or field, in creating its terminology, thereby lays the groundwork for an ontology. Each uses ontological assumptions to frame explicit theories, research and applications. Improved ontologies may improve problem solving within that domain, interoperability of data systems, and discoverability of data. Translating research papers within every field is a problem made easier when experts from different countries maintain a controlled vocabulary of jargon between each of their languages. For instance, the definition and ontology of economics is a primary concern in Marxist economics, but also in other subfields of economics. An example of economics relying on information science occurs in cases where a simulation or model is intended to enable economic decisions, such as determining what capital assets are at risk and by how much (see risk management). What ontologies in both information science and philosophy have in common is the attempt to represent entities, including both objects and events, with all their interdependent properties and relations, according to a system of categories. In both fields, there is considerable work on problems of ontology engineering (e.g., Quine and Kripke in philosophy, Sowa and Guarino in information science), and debates concerning to what extent normative ontology is possible (e.g., foundationalism and coherentism in philosophy, BFO and Cyc in artificial intelligence). Applied ontology is considered by some as a successor to prior work in philosophy. However many current efforts are more concerned with establishing controlled vocabularies of narrow domains than with philosophical first principles, or with questions such as the mode of existence of fixed essences or whether enduring objects (e.g., perdurantism and endurantism) may be ontologically more primary than processes. Artificial intelligence has retained considerable attention regarding applied ontology in subfields like natural language processing within machine translation and knowledge representation, but ontology editors are being used often in a range of fields, including biomedical informatics and industry. Such efforts often use ontology editing tools such as Protégé. == Ontology in philosophy == Ontology is a branch of philosophy and intersects areas such as metaphysics, epistemology, and philosophy of language, as it considers how knowledge, language, and perception relate to the nature of reality. Metaphysics deals with questions like "what exists?" and "what is the nature of reality?". One of five traditional branches of philosophy, metaphysics is concerned with exploring existence through properties, entities and relations such as those between particulars and universals, intrinsic and extrinsic properties, or essence and existence. Metaphysics has been an ongoing topic of discussion since recorded history. == Etymology == The compound word ontology combines onto-, from the Greek ὄν, on (gen. ὄντος, ontos), i.e. "being; that which is", which is the present participle of the verb εἰμί, eimí, i.e. "to be, I am", and -λογία, -logia, i.e. "logical discourse", see classical compounds for this type of word formation. While the etymology is Greek, the oldest extant record of the word itself, the Neo-Latin form ontologia, appeared in 1606 in the work Ogdoas Scholastica by Jacob Lorhard (Lorhardus) and in 1613 in the Lexicon philosophicum by Rudolf Göckel (Goclenius). The first occurrence in English of ontology as recorded by the OED (Oxford English Dictionary, online edition, 2008) came in Archeologia Philosophica Nova or New Principles of Philosophy (1663) by Gideon Harvey. == Formal ontology == Since the mid-1970s, researchers in the field of artificial intelligence (AI) have recognized that knowledge engineering is the key to building large and powerful AI systems. AI researchers argued that they could create new ontologies as computational models that enable certain kinds of automated reasoning, which was only marginally successful. In the 1980s, the AI community began to use the term ontology to refer to both a theory of a modeled world and a component of knowledge-based systems. In particular, David Powers introduced the word ontology to AI to refer to real world or robotic grounding, publishing in 1990 literature reviews emphasizing grounded ontology in association with the call for papers for a AAAI Summer Symposium Machine Learning of Natural Language and Ontology, with an expanded version published in SIGART Bulletin and included as a preface to the proceedings. Some researchers, drawing inspiration from philosophical ontologies, viewed computational ontology as a kind of applied philosophy. In 1993, the widely cited web page and paper "Toward Principles for the Design of Ontologies Used for Knowledge Sharing" by Tom Gruber used ontology as a technical term in computer science closely related to earlier idea of semantic networks and taxonomies. Gruber introduced the term as a specification of a conceptualization: An ontology is a description (like a formal specification of a program) of the concepts and relationships that can formally exist for an agent or a community of agents. This definition is consistent with the usage of ontology as set of concept definitions, but more general. And it is a different sense of the word than its use in philosophy. Attempting to distance ontologies from taxonomies and similar efforts in knowledge modeling that rely on classes and inheritance, Gruber stated (1993): Ontologies are often equated with taxonomic hierarchies of classes, class definitions, and the subsumption relation, but ontologies need not be limited to these forms. Ontologies are also not limited to conservative definitions, that is, definitions in the traditional logic sense that only introduce terminology and do not add any knowledge about the world (Enderton, 1972). To specify a conceptualization, one needs to state axioms that do constrain the possible interpretations for the defined terms. Recent experimental ontology frameworks have also explored resonance-based AI-human co-evolution structures, such as IAMF (Illumination AI Matrix Framework), OntoMotoOS (a meta-operating system concept for ethical and ontological AI–human co-evolution), and PSRT (Phase-Structural Reality Theory across multi-scale ontological layers). Though not yet widely adopted in academic discourse, such models propose phased approaches to ethical harmonization and structural emergence. As refinement of Gruber's definition Feilmayr and Wöß (2016) stated: "An ontology is a formal, explicit specification of a shared conceptualization that is characterized by high semantic expressiveness required for increased complexity." == Formal ontology components == Contemporary ontologies share many structural similarities, regardless of the language in which they are expressed. Most ontologies describe individuals (instances), classes (concepts), attributes and relations. === Types === ==== Domain ontology ==== A domain ontology (or domain-specific ontology) represents concepts which belong to a realm of the world, such as biology or politics. Each domain ontology typically models domain-specific definitions of terms. For example, the word card has many different meanings. An ontology about the domain of poker would model the "playing card" meaning of the word, while an ontology about the domain of computer hardware would model the "punched card" and "video card" meanings. Since domain ontologies are written by different people, they represent concepts in very specific and unique ways, and are often incompatible within the same project. As systems that rely on domain ontologies expand, they often need to merge domain ontologies by hand-tuning each entity or using a combination of software merging and hand-tuning. This presents a challenge to the ontology designer. Different ontologies in the same domain arise due to different languages, different intended usage of the ontologies, and different perceptions of the domain (based on cultural background, education, ideology, etc.). At present, merging ontologies that are not developed from a common upper ontology is a largely manual process and therefore time-consuming and expensive. Domain ontologies that use the same upper ontology to provide a set of basic elements with which to specify the meanings of the domain ontology entities can be merged with less effo

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  Very large database

  A very large database, (originally written very large data base) or VLDB, is a database that contains a very large amount of data, so much that it can require specialized architectural, management, processing and maintenance methodologies. == Definition == The vague adjectives of very and large allow for a broad and subjective interpretation, but attempts at defining a metric and threshold have been made. Early metrics were the size of the database in a canonical form via database normalization or the time for a full database operation like a backup. Technology improvements have continually changed what is considered very large. One definition has suggested that a database has become a VLDB when it is "too large to be maintained within the window of opportunity… the time when the database is quiet". == Sizes of a VLDB database == There is no absolute amount of data that can be cited. For example, one cannot say that any database with more than 1 TB of data is considered a VLDB. This absolute amount of data has varied over time as computer processing, storage and backup methods have become better able to handle larger amounts of data. That said, VLDB issues may start to appear when 1 TB is approached, and are more than likely to have appeared as 30 TB or so is exceeded. == VLDB challenges == Key areas where a VLDB may present challenges include configuration, storage, performance, maintenance, administration, availability and server resources. === Configuration === Careful configuration of databases that lie in the VLDB realm is necessary to alleviate or reduce issues raised by VLDB databases. === Administration === The complexities of managing a VLDB can increase exponentially for the database administrator as database size increases. === Availability and maintenance === When dealing with VLDB operations relating to maintenance and recovery such as database reorganizations and file copies which were quite practical on a non-VLDB take very significant amounts of time and resources for a VLDB database. In particular it typically infeasible to meet a typical recovery time objective (RTO), the maximum expected time a database is expected to be unavailable due to interruption, by methods which involve copying files from disk or other storage archives. To overcome these issues techniques such as clustering, cloned/replicated/standby databases, file-snapshots, storage snapshots or a backup manager may help achieve the RTO and availability, although individual methods may have limitations, caveats, license, and infrastructure requirements while some may risk data loss and not meet the recovery point objective (RPO). For many systems only geographically remote solutions may be acceptable. ==== Backup and recovery ==== Best practice is for backup and recovery to be architectured in terms of the overall availability and business continuity solution. === Performance === Given the same infrastructure there may typically be a decrease in performance, that is increase in response time as database size increases. Some accesses will simply have more data to process (scan) which will take proportionally longer (linear time); while the indexes used to access data may grow slightly in height requiring perhaps an extra storage access to reach the data (sub-linear time). Other effects can be caching becoming less efficient because proportionally less data can be cached and while some indexes such as the B+ automatically sustain well with growth others such as a hash table may need to be rebuilt. Should an increase in database size cause the number of accessors of the database to increase then more server and network resources may be consumed, and the risk of contention will increase. Some solutions to regaining performance include partitioning, clustering, possibly with sharding, or use of a database machine. ==== Partitioning ==== Partitioning may be able assist the performance of bulk operations on a VLDB including backup and recovery., bulk movements due to information lifecycle management (ILM), reducing contention as well as allowing optimization of some query processing. === Storage === In order to satisfy needs of a VLDB the database storage needs to have low access latency and contention, high throughput, and high availability. === Server resources === The increasing size of a VLDB may put pressure on server and network resources and a bottleneck may appear that may require infrastructure investment to resolve. == Relationship to big data == VLDB is not the same as big data, but the storage aspect of big data may involve a VLDB database. That said some of the storage solutions supporting big data were designed from the start to support large volumes of data, so database administrators may not encounter VLDB issues that older versions of traditional RDBMS's might encounter.

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  Semantic heterogeneity

  Semantic heterogeneity is when database schema or datasets for the same domain are developed by independent parties, resulting in differences in meaning and interpretation of data values. Beyond structured data, the problem of semantic heterogeneity is compounded due to the flexibility of semi-structured data and various tagging methods applied to documents or unstructured data. Semantic heterogeneity is one of the more important sources of differences in heterogeneous datasets. Yet, for multiple data sources to interoperate with one another, it is essential to reconcile these semantic differences. Decomposing the various sources of semantic heterogeneities provides a basis for understanding how to map and transform data to overcome these differences. == Classification == One of the first known classification schemes applied to data semantics is from William Kent in the late 80s. Kent's approach dealt more with structural mapping issues than differences in meaning, which he pointed to data dictionaries as potentially solving. One of the most comprehensive classifications is from Pluempitiwiriyawej and Hammer, "Classification Scheme for Semantic and Schematic Heterogeneities in XML Data Sources". They classify heterogeneities into three broad classes: Structural conflicts arise when the schema of the sources representing related or overlapping data exhibit discrepancies. Structural conflicts can be detected when comparing the underlying schema. The class of structural conflicts includes generalization conflicts, aggregation conflicts, internal path discrepancy, missing items, element ordering, constraint and type mismatch, and naming conflicts between the element types and attribute names. Domain conflicts arise when the semantics of the data sources that will be integrated exhibit discrepancies. Domain conflicts can be detected by looking at the information contained in the schema and using knowledge about the underlying data domains. The class of domain conflicts includes schematic discrepancy, scale or unit, precision, and data representation conflicts. Data conflicts refer to discrepancies among similar or related data values across multiple sources. Data conflicts can only be detected by comparing the underlying sources. The class of data conflicts includes ID-value, missing data, incorrect spelling, and naming conflicts between the element contents and the attribute values. Moreover, mismatches or conflicts can occur between set elements (a "population" mismatch) or attributes (a "description" mismatch). Michael Bergman expanded upon this schema by adding a fourth major explicit category of language, and also added some examples of each kind of semantic heterogeneity, resulting in about 40 distinct potential categories . This table shows the combined 40 possible sources of semantic heterogeneities across sources: A different approach toward classifying semantics and integration approaches is taken by Sheth et al. Under their concept, they split semantics into three forms: implicit, formal and powerful. Implicit semantics are what is either largely present or can easily be extracted; formal languages, though relatively scarce, occur in the form of ontologies or other description logics; and powerful (soft) semantics are fuzzy and not limited to rigid set-based assignments. Sheth et al.'s main point is that first-order logic (FOL) or description logic is inadequate alone to properly capture the needed semantics. == Relevant applications == Besides data interoperability, relevant areas in information technology that depend on reconciling semantic heterogeneities include data mapping, semantic integration, and enterprise information integration, among many others. From the conceptual to actual data, there are differences in perspective, vocabularies, measures and conventions once any two data sources are brought together. Explicit attention to these semantic heterogeneities is one means to get the information to integrate or interoperate. A mere twenty years ago, information technology systems expressed and stored data in a multitude of formats and systems. The Internet and Web protocols have done much to overcome these sources of differences. While there is a large number of categories of semantic heterogeneity, these categories are also patterned and can be anticipated and corrected. These patterned sources inform what kind of work must be done to overcome semantic differences where they still reside.

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  Weak supervision

  Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log ⁡ p ( { x i , y i } i = 1 l | θ ) + λ log ⁡ p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign ⁡ f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst

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  Bartels–Stewart algorithm

  In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation A X − X B = C {\displaystyle AX-XB=C} . Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A {\displaystyle A} and B {\displaystyle B} to transform A X − X B = C {\displaystyle AX-XB=C} into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X {\displaystyle X} is of small to moderate size. == The algorithm == Let X , C ∈ R m × n {\displaystyle X,C\in \mathbb {R} ^{m\times n}} , and assume that the eigenvalues of A {\displaystyle A} are distinct from the eigenvalues of B {\displaystyle B} . Then, the matrix equation A X − X B = C {\displaystyle AX-XB=C} has a unique solution. The Bartels–Stewart algorithm computes X {\displaystyle X} by applying the following steps: 1.Compute the real Schur decompositions R = U T A U , {\displaystyle R=U^{T}AU,} S = V T B T V . {\displaystyle S=V^{T}B^{T}V.} The matrices R {\displaystyle R} and S {\displaystyle S} are block-upper triangular matrices, with diagonal blocks of size 1 × 1 {\displaystyle 1\times 1} or 2 × 2 {\displaystyle 2\times 2} . 2. Set F = U T C V . {\displaystyle F=U^{T}CV.} 3. Solve the simplified system R Y − Y S T = F {\displaystyle RY-YS^{T}=F} , where Y = U T X V {\displaystyle Y=U^{T}XV} . This can be done using forward substitution on the blocks. Specifically, if s k − 1 , k = 0 {\displaystyle s_{k-1,k}=0} , then ( R − s k k I ) y k = f k + ∑ j = k + 1 n s k j y j , {\displaystyle (R-s_{kk}I)y_{k}=f_{k}+\sum _{j=k+1}^{n}s_{kj}y_{j},} where y k {\displaystyle y_{k}} is the k {\displaystyle k} th column of Y {\displaystyle Y} . When s k − 1 , k ≠ 0 {\displaystyle s_{k-1,k}\neq 0} , columns [ y k − 1 ∣ y k ] {\displaystyle [y_{k-1}\mid y_{k}]} should be concatenated and solved for simultaneously. 4. Set X = U Y V T . {\displaystyle X=UYV^{T}.} === Computational cost === Using the QR algorithm, the real Schur decompositions in step 1 require approximately 10 ( m 3 + n 3 ) {\displaystyle 10(m^{3}+n^{3})} flops, so that the overall computational cost is 10 ( m 3 + n 3 ) + 2.5 ( m n 2 + n m 2 ) {\displaystyle 10(m^{3}+n^{3})+2.5(mn^{2}+nm^{2})} . === Simplifications and special cases === In the special case where B = − A T {\displaystyle B=-A^{T}} and C {\displaystyle C} is symmetric, the solution X {\displaystyle X} will also be symmetric. This symmetry can be exploited so that Y {\displaystyle Y} is found more efficiently in step 3 of the algorithm. == The Hessenberg–Schur algorithm == The Hessenberg–Schur algorithm replaces the decomposition R = U T A U {\displaystyle R=U^{T}AU} in step 1 with the decomposition H = Q T A Q {\displaystyle H=Q^{T}AQ} , where H {\displaystyle H} is an upper-Hessenberg matrix. This leads to a system of the form H Y − Y S T = F {\displaystyle HY-YS^{T}=F} that can be solved using forward substitution. The advantage of this approach is that H = Q T A Q {\displaystyle H=Q^{T}AQ} can be found using Householder reflections at a cost of ( 5 / 3 ) m 3 {\displaystyle (5/3)m^{3}} flops, compared to the 10 m 3 {\displaystyle 10m^{3}} flops required to compute the real Schur decomposition of A {\displaystyle A} . == Software and implementation == The subroutines required for the Hessenberg-Schur variant of the Bartels–Stewart algorithm are implemented in the SLICOT library. These are used in the MATLAB control system toolbox. == Alternative approaches == For large systems, the O ( m 3 + n 3 ) {\displaystyle {\mathcal {O}}(m^{3}+n^{3})} cost of the Bartels–Stewart algorithm can be prohibitive. When A {\displaystyle A} and B {\displaystyle B} are sparse or structured, so that linear solves and matrix vector multiplies involving them are efficient, iterative algorithms can potentially perform better. These include projection-based methods, which use Krylov subspace iterations, methods based on the alternating direction implicit (ADI) iteration, and hybridizations that involve both projection and ADI. Iterative methods can also be used to directly construct low rank approximations to X {\displaystyle X} when solving A X − X B = C {\displaystyle AX-XB=C} .

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  NoSQL

  NoSQL (originally meaning "not only SQL" or "non-relational") refers to a type of database design that stores and retrieves data differently from the traditional table-based structure of relational databases. Unlike relational databases, which organize data into rows and columns like a spreadsheet, NoSQL databases use a single data structure—such as key–value pairs, wide columns, graphs, or documents—to hold information. Since this non-relational design does not require a fixed schema, it scales easily to manage large, often unstructured datasets. NoSQL systems are sometimes called "Not only SQL" because they can support SQL-like query languages or work alongside SQL databases in polyglot-persistent setups, where multiple database types are combined. Non-relational databases date back to the late 1960s, but the term "NoSQL" emerged in the early 2000s, spurred by the needs of Web 2.0 companies like social media platforms. NoSQL databases are popular in big data and real-time web applications due to their simple design, ability to scale across clusters of machines (called horizontal scaling), and precise control over data availability. These structures can speed up certain tasks and are often considered more adaptable than fixed database tables. However, many NoSQL systems prioritize speed and availability over strict consistency (per the CAP theorem), using eventual consistency—where updates reach all nodes eventually, typically within milliseconds, but may cause brief delays in accessing the latest data, known as stale reads. While most lack full ACID transaction support, some, like MongoDB, include it as a key feature. == Barriers to adoption == Barriers to wider NoSQL adoption include their use of low-level query languages instead of SQL, inability to perform ad hoc joins across tables, lack of standardized interfaces, and significant investments already made in relational databases. Some NoSQL systems risk losing data through lost writes or other forms, though features like write-ahead logging—a method to record changes before they’re applied—can help prevent this. For distributed transaction processing across multiple databases, keeping data consistent is a challenge for both NoSQL and relational systems, as relational databases cannot enforce rules linking separate databases, and few systems support both ACID transactions and X/Open XA standards for managing distributed updates. Limitations within the interface environment are overcome using semantic virtualization protocols, such that NoSQL services are accessible to most operating systems. == History == The term NoSQL was used by Carlo Strozzi in 1998 to name his lightweight Strozzi NoSQL open-source relational database that did not expose the standard Structured Query Language (SQL) interface, but was still relational. His NoSQL RDBMS is distinct from the around-2009 general concept of NoSQL databases. Strozzi suggests that, because the current NoSQL movement "departs from the relational model altogether, it should therefore have been called more appropriately 'NoREL'", referring to "not relational". Johan Oskarsson, then a developer at Last.fm, reintroduced the term NoSQL in early 2009 when he organized an event to discuss "open-source distributed, non-relational databases". The name attempted to label the emergence of an increasing number of non-relational, distributed data stores, including open source clones of Google's Bigtable/MapReduce and Amazon's DynamoDB. == Types and examples == There are various ways to classify NoSQL databases, with different categories and subcategories, some of which overlap. What follows is a non-exhaustive classification by data model, with examples: === Key–value store === Key–value (KV) stores use the associative array (also called a map or dictionary) as their fundamental data model. In this model, data is represented as a collection of key–value pairs, such that each possible key appears at most once in the collection. The key–value model is one of the simplest non-trivial data models, and richer data models are often implemented as an extension of it. The key–value model can be extended to a discretely ordered model that maintains keys in lexicographic order. This extension is computationally powerful, in that it can efficiently retrieve selective key ranges. Key–value stores can use consistency models ranging from eventual consistency to serializability. Some databases support ordering of keys. There are various hardware implementations, and some users store data in memory (RAM), while others on solid-state drives (SSD) or rotating disks (aka hard disk drive (HDD)). === Document store === The central concept of a document store is that of a "document". While the details of this definition differ among document-oriented databases, they all assume that documents encapsulate and encode data (or information) in some standard formats or encodings. Encodings in use include XML, YAML, and JSON and binary forms like BSON. Documents are addressed in the database via a unique key that represents that document. Another defining characteristic of a document-oriented database is an API or query language to retrieve documents based on their contents. Different implementations offer different ways of organizing and/or grouping documents: Collections Tags Non-visible metadata Directory hierarchies Compared to relational databases, collections could be considered analogous to tables and documents analogous to records. But they are different – every record in a table has the same sequence of fields, while documents in a collection may have fields that are completely different. === Graph === Graph databases are designed for data whose relations are well represented as a graph consisting of elements connected by a finite number of relations. Examples of data include social relations, public transport links, road maps, network topologies, etc. Graph databases and their query language == Performance == The performance of NoSQL databases is usually evaluated using the metric of throughput, which is measured as operations per second. Performance evaluation must pay attention to the right benchmarks such as production configurations, parameters of the databases, anticipated data volume, and concurrent user workloads. Ben Scofield rated different categories of NoSQL databases as follows: Performance and scalability comparisons are most commonly done using the YCSB benchmark. == Handling relational data == Since most NoSQL databases lack ability for joins in queries, the database schema generally needs to be designed differently. There are three main techniques for handling relational data in a NoSQL database. (See table join and ACID support for NoSQL databases that support joins.) === Multiple queries === Instead of retrieving all the data with one query, it is common to do several queries to get the desired data. NoSQL queries are often faster than traditional SQL queries, so the cost of additional queries may be acceptable. If an excessive number of queries would be necessary, one of the other two approaches is more appropriate. === Caching, replication and non-normalized data === Instead of only storing foreign keys, it is common to store actual foreign values along with the model's data. For example, each blog comment might include the username in addition to a user id, thus providing easy access to the username without requiring another lookup. When a username changes, however, this will now need to be changed in many places in the database. Thus this approach works better when reads are much more common than writes. === Nesting data === With document databases like MongoDB it is common to put more data in a smaller number of collections. For example, in a blogging application, one might choose to store comments within the blog post document, so that with a single retrieval one gets all the comments. Thus in this approach a single document contains all the data needed for a specific task. == ACID and join support == A database is marked as supporting ACID properties (atomicity, consistency, isolation, durability) or join operations if the documentation for the database makes that claim. However, this doesn't necessarily mean that the capability is fully supported in a manner similar to most SQL databases. == Query optimization and indexing in NoSQL databases == Different NoSQL databases, such as DynamoDB, MongoDB, Cassandra, Couchbase, HBase, and Redis, exhibit varying behaviors when querying non-indexed fields. Many perform full-table or collection scans for such queries, applying filtering operations after retrieving data. However, modern NoSQL databases often incorporate advanced features to optimize query performance. For example, MongoDB supports compound indexes and query-optimization strategies, Cassandra offers secondary indexes and materialized views, and Redis employs custom indexing mechanisms tailored to specific use cases. Systems like El

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