A concordancer is a computer program that automatically constructs a concordance—an alphabetised index of every occurrence of a word or phrase in a body of text, each entry displayed with its surrounding context. Concordancers are primary tools in corpus linguistics, lexicography, computer-assisted translation, and language teaching. The most common display format is the key word in context (KWIC) layout, in which each hit appears centred on a line with a fixed span of words to its left and right, enabling rapid scanning of usage patterns across many occurrences. == History == === Pre-computational concordances === The compilation of concordances predates computers by many centuries. Around 1230, the French Dominican cardinal Hugh of Saint-Cher directed a team of friars in assembling a concordance of the Latin Vulgate Bible, generally regarded as the first systematic concordance of any text. To help readers locate passages, Hugh divided each biblical chapter into lettered sections. Later milestones include a Hebrew Old Testament concordance compiled by Rabbi Mordecai Nathan (1448), Alexander Cruden's Complete Concordance to the Holy Scriptures (1737), and the manuscript Asaf ha-Mazkir, an unfinished concordance to the Babylonian Talmud compiled by Moses Rigotz around the turn of the 19th century. === First computer concordance === The first concordance produced with computing assistance was the Index Thomisticus, a comprehensive lexical index of the writings of and around Thomas Aquinas, totalling approximately 10.6 million Latin words. The Italian Jesuit priest Roberto Busa conceived the project in 1946 and secured the sponsorship of IBM in 1949 after a meeting with chairman Thomas J. Watson. Keypunch operators in Gallarate, Italy, encoded the texts onto punched cards from around 1950. IBM executive Paul Tasman developed the processing methods. The full 56-volume printed edition was completed around 1980, followed by a CD-ROM edition in 1989 and a web-accessible version in 2005. === The KWIC format === The key word in context (KWIC) display was formalised as a computational technique by Hans Peter Luhn, a researcher at IBM, in a 1960 paper in American Documentation. In KWIC output, each instance of the search term (the node word) is centred on a line with a fixed window of words to each side; sorting the resulting lines alphabetically by the immediately adjacent word reveals collocational and phraseological patterns at a glance. === COCOA === One of the first dedicated concordancing programs was COCOA (COunt and COncordance Generation on Atlas), created in 1965 by D. B. Russell at University College London and the Atlas Computer Laboratory in Harwell, Oxfordshire. Written in approximately 4,000 cards of FORTRAN, it processed text annotated with flat, non-hierarchical markup tags and could produce word counts and concordances in multiple languages. Within its first six months COCOA had been applied to texts in at least six languages. A second version designed for multiple mainframe platforms was distributed to British computing centres in the mid-1970s. Growing dissatisfaction with its interface and the eventual withdrawal of Atlas Laboratory support prompted British funding bodies to commission a successor program. === Oxford Concordance Program === The Oxford Concordance Program (OCP) was designed and written in FORTRAN by Susan Hockey and Ian Marriott at Oxford University Computing Services (OUCS) between 1979 and 1980 and first released in 1981. Hockey and Marriott acknowledged that OCP owed much to COCOA and the CLOC system at the University of Birmingham. OCP accepted COCOA-format markup to encode metadata such as author, act, scene, and line number, and was described by its authors as "a machine-independent text analysis program for producing word lists, indices and concordances in a variety of languages and alphabets." By the mid-1980s it had been licensed to approximately 240 institutions in 23 countries. A personal computer version, Micro-OCP, was developed for the IBM PC and sold by Oxford University Press from the late 1980s. Version 2 was rewritten in 1985–86 and documented in the same 1987 article by Hockey and co-author John Martin. === Personal computer era === The availability of affordable personal computers in the 1980s and 1990s enabled standalone concordancing applications that analysts could run locally without specialist computing facilities. MicroConcord, developed by Mike Scott and Tim Johns and published by Oxford University Press in 1993 for MS-DOS, was among the first concordancers designed specifically for classroom language teaching. WordSmith Tools, also developed by Mike Scott, was first released in 1996 and became one of the most widely used corpus analysis suites in academic linguistics research. Other tools from this era include TACT (University of Toronto, 1989), a suite of MS-DOS freeware programs for literary text analysis, and MonoConc, a Windows concordancer created by Michael Barlow. === Web-based concordancers === From the late 1990s onwards, web-based concordancers hosted on remote servers gave researchers browser access to large preloaded corpora without requiring local storage or processing. The Sketch Engine, developed by Adam Kilgarriff and Pavel Rychlý (Masaryk University), was launched commercially in July 2003 by Lexical Computing Limited and introduced word sketches—automatically generated one-page profiles of a word's typical grammatical relations and collocations. AntConc, created by Laurence Anthony at Waseda University, Tokyo, was first released in 2002 as freeware for Windows, macOS, and Linux. == Features == Modern concordancers typically offer a range of analytical functions beyond basic KWIC display. These commonly include: KWIC display with the node word centred and context words in aligned columns, sortable by the word one, two, or three positions to the left or right of the node (L1–L3 and R1–R3) Concordance plots, visualising the distribution of hits as marks along a scaled bar representing each text in the corpus Frequency and word lists, both alphabetical and ranked by frequency Collocation statistics, identifying words that co-occur with the search term more often than chance, quantified by measures such as mutual information, the t-score, or log-likelihood Keyword analysis, comparing word frequencies between a study corpus and a reference corpus to identify statistically distinctive items N-gram analysis, finding frequently recurring word sequences of a specified length Part-of-speech tagging integration, allowing searches filtered to particular grammatical categories Unicode support for multilingual text Bilingual and parallel concordancers additionally display aligned text in two or more languages side by side, enabling comparison of translation equivalents across language pairs. == Notable concordancers == === WordSmith Tools === Created by Mike Scott and first released in 1996, WordSmith Tools is a Windows corpus analysis suite that evolved from MicroConcord. Its three core modules are Concord (KWIC concordances), WordList (frequency and alphabetical word lists), and Keywords (statistical keyword identification relative to a reference corpus). Oxford University Press used WordSmith Tools for dictionary preparation work. Version 4.0 is freely available; later versions are sold by Lexical Analysis Software Limited. === AntConc === AntConc is a freeware, multiplatform concordancing toolkit created by Laurence Anthony, Professor of Applied Linguistics at Waseda University, Tokyo. First released in 2002 and formally described in a 2005 academic paper, it runs on Windows, macOS, and Linux. Its tools include a KWIC concordancer, a concordance plot for visualising distribution across texts, a collocates tool, a keyword list, and an n-gram analysis module. Because it is free and requires only plain text files, AntConc is widely used in linguistics courses and independent research worldwide. === Sketch Engine === The Sketch Engine is a corpus management and query system co-created by Adam Kilgarriff and Pavel Rychlý and launched in 2003 by Lexical Computing Limited. It provides browser-based access to over 800 corpora in more than 100 languages. Beyond concordance searching, it offers word sketches, collocation analysis, distributional thesaurus construction, keyword and terminology extraction, and diachronic analysis. It is used by major publishers including Macmillan and Oxford University Press for lexicographic research. A subset tool, SKELL (Sketch Engine for Language Learning), is freely accessible to individual learners. === Wmatrix === Wmatrix is a web-based corpus processing environment developed by Paul Rayson at the University Centre for Computer Corpus Research on Language (UCREL), Lancaster University. Alongside concordances and frequency lists, Wmatrix integrates CLAWS part-of-speech tagging and the USAS semantic tagger, enabling keyword analysis simultane
Jais (language model)
Jais is an open-source large language model launched in August 2023. Developed as a collaboration between Emirati AI company G42, the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), and US-based Cerebras Systems, Jais was designed to produce high-quality Arabic text and was also trained on English data. The model's creation was motivated by the underrepresentation of the Arabic language in the field of generative artificial intelligence. It aims to provide a more culturally and linguistically accurate model for the world's 400 million Arabic speakers. Its name is a reference to Jebel Jais, the highest mountain in the UAE. == Background and development == Jais was developed in response to the limited availability of advanced generative artificial intelligence models for the Arabic language, despite it being spoken by over 400 million people. Existing models were often trained on limited or low-quality Arabic web content, resulting in poor performance. The project represents a significant investment by the United Arab Emirates in the field of AI as part of its national strategy. The model was created through a partnership between Inception (now Core42), a subsidiary of the Abu Dhabi-based AI company G42; the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI); and Cerebras Systems, a US company specializing in AI hardware. The model is named after Jebel Jais, the highest peak in the UAE. == Training == The initial version of Jais released in August 2023 had 13 billion parameters. In November 2023, Core42 released Jais 30B, an improved version with 30 billion parameters. Both models were trained on a subset of the Cerebras Condor Galaxy 1 supercomputer. The training dataset consisted of a mix of Arabic, English, and computer code. According to Timothy Baldwin, a professor of natural language processing at MBZUAI, training the model on a diverse Arabic dataset allows it to switch between dialects. == Features == Jais is designed to generate text in both English and Arabic. The project has also released instruction-tuned "Chat" variants for both the 13B and 30B models, which are specifically optimized for conversational applications. Additional functionality for working with images, graphs, and tabular data is planned for future releases.
Personoid
Personoid is the concept coined by Stanisław Lem, a Polish science-fiction writer, in Non Serviam, from his book A Perfect Vacuum (1971). His personoids are an abstraction of functions of human mind and they live in computers; they do not need any human-like physical body. In cognitive and software modeling, personoid is a research approach to the development of intelligent autonomous agents. In frame of the IPK (Information, Preferences, Knowledge) architecture, it is a framework of abstract intelligent agent with a cognitive and structural intelligence. It can be seen as an essence of high intelligent entities. From the philosophical and systemics perspectives, personoid societies can also be seen as the carriers of a culture. According to N. Gessler, the personoids study can be a base for the research on artificial culture and culture evolution. == Personoids on TV and cinema == Welt am Draht (1973) The Thirteenth Floor (1999)
.ai
.ai is the Internet country code top-level domain (ccTLD) for Anguilla, a British Overseas Territory in the Caribbean. It is administered by the government of Anguilla. It is a popular domain hack with companies and projects related to the artificial intelligence industry (AI). Google's ad targeting treats .ai as a generic top-level domain (gTLD) because "users and website owners frequently see [the domain] as being more generic than country-targeted." In 2021, Google Search analyst Gary Illyes announced that ".ai" had been added to Google’s list of generic country-code top-level domains, meaning that Google would no longer infer Anguilla-specific targeting from the ccTLD. Identity Digital began managing the domain as of January 2025. == Second and third level registrations == Registrations within off.ai, com.ai, net.ai, and org.ai are available worldwide without restriction. From 15 September 2009, second level registrations within .ai are available to everyone worldwide. == Registration == The minimum registration term allowed for .ai domains is 2 through 10 years for registration and renewal, and a 2-year renewal for domain transfer. Identity Digital is the authority in charge of managing this extension. Registrations began on 16 February 1995. The limits on the number of characters used for the domain name are, at a minimum, from 1 to 3, depending on the registrar, and always at most 63 characters. The character set supported for .ai domain names includes A–Z, a–z, 0–9, and hyphen. As of November 2022, .ai domains cannot accommodate IDN characters. There are no requirements for registering a domain, including local and foreign residents. A .ai domain can be suspended or revoked, if the domain is involved in illegal activity such as violating trademarks or copyrights. Usage must not violate the laws of Anguilla. Anguilla uses the UDRP. Filing a UDRP challenge requires using one of the ICANN Approved Dispute Resolution Service Providers. If the domain is with an ICANN accredited registrar, they should work with the arbitrator. Usually this means either doing nothing or transferring a domain. .ai domains are transferable to any desired registrars as the registration of domain is done maintaining EPP. There used to be a whois.ai-based platform of expired domains in which those could be procured and auctioned every ten days through a standard online process. The last auctions of such kind closed there in December 2024; the platform had been scheduled for shutdown on 30 June 2025, but remained online in the months following that date. == Valuation == Domains cost depends on the registrar, with yearly fees ranging from US$140 (the base fee, as established by Anguilla) to $200. As of July 2025, the highest-valued .ai domain is an undisclosed one sold on 8 November 2023, on Escrow.com, for US$1,500,000—months after an initial $300,000 sale to the same buyer. Among the publicly disclosed ones, the most valued, fin.ai, was sold for $1,000,000 in March 2025. On 16 December 2017, the .ai registry started supporting the Extensible Provisioning Protocol (EPP) and migrated all of its domains onto an EPP system. Consequently, many registrars are allowed to sell .ai domains. Since that date, the .ai ccTLD has also been popular with artificial intelligence companies and organisations. Though such trends are primarily seen among new AI based companies or startups, many established AI and Tech companies preferred not to opt for .ai domains. For example, DeepMind has its domain retained at .com; Meta has redirected its facebook.ai domain to ai.meta.com. == Impact on Anguilla's economy == The registration fees earned from the .ai domains go to the treasury of the Government of Anguilla. As per a 2018 New York Times report, the total revenue generated out of selling .ai domains was $2.9 million. In 2023, Anguilla's government made about US$32 million from fees collected for registering .ai domains; that amounted to over 10% of gross domestic product for the territory. "In the years before the real breakthrough of AI, revenue from .ai domains made up less than 1% of our state income, by 2025 it will be around 47%," explained Jose Vanterpool, Minister of Infrastructure and Communications (MICUHITES), in an interview with BBC. The high 90% renewal rate of .ai domains and the 2025 renewal wave of domains registered in 2023 are driving another surge in state revenues, according to Domaintechnik.
Meta-Labeling
Meta-labeling, also known as corrective AI, is a machine learning (ML) technique utilized in quantitative finance to enhance the performance of investment and trading strategies, developed in 2017 by Marcos López de Prado at Guggenheim Partners and Cornell University. The core idea is to separate the decision of trade direction (side) from the decision of trade sizing, addressing the inefficiencies of simultaneously learning both side and size predictions. The side decision involves forecasting market movements (long, short, neutral), while the size decision focuses on risk management and profitability. It serves as a secondary decision-making layer that evaluates the signals generated by a primary predictive model. By assessing the confidence and likely profitability of those signals, meta-labeling allows investors and algorithms to dynamically size positions and suppress false positives. == Motivation == Meta-labeling is designed to improve precision without sacrificing recall. As noted by López de Prado, attempting to model both the direction and the magnitude of a trade using a single algorithm can result in poor generalization. By separating these tasks, meta-labeling enables greater flexibility and robustness: Enhances control over capital allocation. Reduces overfitting by limiting model complexity. Allows the use of interpretability tools and tailored thresholds to manage risk. Enables dynamic trade suppression in unfavorable regimes. == Applications == Meta-labeling has been applied in a variety of financial ML contexts, including: Algorithmic trading: Filtering and sizing trades to reduce false positives. Portfolio optimization: Scaling exposure across multiple signals with differing confidence levels. Risk management: Dynamically disabling strategies in adverse market conditions. Model validation: Interpreting when and why a model may be underperforming due to regime shifts. == General architecture == Meta-labeling decouples two core components of systematic trading strategies: directional prediction and position sizing. The process involves training a primary model to generate trade signals (e.g., buy, sell, or hold) and then training a secondary model to determine whether each signal is likely to lead to a profitable trade. The second model outputs a probability that is interpreted as the confidence in the forecast, which can be used to adjust the position size or to filter out unreliable trades. Meta-labeling is typically implemented as a three-stage process: Primary model (M1): Predicts the direction or label of a financial outcome using features such as market prices, returns, or volatility indicators. A typical output is directional, e.g., Y ∈ {−1,0,1}, representing short, neutral, or long positions. Secondary model (M2): A binary classifier trained to predict whether the primary model's prediction will be profitable. The target variable is a binary meta-label F ∈ { 0 , 1 } {\displaystyle F\in \{0,1\}} . Inputs can include features used in the primary model, performance diagnostics, or market regime data. Position sizing algorithm (M3): Translates the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 1: Forecasting side === Primary model architecture Figure 1 Figure 1 presents the architecture of a primary model. It focuses on forecasting the side of the trade. Following the example, this model (M1) takes in input data – such as open-high-low-close data and determines the side of the position to take: a negative number is a short position, and positive number is a long position, the range is set between −1 and 1 (the closer it is to −1 or 1, the stronger the models conviction is). When training the model, the labels are −1 and 1, based on the direction of forward returns for some predefined investment horizon. The researcher may decide to apply a recall check (τ: "Tau") by setting a minimum threshold that the initial output needs to be to qualify of a short or long position (if the threshold is not met, no side forecast is predicted, leading to closing of any open positions), this leads to the primary model output which is one of three possible side forecasts: −1, 0, or 1. The primary model also generates evaluation data which can be used by the secondary model, to improve performance of size forecasts. Some examples of evaluation data include rolling accuracy, F1, recall, precision, and AUC scores. === Stage 2: Filtering out false positives === General meta-labeling architecture Figure 2 Next comes the phase of filtering out false positives, by applying a secondary machine learning model (M2), which is a binary classifier trained to determine if the trade will be profitable or not. The model takes as input four general groupings of data: General input data which is predictive of a false positive. For example the last 30 days rolling volatility of the underlying asset. Evaluation data. Market state and regime data, one may find that macro economic data or clustering the market into regimes may help as specific trading strategies are known to perform better in particular regimes. Example: momentum based strategies perform best in periods with low volatility and strong directional moves. Primary models initial input which is a value between −1 and 1. This highlights the strength of the primary models conviction. The output of the model is a value between −1 and 1 (if using a Tanh function) which will indicate the strength of the conviction that a short or long position is profitable, or it could simply be between 0 and 1 (using a sigmoid function) if one only wanted to know if it made money or not. This output allows filtering out trades that are likely to lead to losses. One could stop at this point or use the outputs of the secondary model as inputs to a position sizing algorithm (M3) which could further enhance strategy performance metrics by translating the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 3: Optimizing position sizes === ==== Position sizing methods (M3) ==== Various algorithms have been proposed for transforming predicted probabilities into trade sizes: All-or-nothing: Allocate 100% of capital if the probability exceeds a predefined threshold (e.g., 0.5); otherwise, do not trade. Model confidence: Use the probability score directly as the fraction of capital allocated. Linear scaling: Rescale the model's probabilities using min-max normalization based on the training data. Normal CDF (NCDF): Use a normal cumulative distribution function applied to a z-statistic derived from the predicted probability. Empirical CDF (ECDF): Rank probabilities based on their percentile in the training data to ensure relative allocation. Sigmoid Optimal Position Sizing (SOPS): Applies a smooth non-linear sigmoid transformation optimized to maximize risk-adjusted returns (Sharpe ratio). ==== Model calibration ==== Each machine learning algorithm used in meta-labeling tends to produce outputs with different characteristic distributions; for example, some are approximately normally distributed, whereas others exhibit a pronounced U-shape, concentrating probabilities near the extremes. Due to these varying distributions, simply summing the outputs of different models can inadvertently lead to uneven weighting of signals, biasing trade decisions. To address this, model calibration techniques are essential to adjust the predicted probabilities towards frequentist probabilities, ensuring that model outputs reflect true likelihoods more accurately. Two common calibration techniques are: Platt scaling (Sigmoid scaling): Suitable for correcting S-shaped calibration plots typically produced by models such as support vector machines (SVMs). Isotonic regression: Fits a non-decreasing step function to probabilities and is effective particularly with larger datasets, though it can sometimes lead to overfitting. Transforming predictions to frequentist probabilities is crucial as it provides probabilistic outputs that are directly interpretable as the actual likelihood of an event occurring. Such calibration significantly enhances the effectiveness of fixed position sizing methods, reducing maximum drawdowns and increasing risk-adjusted returns. However, calibration has less impact on position sizing methods that directly estimate parameters from the training data, such as ECDF and SOPS, suggesting that calibration is a critical step mainly for fixed methods that rely heavily on raw model outputs. =
Hexagonal sampling
A multidimensional signal is a function of M independent variables where M ≥ 2 {\displaystyle M\geq 2} . Real world signals, which are generally continuous time signals, have to be discretized (sampled) in order to ensure that digital systems can be used to process the signals. It is during this process of discretization where sampling comes into picture. Although there are many ways of obtaining a discrete representation of a continuous time signal, periodic sampling is by far the simplest scheme. Theoretically, sampling can be performed with respect to any set of points. But practically, sampling is carried out with respect to a set of points that have a certain algebraic structure. Such structures are called lattices. Mathematically, the process of sampling an N {\displaystyle N} -dimensional signal can be written as: w ( t ^ ) = w ( V . n ^ ) {\displaystyle w({\hat {t}})=w(V.{\hat {n}})} where t ^ {\displaystyle {\hat {t}}} is continuous domain M-dimensional vector (M-D) that is being sampled, n ^ {\displaystyle {\hat {n}}} is an M-dimensional integer vector corresponding to indices of a sample, and V is an N × N {\displaystyle N\times N} sampling matrix. == Motivation == Multidimensional sampling provides the opportunity to look at digital methods to process signals. Some of the advantages of processing signals in the digital domain include flexibility via programmable DSP operations, signal storage without the loss of fidelity, opportunity for encryption in communication, lower sensitivity to hardware tolerances. Thus, digital methods are simultaneously both powerful and flexible. In many applications, they act as less expensive alternatives to their analog counterparts. Sometimes, the algorithms implemented using digital hardware are so complex that they have no analog counterparts. Multidimensional digital signal processing deals with processing signals represented as multidimensional arrays such as 2-D sequences or sampled images.[1] Processing these signals in the digital domain permits the use of digital hardware where in signal processing operations are specified by algorithms. As real world signals are continuous time signals, multidimensional sampling plays a crucial role in discretizing the real world signals. The discrete time signals are in turn processed using digital hardware to extract information from the signal. == Preliminaries == === Region of Support === The region outside of which the samples of the signal take zero values is known as the Region of support (ROS). From the definition, it is clear that the region of support of a signal is not unique. === Fourier transform === The Fourier transform is a tool that allows us to simplify mathematical operations performed on the signal. The transform basically represents any signal as a weighted combination of sinusoids. The Fourier and the inverse Fourier transform of an M-dimensional signal can be defined as follows: X a ( Ω ^ ) = ∫ − ∞ + ∞ x a ( t ^ ) e − j Ω ^ T t ^ d t ^ {\displaystyle X_{a}({\hat {\Omega }})=\int _{-\infty }^{+\infty }\!x_{a}({\hat {t}})e^{-j{\hat {\Omega }}^{T}{\hat {t}}}d{\hat {t}}} x a ( t ^ ) = 1 2 π M ∫ − ∞ + ∞ X ( Ω ^ ) e ( j Ω ^ T t ^ ) d Ω ^ {\displaystyle x_{a}({\hat {t}})={\frac {1}{2\pi ^{M}}}\int _{-\infty }^{+\infty }\!X({\hat {\Omega }})e^{(j{\hat {\Omega }}^{T}{\hat {t}})}\,\mathrm {d} {\hat {\Omega }}} The cap symbol ^ indicates that the operation is performed on vectors. The Fourier transform of the sampled signal is observed to be a periodic extension of the continuous time Fourier transform of the signal. This is mathematically represented as: X ( ω ) = 1 | d e t ( V ) | ∑ k X a ( Ω ^ − U k ) {\displaystyle X(\omega )={\frac {1}{|det(V)|}}\sum _{k}\!X_{a}({\hat {\Omega }}-Uk)} where ω = V ~ Ω {\displaystyle \omega ={\tilde {V}}\Omega } and U = 2 π V ~ {\displaystyle U=2\pi {\tilde {V}}} is the periodicity matrix where ~ denotes matrix transposition. Thus sampling in the spatial domain results in periodicity in the Fourier domain. === Aliasing === A band limited signal may be periodically replicated in many ways. If the replication results in an overlap between replicated regions, the signal suffers from aliasing. Under such conditions, a continuous time signal cannot be perfectly recovered from its samples. Thus in order to ensure perfect recovery of the continuous signal, there must be zero overlap multidimensional sampling of the replicated regions in the transformed domain. As in the case of 1-dimensional signals, aliasing can be prevented if the continuous time signal is sampled at an adequate sufficiently high rate. === Sampling density === It is a measure of the number of samples per unit area. It is defined as: S . D = 1 | d e t ( V ) | = | d e t ( U ) | 4 π 2 {\displaystyle S.D={\frac {1}{|det(V)|}}={\frac {|det(U)|}{4\pi ^{2}}}} . The minimum number of samples per unit area required to completely recover the continuous time signal is termed as optimal sampling density. In applications where memory or processing time are limited, emphasis must be given to minimizing the number of samples required to represent the signal completely. == Existing approaches == For a bandlimited waveform, there are infinitely many ways the signal can be sampled without producing aliases in the Fourier domain. But only two strategies are commonly used: rectangular sampling and hexagonal sampling. === Rectangular and Hexagonal sampling === In rectangular sampling, a 2-dimensional signal, for example, is sampled according to the following V matrix: V r e c t = [ T 1 0 0 T 2 ] {\displaystyle V_{rect}={\begin{bmatrix}T1&0\\0&T2\end{bmatrix}}} where T1 and T2 are the sampling periods along the horizontal and vertical direction respectively. In hexagonal sampling, the V matrix assumes the following general form: V h e x = [ T 1 T 1 − T 2 T 2 ] {\displaystyle V_{hex}={\begin{bmatrix}T1&T1\\-T2&T2\end{bmatrix}}} The difference in the efficiency of the two schemes is highlighted using a bandlimited signal with a circular region of support of radius R. The circle can be inscribed in a square of length 2R or a regular hexagon of length 2 R 3 {\displaystyle {\frac {2R}{\sqrt {3}}}} . Consequently, the region of support is now transformed into a square and a hexagon respectively. If these regions are periodically replicated in the frequency domain such that there is zero overlap between any two regions, then by periodically replicating the square region of support, we effectively sample the continuous signal on a rectangular lattice. Similarly periodic replication of the hexagonal region of support maps to sampling the continuous signal on a hexagonal lattice. From U, the periodicity matrix, we can calculate the optimal sampling density for both the rectangular and hexagonal schemes. It is found that in order to completely recover the circularly band-limited signal, the hexagonal sampling scheme requires 13.4% fewer samples than the rectangular sampling scheme. The reduction may appear to be of little significance for a 2-dimensional signal. But as the dimensionality of the signal increases, the efficiency of the hexagonal sampling scheme will become far more evident. For instance, the reduction achieved for an 8-dimensional signal is 93.8%. To highlight the importance of the obtained result [2], try and visualize an image as a collection of infinite number of samples. The primary entity responsible for vision, i.e. the photoreceptors (rods and cones) are present on the retina of all mammals. These cells are not arranged in rows and columns. By adapting a hexagonal sampling scheme, our eyes are able to process images much more efficiently. The importance of hexagonal sampling lies in the fact that the photoreceptors of the human vision system lie on a hexagonal sampling lattice and, thus, perform hexagonal sampling.[3] In fact, it can be shown that the hexagonal sampling scheme is the optimal sampling scheme for a circularly band-limited signal. == Applications == === Aliasing effects minimized by the use of optimal sampling grids === Recent advances in the CCD technology has made hexagonal sampling feasible for real life applications. Historically, because of technology constraints, detector arrays were implemented only on 2-dimensional rectangular sampling lattices with rectangular shape detectors. But the super [CCD] detector introduced by Fuji has an octagonal shaped pixel in a hexagonal grid. Theoretically, the performance of the detector was greatly increased by introducing an octagonal pixel. The number of pixels required to represent the sample was reduced and there was significant improvement in the Signal-to-Noise Ratio (SNR) when compared with that of a rectangular pixel. But the drawback of using hexagonal pixels is that the associated fill factor will be less than 82%. An alternative method would be to interpolate hexagonal pixels in such a manner that we ultimately end up with a rectangular grid. The Spot 5 satellite incorporates a
Hierarchical navigable small world
Hierarchical navigable small world (HNSW) is an algorithm for approximate nearest neighbor search. It is used to find items that are similar to a query item in a large collection, without comparing the query with every item one by one. The algorithm is commonly used for searching vector data. In these systems, an item such as a document, image, song, or user profile is represented by a list of numbers called a vector. Items with similar vectors are treated as similar according to the model that produced the vectors. HNSW provides a way to search these vectors quickly, especially in large datasets. HNSW stores vectors in a graph. Each vector is a node, and links connect it to some nearby vectors. The graph has several layers: upper layers contain fewer nodes and act like a rough map, while the bottom layer contains all nodes and gives a more detailed view. A search starts in an upper layer, follows links toward nodes that are closer to the query, and then repeats the process in lower layers until it finds a set of likely nearest neighbors. == Background == The nearest neighbor search problem asks which items in a dataset are closest to a query item. A direct search can compare the query with every item in the dataset, but this becomes slow when the dataset is large. Exact search methods based on spatial trees, such as the k-d tree and R-tree, can also become less effective for high-dimensional data, a problem often associated with the curse of dimensionality. Approximate nearest neighbor methods trade some exactness for speed or lower resource use. Instead of always guaranteeing the exact closest item, they try to return close items quickly. Other approximate methods include locality-sensitive hashing and product quantization. HNSW builds on research into small-world networks and navigable graphs. In a small-world graph, most nodes can be reached from other nodes through a short chain of links. In a navigable graph, a search procedure can use local information to move toward a target. Jon Kleinberg's work on navigation in small-world networks is an important example of this research area. Later work studied ways to add links that make graphs easier to navigate greedily. The HNSW algorithm extends earlier navigable small world methods for similarity search by adding a hierarchy of graph layers. This hierarchy helps the algorithm find a good region of the graph before doing a more detailed search in the bottom layer. == Algorithm == HNSW is based on a proximity graph. In this graph, nearby vectors are connected by edges. The algorithm uses these edges to move through the dataset, rather than scanning every vector. The graph is hierarchical. Every vector appears in the bottom layer. Some vectors are also placed in higher layers, with fewer vectors appearing as the layers go upward. The upper layers allow long-range movement across the dataset, while the lower layers allow a more detailed search near promising candidates. A typical search proceeds as follows: The search begins from an entry point in the highest layer. At each step, the algorithm looks at neighboring nodes and moves to a neighbor that is closer to the query. When it cannot find a closer neighbor in that layer, it moves down to the next layer. In the bottom layer, it explores a wider set of candidate nodes and returns the nearest candidates found. This search strategy is often described as greedy navigation. The algorithm repeatedly chooses locally better nodes, using the graph structure to approach the query point. == Construction and parameters == The HNSW graph is built incrementally. When a new vector is inserted, the algorithm assigns it a maximum layer, searches for nearby existing nodes, and connects the new node to selected neighbors in each layer where it appears. Implementations usually expose parameters that control the trade-off between speed, accuracy, memory use, and construction time. A higher number of graph connections can improve recall but requires more memory. A larger search candidate list can improve accuracy but makes queries slower. A larger construction candidate list can improve the quality of the graph but makes index building slower. Because HNSW is approximate, its results are not always identical to a full exact search. Its practical performance depends on the dataset, distance measure, implementation, and parameter settings. Benchmarking studies have found HNSW-based libraries to be strong performers among approximate nearest neighbor methods, although worst-case performance can differ from performance on common benchmark datasets. == Use in vector search systems == HNSW is used as an index in systems that store and search high-dimensional vectors. These systems include vector databases, search engines, and database extensions. Typical uses include semantic search, recommender systems, image similarity search, and retrieval-augmented generation. Several software projects implement or support HNSW. Libraries include hnswlib, which is associated with the original HNSW authors, and FAISS. Database and search systems that document HNSW support include Apache Lucene, Chroma, ClickHouse, DuckDB, MariaDB, Milvus, pgvector, Qdrant, and Redis.