Alex James is an Indian scientist who is a professor of AI hardware at School of Electronic Systems and Automation, and Dean at Digital University Kerala (IIITM-K). He is the professor in charge of Maker Village, Kochi, Chief Investigator of the centre for Intelligent IoT Sensors, and India Innovation Centre for Graphene. James features in top 1% scientists list published by Elsevier BV in the world in the field of Electrical and Electronics Engineering. He appeared in the list for the third consecutive time. He specializes in the scientific field of Memristive Systems, AI hardware, Neuromorphic VLSI (very-large-scale integration) system, Intelligent Imaging and Machine learning, and Analogue electronics. == Education and career == James earned his Ph.D. degree from the Queensland Micro and Nanotechnology Centre, Griffith University, Brisbane, Australia. Since 2009, he has been working as a faculty member at different universities in Australia and India. He was a Member of IET Vision and Imaging Network, and is a Member of BCS’ Fellows Technical Advisory Group (F-TAG). He is the founding chair for IEEE Kerala Section Circuits and Systems Society, and is a fellow of British Computer Society (FBCS), and Institution of Engineering and Technology. He was an Editorial Board Member of Information Fusion (2010–2014), Elsevier, and associate editor for HCIS (2015–2020), Springer; and Guest Associate Editor for IEEE Transactions on Emerging Topics in Computational Intelligence (2017). Currently he is serving as an Associate Editor of IEEE Access, Frontiers in Neuroscience, and IEEE Transactions on Circuits and Systems I: Regular Papers journal. == Scientific research == IIITM-K has achieved a breakthrough in developing Analogue Integrated circuit for implementing Generative Adversarial Networks (GAN) in a joint research project with Analogue Circuits and Image Sensors Lab, Siegen university and Fraunhofer, Germany, and Centre for Excellence in Artificial general intelligence and Neuromorphic Systems (neuroAGI). According to A. P. James, professor at the School of Electronics at IIITM-K, this complicated and meticulous AI circuits research can accelerate and operate GAN applications in low power devices. It also can be used to analyze and interpret 2019-nCoV data for a possible solution to the pandemic. An AI Semantic search engine has been created by a research team led by A.P. James to help researchers gain deeper insights into Scientific Investigation, particularly since the COVID-19 issue has necessitated the collection of a significant amount of complex scientific data. The search engine is called "www.vilokana.in, which is Sanskrit for "finding out. == Awards and honors == James is a member of IEEE CASS Technical committee on Nonlinear Circuits and Systems, IEEE CASS Technical committee on Cellular Nanoscale networks and Memristor Array Computing, IEEE Consumer Technology Society Technical Committee on Quantum in Consumer Technology (QCT), Technical Committee on Machine learning, Deep learning and AI in CE (MDA) and Member of BCS’ Fellows Technical Advisory Group (F-TAG). James was awarded best associate editor of IEEE Transactions on Circuits and Systems I: Regular Papers TCAS-I, by the IEEE Circuits and Systems Society (IEEE CASS) for the year 2020–21. He has been an associate editor for the journal since 2017. He is also an editorial board member of PeerJ CS and a Senior Member of IEEE, Life Member of ACM, Senior Fellow of HEA.
Granular computing
Granular computing is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like. At present, granular computing is more a theoretical perspective than a coherent set of methods or principles. As a theoretical perspective, it encourages an approach to data that recognizes and exploits the knowledge present in data at various levels of resolution or scales. In this sense, it encompasses all methods which provide flexibility and adaptability in the resolution at which knowledge or information is extracted and represented. == Types of granulation == As mentioned above, granular computing is not an algorithm or process; there is no particular method that is called "granular computing". It is rather an approach to looking at data that recognizes how different and interesting regularities in the data can appear at different levels of granularity, much as different features become salient in satellite images of greater or lesser resolution. On a low-resolution satellite image, for example, one might notice interesting cloud patterns representing cyclones or other large-scale weather phenomena, while in a higher-resolution image, one misses these large-scale atmospheric phenomena but instead notices smaller-scale phenomena, such as the interesting pattern that is the streets of Manhattan. The same is generally true of all data: At different resolutions or granularities, different features and relationships emerge. The aim of granular computing is to try to take advantage of this fact in designing more effective machine-learning and reasoning systems. There are several types of granularity that are often encountered in data mining and machine learning, and we review them below: === Value granulation (discretization/quantization) === One type of granulation is the quantization of variables. It is very common that in data mining or machine-learning applications the resolution of variables needs to be decreased in order to extract meaningful regularities. An example of this would be a variable such as "outside temperature" (temp), which in a given application might be recorded to several decimal places of precision (depending on the sensing apparatus). However, for purposes of extracting relationships between "outside temperature" and, say, "number of health-club applications" (club), it will generally be advantageous to quantize "outside temperature" into a smaller number of intervals. ==== Motivations ==== There are several interrelated reasons for granulating variables in this fashion: Based on prior domain knowledge, there is no expectation that minute variations in temperature (e.g., the difference between 80–80.7 °F (26.7–27.1 °C)) could have an influence on behaviors driving the number of health-club applications. For this reason, any "regularity" which our learning algorithms might detect at this level of resolution would have to be spurious, as an artifact of overfitting. By coarsening the temperature variable into intervals the difference between which we do anticipate (based on prior domain knowledge) might influence number of health-club applications, we eliminate the possibility of detecting these spurious patterns. Thus, in this case, reducing resolution is a method of controlling overfitting. By reducing the number of intervals in the temperature variable (i.e., increasing its grain size), we increase the amount of sample data indexed by each interval designation. Thus, by coarsening the variable, we increase sample sizes and achieve better statistical estimation. In this sense, increasing granularity provides an antidote to the so-called curse of dimensionality, which relates to the exponential decrease in statistical power with increase in number of dimensions or variable cardinality. Independent of prior domain knowledge, it is often the case that meaningful regularities (i.e., which can be detected by a given learning methodology, representational language, etc.) may exist at one level of resolution and not at another. For example, a simple learner or pattern recognition system may seek to extract regularities satisfying a conditional probability threshold such as p ( Y = y j | X = x i ) ≥ α . {\displaystyle p(Y=y_{j}|X=x_{i})\geq \alpha .} In the special case where α = 1 , {\displaystyle \alpha =1,} this recognition system is essentially detecting logical implication of the form X = x i → Y = y j {\displaystyle X=x_{i}\rightarrow Y=y_{j}} or, in words, "if X = x i , {\displaystyle X=x_{i},} then Y = y j {\displaystyle Y=y_{j}} ". The system's ability to recognize such implications (or, in general, conditional probabilities exceeding threshold) is partially contingent on the resolution with which the system analyzes the variables. As an example of this last point, consider the feature space shown to the right. The variables may each be regarded at two different resolutions. Variable X {\displaystyle X} may be regarded at a high (quaternary) resolution wherein it takes on the four values { x 1 , x 2 , x 3 , x 4 } {\displaystyle \{x_{1},x_{2},x_{3},x_{4}\}} or at a lower (binary) resolution wherein it takes on the two values { X 1 , X 2 } . {\displaystyle \{X_{1},X_{2}\}.} Similarly, variable Y {\displaystyle Y} may be regarded at a high (quaternary) resolution or at a lower (binary) resolution, where it takes on the values { y 1 , y 2 , y 3 , y 4 } {\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}} or { Y 1 , Y 2 } , {\displaystyle \{Y_{1},Y_{2}\},} respectively. At the high resolution, there are no detectable implications of the form X = x i → Y = y j , {\displaystyle X=x_{i}\rightarrow Y=y_{j},} since every x i {\displaystyle x_{i}} is associated with more than one y j , {\displaystyle y_{j},} and thus, for all x i , {\displaystyle x_{i},} p ( Y = y j | X = x i ) < 1. {\displaystyle p(Y=y_{j}|X=x_{i})<1.} However, at the low (binary) variable resolution, two bilateral implications become detectable: X = X 1 ↔ Y = Y 1 {\displaystyle X=X_{1}\leftrightarrow Y=Y_{1}} and X = X 2 ↔ Y = Y 2 {\displaystyle X=X_{2}\leftrightarrow Y=Y_{2}} , since every X 1 {\displaystyle X_{1}} occurs iff Y 1 {\displaystyle Y_{1}} and X 2 {\displaystyle X_{2}} occurs iff Y 2 . {\displaystyle Y_{2}.} Thus, a pattern recognition system scanning for implications of this kind would find them at the binary variable resolution, but would fail to find them at the higher quaternary variable resolution. ==== Issues and methods ==== It is not feasible to exhaustively test all possible discretization resolutions on all variables in order to see which combination of resolutions yields interesting or significant results. Instead, the feature space must be preprocessed (often by an entropy analysis of some kind) so that some guidance can be given as to how the discretization process should proceed. Moreover, one cannot generally achieve good results by naively analyzing and discretizing each variable independently, since this may obliterate the very interactions that we had hoped to discover. A sample of papers that address the problem of variable discretization in general, and multiple-variable discretization in particular, is as follows: Chiu, Wong & Cheung (1991), Bay (2001), Liu et al. (2002), Wang & Liu (1998), Zighed, Rabaséda & Rakotomalala (1998), Catlett (1991), Dougherty, Kohavi & Sahami (1995), Monti & Cooper (1999), Fayyad & Irani (1993), Chiu, Cheung & Wong (1990), Nguyen & Nguyen (1998), Grzymala-Busse & Stefanowski (2001), Ting (1994), Ludl & Widmer (2000), Pfahringer (1995), An & Cercone (1999), Chiu & Cheung (1989), Chmielewski & Grzymala-Busse (1996), Lee & Shin (1994), Liu & Wellman (2002), Liu & Wellman (2004). === Variable granulation (clustering/aggregation/transformation) === Variable granulation is a term that could describe a variety of techniques, most of which are aimed at reducing dimensionality, redundancy, and storage requirements. We briefly describe some of the ideas here, and present pointers to the literature. ==== Variable transformation ==== A number of classical methods, such as principal component analysis, multidimensional scaling, factor analysis, and structural equation modeling, and their relatives, fall under the genus of "variable transformation." Also in this category are more modern areas of study such as dimensionality reduction, projection pursuit, and independent component analysis. The common goal of these methods in general is to find a representation of the data in terms of new variables, which are a linear or nonlinear transformation of the original variables, and in which important stati
Collision problem
The r-to-1 collision problem is an important theoretical problem in complexity theory, quantum computing, and computational mathematics. The collision problem most often refers to the 2-to-1 version: given n {\displaystyle n} even and a function f : { 1 , … , n } → { 1 , … , n } {\displaystyle f:\,\{1,\ldots ,n\}\rightarrow \{1,\ldots ,n\}} , we are promised that f is either 1-to-1 or 2-to-1. We are only allowed to make queries about the value of f ( i ) {\displaystyle f(i)} for any i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} . The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1. == Classical solutions == === Deterministic === Solving the 2-to-1 version deterministically requires n 2 + 1 {\textstyle {\frac {n}{2}}+1} queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires n r + 1 {\textstyle {\frac {n}{r}}+1} queries. This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after n r + 1 {\textstyle {\frac {n}{r}}+1} queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, n r + 1 {\textstyle {\frac {n}{r}}+1} queries suffice. If we are unlucky, then the first n / r {\displaystyle n/r} queries could return distinct answers, so n r + 1 {\textstyle {\frac {n}{r}}+1} queries is also necessary. === Randomized === If we allow randomness, the problem is easier. By the birthday paradox, if we choose (distinct) queries at random, then with high probability we find a collision in any fixed 2-to-1 function after Θ ( n ) {\displaystyle \Theta ({\sqrt {n}})} queries. == Quantum solution == The BHT algorithm, which uses Grover's algorithm, solves this problem optimally by only making O ( n 1 / 3 ) {\displaystyle O(n^{1/3})} queries to f. The matching lower bound of Ω ( n 1 / 3 ) {\displaystyle \Omega (n^{1/3})} was proved by Aaronson and Shi using the polynomial method.
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
Linguistic categories
Linguistic categories include Lexical category, a part of speech such as noun, preposition, etc. Syntactic category, a similar concept which can also include phrasal categories Grammatical category, a grammatical feature such as tense, gender, etc. The definition of linguistic categories is a major concern of linguistic theory, and thus, the definition and naming of categories varies across different theoretical frameworks and grammatical traditions for different languages. The operationalization of linguistic categories in lexicography, computational linguistics, natural language processing, corpus linguistics, and terminology management typically requires resource-, problem- or application-specific definitions of linguistic categories. In Cognitive linguistics it has been argued that linguistic categories have a prototype structure like that of the categories of common words in a language. == Linguistic category inventories == To facilitate the interoperability between lexical resources, linguistic annotations and annotation tools and for the systematic handling of linguistic categories across different theoretical frameworks, a number of inventories of linguistic categories have been developed and are being used, with examples as given below. The practical objective of such inventories is to perform quantitative evaluation (for language-specific inventories), to train NLP tools, or to facilitate cross-linguistic evaluation, querying or annotation of language data. At a theoretical level, the existence of universal categories in human language has been postulated, e.g., in Universal grammar, but also heavily criticized. === Part-of-Speech tagsets === Schools commonly teach that there are 9 parts of speech in English: noun, verb, article, adjective, preposition, pronoun, adverb, conjunction, and interjection. However, there are clearly many more categories and sub-categories. For nouns, the plural, possessive, and singular forms can be distinguished. In many languages words are also marked for their case (role as subject, object, etc.), grammatical gender, and so on; while verbs are marked for tense, aspect, and other things. In some tagging systems, different inflections of the same root word will get different parts of speech, resulting in a large number of tags. For example, NN for singular common nouns, NNS for plural common nouns, NP for singular proper nouns (see the POS tags used in the Brown Corpus). Other tagging systems use a smaller number of tags and ignore fine differences or model them as features somewhat independent from part-of-speech. In part-of-speech tagging by computer, it is typical to distinguish from 50 to 150 separate parts of speech for English. POS tagging work has been done in a variety of languages, and the set of POS tags used varies greatly with language. Tags usually are designed to include overt morphological distinctions, although this leads to inconsistencies such as case-marking for pronouns but not nouns in English, and much larger cross-language differences. The tag sets for heavily inflected languages such as Greek and Latin can be very large; tagging words in agglutinative languages such as Inuit languages may be virtually impossible. Work on stochastic methods for tagging Koine Greek (DeRose 1990) has used over 1,000 parts of speech and found that about as many words were ambiguous in that language as in English. A morphosyntactic descriptor in the case of morphologically rich languages is commonly expressed using very short mnemonics, such as ncmsan for category = noun, type = common, gender = masculine, number = singular, case = accusative, animate = no. The most popular tag set for POS tagging for American English is probably the Penn tag set, developed in the Penn Treebank project. === Multilingual annotation schemes === For Western European languages, cross-linguistically applicable annotation schemes for parts-of-speech, morphosyntax and syntax have been developed with the EAGLES Guidelines. The "Expert Advisory Group on Language Engineering Standards" (EAGLES) was an initiative of the European Commission that ran within the DG XIII Linguistic Research and Engineering programme from 1994 to 1998, coordinated by Consorzio Pisa Ricerche, Pisa, Italy. The EAGLES guidelines provide guidance for markup to be used with text corpora, particularly for identifying features relevant in computational linguistics and lexicography. Numerous companies, research centres, universities and professional bodies across the European Union collaborated to produce the EAGLES Guidelines, which set out recommendations for de facto standards and rules of best practice for: Large-scale language resources (such as text corpora, computational lexicons and speech corpora); Means of manipulating such knowledge, via computational linguistic formalisms, mark up languages and various software tools; Means of assessing and evaluating resources, tools and products. The Eagles guidelines have inspired subsequent work on other regions, as well, e.g., Eastern Europe. A generation later, a similar effort was initiated by the research community under the umbrella of Universal Dependencies. Petrov et al. have proposed a "universal", but highly reductionist, tag set, with 12 categories (for example, no subtypes of nouns, verbs, punctuation, etc.; no distinction of "to" as an infinitive marker vs. preposition (hardly a "universal" coincidence), etc.). Subsequently, this was complemented with cross-lingual specifications for dependency syntax (Stanford Dependencies), and morphosyntax (Interset interlingua, partially building on the Multext-East/Eagles tradition) in the context of the Universal Dependencies (UD), an international cooperative project to create treebanks of the world's languages with cross-linguistically applicable ("universal") annotations for parts of speech, dependency syntax, and (optionally) morphosyntactic (morphological) features. Core applications are automated text processing in the field of natural language processing (NLP) and research into natural language syntax and grammar, especially within linguistic typology. The annotation scheme has it roots in three related projects: The UD annotation scheme uses a representation in the form of dependency trees as opposed to a phrase structure trees. At as of February 2019, there are just over 100 treebanks of more than 70 languages available in the UD inventory. The project's primary aim is to achieve cross-linguistic consistency of annotation. However, language-specific extensions are permitted for morphological features (individual languages or resources can introduce additional features). In a more restricted form, dependency relations can be extended with a secondary label that accompanies the UD label, e.g., aux:pass for an auxiliary (UD aux) used to mark passive voice. The Universal Dependencies have inspired similar efforts for the areas of inflectional morphology, frame semantics and coreference. For phrase structure syntax, a comparable effort does not seem to exist, but the specifications of the Penn Treebank have been applied to (and extended for) a broad range of languages, e.g., Icelandic, Old English, Middle English, Middle Low German, Early Modern High German, Yiddish, Portuguese, Japanese, Arabic and Chinese. === Conventions for interlinear glosses === In linguistics, an interlinear gloss is a gloss (series of brief explanations, such as definitions or pronunciations) placed between lines (inter- + linear), such as between a line of original text and its translation into another language. When glossed, each line of the original text acquires one or more lines of transcription known as an interlinear text or interlinear glossed text (IGT)—interlinear for short. Such glosses help the reader follow the relationship between the source text and its translation, and the structure of the original language. There is no standard inventory for glosses, but common labels are collected in the Leipzig Glossing Rules. Wikipedia also provides a List of glossing abbreviations that draws on this and other sources. === General Ontology for Linguistic Description (GOLD) === GOLD ("General Ontology for Linguistic Description") is an ontology for descriptive linguistics. It gives a formalized account of the most basic categories and relations used in the scientific description of human language, e.g., as a formalization of interlinear glosses. GOLD was first introduced by Farrar and Langendoen (2003). Originally, it was envisioned as a solution to the problem of resolving disparate markup schemes for linguistic data, in particular data from endangered languages. However, GOLD is much more general and can be applied to all languages. In this function, GOLD overlaps with the ISO 12620 Data Category Registry (ISOcat); it is, however, more stringently structured. GOLD was maintained by the LINGUIST List and others from 2007 to 2010. The RELISH project created a mirro
Topological deep learning
Topological deep learning (TDL) is a research field that extends deep learning to handle complex, non-Euclidean data structures. Traditional deep learning models, such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs), excel in processing data on regular grids and sequences. However, scientific and real-world data often exhibit more intricate data domains encountered in scientific computations, including point clouds, meshes, time series, scalar fields graphs, or general topological spaces like simplicial complexes and CW complexes. TDL addresses this by incorporating topological concepts to process data with higher-order relationships, such as interactions among multiple entities and complex hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global dependencies and qualitative spatial properties, offering a more nuanced representation of data. TDL also encompasses methods from computational and algebraic topology that permit studying properties of neural networks and their training process, such as their predictive performance or generalization properties. The mathematical foundations of TDL are algebraic topology, differential topology, and geometric topology. Therefore, TDL can be generalized for data on differentiable manifolds, knots, links, tangles, curves, etc. == History and motivation == Traditional techniques from deep learning often operate under the assumption that a dataset is residing in a highly-structured space (like images, where convolutional neural networks exhibit outstanding performance over alternative methods) or a Euclidean space. The prevalence of new types of data, in particular graphs, meshes, and molecules, resulted in the development of new techniques, culminating in the field of geometric deep learning, which originally proposed a signal-processing perspective for treating such data types. While originally confined to graphs, where connectivity is defined based on nodes and edges, follow-up work extended concepts to a larger variety of data types, including simplicial complexes and CW complexes, with recent work proposing a unified perspective of message-passing on general combinatorial complexes. An independent perspective on different types of data originated from topological data analysis, which proposed a new framework for describing structural information of data, i.e., their "shape," that is inherently aware of multiple scales in data, ranging from local information to global information. While at first restricted to smaller datasets, subsequent work developed new descriptors that efficiently summarized topological information of datasets to make them available for traditional machine-learning techniques, such as support vector machines or random forests. Such descriptors ranged from new techniques for feature engineering over new ways of providing suitable coordinates for topological descriptors, or the creation of more efficient dissimilarity measures. Contemporary research in this field is largely concerned with either integrating information about the underlying data topology into existing deep-learning models or obtaining novel ways of training on topological domains. == Learning on topological spaces == One of the core concepts in topological deep learning is considering the domain upon which this data is defined and supported. In case of Euclidean data, such as images, this domain is a grid, upon which the pixel value of the image is supported. In a more general setting this domain might be a topological domain. Studying and developing deep learning models that are supported ln topological domains constitute the essence of topological deep learning. Next, we introduce the most common topological domains that are encountered in a deep learning setting. These domains include, but not limited to, graphs, simplicial complexes, cell complexes, combinatorial complexes and hypergraphs. Given a finite set S of abstract entities, a neighborhood function N {\displaystyle {\mathcal {N}}} on S is an assignment that attach to every point x {\displaystyle x} in S a subset of S or a relation. Such a function can be induced by equipping S with an auxiliary structure. Edges provide one way of defining relations among the entities of S. More specifically, edges in a graph allow one to define the notion of neighborhood using, for instance, the one hop neighborhood notion. Edges however, limited in their modeling capacity as they can only be used to model binary relations among entities of S since every edge is connected typically to two entities. In many applications, it is desirable to permit relations that incorporate more than two entities. The idea of using relations that involve more than two entities is central to topological domains. Such higher-order relations allow for a broader range of neighborhood functions to be defined on S to capture multi-way interactions among entities of S. Next we review the main properties, advantages, and disadvantages of some commonly studied topological domains in the context of deep learning, including (abstract) simplicial complexes, regular cell complexes, hypergraphs, and combinatorial complexes. ==== Comparisons among topological domains ==== Each of the enumerated topological domains has its own characteristics, advantages, and limitations: Simplicial complexes Simplest form of higher-order domains. Extensions of graph-based models. Admit hierarchical structures, making them suitable for various applications. Hodge theory can be naturally defined on simplicial complexes. Require relations to be subsets of larger relations, imposing constraints on the structure. Cell Complexes Generalize simplicial complexes. Provide more flexibility in defining higher-order relations. Each cell in a cell complex is homeomorphic to an open ball, attached together via attaching maps. Boundary cells of each cell in a cell complex are also cells in the complex. Represented combinatorially via incidence matrices. Hypergraphs Allow arbitrary set-type relations among entities. Relations are not imposed by other relations, providing more flexibility. Do not explicitly encode the dimension of cells or relations. Useful when relations in the data do not adhere to constraints imposed by other models like simplicial and cell complexes. Combinatorial Complexes : Generalize and bridge the gaps between simplicial complexes, cell complexes, and hypergraphs. Allow for hierarchical structures and set-type relations. Combine features of other complexes while providing more flexibility in modeling relations. Can be represented combinatorially, similar to cell complexes. ==== Hierarchical structure and set-type relations ==== The properties of simplicial complexes, cell complexes, and hypergraphs give rise to two main features of relations on higher-order domains, namely hierarchies of relations and set-type relations. ===== Rank function ===== A rank function on a higher-order domain X is an order-preserving function rk: X → Z, where rk(x) attaches a non-negative integer value to each relation x in X, preserving set inclusion in X. Cell and simplicial complexes are common examples of higher-order domains equipped with rank functions and therefore with hierarchies of relations. ===== Set-type relations ===== Relations in a higher-order domain are called set-type relations if the existence of a relation is not implied by another relation in the domain. Hypergraphs constitute examples of higher-order domains equipped with set-type relations. Given the modeling limitations of simplicial complexes, cell complexes, and hypergraphs, we develop the combinatorial complex, a higher-order domain that features both hierarchies of relations and set-type relations. The learning tasks in TDL can be broadly classified into three categories: Cell classification: Predict targets for each cell in a complex. Examples include triangular mesh segmentation, where the task is to predict the class of each face or edge in a given mesh. Complex classification: Predict targets for an entire complex. For example, predict the class of each input mesh. Cell prediction: Predict properties of cell-cell interactions in a complex, and in some cases, predict whether a cell exists in the complex. An example is the prediction of linkages among entities in hyperedges of a hypergraph. In practice, to perform the aforementioned tasks, deep learning models designed for specific topological spaces must be constructed and implemented. These models, known as topological neural networks, are tailored to operate effectively within these spaces. === Topological neural networks === Central to TDL are topological neural networks (TNNs), specialized architectures designed to operate on data structured in topological domains. Unlike traditional neural networks tailored for grid-like structures, TNNs are adept at handling more intricate data representations, such as graphs
Manhattan address algorithm
The Manhattan address algorithm is a series of formulas used to estimate the closest east–west cross street for building numbers on north–south avenues in the New York City borough of Manhattan. == Algorithm == To find the approximate number of the closest cross street, divide the building number by a divisor (generally 20) and add (or subtract) the "tricky number" from the table below: For the north–south avenues, there are typically 20 address numbers between consecutive east–west streets (10 on either side of the avenue). A standard land lot on each avenue was originally 20 feet (6.1 m) wide, and there is about 200 feet (61 m) between each pair of east–west streets, for ten land lots between each pair of streets. The exceptions are Riverside Drive, as well as Fifth Avenue and Central Park West between 59th and 110th streets, which use a divisor of 10. These avenues all have buildings only on one side of the street, with a park on the other side. The "tricky number" often corresponds to a street near the southern end of the avenue. There are some notable exceptions: York Avenue address numbers are continuations of Avenue A address numbers, since the avenue was originally called Avenue A. East End Avenue address numbers are continuations of Avenue B address numbers, since the avenue was originally called Avenue B. Sixth Avenue and Broadway start south of Houston Street, the southern boundary of the Manhattan street numbering system. Although Park Avenue's southern terminus is at 32nd Street, a homeowner at 34th Street wanted the address "1 Park Avenue" (this was later changed). === Examples === For example, if you are at 62 Avenue B, 62 ÷ 20 ≈ 3 {\displaystyle 62\div 20\approx 3} , then add the "tricky number" 3 {\displaystyle 3} to give 6 {\displaystyle 6} . The nearest cross street to 62 Avenue B is East 6th Street. If you are at 78 Riverside Drive, 78 ÷ 10 ≈ 8 {\displaystyle 78\div 10\approx 8} , then add the "tricky number" 72 {\displaystyle 72} to give 80 {\displaystyle 80} . The nearest cross street to 78 Riverside Drive is West 80th Street. If you are at 501 5th Avenue, 501 ÷ 20 ≈ 25 {\displaystyle 501\div 20\approx 25} , then add the "tricky number" 18 {\displaystyle 18} to give 43 {\displaystyle 43} . The nearest cross street to 501 5th Avenue is actually 42nd Street, not 43rd Street, as the Manhattan address algorithm only gives approximate answers.