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Khatri–Rao Product
In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrix, partitioned matrices \mathbf and \mathbf is defined as : \mathbf \ast \mathbf = \left(\mathbf_ \otimes \mathbf_\right)_ in which the ''ij''-th block is the sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both Matrix (mathematics), matrices is equal. The size of the product is then . For example, if A and B both are partitioned matrices e.g.: : \mathbf = \left[ \begin \mathbf_ & \mathbf_ \\ \hline \mathbf_ & \mathbf_ \end \right] = \left[ \begin 1 & 2 & 3 \\ 4 & 5 & 6 \\ \hline 7 & 8 & 9 \end \right] ,\quad \mathbf = \left[ \begin \mathbf_ & \mathbf_ \\ \hline \mathbf_ & \mathbf_ \end \right] = \left[ \begin 1 & 4 & 7 \\ \hline 2 & 5 & 8 \\ 3 & 6 & 9 \end \right] , we obtain: : \mathbf \ast \mathbf = \left[ \begin \mathbf_ \otimes \mathbf_ & \mathbf_ \otimes \mathbf_ \\ \hline \mathbf_ \otimes \mathbf_ & \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Count Sketch
Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms.Woodruff, David P. "Sketching as a Tool for Numerical Linear Algebra." Theoretical Computer S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Functions of a continuous variable Consider two functions u(x) and v(x) with Fourier transforms U and V: :\begin U(f) &\triangleq \mathcal\(f) = \int_^u(x) e^ \, dx, \quad f \in \mathbb\\ V(f) &\triangleq \mathcal\(f) = \int_^v(x) e^ \, dx, \quad f \in \mathbb \end where \mathcal denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically 2\pi or \sqrt) will appear in the convolution theorem below. The convolution of u and v is defined by: :r(x) = \(x) \triangleq \int_^ u(\tau) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Generalized Linear Array Model
In statistics, the generalized linear array model (GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product. Overview The generalized linear array model or GLAM was introduced in 2006. Such models provide a structure and a computational procedure for fitting generalized linear models or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm. Suppose that the data \mathbf Y is arranged in a d-dimensional array with size n_1\times n_2\times\dots\times n_d; thus, the corresponding data vector \mathbf y = \operatorname(\mathbf Y) has size n_1n_2n_3\cdots n_d. Suppose also that the design matrix In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Natural Language Processing
Natural language processing (NLP) is a subfield of computer science and especially artificial intelligence. It is primarily concerned with providing computers with the ability to process data encoded in natural language and is thus closely related to information retrieval, knowledge representation and computational linguistics, a subfield of linguistics. Major tasks in natural language processing are speech recognition, text classification, natural-language understanding, natural language understanding, and natural language generation. History Natural language processing has its roots in the 1950s. Already in 1950, Alan Turing published an article titled "Computing Machinery and Intelligence" which proposed what is now called the Turing test as a criterion of intelligence, though at the time that was not articulated as a problem separate from artificial intelligence. The proposed test includes a task that involves the automated interpretation and generation of natural language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tensor Sketch
In statistics, machine learning and algorithms, a tensor sketch is a type of dimensionality reduction that is particularly efficient when applied to Vector (mathematics and physics), vectors that have tensor structure. Such a sketch can be used to speed up explicit kernel methods, bilinear Pool (computer science), pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms.Woodruff, David P.Sketching as a Tool for Numerical Linear Algebra." Theoretical Computer Science 10.1-2 (2014): 1–157. Mathematical definition Mathematically, a dimensionality reduction or sketching matrix is a matrix M\in\mathbb R^, where k |
Machine Learning
Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task (computing), tasks without explicit Machine code, instructions. Within a subdiscipline in machine learning, advances in the field of deep learning have allowed Neural network (machine learning), neural networks, a class of statistical algorithms, to surpass many previous machine learning approaches in performance. ML finds application in many fields, including natural language processing, computer vision, speech recognition, email filtering, agriculture, and medicine. The application of ML to business problems is known as predictive analytics. Statistics and mathematical optimisation (mathematical programming) methods comprise the foundations of machine learning. Data mining is a related field of study, focusing on exploratory data analysi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Artificial Intelligence
Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of research in computer science that develops and studies methods and software that enable machines to machine perception, perceive their environment and use machine learning, learning and intelligence to take actions that maximize their chances of achieving defined goals. High-profile applications of AI include advanced web search engines (e.g., Google Search); recommendation systems (used by YouTube, Amazon (company), Amazon, and Netflix); virtual assistants (e.g., Google Assistant, Siri, and Amazon Alexa, Alexa); autonomous vehicles (e.g., Waymo); Generative artificial intelligence, generative and Computational creativity, creative tools (e.g., ChatGPT and AI art); and Superintelligence, superhuman play and analysis in strategy games (e.g., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Block Matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block. : \left \begin a_ & a_ & a_ & b_ \\ a_ & a_ & a_ & b_ \\ \hline c_ & c_ & c_ & d \end \right Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transposed Block Face-Splitting Product
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: # Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left mathbf^\operatorname\right = \left mathbf\right. If is an matrix, then is an matrix. In the case of square matrices, may also denote the th power of the matrix . For avoiding a possible confusion, many authors use left upperscripts, that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Johnson–Lindenstrauss Lemma
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. In the classical proof of the lemma, the embedding is a random orthogonal projection. The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, graph embedding, and natural language processing. Much of the data stored and manipulated on computers, including text and images, can be represented as points in a high-dimensional space (see vector space model for the case of text). However, the essential algorithms for working with such data tend to become bogged down very quickly as dimension increases. It is therefore desirable to reduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
