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Integral Probability Metric
In probability theory, integral probability metrics are types of distance functions between probability distributions, defined by how well a class of functions can distinguish the two distributions. Many important statistical distances are integral probability metrics, including the Wasserstein-1 distance and the total variation distance. In addition to theoretical importance, integral probability metrics are widely used in areas of statistics and machine learning. The name "integral probability metric" was given by German statistician Alfred Müller; the distances had also previously been called "metrics with a -structure." Definition Integral probability metrics (IPMs) are distances on the space of distributions over a set \mathcal X, defined by a class \mathcal of real-valued functions on \mathcal as D_(P, Q) = \sup_ \big, \mathbb E_ f(X) - \mathbb E_ f(Y) \big, = \sup_ \big, P f - Q f \big, ; here the notation refers to the expectation of under the distribution . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Embedding Of Distributions
In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS).A. Smola, A. Gretton, L. Song, B. Schölkopf. (2007)A Hilbert Space Embedding for Distributions. ''Algorithmic Learning Theory: 18th International Conference''. Springer: 13–31. A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis.L. Song, K. Fukumizu, F. Dinuzzo, A. Gretton (2013)Kernel Embeddings of Conditional Distributions: A unified kernel framework for non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Neural Network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit a signal to other neurons. An artificial neuron receives signals then processes them and can signal neurons connected to it. The "signal" at a connection is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called ''edges''. Neurons and edges typically have a ''weight'' that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Conference On Machine Learning
The International Conference on Machine Learning (ICML) is the leading international academic conference in machine learning. Along with NeurIPS and ICLR, it is one of the three primary conferences of high impact in machine learning and artificial intelligence research. It is supported by the ( IMLS). Precise dates vary year to year, but paper submissions are generally due at the end of January, and the conference is generally held the following July. The first ICML was held 1980 in Pittsburgh. Locations * ICML 2026 Seoul, South Korea * ICML 2025 Vancouver, Canada * ICML 2024 Vienna, Austria * ICML 2023 Honolulu, Hawaii, United States * ICML 2022 Baltimore, Maryland, United States * ICML 2021 Vienna, Austria (virtual conference) * ICML 2020 Vienna, Austria (virtual conference) * ICML 2019 Los Angeles, United States * ICML 2018 Stockholm, Sweden * ICML 2017 Sydney, Australia * ICML 2016 New York City, United States * ICML 2015 Lille, France * ICML 2014 Beijing, Ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Annals Of Statistics
The ''Annals of Statistics'' is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the ''Annals of Mathematical Statistics (1930)'', which was split into the ''Annals of Statistics'' and the '' Annals of Probability''. The journal CiteScore is 5.8, and its SCImago Journal Rank is 5.877, both from 2020. Articles older than 3 years are available on JSTOR, and all articles since 2004 are freely available on the arXiv. Editorial board The following persons have been editors of the journal: * Ingram Olkin (1972–1973) * I. Richard Savage (1974–1976) * Rupert Miller (1977–1979) * David V. Hinkley (1980–1982) * Michael D. Perlman (1983–1985) * Willem van Zwet (1986–1988) * Arthur Cohen (1988–1991) * Michael Woodroofe (1992–1994) * Larry Brown and John Rice (1995–1997) * Hans-Rudolf Künsch and James O. Berger (1998–2000) * John Marden and Jon A. Wellner (2001–2003) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-sample Hypothesis Testing
In statistical hypothesis testing, a two-sample test is a test performed on the data of two random samples, each independently obtained from a different given population. The purpose of the test is to determine whether the difference between these two populations is statistically significant. There are a large number of statistical tests that can be used in a two-sample test. Which one(s) are appropriate depend on a variety of factors, such as: * Which assumptions (if any) may be made ''a priori'' about the distributions from which the data have been sampled? For example, in many situations it may be assumed that the underlying distributions are normal distributions. In other cases the data are categorical, coming from a discrete distribution over a nominal scale, such as which entry was selected from a menu. * Does the hypothesis being tested apply to the distributions as a whole, or just some population parameter, for example the mean or the variance? * Is the hypothesis being t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generative Adversarial Network
A generative adversarial network (GAN) is a class of machine learning frameworks designed by Ian Goodfellow and his colleagues in June 2014. Two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is another agent's loss. Given a training set, this technique learns to generate new data with the same statistics as the training set. For example, a GAN trained on photographs can generate new photographs that look at least superficially authentic to human observers, having many realistic characteristics. Though originally proposed as a form of generative model for unsupervised learning, GANs have also proved useful for semi-supervised learning, fully supervised learning, and reinforcement learning. The core idea of a GAN is based on the "indirect" training through the discriminator, another neural network that can tell how "realistic" the input seems, which itself is also being updated dynamically. This means that the generator is not tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Besov Space
In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B^s_(\mathbf) is a Complete metric space, complete quasinormed space which is a Banach space when . These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions. Definition Several equivalent definitions exist. One of them is given below. Let : \Delta_h f(x) = f(x-h) - f(x) and define the modulus of continuity by : \omega^2_p(f,t) = \sup_ \left \, \Delta^2_h f \right \, _p Let be a non-negative integer and define: with . The Besov space B^s_(\mathbf) contains all functions such that : f \in W^(\mathbf), \qquad \int_0^\infty \left, \frac \^q \frac < \infty. Norm The Besov space is equipped with the norm : |
International Conference On Learning Representations
The International Conference on Learning Representations (ICLR) is a machine learning conference typically held in late April or early May each year. The conference includes invited talks as well as oral and poster presentations of refereed papers. Since its inception in 2013, ICLR has employed an open peer review process to referee paper submissions (based on models proposed by Yann LeCun). In 2019, there were 1591 paper submissions, of which 500 accepted with poster presentations (31%) and 24 with oral presentations (1.5%).. In 2021, there were 2997 paper submissions, of which 860 were accepted (29%).. Along with ICML and NeurIPS, ICLR is one of the three major machine learning and artificial intelligence conferences, and has the highest impact of the three. Locations * ICLR 2023, Kigali * ICLR 2022 (virtual conference) * ICLR 2021, Vienna, Austria (virtual conference) * ICLR 2020, Addis Ababa, Ethiopia (virtual conference) * ICLR 2019, New Orleans, Louisiana, United S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generative Artificial Intelligence
Generative artificial intelligence (generative AI, GenAI, or GAI) is a subset of artificial intelligence that uses generative models to produce text, images, videos, or other forms of data. These models machine learning, learn the underlying patterns and structures of their training data set, training data and use them to produce new data based on the input, which often comes in the form of natural language Prompt (natural language), prompts. Improvements in transformer (machine learning model), transformer-based deep learning, deep neural networks, particularly large language model, large language models (LLMs), enabled an AI boom of generative AI systems in the early 2020s. These include chatbots such as ChatGPT, Microsoft Copilot, Copilot, Gemini (chatbot), Gemini, and LLaMA; text-to-image artificial intelligence art, artificial intelligence image generation systems such as Stable Diffusion, Midjourney, and DALL-E; and Text-to-video model, text-to-video AI generators such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy Distance
Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in ''R''d with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of : D^2(F, G) = 2\operatorname E\, X - Y\, - \operatorname E\, X - X'\, - \operatorname E\, Y - Y'\, \geq 0, where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, \operatorname E is the expected value, and , , . , , denotes the length of a vector. Energy distance satisfies all axioms of a metric thus energy distance characterizes the equality of distributions: D(F,G) = 0 if and only if F = G. Energy distance for statistical applications was introduced in 1985 by Gábor J. Székely, who proved that for real-valued random variables D^2(F, G) is exactly twice Harald Cramér's distance: : \int_^\infty (F(x) - G(x))^2 \, dx. For a simple proof of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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