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Bernhard Schölkopf
Bernhard Schölkopf is a German computer scientist (born 20 February 1968) known for his work in machine learning, especially on kernel methods and causality. He is a director at the Max Planck Institute for Intelligent Systems in Tübingen, Germany, where he heads the Department of Empirical Inference. He is also an affiliated professor at ETH Zürich, honorary professor at the University of Tübingen and the Technical University Berlin, and chairman of the European Laboratory for Learning and Intelligent Systems (ELLIS). Research Kernel methods Schölkopf developed SVM methods achieving world record performance on the MNIST pattern recognition benchmark at the time. With the introduction of kernel PCA, Schölkopf and coauthors argued that SVMs are a special case of a much larger class of methods, and all algorithms that can be expressed in terms of dot products can be generalized to a nonlinear setting by means of what is known as reproducing kernels. Another significant obser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of London
The University of London (UoL; abbreviated as Lond or more rarely Londin in post-nominals) is a federal public research university located in London, England, United Kingdom. The university was established by royal charter in 1836 as a degree-awarding examination board for students holding certificates from University College London and King's College London and "other such other Institutions, corporate or unincorporated, as shall be established for the purpose of Education, whether within the Metropolis or elsewhere within our United Kingdom". This fact allows it to be one of three institutions to claim the title of the third-oldest university in England, and moved to a federal structure in 1900. It is now incorporated by its fourth (1863) royal charter and governed by the University of London Act 2018. It was the first university in the United Kingdom to introduce examinations for women in 1869 and, a decade later, the first to admit women to degrees. In 1913, it appointe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Technical University Berlin
The Technical University of Berlin (official name both in English and german: link=no, Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public research university located in Berlin, Germany. It was the first German university to adopt the name "Technische Universität" (Technical University). The university alumni and professor list includes several US National Academies members, two National Medal of Science laureates and ten Nobel Prize laureates. TU Berlin is a member of TU9, an incorporated society of the largest and most notable German institutes of technology and of the Top International Managers in Engineering network, which allows for student exchanges between leading engineering schools. It belongs to the Conference of European Schools for Advanced Engineering Education and Research. The TU Berlin is home of two innovation centers designated by the European Institute of Innovation and Technology. The university is label ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program ''P'' computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than ''P'''s own leng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Inference
Causal inference is the process of determining the independent, actual effect of a particular phenomenon that is a component of a larger system. The main difference between causal inference and inference of association is that causal inference analyzes the response of an effect variable when a cause of the effect variable is changed. The science of why things occur is called etiology. Causal inference is said to provide the evidence of causality theorized by causal reasoning. Causal inference is widely studied across all sciences. Several innovations in the development and implementation of methodology designed to determine causality have proliferated in recent decades. Causal inference remains especially difficult where experimentation is difficult or impossible, which is common throughout most sciences. The approaches to causal inference are broadly applicable across all types of scientific disciplines, and many methods of causal inference that were designed for certain discipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraunhofer Diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer condition) from the object (in the far-field region), and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the diffracting object (in the near field region) is given by the Fresnel diffraction equation. The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory. This article explains where the Fraunhofer equation can be applied, and shows Fraunhofer diffraction patterns for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation. Equation When a beam of light is partly blocked by an obstacle, some of the light is scattered around the o ... [...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] |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representer Theorem
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f^ of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. Formal statement The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : \mathcal \times \mathcal \to \R on a non-empty set \mathcal with a corresponding reproducing kernel Hilbert space H_k. Let there be given * a training sample (x_1, y_1), \dotsc, (x_n, y_n) \in \mathcal \times \R, * a strictly increasing real-valued function g \colon _0.__Schölkopf,_Herbrich,_and_Smola_generalized_this_result_by_relaxing_the_assumption_of_the_squared-loss_cost_and_allowing_the_regularizer_to_be_any_strictly_monotonically_increasing_function_g(\c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Dimensionality Reduction
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. Applications of NLDR Consider a dataset represented as a matrix (or a database table), such that each row represents a set of attributes (or features or dimensions) that describe a particular instance of something. If the number of attributes is large, then the space of unique possible rows is exponentially large. Thus, the larger the dimensionality, the more difficult it becomes to sample the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sara Solla
Sara A. Solla is an Argentine-American physicist and neuroscientist whose research applies ideas from statistical mechanics to problems involving neural networks, machine learning, and neuroscience. She is a professor of physics and of physiology at Northwestern University. Education and career Solla is originally from Buenos Aires, and earned a licenciatura in physics in 1974 from the University of Buenos Aires. She completed a Ph.D. in physics in 1982 at the University of Washington. She became a postdoctoral researcher at Cornell University and at the Thomas J. Watson Research Center of IBM Research. Influenced to work in neural networks by a talk from John Hopfield at Cornell, she became a researcher in the neural networks group at Bell Labs. She took her present position at Northwestern University in 1997. Recognition Solla is a Member of the American Academy of Arts and Sciences (AAAS), and a Fellow of the American Physical Society The American Physical Society honors memb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klaus-Robert Müller
Klaus-Robert Müller (born 1964 in Karlsruhe, Germany) is a German computer scientist and physicist, most noted for his work in machine learning and brain–computer interfaces. Career Klaus-Robert Müller received his Diplom in mathematical physics and PhD in theoretical computer science from the University of Karlsruhe. Following his Ph.D. he went to Berlin as a postdoctoral fellow at GMD (German National Research Center for Computer Science) Berlin (now part of Fraunhofer Institute for Open Communication Systems), where he started building up the Intelligent Data Analysis (IDA) group. From 1994 to 1995 he was a research fellow at Shun'ichi Amari's lab at the University of Tokyo. 1999 Müller became an associate professor for neuroinformatics at the University of Potsdam, transitioning to the full professorship for Neural Networks and Time Series Analysis in 2003. Since 2006 he holds the chair for Machine Learning at the Technical University Berlin. Since 2012 he holds a di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram Matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^* X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose rows are the vectors v_k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
