138,463 research outputs found
An Introduction to Wishart Matrix Moments
These lecture notes provide a comprehensive, self-contained introduction to
the analysis of Wishart matrix moments. This study may act as an introduction
to some particular aspects of random matrix theory, or as a self-contained
exposition of Wishart matrix moments. Random matrix theory plays a central role
in statistical physics, computational mathematics and engineering sciences,
including data assimilation, signal processing, combinatorial optimization,
compressed sensing, econometrics and mathematical finance, among numerous
others. The mathematical foundations of the theory of random matrices lies at
the intersection of combinatorics, non-commutative algebra, geometry,
multivariate functional and spectral analysis, and of course statistics and
probability theory. As a result, most of the classical topics in random matrix
theory are technical, and mathematically difficult to penetrate for non-experts
and regular users and practitioners. The technical aim of these notes is to
review and extend some important results in random matrix theory in the
specific context of real random Wishart matrices. This special class of
Gaussian-type sample covariance matrix plays an important role in multivariate
analysis and in statistical theory. We derive non-asymptotic formulae for the
full matrix moments of real valued Wishart random matrices. As a corollary, we
derive and extend a number of spectral and trace-type results for the case of
non-isotropic Wishart random matrices. We also derive the full matrix moment
analogues of some classic spectral and trace-type moment results. For example,
we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and
full matrix cases. Laplace matrix transforms and matrix moment estimates are
also studied, along with new spectral and trace concentration-type
inequalities
Concise computer mathematics: tutorials on theory and problems
Adapted from a modular undergraduate course on computational mathematics, Concise Computer Mathematics delivers an easily accessible, self-contained introduction to the basic notions of mathematics necessary for a computer science degree. The text reflects the need to quickly introduce students from a variety of educational backgrounds to a number of essential mathematical concepts. The material is divided into four units: discrete mathematics (sets, relations, functions), logic (Boolean types, truth tables, proofs), linear algebra (vectors, matrices and graphics), and special topics (graph theory, number theory, basic elements of calculus). The chapters contain a brief theoretical presentation of the topic, followed by a selection of problems (which are direct applications of the theory) and additional supplementary problems (which may require a bit more work). Each chapter ends with answers or worked solutions for all of the problems
An introduction to Wishart matrix moments
© 2018 Now Publishers Inc. All rights reserved. These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory.We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
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