Matrix decomposition

Summary

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A \mathbf{x} = \mathbf{b}, the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L(U \mathbf{x}) = \mathbf{b} and U \mathbf{x} = L^{-1} \mathbf{b} require fewer additions and multiplications to solve, compared with the original system A \mathbf{x} = \mathbf{b}, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition

Official source

https://en.wikipedia.org/wiki/Matrix_decomposition

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (23)

Related publications

Related people (4)

Related people

Related units (2)

Related units

Related concepts (34)

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbas

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm{T}

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin{cases} 3x+2y-z=1\ 2x-2y+4z=

Related concepts

Related courses (45)

This course teaches the students practical skills needed for solving modern physics problems by means of computation. A number of examples illustrate the utility of numerical computations in various domains of physics.

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

Students extend their knowledge on wireless communication systems to spread-spectrum communication and to multi-antenna systems. They also learn about the basic information theoretic concepts, about channel coding, and bit-interleaved coded modulation.

Related courses

Related lectures (90)

Related lectures

Unique decomposition of homogeneous languages and application to isothetic regions

Nicolas René Jean Ninin

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, where it has been proven that no barcode-like decomposition exists. To tackle this problem, algebraic invariants have been proposed to summarize multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of their stability has raised little attention so far, and many of the proposed invariants do not satisfy a naive form of stability. In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance. As an application, this explains why the graded-Betti numbers of a persistence module do not satisfy a naive form of stability. This opens the door to performing homological algebra operations while keeping track of stability. We believe this approach can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in [2]).

SPRINGER INTERNATIONAL PUBLISHING AG2021

Variational Gaussian Inference for Bilinear Models of Count Data

Mohammad Emtiyaz Khan, Young Jun Ko

Bilinear models of count data with Poisson distribution are popular in applications such as matrix factorization for recommendation systems, modeling of receptive fields of sensory neurons, and modeling of neural-spike trains. Bayesian inference in such models remains challenging due to the product term of two Gaussian random vectors. In this paper, we propose new algorithms for such models based on variational Gaussian (VG) inference. We make two contributions. First, we show that the VG lower bound for these models, previously known to be intractable, is available in closed form under certain non-trivial constraints on the form of the posterior. Second, we show that the lower bound is biconcave and can be efficiently optimized for mean-field approximations. We also show that bi-concavity generalizes to the larger family of log-concave likelihoods, that subsume the Poisson distribution. We present new inference algorithms based on these results and demonstrate better performance on real-world problems at the cost of a modest increase in computation. Our contributions in this paper, therefore, provide more choices for Bayesian inference in terms of a speed-vs-accuracy tradeoff.

2014