Bayesian Matrix Decomposition and Applications

• URL: http://arxiv.org/abs/2302.11337v3
• Date: Thu, 26 Sep 2024 03:51:19 GMT
• Title: Bayesian Matrix Decomposition and Applications
• Authors: Jun Lu,
• Abstract summary: The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.
• Score: 8.034728173797953
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.

Related papers

• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
  Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.
  arXiv  Detail & Related papers  (2024-07-15T07:11:44Z)
• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• Concentration properties of fractional posterior in 1-bit matrix completion [0.0]
  This work specifically addresses the scenario of binary observations, often termed as 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our results are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.
  arXiv  Detail & Related papers  (2024-04-13T11:22:53Z)
• Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson [0.0]
  We present four important matrix decompositions commonly used in quantum optics. The first two of these decompositions are specialized versions of the singular-value decomposition. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group.
  arXiv  Detail & Related papers  (2024-03-07T15:43:17Z)
• Synergistic eigenanalysis of covariance and Hessian matrices for enhanced binary classification [72.77513633290056]
  We present a novel approach that combines the eigenanalysis of a covariance matrix evaluated on a training set with a Hessian matrix evaluated on a deep learning model. Our method captures intricate patterns and relationships, enhancing classification performance.
  arXiv  Detail & Related papers  (2024-02-14T16:10:42Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Generalized Leverage Scores: Geometric Interpretation and Applications [15.86621510551207]
  We extend the definition of leverage scores to relate the columns of a matrix to arbitrary subsets of singular vectors. We employ this result to design approximation algorithms with provable guarantees for two well-known problems.
  arXiv  Detail & Related papers  (2022-06-16T10:14:08Z)
• Matrix Decomposition and Applications [8.034728173797953]
  In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition. matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network.
  arXiv  Detail & Related papers  (2022-01-01T08:13:48Z)
• Robust Low-rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method [47.80060761046752]
  Robust low-rank matrix completion (RMC) has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-
  arXiv  Detail & Related papers  (2020-08-18T04:46:22Z)
• Nonconvex Matrix Completion with Linearly Parameterized Factors [10.163102766021373]
  Parametric Factorization holds for important examples including subspace and completion simulations. The effectiveness of our unified nonconstrained matrix optimization method is also illustrated.
  arXiv  Detail & Related papers  (2020-03-29T22:40:47Z)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.