6 research outputs found
The inertia of the symmetric approximation for low-rank matrices
© 2017 Informa UK Limited, trading as Taylor & Francis Group In many areas of applied linear algebra, it is necessary to work with matrix approximations. A usual situation occurs when a matrix obtained from experimental or simulated data is needed to be approximated by a matrix that lies in a corresponding statistical model and satisfies some specific properties. In this short note, we focus on symmetric and positive-semidefinite approximations and we show that the positive and negative indices of inertia of the symmetric approximation and the rank of the positive-semidefinite approximation are always bounded from above by the rank of the original matrix.Peer ReviewedPostprint (published version
Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences
This paper introduces sparse coding and dictionary learning for Symmetric
Positive Definite (SPD) matrices, which are often used in machine learning,
computer vision and related areas. Unlike traditional sparse coding schemes
that work in vector spaces, in this paper we discuss how SPD matrices can be
described by sparse combination of dictionary atoms, where the atoms are also
SPD matrices. We propose to seek sparse coding by embedding the space of SPD
matrices into Hilbert spaces through two types of Bregman matrix divergences.
This not only leads to an efficient way of performing sparse coding, but also
an online and iterative scheme for dictionary learning. We apply the proposed
methods to several computer vision tasks where images are represented by region
covariance matrices. Our proposed algorithms outperform state-of-the-art
methods on a wide range of classification tasks, including face recognition,
action recognition, material classification and texture categorization
Learning Discriminative Stein Kernel for SPD Matrices and Its Applications
Stein kernel has recently shown promising performance on classifying images
represented by symmetric positive definite (SPD) matrices. It evaluates the
similarity between two SPD matrices through their eigenvalues. In this paper,
we argue that directly using the original eigenvalues may be problematic
because: i) Eigenvalue estimation becomes biased when the number of samples is
inadequate, which may lead to unreliable kernel evaluation; ii) More
importantly, eigenvalues only reflect the property of an individual SPD matrix.
They are not necessarily optimal for computing Stein kernel when the goal is to
discriminate different sets of SPD matrices. To address the two issues in one
shot, we propose a discriminative Stein kernel, in which an extra parameter
vector is defined to adjust the eigenvalues of the input SPD matrices. The
optimal parameter values are sought by optimizing a proxy of classification
performance. To show the generality of the proposed method, three different
kernel learning criteria that are commonly used in the literature are employed
respectively as a proxy. A comprehensive experimental study is conducted on a
variety of image classification tasks to compare our proposed discriminative
Stein kernel with the original Stein kernel and other commonly used methods for
evaluating the similarity between SPD matrices. The experimental results
demonstrate that, the discriminative Stein kernel can attain greater
discrimination and better align with classification tasks by altering the
eigenvalues. This makes it produce higher classification performance than the
original Stein kernel and other commonly used methods.Comment: 13 page
Generalized Dictionary Learning for Symmetric Positive Definite Matrices with Application to Nearest Neighbor Retrieval
Abstract. We introduceGeneralized Dictionary Learning (GDL), a sim-ple but practical framework for learning dictionaries over the manifold of positive definite matrices. We illustrate GDL by applying it to Nearest Neighbor (NN) retrieval, a task of fundamental importance in disciplines such as machine learning and computer vision. GDL distinguishes itself from traditional dictionary learning approaches by explicitly taking into account the manifold structure of the data. In particular, GDL allows performing “sparse coding ” of positive definite matrices, which enables better NN retrieval. Experiments on several covariance matrix datasets show that GDL achieves performance rivaling state-of-the-art techniques.