Riemannian Metric Learning for Symmetric Positive Definite Matrices
نویسندگان
چکیده
Over the past few years, symmetric positive definite matrices (SPD) have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced by Riemannian geometry. In this work, we focus on the log-Euclidean Riemannian geometry and propose a data-driven approach for learning Riemannian metrics/geodesic distances for SPD matrices. We show that the geodesic distance learned using the proposed approach performs better than various existing distance measures when evaluated on face matching and clustering tasks. Notations – I denotes the identity matrix of appropriate size. – 〈 , 〉 denotes an inner product. – Sn denotes the set of n× n symmetric matrices. – S n denotes the set of n× n symmetric positive definite matrices. – TpM denotes the tangent space to the manifold M at the point p ∈ M. – ‖ ‖F denotes the matrix Frobenius norm. – Chol(P) denotes the lower triangular matrix obtained from the Cholesky decomposition of a matrix P. – exp() and log() denote matrix exponential and logarithm respectively. – ∂ ∂x and ∂ 2 ∂x represent partial derivatives. 2 Raviteja Vemulapalli, David W. Jacobs
منابع مشابه
CMSC 828J: Linear Subspaces and Manifolds for Computer Vision and Machine Learning Riemannian Metric Learning for Symmetric Positive Definite Matrices
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ورودعنوان ژورنال:
- CoRR
دوره abs/1501.02393 شماره
صفحات -
تاریخ انتشار 2015