604 research outputs found
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
Total Jensen divergences: Definition, Properties and k-Means++ Clustering
We present a novel class of divergences induced by a smooth convex function
called total Jensen divergences. Those total Jensen divergences are invariant
by construction to rotations, a feature yielding regularization of ordinary
Jensen divergences by a conformal factor. We analyze the relationships between
this novel class of total Jensen divergences and the recently introduced total
Bregman divergences. We then proceed by defining the total Jensen centroids as
average distortion minimizers, and study their robustness performance to
outliers. Finally, we prove that the k-means++ initialization that bypasses
explicit centroid computations is good enough in practice to guarantee
probabilistically a constant approximation factor to the optimal k-means
clustering.Comment: 27 page
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