8,150 research outputs found
Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes
Given a multivariate data set, sparse principal component analysis (SPCA)
aims to extract several linear combinations of the variables that together
explain the variance in the data as much as possible, while controlling the
number of nonzero loadings in these combinations. In this paper we consider 8
different optimization formulations for computing a single sparse loading
vector; these are obtained by combining the following factors: we employ two
norms for measuring variance (L2, L1) and two sparsity-inducing norms (L0, L1),
which are used in two different ways (constraint, penalty). Three of our
formulations, notably the one with L0 constraint and L1 variance, have not been
considered in the literature. We give a unifying reformulation which we propose
to solve via a natural alternating maximization (AM) method. We show the the AM
method is nontrivially equivalent to GPower (Journ\'{e}e et al; JMLR
11:517--553, 2010) for all our formulations. Besides this, we provide 24
efficient parallel SPCA implementations: 3 codes (multi-core, GPU and cluster)
for each of the 8 problems. Parallelism in the methods is aimed at i) speeding
up computations (our GPU code can be 100 times faster than an efficient serial
code written in C++), ii) obtaining solutions explaining more variance and iii)
dealing with big data problems (our cluster code is able to solve a 357 GB
problem in about a minute).Comment: 29 pages, 9 tables, 7 figures (the paper is accompanied by a release
of the open-source code '24am'
Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition
This paper deals with the rotation synchronization problem, which arises in
global registration of 3D point-sets and in structure from motion. The problem
is formulated in an unprecedented way as a "low-rank and sparse" matrix
decomposition that handles both outliers and missing data. A minimization
strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against
state-of-the-art algorithms on simulated and real data. The results show that
R-GoDec is the fastest among the robust algorithms.Comment: The material contained in this paper is part of a manuscript
submitted to CVI
- …