19,077 research outputs found
Theory and Algorithms for Reliable Multimodal Data Analysis, Machine Learning, and Signal Processing
Modern engineering systems collect large volumes of data measurements across diverse sensing modalities. These measurements can naturally be arranged in higher-order arrays of scalars which are commonly referred to as tensors. Tucker decomposition (TD) is a standard method for tensor analysis with applications in diverse fields of science and engineering. Despite its success, TD exhibits severe sensitivity against outliers —i.e., heavily corrupted entries that appear sporadically in modern datasets. We study L1-norm TD (L1-TD), a reformulation of TD that promotes robustness. For 3-way tensors, we show, for the first time, that L1-TD admits an exact solution via combinatorial optimization and present algorithms for its solution. We propose two novel algorithmic frameworks for approximating the exact solution to L1-TD, for general N-way tensors. We propose a novel algorithm for dynamic L1-TD —i.e., efficient and joint analysis of streaming tensors. Principal-Component Analysis (PCA) (a special case of TD) is also outlier responsive. We consider Lp-quasinorm PCA (Lp-PCA) for
Fast Robust PCA on Graphs
Mining useful clusters from high dimensional data has received significant
attention of the computer vision and pattern recognition community in the
recent years. Linear and non-linear dimensionality reduction has played an
important role to overcome the curse of dimensionality. However, often such
methods are accompanied with three different problems: high computational
complexity (usually associated with the nuclear norm minimization),
non-convexity (for matrix factorization methods) and susceptibility to gross
corruptions in the data. In this paper we propose a principal component
analysis (PCA) based solution that overcomes these three issues and
approximates a low-rank recovery method for high dimensional datasets. We
target the low-rank recovery by enforcing two types of graph smoothness
assumptions, one on the data samples and the other on the features by designing
a convex optimization problem. The resulting algorithm is fast, efficient and
scalable for huge datasets with O(nlog(n)) computational complexity in the
number of data samples. It is also robust to gross corruptions in the dataset
as well as to the model parameters. Clustering experiments on 7 benchmark
datasets with different types of corruptions and background separation
experiments on 3 video datasets show that our proposed model outperforms 10
state-of-the-art dimensionality reduction models. Our theoretical analysis
proves that the proposed model is able to recover approximate low-rank
representations with a bounded error for clusterable data
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'
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