38 research outputs found
Weighted Low Rank Approximation for Background Estimation Problems
Classical principal component analysis (PCA) is not robust to the presence of
sparse outliers in the data. The use of the norm in the Robust PCA
(RPCA) method successfully eliminates the weakness of PCA in separating the
sparse outliers. In this paper, by sticking a simple weight to the Frobenius
norm, we propose a weighted low rank (WLR) method to avoid the often
computationally expensive algorithms relying on the norm. As a proof
of concept, a background estimation model has been presented and compared with
two norm minimization algorithms. We illustrate that as long as a
simple weight matrix is inferred from the data, one can use the weighted
Frobenius norm and achieve the same or better performance
On a Problem of Weighted Low-Rank Approximation of Matrices
We study a weighted low rank approximation that is inspired by a problem of
constrained low rank approximation of matrices as initiated by the work of
Golub, Hoffman, and Stewart (Linear Algebra and Its Applications, 88-89(1987),
317-327). Our results reduce to that of Golub, Hoffman, and Stewart in the
limiting cases. We also propose an algorithm based on the alternating direction
method to solve our weighted low rank approximation problem and compare it with
the state-of-art general algorithms such as the weighted total alternating
least squares and the EM algorithm
A Nonconvex Projection Method for Robust PCA
Robust principal component analysis (RPCA) is a well-studied problem with the
goal of decomposing a matrix into the sum of low-rank and sparse components. In
this paper, we propose a nonconvex feasibility reformulation of RPCA problem
and apply an alternating projection method to solve it. To the best of our
knowledge, we are the first to propose a method that solves RPCA problem
without considering any objective function, convex relaxation, or surrogate
convex constraints. We demonstrate through extensive numerical experiments on a
variety of applications, including shadow removal, background estimation, face
detection, and galaxy evolution, that our approach matches and often
significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial
Intelligence (AAAI-19
Online and Batch Supervised Background Estimation via L1 Regression
We propose a surprisingly simple model for supervised video background
estimation. Our model is based on regression. As existing methods for
regression do not scale to high-resolution videos, we propose several
simple and scalable methods for solving the problem, including iteratively
reweighted least squares, a homotopy method, and stochastic gradient descent.
We show through extensive experiments that our model and methods match or
outperform the state-of-the-art online and batch methods in virtually all
quantitative and qualitative measures
Chronic Obstructive Pulmonary Disease (COPD): Making Sense of Existing GWAS Findings in Indian Context
To date, more than 1456 associations have been identified for Chronic Obstructive Pulmonary Disease (COPD) risk through Genome-Wide Association Studies (GWAS). However, target genes for COPD susceptibility in the Indian population and the mechanism underlying remains largely unexplored and no GWAS studies on COPD are available on the Indian population till now. This study was conducted using the existing public data on GWAS of different parts of the world, and the genetic polymorphisms to understand the possible mechanisms of these polymorphisms using available data from the Genotype-Tissue Expression (GTEx) project. We jotted down 16 important genes and 28 Single Nucleotide Polymorphisms (SNPs) in the Indian population from 1456 variants. Pathway analysis showed that these relevant genes are mostly associated with immune responses and activation, which is a key factor in COPD development. Our investigation revealed possible target genes associated with COPD in the context of the Indian population
Weighted Low-Rank Approximation of Matrices and Background Modeling
We primarily study a special a weighted low-rank approximation of matrices
and then apply it to solve the background modeling problem. We propose two
algorithms for this purpose: one operates in the batch mode on the entire data
and the other one operates in the batch-incremental mode on the data and
naturally captures more background variations and computationally more
effective. Moreover, we propose a robust technique that learns the background
frame indices from the data and does not require any training frames. We
demonstrate through extensive experiments that by inserting a simple weight in
the Frobenius norm, it can be made robust to the outliers similar to the
norm. Our methods match or outperform several state-of-the-art online
and batch background modeling methods in virtually all quantitative and
qualitative measures.Comment: arXiv admin note: text overlap with arXiv:1707.0028
Shrinkage Function And Its Applications In Matrix Approximation
The shrinkage function is widely used in matrix low-rank approximation,
compressive sensing, and statistical estimation. In this article, an elementary
derivation of the shrinkage function is given. In addition, applications of the
shrinkage function are demonstrated in solving several well-known problems,
together with a new result in matrix approximation
Weighted Low-Rank Approximation of Matrices:Some Analytical and Numerical Aspects
This dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing state-of-the-art unweighted and weighted low-rank approximation algorithms. Classical principal component analysis (PCA) is constrained to have equal weighting on the elements of the matrix, which might lead to a degraded design in some problems. To address this fundamental flaw in PCA, Golub, Hoffman, and Stewart proposed and solved a problem of constrained low-rank approximation of matrices: For a given matrix , find a low rank matrix such that is less than , a prescribed bound, and is small.~Motivated by the above formulation, we propose a weighted low-rank approximation problem that generalizes the constrained low-rank approximation problem of Golub, Hoffman and Stewart.~We study a general framework obtained by pointwise multiplication with the weight matrix and consider the following problem:~For a given matrix solve: \begin{eqnarray*}\label{weighted problem} \min_{\substack{X}}\|\left(A-X\right)\odot W\|_F^2~{\rm subject~to~}{\rm rank}(X)\le r, \end{eqnarray*} where denotes the pointwise multiplication and is the Frobenius norm of matrices. In the first part, we study a special version of the above general weighted low-rank approximation problem.~Instead of using pointwise multiplication with the weight matrix, we use the regular matrix multiplication and replace the rank constraint by its convex surrogate, the nuclear norm, and consider the following problem: \begin{eqnarray*}\label{weighted problem 1} \hat{X} &=& \arg \min_X \{\frac{1}{2}\|(A-X)W\|_F^2 +\tau\|X\|_\ast\}, \end{eqnarray*} where denotes the nuclear norm of .~Considering its resemblance with the classic singular value thresholding problem we call it the weighted singular value thresholding~(WSVT)~problem.~As expected,~the WSVT problem has no closed form analytical solution in general,~and a numerical procedure is needed to solve it.~We introduce auxiliary variables and apply simple and fast alternating direction method to solve WSVT numerically.~Moreover, we present a convergence analysis of the algorithm and propose a mechanism for estimating the weight from the data.~We demonstrate the performance of WSVT on two computer vision applications:~background estimation from video sequences~and facial shadow removal.~In both cases,~WSVT shows superior performance to all other models traditionally used. In the second part, we study the general framework of the proposed problem.~For the special case of weight, we study the limiting behavior of the solution to our problem,~both analytically and numerically.~In the limiting case of weights,~as (W_1)_{ij}\to\infty, W_2=\mathbbm{1}, a matrix of 1,~we show the solutions to our weighted problem converge, and the limit is the solution to the constrained low-rank approximation problem of Golub et. al. Additionally, by asymptotic analysis of the solution to our problem,~we propose a rate of convergence.~By doing this, we make explicit connections between a vast genre of weighted and unweighted low-rank approximation problems.~In addition to these, we devise a novel and efficient numerical algorithm based on the alternating direction method for the special case of weight and present a detailed convergence analysis.~Our approach improves substantially over the existing weighted low-rank approximation algorithms proposed in the literature.~Finally, we explore the use of our algorithm to real-world problems in a variety of domains, such as computer vision and machine learning. Finally, for a special family of weights, we demonstrate an interesting property of the solution to the general weighted low-rank approximation problem. Additionally, we devise two accelerated algorithms by using this property and present their effectiveness compared to the algorithm proposed in Chapter 4