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 $\ell_1$ 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 $\ell_1$ norm. As a proof of concept, a background estimation model has been presented and compared with two $\ell_1$ 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 $\ell_1$ regression. As existing methods for $\ell_1$ 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 $\ell_1$ 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

A Batch-Incremental Video Background Estimation Model using Weighted Low-Rank Approximation of Matrices

Principal component pursuit (PCP) is a state-of-the-art approach for background estimation problems. Due to their higher computational cost, PCP algorithms, such as robust principal component analysis (RPCA) and its variants, are not feasible in processing high definition videos. To avoid the curse of dimensionality in those algorithms, several methods have been proposed to solve the background estimation problem in an incremental manner. We propose a batch-incremental background estimation model using a special weighted low-rank approximation of matrices. Through experiments with real and synthetic video sequences, we demonstrate that our method is superior to the state-of-the-art background estimation algorithms such as GRASTA, ReProCS, incPCP, and GFL