Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFast and Robust Fixed-Rank Matrix Recovery
123
0
0.0
(
0
)
Added by
German Ros
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually imposed by the physical restrictions of the system under study. Here we propose a method to perform accurate and very efficient S-FR decomposition that is more suitable for large-scale problems than existing approaches. Our method is a grateful combination of geometrical and algebraical techniques, which avoids the bottleneck caused by the Truncated SVD (TSVD). Instead, a polar factorization is used to exploit the manifold structure of fixed-rank problems as the product of two Stiefel and an SPD manifold, leading to a better convergence and stability. Then, closed-form projectors help to speed up each iteration of the method. We introduce a novel and fast projector for the $text{SPD}$ manifold and a proof of its validity. Further acceleration is achieved using a Nystrom scheme. Extensive experiments with synthetic and real data in the context of robust photometric stereo and spectral clustering show that our proposals outperform the state of the art.
rate research
Read More
We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in ~O(|A| + r^omega) field operations, where |A| denotes the number of nonzero entries in A and omega < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with running time O(mnr^{omega-2}). Our algorithm is faster when r < max(m,n), for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in ~O(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in numerical linear algebra, combinatorial optimization and dynamic data structure.
Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.
Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix.
Low-rank Multi-view Subspace Learning (LMvSL) has shown great potential in cross-view classification in recent years. Despite their empirical success, existing LMvSL based methods are incapable of well handling view discrepancy and discriminancy simultaneously, which thus leads to the performance degradation when there is a large discrepancy among multi-view data. To circumvent this drawback, motivated by the block-diagonal representation learning, we propose Structured Low-rank Matrix Recovery (SLMR), a unique method of effectively removing view discrepancy and improving discriminancy through the recovery of structured low-rank matrix. Furthermore, recent low-rank modeling provides a satisfactory solution to address data contaminated by predefined assumptions of noise distribution, such as Gaussian or Laplacian distribution. However, these models are not practical since complicated noise in practice may violate those assumptions and the distribution is generally unknown in advance. To alleviate such limitation, modal regression is elegantly incorporated into the framework of SLMR (term it MR-SLMR). Different from previous LMvSL based methods, our MR-SLMR can handle any zero-mode noise variable that contains a wide range of noise, such as Gaussian noise, random noise and outliers. The alternating direction method of multipliers (ADMM) framework and half-quadratic theory are used to efficiently optimize MR-SLMR. Experimental results on four public databases demonstrate the superiority of MR-SLMR and its robustness to complicated noise.
We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m asymp r s ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m asymp r s^gamma ln(en/s)$ for some exponent $gamma > 0$. We show that this is feasible for $gamma = 2$, and that the proposed analysis cannot cover the case $gamma leq 1$. The precise value of the optimal exponent $gamma in [1,2]$ is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
