Unified framework for solving low-rank plus sparse matrix recovery problems.
problem Solving general low-rank plus sparse matrix recovery problems.
method Unified framework based on matrix factorization, projected gradient descent, and double thresholding operator.
result Our algorithm converges to the unknown low-rank and sparse matrices at a locally linear rate, matching robustness guarantees.
Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with tra…
New method improves robust low-rank matrix completion for computer vision.
problem Robust low-rank matrix completion for partially observed data.
method Formulated as a nonsmooth Riemannian optimization problem over Grassmann manifold, solved with an alternating manifold proximal gradient continuation method.
result Demonstrated advantages over existing approaches in background extraction from surveillance videos.
New nonconvex regularizers improve low-rank matrix recovery efficiency and accuracy.
problem Efficiently recover low-rank matrices from incomplete data.
method Factor group-sparse regularization, related to Schatten-p norms.
result Improved generalization error bounds for Schatten-p norms as p decreases.
Paper proposes a new method to separate low rank and sparse matrices without bias.
problem Recovering low rank and sparse matrices from measurements.
method Uses nonconvex regularizers and alternating proximal gradient descent.
result Error bounds for the algorithm applied to sparse optimization, matrix completion, and robust PCA.
Proposes a method to estimate sparse low-rank matrices from noisy data.
problem Estimating sparse low-rank matrices from noisy observations.
method Objective function with non-convex penalties, ADMM algorithm.
result Proposed method outperforms convex methods in estimating sparse low-rank matrices.
Algorithm recovers sparse and low rank matrix components efficiently.
problem Recovery of sparse and low rank components of matrices.
method Iterative method with adaptive thresholding.
result Algorithm performs well with low run-time and suitable for non-sparse noise.
Paper proposes a faster method for sparse parameter recovery from noisy linear combinations with low-rank matrices.
problem Recovering sparse parameters from noisy linear combinations with partial matrix information.
method Unified four-step problem combining partial matrix completion and sparse vector recovery, ignoring zero elements in the sparse vector.
result The unified approach achieves best performance with less computational requirements.
ReFACTor improves low-rank matrix recovery from noisy data.
problem Recovering low-rank matrices from noisy column-sparse data.
method A simple variation of TSVD, leveraging column-sparsity.
result ReFACTor outperforms TSVD and PCA in various scenarios.
New robust PCA algorithm for matrices with both sparse and outlying elements.
problem Simultaneous sparse and outlying corruption in matrices.
method Sparse approximation of a sparsely corrupted column to distinguish inliers from outliers.
result Robust PCA algorithm can handle both sparse and outlying corruptions.
The article develops a method to learn sparse and low rank PARAFAC decomposition robust to noise.
problem Learning sparse and low rank PARAFAC decomposition for tensors with missing values.
method Bayesian model with elastic net regularization, efficient algorithms for large scale problems.
result The method finds true rank and sparse factor matrix robust to noise.
New method decomposes corrupted data matrices into sparse and low-rank components.
problem Decomposing corrupted data matrices into sparse and low-rank components.
method Discrete optimization approach with alternating minimization, semidefinite relaxation, and branch-and-bound algorithm.
result High-quality solutions and meaningful bounds for SLR problems.
New algorithm speeds up LVGGM estimation by solving nonconvex optimization.
problem Estimating the latent variable Gaussian graphical model with sparse and low-rank components.
method Sparsity constrained maximum likelihood estimator with alternating gradient descent and hard thresholding.
result Our algorithm converges linearly to the optimal components up to statistical precision.
New guarantees for recovering matrices as low-rank plus sparse from fewer measurements.
problem Recovering matrices as the sum of a low-rank and sparse matrix from a limited number of measurements.
method Developed guarantees for recovery of low-rank plus sparse matrices from O(r(m+n−r)+s)log(mn/s) measurements, using semidefinite programming and gradient descent algorithms. result Guarantees for recovery of low-rank plus sparse matrices from fewer measurements than previously possible.
Tensor method robustly decomposes tensors with sparse perturbations.
problem Robust tensor decomposition under block sparse perturbations.
method Non-convex iterative algorithm alternating low-rank CP decomposition and hard thresholding.
result Proves convergence to globally optimal solution under natural conditions.
Paper analyzes robust matrix completion with efficient nonconvex method and leave-one-out analysis.
problem Robust matrix completion with sparse noise.
method Alternates between projected gradient step for low-rank and thresholding step for sparse noise.
result Achieves linear convergence for general thresholding functions.
Improved Frank-Wolfe method tackles nonsmooth functions.
problem Efficiently solving large nonsmooth problems with sparse structures.
method Optimizes for approximation quality over all affine approximations.
result Overcomes issues with existing nonsmooth methods in low-rank matrix estimation.
Proposes a new method for high-dimensional data analysis.
problem Sparse PCA limitations in high-dimensional data analysis.
method Low-rank principal eigenmatrix analysis, matricized rank-truncated power method.
result Competitive empirical performance in synthetic data sets.
Unified model for tensor completion using low-rank and sparse Tucker decomposition.
problem Estimating missing data from incomplete tensor measurements.
method Unified low-rank and sparse enhanced Tucker decomposition model with ADMM.
result Our model achieves higher recovery accuracy on various real-world data sets.
This work tackles sparse coding in DLRA for interpretable multiway data.
problem Sparse coding in DLRA for interpretable multiway data.
method Proposes a new sparse-coding subproblem (MSC) and several algorithms to solve it.
result DLRA extends low-rank approximations, reducing variance and enhancing interpretability.
SEED method finds sparse low-rank representations of data.
problem Finding low-rank representations of data efficiently.
method Greedy selection of incoherent vectors to form a basis.
result SEED can exactly represent low-rank matrices and vectors.
We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of …
HERA improves PLL by integrating heterogeneous loss and sparse-low-rank regularization.
problem Learning from data with partial labels.
method Combines heterogeneous loss and sparse-low-rank regularization.
result Achieves superior performance on artificial and real-world data.
New method for factor analysis using nuclear and ℓ0 norms.
problem Finding a low-rank plus sparse decomposition from noisy covariance matrix.
method Formulated an optimization problem with nuclear norm, ℓ0 norm, and KL divergence. Used alternating minimization algorithm. result Algorithm effectively decomposes covariance matrices in synthetic and real datasets.
New method solves robust matrix completion using nonlinear equations.
problem Recover low rank and sparse matrices from incomplete observations.
method Transforms problem into solving a system of nonlinear equations, then uses the alternative direction method.
result Algorithm converges linearly to the true solution under proper assumptions.
This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables an…
New method improves matrix completion accuracy, especially in noisy data.
problem Noisy matrix completion in recommendation systems and signal processing.
method Residual Spectral Matching criterion and pseudo-gradient algorithms.
result Improved numerical performance in noisy data environments.
We solve robust regression and matrix completion problems with sparse and low-rank models.
problem Adversarial contamination and noisy matrix completion in high-dimensional settings.
method Subgaussian statistical learning framework, trace-regression with matrix decomposition, novel Huber-type loss.
result Near-optimal estimation rates for robust regression and matrix completion.
Recovery of low-rank matrices has recently seen significant activity in many areas of science and engineering, motivated by recent theoretical results for exact reconstruction guarantees and interesting practical applications. A number of methods have been developed for this recovery problem. However, a principled meth…
The paper tackles transfer learning for growing matrix representations, improving estimation accuracy.
problem Structured matrix estimation under growing ambient dimensions and latent representations.
method Proposes a general transfer framework decomposing target parameters into embedded source components, low-rank innovations, and sparse edits. Develops an anchored alternating projection estimator.
result Establishes deterministic error bounds that separate target noise, representation growth, and source estimation error, yielding improved rates.
We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation X of the sum of an approximately) low rank matrix Θ⋆ with a second matrix Γ⋆ endowed with a complementary …
Paper tackles matrix estimation under arbitrary noise, achieving minimax optimality.
problem Noisy low-rank-plus-sparse matrix recovery under arbitrary dependence.
method Incoherent-constrained least-square estimator, novel energy spreading result.
result Achieves minimax optimality in estimating structured Markov transition kernels.
Paper detects communities from graph signals using low-rank excitation modeling.
problem Detect communities in graphs from noisy signals.
method Model signals as graph filter outputs, apply spectral method to covariance matrix.
result Community structure can be retrieved directly from graph signals.
Study on signal recovery from low-rank matrix with sparse noise.
problem Inference of a rank-one signal in the presence of sparse noise.
method Replica method from statistical physics, recursive distributional equations, population dynamics algorithm.
result Critical signal strength for recovery via top eigenvector identified.
As surrogate functions of L0-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex l…
Sparse R-LSSVM improves robustness and sparsity of LSSVM.
problem Robustness and sparsity issues in LSSVM.
method Interpreting robustness as a re-weighted problem, proposing a sparse R-LSSVM algorithm using low-rank approximation and entropy penalty.
result Proposed SR-LSSVM achieves sparse solutions efficiently for large-scale problems.
Survey on nonconvex penalties for sparse and low-rank recovery in various fields.
problem Achieving sparsity and low-rankness in signal processing, statistics, and machine learning.
method Analysis of nonconvex penalties and their applications.
result Nonconvex penalties can significantly improve performance in various applications.
Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in da…
Sparse group matrix completion reduces complexity and improves performance.
problem Matrix completion with non-informative side features.
method Group-Lasso regularization for feature selection in matrix factorization.
result Theoretical sample complexity is significantly lower than competitors.
Develops PRPCA for smooth image recovery combining low-rank and smoothness.
problem Image matrix recovery under low-rank and smoothness assumptions.
method Projected Robust PCA (PRPCA) framework combining low-rank and smoothness.
result Explicit statistical guarantees for PRPCA, reducing matrix dimensionality.
Review of robust PCA and matrix completion methods.
problem Robust Principal Component Analysis and matrix completion with outliers.
method Various provably correct, fast, and practical solutions to RPCA and matrix completion.
result Exhaustive review of recent literature on RPCA and dynamic RPCA.
Unified framework HASSLE-free decomposes large model weights into sparse and low-rank components.
problem Efficiently compress large foundation models to reduce inference costs.
method Designs a unified framework for sparse plus low-rank matrix decomposition with a local layer-wise reconstruction error objective.
result HASSLE-free framework significantly outperforms state-of-the-art methods in compression and evaluation benchmarks.
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint/mixed relaxations (i.e., l_{1}-norm and tr…
This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which…
New algorithms mix spatial and spectral data to improve unmixing of hyperspectral images.
problem Improving spectral unmixing in hyperspectral images.
method Introduced a novel convex mixed penalty term combining ℓ1 and nuclear norm regularization, applied to a sliding window of the image. result Demonstrated enhanced estimation results for abundance matrix in hyperspectral images.
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse v…
New algorithms improve RPCA for large matrices with upper rank bounds.
problem Efficiently decompose large matrices into low-rank and sparse parts.
method Combine regularization and matrix multiplication approaches with upper rank bounds.
result Proposed algorithms are faster and more robust than existing methods.
Improved matrix completion for non-uniformly sampled data.
problem Estimating unobserved entries in a matrix with varying sampling probabilities.
method Developed entry-specific bounds for low-rank matrix completion under structured non-uniform sampling.
result Error bounds for each entry match minimax lower bounds under certain conditions.