Matrix approximation is a common tool in machine learning for building accurate prediction models for recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is of low-rank. We propose a new matrix approximation model w…
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48 results for “Low-Rank Matrix”
Robust Low-rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Methodcs.LG
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 spectral methods improve matrix estimation in RL with low-rank structure.
problem Estimating matrices with low-rank structure in reinforcement learning.
method Spectral-based matrix estimation approaches.
result Spectral methods efficiently recover singular subspaces and minimize entry-wise error.
Efficient Low-Rank Matrix Estimation, Experimental Design, and Arm-Set-Dependent Low-Rank Banditsstat.ML
Novel method for efficient low-rank matrix estimation and bandit algorithms.
problem Low-rank matrix estimation and bandit problems.
method LowPopArt method for low-rank matrix estimation and novel experimental design criterion.
result Improved recovery guarantees and regret bounds for low-rank bandit algorithms.
New approach to convex hulls for low-rank problems.
problem Characterizing convex hulls for low-rank sets.
method Matrix perspective function and orthogonal projection matrices.
result Strong relaxations for various low-rank problems.
New nonconvex regularizer speeds up low-rank matrix completion.
problem Low-rank matrix completion with good theoretical and empirical performance.
method Proposes a new nonconvex regularizer with adaptive shrinkage, scalable, and fast optimization.
result Proposed method achieves state-of-the-art recovery performance and is the fastest.
The problem of low rank matrix completion is considered in this paper. To exploit the underlying low-rank structure of the data matrix, we propose a hierarchical Gaussian prior model, where columns of the low-rank matrix are assumed to follow a Gaussian distribution with zero mean and a common precision matrix, and a W…
SGD with mini-batches can solve convex low-rank matrix problems efficiently.
problem Solving large-scale convex low-rank matrix problems efficiently.
method Stochastic Gradient Descent with mini-batches and low-rank projections.
result SGD with mini-batches produces low-rank iterates with high probability.
Matrix completion is a problem that arises in many data-analysis settings where the input consists of a partially-observed matrix (e.g., recommender systems, traffic matrix analysis etc.). Classical approaches to matrix completion assume that the input partially-observed matrix is low rank. The success of these methods…
Paper develops a new weighted low-rank matrix approximation technique.
problem Matrix completion with missing data.
method Element-wise weighted generalization of low-rank matrix approximation.
result Proposes an algorithm and acceleration techniques for solving the weighted problem.
ScaledGD improves gradient descent for ill-conditioned low-rank matrix estimation.
problem Efficiently solving ill-conditioned low-rank matrix estimation problems.
method Scaled Gradient Descent (ScaledGD) with adaptive pre-conditioners.
result Linear convergence rate independent of condition number, low per-iteration cost.
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.
Implicit Regularization in Perturbed Deep Matrix Factorization: Spectral Conditions and Stabilitymath.OC
The paper examines how gradient descent stabilizes low-rank matrix factorization in noisy conditions.
problem Stability of low-rank implicit regularization in perturbed deep matrix factorization.
method Derives spectral conditions for gradient descent to exhibit a low-rank phase in noiseless settings and analyzes perturbed dynamics.
result Gradient descent converges to a low-rank solution under perturbation, with explicit dependence on perturbation size.
Unified approach for robust low rank matrix estimation with adversaries.
problem Robust low rank matrix estimation in the presence of adversaries.
method Unified approach combining Huber loss and nuclear norm penalization.
result Sharp estimation error bounds for matrix compressed sensing and completion.
Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matr…
We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general case with noisy observations, we show that our algorithm is guaranteed to linearl…
In this paper, we consider the problem of low-rank phase retrieval whose objective is to estimate a complex low-rank matrix from magnitude-only measurements. We propose a hierarchical prior model for low-rank phase retrieval, in which a Gaussian-Wishart hierarchical prior is placed on the underlying low-rank matrix to …
New framework explains why nonconvex methods work well in low-rank matrix estimation.
problem Nonconvex low-rank matrix estimation problems in machine learning.
method Developed a theoretical framework revealing a benign regularizer.
result Nonconvex procedures can behave well due to a disguised convexity.
We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we rigorously show that under a certain condition on the magnitude of t…
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the…
Proposes a new model for image restoration combining deep learning and total variation.
problem Restoring images from limited data with low-rank constraints insufficient.
method Regularized Deep Matrix Factorized (RDMF) model using deep neural network's low-rank bias and total variation.
result Outperforms state-of-the-art models in image restoration from few observations.
Paper proposes fast, robust methods for low-rank matrix recovery.
problem Estimating low-rank matrices from incomplete or corrupted data.
method Scaled subgradient methods for nonsmooth, nonconvex formulations.
result Methods converge almost dimension-free and condition-number independent.
New method solves nonsmooth low-rank matrix optimization problems efficiently.
problem Nonsmooth and low-rank matrix optimization problems in statistics and machine learning.
method Low-rank Extragradient Method with warm-start initialization.
result The extragradient method converges to an optimal solution with rate and requires only two low-rank SVDs per iteration.
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure ap…
Gradient descent solves asymmetric low-rank matrix sensing without balancing.
problem Recovering asymmetric low-rank matrices from linear measurements.
method Gradient descent with spectral initialization, avoiding balancing term.
result Gradient descent converges linearly without balancing, factors stay balanced.
CoreFlow models matrix-valued distributions efficiently, preserving shared low-rank structure.
problem Learning matrix-valued distributions from high-dimensional and incomplete data.
method Low-rank flow model that learns shared row/column subspaces and trains a normalizing flow on the core.
result CoreFlow improves generation quality in few-sample regimes and remains competitive in data-rich settings.
This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both case…
New algorithm for weighted low rank approximation with provable guarantees.
problem Weighted low rank approximation (WLRA) is computationally hard.
method Reweights the low rank solution using the weight matrix itself.
result Provably optimal approximation guarantees for WLRA.
Rank-one measurements limit feasible sets for low-rank PSD matrices.
problem Feasibility of PSD matrices under rank-one measurements.
method Characterization of feasible sets for PSD matrices given rank-one projections.
result Radius of feasible sets determines singleton solution sets for low-rank matrices.
Paper finds a lower bound for estimating low-rank matrices in logistic regression.
problem Estimating low-rank coefficient matrices in logistic regression.
method Derives a minimax lower bound on the risk.
result The bound depends on matrix dimensions, rank, and sample size.
Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomaliescs.IT
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 for robust matrix completion with mixed data types.
problem Recovering a structured low rank matrix with mixed data types.
method Proposes a computationally feasible statistical approach with strong recovery guarantees for mixed data types.
result Strong recovery guarantees for low rank matrix completion with mixed data types.
Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. In the last decade, numerous statistical methodologies have been developed for efficiently recovering the unknown low-rank matrices. However, in some applicat…
Low-rank modeling generally refers to a class of methods that solve problems by representing variables of interest as low-rank matrices. It has achieved great success in various fields including computer vision, data mining, signal processing and bioinformatics. Recently, much progress has been made in theories, algori…
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on ma…
Study the distribution for low-rank matrix learning, improving inference methods.
problem Lack of understanding of underlying probability distributions in low-rank matrix learning.
method Analyze the distribution , using differential geometry to design an improved MCMC algorithm and learn penalty parameter λ.
result Improved MCMC algorithm and penalty parameter learning for low-rank Bayesian inference.
In this paper we consider the low-rank matrix completion problem with specific application to forecasting in time series analysis. Briefly, the low-rank matrix completion problem is the problem of imputing missing values of a matrix under a rank constraint. We consider a matrix completion problem for Hankel matrices an…
We consider the problem of estimation of a low-rank matrix from a limited number of noisy rank-one projections. In particular, we propose two fast, non-convex \emph{proper} algorithms for matrix recovery and support them with rigorous theoretical analysis. We show that the proposed algorithms enjoy linear convergence a…
Paper develops methods for non-quadratic loss low-rank matrix recovery.
problem Recovery of low-rank matrices with non-quadratic losses.
method Projected gradient method with a regularity projection oracle.
result Projected gradient method converges globally and linearly.
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…
New method solves matrix completion problems to certifiable optimality.
problem Certifying optimality in low-rank matrix completion.
method Disjunctive branch-and-bound scheme for convex relaxation.
result Decreases optimality gap by two orders of magnitude.
Paper studies asymmetric matrix sensing, proving gradient descent converges to low-rank solutions.
problem Reconstructing asymmetric low-rank matrices from linear measurements.
method Factorized gradient descent with coupling and regularization properties.
result Gradient descent from small random initialization converges to globally optimal and generalizing solutions.
Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study ma…
This paper tackles fitting multilevel low rank matrices by addressing three problems.
problem Fitting a given matrix by an MLR matrix in the Frobenius norm.
method Factor fitting, rank allocation, and hierarchical partitioning.
result The proposed methods can fit a given matrix by an MLR matrix in the Frobenius norm.
Estimates low-rank distributional matrices from incomplete samples.
problem Matrix completion for distributional entries with limited observed data.
method Kernel mean embeddings, Tucker rank, functional unfolding operators.
result Effective estimator for distributional matrix completion established.
Solves weakly supervised regression using low-rank approximations and manifold regularization.
problem Weakly supervised regression with known, unknown, and uncertain labels.
method Combines manifold regularization and low-rank matrix decomposition for optimization.
result Improves solution quality and stability for large datasets.
In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and the Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank m…
Flat minima lead to better generalization in low-rank matrix recovery models.
problem Understanding why flat minima generalize well in overparameterized models.
method Analysis of overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and neural networks with quadratic activation functions.
result Flat minima, measured by the trace of the Hessian, exactly recover the ground truth in low-rank matrix recovery models under standard statistical assumptions.