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.
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 O ( 1 / t ) O(1/t) O ( 1/ t ) and requires only two low-rank SVDs per iteration. Distributed algorithm finds global solutions for low-rank matrices.
problem Finding global solutions for low-rank matrices in distributed systems.
method Distributed Gradient Descent (DGD+) with LOCAL variables.
result DGD+LOCAL converges to global minimizer with exact consensus.
Paper introduces new norms for rank-constrained optimization problems.
problem Rank-constrained optimization problems in various fields.
method Introduces a family of low-rank inducing norms and regularizers.
result Other low-rank inducing norms outperform nuclear norm in matrix completion problems.
This work simplifies proximal mapping for low-rank norms.
problem Efficient computation of proximal mappings for low-rank inducing norms.
method Reduces proximal mapping to nested binary search, solving simpler problems analytically.
result Simplified computation of proximal mappings for various norms.
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.
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…
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.
Robust PCA method optimizes low-rank matrices with corrupted data.
problem Recover a low-rank matrix from grossly corrupted observations.
method Nonconvex optimization on the manifold of low-rank matrices, using manifold optimization algorithms.
result Proposed algorithms converge to the underlying low-rank matrix linearly with proper initialization.
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.
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.
Algorithm recovers multiple low-rank matrices from unlabeled data.
problem Learning mixtures of low-rank models from unlabelled data.
method Three-stage meta-algorithm that copes with non-convexity and noise.
result Near-optimal sample and computational complexities under Gaussian designs.
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.
This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We r…
Solves low-rank approximation problems in Hilbert spaces.
problem Low-rank approximation in Hilbert spaces.
method Closed-form solutions and error bounds for bounded linear operators.
result Generalization to bounded linear operators from finite dimensions.
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.
Novel factorization for low-rank matrices in subspaces, improving efficiency.
problem Learning low-rank matrices constrained to subspaces.
method Riemannian manifold optimization with conjugate gradient and trust-region algorithms.
result Efficient algorithms for structured low-rank matrix learning.
Paper improves MVSC using tensor low-rank modeling.
problem Improving multi-view spectral clustering.
method Structured tensor low-rank norm for MVSC optimization.
result Proposed method outperforms state-of-the-art methods.
The paper tackles matrix estimation from noisy data, focusing on low-rank matrices.
problem Estimating a low-rank matrix from noisy observations.
method The paper analyzes several estimators, including constrained nuclear-norm minimization, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem.
result The estimators provide upper error bounds that depend on matrix rank, observed fraction, and matrix sums, and are minimax optimal.
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.
SketchyCGM optimizes matrices with optimal storage and low-rank solutions.
problem Optimizing matrices with low-rank solutions efficiently.
method Modifies conditional gradient method to use a small randomized sketch of the matrix variable.
result SketchyCGM converges to a low-rank solution with optimal storage.
Paper proposes a method for estimating complex low-rank matrices from phase-only measurements.
problem Estimating complex low-rank matrices from magnitude-only measurements.
method A hierarchical prior model with a Gaussian-Wishart distribution is used to promote low-rankness. A variational EM algorithm is developed to solve the problem.
result The proposed method is less sensitive to initialization and performs well with random initialization.
New algorithms solve large-scale low-rank and nonsmooth optimization problems efficiently.
problem Solving large-scale composite convex optimization problems with nonsmooth and low-rank terms.
method Stochastic optimization algorithms combining variance reduction and weak proximal oracle.
result First algorithm with nearly optimal sample complexity, single low-rank SVD per iteration, and log 1 / ε \log{1/ε} log 1/ ε thin-SVD computations. 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.
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.
Bundle method solves low rank SDP problems without full matrix construction.
problem Solving semidefinite programming problems with low rank solutions.
method Applying bundle method to randomly sketch matrix optimization problems and using recent results on bundle methods.
result Algorithm produces solutions with low rank representation and convergence rates.
New model enhances SPIM for solving low-rank combinatorial optimization and statistical learning problems.
problem Solving large-scale combinatorial optimization problems efficiently.
method Proposed a new computing model for SPIM that can handle low-rank interaction matrices.
result Demonstrated efficient learning, classification, and sampling of MNIST images using the model.
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.
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.
New algorithm tackles low-rank constraints in optimal transport problems.
problem Optimal transport problems with low-rank constraints.
method Explicit factorization of low-rank couplings as a product of sub-coupling factors linked by a common marginal.
result Stationary convergence of the algorithm proved.
An algorithm tackles low-rank linear bandit problems with improved regret bounds.
problem Low-rank linear bandit problems where rewards are inner products with an unknown low-rank matrix.
method Combines online-to-confidence-set conversion and exponentially weighted average forecaster with a covering of low-rank matrices.
result Achieves O ~ ( ( d 1 + d 2 ) 3 / 2 r T ) \widetilde{O}((d_1+d_2)^{3/2}\sqrt{rT}) O (( d 1 + d 2 ) 3/2 r T ) regret, improving over standard bounds when r ≪ min { d 1 , d 2 } r \ll \min\{d_1,d_2\} r ≪ min { d 1 , d 2 } . New framework for low-rank tensor analysis on graphs.
problem Low-rank tensor analysis on non-Euclidean domains.
method Graph-based low-rank decomposition and convex optimization.
result Significant speed-up and performance enhancement at low SNR.
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.
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…
New framework solves low-rank optimization problems to certifiable optimality.
problem Low-rank optimization problems with certifiable solutions.
method Mixed-Projection Conic Optimization framework using symmetric projection matrices and outer-approximation algorithms.
result Solves low-rank problems to certifiable optimality, outperforming existing methods.
Paper presents optimal low-rank DMD for better system analysis.
problem Improving DMD for low-rank approximations in non-linear systems.
method Developed a closed-form optimal solution using SVD.
result Demonstrated superior performance compared to existing methods.
This work learns low-rank hyperbolic embeddings for tasks with hierarchical structures.
problem Learning hyperbolic embeddings of tasks with hierarchical structures.
method Formulated as manifold optimization problems and proposed computationally efficient algorithms.
result Efficacy of the proposed approach demonstrated through empirical results.
As surrogate functions of L 0 L_0 L 0 -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…
Simplifies solving noisy SDPs for low rank matrix recovery problems.
problem Solving SDPs with noisy data for low rank matrix recovery problems.
method Identifies conditions called simplicity to limit error in noisy SDP solutions.
result Simple SDPs can be efficiently solved and their approximate solutions trusted.
New framework shows all local minima are globally optimal in non-convex low-rank problems.
problem Non-convex low-rank problems, including matrix sensing, completion, and robust PCA.
method Developed a new framework to analyze the optimization landscapes of these problems.
result All local minima are also globally optimal and no high-order saddle points exist.
Greedy method improves low rank matrix estimation with new approximation guarantees.
problem Low rank matrix estimation under restricted strong convexity and smoothness.
method Novel greedy algorithm analysis linking to combinatorial optimization.
result Improved approximation guarantees and statistical recovery.
New method for initializing low-rank neural networks improves performance.
problem Training low-rank neural networks efficiently and accurately.
method Inspired by function approximation, proposes a novel low-rank initialization framework.
result Demonstrates significant gap between spectral and low-rank initialization approaches.
We study the problem of prediction for evolving graph data. We formulate the problem as the minimization of a convex objective encouraging sparsity and low-rank of the solution, that reflect natural graph properties. The convex formulation allows to obtain oracle inequalities and efficient solvers. We provide empirical…
New algorithm for low-rank optimal transport with improved interpretability and efficiency.
problem Quadratic scaling of optimal transport coupling matrix for massive datasets.
method Factor Relaxation with Latent Coupling (FRLC) algorithm.
result Superior performance on diverse applications including graph clustering and spatial transcriptomics.
Paper solves low-rank Boolean matrix approximation using integer programming.
problem Finding low-rank approximations to Boolean matrices.
method Integer programming formulation with polynomial variables and constraints.
result First computationally tractable integer programming approach.
Improved Frank-Wolfe for sparse/low-rank problems.
problem Sparse/low-rank optimization problems.
method Primal-Dual Block Frank-Wolfe algorithm.
result Empirically outperforms state-of-the-art methods in classification tasks.
Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.
problem Interpolating and identifying covariance matrices in high dimensions with limited data.
method Differential geometric construction of low-rank covariance families, interpolation on manifolds, and distance minimization for identification.
result Differential geometric covariance families offer significant flexibility and computational tractability.
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.