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) and requires only two low-rank SVDs per iteration. 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.
This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modi…
We introduce a new framework for optimal transport using Schatten-p regularization to recover low-rank structures.
problem Optimal transport problems with low-rank structure recovery.
method Schatten-p norm regularization to promote low-rank structure in transport maps and plans.
result Unified convex programs for low-rank structure recovery with theoretical guarantees and efficient algorithms.
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…
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.
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.
A low-rank tensor model simplifies multi-dimensional Markov chains.
problem Simplifying the dynamics of multi-dimensional Markov chains.
method Low-rank tensor decomposition for multi-dimensional state spaces.
result Our tensor model requires fewer parameters and samples than conventional methods.
Equivalent formulations for low-rank matrix optimization are proven.
problem Low-rank matrix optimization with rank constraints.
method Established geometric landscape connections between manifold and factorization formulations.
result Equivalence between manifold and factorization formulations at FOSPs, SOSPs, and strict saddles.
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.
Research reveals deep networks often learn low-rank structures, leading to more efficient training and fine-tuning.
problem Efficient training and deployment of large-scale deep learning models.
method Complementary theoretical perspectives on low-rank structures during training and convergence, and practical applications of LoRA and masked training.
result Understanding and exploiting low-rank structures can improve efficiency and effectiveness of training and fine-tuning.
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.
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization variables appear only locally at a single node in the network. We term the resulting a…
Efficiently implements MEG for low-rank matrix optimization problems.
problem Optimization over spectrahedron with low-rank matrices.
method Matrix Exponentiated Gradient (MEG) method with efficient implementations.
result Methods converge from a warm-start initialization with similar rates to full-SVD-based counterparts.
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.
The paper reviews Hankel low-rank methods for time series analysis and forecasting.
problem Developing efficient methods for time series analysis and forecasting.
method Hankel low-rank approximation and completion techniques.
result Discussion of methods and challenges in obtaining optimal solutions.
Optimizes wide low-rank neural networks for reduced parameters and cost.
problem Reducing the number of learnable parameters in wide neural networks.
method Analyzed edge-of-chaos dynamics and derived formulae for optimal weight and bias variances.
result Optimal weight and bias variances for low-rank networks follow from multiplicative scaling.
Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing …
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…
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.
A new Riemannian framework optimizes LoRA for faster convergence and better performance.
problem Optimizing low-rank adapters in neural networks to improve convergence and performance.
method Integrates Riemannion optimizer, LoRA initialization, and efficient implementation for geometrically treating low-rank adapters.
result Consistent and noticeable improvements in convergence speed and final task performance over standard LoRA and its modifications.
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank estimation and combinatorial optimization, so much so that our bounds are reminisce…
Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging t…
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.
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.
TensorGuide improves LoRA efficiency and expressivity through joint tensor-train optimization.
problem Limited expressivity and generalization of standard LoRA.
method TensorGuide uses a unified tensor-train structure with controlled Gaussian noise to generate correlated low-rank matrices.
result TensorGuide achieves superior accuracy and scalability with fewer parameters compared to standard LoRA and TT-LoRA.
Paper explores statistical and computational limits of estimating low-rank Gaussian mixtures.
problem Estimating low-rank matrix-variate observations with optimal statistical and computational limits.
method Low-rank Gaussian mixture model (LrMM) and minimax lower bounds.
result Minimax optimality of maximum likelihood estimator and spectral aggregation method.
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 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.
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex optimization problem of matrix sensing. Our algorithm is applicable to both noisy and…
Flora uses random projections to achieve high-rank updates with low memory usage.
problem Excessive memory usage in large neural networks during training.
method Flora approximates LoRA using random projections to enable high-rank updates with sublinear space complexity.
result Flora achieves high-rank updates with significantly reduced memory usage compared to LoRA.
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.
New findings show DNC is not optimal for deep models, revealing a low-rank bias.
problem Theoretical limitations of DNC in non-linear models and multi-class classification.
method Analysis of non-linear models of arbitrary depth in multi-class classification.
result DNC stops being optimal for DUFM when going beyond two layers or two classes, due to a low-rank bias.
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…
Unified approach tackles high-dimensional tensor bandits with convex optimization and weakly decomposable regularizers.
problem Challenges in high-dimensional generalized tensor bandits where existing algorithms fail.
method Proposes a generalized linear tensor bandits algorithm with a unified analytical framework using convex optimization and weakly decomposable regularizers.
result Unified analytical framework provides better bounds and broader applicability compared to existing methods.
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.
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigensp…
Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face" directions at each iteration, as well as t…
New method for online low-rank matrix completion with improved regret.
problem Designing an efficient algorithm for online recommendation systems with low regret.
method Explore-then-commit (ETC) approach and iterative user clustering (OCTAL) for rank-1 setting.
result Nearly optimal regret bounds for online low-rank matrix completion.
New method normalizes matrix features for robust low-rank approximation.
problem Robust feature normalization for low-rank matrix approximation.
method Learn quantile normalization operators jointly with matrix factorization.
result Improves quality of low-rank representation of data.
Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.
problem Reconstructing a low-rank matrix from few measurements.
method Gradient descent with small random initialization followed by a few iterations.
result Gradient descent from small random init converges to a well-generalizing solution.
Partial convexification improves tractability of low-rank spectral optimization problems.
problem Minimizing linear objectives subject to matrix inequalities and low-rank constraints.
method Partial convexification of the domain set, deriving rank bounds, and developing a column generation algorithm.
result The partial convexification LSOP-R is equivalent to the original LSOP under certain conditions and yields high-quality solutions.
In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong d…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algor…
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…
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.
Low-rank metric learning aims to learn better discrimination of data subject to low-rank constraints. It keeps the intrinsic low-rank structure of datasets and reduces the time cost and memory usage in metric learning. However, it is still a challenge for current methods to handle datasets with both high dimensions and…
We propose a generic framework based on a new stochastic variance-reduced gradient descent algorithm for accelerating nonconvex low-rank matrix recovery. Starting from an appropriate initial estimator, our proposed algorithm performs projected gradient descent based on a novel semi-stochastic gradient specifically desi…