Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Abstract

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization.

References in corpus (1)

• Sparsity and Incoherence in Compressive Sampling

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• Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm
• Vector-output ReLU Neural Network Problems are Copositive Programs: Convex Analysis of Two Layer Networks and Polynomial-time Algorithms
• Lower and Upper Bounds on the VC-Dimension of Tensor Network Models
• Multi-Dimensional Wireless Tomography with Tensor-Based Compressed Sensing
• Block-Diagonal Sparse Representation by Learning a Linear Combination Dictionary for Recognition
• Linear models based on noisy data and the Frisch scheme
• Compressed Sensing Tomography for qudits in Hilbert spaces of non-power-of-two dimensions
• Convergence Rates with Inexact Non-expansive Operators
• Checking the strict positivity of Kraus maps is NP-hard
• Compressed Sensing on the Image of Bilinear Maps
• Convex Relaxation Regression: Black-Box Optimization of Smooth Functions by Learning Their Convex Envelopes
• Learning Non-Parametric Basis Independent Models from Point Queries via Low-Rank Methods
• Distributed Localization of Tree-structured Scattered Sensor Networks
• Sufficient Conditions for Low-rank Matrix Recovery, Translated from Sparse Signal Recovery
• Efficient Neural Interaction Function Search for Collaborative Filtering
• Stable rank one matrix completion is solved by two rounds of semidefinite programming relaxation
• RIP-based performance guarantee for low-tubal-rank tensor recovery
• Guarantees of Augmented Trace Norm Models in Tensor Recovery
• Top-N Recommendation with Novel Rank Approximation
• Unit balls of constant volume: which one has optimal representation?
• Balanced Augmented Lagrangian Method for Convex Programming
• A Rank-Corrected Procedure for Matrix Completion with Fixed Basis Coefficients
• An Efficient Primal-Dual Prox Method for Non-Smooth Optimization
• Matrix Completion with Nonconvex Regularization: Spectral Operators and Scalable Algorithms
• On the Tightness of Semidefinite Relaxations for Certifying Robustness to Adversarial Examples
• A Greedy Algorithm for Matrix Recovery with Subspace Prior Information
• Accelerated Linearized Bregman Method
• Alternating Iteratively Reweighted Minimization Algorithms for Low-Rank Matrix Factorization
• An algorithm for online tensor prediction
• From Blind deconvolution to Blind Super-Resolution through convex programming
• Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization
• Stochastic Variance-reduced Gradient Descent for Low-rank Matrix Recovery from Linear Measurements
• An Analysis of Dropout for Matrix Factorization
• Sparse Subspace Decomposition for Millimeter Wave MIMO Channel Estimation
• How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?
• Quasi-Hankel low-rank matrix completion: a convex relaxation
• A new graph parameter related to bounded rank positive semidefinite matrix completions
• Convergence bounds for nonlinear least squares and applications to tensor recovery
• How well can we learn large factor models without assuming strong factors?
• Explicit Frames for Deterministic Phase Retrieval via PhaseLift
• Decentralized Dictionary Learning Over Time-Varying Digraphs
• Low rank tensor completion with sparse regularization in a transformed domain
• A Geometric Approach to Low-Rank Matrix Completion
• Every critical point of an L0 composite minimization problem is a local minimizer
• Compressed Subspace Matching on the Continuum
• Performance Analysis of Convex LRMR based Passive SAR Imaging
• Low-rank Matrix Completion in a General Non-orthogonal Basis
• The Gram dimension of a graph
• Moving Object Detection by Detecting Contiguous Outliers in the Low-Rank Representation
• Global and Local Analyses of Nonlinear Low-Rank Matrix Recovery Problems
• A Super-Resolution Framework for Tensor Decomposition
• Consistent Collective Matrix Completion under Joint Low Rank Structure
• Poisson Matrix Completion
• Low-Rank and Total Variation Regularization and Its Application to Image Recovery
• Social Trust Prediction via Max-norm Constrained 1-bit Matrix Completion
• Subspace clustering based on low rank representation and weighted nuclear norm minimization
• Multiple Hankel matrix rank minimization for audio inpainting
• The Minimizer of the Sum of Two Strongly Convex Functions
• Primal-Dual Interior-Point Methods for Domain-Driven Formulations
• Low Rank Forecasting
• Efficient iterative thresholding algorithms with functional feedbacks and convergence analysis
• An equivalence between critical points for rank constraints versus low-rank factorizations
• Orthogonal Rank-One Matrix Pursuit for Low Rank Matrix Completion
• Efficient Online Minimization for Low-Rank Subspace Clustering
• Regularizers versus Losses for Nonlinear Dimensionality Reduction: A Factored View with New Convex Relaxations
• Recovering Multiple Nonnegative Time Series From a Few Temporal Aggregates
• Low rank estimation of smooth kernels on graphs
• Finding a low-rank basis in a matrix subspace
• Implicit Regularization in Tensor Factorization
• Shape and Spatially-Varying Reflectance Estimation From Virtual Exemplars
• Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction
• Alternating Strategies Are Good For Low-Rank Matrix Reconstruction
• Subspace Expanders and Matrix Rank Minimization
• A Unified Convex Surrogate for the Schatten-$p$ Norm
• Low Phase-Rank Approximation
• Gray Box Identification of State-Space Models Using Difference of Convex Programming
• Compressed sensing of block-sparse positive vectors
• Sample Complexity of Power System State Estimation using Matrix Completion
• Matrix Completion with Prior Subspace Information via Maximizing Correlation
• A Line-Search Descent Algorithm for Strict Saddle Functions with Complexity Guarantees
• Robust Matrix Elastic Net based Canonical Correlation Analysis: An Effective Algorithm for Multi-View Unsupervised Learning
• Rank-one Solutions for Homogeneous Linear Matrix Equations over the Positive Semidefinite Cone
• Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery
• Low-Rank Matrix Completion: A Contemporary Survey
• Distributed Computation for Solving the Sylvester Equation Based on Optimization
• Factorization Approach for Low-complexity Matrix Completion Problems: Exponential Number of Spurious Solutions and Failure of Gradient Methods
• Approximate matrix completion based on cavity method
• On Convex Duality in Linear Inverse Problems
• Robust Ordinal Embedding from Contaminated Relative Comparisons
• ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching
• Fine-grained Classification using Heterogeneous Web Data and Auxiliary Categories
• Efficient Proximal Mapping Computation for Unitarily Invariant Low-Rank Inducing Norms
• Adaptive Low-Nonnegative-Rank Approximation for State Aggregation of Markov Chains
• Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications
• Robust Sparse Reduced Rank Regression in High Dimensions
• Truncated Sparse Approximation Property and Truncated $q$-Norm Minimization
• Implicit regularization and solution uniqueness in over-parameterized matrix sensing
• Nonconvex Matrix Completion with Linearly Parameterized Factors
• Network Reconstruction and Controlling Based on Structural Regularity Analysis
• An Overview of Lead and Accompaniment Separation in Music
• A Parallelizable Acceleration Framework for Packing Linear Programs
• Novel min-max reformulations of Linear Inverse Problems
• Low-rank matrix recovery with non-quadratic loss: projected gradient method and regularity projection oracle
• Semidefinite Programming and Nash Equilibria in Bimatrix Games
• A Non-monotone Alternating Updating Method for A Class of Matrix Factorization Problems
• Low-rank matrix estimation in multi-response regression with measurement errors: Statistical and computational guarantees
• Blind Demixing of Diffused Graph Signals
• The Nonconvex Geometry of Linear Inverse Problems
• Sign-RIP: A Robust Restricted Isometry Property for Low-rank Matrix Recovery
• Linear Matrix Inequality Approaches to Koopman Operator Approximation
• Distributable Consistent Multi-Object Matching
• Affine matrix rank minimization problem via non-convex fraction function penalty
• Implicit Regularization in Matrix Sensing via Mirror Descent
• Learning Low-Complexity Autoregressive Models via Proximal Alternating Minimization
• Nonparametric Trace Regression in High Dimensions via Sign Series Representation
• Learning of Generalized Low-Rank Models: A Greedy Approach
• On the Optimality of Nuclear-norm-based Matrix Completion for Problems with Smooth Non-linear Structure
• Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion
• Landscape Correspondence of Empirical and Population Risks in the Eigendecomposition Problem
• A low-rank approach to image defringing
• Robust On-line Matrix Completion on Graphs
• Confidence-Constrained Maximum Entropy Framework for Learning from Multi-Instance Data
• Coded Illumination for 3D Lensless Imaging
• Dynamic Matrix Recovery
• Variational Analysis of the Ky Fan $k$-norm
• Efficient Low Rank Matrix Recovery With Flexible Group Sparse Regularization
• Large-scale Kernel-based Feature Extraction via Budgeted Nonlinear Subspace Tracking
• Fast Optimization Algorithm on Riemannian Manifolds and Its Application in Low-Rank Representation
• Similarity Learning via Adaptive Regression and Its Application to Image Retrieval
• Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting
• Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms
• Bilinear Sequence Regression: A Model for Learning from Long Sequences of High-dimensional Tokens
• Simultaneous Heterogeneity and Reduced-rank Learning for Multivariate Response Regression
• Lecture notes on non-convex algorithms for low-rank matrix recovery
• Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming
• Asymptotic Log-Det Rank Minimization via (Alternating) Iteratively Reweighted Least Squares
• State-Of-The-Art Algorithms For Low-Rank Dynamic Mode Decomposition
• A Trace-restricted Kronecker-Factored Approximation to Natural Gradient
• Scalable Nuclear-norm Minimization by Subspace Pursuit Proximal Riemannian Gradient
• Sparse Array Beamformer Design for Active and Passive Sensing
• Capacity Region of MISO Broadcast Channel for Simultaneous Wireless Information and Power Transfer
• A Global Approach for Solving Edge-Matching Puzzles
• The perturbation analysis of nonconvex low-rank matrix robust recovery
• Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets
• Rank Constrained Diffeomorphic Density Motion Estimation for Respiratory Correlated Computed Tomography
• Spectral Algorithm for Low-rank Multitask Regression
• Matrix recovery using Split Bregman
• Sparse recovery via nonconvex regularized $M$-estimators over ${\ell }_q$-balls
• Multiplicative Noise Removal: Nonlocal Low-Rank Model and Its Proximal Alternating Reweighted Minimization Algorithm
• Autocalibrating and Calibrationless Parallel Magnetic Resonance Imaging as a Bilinear Inverse Problem
• Tensor Principal Component Analysis via Convex Optimization
• Stable Blind Deconvolution over the Reals from Additional Autocorrelations
• Online Forecasting Matrix Factorization
• Efficient Algorithms for Positive Semi-Definite Total Least Squares Problems, Minimum Rank Problem and Correlation Matrix Computation
• Restricted Isometry Property of Subspace Projection Matrix Under Random Compression
• Solving ptychography with a convex relaxation
• SPRITE: A Response Model For Multiple Choice Testing
• Index Coding and Network Coding via Rank Minimization
• Information Theoretic Bounds for Low-Rank Matrix Completion
• Sampling and reconstructing signals from a union of linear subspaces
• Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensembles
• Low Tensor Train- and Low Multilinear Rank Approximations for De-speckling and Compression of 3D Optical Coherence Tomography Images
• Learning Data-adaptive Nonparametric Kernels
• Convex Formulation for Planted Quasi-Clique Recovery
• Relaxation algorithms for matrix completion, with applications to seismic travel-time data interpolation
• Enhancing the Spatio-Temporal Observability of Residential Loads
• Stochastic Iterative Hard Thresholding for Low-Tucker-Rank Tensor Recovery
• Convex Latent Effect Logit Model via Sparse and Low-rank Decomposition
• Clipped Matrix Completion: A Remedy for Ceiling Effects
• Equivalence and Strong Equivalence between Sparsest and Least ${\ell }_1$-Norm Nonnegative Solutions of Linear Systems and Their Application
• Blind Identification via Lifting
• Bundle methods for dual atomic pursuit
• Stochastic continuum armed bandit problem of few linear parameters in high dimensions
• Understanding Modern Techniques in Optimization: Frank-Wolfe, Nesterov's Momentum, and Polyak's Momentum
• Group-Structured Adversarial Training
• Minimum $n$-Rank Approximation via Iterative Hard Thresholding
• Efficient Rank Minimization via Solving Non-convexPenalties by Iterative Shrinkage-Thresholding Algorithm
• Lectures on Nonnegative Polynomials and Sums of Squares
• Theoretical and Experimental Analyses of Tensor-Based Regression and Classification
• A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery
• Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models
• Correlated Purification for Restoring $N$-Representability in Quantum Simulation
• Balancing Accuracy and Diversity in Recommendations using Matrix Completion Framework
• High-Dimensional Dynamic Systems Identification with Additional Constraints
• Generalized low rank approximation to the symmetric positive semidefinite matrix
• Sparse Optimization on General Atomic Sets: Greedy and Forward-Backward Algorithms
• A Sparsity Inducing Nuclear-Norm Estimator (SpINNEr) for Matrix-Variate Regression in Brain Connectivity Analysis
• Adaptive iterative singular value thresholding algorithm to low-rank matrix recovery
• Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery
• Constrained Shadow Tomography for Molecular Simulation on Quantum Devices
• Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices
• Identifiability Conditions for Multi-channel Blind Deconvolution with Short Filters
• Probabilistic Segmentation via Total Variation Regularization
• Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel
• Quantum Tomography by Regularized Linear Regression
• Blind Image Denoising via Dependent Dirichlet Process Tree
• Improved Algorithms for White-Box Adversarial Streams
• Pursuits in Structured Non-Convex Matrix Factorizations
• Provable Low Rank Phase Retrieval
• Error bounds for rank constrained optimization problems and applications
• Joint Space Decomposition-and-Synthesis Theory for K-User MIMO Channels: Interference Alignment and DoF Region
• Sketching sparse low-rank matrices with near-optimal sample- and time-complexity using message passing
• Sweep Distortion Removal from THz Images via Blind Demodulation
• Multiple Images Recovery Using a Single Affine Transformation
• A Non-Convex Relaxation for Fixed-Rank Approximation
• Sparse Functional Identification of Complex Cells from Spike Times and the Decoding of Visual Stimuli
• On the analysis of optimization with fixed-rank matrices: a quotient geometric view
• Matrix Factorization Method for Decentralized Recommender Systems
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization
• A Nonconvex Nonsmooth Regularization Method for Compressed Sensing and Low-Rank Matrix Completion
• Towards Understanding Generalization via Decomposing Excess Risk Dynamics
• A problem dependent analysis of SOCP algorithms in noisy compressed sensing
• Deep learned SVT: Unrolling singular value thresholding to obtain better MSE
• Bridging and Improving Theoretical and Computational Electric Impedance Tomography via Data Completion
• Kernel based low-rank sparse model for single image super-resolution
• A Parallel Best-Response Algorithm with Exact Line Search for Nonconvex Sparsity-Regularized Rank Minimization
• A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery
• Matrix completion based on Gaussian parameterized belief propagation
• Enhanced image approximation using shifted rank-1 reconstruction
• Optimal Sample Complexity for Stable Matrix Recovery
• On the subdifferential of symmetric convex functions of the spectrum for symmetric and orthogonally decomposable tensors
• Multifrequency interferometric imaging with intensity-only measurements
• Enhancing the Spatio-temporal Observability of Grid-Edge Resources in Distribution Grids
• Stable Manifold Embeddings with Structured Random Matrices
• Maximum a Posteriori Inference of Random Dot Product Graphs via Conic Programming
• Learning Semidefinite Regularizers
• Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance
• Low rank solutions to differentiable systems over matrices and applications
• Sharp Oracle Inequalities for Low-complexity Priors
• Tensor Completion via Convolutional Sparse Coding Regularization
• Defect Detection by MIMO Wireless Sensing based on Weighted Low-Rank plus Sparse Recovery
• Rank Restricted Semidefinite Matrices and Image Closedness
• Policy Search with High-Dimensional Context Variables
• Low-rank matrix recovery via rank one tight frame measurements
• GFB-MRF: A parallel spatial and Bloch manifold regularized iterative reconstruction method for MR Fingerprinting
• Infrared target tracking based on proximal robust principal component analysis method
• Paying down metadata debt: learning the representation of concepts using topic models
• Revisiting L21-norm Robustness with Vector Outlier Regularization
• Generalized singular value thresholding operator to affine matrix rank minimization problem
• A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery
• Painless Breakups -- Efficient Demixing of Low Rank Matrices
• Learning Mixtures of Low-Rank Models
• Speeding up finite-time consensus via minimal polynomial of a weighted graph - a numerical approach
• A Low-rank Spline Approximation of Planar Domains
• Convex Total Least Squares
• Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction
• $k$FW: A Frank-Wolfe style algorithm with stronger subproblem oracles
• Implicit Convex Regularizers of CNN Architectures: Convex Optimization of Two- and Three-Layer Networks in Polynomial Time
• Lower Memory Oblivious (Tensor) Subspace Embeddings with Fewer Random Bits: Modewise Methods for Least Squares
• Estimating Traffic and Anomaly Maps via Network Tomography
• Efficient MCMC Sampling for Bayesian Matrix Factorization by Breaking Posterior Symmetries
• Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression
• Upper Bounds on the Error of Sparse Vector and Low-Rank Matrix Recovery
• Nonparametric Estimation of Low Rank Matrix Valued Function
• On the construction of general cubature formula by flat extensions
• Illumination strategies for intensity-only imaging
• A Sequential Subspace Method for Millimeter Wave MIMO Channel Estimation
• Error bound of critical points and KL property of exponent $1/2$ for squared F-norm regularized factorization
• Learning Parameters for Weighted Matrix Completion via Empirical Estimation
• High-dimensional covariance matrix estimation using a low-rank and diagonal decomposition
• The Minimum-Rank Gram Matrix Completion via Modified Fixed Point Continuation Method
• Efficient nuclear norm approximation via the randomized UTV algorithm
• Solving the Robust Matrix Completion Problem via a System of Nonlinear Equations
• Enhancing low-rank solutions in semidefinite relaxations of Boolean quadratic problems
• A New Nonconvex Strategy to Affine Matrix Rank Minimization Problem
• A Convex Approach to Frisch-Kalman Problem
• An analysis of noise folding for low-rank matrix recovery
• Image Segmentation Using Subspace Representation and Sparse Decomposition
• Stop memorizing: A data-dependent regularization framework for intrinsic pattern learning
• Leveraging Subspace Information for Low-Rank Matrix Reconstruction
• The Stability of Low-Rank Matrix Reconstruction: a Constrained Singular Value View
• Provably convergent acceleration in factored gradient descent with applications in matrix sensing
• A Convergent Semi-Proximal Alternating Direction Method of Multipliers for Recovering Internet Traffics from Link Measurements
• Online Optimization for Large-Scale Max-Norm Regularization
• Exact Sparse Orthogonal Dictionary Learning
• OAAE: Adversarial Autoencoders for Novelty Detection in Multi-modal Normality Case via Orthogonalized Latent Space
• Factor Analysis of Moving Average Processes
• A Comparison of Clustering and Missing Data Methods for Health Sciences
• Low-M-Rank Tensor Completion and Robust Tensor PCA
• Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion
• Speaker diarization with session-level speaker embedding refinement using graph neural networks
• Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets
• Riemannian Conjugate Gradient Descent Method for Third-Order Tensor Completion
• Discrete-Aware Matrix Completion via Proximal Gradient
• Efficient Optimization Algorithms for Robust Principal Component Analysis and Its Variants
• Multi-weight Matrix Completion with Arbitrary Subspace Prior Information
• Alternating Energy Minimization Methods for Multi-term Matrix Equations
• Subspace Change-Point Detection via Low-Rank Matrix Factorisation
• Performance analysis of weighted low rank model with sparse image histograms for face recognition under lowlevel illumination and occlusion
• Complexity Aspects of Fundamental Questions in Polynomial Optimization
• Multi-Tensor Network Representation for High-Order Tensor Completion