Paper analyzes structured matrix recovery using generalized Dantzig selector.
problem Structured matrix recovery for applications like recommender systems and computer vision.
method Non-asymptotic analysis of generalized Dantzig selector for estimation of generally structured matrices.
result Estimation error can be expressed in terms of geometric measures of suitable sets.
Study reveals efficient recovery of multi-modal signals via Bayesian methods and sequential learning.
problem Recovering multiple high-dimensional signals from correlated modalities.
method Bayesian Approximate Message Passing and Sequential Curriculum Learning.
result Sequential learning strategy optimally recovers weak signals in multi-modal settings.
HSNLD solves robust Hankel recovery efficiently and robustly.
problem Robust Hankel recovery of sparse outliers and missing entries.
method Hankel Structured Newton-Like Descent (HSNLD) algorithm.
result HSNLD achieves linear convergence independent of the condition number.
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.
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.
Survey of structured low-rank algorithms for MR signal recovery.
problem Recovering multidimensional signals from few non-uniform measurements.
method Structured low-rank matrix completion formulation.
result Performance guarantees and fast algorithms for large-scale MR problems.
New method recovers matrices with nonlinear structures using optimization on Grassmann manifold.
problem Recovering high-rank matrices with nonlinear structures like subspaces or clusters.
method Formulated as rank minimization of a nonlinear feature map, approximated by constrained non-convex optimization on the Grassmann manifold, using Riemannian and alternating minimization schemes.
result Global convergence and worst-case complexity bounds for alternating minimization scheme, leading to unique limit point.
GAME improves matrix completion by considering subgroup-specific latent structures.
problem Heterogeneous data with overlapping categories, smoothing away subgroup-specific variation.
method Group-Aware Matrix Estimation (GAME) with overlapping nuclear-norm penalties.
result GAME outperforms global low-rank estimators in structured missingness regimes.
Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.
problem Recovering a rank-one signal matrix from noisy data with additional prior information.
method Analysis of a nonlinear least squares objective with a favorable global optimization landscape.
result Established optimal sample complexity for generative priors in rank-one matrix recovery.
The study uses Random Matrix Theory to identify structural changes in stock markets during shocks.
problem Understanding structural changes in stock markets during exogenous shocks.
method Random Matrix Theory and complexity gap analysis.
result The complexity gap collapses during shocks, indicating strong synchronization, and widens before shocks, signaling a rich structure.
For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation matrix completion in various fields such as the financial market and the low-ra…
New framework inscribes maximum volume ellipsoid for structured matrix factorization.
problem Structured matrix factorization with columns in unit simplex.
method Maximum volume inscribed ellipsoid (MVIE) via facet enumeration and convex optimization.
result MVIE framework guarantees exact recovery under certain conditions.
Matrix completion has attracted significant recent attention in many fields including statistics, applied mathematics and electrical engineering. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by a…
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.
Paper refines null space conditions for nuclear norm minimization in low-rank matrix recovery.
problem Establishing conditions for successful nuclear norm minimization recovery of low-rank matrices.
method Developed new null space conditions for nuclear norm minimization, proving their necessity and sufficiency.
result Weak null space condition is sufficient but not necessary for nuclear norm minimization recovery, providing a new necessary and sufficient condition.
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.
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.
Global optimization for low-rank matrix recovery from noisy measurements.
problem Low-rank matrix recovery from noisy measurements.
method Factorized parametrization, curvature bound, stochastic gradient descent.
result Global convergence guarantee for stochastic gradient descent from random initialization.
This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when the natura…
StrTransformer recovers sources without labels by optimizing latent matrices and enforcing structural constraints.
problem Unsupervised blind source recovery in signal processing.
method Source-wise structured Transformer framework with latent source matrix optimization, structural regularization, and branch-specific weights.
result StrTransformer learns distinct temporal-scale structures and recovers source-aligned latent trajectories.
The paper tackles dynamic matrix recovery from incomplete data, improving accuracy and sample complexity.
problem Improving matrix recovery from incomplete data using dynamic constraints.
method Proposes LOWEMS framework for dynamic matrix recovery and establishes error bounds.
result Establishes error bounds for LOWEMS in matrix sensing and completion models, quantifying benefits of dynamic constraints.
New criterion ensures recovery of latent factors in NMF with mild conditions.
problem Identifying latent factors in nonnegative matrix factorization (NMF) under mild conditions.
method Proposed a new identification criterion based on the scatteredness of one factor's rows in the nonnegative orthant.
result Latent factors can be provably identified from the NMF model with minimal structural assumptions.
We consider the problem of signal recovery on graphs as graphs model data with complex structure as signals on a graph. Graph signal recovery implies recovery of one or multiple smooth graph signals from noisy, corrupted, or incomplete measurements. We propose a graph signal model and formulate signal recovery as a cor…
New method avoids spurious critical points for low-rank matrix recovery.
problem Low-rank matrix recovery problems on Riemannian manifold.
method Riemannian gradient descent with random initialization.
result Riemannian gradient descent avoids spurious critical points and converges nearly linearly.
Paper proves conditions for nonconvex matrix recovery to avoid spurious local minima.
problem Ensuring no spurious local minima in nonconvex matrix recovery.
method Sharp restricted isometry bounds proof technique.
result RIP constant of δ < 1/2 is necessary and sufficient for exact recovery.
Sign-RIP improves robust low-rank matrix recovery by preserving norms even with corrupted measurements.
problem Robust low-rank matrix recovery in the presence of corrupted measurements.
method Proposed Sign-RIP, a robust restricted isometry property.
result Sign-RIP guarantees uniform convergence of subdifferentials in robust low-rank matrix recovery.
Proposes a new tensor grid method for image completion.
problem Image completion from missing data.
method Low-rank tensor grid with two-stage density matrix renormalization group initialization and alternating least squares factorization.
result The proposed tensor grid method outperforms existing methods in image recovery accuracy.
Improved stability for matrix recovery from rank-one measurements.
problem Phase retrieval problem of recovering rank-one positive semidefinite matrices.
method Developed a smoothing Newton method based on Bures-Wasserstein gradient descent.
result Superlinear convergence with rigorous guarantees and stable implementation.
Paper proposes a new matrix recovery method relaxing uniform sampling assumptions.
problem Matrix completion under arbitrary sampling schemes.
method Max-norm and nuclear-norm regularization, alternating direction method of multipliers.
result The proposed method achieves fast rates of convergence and is computationally efficient.
Paper shows moderate RIP is insufficient for avoiding spurious local minima in matrix recovery.
problem The need for moderate RIP to avoid spurious local minima in matrix recovery.
method Analyzes the necessity of RIP constants and provides counterexamples.
result Counterexamples show spurious local minima exist even with moderate RIP.
New algorithm speeds up recovery of low-rank matrices.
problem Nonconvex low-rank matrix recovery problems.
method Stochastic variance-reduced gradient descent with semi-stochastic gradient.
result Linear convergence rate to unknown low-rank matrix.
The paper explores the problem of \emph{spectral compressed sensing}, which aims to recover a spectrally sparse signal from a small random subset of its n time domain samples. The signal of interest is assumed to be a superposition of r multi-dimensional complex sinusoids, while the underlying frequencies can assum…
Study robust recovery of low-rank matrices from corrupted measurements without rank prior.
problem Robust recovery of low-rank matrices from corrupted Gaussian measurements with unknown rank.
method Subgradient method with diminishing stepsizes for nonconvex nonsmooth problem.
result Subgradient method converges to exact low-rank solution at sublinear rate under RDPP condition.
This paper describes a simple framework for structured sparse recovery based on convex optimization. We show that many structured sparsity models can be naturally represented by linear matrix inequalities on the support of the unknown parameters, where the constraint matrix has a totally unimodular (TU) structure. For …
Extends IBP for non-diagonal latent covariance structures, improving feature recovery and denoising.
problem Modeling latent features with smoothness characteristics.
method Extend Indian Buffet Process to include non-diagonal latent covariance structures.
result Smoothness prior improves feature recovery and denoising under appropriate conditions.
New method recovers matrix column space with active sampling for better results.
problem Recovering column space of partially observed matrices with limited data.
method Alternating minimization with active sampling strategy.
result Active sampling improves convergence to true column space with higher probability.
3d-SMRnet speeds up MPI system matrix recovery to 1 minute with high quality.
problem Slow system matrix recovery in MPI due to recalibration.
method 3d-System Matrix Recovery Network using deep learning.
result 3d-SMRnet recovers 3d system matrix with 64x subsampling in 1 minute.
Researchers use shape analysis to recover protein structures from Cryo-EM data.
problem Recovering the three-dimensional backbone structure of single polypeptide proteins from noisy tomographic projections.
method Shape analysis and matrix Lie group actions to deform point clouds to match 2D tomography data.
result Optimal deformations are computed to recover the three-dimensional backbone structure of proteins.
We recover matrix networks from incomplete observations using low-rank graph Fourier transform.
problem Recovering a partially observed matrix network from incomplete observations.
method We propose a convex optimization problem with a structural assumption of low-rank graph Fourier transform. We prove an exact recovery guarantee and provide an iterative imputation algorithm.
result We discover a new phase transition phenomenon and demonstrate the algorithm's effectiveness on large-scale matrix networks.
The paper analyzes sparse PCA for incomplete data and proves support recovery conditions.
problem Support recovery in sparse PCA with non-random missing data.
method Semidefinite relaxation of the ℓ1-regularized PCA problem. result Support of the sparse leading eigenvector can be recovered with high probability.
The problem of finding the missing values of a matrix given a few of its entries, called matrix completion, has gathered a lot of attention in the recent years. Although the problem under the standard low rank assumption is NP-hard, Candès and Recht showed that it can be exactly relaxed if the number of observed entrie…
New algorithms recover low-rank matrices from few noisy projections.
problem Estimating low-rank matrices from rank-one projections with noise.
method Two fast, non-convex algorithms for matrix recovery.
result Proposed algorithms achieve linear convergence and independent sample complexity of condition number.
Paper recovers multi-subspace matrices from permuted data.
problem Recovering a multi-subspace matrix from permuted data with corrupted columns.
method Four-stage algorithm pipeline: outlier identification, subspace reconstruction, outlier classification, unsupervised sensing.
result The pipeline provides theoretical guarantees for reliable multi-subspace matrix recovery.
We introduce a general framework to handle structured models (sparse and block-sparse with possibly overlapping blocks). We discuss new methods for their recovery from incomplete observation, corrupted with deterministic and stochastic noise, using block-ℓ1 regularization. While the current theory provides promis…
A hierarchical model shows how scaling laws emerge from sequential feature recovery.
problem Emergence of scaling laws from feature learning in multi-layer networks.
method Layer-wise spectral algorithm adapted to compositional structure, sequential feature detection.
result Sequential detection of latent features, leading to explicit power-law decay of prediction error.
The paper studies the problem of recovering a spectrally sparse object from a small number of time domain samples. Specifically, the object of interest with ambient dimension n is assumed to be a mixture of r complex multi-dimensional sinusoids, while the underlying frequencies can assume any value in the unit disk…
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.
New algorithm recovers matrices with unknown correspondences.
problem Recovering matrices from observations with unknown correspondences.
method Solves a nuclear norm minimization problem via proximal gradient with a Max-Oracle.
result Achieves state-of-the-art performance and high accuracy in recovering ground-truth correspondences.