We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadrat…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “matrix condition number”
Improved perturbation reduces matrix condition number to O(n) with minimal storage.
problem Reducing the condition number of deterministic matrices for efficient algorithmic use.
method Introduced pattern matrices and sparse perturbations with dependent entries.
result Condition number reduced to O(n) with O(n) random numbers in O(log n) precision.
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.
Improved convergence for overparameterized low-rank matrix sensing.
problem Overparameterized low-rank matrix sensing with unknown rank and ill-conditioning.
method ScaledGD() - preconditioned gradient descent method.
result ScaledGD() converges at a constant linear rate after a logarithmic number of iterations.
We analyze the condition number of random feature matrices and prove their well-conditioned nature.
problem Understanding the condition number of random feature matrices and its impact on generalization error.
method Established concentration bounds and derived risk bounds for regression problems using random feature matrices.
result The risk associated with random feature matrices exhibits the double descent phenomenon, improving even with noise.
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.
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.
This paper optimizes diagonal preconditioning to improve matrix condition numbers.
problem Optimizing diagonal preconditioning to reduce matrix condition numbers.
method Reformulated as a quasi-convex problem, solved with bisection and Newton updates.
result Optimal diagonal preconditioners can significantly improve iterative methods.
The stability and robustness of compact schemes for parabolic PDEs are analyzed.
problem Stability and robustness of compact schemes for solving parabolic PDEs.
method Compact spatial discretization, Crank-Nicolson temporal discretization, eigenvalue analysis of amplification matrix.
result An upper bound on the condition number of the amplification matrix is derived, showing stability.
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…
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…
Scalable method completes ill-conditioned matrices from few samples.
problem Matrix completion from few samples for ill-conditioned matrices.
method Iterative algorithm combining IRLS, smoothing Newton, and proximal gradient methods.
result Local quadratic convergence rate and well-conditioned linear systems.
Scaled gradient descent improves matrix recovery for ill-conditioned matrices with optimal sampling complexity.
problem Recovering low-rank matrices from limited measurements efficiently and accurately.
method Scaled gradient descent (ScaledGD) with optimal sample complexity and improved iteration complexity.
result ScaledGD achieves optimal sample complexity and improved iteration complexity for ill-conditioned matrices.
Method selects number of communities in weighted networks.
problem Selecting the number of communities in weighted networks.
method Proposes a novel weighted DCSBM and uses a sequential testing framework with spectral clustering and matrix scaling.
result Method is consistent in estimating the true number of communities under mild conditions.
Investigates portfolio optimization with and without gearing constraints.
problem Improving portfolio weights for better alignment with expected returns.
method Extends the alpha-weight angle bound to include gearing constraints and uses theoretical arguments and simulations.
result Equally weighted portfolios are not preferable to mean-variance portfolios even with poor forecast ability and a badly conditioned covariance matrix.
Many modern statistical applications ask for the estimation of a covariance (or precision) matrix in settings where the number of variables is larger than the number of observations. There exists a broad class of ridge-type estimators that employs regularization to cope with the subsequent singularity of the sample cov…
This paper improves linear system solving by optimizing matrix diagonal scaling.
problem Improving the condition number of a matrix for faster iterative methods.
method Left or right diagonal rescaling of the matrix A, with new bounds and algorithms.
result Jacobi preconditioning reduces A's condition number to within a quadratic factor of the best possible scaling.
The study characterizes the conditioning of the Gauss-Newton matrix in neural networks.
problem Understanding the conditioning of the Gauss-Newton matrix in neural networks.
method Theoretical analysis of the GN matrix in deep linear and ReLU networks, extending to residual connections and convolutional layers.
result Established tight bounds on the condition number of the GN matrix in neural networks.
Provably convergent acceleration in factored gradient descent with applications in matrix sensingcs.LG
We present theoretical results on the convergence of \emph{non-convex} accelerated gradient descent in matrix factorization models with -norm loss. The purpose of this work is to study the effects of acceleration in non-convex settings, where provable convergence with acceleration should not be considered a \em…
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With random observations of a $n_1 \times n…
Recently, Neural networks have seen a huge surge in its adoption due to their ability to provide high accuracy on various tasks. On the other hand, the existence of adversarial examples have raised suspicions regarding the generalization capabilities of neural networks. In this work, we focus on the weight matrix learn…
The study proposes a method for risk reduction without relying on risk measurement.
problem Theoretical utopia of risk minimization vs. practical risk reduction.
method Generalization of matrix rank and condition number for identifying riskiest scenarios.
result Risk reduction achieved without risk measurement, validated by real data.
We consider the following signal recovery problem: given a measurement matrix and a noisy observation vector constructed from where is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover …
High-dimensional settings, where the data dimension () far exceeds the number of observations (), are common in many statistical and machine learning applications. Methods based on -relaxation, such as Lasso, are very popular for sparse recovery in these settings. Restricted Eigenvalue (RE) condition is a…
A new method for robust matrix completion overcomes limitations of existing approaches.
problem Robust matrix completion from corrupted entries, especially under overparameterization and ill-conditioning.
method Factorization-based iterative algorithm combining Gauss-Newton linearization and outlier removal.
result Theoretical guarantees of exact recovery for suitable assumptions.
In this paper, we study the general problem of optimizing a convex function over the set of matrices, subject to rank constraints on . However, existing first-order methods for solving such problems either are too slow to converge, or require multiple invocations of singular value decompositions.…
Preconditioned SGD accelerates convergence for ill-conditioned huge-scale matrix completion.
problem Recovering a low-rank matrix from incomplete data with high condition number.
method Preconditioned Stochastic Gradient Descent (SGD) for huge-scale online optimization.
result Preconditioned SGD converges to ε-accuracy in O(log(1/ε)) iterations, compared to O(κlog(1/ε)) for unpreconditioned SGD.
Multivariate regression model is a natural generalization of the classical univari- ate regression model for fitting multiple responses. In this paper, we propose a high- dimensional multivariate conditional regression model for constructing sparse estimates of the multivariate regression coefficient matrix that accoun…
AMP algorithm for matrix tensor product model provides recovery conditions.
problem Generalization of standard spiked matrix models with multiple pairwise observations.
method Approximate message passing with optimal weighing and combining of estimates.
result Asymptotically exact performance description and necessary/sufficient recovery conditions.
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-…
Paper introduces a differentiable regularizer for condition number to improve neural network stability.
problem Maintaining numerical stability in neural networks to ensure reliable and performant models.
method Introduces a novel differentiable regularizer for the condition number of weight matrices.
result Derives a differentiable formula for the gradient of the regularizer, promoting matrices with low condition numbers.
In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.
We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several e…
New algorithms achieve logarithmic regret in learning linear quadratic control systems.
problem Learning in Linear Quadratic Control systems with unknown parameters.
method Efficient algorithms for two scenarios: unknown or with certain conditions.
result Regret scales logarithmically with the number of steps, not square root.
Study shows how leveraging hierarchical similarity graphs improves matrix completion in recommender systems.
problem Improving matrix completion in recommender systems using hierarchical similarity graphs.
method Characterizes the optimal sample complexity using hierarchical stochastic block models and low-rank rating matrices.
result Exploiting hierarchical structure of social graphs significantly reduces the number of observed entries needed for accurate matrix completion.
SDP approach recovers communities in multilayer hypergraphs from aggregated similarity matrices.
problem Community recovery in multilayer hypergraphs using aggregated similarity matrices.
method Semidefinite programming (SDP) approach.
result Information-theoretic conditions for exact recovery in both assortative and disassortative cases.
Improved covariance matrix estimation for portfolio optimization with guaranteed PSD and controlled conditioning.
problem Guaranteeing positive semidefinite ness and controlling spectral conditioning in IQ estimators.
method Introducing squeezing identity and atomic-IQ parameterization to construct structured channel matrices with PSD guarantees and analytic eigen floor for conditioning control.
result Atomic-IQ improves Sharpe ratios and delivers a more stable risk profile compared to standard estimators.
New methods improve solving linear systems and preconditioning with reduced complexity.
problem Efficiently solving linear systems and preconditioning matrices.
method Developed structured semidefinite programming algorithms.
result Improved runtimes for preconditioning and solving linear systems.
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
A Bayesian Boolean Matrix Factorization for cancer genomics
problem Identifying coordinated feature changes in cancer
method Bayesian Boolean Matrix Factorization
result Captures widespread, near-simultaneous chromosome-number changes
New method improves efficiency of non-convex matrix reconstruction.
problem Reconstructing a low-rank matrix from few linear measurements.
method Factorized gradient descent with spectral initialization.
result Non-convex approaches can match the sample complexity of convex methods.
GD and NAG accelerate matrix factorization and neural networks.
problem Optimizing rectangular matrix factorization and linear neural networks.
method Gradient descent and Nesterov's accelerated gradient with specific initialization.
result NAG achieves the best-known iteration complexity for these problems.
Method reduces categorical data to lower dimensions using density matrices.
problem Dimensionality reduction for categorical data.
method Density-matrix construction from class-conditional frequencies; spectral embedding.
result Low-dimensional spectral embeddings with controlled rank.
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this problem. However, the sample complexity in all these results is sub-optimal in it…
New algorithms for SSMF with weaker identifiability conditions than SSC.
problem Identifying unique decompositions in simplex-structured matrix factorization.
method Extracting facets containing the largest number of points to ensure identifiability.
result Our algorithms recover unique decompositions under weaker conditions than SSC.
The paper tackles matrix completion in ultra-sparse sampling, improving imputation accuracy.
problem Matrix completion in ultra-sparse sampling, where each row has only a few entries.
method Estimate row span of matrix or averaged second-moment matrix, normalize and impute missing entries.
result Gradient descent method normalizes and imputes missing entries, achieving low variance and unbiased estimation.
A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In…