arXiv research

New algorithm approximates large psd matrices from sketches.

New inequalities for matrix supermartingales converge under various conditions.

Study nonconvex matrix completion for low-rank approximation without rank assumptions.

This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatil…

New PSDMF algorithms derived from PR and ARM methods.

The paper introduces new processes for modeling multivariate volatility.

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial …

Introduces a new model for mapping matrices to matrices, subsuming linear regression.

An algorithm for computing positive semidefinite factorizations of matrices.

Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, have enjoye…

A fast method estimates correlations in hybrid systems using observable market data.

A new matrix concentration inequality for random products of matrices.

Denise learns a function to quickly decompose covariance matrices robustly.

Global stability bounds for matrix frames in phase retrieval problems.

Paper analyzes convergence of distributed inference using BP in linear Gaussian models.

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 $O( μr^2 κ^2 n \max(μ, \log n))$ random observations of a $n_1 \times n…

Improved covariance matrix estimation for portfolio optimization with guaranteed PSD and controlled conditioning.

Accelerated RPCholesky speeds up kernel matrix approximations.

The paradigm of multi-task learning is that one can achieve better generalization by learning tasks jointly and thus exploiting the similarity between the tasks rather than learning them independently of each other. While previously the relationship between tasks had to be user-defined in the form of an output kernel, …

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-speci…

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…

Improved stability for matrix recovery from rank-one measurements.

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…

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its …

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r^3 κ^2 n \log n)$ random measurements of a positive semidefinite $n \times n$ matrix of rank $r$ and condition number $κ$…

We propose a new input perturbation mechanism for publishing a covariance matrix to achieve $(ε,0)$-differential privacy. Our mechanism uses a Wishart distribution to generate matrix noise. In particular, We apply this mechanism to principal component analysis. Our mechanism is able to keep the positive semi-definitene…

We present a hybrid algorithm for optimizing a convex, smooth function over the cone of positive semidefinite matrices. Our algorithm converges to the global optimal solution and can be used to solve general large-scale semidefinite programs and hence can be readily applied to a variety of machine learning problems. We…

Most existing word embedding methods can be categorized into Neural Embedding Models and Matrix Factorization (MF)-based methods. However some models are opaque to probabilistic interpretation, and MF-based methods, typically solved using Singular Value Decomposition (SVD), may incur loss of corpus information. In addi…

RPCholesky approximates kernel matrices with few evaluations.

Improved guarantees for nonconvex matrix factorization with rank overparameterization.

New geometric framework for positive semidefinite matrices of fixed rank.

Improved gradient descent for rectangular matrix completion without $\ell_{2,\infty}$ regularization.

Gradient descent implicitly regularizes over-parameterized matrix factorization and neural networks with quadratic activations.

Paper tackles multi-label learning by improving SVR for positive semidefinite metrics.

New methods for sketching non-PSD matrices improve regression and optimization tasks.

Rank-one measurements limit feasible sets for low-rank PSD matrices.

A new test statistic measures discrepancy between conditional distributions.

Efficient private matrix analysis algorithms for recent variants.

Unified approach for learning quantum operations from measurements.

CDP reduces point cloud dimensions by preserving detour-induced local non-convexity.

Mirror descent algorithm recovers low-rank matrices in matrix sensing.

Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.

Semidefinite tests detect latent causal structures efficiently.

This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence paramete…

Paper finds formulas for mutual information and MMSE in matrix tensor product problems.

We solve a challenging factor analysis problem using ML principle and scalable algorithms.

Wider neural networks have predominantly positive curvature, aiding optimization.

Generates correlation matrices with specific graph structures using convex optimization.