arXiv research

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

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.

Convex optimization method recovers low-rank matrices from rank-one projections efficiently.

Improved stability for matrix recovery from rank-one measurements.

Stochastic Rank-One Bandits (Katarya et al, (2017a,b)) are a simple framework for regret minimization problems over rank-one matrices of arms. The initially proposed algorithms are proved to have logarithmic regret, but do not match the existing lower bound for this problem. We close this gap by first proving that rank…

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…

We propose a geometric assumption on nonnegative data matrices such that under this assumption, we are able to provide upper bounds (both deterministic and probabilistic) on the relative error of nonnegative matrix factorization (NMF). The algorithm we propose first uses the geometric assumption to obtain an exact clus…

This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.

Estimates the probability of a random symmetric tensor being close to rank-one.

AMP method reconstructs rank-one matrices from noisy data efficiently.

A Kleinian manifold Y is a quotient of a rank-one symmetric space of non-compact type by a convex-cocompact discrete group of isometries. We describe the spectral decomposition of the space of square integrable sections of locally homogeneous bundles on Y with respect to locally invariant differential operators. In the…

Study on estimating rank-one tensors in noisy data with heavy tails.

Study on signal-plus-noise decomposition in nonlinear spiked random matrices.

The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.

We consider worker skill estimation for the single-coin Dawid-Skene crowdsourcing model. In practice, skill-estimation is challenging because worker assignments are sparse and irregular due to the arbitrary and uncontrolled availability of workers. We formulate skill estimation as a rank-one correlation-matrix completi…

Improves signal detection in non-Gaussian noise using transformed data.

New analysis shows how attention masks and LayerNorm prevent rank collapse in transformers.

We formulate and solve a tensor model using a latent-variable approach.

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does no…

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic acc…

New method solves matrix completion problems to certifiable optimality.

We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor …

Extended Rank-One Theorem to special metric spaces.

We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless…

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

Study detects signals in spiked Wigner models using log likelihood ratio.

Constructs explicit p-harmonic functions on specific Lie groups.

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for Hilbert geometries and characterize them as being equivalent to contracting elem…

The paper reveals low-rank structure in neural network gradients, influenced by data and model parameters.

The study establishes uncertainty principles on harmonic manifolds of rank one.

Classifies foliations on specific symmetric spaces.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the jo…

New methods estimate covariance for matrix data without assuming fixed size or specific distributions.

Classifies rank-one submanifolds in Euclidean space.

New model enhances SPIM for solving low-rank combinatorial optimization and statistical learning problems.

Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.

No Einstein hypersurfaces found in Damek-Ricci spaces.

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study th…

New insights into compact rank-one ECS manifolds, proving they are bundles over circles.

Compact rank one symmetric spaces are rigid under certain curvature conditions.

Study closed manifolds with rank one ray structures, proving completeness or covering properties.

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

In recent years, a class of dictionaries have been proposed for multidimensional (tensor) data representation that exploit the structure of tensor data by imposing a Kronecker structure on the dictionary underlying the data. In this work, a novel algorithm called "STARK" is provided to learn Kronecker structured dictio…

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

Researchers describe the metric structure of compact ECS manifolds.