arXiv research

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and $\mathcal{R}$-matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{…

Researchers develop geodesics for a new metric on correlation matrices.

Estimating fundamental matrices is a classic problem in computer vision. Traditional methods rely heavily on the correctness of estimated key-point correspondences, which can be noisy and unreliable. As a result, it is difficult for these methods to handle image pairs with large occlusion or significantly different cam…

Study embeds PC matrices into Grassmannian manifold for geometric interpretation.

Efficiently approximates eigenspaces for symmetric and general matrices.

Study optimizes shared singular subspace estimation from noisy matrices.

New metric tensor field on symmetric matrices simplifies eigenvector computation.

Complex systems are typically represented by large ensembles of observations. Correlation matrices provide an efficient formal framework to extract information from such multivariate ensembles and identify in a quantifiable way patterns of activity that are reproducible with statistically significant frequency compared…

New technique stabilizes singular values in concatenated matrices.

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions. In this way, we show phenomenons of duality between Subman…

The paper presents two schemes for sampling matrices from specific distributions on a manifold.

Analyzes tt*-structures from $ADE$-type Stokes data.

If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equa…

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundament…

Direct proof of Alexander polynomial scaling for L-shaped representations.

The paper tackles joint learning of linear systems, improving accuracy with pooled data.

Proteins are the major building blocks of life, and actuators of almost all chemical and biophysical events in living organisms. Their native structures in turn enable their biological functions which have a fundamental role in drug design. This motivates predicting the structure of a protein from its sequence of amino…

The fundamental group of every surface that is not the projective plane or Klein bottle has a representation to a torsion-free group of upper-triangular matrices in SL(2,R) with no simple loop (i.e. a nontrivial element representing a simple closed curve) in the kernel.

This paper analyzes AJIVE for estimating shared subspace across multiple datasets, revealing its strengths and limitations.

New algorithm optimizes matrix reordering for noisy disordered matrices.

Spectral regularization simplifies sequence models by focusing on grammatical simplicity.

Novel algorithm speeds up log-determinant estimation for large matrices.

Study heavy-tailed weights' impact on neural network's spectral distribution.

The paper studies matrix normalization and graph balancing using a new functional and gradient descent.

GeoHNN models physics laws for stable, accurate predictions.

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian s…

In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries,…

We give an explicit algorithm and source code for constructing risk models based on machine learning techniques. The resultant covariance matrices are not factor models. Based on empirical backtests, we compare the performance of these machine learning risk models to other constructions, including statistical risk mode…

A new algorithm speeds up matrix multiplication without actual multiplication.

This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the charac…

Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an $n\times n$ SDDM matrix ${\bf \mathbf{M}}$, and a constant $-1 \leq p \leq 1$, our algorithm gives efficient access to a…

According to recent findings [1,2], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [1], e.g., it is reported that about 94% of th…

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to…

Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to recover…

A new mechanism for differentially private Fréchet mean on SPD matrices.

Enhances power of covariance matrix tests for high-dimensional data.

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…

Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distrib…

The abstract discusses resurgent functions in quantum knot invariants.

In this paper, we examine the problem of approximating a general linear dimensionality reduction (LDR) operator, represented as a matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$, by a partial circulant matrix with rows related by circular shifts. Partial circulant matrices admit fast implementations via Fourier tra…

Graph energy helps detect communities in networks better than traditional methods.

New tools in nonlinear random matrices improve understanding of the Sum of Squares hierarchy.

Develops PRPCA for smooth image recovery combining low-rank and smoothness.

Simple matrix formulas for Grassmannian curvatures.

Random representations of surface groups approach asymptotic freeness in large $n$ limit.

Paper finds a lower bound for estimating low-rank matrices in logistic regression.