arXiv research

Constructs orthogonal coordinates in curved spaces.

Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We u…

Polynomial time algorithm matches correlated Gaussian matrices without vanishing correlation.

The paper studies minimal 2-spheres of constant curvature in complex hyperquadrics using matrix theory.

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the $\ell_{2}$-prediction error and the $\ell_{1}$-risk of the lasso and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus we can estimate a high-dim…

This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics a…

In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\e…

We analyze the condition number of random feature matrices and prove their well-conditioned nature.

We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of s…

The paper uses Betti curves to confirm hyperbolic geometry in brain, climate, and financial networks.

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…

Develops log-Euclidean Lie groups for SPD and correlation matrices.

The paper addresses portfolio allocation with uncertain covariance matrices, finding a logarithmic risk dependence.

New structures on symplectic manifolds derived from convex functions and matrices.

Analog arrays are a promising upcoming hardware technology with the potential to drastically speed up deep learning. Their main advantage is that they compute matrix-vector products in constant time, irrespective of the size of the matrix. However, early convolution layers in ConvNets map very unfavorably onto analog a…

New method tightens Lipschitz bounds for CNNs efficiently.

New GMM models fit high-dimensional data with fewer parameters.

New estimators reduce computation for Kendall's tau and conditional Kendall's tau matrices under structural assumptions.

This note improves correlation stress tests using geodesic distance.

Constructs finite element spaces for $(p,q)$-forms, excluding one subspace.

Sharp inequalities for matrix means with unknown variance.

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove …

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networ…

Improved concentration inequalities for sub-Weibull variables enhance statistical and machine learning applications.

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…

Unified determinants via a single equation.

This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for…

New algorithm reduces matrix multiplication time for sparse matrices.

New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Ri…

Deviance-style normalization for sparse, jointly overdispersed count matrices

InQMAD detects anomalies in streaming data using quantum measurements and density matrices.

We propose an L-BFGS optimization algorithm on Riemannian manifolds using minibatched stochastic variance reduction techniques for fast convergence with constant step sizes, without resorting to linesearch methods designed to satisfy Wolfe conditions. We provide a new convergence proof for strongly convex functions wit…

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which cover all elliptic, overdetermined elliptic, subelliptic and parabolic…

The paper calculates bounds on the local Lipschitz constants of neural network layers.

Paper tackles robust graph matching in dense graphs with AMP type algorithm.

Estimates matrix trace optimization with statistical learning theory.

Study finds the minimum number of finite Gaussian mixtures for best approximation.

Random surfaces have a strong spectral gap with polynomial rate.

This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse v…

New algorithms learn graph structures privately, matching best results.

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

Kernel matrix concentration leads to KSC consistency.

This paper tightens bounds on the smallest eigenvalue of NTK for deep ReLU networks.

This paper investigates the average-case time complexity of certifying RIP matrices.

Extends covariance estimation with multiple targets for better performance.

Many problems in computer vision and recommender systems involve low-rank matrices. In this work, we study the problem of finding the maximum entry of a stochastic low-rank matrix from sequential observations. At each step, a learning agent chooses pairs of row and column arms, and receives the noisy product of their l…