arXiv research

Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.

The paper constructs Goeritz matrices from Dehn colorings.

Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. Fo…

Classification of SL(n) covariant valuations on Orlicz spaces.

We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.

New method recovers matrix column space with active sampling for better results.

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…

Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However, these techniques require tensor decomposition which could lead to the loss of infor…

Robust method learns nonlinear structures robustly to noise.

In this paper, we consider the streaming memory-limited matrix completion problem when the observed entries are noisy versions of a small random fraction of the original entries. We are interested in scenarios where the matrix size is very large so the matrix is very hard to store and manipulate. Here, columns of the o…

A new method for unfolding histograms without matrix inversion.

Paper builds neural networks on matrix manifolds using gyrovector spaces.

Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

New method reduces computational cost for nonnegative low rank matrix approximation.

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…

We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense …

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.

Paper proposes new costs for learning multiple centers in MDNs.

We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.

In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and each remaining column is partially observed with uniform sampling. To recover the …

Paper analyzes AIRL in high-dimensional spaces using random matrix theory.

The study uses a ReLU network to discern geometric structure in data via the Data Information Matrix.

We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results …

Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to compute the inverse or inverse square root of a matrix whose size is quadratic of …

Bayesian deep learning avoids underfitting by projecting onto null space of generalized Gauss-Newton matrix.

We consider the problem of finding anomalies in high-dimensional data using popular PCA based anomaly scores. The naive algorithms for computing these scores explicitly compute the PCA of the covariance matrix which uses space quadratic in the dimensionality of the data. We give the first streaming algorithms that use …

Minimal submanifolds in matrix spaces proven for specific ranks.

We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion…

Method uses optimal transport to complete distributional matrices.

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

UA-LQE improves value function learning by selectively erasing uncertain entries in Q-matrix.

The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to…

Optimal transfer learning for missing not-at-random matrix completion using source data.

Improved perturbation reduces matrix condition number to O(n) with minimal storage.

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…

Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets…

A new method for state estimation in state-space models using incomplete data.

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

Paper presents a new framework for covariance matrix estimation with geometric insights.

The paper analyzes stability of random matrix products with Markovian noise.

New method proves asymptotic normality for matrix sensing problems.

FREDE efficiently embeds graphs using linear space and guarantees quality.

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…

Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.

Finding a new mathematical representations for graph, which allows direct comparison between different graph structures, is an open-ended research direction. Having such a representation is the first prerequisite for a variety of machine learning algorithms like classification, clustering, etc., over graph datasets. In…