arXiv research

Generates valid Euclidean distance matrices for molecular structures.

A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.

Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.

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

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

We investigate some geometric properties of the real algebraic variety $Δ$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $Δ$. We exhibit conne…

New Sliced-Wasserstein distances for non-Euclidean data.

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…

WE constructs GP kernels for mixed inputs using weighted EDMs.

Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…

Python package for SPD matrix distances, reproducible and extensible.

New method classifies manifold-valued data using Riemannian geometry.

Extends metrics for SPD matrices to infinite dimensions.

Unified framework for hyperbolic embeddings from mixed data types.

A fast binary embedding method preserves Euclidean distances in high-dimensional data.

This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.

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 …

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP…

We report on experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced mac…

We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerica…

Efficiently implements MEG for low-rank matrix optimization problems.

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

Study on rotating surfaces in 4D space with matrices.

This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associa…

Bounds on geodesic distances on Stiefel manifold derived from new metrics.

We calculate Euclidean distance degrees for common manifold optimization types.

Study infinite Euclidean distance discriminants of algebraic varieties.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…

Proposes a new Sliced-Wasserstein distance for covariance matrices in M/EEG signals.

Vectors of data are at the heart of machine learning and data mining. Recently, vector quantization methods have shown great promise in reducing both the time and space costs of operating on vectors. We introduce a vector quantization algorithm that can compress vectors over 12x faster than existing techniques while al…

Paper introduces S-SSE for stable sparse subspace embedding.

Study of strictly accretive matrices using Finsler geometry.

Paper proposes a method to recover point configurations from noisy distance data.

Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequ…

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the di…

The study compares Euclidean and cosine distances in medical drug prescription prediction.

Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so through pairwise kernel evaluations between pdfs (or sample sets). While such an appr…

Model tracks structural changes in Brownian particle configurations on a sphere.

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exac…

The paper calculates period matrices for specific algebraic curves.

Study of metrics on positive-definite matrices from power potential, linking to power means.

We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…

Revisits Isomap, showing it constructs Euclidean representations of geodesic structure.

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…

Covariance and histogram image descriptors provide an effective way to capture information about images. Both excel when used in combination with special purpose distance metrics. For covariance descriptors these metrics measure the distance along the non-Euclidean Riemannian manifold of symmetric positive definite mat…

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

A method for learning embeddings from multi-view data using Gromov-Wasserstein.

Extends manifold learning to non-Euclidean metrics.