arXiv research

Improved random forest proximities capture data geometry.

Surveying probabilistic real algebraic geometry.

Study on volumes of random inscribed polytopes in projective geometries.

Surveying random sections on Kähler manifolds, leading to metrics.

Survey on spectral gaps of random hyperbolic surfaces.

High-dimensional random geometry shows phase transitions in various problems.

Study large deviations in random walks on Lie groups.

Study on connectivity and geometry of random Coxeter groups.

The paper characterizes the geometry and topology of spin random fields.

Survey on strong convergence in random matrices and its applications.

Study finds significant instability in node embeddings due to randomness.

Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.

Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on this theory, we demonstrate the efficiency of Subspace Clustering in fairly broad…

The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.

Study reveals limits of detecting local geometry in random graphs.

Researchers study the conformal geometry of bivariate Gaussian manifolds.

New framework models neural systems with random architecture on manifolds.

We consider the problem of efficient randomized dimensionality reduction with norm-preservation guarantees. Specifically we prove data-dependent Johnson-Lindenstrauss-type geometry preservation guarantees for Ho's random subspace method: When data satisfy a mild regularity condition -- the extent of which can be estima…

Paper proposes a new method for supervised manifold learning using random forest proximities.

Researchers use information geometry to analyze and improve DRWs for node classification.

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.

Quantum field theory connects Riemannian geometry to quantum fluctuations.

Random hyperbolic surfaces have nearly optimal spectral gaps.

New random forest variants achieve optimal performance in high dimensions.

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…

Random hyperbolic surfaces have low Cheeger constants.

Geodesic walks converge to Brownian motion on Finsler manifolds.

We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. We derive also the information geometry of the 3-manifold of McKay bivariate gamma dist…

Proposes a new gauge theory for fuzzy geometries using finite-dimensional algebras.

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these man…

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences o…

In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately …

New method speeds up optimization over probability measures.

Study of random multicurves and square-tiled surfaces on large genus surfaces.

This work uses stochastic geometry to improve STIT processes in machine learning.

Muons and random optimizers perform similarly, challenging geometric optimization theory.

The study explores generalized divergences and exponential families with a focus on sufficient conditions and laws of large numbers.

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $\varepsilon>0$. We first prove that as $\varepsilon\to 0$ the ex…

A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a certain class. In this paper, we forge a connection between random forests and ke…

A simple geometrical proof shows that any target function can be found in a random network's neighborhood.

Formula found for probability of random triangles on flat tori being homotopically trivial.

Develops calculus for random submanifolds using zonoids.

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…

The study finds effective lower bounds for spectra of random surfaces and bundles.

A new method for approximating softmax and Gaussian kernels with reduced error.

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

Generative model for joint discrete distributions using randomized assignment flows.