Proposes a method for modeling random objects in metric spaces using random effects.
problem Modeling random objects in non-Euclidean spaces with random effects.
method Nonlinear Fréchet-based algorithm for M-estimation.
result Consistent estimation of prediction target under random-effects formulation.
Surveying random sections on Kähler manifolds, leading to metrics.
problem Understanding statistics of random sections on Kähler manifolds.
method Analyzing tensor powers of line bundles.
result Induced metrics from random sections.
Random walks on metric spaces embed quasi-isometrically into the space.
problem Embedding random subgroups of metric spaces quasi-isometrically.
method Analyzing random walks and contracting elements in metric spaces.
result Random subgroups of isometry groups are quasi-isometrically embedded.
Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert space…
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.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
Study of lengths of cycles in large genus random maps converging to Poisson process.
problem Understanding the distribution of cycle lengths in large genus random maps.
method Teichmüller theory approach for uniformly random metric maps (ribbon graphs).
result The length spectrum converges to a Poisson point process with an explicit intensity as genus tends to infinity.
New framework for choosing optimal proxy metrics from past experiments.
problem Difficult to measure long-term treatment effects in experiments.
method Statistical framework to define and construct optimal proxy metrics.
result Optimal proxy metric depends on experiment's sample size.
Estimates metric tensor on neuromanifolds using Fisher information and random methods.
problem Computing the metric tensor on high-dimensional neuromanifolds efficiently and accurately.
method Deterministic bounds and unbiased random estimators based on Hutchinson's trace method.
result An efficient random estimator with bounded standard deviation.
New methods link Calabi-Yau metrics to random matrices.
problem Lack of explicit metrics on Calabi-Yau manifolds hinders particle physics computations.
method Numerical approximations of the Laplacian spectrum on Calabi-Yau spaces.
result Surprising link found between Calabi-Yau metrics and random matrix theory.
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…
We introduce a new framework for comparing parametric network families.
problem Comparing and analyzing data modeled as parameterized families of networks.
method A Gromov-Wasserstein variant of optimal transport for defining distances.
result Established foundational properties and theoretical approximation guarantees for the new distances.
Study large deviations and speed of random walks in hyperbolic spaces.
problem Understanding the speed of random walks in hyperbolic spaces.
method Large deviations analysis for random walks with a non-elementary semi-group.
result Established large deviations results for random walk distances.
Invites probabilistic approach to Kähler-Einstein metrics via random point processes.
problem Constructing Kähler-Einstein metrics on complex projective algebraic manifolds.
method Large N-limit from random point processes defined by algebro-geometric data; variational approach for positive Ricci curvature.
result Convergence of metrics to Kähler-Einstein metrics under specific conditions.
Quantum RNG improves financial risk metrics estimation.
problem Estimating financial risk metrics with high precision.
method Quantum-Enhanced Monte Carlo using QRNG.
result Improved accuracy in VaR and CVaR estimation.
New k-means method handles random data better than traditional techniques.
problem Limitations of traditional clustering methods in random data.
method Probabilistic metric space with random normed k-means (RNKM).
result RNKM outperforms traditional methods in complex clustering scenarios.
We study random Morse functions on a Riemann manifold (Mm,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 ε>0. We first prove that as ε→0 the ex…
We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolut…
The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…
Study categorizes time series anomaly detection metrics based on evaluation challenges.
problem Challenges in evaluating time series anomaly detection due to diverse application objectives and metric assumptions.
method Problem-oriented framework categorizing metrics into six dimensions based on evaluation challenges.
result Quantifies each metric's discriminative ability and reveals limitations of widely used metrics.
Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in Rd. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that …
For any pseudo-Anosov diffeomorphism on a closed orientable surface S of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this paper, we consider random walks on th…
We prove that the Euler form of a metric connection on real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…
Random hyperbolic surfaces with punctures converge to the Brownian sphere.
problem Understanding the geometry of random hyperbolic surfaces with punctures.
method Rescaling and encoding via plane trees with continuous labels.
result Rescaled random hyperbolic surfaces converge to the Brownian sphere.
Paper introduces a medoid-based approach for efficient Fréchet regression.
problem Regression in metric spaces with random objects.
method Adapted random forest algorithm with medoid-based splitting rule.
result Asymptotic equivalence and consistency of the regression estimator.
The study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.
problem Understanding spectral gaps of random hyperbolic surfaces with many cusps.
method Analysis of moduli spaces of hyperbolic surfaces with Weil-Petersson metric.
result Arbitrarily small spectral gaps are observed as the number of cusps grows slower than the genus.
Random covers of hyperbolic surfaces follow a specific probability measure.
problem Understanding the distribution of random covers of hyperbolic surfaces.
method Analyzing random covers subject to specific group isomorphism conditions.
result Asymptotic distribution of random covers according to a probability measure on moduli space of metric graphs.
Estimates means in metric spaces using quantization.
problem No practical estimator for Fréchet means in all metric spaces.
method Introduced estimators based on random quantization and data-driven partitioning.
result Universal consistency of estimators across separable metric spaces and Banach spaces.
A stability metric compares feature selection algorithms in machine learning.
problem Stability of feature selection algorithms in machine learning.
method Rank-based instability index to compare MDA, LIME, and SHAP algorithms.
result LIME and SHAP are more stable than MDA, with LIME being best for human interpretability.
A faster graph kernel using optical random features.
problem High computation cost of graphlet kernel due to isomorphism test.
method Kernel random features, optical random features, mean kernel metric.
result The proposed method is orders of magnitude faster with similar or better accuracy.
Study of random sections on complex spaces converging to equilibrium metrics.
problem Understanding the behavior of random holomorphic sections on complex spaces.
method Analyzing the convergence of normalized Fubini-Study currents and integration currents to the equilibrium metric's curvature.
result The normalized currents of integration along zero divisors converge almost surely to the curvature current of the equilibrium metric.
We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results…
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…
The paper analyzes Random Search and introduces BLiN-MOS for bandit learning.
problem Understanding and optimizing hyperparameter tuning in metric measure spaces.
method Introducing scattering dimension to quantify performance, and developing BLiN-MOS for bandit learning.
result Random Search converges to optimal values with specific rates in noise-free and noisy environments.
Continuous vector representations of words and objects appear to carry surprisingly rich semantic content. In this paper, we advance both the conceptual and theoretical understanding of word embeddings in three ways. First, we ground embeddings in semantic spaces studied in cognitive-psychometric literature and introdu…
Paper reinterprets majorizing measure theorem in terms of coding theory.
problem Understanding boundedness of random processes.
method Information-theoretic perspective using variable-length codes.
result Boundedness of random processes linked to efficient coding.
To improve the classification performance in the context of hyperspectral image processing, many works have been developed based on two common strategies, namely the spatial-spectral information integration and the utilization of neural networks. However, both strategies typically require more training data than the cl…
Paper defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
problem Defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
method Defines Fisher co-metric directly from Fisher metric without going through tangent bundle, using a natural correspondence between cotangent vectors and random variables.
result Clarifies the relation between Fisher co-metric and variance/covariance, trivializing the Cramér-Rao inequality.
We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Rie…
Revises individual fairness by finding a fair metric for a model.
problem Difficulties in specifying a suitable fairness metric a priori.
method Introduces minimal metrics and applies randomized smoothing from adversarial robustness.
result Adapting minimal metrics to complex models yields interpretable fairness guarantees.
Proposes a new random forest weighted local Fréchet regression method.
problem Complex metric space valued responses and curse of dimensionality in Fréchet regression.
method Locally adaptive kernel generated by random forests for local average and local linear Fréchet regression.
result Significantly improves existing Fréchet regression methods with theoretical guarantees.
Random hyperbolic surfaces have a spectral gap that approaches 1/4 as genus grows.
problem Estimating the spectral gap of random hyperbolic surfaces.
method Analyzing the Weil-Petersson measure on moduli spaces of metrics.
result The spectral gap of random hyperbolic surfaces converges to 1/4 as the genus increases.
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 …
Study geodesics on graphs with random lengths, proving bi-infinite paths exist.
problem Existence of bi-infinite geodesic paths on graphs with random edge lengths.
method Sublinear Morse geodesics and first passage percolation analysis.
result Proves the existence of bi-infinite geodesic paths in graphs with specific properties.
The paper studies Fubini-Study metrics and zero distributions on CR manifolds.
problem Understanding Fubini-Study metrics on CR manifolds.
method Asymptotic analysis of Toeplitz operators and pull-back metrics.
result Established the distribution of zero divisors of random CR functions.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.