Mathematics

We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression…

A central question in high-dimensional statistics is to understand statistical--computational gaps: regimes in which recovering a hidden signal is information-theoretically possible but conjectured to be computationally intractable. The…

We study estimation in the low signal-to-noise ratio (SNR) regime for a broad class of Gaussian latent-variable models, including Gaussian mixtures and orbit recovery problems. We show that, in this regime, the generalized method-of-moments…

Hjort and Glad (1995) present a method for semiparametric density estimation. Relative to the ordinary kernel density estimator, this technique performs much better when a parametric vehicle distribution fits the data, and otherwise…

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a…

We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized…

We introduce an Indian-buffet-type model for multi-factorial innovation in which each arriving agent may exhibit both previously observed and new features. The number of new features follows a power-law behavior, while the probability of…

Let $(R,\frak m)$ be a generalized Cohen-Macaulay local ring of prime characteristic $p$. In this paper we give a sharp bound for the Frobenius test exponent of parameter ideals. Namely, we prove that $$\mathrm{Fte}(R) \le \lceil…

We develop a natural Bayesian multiplicity-correcting prior distribution within the probabilistic forward stepwise representation of model space priors for regression problems. The proposed prior, obtained from making an analogy to the Holm…

Let $R$ be a local or positively graded ring with a regular presentation $R \cong Q/I$ where $I$ is a monomial ideal generated by $n$ elements on a regular sequence. In Briggs-Grifo-Pollitz (2025), the authors classify the cohomological…

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is…

Being encouraged by [AKRS] that provides an amazing bridge between Statistics and Invariant Theory, and especially by [FM], where quiver semi-invariant techniques apply to verify the existence of MLE for a recent iPCA model, we provide an…

In this paper, we consider the problem of simultaneous testing of multivariate normal means under arbitrary covariance dependence. Specifically, let $\boldsymbol{X}\sim N_n(\boldsymbol{\theta},\boldsymbol{\Sigma})$, where…

We report on a collection of open problems in commutative algebra and related areas that have been resolved (proved or disproved) using the Rethlas natural-language automated reasoning system. The problems are drawn from several published…

This paper gives a structural explanation for the Z-relation by modelling pitch-class sets as complete weighted graphs and encoding their interval content in a composition of $n$ via an additivity rule. We introduce the realization number…

We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $\Theta$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the…

Many Bayesian inference problems involve high-dimensional models where the performance of standard importance sampling (IS) methods often degrades rapidly as the dimensionality increases. Classical analyses of IS typically rely on the…

We study a generalization of the classical hidden clique problem to graphs with real-valued edge weights. Formally, we define a hypothesis testing problem. Under the null hypothesis, edges of a complete graph on $n$ vertices are associated…

This article explores several fundamental aspects of fuzzy $\mathscr{F}$-metric spaces and their applications in mathematical analysis. We investigate some essential properties concerning compactness and total boundedness in fuzzy…

We introduce a new version of the KL-divergence for Gaussian distributions which is based on Wasserstein geometry and referred to as WKL-divergence. We show that this version is consistent with the geometry of the sample space ${\Bbb R}^n$.…