arXiv subjects

Susovan Pal

Publications and source records attributed to Susovan Pal.

10 recordsLinked to original sources

arXiv2026

Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres

We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere $\mathbb{S}^m$. The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fr\'echet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fr\'echet means and the corresponding Wald-type $\chi^2$ inference. For rotationally symmetric densities, we show that full and directional smeariness are equivalent. The Hessian and fourth-order terms are governed by two explicit geometry-dependent radii $R_m 0$ with $S_m+\varepsilon<\pi$, we construct examples of rotationally symmetric $2$-smeary densities supported in the ball of radius $S_m+\varepsilon$. For general densities, closed hemispherical support rules out both full and directional smeariness. Support contained in the closed ball of radius $S_m$ rules out full smeariness, while we construct explicit, directionally $2$-smeary examples supported in balls of radius $\pi/2+\varepsilon$. As a byproduct, the explicit Hessian formulas in this paper also provide a practical diagnostic for detecting proximity to the Hessian-degenerate, non-classical regime.

math.ST↗

arXiv2026

Finite Sample Smeariness on Spheres and Modulation-aware Tests

Directional and shape data often live on manifolds, where standard central limit theorems (CLT) and the associated Wald-type $\chi^2$-tests require curvature-dependent recalibration: the limiting normal covariance is modified by the Hessian of the Fr\'echet function. On spheres, under the positive-Hessian assumptions considered here, this curvature correction yields Type~I finite sample smeariness (FSS), meaning that the asymptotic modulation exceeds $1$. This paper studies FSS on $\mathbb{S}^m$ and develops modulation-aware tests. We first show that, for absolutely continuous distributions on $\mathbb{S}^m$ with positive definite Hessians, Type~I FSS is unavoidable under the stated classical CLT and moment assumptions. The geometric mechanism is that, for the absolutely continuous spherical distributions considered here, positive curvature of the sphere gives the strict spectral bound $H \prec 2I$, so that the limiting modulation is strictly larger than one. We then prove a curse-of-dimensionality result for dimensionally comparable rotationally symmetric families: their asymptotic modulation is monotone increasing with dimension and can become arbitrarily large as the support approaches a hemisphere. Motivated by these results, we derive \textit{explicit} consistent plug-in estimators of the Fr\'echet-function Hessian and use them to construct one-sample and two-sample Hessian-corrected Hotelling-type statistics with $\chi^2$ limits. Numerical experiments on low- and high-dimensional spheres show that the proposed modulation-aware tests have rejection behavior comparable to bootstrap-based FSS corrections, while being substantially faster due to the explicit Hessian formulas.

math.ST↗

arXiv2026

Construction of a Closed Hyperbolic Surface of Arbitrarily Small Eigenvalue of Prescribed Serial Number

In this paper we construct, for given any small positive number $\epsilon$ and given natural number $n$, and given any closed hyperbolic surface $M$, a closed hyperbolic covering surface $\widetilde{M}$, such that its $n$-th eigenvalue is less than $\epsilon$. An application of this result will also be discussed. The main result follows from the techniques used in B.Randol's paper in 1974 [Ran]. Here I give a new and geometric proof of the main result.

math.GT↗

arXiv2026

Identifiability of the Unnormalized Graph Laplace Operators

In this short note, we show that the continuous intrinsic graph Laplace operator with Gaussian kernel on a compact Riemannian manifold without boundary uniquely determines both the Riemannian metric and the sampling density, provided the latter is positive. In contrast, the corresponding continuous extrinsic graph Laplace operator uniquely determines the sampling measure; moreover, when the operator is defined via an embedding into Euclidean space, it also uniquely determines the induced Riemannian metric and the sampling density.

math.DG↗

arXiv2025

Asymptotics of the graph Laplace operator near an isolated singularity

In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $\kappa$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $\kappa$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results.

math.DG↗

arXiv2025

A Riemannian Framework for Linear and Quadratic Discriminant Analysis on the Tangent Space of Shapes

We present a Riemannian framework for linear and quadratic discriminant classification on the tangent plane of the shape space of curves. The shape space is infinite dimensional and is constructed out of square root velocity functions of curves. We introduce the idea of mean and covariance of shape-valued random variables and samples from a tangent space to the pre-shape space (invariant to translation and scaling) and then extend it to the full shape space (rotational invariance). The shape observations from the population are approximated by coefficients of a Fourier basis of the tangent space. The algorithms for linear and quadratic discriminant analysis are then defined using reduced dimensional features obtained by projecting the original shape observations on to the truncated Fourier basis. We show classification results on synthetic data and shapes of cortical sulci, corpus callosum curves, as well as facial midline curve profiles from patients with fetal alcohol syndrome (FAS).

stat.ME↗

arXiv2025

Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.

math.DG↗

arXiv2025

Convergence and clustering analysis for Mean Shift with radially symmetric, positive definite kernels

The mean shift (MS) is a non-parametric, density-based, iterative algorithm with prominent usage in clustering and image segmentation. A rigorous proof for the convergence of its mode estimate sequence in full generality remains unknown. In this paper, we show that for\textit{ sufficiently large bandwidth} convergence is guaranteed in any dimension with \textit{any radially symmetric and strictly positive definite kernels}. Although the author acknowledges that our result is partially more restrictive than that of \cite{YT} due to the lower limit of the bandwidth, our kernel class is not covered by the kernel class in \cite{YT}, and the proof technique is different. Moreover, we show theoretically and experimentally that while for Gaussian kernel, accurate clustering at \textit{large bandwidths} is generally impossible, it may still be possible for other radially symmetric, strictly positive definite kernels.

stat.ML↗

arXiv2025

Two-Sample Tests for Optimal Lifts, Manifold Stability and Reverse Labeling Reflection Shap

We consider a quotient of a complete Riemannian manifold modulo an isometrically and properly acting Lie group and lifts of the quotient to the manifolds in optimal position to a reference point on the manifold. With respect to the pushed forward Riemannian volume onto the quotient we derive continuity and uniqueness a.e. and smoothness to large extents also with respect to the reference point. In consequence we derive a general manifold stability theorem: the Fr\'echet mean lies in the highest dimensional stratum assumed with positive probability, and a strong law for optimal lifts. This allows to define new two-sample tests utilizing individual optimal lifts which outperform existing two-sample tests on simulated data. They also outperform existing tests on a newly derived reverse labeling reflection shape space, that is used to model filament data of microtubules within cells in a biological application.

math.ST↗

arXiv2016

Optimal bandwidth estimation for a fast manifold learning algorithm to detect circular structure in high-dimensional data

We provide a way to infer about existence of topological circularity in high-dimensional data sets in $\mathbb{R}^d$ from its projection in $\mathbb{R}^2$ obtained through a fast manifold learning map as a function of the high-dimensional dataset $\mathbb{X}$ and a particular choice of a positive real $σ$ known as bandwidth parameter. At the same time we also provide a way to estimate the optimal bandwidth for fast manifold learning in this setting through minimization of these functions of bandwidth. We also provide limit theorems to characterize the behavior of our proposed functions of bandwidth.

stat.ML↗