arXiv subjects

Amit Vishwakarma

Publications and source records attributed to Amit Vishwakarma.

4 recordsLinked to original sources

arXiv2026

A Tractable Pseudo-Metric on Non-Parametric Exponential Statistical Manifolds via SPD Geometry

Computing distances between probability distributions on non-parametric statistical manifolds is fundamentally intractable. The geodesicequations live in infinite-dimensional function spaces and admit no general closed-form solution. We develop a two-stage framework that produces a computable pseudo-metric on the Pistone--Sempi exponential manifold and apply it to two-sample hypothesis testing. In the first stage, an arbitrary distribution is projected onto a chosen finite-dimensional parametric exponential family via moment-matching. This projection is many-to-one, so the resulting object is a pseudo-metric rather than a true metric. In the second stage, the parametric family is embedded into the manifold of symmetric positive definite matrices via the expected outer product of the augmented sufficient statistics vector. The embedding is a smooth diffeomorphism. The induced pullback metric differs from the Fisher--Rao metric by an explicit correction involving third-order joint cumulants of the sufficient statistics; the correction vanishes for the Gaussian family, recovering the Calvo--Oller embedding as a special case. The affine-invariant Riemannian metric on the ambient matrix manifold then provides a closed-form lower bound for the pseudo-metric, computable directly from sample moments. Applied to two-sample testing, the framework produces a test statistic that is affine-invariant and requires no continuous tuning parameters such as a bandwidth. The choice of target exponential family determines which moments are compared. Critical values are obtained by permutation.

math.ST↗

arXiv2025

Enhanced 3D Shape Analysis via Information Geometry

Three-dimensional point clouds provide highly accurate digital representations of objects, essential for applications in computer graphics, photogrammetry, computer vision, and robotics. However, comparing point clouds faces significant challenges due to their unstructured nature and the complex geometry of the surfaces they represent. Traditional geometric metrics such as Hausdorff and Chamfer distances often fail to capture global statistical structure and exhibit sensitivity to outliers, while existing Kullback-Leibler (KL) divergence approximations for Gaussian Mixture Models can produce unbounded or numerically unstable values. This paper introduces an information geometric framework for 3D point cloud shape analysis by representing point clouds as Gaussian Mixture Models (GMMs) on a statistical manifold. We prove that the space of GMMs forms a statistical manifold and propose the Modified Symmetric Kullback-Leibler (MSKL) divergence with theoretically guaranteed upper and lower bounds, ensuring numerical stability for all GMM comparisons. Through comprehensive experiments on human pose discrimination (MPI-FAUST dataset) and animal shape comparison (G-PCD dataset), we demonstrate that MSKL provides stable and monotonically varying values that directly reflect geometric variation, outperforming traditional distances and existing KL approximations.

cs.CV↗

arXiv2025

Distance Measure Based on an Embedding of the Manifold of K-Component Gaussian Mixture Models into the Manifold of Symmetric Positive Definite Matrices

In this paper, a distance between the Gaussian Mixture Models(GMMs) is obtained based on an embedding of the K-component Gaussian Mixture Model into the manifold of the symmetric positive definite matrices. Proof of embedding of K-component GMMs into the manifold of symmetric positive definite matrices is given and shown that it is a submanifold. Then, proved that the manifold of GMMs with the pullback of induced metric is isometric to the submanifold with the induced metric. Through this embedding we obtain a general lower bound for the Fisher-Rao metric. This lower bound is a distance measure on the manifold of GMMs and we employ it for the similarity measure of GMMs. The effectiveness of this framework is demonstrated through an experiment on standard machine learning benchmarks, achieving accuracy of 98%, 92%, and 93.33% on the UIUC, KTH-TIPS, and UMD texture recognition datasets respectively.

math.DG↗

arXiv2024

Information Geometric Framework For Point Cloud Data

In this paper, we introduce a novel method for comparing 3D point clouds, a critical task in various machine learning applications. By interpreting point clouds as samples from underlying probability density functions, the statistical manifold structure is given to the space of point clouds. This manifold structure will help us to use the information geometric tools to analyze the point clouds. Our method uses the Gaussian Mixture Model (GMM) to find the probability density functions and the Modified Symmetric KL divergence to measure how similar the corresponding probability density functions are. This method of comparing the point clouds takes care of the geometry of the objects represented by the point clouds. To demonstrate the effectiveness of our approach, we take up five distinct case studies:(i) comparison of basic geometric shapes, (ii) comparison of 3D human body shapes within the MP FAUST dataset, (iii) comparison of animal shapes, (iv) comparison of human and animal datasets and (v) comparison of audio signals.

math.DG↗