arXiv subjects

Sunder Ram Krishnan

Publications and source records attributed to Sunder Ram Krishnan.

12 recordsLinked to original sources

arXiv2026

A Hyperfinite Framework for Score-Based Generative Modeling

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker--Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $\kappa=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling--while laying foundations for further extensions--that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

stat.ML↗

arXiv2026

Rao-Blackwellized Coverage Estimation in Poisson Networks: A High-Fidelity Hybrid Framework

While stochastic geometry provides a powerful framework for the analysis of cellular networks, standard Monte Carlo simulations often suffer from slow convergence due to the stochasticity of the infinite far-field. This work introduces the \textit{Rao-Blackwellized Hybrid Estimator} (RBHE), which enhances simulation efficiency by analytically marginalizing the residual far-field interference via the conditional Laplace functional. By partitioning the interference field into $K$ dominant interferers and an infinite tail, we derive an estimator that combines exact spatial sampling with a rigorous analytical representation. We prove that the RBHE is an unbiased estimator for any finite truncation, while its systematic bias relative to the infinite-plane benchmark decays at a rate of $\mathcal{O}(K^{1-η/2})$. Numerical results demonstrate significant sample parsimony; in the high-reliability regime ($T = -10$ dB) with $K=2$, the RBHE yields a variance reduction gain of $90.75\times$, enabling a $98.90\%$ reduction in the spatial realizations required to reach a target precision. This framework effectively bridges the gap between tractable analytical models and high-fidelity simulations.

cs.IT↗

arXiv2026

Improving Cramér-Rao Bound And Its Variants: An Extrinsic Geometry Perspective

This work presents a geometric refinement of the classical Cramér--Rao bound (CRB) in the non-asymptotic regime by incorporating curvature-aware corrections based on the second fundamental form associated with the statistical model manifold. That is, our formulation shows that relying on the extrinsic geometry of the square root embedding of the manifold in the ambient Hilbert space comprising square integrable functions with respect to a fixed base measure offers a rigorous (and intuitive) way to improve upon the CRB and some of its variants, such as the Bhattacharyya-type bounds, that use higher-order derivatives of the log-likelihood. Precisely, the improved bounds in the latter case make explicit use of the elegant framework offered by employing the Faà di Bruno formula and exponential Bell polynomials in expressing the jets associated with the square root embedding in terms of the raw scores. The interplay between the geometry of the statistical embedding and the behavior of the estimator variance is quantitatively analyzed in concrete examples, showing that our corrections can meaningfully tighten the lower bound, suggesting further exploration into connections with estimator efficiency in more general situations.

math.ST↗

arXiv2026

Refining Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective

We derive a vector generalization of the curvature-corrected Cramér--Rao bound (CRB) in the nonasymptotic regime using a Hilbert space square-root embedding. Building on previous scalar results, we establish a \emph{directional} curvature correction derived from the second fundamental form of the model manifold. To obtain matrix-valued refinements, we formulate sufficient conditions for a conservative matrix-level correction using a semidefinite program (SDP) based on sum-of-squares (SOS) relaxations. The framework is rigorously illustrated with two distinct geometries: (i) a curved Gaussian location model, which reveals a characteristic \textit{pinching effect} where directional bounds vanish along principal axes despite non-zero extrinsic curvature and classical subspace-based bounds using the second-order Bhattacharyya matrix provide overly optimistic variance predictions that fail to account for the manifold's directional topology, and (ii) a spherical multinomial model where the curvature is isotropic. Our results demonstrate that while classical second-order corrections using the Bhattacharyya matrix provide useful benchmarks derived from the local coordinate basis, the proposed directional and SOS-certified bounds offer a more faithful and geometry-consistent representation of the directional sensitivity and fundamental limits of estimation in curved statistical families.

math.ST↗

arXiv2025

Cartan meets Cramér-Rao

A Cartan-geometric, jet bundle formulation of curvature-aware variance bounds in parametric statistical estimation is developed. Building on our earlier extrinsic Hilbert space approach to the Cramér-Rao and Bhattacharyya-type inequalities, we show that the curvature corrections induced by the square root embedding of a statistical model admit a canonical intrinsic interpretation via jet geometry and Cartan's prolongation theory. For a scalar-parameter family with square root map $s_θ=\sqrt{f(\cdot;θ)}\in L^2(μ)$, we regard $s_θ$ as a section of the statistical bundle $E=Θ\times L^2(μ)$ and study its finite-order prolongations. We point out that the classical algebraic efficiency condition--that the estimator residual $(T-θ)s_θ$ lies in the span of derivatives of $s_θ$ up to order $m$--is equivalent to the existence of a linear ordinary differential equation (ODE) of order $m$ satisfied by the square root map. Geometrically, this means the prolonged section lies in an ODE-defined submanifold of the jet bundle and is an integral curve of the restricted Cartan vector field. The obstruction to such finite-order integrability is identified with the vertical component of the canonical Ehresmann connection on the jet tower, which coincides with the curvature correction term in variance bounds. This establishes a direct correspondence between algebraic projection conditions in $L^2(μ)$ and intrinsic holonomy properties of statistical sections, yielding a unified geometric interpretation of higher-order information inequalities.

math.ST↗

arXiv2025

Coherent Source Enumeration with Compact ULAs

Source enumeration typically relies on subspace-based techniques that require accurate separation of signal and noise subspaces. However, prior works do not address coherent sources in small uniform linear arrays, where ambiguities arise in the spatial spectrum. We address this by decomposing the forward-backward smoothed covariance matrix into a sum of a rank-constrained Toeplitz matrix and a diagonal matrix with non-negative entries representing the signal and noise subspaces, respectively. The resulting non-convex optimization problem is solved by proposing Toeplitz approach for rank-based target estimation (TARgEt) that employs the alternating direction method of multipliers. Numerical results on both synthetic and real-world datasets demonstrate the effectiveness and robustness of TARgEt over the state-of-the-art.

eess.SP↗

arXiv2025

Restricted Quasiconvexity Isometry Property for Symmetric $α$-Stable Random Matrices

We formulate a generalization of the Restricted Isometry Property (RIP) referred to as the Restricted Quasiconvexity Isometry Property (RQIP) for alpha stable random projections with $0<α<1$. A lower bound on the number of rows for RQIP to hold for random matrices whose entries are drawn from a symmetric $α$-stable ($SαS$) distribution is derived. The proof leverages two key components: a concentration inequality for empirical fractional moments of $SαS$ variables and a covering number bound for sparse $\ell_α$ balls. The resulting sample complexity reflects the polynomial tail behavior of the concentration and reinforces an observation made in the literature that the RIP framework may have to be replaced with other sparse recovery formulations in practice, such as those based on the null space property.

math.PR↗

arXiv2018

Betti Numbers of Gaussian Excursions in the Sparse Regime

Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further obtain a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools.

math.PR↗

arXiv2016

Convergence of the Reach for a Sequence of Gaussian-Embedded Manifolds

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of $M$. Roughly speaking, the reach is a measure of a manifold's departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.

math.PR↗

arXiv2016

Crossing Numbers of Random Two-Bridge Knots

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work.

math.GT↗

arXiv2015

The Intrinsic Geometry of Some Random Manifolds

We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

math.PR↗

arXiv2015

Random knots using Chebyshev billiard table diagrams

We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model decays to zero as the number of crossings goes to infinity.

math.GT↗