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 $κ=3$ 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.

Cartan meets Cramér-Rao

This paper develops a jet bundle and Cartan geometric foundation for the curvature-aware refinements of the Cramér-Rao bound (CRB) introduced in our earlier work. We show that the extrinsic corrections to variance bounds, previously derived from the second fundamental form of the square root embedding $s_θ=\sqrt{f(\cdot;θ)}\in L^2(μ)$ for model density $f(\cdot;θ)$ with scalar parameter $θ$, admit an intrinsic formulation within the Cartan prolongation framework. Starting from the canonical contact forms and total derivative on the finite jet bundle $J^m(\mathbb{R}\times \mathbb{R})$, we construct the Cartan distribution and the associated Ehresmann connection, whose non-integrability and torsion encode the geometric source of curvature corrections in statistical estimation. In the statistical jet bundle $E=\mathbb{R}\times L^2(μ)$, we point out that the condition for an estimator error to lie in the span of derivatives of $s_θ$ up to order $m$ is equivalent to the square root map satisfying a linear differential equation of order~$m$. The corresponding submanifold of $J^m(E)$ defined by this equation represents the locus of $m$-th order efficient models, and the prolonged section must form an integral curve of the restricted Cartan vector field. This establishes a one-to-one correspondence between algebraic projection conditions underlying CRB and Bhattacharyya-type bounds and geometric integrability conditions for the statistical section in the jet bundle hierarchy. The resulting framework links variance bounds, curvature, and estimator efficiency through the geometry of Cartan distributions, offering a new differential equation and connection-theoretic interpretation of higher-order information inequalities.

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 $0<\alpha<1$. A lower bound on the number of rows for RQIP to hold for random matrices whose entries are drawn from a symmetric $\alpha$-stable ($S\alpha S$) distribution is derived. The proof leverages two key components: a concentration inequality for empirical fractional moments of $S\alpha S$ variables and a covering number bound for sparse $\ell_\alpha$ 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.

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 subspace, respectively. We solve the resulting non-convex optimization problem 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 a recent subspace matching method and a related covariance matrix reconstruction approach.