skip to main content

Explore Research Products in the NSF-PAR

This content will become publicly available on January 1, 2025

Title: Geometric sharp large deviations for random projections of ℓpn spheres and balls
Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the “prefactor” in front of the exponentially decaying term. For fixed p ∈ (1, ∞), consider independent sequences (X(n,p))_{n∈N} and (Θ_n)_{n∈N} of random vectors with Θn distributed according to the normalized cone measure on the unit l^n_2 sphere, and X(n,p) distributed according to the normalized cone measure on the unit lnp sphere. For almost every realization (θn)_{n∈N} of (Θ_n)_{n∈N}, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of X(n,p) onto θ_n, that are asymptotically exact (as the dimension n tends to infinity). Furthermore, the case when (X(n,p))_{n∈N} is replaced with (X(n,p))_{n∈N}, where X(n,p) is distributed according to the uniform (or normalized volume) measure on the unit lnp ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions (θ_n)_{n∈N} that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of n. The results on the one hand provide more accurate quantitative estimates of tail probabilities of random projections of \ell^n_p spheres than logarithmic asymptotics, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques. Along the way, several results of independent interest are obtained including a simpler representation for the quenched large deviation rate function that shows that it is strictly convex, a central limit theorem for random projections under a certain family of tilted measures, and multidimensional generalized Laplace asymptotics.  more » « less
Award ID(s):
1954351
NSF-PAR ID:
10484620
Author(s) / Creator(s):
;
Publisher / Repository:
IMS
Date Published:
Journal Name:
Electronic Journal of Probability
Volume:
29
Issue:
1
ISSN:
1083-6489
Page Range / eLocation ID:
1--56
Subject(s) / Keyword(s):
["sharp large deviations","random projections","rate function","prefactor","Bahadur-Rao correction","lnp spheres","cone measure","spherical caps","central limit theorem for convex sets."]
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ( , Journal of Applied Probability)
    null (Ed.)
    Abstract The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓ p -spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓ p -spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓ p -balls in a high-dimensional Euclidean space. 
    more » « less
  2. ; ; ( , Journal of Mathematical Physics)
    Using precise random matrix theory tools and the Kac–Rice formula, we provide sharp O(1) asymptotics for the average number of deep minima of the ( p, k) spiked tensor model. These sharp estimates allow us to prove that, when the signal-to-noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and to establish the occurrence of topological trivialization by showing that this number vanishes when the strength of the signal-to-noise ratio diverges. We also derive an explicit formula for the value of the absolute minimum (the limiting ground state energy) on the N-dimensional sphere, similar to the recent work of Jagannath, Lopatto, and Miolane [Ann. Appl. Probab. 4, 1910–1933 (2020)]. 
    more » « less
  3. ; ( , Random Structures & Algorithms)

    We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

     
    more » « less
  4. ; ; ( , Communications in Mathematical Physics)
    Abstract

    We analyse the eigenvectors of the adjacency matrix of the Erdős–Rényi graph$${\mathbb {G}}(N,d/N)$$G(N,d/N)for$$\sqrt{\log N} \ll d \lesssim \log N$$logN≪d≲logN. We show the existence of a localized phase, where each eigenvector is exponentially localized around a single vertex of the graph. This complements the completely delocalized phase previously established in Alt et al. (Commun Math Phys 388(1):507–579, 2021). For large enoughd, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. We derive explicit asymptotics for the localization length up to the mobility edge and characterize its divergence near the phase boundary. The proof is based on a rigorous verification of Mott’s criterion for localization, comparing the tunnelling amplitude between localization centres with the eigenvalue spacing. The first main ingredient is a new family of global approximate eigenvectors, for which sharp enough estimates on the tunnelling amplitude can be established. The second main ingredient is a lower bound on the spacing of approximate eigenvalues. It follows from an anticoncentration result for these discrete random variables, obtained by a recursive application of a self-improving anticoncentration estimate due to Kesten.

     
    more » « less
  5. ; ; ; ; ( , Journal of Microscopy)
    Summary

    An accurate knowledge of the complex microstructure of a heterogeneous material is crucial for its performance prediction, prognosis and optimization. X‐ray tomography has provided a nondestructive means for microstructure characterization in 3D and 4D (i.e. structural evolution over time), in which a material is typically reconstructed from a large number of tomographic projections using filtered‐back‐projection (FBP) method or algebraic reconstruction techniques (ART). Here, we present in detail a stochastic optimization procedure that enables one to accurately reconstruct material microstructure from a small number of absorption contrast x‐ray tomographic projections. This discrete tomography reconstruction procedure is in contrast to the commonly used FBP and ART, which usually requires thousands of projections for accurate microstructure rendition. The utility of our stochastic procedure is first demonstrated by reconstructing a wide class of two‐phase heterogeneous materials including sandstone and hard‐particle packing from simulated limited‐angle projections in both cone‐beam and parallel beam projection geometry. It is then applied to reconstruct tailored Sn‐sphere‐clay‐matrix systems from limited‐angle cone‐beam data obtained via a lab‐scale tomography facility at Arizona State University and parallel‐beam synchrotron data obtained at Advanced Photon Source, Argonne National Laboratory. In addition, we examine the information content of tomography data by successively incorporating larger number of projections and quantifying the accuracy of the reconstructions. We show that only a small number of projections (e.g. 20–40, depending on the complexity of the microstructure of interest and desired resolution) are necessary for accurate material reconstructions via our stochastic procedure, which indicates its high efficiency in using limited structural information. The ramifications of the stochastic reconstruction procedure in 4D materials science are also discussed.

     
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email