Abstract:Spectral estimation is an area of research within signal processing concerned with determining the frequencies of a signal based on finite uniform samples disrupted by noise. There exists a family of highly celebrated spectral estimation algorithms referred to as subspace methods which are widely known and readily available to the general public. Root-MUSIC is one such algorithm that approximates the signal's frequencies by constructing a high-degree polynomial and finding a subset of roots which are closest to the complex unit circle. We prove that the selection process chooses the relevant roots of the polynomial, and provide sharp, non-asymptotic, and explicit error bounds for the accuracy of the selected roots in terms of fundamental model parameters. All results hold under a natural separation condition on the correct signal frequencies and are applicable to several versions of the problem which are used in practice.
September 11
(CUNY closed, no seminar)
September 18
Speaker: Junren Chen (Columbia University)
Title: Finite-sample guarantees for logistic regression with Gaussian design
Abstract: In this talk, we discuss the finite-sample parameter estimation problem in logistic regression with Gaussian design. Specifically, we ask two central questions: (i) what is the minimax optimal error rate? (ii) what is the estimation performance of the maximum likelihood estimator (MLE)? Let $n$ be the measurement number, $d$ be the dimension, and $R$ be the $\ell_2$ norm of the underlying parameter, and consider the regime $n\gtrsim Rd$ where the MLE exists with high probability (Chardon et al, 2024). We show that the minimax optimal error rate is at the order of $\sqrt(Rd/n)+\sqrt(R^3/n)$, with the upper bound being attained by a debiased estimator. We also show that MLE achieves the error rate $\tilde{O}(\sqrt(Rd/n)+\sqrt(R^3/n)+R^2d/n)$. This improves on prior results but is suboptimal due to the additional term $R^2d/n$, which appears to be intrinsic to the MLE. Finally, we provide some novel results for gradient descent (GD) on logistic loss that go beyond existing theories. For instance, under $n=\tilde{\Omega}(R^6d)$, GD with small stepsize achieves a fast global linear convergence and $O(\sqrt{R^5d/n})$ statistical estimation error.
September 25
Speaker: Alan Chang (Washington University, St. Louis)
Title: Projections of random Cantor sets
Abstract: The four-corner Cantor set is a planar analogue of the classical Cantor set and arises in several areas of analysis, including the study of Kakeya sets and removable singularities for analytic functions. A central problem is to understand how this set behaves when projected onto lines. This turns out to be a very difficult question, so we study a random variant of the Cantor set, where we are able to obtain sharp estimates. This is joint work with Pablo Shmerkin and Ville Suomala.
October 2
Speaker: Ludovick Bouthat (University of Delaware)
Title:The sharp constant in the Mashreghi--Ransford inequality
Abstract: Let a[0], a[1], a[2], ... be a sequence of complex numbers. For each nonnegative integer n, define
b[n] = sum from k = 0 to n of (n choose k) a[k],
c[n] = sum from k = 0 to n of (n choose k) (-1)^(n-k) a[k].
Let beta > 1 and set alpha = sqrt(beta^2 - 1). Suppose that both b[n] and c[n] are O(beta^n). Mashreghi and Ransford proved in 2005 that
where all three limit superiors are taken as n tends to infinity, and 2/sqrt(3) <= kappa <= 2. We prove that the optimal constant is kappa = 2/sqrt(3). Our proof uses exponential generating functions, Phragmén–Lindelöf estimates, and Cauchy’s formula, and we compare it with the original proof of Mashreghi and Ransford.
October 9
(No seminar)
October 16
Speaker: Istvan Kadar (ETH Zürich)
Title: Smooth singularity formation for the energy critical wave equation without quantization
Abstract: The energy-critical nonlinear wave equation (NW) in three dimensions admits a stationary solution. In their foundational work, Krieger-Schlag-Tataru constructed (non-smooth) solutions to (NW) that develop singularities with arbitrarily polynomial blow-up rates exceeding the self-similar rate by adiabatically shrinking to a single soliton in spherical symmetry. For comparison, their analogous results for the wave map equation stand in sharp contrast to the quantized blow-up rates for smooth solutions obtained by Raphaël-Rodnianski. In this talk, I discuss a new result establishing the existence of a family of smooth solutions to (NW) that develop singularities at a continuum of polynomial rates, achieved through the collapse of multiple solitons.
October 23
Speaker: Giulia Carigi (Indiana University Bloomington)
Title: Mixing properties of Markov processes in Hilbert Spaces: a MCMC application
Abstract:In this talk, we explore Markov Chain Monte Carlo (MCMC) methods for sampling high-dimensional probability measures, such as those arising in Bayesian formulations of PDE inverse problems. Our focus is on a promising class of parallel-friendly algorithms: multiproposal MCMC schemes, specifically multiproposal preconditioned Crank-Nicolson. We rigorously analyze their mixing properties in infinite-dimensional spaces using the weak Harris theorem. This approach provides dimension-independent mixing rates, ensuring the algorithms are effective in high-dimensional settings. This work is in collaboration with N. Glatt-Holtz, C. Mondaini, and G. Senn.
October 30
(No seminar)
November 6
Speaker: Ioakeim Ampatzoglou (CUNY Baruch and GC)
Title:The use of collisional averaging in the study of hard-potential kinetic equations
Abstract: In this talk, we summarize recent progress made in the understanding of the well-posedness and regularity of hard-potential kinetic equations through the means of collisional averaging. We will present results related to the global well-posedness of the space inhomogeneous 4-wave KWE, a well-posedness/illposedness threshold for quasilinear KWE of MMT-type, as well as moment preserving weighted Young’s inequalities for the Boltzmann gain operator.
Title: A functional analytic framework for computer-assisted proofs in the axisymmetric Navier-Stokes equations with swirl
Abstract: The global regularity of solutions to the 3D Navier–Stokes equations is famously difficult. At the time of writing, the axisymmetric case with swirl is perhaps the simplest geometry in which the existence of blowup is unknown. We study this setting on a periodic cylinder with no-slip boundary conditions and time-independent body forcing, where weak solutions are guaranteed to have a global attractor and computer-assisted proofs are uniquely positioned to give rigorous results in a non-perturbative regime. In this geometry, the equations reduce to a system in a periodic axial variable and a radial variable. Our approach for developing computer-assisted proofs of steady states of the Navier-Stokes equations in a cylindrical domain uses a Fourier–Jacobi (Zernike) basis.
In any problem one wants the basis used to express solutions to have several desirable properties: easy to express differential operators, easy to compute products, and rapid convergence properties. Fourier series are best for periodic domains, but there is no uniformly best option for radial variables. A Fourier–Jacobi basis has its advantages --- the differential operators act as sparse ladder and conversion operators between Jacobi families; multiplication via interpolation may be computed accurately and relatively quickly; the spectral coefficients of solutions converge geometrically --- however laborious quantitative estimates are required for the computer-assisted proof.