Abstract: We are going to discuss some recent developments in the study of finite point configuration in sets of a given Hausdorff dimension. We shall also survey some applications of the finite point configuration machinery to the problems of existence and non-existence of exponential/Gabor bases and frames.
Abstract: Our goal here is to introduce recent developments of analysis of highly oscillatory functions. In particular we will sketch methods extending conventional Fourier analysis, exploiting both phase and amplitudes of holomorphic functions. The miracles of nonlinear complex holomorphic analysis, such as factorization and composition of functions lead to new versions of holomorphic orthonormal bases , relating them to multiscale dynamical systems, obtained by composing Blaschke factors.
We also, remark, that the phase of a Blaschke product is a one layer neural net with ($\arctan$ as an activation sigmoid) and that the composition is a "Deep Neural Net" whose depth is the number of compositions, our results provide a wealth of related libraries of orthogonal bases . We will also indicate a number of applications in medical signal processing , as well in precision Doppler. Each droplet in the phase image below represent a unit of a two layers deep net and gives rise to an orthonormal basis the Hardy space
Abstract: Formulating and solving boundary value problems for divergence form real elliptic equations has been an active and productive area of research ever since the foundational work of De Giorgi - Nash - Moser established Holder continuity of solutions when the coefficients are merely bounded and measurable. The solutions to such real-valued equations share some important properties with harmonic functions: maximum principles, Harnack principles, and estimates up to the boundary that enable one to solve Dirichlet problems in the classical sense of nontangential convergence. Solutions to complex elliptic equations and elliptic systems do not necessarily share these good properties of continuity or maximum principles. In joint work with M. Dindos, we introduce in 2017 a structural condition (p-ellipticity) on divergence form elliptic equations with complex valued matrices which was inspired by a condition related to Lp contractivity due to Cialdea and Maz'ya. The p-ellipticity condition that generalizes Cialdea-Maz'ya was also simultaneously discovered by Carbonaro-Dragicevic, who used it to prove a bilinear embedding result. Subsequently, Feneuil - Mayboroda - Zhao have used p-ellipticity to study well-posedness of a degenerate elliptic operator associated with domains with lower-dimensional boundary. In this seminar, we discuss p-ellipticity for complex divergence form equations, and then describe recent work, joint with J. Li and M. Dindos, extending this condition to elliptic systems. In particular, we can give applications to solvability of Dirichlet problems for the Lame systems.
Abstract: An open neighborhood U of 0 in Euclidean space is called symmetric if -U=U. Let PD(U) be the class of continuous positive definite functions supported on U and taking the value 1 at the origin. The Turan problem for U consists in computing the Turan constant of U, which is the supremum of the integrals of the functions in PD(U). Clearly, this problem can also be stated on any locally compact abelian group. In this talk, we will introduce the notion of "dual" Turan problem. In the case of a finite abelian group G, the Turan problem for a symmetric set S consists thus in maximizing the integral (which is just a finite sum) over G of the positive definite functions taking the value 1 at 0 and supported on S, while its dual is just the Turan problem for the set consisting of the complement of S together with the origin. We will show a surprising relationship between the maximizers of the Turan problem and those of the dual problem. In particular, their convolution product must be identically 1 on G. We then extend those results to Euclidean space by first finding an appropriate notion of dual Turan problem in this context. We will also point out an interesting connection between the Turan problem and frame theory by characterizing so-called Turan domains as domains admitting Parseval frames of (weighted) exponentials of a special kind.
Abstract: Very recently, square loss has been observed to perform well in classification tasks with deep networks. However, a theoretical justification is lacking, unlike the cross-entropy case for which an asymptotic analysis is available. Here we discuss several observations on the dynamics of gradient flow under the square loss in ReLU networks. We show how convergence to a local minimum norm solution is expected when normalization techniques such as Batch Normalization (BN) or Weight Normalization (WN) are used, in a way which is similar to the behavior of linear degenerate networks under gradient descent (GD), though the reason for zero-initial conditions is different. The main property of the minimizer that bounds its expected error is its norm: we prove that among all the interpolating solutions, the ones associated with smaller Frobenius norms of the weight matrices have better margin and better bounds on the expected classification error. The theory yields several predictions, including aspects of Donoho's Neural Collapse and the bias induced by BN on the weight matrices towards orthogonality.
Abstract: A result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem. The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)). There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in [1 ,..., N]^2, for example.
Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting. We will survey this area. Joint work with Rui Han and Fan Yang.
Abstract: The sparse matrix normal approximation to a random tensor valued variable is a sparse low rank approximation to the population covariance matrix. The approximation fits low dimensional Kronecker factors that can be interpreted as modal covariances of each mode of the tensor. The approximation has been formulated for approximating the covariance matrix and for approximating its inverse with both Kronecker products and Kronecker sums of factors. After introducing and illustrating the general framework, we will present a new approximation that is based on a generalized Sylvester representation that is exact for spatio-temporal random fields obeying a Poisson equation. These approximations will be illustrated for the application of predicting the spatio-temporal evolution of solar active regions (sunspots) and flares from NASA SDO image sensor data.
Abstract: It is known that the compactness of the commutators of bilinear homogeneous Caldero ÃÂn-Zygmund operators with pointwise multiplication when acting on product of Lebesgue spaces is characterized by the membership of the multiplying function in CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that, for bilinear Caldero ÃÂn-Zygmund operators arising from smooth (inhomogeneous) bilinear Fourier multipliers or bilinear pseudodifferential operators, one can actually consider multiplying functions in a new subspace of BMO larger than CMO. In this talk we will recall some introductory material to put the work in context and then present the new results. The new results are joint work with Qingying Xue from Beijing Normal University.
Abstract: This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday, and this talk is dedicated to him. Years ago, John wrote a text "Real Variable and Integration", published in 1976. This was not the text that I first learned real analysis from, but it became an important reference for me. A later revision and expansion by John and Wojtek Czaja appeared in 2009. Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory. My goal was that each proof was to be both rigorous and enlightening. I failed (in the chapters on differentiation and absolute continuity). I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening. But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text. This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult. This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.
Abstract: Let g be a function in L2(R). By G_ÃÂ, àâ R^2 we denote the system of time-frequency shifts of g, G_à= {e 2 pi iÃÂxg(x â t)}(t,ÃÂ)âÂÂÃÂ.
A typical model set àis the rectangular lattice ÃÂ(ñ,ò) := ñZÃÂòZ and one of the basic problems of the Gabor analysis is the description of the frame set of g i.e., all pairs ñ, ò such that GÃÂñ,ò is a frame in L2(R).
It follows from the general theory that ñò ⤠1 is a necessary condition (we assume ñ, ò > 0, of course). Do all such ñ, ò belong to the frame set of g? Up to 2011 only few such functions g (up to translation, modulation, dilation and Fourier transform) were known. In 2011 K. Grochenig and J. Stockler extended this class by including the totally positive functions of finite type (uncountable family yet depending on finite number of parameters) and later added the Gaussian finite type totally positive functions.
We suggest another approach to the problem and prove that all Herglotz rational functions with imaginary poles also belong to this class. This approach also gives new results for general rational functions.
Abstract: Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This talk describes a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment problems. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over 10^14 entries.
This talk will highlight the ideas behind the algorithm in a streamlined setting. The insights include a careful problem formulation, design of a bespoke optimization method, and use of randomized matrix computations.
Joint work with Alp Yurtsever, Olivier Fercoq, Madeleine Udell, and Volkan Cevher. Based on arXiv 1912.02949 (Scalable SDP, SIMODS 2021) and other papers (SketchyCGM in AISTATS 2017, Nyström sketch in NeurIPS 2017).
Seminar Link: https://umd.zoom.us/j/98101606043
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