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.
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