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Series: Analysis Seminar

Recently, Awasthi et al proved that a semidefinite relaxation of the k-means clustering problem is tight under a particular data model called the stochastic ball model. This result exhibits two shortcomings: (1) naive solvers of the semidefinite program are computationally slow, and (2) the stochastic ball model prevents outliers that occur, for example, in the Gaussian mixture model. This talk will cover recent work that tackles each of these shortcomings. First, I will discuss a new type of algorithm (introduced by Bandeira) that combines fast non-convex solvers with the optimality certificates provided by convex relaxations. Second, I will discuss how to analyze the semidefinite relaxation under the Gaussian mixture model. In this case, outliers in the data obstruct tightness in the relaxation, and so fundamentally different techniques are required. Several open problems will be posed throughout.This is joint work with Takayuki Iguchi and Jesse Peterson (AFIT), as well as Soledad Villar and Rachel Ward (UT Austin).

Series: Analysis Seminar

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions,
pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood.
Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

Series: Analysis Seminar

We use Fourier analysis to establish $L^p$ bounds for Stein's spherical maximal theorem in the setting of compactly supported Borel measures $\mu, \nu$ satisfying natural local size assumptions $\mu(B(x,r)) \leq Cr^{s_{\mu}}, \nu(B(x,r)) \leq Cr^{s_{\nu}}$. As an application, we address the following geometric problem: Suppose that $E\subset \mathbb{R}^d$ is a union of translations of the unit circle, $\{z \in \mathbb{R}^d: |z|=1\}$, by points in a set $U\subset \mathbb{R}^d$. What are the minimal assumptions on the set $U$ which guarantee that the $d-$dimensional Lebesgue measure of $E$ is positive?

Series: Analysis Seminar

This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calder\'on's andCoifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset\mathbb C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel:\vskip-1.0em$$H(w, z) = \frac{1}{2\pi i}\frac{dw}{w-z}$$\smallskip\vskip-0.7em\noindent is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogueof $H(w, z)$. This is because of geometric obstructions (the Levi problem) that in dimension 1 are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Lerayin the context of a $C^\infty$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (much less Lipschitz!) Leray's construction becomes conceptually problematic.In this talk I will present {\em(a)}, the construction of theCauchy-Leray kernel and {\em(b)}, the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called ``\,$T(1)$-theorem technique'' from real harmonic analysis.Time permitting, I will describe applications of this work to complex function theory -- specifically, to the Szeg\H o and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

Series: Analysis Seminar

Consistent reconstruction is a method for estimating a signal from a
collection of noisy linear measurements that are corrupted by uniform
noise. This problem arises, for example, in analog-to-digital
conversion under the uniform noise model for memoryless scalar
quantization. We shall give an overview of consistent reconstruction
and prove optimal mean squared error bounds for the quality of
approximation. We shall also discuss an iterative alternative (due to
Rangan and Goyal) to consistent reconstruction which is also able to
achieve optimal mean squared error; this is closely related to the
classical Kaczmarz algorithm and provides a simple example of the power
of randomization in numerical algorithms.

Series: Analysis Seminar

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L_\alpha^\Phi(\mathbb B^n)$, which are generalizations of classical Bergman spaces. Weobtain their atomic decomposition and then prove weak factorization theorems involving the Bloch space and Bergman-Orlicz space and also weak factorization involving two Bergman-Orlicz spaces. This talk is based on joint work with D. Bekolle and A. Bonami.

Series: Analysis Seminar

In this talk we will discuss the connection between functions
with bounded mean oscillation (BMO) and commutators of Calderon-Zygmund
operators. In particular, we will discuss how to characterize certain BMO
spaces related to second order differential operators in terms of Riesz
transforms adapted to the operator and how to characterize commutators when
acting on weighted Lebesgue spaces.

Series: Analysis Seminar

We generalize the Calderon commutator to the higher-dimensional multicommutator with more input functions in higher dimensions. For this new multilinear operator, we establish the strong boundedness of it in all possible open points by a new multilinear multiplier theorem utilizing a new type of Sobolev spaces.

Series: Analysis Seminar

The asymptotic distribution of the zeros of two families of multiple
orthogonal polynomials will be given, namely the Jacobi-Pineiro polynomials (which
are an extension of the Jacobi polynomials) and the multiple Laguerre polynomials
of the first kind (which are an extension of the Laguerre polynomials). We use the
nearest neighbor recurrence relations for these polynomials and a recent result on
the ratio asymptotics of multiple orthogonal polynomials. We show how these
asymptotic zero distributions are related to the Fuss-Catalan distribution.

Series: Analysis Seminar

Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.