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# Upcoming Seminars

The reducibility and structure of parabolic inductions is a basic problem in the representation theory of p-adic groups. Of particular interest are principal series and degenerate principal series representations, that is parabolic induction of 1-dimensional representations of Levi subgroups.

In this talk, I will start by describing the functor of normalized induction and its left adjoint, the Jacquet functor, and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations.

This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).

The talk will be about two classical problems in the theory of Carleman classes of smooth functions. The first one is to describe the image of a Carleman class under the Borel map (which maps a smooth function to its jet of Taylor coefficients at a given point). The second one concerns possible ways to construct a function in a given Carleman class with prescribed Taylor coefficients. I will present solutions to both problems. If time permits, I will also discuss related problems which originate in the singularity theory of Carleman classes.

The Blind Source Separation (BSS) problem consists of retrieving signals of interest, termed the sources, from a dataset consisting of their mixtures. One of the most popular and common paradigms for solving the BSS problem is Independent Component Analysis (ICA), where the sources are assumed to be (only) mutually statistically independent random processes, and the mixtures are assumed to be linear combinations thereof, where the linear mixing operator is unknown. In this talk, we shall start with the Gaussian Maximum Likelihood (GML) approach for the semi-blind problem, in which the sources are assumed to be temporally-dive

I will talk about my joint work with Dave Benson which constructs new symmetric tensor categories in characteristic 2 arising from modular representation theory of elementary abelian 2-groups, and about its conjectural generalization to characteristic p>2. I will also discuss my work with Gelaki and Coulembier which shows that integral symmetric tensor categories in characteristic p>2 whose simple objects form a fusion category are super-Tannakian (i.e., representation categories of a supergroup scheme).

In recent years, deep neural networks (DNNs) achieved unprecedented performance in many vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of millions of parameters. To make DNNs implementable on platforms with limited resources, it is necessary to weaken the tradeoff between performance and efficiency. In this work, we propose a new activation unit, which is suitable for both high and low level vision problems. In contrast to the widespread per-pixel activation units, like ReLUs and sigmoids, our unit implements a learnable nonlinear function with spatial connections. This enables the net to capture much more complex features, thus requiring a significantly smaller number of layers in order to reach the same performance. We illustrate the effectiveness of our units through experiments with state-of-the-art nets for classification, denoising, and super resolution. With our approach, we are able to reduce the size of these models by nearly 50% without incurring any degradation in performance.

*Spotlight presentation at CVPR'18 (+ submitted extension)

A joint work with Tamar Rott Shaham and Tomer Michaeli

The sharp threshold phenomenon is a central topic of research in the analysis of Boolean functions. Here, one aims to give sufficient conditions for a monotone Boolean function f to satisfy $\mu p(f)=o(\mu q(f))$, where $q = p + o(p)$, and $\mu p(f)$ is the probability that $f=1$ on an input with independent coordinates, each taking the value $1$ with probability $p$. The dense regime, where $\mu p(f)=\Theta(1)$, is somewhat understood due to seminal works by Bourgain, Friedgut, Hatami, and Kalai. On the other hand, the sparse regime where $\mu p(f)=o(1)$ was out of reach of the available methods. However, the potential power of the sparse regime was envisioned by Kahn and Kalai already in 2006. In this talk we show that if a monotone Boolean function $f$ with $\mu p(f)=o(1)$ satisfies some mild pseudo-randomness conditions then it exhibits a sharp threshold in the interval $[p,q]$, with $q = p+o(p)$. More specifically, our mild pseudo-randomness hypothesis is that the $p$-biased measure of $f$ does not bump up to $\Theta(1)$ whenever we restrict $f$ to a sub-cube of constant co-dimension, and our conclusion is that we can find $q=p+o(p)$, such that $\mu p(f)=o(\mu q(f))$ At its core, this theorem stems from a novel hypercontactive theorem for Boolean functions satisfying pseudorandom conditions, which we call `small generalized influences'. This result takes on the role of the usual hypercontractivity theorem, but is significantly more effective in the regime where $p = o(1)$. We demonstrate the power of our sharp threshold result by reproving the recent breakthrough result of Frankl on the celebrated Erdos matching conjecture, and by proving conjectures of Huang--Loh--Sudakov and Furedi--Jiang for a new wide range of the parameters. Based on a joint work with Keevash, Long, and Minzer.

We introduce a 3D shape generative model based on deep neural networks. A new image-like (i.e., tensor) data representation for genus-zero 3D shapes is devised. It is based on the observation that complicated shapes can be well represented by multiple parameterizations (charts), each focusing on a different part of the shape.

The new tensor data representation is used as input to Generative Adversarial Networks for the task of 3D shape generation. The 3D shape tensor representation is based on a multichart structure that enjoys a shape covering property and scale-translation rigidity. Scale-translation rigidity facilitates high quality 3D shape learning and guarantees unique reconstruction. The multi-chart structure uses as input adataset of 3D shapes (with arbitrary connectivity) and a sparse correspondence between them.

The output of our algorithm is a generative model that learns the shape distribution and is able to generate novel shapes, interpolate shapes, and explore the generated shape space. The effectiveness of the method is demonstrated for the task of anatomic shape generation including human body and bone (teeth) shape generation.

We explore the power of interactive proofs with a distributed verifier. In this setting, the verifier consists of n nodes and a graph G that defines their communication pattern. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph G belongs to some language in a small number of rounds, and with small communication bound, i.e., the proof size.

This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -- both of which require proofs of (n^2)-bits without interaction.

In this work, we provide a new general framework for distributed interactive proofs that allows one to translate standard interactive protocols (i.e., with a centralized verifier) to ones where the verifier is distributed with a proof size that depends on the computational complexity of the verification algorithm run by the centralized verifier. We show the following:

* Every (centralized) computation performed in time O(n) on a RAM can be translated into three-round distributed interactive protocol with O(log n) proof size. This implies that many graph problems for sparse graphs have succinct proofs (e.g., testing planarity).

* Every (centralized) computation implemented by either a small space or by uniform NC circuit can be translated into a distributed protocol with O(1) rounds and O(log n) bits proof size for the low space case and polylog(n) many rounds and proof size for NC.

* We show that for Graph Non-Isomorphism, one of the striking demonstrations of the power of interaction, there is a 4-round protocol with O(log n) proof size, improving upon the O(n*log n) proof size of Kol et al.

* For many problems, we show how to reduce proof size below the seemingly natural barrier of log n. By employing our RAM compiler, we get a 5-round protocol with proof size O(loglog n) for a family of problems including Fixed Automorphism, Clique and Leader Election (for the latter two problems we actually get O(1) proof size).

* Finally, we discuss how to make these proofs non-interactive {\em arguments} via random oracles.

Our compilers capture many natural problems and demonstrate the difficulty in showing lower bounds in these regimes.

Joint work with Moni Naor and Merav Parter.