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Series: Stochastics Seminar

The focus of my talk will be stochastic forms of isoperimetric
inequalities for convex sets. I will review some fundamental
inequalities including the classical isoperimetric inequality and
those of Brunn-Minkowski and Blaschke-Santalo on the product of
volumes of a convex body and its polar dual. I will show how one can
view these as global inequalities that arise via random approximation
procedures in which stochastic dominance holds at each stage. By laws
of large numbers, these randomized versions recover the classical
inequalities. I will discuss when such stochastic dominance arises
and its applications in convex geometry and probability. The talk
will be expository and based on several joint works with G. Paouris,
D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.

Series: Stochastics Seminar

In the late 80's, several relationships have been established
between the Information Theory and Convex Geometry, notably
through the pioneering work of Costa, Cover, Dembo and Thomas.
In this talk, we will focus on one particular relationship. More
precisely, we will focus on the following conjecture of Bobkov,
Madiman, and Wang (2011), seen as the analogue of the
monotonicity of entropy in the Brunn-Minkowski theory:
The inequality
$$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k
|\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$
holds for every compact sets $A_1, \dots, A_k \subset
\mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in
$\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes
the Minkowski sum of $A$ and $B$.
(Based on a joint work with M. Fradelizi, M. Madiman, and A.
Zvavitch.)

Series: Stochastics Seminar

Paper available on arXiv:1412.3661

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities Pr(n−1/2∑ni=1Xi∈A) where X1,…,Xn are independent random vectors in ℝp and Ais a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞ as n→∞ and p≫n; in particular, p can be as large as O(eCnc) for some constants c,C>0. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

Series: Stochastics Seminar

Log Brunn-Minkowski conjecture was proposed by Boroczky, Lutwak,
Yang and Zhang in 2013. It states that in the case of symmetric convex sets
the classical Brunn-MInkowski inequality may be improved. The Gaussian
Brunn-MInkowski inequality was proposed by Gardner and Zvavitch in 2007. It
states that for the standard Gaussian measure an inequality analogous to the
additive form of Brunn_minkowski inequality holds true for symmetric convex
sets. In this talk we shall discuss a derivation of an equivalent
infinitesimal versions of these inequalities for rotation invariant measures
and a few partial results related to both of them as well as to the
classical Alexander-Fenchel inequality.

Series: Stochastics Seminar

Series: Stochastics Seminar

For a random (complex) entire function, what can we say about the
behavior of the zero set of its N-th derivative, as N goes to infinity?
In this talk, we shall discuss the result of repeatedly differentiating a
certain class of random entire functions whose zeros are the points of a
Poisson process of intensity 1 on the real line. We shall also discuss the
asymptotic behavior of the coefficients of these entire functions. Based on
joint work with Robin Pemantle.

Series: Stochastics Seminar

We prove a central limit theorem for a class of additive processes that
arise naturally in the theory of finite horizon Markov decision problems.
The main theorem generalizes a classic result of Dobrushin (1956) for
temporally non-homogeneous Markov chains, and the principal innovation is
that here the summands are permitted to depend on both the current state
and a bounded number of future states of the chain. We show through several
examples that this added flexibility gives one a direct path to asymptotic
normality of the optimal total reward of finite horizon Markov decision
problems. The same examples also explain why such results are not easily
obtained by alternative Markovian techniques such as enlargement of the
state space. (Joint work with J. M. Steele.)

Series: Stochastics Seminar

Detecting change-points from high-dimensional streaming data is a fundamental problem that arises in many big-data applications such as video processing, sensor networks, and social networks. Challenges herein include developing algorithms that have low computational complexity and good statistical power, that can exploit structures to detecting weak signals, and that can provide reliable results over larger classes of data distributions. I will present two aspects of our recent work that tackle these challenges: (1) developing kernel-based methods based on nonparametric statistics; and (2) using sketching of high-dimensional data vectors to reduce data dimensionality. We also provide theoretical performance bounds and demonstrate the performance of the algorithms using simulated and real data.

Series: Stochastics Seminar

A significant achievement of modern probability theory is the
development of sharp connections between the boundedness of random
processes and the geometry of the underlying index set. In particular, the
generic chaining method of Talagrand provides in principle a sharp
understanding of the suprema of Gaussian processes. The multiscale
geometric structure that arises in this method is however notoriously
difficult to control in any given situation. In this talk, I will exhibit a
surprisingly simple but very general geometric construction, inspired by
real interpolation of Banach spaces, that is readily amenable to explicit
computations and that explains the behavior of Gaussian processes in various
interesting situations where classical entropy methods are known to fail.
(No prior knowledge of this topic will be assumed in the talk.)

Series: Stochastics Seminar

We consider the problem of recovering a set
of locations given observations of the direction between pairs of these
locations. This recovery task arises from the Structure from Motion
problem, in which a three-dimensional structure is sought from a
collection of two-dimensional images. In this context, the locations of
cameras and structure points are to be found from epipolar geometry and
point correspondences among images. These correspondences are often
incorrect because of lighting, shadows, and the effects of perspective.
Hence, the resulting observations of relative directions contain
significant corruptions. To solve the location recovery problem in the
presence of corrupted relative directions, we introduce a tractable
convex program called ShapeFit. Empirically, ShapeFit can succeed on
synthetic data with over 40% corruption. Rigorously, we prove that
ShapeFit can recover a set of locations exactly when a fraction of the
measurements are adversarially corrupted and when the data model is
random. This and subsequent work was done in collaboration with
Choongbum Lee, Vladislav Voroninski, and Tom Goldstein.