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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.

Series: Stochastics Seminar

Consider a sample of a centered random vector with unit covariance matrix.
We show that under certain regularity assumptions, and up to a natural
scaling, the smallest and the largest eigenvalues of the empirical
covariance matrix converge, when the dimension and the sample size both
tend to infinity, to the left and right edges of the Marchenko-Pastur
distribution. The assumptions are related to tails of norms of orthogonal
projections. They cover isotropic log-concave random vectors as well as
random vectors with i.i.d. coordinates with almost optimal moment
conditions. The method is a refinement of the rank one update approach used
by Srivastava and Vershynin to produce non-asymptotic quantitative
estimates. In other words we provide a new proof of the Bai and Yin theorem
using basic tools from probability theory and linear algebra, together with
a new extension of this theorem to random matrices with dependent entries.
Based on joint work with Djalil Chafai.

Series: Stochastics Seminar

Low-rank structure commonly arises in many applications including genomics, signal processing, and portfolio allocation. It is also used in many statistical inference methodologies such as principal component analysis. In this talk, I will present some recent results on recovery of a high-dimensional low-rank matrix with rank-one measurements and related problems including phase retrieval and optimal estimation of a spiked covariance matrix based on one-dimensional projections. I will also discuss structured matrix completion which aims to recover a low rank matrix based on incomplete, but structured observations.

Series: Stochastics Seminar

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the
Lebesgue measure in dimension n
would imply the log-BMI and, therefore, the B-conjecture for any even
log-concave measure in dimension n. As a consequence,
we prove the log-BMI and the B-conjecture for any even log-concave
measure,
in the plane. Moreover, we prove that the log-BMI
reduces to the following: For each dimension n, there is a density
f_n,
which satisfies an integrability assumption, so that the
log-BMI holds for parallelepipeds with parallel facets, for the density
f_n. As byproduct of our methods, we study possible
log-concavity of the function t -> |(K+_p\cdot e^tL)^{\circ}|,
where
p\geq 1 and K, L are symmetric convex bodies,
which we are able to prove in some instances and as a further
application,
we confirm the variance conjecture in a special class of convex bodies.
Finally, we establish
a non-trivial dual form of the log-BMI.

Series: Stochastics Seminar

Recently, new general bounds for the distance to the normal of a non-linear
functional have been obtained, both with Poisson input and with IID points
input. In the Poisson case, the results have been obtained by combining
Stein's method with Malliavin calculus and a 'second-order Poincare
inequality', itself obtained through a coupling involving Glauber's
dynamics. In the case where the input consists in IID points, Stein's
method is again involved, and combined with a particular inequality
obtained by Chatterjee in 2008, similar to the second-order Poincar?
inequality. Many new results and optimal speeds have been obtained for some
Euclidean geometric functionals, such as the minimal spanning tree, the
Boolean model, or the Voronoi approximation of sets.

Series: Stochastics Seminar

This is survey talk where, both for random words and random permutations, I
will present a panoramic view of the subject ranging from classical results to recent
breakthroughs. Throughout, equivalencies with some directed last passage percolation
models with dependent weights will be pointed out.

Series: Stochastics Seminar

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly
common task, tensor recovery remains a challenging problem because of the delicacy
associated with the decomposition of higher order tensors. We introduce a general
framework of convex regularization for low rank tensor estimation.