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Series: Stochastics Seminar

Deducing the state or structure of a system from partial, noisy
measurements is a fundamental task throughout the sciences and
engineering. The resulting inverse problems are often ill-posed because
there are fewer measurements available than the ambient dimension of the
model to be estimated. In practice, however, many interesting signals
or models contain few degrees of freedom relative to their ambient
dimension: a small number of genes may constitute the signature of a
disease, very few parameters may specify the correlation structure of a
time series, or a sparse collection of geometric constraints may
determine a molecular configuration. Discovering, leveraging, or
recognizing such low-dimensional structure plays an important role in
making inverse problems well-posed.
In this talk, I will propose a unified approach to transform notions of
simplicity and latent low-dimensionality into convex penalty functions.
This approach builds on the success of generalizing compressed sensing
to matrix completion, and greatly extends the catalog of objects and
structures that can be recovered from partial information. I will focus
on a suite of data analysis algorithms designed to decompose general
signals into sums of atoms from a simple---but not necessarily
discrete---set. These algorithms are derived in a convex optimization
framework that encompasses previous methods based on l1-norm
minimization and nuclear norm minimization for recovering sparse vectors
and low-rank matrices. I will provide sharp estimates of the number of
generic measurements required for exact and robust recovery of a variety
of structured models. These estimates are based on computing certain
Gaussian statistics related to the latent model geometry. I will detail
several example applications and describe how to scale the corresponding
inference algorithms to very large data sets.
(Joint work with Venkat Chandrasekaran, Pablo Parrilo, and Alan Willsky)

Series: Stochastics Seminar

In this talk we re-visit Fisher's controversial fiducial technique for
conducting statistical inference. In particular, a generalization of
Fisher's technique, termed generalized fiducial inference, is
introduced. We illustrate its use with wavelet regression. Current
and future work for generalized fiducial inference will also be
discussed.
Joint work with Jan Hannig and Hari Iyer

Series: Stochastics Seminar

We study the problem of testing for the significance of a subset of
regression coefficients in a linear model under the assumption that the
coefficient vector is sparse, a common situation in modern
high-dimensional settings. Assume there are p variables and let S be
the number of nonzero coefficients. Under moderate sparsity levels,
when we may have S > p^(1/2), we show that the analysis of variance
F-test is essentially optimal. This is no longer the case under the
sparsity constraint S < p^(1/2). In such settings, a multiple
comparison procedure is often preferred and we establish its optimality
under the stronger assumption S < p^(1/4). However, these two very
popular methods are suboptimal, and sometimes powerless, when p^(1/4)
< S < p^(1/2). We suggest a method based on the Higher Criticism
that is essentially optimal in the whole range S < p^(1/2). We
establish these results under a variety of designs, including the
classical (balanced) multi-way designs and more modern `p > n'
designs arising in genetics and signal processing.
(Joint work with Emmanuel Candès and Yaniv Plan.)

Series: Stochastics Seminar

For general graphs, approximating the maximum clique is a notoriously hard
problem even to approximate to a factor of nearly n, the number of vertices.
Does the situation get better with random graphs? A random graph on n
vertices where each edge is chosen with probability 1/2 has a clique of size
nearly 2\log n with high probability. However, it is not know how to find one
of size 1.01\log n in polynomial time. Does the problem become easier if a
larger clique were planted in a random graph? The current best algorithm can
find a planted clique of size roughly n^{1/2}. Given that any planted clique
of size greater than 2\log n is unique with high probability, there is a
large gap here. In an intriguing paper, Frieze and Kannan introduced a
tensor-based method that could reduce the size of the planted clique to as
small as roughly n^{1/3}. Their method relies on finding the spectral norm
of a 3-dimensional tensor, a problem whose complexity is open. Moreover,
their combinatorial proof does not seem to extend beyond this threshold. We
show how to recover the Frieze-Kannan result using a purely probabilistic
argument that generalizes naturally to r-dimensional tensors and allows us
recover cliques of size as small as poly(r).n^{1/r} provided we can find the
spectral norm of r-dimensional tensors. We highlight the algorithmic
question that remains open.
This is joint work with Charlie Brubaker.

Series: Stochastics Seminar

If $X_1,...,X_n$ are a random sample from a density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique log-concave maximum likelihood estimator $\hat{f}_n$ of $f$. The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. We exhibit an iterative algorithm for computing the estimator and show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities. Applications to classification, clustering and functional estimation problems will be discussed, as well as recent theoretical results on the performance of the estimator. The talk will be illustrated with pictures from the R package LogConcDEAD.
Co-authors: Yining Chen, Madeleine Cule, Lutz Duembgen (Bern), RobertGramacy (Cambridge), Dominic Schuhmacher (Bern) and Michael Stewart

Series: Stochastics Seminar

I will discuss certain geometric properties of random matrices
with independent logarithmically concave columns, obtained in the last
several years jointly with O. Guedon, A. Litvak, A. Pajor and N.
Tomczak-Jaegermann. In particular I will discuss estimates on the largest
and smallest singular values of such matrices and rates on convergence of
empirical approximations to covariance matrices of log-concave measures
(the Kannan-Lovasz-Simonovits problem).

Series: Stochastics Seminar

We study the spectral properties of matrices of long-range percolation model. These are N*N random real symmetric matrices H whose elements are independent random variables taking zero value with probability 1-\psi((i-j)/b), b\in \R^{+}, where \psi is an even positive function with \psi(t)<1 and vanishing at infinity. We show that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the ensemble we study in the limit N,b\to\infty. In the second part, we study the leading term of the correlation function of the resolvent G(z)=(H-z)^{-1} with large enough |Imz| in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3<\alpha<1. We show that this leading term, when considered in the local spectral scale leads to an expression found earlier by other authors for band random matrix ensembles. This shows that the ensemble we study and that of band random matrices belong to the same class of spectral universality.

Series: Stochastics Seminar

The G-equation is a Hamilton-Jacobi level-set equation, that is used in turbulent combustion theory. Level sets of the solution represent a ﬂame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. In this work I will discuss the large-scale long-time asymptotics of these solutions when the fluid velocity is modeled as a stationary incompressible random field. The main challenge of this work comes from the fact that our Hamiltonian is noncoercive. This is a joint work with J.Nolen.

Series: Stochastics Seminar

A two-player zero-sum stochastic differential game, defined in terms of an m-dimensional state process that is driven by a one-dimensional Brownian motion, played until the state exits the domain, is studied.The players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process. We show that the game has value, and characterize the value function as the unique viscosity solution of an inhomogeneous infinity Laplace equation.Joint work with R. Atar.

Series: Stochastics Seminar

We study a problem of estimation of a large Hermitian nonnegatively definite
matrix S of unit trace based on n independent measurements
Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n,
where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean
zero random variables independent of {X_j}. Problems of this nature are
of interest in quantum state tomography, where S is an unknown density
matrix of a quantum system. The estimator is based on penalized least
squares method with
complexity penalty defined in terms of von Neumann entropy.
We derive oracle inequalities showing how the estimation error depends on the
accuracy of approximation of the unknown state S by low-rank matrices.
We will discuss these results as well as some of the tools used in their
proofs (such as generic chaining bounds for empirical processes and
noncommutative Bernstein type inequalities).