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Series: Stochastics Seminar

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the ultimate goal of understanding the sample complexity of Cryo-EM. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations. This enables us to derive matching lower and upper bounds for the optimal rate of statistical estimation for the underlying signal. These bounds show a striking dependence on the signal-to-noise ratio of the problem. We also show how a tensor based method of moments can solve the problem efficiently. Based on joint work with Afonso Bandeira (NYU), Amelia Perry (MIT), Amit Singer (Princeton) and Jonathan Weed (MIT).

Series: Stochastics Seminar

We obtain an extension of
the Ito-Nisio theorem to certain non separable Banach spaces and apply
it to the continuity of the Ito map and Levy processes. The Ito map
assigns a rough path input of an ODE to its solution (output).
Continuity of this map usually
requires strong, non separable, Banach space norms on the path space.
We consider as an input to this map a series expansion a Levy process
and study the mode of convergence of the corresponding series of
outputs. The key to this approach is the validity of
Ito-Nisio theorem in non separable Wiener spaces of certain functions
of bounded p-variation.
This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.

Series: Stochastics Seminar

I will discuss two projects concerning Mallows permutations, with Ander
Holroyd, Tom Hutchcroft and Avi Levy. First, we relate the Mallows
permutation to stable matchings, and percolation on bipartite graphs.
Second, we study the scaling limit of the cycles in the Mallows
permutation, and relate it to diffusions and continuous trees.

Series: Stochastics Seminar

Cars are placed with density p on the lattice. The remaining vertices are parking spots that can fit one car. Cars then drive around at random until finding a parking spot. We study the effect of p on the availability of parking spots and observe some intriguing behavior at criticality. Joint work with Michael Damron, Janko Gravner, Hanbeck Lyu, and David Sivakoff. arXiv id: 1710.10529.

Series: Stochastics Seminar

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications.
Joint work with Pascal Maillard.

Series: Stochastics Seminar

The Sherrington-Kirkpatirck (SK) model is
a mean-field spin glass introduced by theoretical physicists in order
to explain the strange behavior of certain alloys, such as CuMn. Despite
of its seemingly simple formulation, it was conjectured to possess a
number of profound properties. This talk will be focused on the energy
landscapes of the SK model and the mixed p-spin model with both Ising
and spherical configuration spaces. We will present Parisi formule for
their maximal energies followed by descriptions of the energy landscapes
near the maximum energy. Based on joint works with A. Auffinger, M. Handschy, G. Lerman, and A. Sen.

Series: Stochastics Seminar

When considering smooth functionals of dependent data, block bootstrap methods have enjoyed considerable success in theory and application. For nonsmooth functionals of dependent data, such as sample quantiles, the theory is less well-developed. In this talk, I will present a general theory of consistency and optimality, in terms of achieving the fastest convergence rate, for block bootstrap distribution estimation for sample quantiles under mild strong mixing assumptions. The case of density estimation will also be discussed. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the subsampling bootstrap and the moving block bootstrap (MBB). Examples of `time series’ models illustrate the benefits of optimally choosing the number of blocks. This is joint work with Stephen M.S. Lee (University of Hong Kong) and Alastair Young (Imperial College London).

Series: Stochastics Seminar

We present a unified framework for sequential low-rank matrix completion and estimation, address the joint goals of uncertainty quantification (UQ) and statistical design. The first goal of UQ aims to provide a measure of uncertainty of estimated entries in the unknown low-rank matrix X, while the second goal of statistical design provides an informed sampling or measurement scheme for observing the entries in X. For UQ, we adopt a Bayesian approach and assume a singular matrix-variate Gaussian prior the low-rank matrix X which enjoys conjugacy. For design, we explore deterministic design from information-theoretic coding theory. The effectiveness of our proposed methodology is then illustrated on applications to collaborative filtering.

Series: Stochastics Seminar

The study of graph-partition problems such as Maxcut, max-bisection and
min-bisection have a long and rich history in combinatorics and theoretical
computer science. A recent line of work studies these problems on sparse random
graphs, via a connection with mean field spin glasses. In this talk, we will look
at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erdös-Rényi and random regular graphs.
Based on joint work with Aukosh Jagannath.

Series: Stochastics Seminar

We consider the $\textit{linearly transformed spiked model}$, where observations $Y_i$ are noisy linear transforms of unobserved signals of interest $X_i$: $$Y_i = A_i X_i + \varepsilon_i,$$ for $i=1,\ldots,n$. The transform matrices $A_i$ are also observed. We model $X_i$ as random vectors lying on an unknown low-dimensional space. How should we predict the unobserved signals (regression coefficients) $X_i$? The naive approach of performing regression for each observation separately is inaccurate due to the large noise. Instead, we develop optimal linear empirical Bayes methods for predicting $X_i$ by "borrowing strength'' across the different samples. Our methods are applicable to large datasets and rely on weak moment assumptions. The analysis is based on random matrix theory. We discuss applications to signal processing, deconvolution, cryo-electron microscopy, and missing data in the high-noise regime. For missing data, we show in simulations that our methods are faster, more robust to noise and to unequal sampling than well-known matrix completion methods. This is joint work with William Leeb and Amit Singer from Princeton, available as a preprint at arxiv.org/abs/1709.03393.