Georgia Institute of Technology College of Sciences

In this talk, I will discuss some examples of sparse signal detection problems in the context of binary outcomes. These will be motivated by examples from next generation sequencing association studies, understanding heterogeneities in large scale networks, and exploring opinion distributions over networks. Moreover, these examples will serve as templates to explore interesting phase transitions present in such studies. In particular, these phase transitions will be aimed at revealing a difference between studies with possibly dependent binary outcomes and Gaussian outcomes. The theoretical developments will be further complemented with numerical results.

High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning due to the well-known curse of dimensionality. Many data sets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees that are only cursed by the intrinsic dimension of data. Specifically, in the setting where a data set in $R^D$ consists of samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$, we consider two sets of problems: low-dimensional geometric approximation to the manifold and regression of a function on the manifold. In the first case we construct multiscale low-dimensional empirical approximations to the manifold and give finite-sample performance guarantees. In the second case we exploit these empirical geometric approximations of the manifold to construct multiscale approximations to the function. We prove finite-sample guarantees showing that we attain the same learning rates as if the function was defined on a Euclidean domain of dimension $d$. In both cases our approximations can adapt to the regularity of the manifold or the function even when this varies at different scales or locations. All algorithms have complexity $C n\log (n)$ where $n$ is the number of samples, and the constant $C$ is linear in $D$ and exponential in $d$.

Asymptotic equivalence between two statistical models means that they have the same asymptotic (large sample) properties with respect to all decision problems with bounded loss. In nonparametric (infinite-dimensional) statistical models, asymptotic equivalence has been found to be useful since it can allow one to derive certain results by studying simpler models. One of the key results in this area is Nussbaum’s theorem, which states that nonparametric density estimation is asymptotically equivalent to a Gaussian shift model, provided that the densities are smooth enough and uniformly bounded away from zero.We will review the notion of asymptotic equivalence and existing results, before presenting recent work on the extent to which one can relax the assumption of being bounded away from zero. We further derive the optimal (Le Cam) distance between these models, which quantifies how close they are for finite-samples. As an application, we also consider Poisson intensity estimation with low count data. This is joint work with Johannes Schmidt-Hieber.

Hybrid particle/grid numerical methods have been around for a long time, andtheir usage is common in some fields, from plasma physics to artist-directedfluids. I will explore the use of hybrid methods to simulate many differentcomplex phenomena occurring all around you, from wine to shaving foam and fromsand to the snow in Disney's Frozen. I will also talk about some of thepractical advantages and disadvantages of hybrid methods and how one of theweaknesses that has long plagued them can now be fixed.