Georgia Institute of Technology College of Sciences

Several new results on asymptotic normality and other asymptotic properties of sample covariance operators for Gaussian observations in a high-dimensional setting will be discussed. Such asymptotics are of importance in various problems of high-dimensional statistics (in particular, related to principal component analysis). The proofs of these results rely on Gaussian concentration inequality. This is a joint work with Karim Lounici.

Motivated by the ubiquity of signal-plus-noise type models in high-dimensional statistical signal processing and machine learning, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Applications in mind are as diverse as radar, sonar, wireless communications, spectral clustering, bio-informatics and Gaussian mixture cluster analysis in machine learning. We provide an application-independent approach that brings into sharp focus a fundamental informational limit of high-dimensional eigen-analysis. Building on this success, we highlight the random matrix origin of this informational limit, the connection with "free" harmonic analysis and discuss how to exploit these insights to improve low-rank signal matrix denoising relative to the truncated SVD.

A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics model in Euclidean geometry and its counterpart in the random geometry. More recently, Duplantier and Sheffield (2011) used the 2-dim Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. Inspired by the work of Duplantier and Sheffield, we apply a similar approach to extend their results and techniques to higher even dimensions R^(2n) for n>=2. This talk mainly focuses on the case of R^4. I will briefly introduce the notion of Gaussian free field (GFF). In our work we adopt a specific 4-dim GFF to construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) being the exponential of an instance of the GFF. Further we establish a 4-dim KPZ relation corresponding to this random measure. This work is joint with Dmitry Jakobson (McGill University).

The 2-core of a hypergraph is the unique subgraph where all vertices have degree at least 2 and which is the maximal induced subgraph with this property. This talk will be about the investigation of the 2-core for a particular random hypergraph model --- a model which differs from the usual random uniform hypergraph in that the vertex degrees are not identically distributed. For this model the main result proved is that as the size of the vertex set, n, tends to infinity then the number of hyperedges in the 2-core obeys a limit law, and this limit exhibits a threshold where the number of hyperedges in the 2-core transitions from o(n) to Theta(n). We will discuss aspects of the ideas involved and discuss the background motivation for the hypergraph model: factoring random integers into primes.