Georgia Institute of Technology College of Sciences

One of the foundational ideas in modern machine learning is the scaling hypothesis: that machine learning models will improve in a predictable manner, with each doubling of resources leading to a commensurate improvement in abilities.  These were formalized for large language models in the Kaplan et al. scaling laws.

This is an almost entirely empirically observed law, which motivates the development probabilistic models that can explain these laws and to ultimately inform how to answer fundamental questions, such as: what can improve these laws? Or what causes them to break?

In this talk I’ll focus on a simple random matrix model of these scaling laws, the power law random features model, which motivates new iteration of stochastic algorithms which have the potential to change these scaling laws.  This random matrix model is not fully solved, and there are many open questions, both in pure probability and machine learning that rise in this study.

We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$, extending the work of Duke, Imamo\={g}lu and T\'{o}th in the level $4$ setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the $\xi_{\frac{1}{2}}$-operator.  We deduce a linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers, giving rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla--Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo\={g}lu. I wil lbe discussing joint work with Andreas Mono and Ngoc Trinh Le.

The theory of dynamical frames arose from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of future states of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations,  whose  frame generators are unique (up to equivalence). In this talk, we will address the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han and Larson in 2000 for group representation frames, but also proves that the frame vectors for any system of the form $\{A_1^{n_1}\cdots A_k^{n_k}: n_j\geq 0\}$, where  $A_1,...,A_k \in B(H)$ commute,  are equivalent. This is joint work with Deguang Han, Keri Kornelson, David Larson, and Rui Liu.

An $S_k$-set is a subset of a group whose $k$-tuples have distinct products. We give explicit constructions of large $S_k$-sets in the groups $\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)$ and of large $S_2$-sets in $\mathrm{Sym}(n)\times\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)\times\mathrm{Alt}(n)$, as well as some probabilistic constructions for 'nice' groups. We show that $k$ is even and $\Gamma$ has a normal abelian subgroup with bounded index then any $S_k$-set has size at most $(1-\varepsilon)|\Gamma|^{1/k}$. We describe some connections between $S_k$-sets and extremal graph theory. We determine up to a constant factor the largest possible minimum outdegree in a digraph with no subgraph in $\{C_{2,2},\ldots,C_{k,k}\}$, where $C_{\ell,\ell}$ is the orientation of $C_{2\ell}$ with two maximal directed $\ell$-paths. Contrasting with undirected cycles, the extremal minimum outdegree for $\{C_{2,2},\ldots,C_{k,k}\}$ is much smaller than that for any $C_{\ell,\ell}$. We count the directed Hamilton cycles in one of our constructions to improve the upper bound for a problem on Hamilton paths introduced by Cohen, Fachini, and Körner. Joint work with Michael Tait.