Motivated by this challenge, we propose to construct surrogate models able to approximate the state of a graph at a given time from the knowledge of the diffusivity parameters. Specifically, we consider recently introduced high-dimensional approximation methods based on sparse polynomial expansions, which are known to produce accurate, sample-efficient approximations when the function to be approximated has holomorphic regularity. Hence, to justify our methodology, we will theoretically show that solution maps arising from a certain class of parametric graph diffusion processes are indeed holomorphic. Then, we will numerically illustrate that it is possible to efficiently compute accurate sparse polynomial surrogate models from a few random samples, hence empirically showing the validity of our approach.

Motivated by a desire to derive the information-theoretic lower bound on the number of bits needed to encode an AVL tree, we develop a new method for the study of combinatorial classes whose generating functions satisfy certain functional equations and use this tool to derive the growth rate of AVL trees and related structures. We also describe a new encoding for AVL trees that uses less than one bit per node.

Joint work with Stephen Melczer, J. Ian Munro, and Ava Pun.

In this poster, we generalise these results in the case of symplectic vector bundles. The essential dimension of a symplectic vector bundle can be broken into two terms, namely the transcendence degree of its field of moduli and the essential dimension of the bundle over its field of moduli. We show that the first part can be related to the essential dimension of hermitian modules over an algebra, while the second part can be bounded by the dimension of moduli stack of symplectic vector bundles equipped with nilpotent morphism of symplectic type. We prove that the latter object is smooth and hence its dimension can be computed by a Riemann-Roch formula.

To address this, our work introduces a novel approach, termed `greedy deep unfolding'. We utilize \emph{soft sorting} to approximate the argsort operator in a differentiable manner. Additionally, we reinterpret greedy algorithms through a projection-based lens and approximate the permutation matrices from argsort with stochastic matrices derived from soft sorting. Our numerical and theoretical analyses show that under certain conditions, the approximation error is minimal, and the performance of the approximated greedy algorithm closely matches the original. We then incorporate this approximate algorithm into a feedforward neural network's layers, integrating learnable \emph{weight} parameters to connect to weighted sparse recovery. Our numerical results demonstrate that this network is trainable and can surpass the performance of the traditional approximate greedy algorithm in certain scenarios.

The result we present is a degree threshold for $G_P$ that allows a fractional tile-decomposition, implying the existence of a fractional completion of $P$. The proof employs spectral decomposition, the properties of coherent configurations, and perturbation theory to estimate a generalized inverse for the matrix representation of a partial Sudoku puzzle in order to find a solution for the relaxed linear system. Improving on this result by finding a minimum degree threshold for an exact tile-decomposition is an interesting open question in this research area.

In this poster, we describe ongoing work to obtain a minimal presentation for the group $G$ of 3-qubit ancilla-free Toffoli+K circuits (where K is the two-fold tense of a Hadamard gate). From prior results in sphere packings, it follows that $G$ is isomorphic to the E8 Weyl group. We start from the Coxeter presentation of this Weyl group and obtain a circuit presentation using a semantic variation of Tietze transformations. We use commutativity relations to prove generator minimality and implement a proof-assistant to validate each semantic transformation. Directions for future work, such as minimizing the set of relations and generalizing to the 3-qubit ancilla-free Toffoli+K circuits, are outlined.