Modelling higher-order dependence with cumulants, 11am in AP&M 2402

Models and estimators for covariance matrices are very well studied. For non-Gaussian distributions, simply studying covariance gives an incomplete picture. Extending the Edgeworth series gives the pxpxp skewness tensor, the pxpxpxp kurtosis tensor, and so on. We describe a strategy for building multilinear factor models of cumulant tensors using subspace varieties. This leads to a difficult optimization problem and a fully implicit, gradient-based numerical optimization method using parallel transport on the Grassmannian to perform estimation. We also discuss some of the associated statistical challenges and applications.

Pfaffian circuits, 2pm in AP&M 6402

Pfaffian circuits are a new, geometrically motivated, and simplified construction of Valiant's holographic algorithms. These algorithms exploit dual Spinor varieties to simulate certain quantum computations (fermionic linear optics) classically, and provide a means to probe the conjectured classical-quantum boundary. Combinatorial problems addressed include planar NAE-SAT, lattice path problems and evaluation of certain Tutte polynomials. Basis change is one route to superposition-like effects, and some of the geometric considerations in analyzing Pfaffi an circuits under arbitrary basis change will be discussed. Connections are made to the sum-product algorithm, SLOCC equivalent entangled states, and monoidal categories.

This workshop will bring together computer scientists and mathematicians to examine the P v. NP problem and its variants from the perspectives of algebra, geometry, and number theory, and to introduce the mathematical aspects of these questions to a larger audience. Diverse researchers working on different aspects of these problems will clarify connections between different approaches.

There will be two main topics: Analogues of P v. NP (e.g., Valiant's conjectures, the Mulmuley-Sohoni Conjecture, the BSS model, and other computational models); and Algebraic, Number Theoretic, and Geometric Aspects of P v. NP (e.g., Holographic algorithms, characterizations of NP in terms of sheaf cohomology, sparse polynomials, and other arithmetic approaches).