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In many cases classical catalecticants fail to reveal the apolar scheme of a form. In this talk we show how common eigenvectors and higher-order generalized eigenvectors encode the geometry of apolar schemes, leading from Waring to tangential...
Vincent Cohen-Addad is a Research Scientist at Google Research based in New-York. His work focuses on the design of inference methods for foundation models, and algorithms for optimization problems stemming from arising in data analysis and machine...
In this talk, we give an overview of different algebraic techniques used in the design and analysis of machine learning models. We show how we can use invariant theory to design symmetry-preserving machine learning models. We explain how Galois theory can deliver efficient, almost universal, machine learning models when the complexity of generating the ring of invariants is too large for practical purposes. And we describe how results in representation stability are related to any-dimensional machine learning models.
Moment maps provide a unifying language for functional theories in electronic structure theory. In this talk, we focus on one-particle reduced density matrix functional theory (RDMFT), which is conceptually well suited to describe the properties of strongly correlated many electron systems. First, we introduce the key tools for incorporating Lie group symmetries into functional theories and, in the case of spin symmetry, show how this identifies the natural variable in a spin-adapted RDMFT via the moment map. For spin-1/2 fermions, the domain of the universal functional is characterized by the moment polytope for the U(d) action on the N-fermion sector of the Hilbert space with fixed total spin S (and, optionally, magnetization M). We present new results on the symmetry-adapted pure and ensemble one-body N-representability problem that characterizes this domain.
In the second part of the talk, we introduce an ensemble formalism to target low-lying excited states within RDMFT. In this setting, the partial trace image of the convex hull of a suitable unitary orbit yields a polyhedral spectral domain describing the set of admissible one-particle reduced density matrices (1RDMs). We show that for these restricted ensembles, the resulting spectral constraints on the eigenvalues of the 1RDM constitute mixed-state generalizations of Pauli's exclusion principle. We then discuss the key structural properties of the facet-defining inequalities of the corresponding moment polytopes and highlight their implications for applications in quantum chemistry.
We will review a few recent developments around the foundations of Weingarten calculus. This includes theoretical formulas for the calculation of moments of centered random permutations, and new techniques to compute expectations of integrals on the unitary group, of polynomials of high degree. This is based on joint works with Manasa Nagatsu, and Sho Matsumoto.