[Aalto AGC Seminar] Nataliia Kushnerchuk | Posets of trek polynomials for directed graphical models | Mon 07.09. at 14:15

[Aalto Algebra, Geometry and Combinatorics Seminar]

We would like to invite you to the next talk of autumn in the Aalto AGC (Algebra, Geometry and Combinatorics) Seminar!

The talk is on
Mon 07.9. at 14:15-15:00
in
M3 (M234), Otakaari 1.

The speaker is
Nataliia Kushnerchuk (Aalto)
giving the talk
Posets of trek polynomials for directed graphical models

Abstract:
When a variety V𝜑 equals the image of a polynomial map 𝜑 whose coordinate
functions are combinatorial generating polynomials (i.e. polynomials enumerating combinatorial
objects), the geometry of V𝜑 reflects identities satisfied by the generating polynomials. The
resulting interplay between combinatorics and algebraic geometry can be used to answer questions about V𝜑 . A recent technique proposes to do so using a partially ordered set (poset) 𝑃𝜑 defined via the coefficient vectors of the polynomials defining 𝜑.
In this talk I am going to talk about a subfamily of Gaussian directed graphical models that we have studied with this new technique. For this family, the generating polynomials of 𝜑 defining a graphical model enumerate certain subgraphs known as treks. We characterized the poset 𝑃𝜑, and used the characterization to compute the linear span of V𝜑 , prove it is toric and deduce
a basis for its vanishing ideal. As an additional consequence, it is shown that the varieties for two distinct directed trees intersect in a strictly lower-dimensional variety. This solves an instance of the structural identifiability problem in the graphical models program from statistics.
Based on https://arxiv.org/abs/2608.08325

The webpage of the seminar is at
math.aalto.fi/en/research/discrete/seminars/agc/<https://math.aalto.fi/en/research/discrete/seminars/agc/>
and we have a mailing list at
https://list.aalto.fi/mailman/listinfo/agc-math-aalto



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