Anna Seigal

Chandler Squires, Anna Seigal, Salil Bhate et al.

Causal disentanglement seeks a representation of data involving latent variables that relate to one another via a causal model. A representation is identifiable if both the latent model and the transformation from latent to observed variables are unique. In this paper, we study observed variables that are a linear transformation of a linear latent causal model. Data from interventions are necessary for identifiability: if one latent variable is missing an intervention, we show that there exist distinct models that cannot be distinguished. Conversely, we show that a single intervention on each latent variable is sufficient for identifiability. Our proof uses a generalization of the RQ decomposition of a matrix that replaces the usual orthogonal and upper triangular conditions with analogues depending on a partial order on the rows of the matrix, with partial order determined by a latent causal model. We corroborate our theoretical results with a method for causal disentanglement that accurately recovers a latent causal model.

Paula Leyes Carreno, Chiara Meroni, Anna Seigal

Linear causal disentanglement is a recent method in causal representation learning to describe a collection of observed variables via latent variables with causal dependencies between them. It can be viewed as a generalization of both independent component analysis and linear structural equation models. We study the identifiability of linear causal disentanglement, assuming access to data under multiple contexts, each given by an intervention on a latent variable. We show that one perfect intervention on each latent variable is sufficient and in the worst case necessary to recover parameters under perfect interventions, generalizing previous work to allow more latent than observed variables. We give a constructive proof that computes parameters via a coupled tensor decomposition. For soft interventions, we find the equivalence class of latent graphs and parameters that are consistent with observed data, via the study of a system of polynomial equations. Our results hold assuming the existence of non-zero higher-order cumulants, which implies non-Gaussianity of variables.

Alvaro Ribot, Anna Seigal, Piotr Zwiernik

Independent Component Analysis (ICA) is a classical method for recovering latent variables with useful identifiability properties. For independent variables, cumulant tensors are diagonal; relaxing independence yields tensors whose zero structure generalizes diagonality. These models have been the subject of recent work in non-independent component analysis. We show that pairwise mean independence answers the question of how much one can relax independence: it is identifiable, any weaker notion is non-identifiable, and it contains the models previously studied as special cases. Our results apply to distributions with the required zero pattern at any cumulant tensor. We propose an algebraic recovery algorithm based on least-squares optimization over the orthogonal group. Simulations highlight robustness: enforcing full independence can harm estimation, while pairwise mean independence enables more stable recovery. These findings extend the classical ICA framework and provide a rigorous basis for blind source separation beyond independence.

Rishi Sonthalia, Anna Seigal, Guido Montufar

We define the supermodular rank of a function on a lattice. This is the smallest number of terms needed to decompose it into a sum of supermodular functions. The supermodular summands are defined with respect to different partial orders. We characterize the maximum possible value of the supermodular rank and describe the functions with fixed supermodular rank. We analogously define the submodular rank. We use submodular decompositions to optimize set functions. Given a bound on the submodular rank of a set function, we formulate an algorithm that splits an optimization problem into submodular subproblems. We show that this method improves the approximation ratio guarantees of several algorithms for monotone set function maximization and ratio of set functions minimization, at a computation overhead that depends on the submodular rank.

Max Pfeffer, Anna Seigal, Bernd Sturmfels

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.

Elina Robeva, Anna Seigal

In this article we show the duality between tensor networks and undirected graphical models with discrete variables. We study tensor networks on hypergraphs, which we call tensor hypernetworks. We show that the tensor hypernetwork on a hypergraph exactly corresponds to the graphical model given by the dual hypergraph. We translate various notions under duality. For example, marginalization in a graphical model is dual to contraction in the tensor network. Algorithms also translate under duality. We show that belief propagation corresponds to a known algorithm for tensor network contraction. This article is a reminder that the research areas of graphical models and tensor networks can benefit from interaction.

Anna Seigal, Guido Montufar

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.