Research

2021	Third-Order Moment Varieties of Linear Non-Gaussian Graphical Models	arXiv, mathrepo
with Carlos Améndola, Mathias Drton, Roser Homs, Elina Robeva	A directed graph corresponds to a statistical model where the nodes represent random variables and arrows encode relations between them. The polynomial relations between entries of the covariance matrices have been previously studied from an algebraic and combinatorial point of view. Dropping Gaussianity makes moment tensors of higher order meaningful. Extending the combinatorial description from covariance matrices to higher-order tensors, we can use computational algebra to obtain a description for the third-order moment variety for trees. We also describe the polytopes coming from the moment parametrisation and explain the situation for graphs with hidden variables.
2020	Exact Solutions in Log-Concave MLE	arXiv, journal, github
with Kaie Kubjas, Olga Kuznetsova, Alexander Heaton, Georgy Scholten, Miruna-Stefana Sorea	In nonparametric statistics one abandons the requirement that a probability density function belongs to a statistical model. Given a sample of points, we want to find the best log-concave distribution that fits this data. This problem was originally studied by statisticians, who found that the optimal solutions have probability density functions whose logarithm is piecewise-linear and used numerical methods to approximate the pdf. In this work we use algebraic and combinatorial methods to provide exact solutions to this problem. At the same time we use tools from algebraic geometry to test if the solutions provided by statistical software can be certified to be correct. How many regions of linearity does the depicted tent function have?
2018	Moment ideals of local Dirac mixtures	arXiv, journal
with Markus Wageringel	Algebraic methods can find wide use in parameter estimation from statistical moments. In this work we use tools from elimination theory, the extended Prony method, and numerical algebra to reconstruct parameters from mixtures of local Diracs. Computational algebra mixes successfully with applications in statistics and signal processing. The picture shows the moment hypersurface of a first order local Dirac mixture, i.e. the variety parametrized by the moments. It is the tangent to the twisted cubic, in the same way that regular mixing gives rise to secants.

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