Regression for sets of polynomial equations

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Franz Kiraly, Paul Von Buenau, Jan Muller, Duncan Blythe, Frank Meinecke, Klaus-Robert Muller ;
Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, PMLR 22:628-637, 2012.

Abstract

We propose a method called ideal regression for approximating an arbitrary system of polynomial equations by a system of a particular type. Using techniques from approximate computational algebraic geometry, we show how we can solve ideal regression directly without resorting to numerical optimization. Ideal regression is useful whenever the solution to a learning problem can be described by a system of polynomial equations. As an example, we demonstrate how to formulate Stationary Subspace Analysis (SSA), a source separation problem, in terms of ideal regression, which also yields a consistent estimator for SSA. We then compare this estimator in simulations with previous optimization-based approaches for SSA.

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