Blow-up Algorithm for Sum-of-Products Polynomials and Real Log Canonical Thresholds
This work addresses a theoretical gap in algebraic geometry for statisticians and machine learning researchers, but it appears incremental as it focuses on a specific polynomial class rather than a general breakthrough.
The paper tackles the problem of resolving singularities for sum-of-products polynomials using a specific blow-up algorithm to compute real log canonical thresholds, which relate to Bayesian generalization errors, and shows that this algorithm can effectively handle these polynomials.
When considering a real log canonical threshold (RLCT) that gives a Bayesian generalization error, in general, papers replace a mean error function with a relatively simple polynomial whose RLCT corresponds to that of the mean error function, and obtain its RLCT by resolving its singularities through an algebraic operation called blow-up. Though it is known that the singularities of any polynomial can be resolved by a finite number of blow-up iterations, it is not clarified whether or not it is possible to resolve singularities of a specific polynomial by applying a specific blow-up algorithm. Therefore this paper considers the blow-up algorithm for the polynomials called sum-of-products (sop) polynomials and its RLCT.