# Blow-up Algorithm for Sum-of-Products Polynomials and Real Log Canonical Thresholds

arXiv: https://arxiv.org/abs/2303.11619 · HTML: https://mlsanity.com/paper/2303.11619/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 3 of 3)

- [The Exact Asymptotic Form of Bayesian Generalization Error in Latent Dirichlet Allocation](https://mlsanity.com/paper/2008.01304/) (2020)
- [Upper Bound of Bayesian Generalization Error in Non-negative Matrix Factorization](https://mlsanity.com/paper/1612.04112/) (2016)
- [Asymptotic Behavior of Bayesian Generalization Error in Multinomial Mixtures](https://mlsanity.com/paper/2203.06884/) (2022)

## Built on by (most cited first, 1 of 1)

- [Dead Directions: Geometric Singular Learning](https://mlsanity.com/paper/2606.05957/) (2026)

## Abstract (arXiv)

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.