# Cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents

arXiv: https://arxiv.org/abs/1804.03154 · HTML: https://mlsanity.com/paper/1804.03154/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2018.

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

- [Free Random Projection for In-Context Reinforcement Learning](https://mlsanity.com/paper/2504.06983/) (2025)

## Abstract (arXiv)

For random matrix models, the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix. We introduce a new parameter optimization method for random matrix models which works even in such a case. The method is based on the spectral distribution instead of the traditional likelihood. In the method, the Cauchy noise has an essential role because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate the spectral distribution perturbed by Cauchy noises by a smooth and accessible density function. Moreover, we study an asymptotic property of determination gap, which has a similar role as generalization gap. Besides, we propose a new dimensionality recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the rank of the signal part even if the true rank is not small. It is a simultaneous rank selection and parameter estimation procedure.