# Singular leaning coefficients and efficiency in learning theory

arXiv: https://arxiv.org/abs/2501.12747 · HTML: https://mlsanity.com/paper/2501.12747/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2025.

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

- [An Asymptotic Equation Linking WAIC and WBIC in Singular Models](https://mlsanity.com/paper/2505.13902/) (2025)

## Abstract (arXiv)

Singular learning models with non-positive Fisher information matrices include neural networks, reduced-rank regression, Boltzmann machines, normal mixture models, and others. These models have been widely used in the development of learning machines. However, theoretical analysis is still in its early stages. In this paper, we examine learning coefficients, which indicate the general learning efficiency of deep linear learning models and three-layer neural network models with ReLU units. Finally, we extend the results to include the case of the Softmax function.