# On the Stability of a non-hyperbolic nonlinear map with non-bounded set of non-isolated fixed points with applications to Machine Learning

arXiv: https://arxiv.org/abs/2401.03051 · HTML: https://mlsanity.com/paper/2401.03051/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 32 references that are in the map.
- First posted on arXiv 5 Jan 2024.

## Builds on (most cited first, 7 of 7)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [Classifier Calibration: A survey on how to assess and improve predicted class probabilities](https://mlsanity.com/paper/2112.10327/) (2021)
- [Deployment of Image Analysis Algorithms under Prevalence Shifts](https://mlsanity.com/paper/2303.12540/) (2023)
- [An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms](https://mlsanity.com/paper/1811.06703/) (2018)
- [Unsupervised Calibration through Prior Adaptation for Text Classification using Large Language Models](https://mlsanity.com/paper/2307.06713/) (2023)

## Abstract (arXiv)

This paper deals with the convergence analysis of the SUCPA (Semi Unsupervised Calibration through Prior Adaptation) algorithm, defined from a first-order non-linear difference equations, first developed to correct the scores output by a supervised machine learning classifier. The convergence analysis is addressed as a dynamical system problem, by studying the local and global stability of the nonlinear map derived from the algorithm. This map, which is defined by a composition of exponential and rational functions, turns out to be non-hyperbolic with a non-bounded set of non-isolated fixed points. Hence, a non-standard method for solving the convergence analysis is used consisting of an ad-hoc geometrical approach. For a binary classification problem (two-dimensional map), we rigorously prove that the map is globally asymptotically stable. Numerical experiments on real-world application are performed to support the theoretical results by means of two different classification problems: Sentiment Polarity performed with a Large Language Model and Cat-Dog Image classification. For a greater number of classes, the numerical evidence shows the same behavior of the algorithm, and this is illustrated with a Natural Language Inference example. The experiment codes are publicly accessible online at the following repository: https://github.com/LautaroEst/sucpa-convergence