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

翻译：本文研究了SUCPA（通过先验自适应进行半无监督校准）算法的收敛性分析，该算法由一阶非线性差分方程定义，最初用于校正由监督机器学习分类器输出的分数。收敛性分析被作为动力系统问题处理，通过研究算法导出的非线性映射的局部与全局稳定性。该映射由指数函数与有理函数复合定义，呈现非双曲特性，且具有非孤立不动点的无界集。因此，本文采用了一种非标准方法进行收敛性分析，即一种特设的几何方法。针对二分类问题（二维映射），我们严格证明了该映射具有全局渐近稳定性。通过两种不同分类问题进行的真实世界应用数值实验支持了理论结果：使用大型语言模型进行的情感极性分类，以及猫狗图像分类。对于更多类别的情况，数值证据显示算法呈现相同行为，并通过自然语言推理示例加以说明。实验代码可通过以下仓库公开获取：https://github.com/LautaroEst/sucpa-convergence