Abstract: To address the mismatch between the output form of topological features extracted from persistent homology and the common input form of machine learning algorithms,this paper proposes a new algorithmic framework－Spatial Pyramid Bag of Words Algorithm Based on Persistent Homology(PHSBoW).This algorithm transforms the persistent diagrams(PDs) generated by persistent homology into fixed-length vectors while maximizing the retention of the topological features contained within the PD diagrams.To improve accuracy and reduce runtime,this paper further develops three algorithms－PHSsBoW,PHSwBoW,and PHSVLAD—based on the PHSBoW algorithm through enhancements like weight optimization,substitution with clustering mo-dels,and expansion of the bag of words model.By conducting experiments on nine datasets of varying types and scales,it combines these four algorithms with support vector machines for classification.The experimental results indicate that,compared to traditional kernel function algorithms(SWK,PSSK,PWGK) and vectorization algorithms(PBoW,PI,PL),classification accuracy is improved on average by 3.29 percentage points to 17.98 percentage points,and runtime is significantly reduced relative to kernel function algorithms.This demonstrates that these algorithms effectively address the challenges of integrating persistent homology into machine learning while significantly enhancing classification accuracy and algorithm execution speed.

Key words: Persistence homology, Bag of words, Spatial pyramid matching, Machine learning, Persistence diagrams