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Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an exclusive topological imprint of data by tracing the evolution of latent structures as a scale parameter changes. Present PH tools are confined to analyzing data through a single filter parameter. However, many scenarios necessitate the consideration of multiple relevant parameters to attain finer insights into the data. We address this issue by introducing the Effective Multidimensional Persistence (EMP) framework. This framework empowers the exploration of data by simultaneously varying multiple scale parameters. The framework integrates descriptor functions into the analysis process, yielding a highly expressive data summary. It seamlessly integrates established single PH summaries into multidimensional counterparts like EMP Landscapes, Silhouettes, Images, and Surfaces. These summaries represent data’s multidimensional aspects as matrices and arrays, aligning effectively with diverse ML models. We provide theoretical guarantees and stability proofs for EMP summaries. We demonstrate EMP’s utility in graph classification tasks, showing its effectiveness. Results reveal EMP enhances various single PH descriptors, outperforming cutting-edge methods on multiple benchmark datasets. 

Biased AI models result in unfair decisions. In response, a number of algorithmic solutions have been engineered to mitigate bias, among which the Synthetic Minority Oversampling Technique (SMOTE) has been studied, to an extent. Although the SMOTE technique and its variants have great potentials to help improve fairness, there is little theoretical justification for its success. In addition, formal error and fairness bounds are not clearly given. This paper attempts to address both issues. We prove and demonstrate that synthetic data generated by oversampling underrepresented groups can mitigate algorithmic bias in AI models, while keeping the predictive errors bounded. We further compare this technique to the existing state-of-the-art fair AI techniques on five datasets using a variety of fairness metrics. We show that this approach can effectively improve fairness even when there is a significant amount of label and selection bias, regardless of the baseline AI algorithm.