Dept. of Mathematics, Warren Center for Network and Data Sciences, University of Pennsylvania

The usual framework for TDA takes as its starting point that a data set is sampled (noisily) from a manifold embedded in a high dimensional space, and provides a reconstruction of topological features of that manifold. However, the underlying algebraic topology can be applied to data in a much broader sense, carries much richer information about the system than just the barcodes, and can be fine-tuned so it sees only features of the data we want it to see. I will discuss this framework broadly, with focus on few of these alternative viewpoints, including applications to neuroscience and matrix factorization.

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