TDA for Neuroscience: Topology of Brain Connectivity
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Open Problems in Topological Machine Learning
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Topological Machine Learning is a young and rapidly growing field. Open problems include: multiparameter persistence (no complete discrete invariant yet), scalability to million-point clouds, theoretical understanding of what PH features encode in learned representations, unification with sheaf theory, and the design of truly end-to-end differentiable TDA architectures.
TDA for Biology and Genomics: Shape in the Life Sciences
less than 1 minute read
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Biology is full of shape: protein folding, cell morphology, gene expression manifolds. TDA has found major applications in single-cell RNA-seq analysis (Mapper for trajectory inference), protein structure comparison (PH-based structural fingerprints), and cancer genomics (topological signatures of tumour heterogeneity). This post surveys the key results and tools.
TDA for Graphs: Persistent Homology Meets GNNs
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Persistent homology can be applied directly to graphs by defining filtrations on nodes or edges (e.g., by WL colours, degree, or learned scalars). The resulting persistence diagrams encode global graph topology — connectivity, cycles, cliques — beyond what standard 1-WL GNNs can detect. This post covers WL-filtrations, extended persistence on graphs, and hybrid GNN+PH architectures.
Topological Layers in Neural Networks: Differentiable TDA
less than 1 minute read
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To train neural networks end-to-end with topological loss terms, we need to differentiate through the persistent homology computation. This post covers the gradient of persistence diagrams with respect to inputs (via sub-gradients through the reduction algorithm), differentiable vectorisations (PLLay, TopNet), and how to write a topological regulariser that encourages or discourages specific shape features during training.