TDA for Graphs: Persistent Homology Meets GNNs
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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 Neuroscience: Topology of Brain Connectivity
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The brain is a network with rich topological structure. Clique topology (Reimann et al., 2017) used persistent homology to show that neocortical microcircuits form high-dimensional simplicial structures, far beyond random. PH-based connectivity fingerprints distinguish subjects, tasks, and disease states from fMRI. This post covers the key findings and open questions in topological neuroscience.
TDA for Biology and Genomics: Shape in the Life Sciences
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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.
Topological Layers in Neural Networks: Differentiable TDA
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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.