Vidit Nanda

Uzu Lim, Harald Oberhauser, Vidit Nanda

We introduce Hades, an unsupervised algorithm to detect singularities in data. This algorithm employs a kernel goodness-of-fit test, and as a consequence it is much faster and far more scaleable than the existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that Hades correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, Hades recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data.

Otto Sumray, Heather A. Harrington, Vidit Nanda

The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features.

Alexander M. Tanaka, Aras T. Asaad, Richard Cooper et al.

We introduce a new ligand-based virtual screening (LBVS) framework that uses piecewise linear (PL) Morse theory to predict ligand binding potential. We model ligands as simplicial complexes via a pruned Delaunay triangulation, and catalogue the critical points across multiple directional height functions. This produces a rich feature vector, consisting of crucial topological features -- peaks, troughs, and saddles -- that characterise ligand surfaces relevant to binding interactions. Unlike contemporary LBVS methods that rely on computationally-intensive deep neural networks, we require only a lightweight classifier. The Morse theoretic approach achieves state-of-the-art performance on standard datasets while offering an interpretable feature vector and scalable method for ligand prioritization in early-stage drug discovery.

Alexandros Dimitrios Keros, Vidit Nanda, Kartic Subr

Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes. By spectrally manipulating their combinatorial $k$-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each $k$-simplex from the nearest "optimal" $k$-th homology generator, effectively providing an alternative to homology localization.

Uzu Lim, Harald Oberhauser, Vidit Nanda

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure.

Ilya Chevyrev, Vidit Nanda, Harald Oberhauser

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.