In a recent preprint, together with Bailleul and Chevyrev we introduced a class of random fields which try to model the basic properties of quantum fields. I will try to explain the basic ideas and some of the many open problems.
To read the preprint, please click here.
The signature kernel is one of the most powerful measures of similarity for sequences of arbitrary length accompanied with attractive theoretical guarantees from stochastic analysis. Previous algorithms to compute the signature kernel scale quadratically in terms of the length and the number of the sequences. To mitigate this severe computational bottleneck, we develop a random Fourier feature-based acceleration of the signature kernel acting on the inherently non-Euclidean domain of sequences. We show uniform approximation guarantees for the proposed unbiased estimator of the signature kernel, while keeping its computation linear in the sequence length and number. In addition, combined with recent advances on tensor projections, we derive two even more scalable time series features with favourable concentration properties and computational complexity both in time and memory. Our empirical results show that the reduction in computational cost comes at a negligible price in terms of accuracy on moderate-sized datasets, and it enables one to scale to large datasets up to a million time series.
A representation on the space of paths is a map which is compatible with the concatenation operation of paths, such as the path signature and Cartan development (or equivalently, parallel transport), and has been used to define characteristic functions for the law of stochastic processes. In this talk, we consider representations of surfaces which are compatible with the two distinct algebraic operations on surfaces: horizontal and vertical concatenation. To build these representations, we use the notion of higher parallel transport, which was first introduced to develop higher gauge theories. We will not assume any background in geometry or category theory. Based on a preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.
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High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.
By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.
Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades
Hypergraphs for multiscale cycles in structured data
Iris Yoon
Understanding the spatial structure of data from complex systems is a challenge of rapidly increasing importance. Even when data is restricted to curves in three-dimensional space, the spatial structure of data provides valuable insight into many scientific disciplines, including finance, neuroscience, ecology, biophysics, and biology. Motivated by concrete examples arising in nature, I will introduce hyperTDA, a topological pipeline for analyzing the structure of spatial curves that combines persistent homology, hypergraph theory, and network science. I will show that the method highlights important segments and structural units of the data. I will demonstrate hyperTDA on both simulated and experimental data. This is joint work with Agnese Barbensi, Christian Degnbol Madsen, Deborah O. Ajayi, Michael Stumpf, and Heather Harrington.
Minmax Connectivity and Persistent Homology
Ambrose Yim
We give a pipeline for extracting features measuring the connectivity between two points in a porous material. For a material represented by a density field f, we derive persistent homology related features by exploiting the relationship between dimension zero persistent homology of the density field and the min-max connectivity between two points. We measure how the min-max connectivity varies when spurious topological features of the porous material are removed under persistent homology guided topological simplification. Furthermore, we show how dimension one persistent homology encodes a relaxed notion of min-max connectivity, and demonstrate how we can summarise the multiplicity of connections between a pair of points by associating to the pair a sub-diagram of the dimension one persistence diagram.