Unsupervised Features Learning for Sampled Vector Fields

In this paper we introduce a new approach to computing hidden features of sampled vector fields. The basic idea is to convert the vector field data to a graph structure and use tools designed for automatic, unsupervised analysis of graphs. Using a few data sets we show that the collected features of the vector fields are correlated with the dynamics known for analytic models which generates the data. In particular the method may be useful in analysis of data sets where the analytic model is poorly understood or not known

CAPD::RedHom - Reduction heuristics for homology algorithms

We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined with smart implementation

Persistence Bag-of-Words for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.Comment: Accepted for the Twenty-Eight International Joint Conference on Artificial Intelligence (IJCAI-19). arXiv admin note: substantial text overlap with arXiv:1802.0485

Computing Fundamental Group via Forman’s Discrete Morse Theory Extended abstract

We present research in progress on the algorithmic computation of the fundamental group of a CW complex. We use the algorithm to compute certain algebraic invariants of the fundamental group of the complement of a knot. We show that the invariants classify the prime knots up to 13 crossings. The long term goal is an automated classification of knots in 3D images, in particular images of proteins

Persistence codebooks for topological data analysis

Persistent homology is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence codebooks, a novel expressive and discriminative fixed-size vectorized representation of PDs that adapts to the inherent sparsity of persistence diagrams. To this end, we adapt bag-of-words, vectors of locally aggregated descriptors and Fischer vectors for the quantization of PDs. Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability. We evaluate the presented representations on several heterogeneous datasets and show their (high) discriminative power. Our approach yields comparable-and partly even higher-performance in much less time than alternative approaches

The efficiency of a homology algorithm based on discrete morse theory and coreductions (extended abstract)

Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms

Z_2-homology of weak (p-2)-faceless p-pseudomanifolds may be computed in O(n) time

We consider the class of weak $(p-2)$-faceless $p$-pseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with $\mathbb{Z}_2$ coefficients may be computed in time $O(n)$ and the $\mathbb{Z}_2$ homology generators in time $O(nm)$ where $n$ denotes the cardinality of the $p$-pseudomanifold on input and $m$ is the number of homology generators