Related papers: Persistence Fisher Kernel: A Riemannian Manifold K…
Topological methods for data analysis present opportunities for enforcing certain invariances of broad interest in computer vision, including view-point in activity analysis, articulation in shape analysis, and measurement invariance in…
The Symmetric Positive Definite (SPD) matrices have received wide attention for data representation in many scientific areas. Although there are many different attempts to develop effective deep architectures for data processing on the…
Distributional comparison is a fundamental problem in statistical data analysis with numerous applications in a variety of scientific and engineering fields. Numerous methods exist for distributional comparison but kernel Stein's method has…
Wasserstein geometry and information geometry are two important structures introduced in a manifold of probability distributions. The former is defined by using the transportation cost between two distributions, so it reflects the metric…
We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework…
The Fisher information metric is an important foundation of information geometry, wherein it allows us to approximate the local geometry of a probability distribution. Recurrent neural networks such as the Sequence-to-Sequence (Seq2Seq)…
The Poisson-Nernst-Planck (PNP) equations are one of the most effective model for describing electrostatic interactions and diffusion processes in ion solution systems, and have been widely used in the numerical simulations of biological…
Persistent Homology (PH) has been successfully used to train networks to detect curvilinear structures and to improve the topological quality of their results. However, existing methods are very global and ignore the location of topological…
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…
State-of-the-art image-set matching techniques typically implicitly model each image-set with a Gaussian distribution. Here, we propose to go beyond these representations and model image-sets as probability distribution functions (PDFs)…
Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for…
We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward Euclidean representation exists. Typically, these tasks are performed by first reducing the high-dimensional data to…
An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The…
Domain specific (dis-)similarity or proximity measures used e.g. in alignment algorithms of sequence data, are popular to analyze complex data objects and to cover domain specific data properties. Without an underlying vector space these…
Data depth is a statistical function that generalizes order and quantiles to the multivariate setting and beyond, with applications spanning over descriptive and visual statistics, anomaly detection, testing, etc. The celebrated halfspace…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on…
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically use…
Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital…
Data sets tend to live in low-dimensional non-linear subspaces. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The symmetric Riemannian geometry setting can be suitable for a variety of…