Lieb-Robinson Bounds for the Toda Lattice

We establish locality estimates, known as Lieb-Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb-Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems

Lieb-Robinson Bounds for the Toda Lattice

We study locality properties of the Toda lattice in terms of Lieb-Robinson bounds. The estimates we prove produce a finite Lieb-Robinson velocity depending on the initial condition. Then we establish analogous results for certain perturbations of the Toda system. Finally, we obtain generalizations of our main results in the setting of the Toda hierarchy

A Computationally Efficient Framework for Vector Representation of Persistence Diagrams

In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in R2 computed using tools of algebraic topology. However, this multi-set structure limits the utility of PDs in applications. Therefore, in recent years efforts have been directed towards extracting informative and efficient summaries from PDs to broaden the scope of their use for machine learning tasks. We propose a computationally efficient framework to convert a PD into a vector in Rn, called a vectorized persistence block (VPB). We show that our representation possesses many of the desired properties of vector-based summaries such as stability with respect to input noise, low computational cost and flexibility. Through simulation studies, we demonstrate the effectiveness of VPBs in terms of performance and computational cost for various learning tasks, namely clustering, classification and change point detection

A fast topological approach for predicting anomalies in time-varying graphs

Large time-varying graphs are increasingly common in financial, social and biological settings. Feature extraction that efficiently encodes the complex structure of sparse, multi-layered, dynamic graphs presents computational and methodological challenges. In the past decade, a persistence diagram (PD) from topological data analysis (TDA) has become a popular descriptor of shape of data with a well-defined distance between points. However, applications of TDA to graphs, where there is no intrinsic concept of distance between the nodes, remain largely unexplored. This paper addresses this gap in the literature by introducing a computationally efficient framework to extract shape information from graph data. Our framework has two main steps: first, we compute a PD using the so-called lower-star filtration which utilizes quantitative node attributes, and then vectorize it by averaging the associated Betti function over successive scale values on a one-dimensional grid. Our approach avoids embedding a graph into a metric space and has stability properties against input noise. In simulation studies, we show that the proposed vector summary leads to improved change point detection rate in time-varying graphs. In a real data application, our approach provides up to 22% gain in anomalous price prediction for the Ethereum cryptocurrency transaction networks