Magnitude of metric spaces has been shown to relate closely to more familiar notions such as volume, dimension, total curvature, and so forth. 

On the other hand, theoretical open questions are raised by counterintuitive phenomena such as the non-continuity of magnitude and irregularity in the coefficients of the asymptotic expansion of magnitude functions. 

One interesting point is that such unexpected cases occur even under mild hypotheses, e.g., for finite metric spaces or Euclidean subsets. 

At the same time, the distinctive character of magnitude as an ``effective count of elements'' has motivated researchers to explore its usefulness on finite point sets arising naturally from mathematical data analysis, including analysis of neural network weights and the detection of point cloud boundary.

Thus, the magnitude of finite metric spaces deserves attention in both theoretical and applied directions.

In this vein, this dissertation focuses on the continuity of magnitude, weighting, and their variations and their application to certain signal processing tasks. 

We first establish theoretical results, including an adaptation of weighting and maximum diversity to construct numerical invariants for time series.

The first part also raises and answers a theoretical problem motivated by applications, namely how the cardinality of metric spaces affects the convergence of magnitude, i.e., whether $\textrm{Mag\ } X_n \rightarrow \textrm{Mag\ }X$ when $X_n \rightarrow X$.

Next, we utilize the theory developed in the first part to define invariants of periodic time series that are stable under perturbations occurring in practical situations. 

As a proof-of-concept, we examine the invariants on real-world periodic time series data and especially show that they improve the performance of identification tasks. 

발표시간: 13:00 ~ 14:00