Abstract. Persistent entropy summarizes persistence values within a single diagram, but many tasks require comparing how one diagram explains another. We introduce a reference-space construction that induces a probability measure on a reference diagram using persistence-weighted kernel similarities from an explaining diagram. This enables information-theoretic comparison without matching points directly. We define persistent cross entropy and show that its entropy excess is a Kullback-Leibler divergence, has a Jensen-gap interpretation, and is stable under Wasserstein perturbations.

Abstract. Zomoroidan and Carlsson (2005) described persistence modules in terms of representations, highlighting a one-to-one correspondence between persistence modules and graded modules over the polynomial ring k[t] by utilizing the structure theorem for finitely generated modules over a principal ideal domain. Carlsson and de Silva (2009) introduce zigzag modules, linking them to representations of A-type quivers. They propose a new constructive proof of Gabriel's theorem in the special case of A-type quivers, providing a practical algorithm for computing decompositions of zigzag modules. As a Topological Data Analysis community member, studying quiver theory appears inevitable due to its connection to persistence modules. In this talk, I will introduce the quiver theory and explain what Gabriel did.

Abstract. Category theory, a powerful mathematical framework, has transformed the way we approach complex systems. This seminar explores why we should study category theory in mathematics and how it helps us better understand diverse concepts by providing a new language for expressing relationships. Category theory looks at the big picture, ignoring specific details to uncover common structures and patterns in different areas. This approach gives us valuable tools to make sense of complex phenomena. Moreover, category theory acts as a bridge between seemingly unrelated subjects, showing how they are connected. This seminar makes a compelling case for the relevance of category theory, illustrating how it enriches our understanding and brings together different ideas, making mathematics and other fields more accessible.

Abstract. We introduce how to build binary trees from data and how to choose dimension cuts and split values. Using a density measure, we propose weighted partitioning process and we will examine it applications.

Abstract. We propose a new partitioning process (PP), so-called the weighted partitioning process (WPP). In order to introduce the WPP, we first introduce a new geometric measure, density measure, which is crucial for WPP. In order to introduce the WPP, we first introduce a new geometric measure, density measure, which is crucial for the construction of the WPP. We provide various mathematical properties of the density measure. The proposed WPP also cuts the tree network adaptively, but with the consideration of the denseness of the data. Anomaly detection algorithms using WPP is more efficient when the data is non-uniformly structured and achieves the desired anomaly scores more rapidly than the ones using PP.

Abstract. We propose a new partitioning process (PP), so-called the weighted partitioning process (WPP). In order to introduce the WPP, we first introduce a new geometric measure, density measure, which is crucial for the construction of the WPP. We provide various mathematical properties of the density measure. The proposed WPP also cuts the tree network adaptively,  but with the consideration of the denseness of the data. Anomaly detection algorithms using WPP is more efficient when the data is non-uniformly structured and achieves the desired anomaly scores more rapidly than the  ones using PP.

Abstract.  The Königsberg bridge problem, which ask "can all the seven bridges be traversed in a single tour?",  is the first step of the graph theory and the graph theory is also the first step of topology. We prove the Euler theorem which tells you that the answer is 'No' and will talk about my subjective impressions.

Abstract. Random cut forest (RCF) algorithms have been developed for anomaly detection, particularly for anomaly detection in time-series data. The RCF algorithm is the improved version of the isolation forest algorithm. Unlike the isolation forest algorithm, the RCF algorithm has the power of determining whether the real-time input has anomaly by inserting the input in the constructed tree network. There have been developed various RCF algorithms including Robust RCF (RRCF) with which the cutting procedure is adaptively chosen probabilistically. The RRCF algorithm shows better performance than the isolation forest as the cutting dimension is decided based on the geometric range of the data. The overall data structure is, however, not considered in the adaptive cutting algorithm with the RRCF algorithm. In this paper, we propose a new RCF algorithm, so-called the weighted RCF (WRCF) algorithm. In order to introduce the WRCF, we first introduce a new geometric measure, a density measure which is crucial for the construction of the WRCF. We provide various mathematical properties of the density measure. The proposed WRCF also cuts the tree network adaptively, but with the consideration of the denseness of the data. The proposed algorithm is more efficient when the data is non-uniformly structured and achieves the desired anomaly scores more rapidly than the RRCF. We provide theorems that prove our claims with numerical examples. 

Abstract.  A Robust Random Cut Tree (RRCT) from a data set is obtained by applying a random cut method repeatedly. The random cut method is an algorithm that partitions a set into two disjoint sets by choosing two random variables p and q.  Instead of treating p as a uniformly distributed random cut variable, we give p an appropriate distribution to improve the RRCF algorithm.

Abstract.  Hilbert's basis theorem says that if a ring is Notherian, the polynomial ring over the ring is also Noetherian. Hence, we only need finite number of polynomials to generate the polynomial ring.