First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper presents a novel approach for column sampling when the data point clusters comprise of non-convex hulls. Column sampling is important in selecting a small subset of data that represents the properties of the original dataset. The presented approach is based upon the computation of Zeta hulls. The authors model the graph cycles by means of the sum-product rule and integrate them using the Zeta function. The authors set up the optimization problem as finding the subset of points with the strongest point extremenesses.

Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured data information into column sampling, research efforts were devoted to the cases where data points are fitted with clusters, simplices, or general convex hulls. This paper aims to study nonconvex hull learning which has rarely been investigated in the literature. In order to learn data-adaptive nonconvex hulls, a novel approach is proposed based on a graph-theoretic measure that leverages graph cycles to characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform structured column sampling. The process of pursuing a Zeta hull involves the computation of matrix inverse. To accelerate the matrix inversion computation and reduce its space complexity as well, we exploit a low-rank approximation to the graph adjacency matrix by using an efficient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can achieve state-of-the-art accuracy in text and image classification tasks.

In network analysis, various matrix representations have been developed to investigate the structural properties of the corresponding network, among which the non-backtracking (NBT) matrix is one such representation. The NBT matrix is well-known for its relationship with the Ihara zeta function [15] defined as an infinite product over equivalence classes of primitive cycles. It was shown in [15, 36] that, for regular graphs, the reciprocal of the Ihara zeta function can be expressed as a polynomial related to the adjacency matrix. The relation between the zeta function and the determinant of the NBT matrix was elucidated in [12], extended to irregular graphs in [3], and an elementary proof was provided in [35]. The connection between the polynomial and the determinant of the NBT matrix, via the Ihara zeta function, is known as the Ihara's formula.

Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured data information into column sampling, research efforts were devoted to the cases where data points are fitted with clusters, simplices, or general convex hulls. This paper aims to study nonconvex hull learning which has rarely been investigated in the literature. In order to learn data-adaptive nonconvex hulls, a novel approach is proposed based on a graph-theoretic measure that leverages graph cycles to characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform structured column sampling. The process of pursuing a Zeta hull involves the computation of matrix inverse. To accelerate the matrix inversion computation and reduce its space complexity as well, we exploit a low-rank approximation to the graph adjacency matrix by using an efficient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can achieve state-of-the-art accuracy in text and image classification tasks.

In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].

The Metropolis algorithm is one of the Markov chain Monte Carlo (MCMC) methods that realize sampling from the target probability distribution. In this paper, we are concerned with the sampling from the distribution in non-identifiable cases that involve models with Fisher information matrices that may fail to be invertible. The theoretical adjustment of the step size, which is the variance of the candidate distribution, is difficult for non-identifiable cases. In this study, to establish such a principle, the average acceptance rate, which is used as a guideline to optimize the step size in the MCMC method, was analytically derived in non-identifiable cases. The optimization principle for the step size was developed from the viewpoint of the average acceptance rate. In addition, we performed numerical experiments on some specific target distributions to verify the effectiveness of our theoretical results.

While loopy Belief Propagation (LBP) has been utilized in a wide variety of applications with empirical success, it comes with few theoretical guarantees. Especially, if the interactions of random variables in a graphical model are strong, the behaviors of the algorithm can be difficult to analyze due to underlying phase transitions. In this paper, we develop a novel approach to the uniqueness problem of the LBP fixed point; our new "necessary and sufficient" condition is stated in terms of graphs and signs, where the sign denotes the types (attractive/repulsive) of the interaction (i.e., compatibility function) on the edge. In all previous works, uniqueness is guaranteed only in the situations where the strength of the interactions are "sufficiently" small in certain senses. In contrast, our condition covers arbitrary strong interactions on the specified class of signed graphs. The result of this paper is based on the recent theoretical advance in the LBP algorithm; the connection with the graph zeta function.

Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured data information into column sampling, research efforts were devoted to the cases where data points are fitted with clusters, simplices, or general convex hulls. This paper aims to study nonconvex hull learning which has rarely been investigated in the literature. In order to learn data-adaptive nonconvex hulls, a novel approach is proposed based on a graph-theoretic measure that leverages graph cycles to characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform structured column sampling. The process of pursuing a Zeta hull involves the computation of matrix inverse. To accelerate the matrix inversion computation and reduce its space complexity as well, we exploit a low-rank approximation to the graph adjacency matrix by using an efficient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can achieve state-of-the-art accuracy in text and image classification tasks.

Detecting underlying clusters from large-scale data plays a central role in machine learning research. In this paper, we attempt to tackle clustering problems for complex data of multiple distributions and large multi-scales. To this end, we develop an algorithm named Zeta l -links, or Zell which consists of two parts: Zeta merging with a similarity graph and an initial set of small clusters derived from local l -links of the graph. More specifically, we propose to structurize a cluster using cycles in the associated subgraph. A mathematical tool, Zeta function of a graph, is introduced for the integration of all cycles, leading to a structural descriptor of the cluster in determinantal form.

This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive \textit{Sequence Extension Mixture Model} (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann $\zeta$ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.