Learn2Extend: Extending sequences by retaining their statistical properties with mixture models (2312.01507v1)

Abstract: 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.

• The paper presents an auto-regressive Sequence Extension Mixture Model (SEMM) that preserves statistical features such as gap distribution and pair correlation.
• The model outperforms basic neural network architectures, demonstrating superior multi-step prediction on diverse point processes including Poisson and CUE sequences.
• Its robust methodology suggests practical applications in fields like astronomy, finance, and the study of the Riemann zeta function's zeros.

The paper presents a novel machine learning method for extending sequences of real numbers while preserving their inherent statistical characteristics. Specifically, the focus is on maintaining the gap distribution and pair correlation function of point sets. Point processes play an essential role in diverse fields such as astronomy, social media analysis, finance, and neuroscience due to their ability to model events in space and time. However, extending existing point samples while keeping their key statistical features intact can be challenging and computationally expensive.

This challenge is tackled by using an auto-regressive Sequence Extension Mixture Model (SEMM), which estimates the conditional density rather than the intensity function of point processes, offering a more flexible approach that can cater to multiple types of sequences. Comparative experiments conclude that this mixture model outperforms basic neural network architectures in sequence extension tasks, achieving better retention of statistical properties such as gap distribution and pair correlation.

The mixture model’s generalizability was tested using various types of point sequences, including Poisson sequences, eigenvalues of the Circular Unit Ensemble (CUE), and zeroes of the ζ function, demonstrating its capability to effectively extend these sequences while preserving their statistical properties. Furthermore, the model’s performance on multi-step term prediction was evaluated, indicating its potential application to real-world scenarios such as extending the sequence of zeroes of the Riemann zeta function—a function deeply connected with number theory and known for its complex zero distribution.

In conclusion, the paper makes a significant contribution to the area of point process extension by proposing a machine learning-based method capable of predicting future terms of a sequence while retaining key statistical aspects. This approach is versatile, extending beyond traditional neural networks and offering new possibilities for mathematical and real-world applications where understanding and maintaining the structure of point processes is fundamental.