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We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai-R\'enyi map.

The application of deep learning to non-stationary temporal datasets can lead to overfitted models that underperform under regime changes. In this work, we propose a modular machine learning pipeline for ranking predictions on temporal panel datasets which is robust under regime changes. The modularity of the pipeline allows the use of different models, including Gradient Boosting Decision Trees (GBDTs) and Neural Networks, with and without feature engineering. We evaluate our framework on financial data for stock portfolio prediction, and find that GBDT models with dropout display high performance, robustness and generalisability with reduced complexity and computational cost. We then demonstrate how online learning techniques, which require no retraining of models, can be used post-prediction to enhance the results. First, we show that dynamic feature projection improves robustness by reducing drawdown in regime changes. Second, we demonstrate that dynamical model ensembling based on selection of models with good recent performance leads to improved Sharpe and Calmar ratios of out-of-sample predictions. We also evaluate the robustness of our pipeline across different data splits and random seeds with good reproducibility.

Text summarization is an essential task in natural language processing, and researchers have developed various approaches over the years, ranging from rule-based systems to neural networks. However, there is no single model or approach that performs well on every type of text. We propose a system that recommends the most suitable summarization model for a given text. The proposed system employs a fully connected neural network that analyzes the input content and predicts which summarizer should score the best in terms of ROUGE score for a given input. The meta-model selects among four different summarization models, developed for the Slovene language, using different properties of the input, in particular its Doc2Vec document representation. The four Slovene summarization models deal with different challenges associated with text summarization in a less-resourced language. We evaluate the proposed SloMetaSum model performance automatically and parts of it manually. The results show that the system successfully automates the step of manually selecting the best model.

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

Kinetic approaches are generally accurate in dealing with microscale plasma physics problems but are computationally expensive for large-scale or multiscale systems. One of the long-standing problems in plasma physics is the integration of kinetic physics into fluid models, which is often achieved through sophisticated analytical closure terms. In this paper, we successfully construct a multi-moment fluid model with an implicit fluid closure included in the neural network using machine learning. The multi-moment fluid model is trained with a small fraction of sparsely sampled data from kinetic simulations of Landau damping, using the physics-informed neural network (PINN) and the gradient-enhanced physics-informed neural network (gPINN). The multi-moment fluid model constructed using either PINN or gPINN reproduces the time evolution of the electric field energy, including its damping rate, and the plasma dynamics from the kinetic simulations. In addition, we introduce a variant of the gPINN architecture, namely, gPINN$p$ to capture the Landau damping process. Instead of including the gradients of all the equation residuals, gPINN$p$ only adds the gradient of the pressure equation residual as one additional constraint. Among the three approaches, the gPINN$p$-constructed multi-moment fluid model offers the most accurate results. This work sheds light on the accurate and efficient modeling of large-scale systems, which can be extended to complex multiscale laboratory, space, and astrophysical plasma physics problems.

The power of Clifford or, geometric, algebra lies in its ability to represent geometric operations in a concise and elegant manner. Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into non-commutative multivectors. The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension. The algorithm is a variation of the Faddeev-LeVerrier-Souriau algorithm and is implemented in the open-source Computer Algebra System Maxima. Symbolic and numerical examples in different Clifford algebras are presented.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.

Deep learning constitutes a recent, modern technique for image processing and data analysis, with promising results and large potential. As deep learning has been successfully applied in various domains, it has recently entered also the domain of agriculture. In this paper, we perform a survey of 40 research efforts that employ deep learning techniques, applied to various agricultural and food production challenges. We examine the particular agricultural problems under study, the specific models and frameworks employed, the sources, nature and pre-processing of data used, and the overall performance achieved according to the metrics used at each work under study. Moreover, we study comparisons of deep learning with other existing popular techniques, in respect to differences in classification or regression performance. Our findings indicate that deep learning provides high accuracy, outperforming existing commonly used image processing techniques.