While current works account for the importance ofdepth for the expressivepower ofneural-networks, itremains an open question whether these benefits are exploited during a gradient-based optimization process.

Ultimately, we find that neural networks' behavior on this

Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative optimization algorithms can possess \emph{fractal structures}, and their generalization error can be formally linked to the complexity of such fractals. This complexity is measured by the fractal's \emph{intrinsic dimension}, a quantity usually much smaller than the number of parameters in the network. Even though this perspective provides an explanation for why overparametrized networks would not overfit, computing the intrinsic dimension (\eg, for monitoring generalization during training) is a notoriously difficult task, where existing methods typically fail even in moderate ambient dimensions. In this study, we consider this problem from the lens of topological data analysis (TDA) and develop a generic computational tool that is built on rigorous mathematical foundations. By making a novel connection between learning theory and TDA, we first illustrate that the generalization error can be equivalently bounded in terms of a notion called the'persistent homology dimension' (PHD), where, compared with prior work, our approach does not require any additional geometrical or statistical assumptions on the training dynamics. Then, by utilizing recently established theoretical results and TDA tools, we develop an efficient algorithm to estimate PHD in the scale of modern deep neural networks and further provide visualization tools to help understand generalization in deep learning. Our experiments show that the proposed approach can efficiently compute a network's intrinsic dimension in a variety of settings, which is predictive of the generalization error.

The emergence of 5G and 6G networks has established network slicing as a significant part of future service-oriented architectures, demanding refined identification methods supported by robust datasets. The article presents SliceVision-F2I, a dataset of synthetic samples for studying feature visualization in network slicing for next-generation networking systems. The dataset transforms multivariate Key Performance Indicator (KPI) vectors into visual representations through four distinct encoding methods: physically inspired mappings, Perlin noise, neural wallpapering, and fractal branching. For each encoding method, 30,000 samples are generated, each comprising a raw KPI vector and a corresponding RGB image at low-resolution pixels. The dataset simulates realistic and noisy network conditions to reflect operational uncertainties and measurement imperfections. SliceVision-F2I is suitable for tasks involving visual learning, network state classification, anomaly detection, and benchmarking of image-based machine learning techniques applied to network data. The dataset is publicly available and can be reused in various research contexts, including multivariate time series analysis, synthetic data generation, and feature-to-image transformations.

Recent developments in applied mathematics increasingly employ machine learning (ML)-particularly supervised learning-to accelerate numerical computations, such as solving nonlinear partial differential equations. In this work, we extend such techniques to objects of a more theoretical nature: the classification and structural analysis of fractal sets. Focusing on the Mandelbrot and Julia sets as principal examples, we demonstrate that supervised learning methods-including Classification and Regression Trees (CART), K-Nearest Neighbors (KNN), Multilayer Perceptrons (MLP), and Recurrent Neural Networks using both Long Short-Term Memory (LSTM) and Bidirectional LSTM (BiLSTM), Random Forests (RF), and Convolutional Neural Networks (CNN)-can classify fractal points with significantly higher predictive accuracy and substantially lower computational cost than traditional numerical approaches, such as the thresholding technique. These improvements are consistent across a range of models and evaluation metrics. Notably, KNN and RF exhibit the best overall performance, and comparative analyses between models (e.g., KNN vs. LSTM) suggest the presence of novel regularity properties in these mathematical structures. Collectively, our findings indicate that ML not only enhances classification efficiency but also offers promising avenues for generating new insights, intuitions, and conjectures within pure mathematics.

Previous works have proved that recurrent neural networks (RNNs) are Turing-complete.