Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap.

Its central concept is the fractal dimension that measures the distributive irregularity of a spatial point set.

Multifractal analysis has revealed regularities in many self-seeding phenomena, yet its use in modern deep learning remains limited. Existing end-to-end multifractal methods rely on heavy pooling or strong feature-space decimation, which constrain tasks such as semantic segmentation. Motivated by these limitations, we introduce two inductive priors: Monofractal and Multifractal Recalibration. These methods leverage relationships between the probability mass of the exponents and the multifractal spectrum to form statistical descriptions of encoder embeddings, implemented as channel-attention functions in convolutional networks. Using a U-Net-based framework, we show that multifractal recalibration yields substantial gains over a baseline equipped with other channel-attention mechanisms that also use higher-order statistics. Given the proven ability of multifractal analysis to capture pathological regularities, we validate our approach on three public medical-imaging datasets: ISIC18 (dermoscopy), Kvasir-SEG (endoscopy), and BUSI (ultrasound). Our empirical analysis also provides insights into the behavior of these attention layers. We find that excitation responses do not become increasingly specialized with encoder depth in U-Net architectures due to skip connections, and that their effectiveness may relate to global statistics of instance variability.

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erdős/Falconer type problems.

Modern datasets often contain high-dimensional features exhibiting complex dependencies. To effectively analyze such data, dimensionality reduction methods rely on estimating the dataset's intrinsic dimension (id) as a measure of its underlying complexity. However, estimating id is challenging due to its dependence on scale: at very fine scales, noise inflates id estimates, while at coarser scales, estimates stabilize to lower, scale-invariant values. This paper introduces a novel, scalable, and parallelizable method called eDCF, which is based on Connectivity Factor (CF), a local connectivity-based metric, to robustly estimate intrinsic dimension across varying scales. Our method consistently matches leading estimators, achieving comparable values of mean absolute error (MAE) on synthetic benchmarks with noisy samples. Moreover, our approach also attains higher exact intrinsic dimension match rates, reaching up to 25.0% compared to 16.7% for MLE and 12.5% for TWO-NN, particularly excelling under medium to high noise levels and large datasets. Further, we showcase our method's ability to accurately detect fractal geometries in decision boundaries, confirming its utility for analyzing realistic, structured data.

Nature and geometry furnish us with many such patterns, such as coastlines, snowflakes, the Cantor set and the Kuch curve.

While Graph Contrastive Learning (GCL) has attracted considerable attention in the field of graph self-supervised learning, its performance heavily relies on data augmentations that are expected to generate semantically consistent positive pairs. Existing strategies typically resort to random perturbations or local structure preservation, yet lack explicit control over global structural consistency between augmented views. To address this limitation, we propose Fractal Graph Contrastive Learning (FractalGCL), a theory-driven framework introducing two key innovations: a renormalisation-based augmentation that generates structurally aligned positive views via box coverings; and a fractal-dimension-aware contrastive loss that aligns graph embeddings according to their fractal dimensions, equipping the method with a fallback mechanism guaranteeing a performance lower bound even on non-fractal graphs. While combining the two innovations markedly boosts graph-representation quality, it also adds non-trivial computational overhead. To mitigate the computational overhead of fractal dimension estimation, we derive a one-shot estimator by proving that the dimension discrepancy between original and renormalised graphs converges weakly to a centred Gaussian distribution. This theoretical insight enables a reduction in dimension computation cost by an order of magnitude, cutting overall training time by approximately 61\%. The experiments show that FractalGCL not only delivers state-of-the-art results on standard benchmarks but also outperforms traditional and latest baselines on traffic networks by an average margin of about remarkably 4\%. Codes are available at (https://anonymous.4open.science/r/FractalGCL-0511/).

This enables a CNN to encode the regularity on the spatial arrangement of image features, leading to a robust yet discriminative spectrum descriptor.