Fractal Dimension

Brain-age prediction from resting-state fMRI provides a quantitative framework for characterizing age-related changes in spontaneous brain dynamics and for identifying functional signatures. Existing studies have linked fractal and multifractal scaling to age and examined the reliability of individual features. However, prediction repeatability depends on how features fluctuate jointly and how a predictor combines them, which feature-wise reliability assessments do not capture. To address this problem, we propose Repeat-informed Multifractal Curve Regression (RMCR), a structured framework for learning stable age-predictive patterns from multifractal curves. By jointly modeling curve structure and repeat-scan variability, RMCR learns predictive combinations of fluctuation orders that target both accuracy and within-subject consistency. Relative to a matched run-level ridge baseline, RMCR reduces single-run MAE by 6.1% on HCP-A and 7.9% on an external Cam-CAN cohort, and within-visit repeat absolute difference by 18.5% on HCP-A, using a single scan at inference.

The heightened prevalence of respiratory disorders, particularly exacerbated by a significant upswing in fatalities due to the novel coronavirus, underscores the critical need for early detection and timely intervention. This imperative is paramount, possessing the potential to profoundly impact and safeguard numerous lives. Medically, chest radiography stands out as an essential and economically viable medical imaging approach for diagnosing and assessing the severity of diverse Respiratory Disorders. However, their detection in Chest X-Rays is a cumbersome task even for well-trained radiologists owing to low contrast issues, overlapping of the tissue structures, subjective variability, and the presence of noise. To address these issues, a novel analytical model termed Exponential Pixelating Integral is introduced for the automatic detection of infections in Chest X-Rays in this work. Initially, the presented Exponential Pixelating Integral enhances the pixel intensities to overcome the low-contrast issues that are then polar-transformed followed by their representation using the locally invariant Mandelbrot and Julia fractal geometries for effective distinction of structural features. The collated features labeled Exponential Pixelating Integral with dually characterized fractal features are then classified by the non-parametric multivariate adaptive regression splines to establish an ensemble model between each pair of classes for effective diagnosis of diverse diseases. Rigorous analysis of the proposed classification framework on large medical benchmarked datasets showcases its superiority over its peers by registering a higher classification accuracy and F1 scores ranging from 98.46 to 99.45% and 96.53-98.10% respectively, making it a precise and interpretable automated system for diagnosing respiratory disorders.

Angle-encoded quantum kernels on tabular data collapse when the feature map is wider than the intrinsic dimension of the data. We propose the correlation fractal dimension D2 as an a priori qubit budget: encode D2 coordinates chosen by FD-ASE instead of the PCA-95% width or all E attributes. On nine data sets and a statevector simulator (n= 32), a one-layer ZZ fidelity kernel at q=D2 stays geometrically alive while the same kernel at the PCA-95% width has already collapsed. The budget is map-dependent: product-state and IQP maps overshoot it; a second ZZ layer undershoots it. Packed dense-angle and re-uploading encodings still live at the fractal q, but not when PCA-95% features are stacked onto those qubits. Shrinking the angle bandwidth moves the ZZ knee later; stretching it kills the kernel earlier. On IBM Quantum (ibm_fez, 256 shots, n=8) the one-layer ZZ kernel at the fractal width matches the exact kernel (MAE 0.021); past that width both hardware and simulator have collapsed. The ceiling is a property of the map-data pair at a stated bandwidth, not of the classical table alone.

This study identifies new depression biomarkers based on the dynamical properties of tract variables, which represent geometric features describing the configuration of the speech articulators. A key advantage of this approach lies in its ability to quantify aspects of the articulatory process that have not been previously explored in the context of depression, namely predictability, complexity, and randomness. These properties are respectively characterised using the Largest Lyapunov Exponent, the Correlation Dimension, and the Sample Entropy. Thorough experiments were conducted on the Androids Corpus, a publicly available dataset comprising 64 speakers diagnosed with depression by clinicians and 54 control speakers with no reported history of mental health conditions. The results indicate that the proposed biomarkers effectively discriminate between the depressed and control speakers, as evidenced by the high Cliffs delta values across both read and spontaneous speech.

Reconstructing the pulmonary vascular tree from computed tomography (CT) images is essential for quantitative lung analysis, vascular morphology assessment, and patient-specific modeling, yet it remains challenging because vessels span multiple scales, from proximal arteries to distal microvasculature. Clinical chest CT is further affected by limited spatial resolution, partial volume effects, heterogeneous image quality, and respiratory motion artifacts. Unlike deep learning-based pulmonary vessel segmentation methods that require large annotated datasets, we propose a deterministic, training-free, and explainable pipeline for CT-based pulmonary vascular tree reconstruction. The method fuses multiscale Hessian-based Frangi and Sato vesselness filters using a weighted maximum response across 12 spatial scales from 1 to 8 mm, enabling detection of large pulmonary arteries and peripheral branches. Lung parenchyma is segmented by Hounsfield unit thresholding, morphological post-processing, and Chan-Vese active contour refinement. Vascular centerlines are extracted using the Kimimaro implementation of the TEASAR algorithm; separate left- and right-lung vascular graphs are then constructed, pruned, and verified for acyclicity. Geometric plausibility is assessed using volumetric fractal dimension, Strahler order analysis, Horton ratios, and Murray's law. The resulting fractal dimension of approximately 2.3 is consistent with reported values for the human pulmonary vasculature. At the same time, residual deviations in branching metrics reflect distal-vessel truncation caused by finite CT resolution. These results indicate that the proposed explainable pipeline can generate geometrically plausible pulmonary vascular tree models and may support quantitative pulmonary imaging, vascular morphometry, and computational lung modeling.

Neural networks learn features that reflect the hierarchical, multi-scale structure of natural data. Synthetic datasets used to evaluate interpretability methods typically lack this structure, limiting their value as realistic toy models. To close this gap, we introduce a family of synthetic datasets consisting of hierarchical functions defined on critical mean-field percolation clusters embedded in a high-dimensional data space. The percolation data consists of sparse, low-dimensional fractal clusters with a power-law size distribution. Latent variables modeling a taxonomic hierarchy generate each data point's target value. The data model is analytically tractable with known critical exponents that fix its properties without requiring hyperparameter tuning. We leverage a mapping between percolation clusters, random trees, and additive coalescence to propose an almost linear-time algorithm to jointly sample a random tree and its hierarchical latent decomposition, enabling data generation at arbitrary scale. Using probing experiments, we find that the model's ground-truth latent variables can be linearly decoded from neural network activations. Together, sparsity, self-similarity, power-law statistics, and analytical tractability make critical percolation a principled testbed for interpretability research.

Predicting the generalization performance of deep neural networks without relying on hold-out validation data is a fundamental challenge in machine learning. While Stochastic Gradient Descent (SGD) drives the optimization of these highly parameterized models, its heavy-tailed, non-Gaussian dynamics induce complex, scale-invariant trajectories in the parameter space. In this paper, we propose a novel generalization measure based on the Fourier fractal dimension of the network's weight variations. By analyzing the characteristic function of the Lévy-driven stochastic differential equations in the frequency domain, we extract a metric that robustly captures the geometric complexity of the learning process. Furthermore, we introduce a customized Fourier-based optimizer designed to actively regularize this fractal dimension during training. Extensive empirical evaluations on the CIFAR-10, SVHN, and MNIST datasets demonstrate that our proposed Fourier generalization measure exhibits a strong correlation with the actual generalization gap. Our method achieves state-of-the-art Kendall rank correlation coefficients, outperforming a wide array of existing norm-based, margin-based, and PAC-Bayesian measures. Ultimately, this work highlights the potential of frequency-domain fractal analysis as both a powerful predictor for model generalizability and a principled foundation for developing more stable optimization algorithms.

Masked autoencoders (MAE) have shown great promise in medical image classification. However, the random masking strategy employed by traditional MAEs may overlook critical areas in medical images, where even subtle changes can indicate disease. To address this limitation, we propose a novel approach that utilizes a multifractal measure (Renyi entropy) to optimize the masking strategy. Our method, termed Multifractal-Optimized Masked Autoencoder (MO-MAE), employs a multifractal analysis to identify regions of high complexity and information content. By focusing the masking process on these areas, MO-MAE ensures that the model learns to reconstruct the most diagnostically relevant features. This approach is particularly beneficial for medical imaging, where fine-grained inspection of tissue structures is crucial for accurate diagnosis. We evaluate MO-MAE on several medical datasets covering various diseases, including MedMNIST and COVID-CT. Our results demonstrate that MO-MAE achieves promising performance, surpassing other basiline and state-of-the-art models. The proposed method also adds minimum computational overhead as the computation of the proposed measure is straightforward. Our findings suggest that the multifractal-optimized masking strategy enhances the model's ability to capture and reconstruct complex tissue structures, leading to more accurate and efficient medical image representation. The proposed MO-MAE framework offers a promising direction for improving the accuracy and efficiency of deep learning models in medical image analysis, potentially advancing the field of computer-aided diagnosis.

Vision-language models (VLMs) have achieved strong performance across diverse multimodal tasks, but their adversarial robustness in visible-infrared (VIS-IR) scenarios remains underexplored. This gap is critical because VIS-IR sensing is widely used in real-world perception systems to support reliable understanding under challenging imaging conditions. To address this cross-modal threat setting, we propose CFGPatch, a curved-edge fractal geometric adversarial patch framework for attacking VIS-IR VLMs. CFGPatch builds on triangular fractal geometry and replaces rigid straight-edged primitives with Bezier-curved elements, preserving multi-scale fractal self-similarity while introducing smoother contours, richer directional variation, and more flexible shape deformation. In addition, we design a modality-specific Fraser-spiral rendering mechanism to inject fine-grained texture distortions and misleading perceptual cues into visible and infrared images. By coupling global curved-fractal geometry with local spiral-based appearance interference, CFGPatch disrupts both shape perception and texture interpretation. We further adopt expectation over transformation (EOT) to improve robustness against common image-level transformations. Extensive experiments show that CFGPatch effectively fools VIS-IR VLMs and consistently outperforms standard patch baselines in attack effectiveness and robustness. Moreover, adversarial samples optimized for zero-shot classification transfer well to image captioning and visual question answering, demonstrating strong cross-task transferability and generalizability across downstream tasks.

AI-generated imagery has reached near-photorealistic fidelity, yet this technology poses significant threats to information security and societal trust. Existing deepfake detection methods often exhibit limited robustness in open-world scenarios. To address this limitation, this paper investigates intrinsic discrepancies between synthetic and authentic images from a signal-level perspective. Our analysis reveals that low-correlation signals serve as distinctive markers for differentiating AI-generated imagery from real photographs. Building on this insight, we introduce a novel method for quantifying these signals based on fractal theory. By analyzing the fractal characteristics of low-correlation signals, our method effectively captures the subtle statistical anomalies inherent to the synthesis process. Extensive experimental results demonstrate the method's robustness and superior detection performance. This work emphasizes the need to shift research focus to a new signal-level direction for deepfake detection. Theoretically, this proposed approach is not limited to face image identification but can be applied to all AI-generated image detection tasks. This study provides a new research direction for deepfake detection.

Traditional distributed backdoor attacks (DBA) in federated learning improve stealthiness by decomposing global triggers into sub-triggers, which however requires more poisoned data to maintian the attck strength and hence increases the exposure risk. To overcome this defect, This paper proposes a novel method, namely Fractal-Triggerred Distributed Backdoor Attack (FTDBA), which leverages the self-similarity of fractals to enhance the feature strength of sub-triggers and hence significantly reduce the required poisoning volume for the same attack strength. To address the detectability of fractal structures in the frequency and gradient domains, we introduce a dynamic angular perturbation mechanism that adaptively adjusts perturbation intensity across the training phases to balance efficiency and stealthiness. Experiments show that FTDBA achieves a 92.3% attack success rate with only 62.4% of the poisoning volume required by traditional DBA methods, while reducing the detection rate by 22.8% and KL divergence by 41.2%. This study presents a low-exposure, high-efficiency paradigm for federated backdoor attacks and expands the application of fractal features in adversarial sample generation.

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.