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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning Linear Response Estimators for Singular Statistical Models Statistical inference with belief functions: A survey Robust stochastic first order methods in heavy-tailed noise via medoid mini-batch gradient sampling Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity Adaptive auditing of AI systems with anytime-valid guarantees Locally Near Optimal Piecewise Linear Regression in High Dimensions via Difference of Max-Affine Functions Risk-Controlled Post-Processing of Decision Policies Covariate Balancing and Riesz Regression Should Be Guided by the Neyman Orthogonal Score in Debiased Machine Learning A Unified Pair-GRPO Family: From Implicit to Explicit Preference Constraints for Stable and General RL Alignment Time-Inhomogeneous Preconditioned Langevin Dynamics A Fine-Grained Understanding of Uniform Convergence for Halfspaces CITE: Anytime-Valid Statistical Inference in LLM Self-Consistency Ratio-based Loss Functions Optimal Confidence Band for Kernel Gradient Flow Estimator A renormalization-group inspired lattice-based framework for piecewise generalized linear models
Exploring the Multifractal Behavior of the Human Genome T...
Yulián A. Alvarez-Ballesteros, Mario A. Quiroz-Juarez, José L. D · 2024-12-22 · via math.ST updates on arXiv.org

In this work, we applied the Chaos Game Representation (CGR) to the complete human genomic sequence T2T-CHM13v2.0, analyzing the entire chromosome assembly and each chromosome separately, including mitochondrial DNA. Multifractal spectra were determined using two types of box-counting coverage, revealing slight variations across most chromosomes. While the geometric support remained consistent, distinct distributions were observed for each chromosome. Chromosomes 9 and Y exhibited the greatest differences in singularity (Hölder exponent), with minor variations in their fractal support. The CGR distributions generally demonstrated an approximate separation between coding and non-coding sections, as well as CpG or GpC islands. A base-by-base analysis of the fractal support of the CGR uncovered characteristic structural bands in chromosome sequences, which align with patterns identified in cytogenetic studies. Using the complete assembly as a reference, we compared two alternative representations: the Binary Genomic Representation (RGB) and the Markov Chain (MC) representation. Both methods tended toward the same fractal support but displayed differing distributions based on the assigned length parameter. Multifractal analysis highlighted quantitative differences between these representations: RGB aligned more closely with high-frequency components, while MC showed better correspondence with low frequencies. The optimal fit was achieved using MC for twelve-base chains, yielding an average percentage error of 2% relative to the full genomic assembly.