Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces

Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $ρ$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.

Convergence Analysis of Statistical Inverse Problems on Reproducing Kernel Banach Spaces

Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $ρ$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.

To Describe or Construct Statistical Learning Models Using the Category-theoretical Language

Statistical learning is a fascinating field that has long been the mainstream of machine learning/artificial intelligence. A large number of results have been produced which can be widely applied to real-world problems. It also leads to many research topics and also stimulates new research. This report summarizes some classical statistical learning models and well-known algorithms, especially for amateurs, and provides a category-theoretic perspective on understanding statistical learning models. The aim is to attract researchers from other fields, including basic mathematics, to participate in the research related to statistical learning.

Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.

Developmental approach reveals the statistical learning of Neural Language Models: Transformers generalize from the most abstract statistical patterns

In this study, we use a developmental approach to investigate the statistical learning and mental representation of neural language models (NLM). A series of Generative Transformer models are trained on a synthetic grammar. The model states are saved at multiple stages in the course of training. Through analyzing how the internal representations of these models change in the developmental path, we found that NLMs acquire the most abstract global statistical knowledge at the beginning of learning and later acquire the relatively local statistical dependencies. This learning path contains many over-generalizations from the very beginning and these over-generalizations are gradually constrained in the later stage of learning. Based on this observation, we propose a new framework to explain the statistical learning and language cognition of NLMs.

Synthetic Network Packet Generation through Statistical Learning and Genetic Algorithms

Developing robust intrusion detection systems (IDS) for IoT environments requires large, labeled datasets capturing realistic traffic distributions across both benign and malicious activity. Existing public datasets suffer from fixed activity distributions and extreme class imbalance, while deep generative models (GANs, VAEs) provide no mechanism to enforce that synthetic packets remain within physically valid feature ranges. This paper proposes and compares two constraint-enforcing approaches for synthetic IoT network packet generation: (i) a statistical learning method combining PCA-based latent space sampling with dual One-Class SVM (OCSVM) and Isolation Forest (IF) boundary enforcement, and (ii) a genetic algorithm (GA) method that treats packet generation as a multi-objective optimization problem with explicit fitness criteria for anomaly model acceptance and distributional fidelity. Both methods embed hard validity constraints -- dual anomaly-detection gating, feature-range clamping, and independent validation -- directly into the synthesis pipeline. Evaluation on the complete ACI IoT 2023 dataset (1,231,411 packets, 12 attack categories, class imbalance up to 175,805:1) demonstrates that both methods achieve PASS status across all categories under independently trained validators with a 30% anomaly rate threshold: the statistical method attains 1.20% average anomaly rate with ~1,091 packets/s throughput, while the GA attains 0.62% average anomaly rate with organic per-class variance (0.00%-2.50%) at ~5.7 packets/s. Both methods successfully amplify the 5-sample ARP Spoofing category by 200x to 1,000 validated packets. The ~190:1 throughput ratio between methods, combined with their complementary quality profiles, provides evidence-based selection criteria for deployment contexts ranging from rapid dataset augmentation to adversarial robustness testing.