Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations

Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields.

On the Relation of State Space Models and Hidden Markov Models

State Space Models (SSMs) and Hidden Markov Models (HMMs) are foundational frameworks for modeling sequential data with latent variables and are widely used in signal processing, control theory, and machine learning. Despite their shared temporal structure, they differ fundamentally in the nature of their latent states, probabilistic assumptions, inference procedures, and training paradigms. Recently, deterministic state space models have re-emerged in natural language processing through architectures such as S4 and Mamba, raising new questions about the relationship between classical probabilistic SSMs, HMMs, and modern neural sequence models. In this paper, we present a unified and systematic comparison of HMMs, linear Gaussian state space models, Kalman filtering, and contemporary NLP state space models. We analyze their formulations through the lens of probabilistic graphical models, examine their inference algorithms -- including forward-backward inference and Kalman filtering -- and contrast their learning procedures via Expectation-Maximization and gradient-based optimization. By highlighting both structural similarities and semantic differences, we clarify when these models are equivalent, when they fundamentally diverge, and how modern NLP SSMs relate to classical probabilistic models. Our analysis bridges perspectives from control theory, probabilistic modeling, and modern deep learning.

Wittgenstein's Family Resemblance Clustering Algorithm

This paper, introducing a novel method in philomatics, draws on Wittgenstein's concept of family resemblance from analytic philosophy to develop a clustering algorithm for machine learning. According to Wittgenstein's Philosophical Investigations (1953), family resemblance holds that members of a concept or category are connected by overlapping similarities rather than a single defining property. Consequently, a family of entities forms a chain of items sharing overlapping traits. This philosophical idea naturally lends itself to a graph-based approach in machine learning. Accordingly, we propose the Wittgenstein's Family Resemblance (WFR) clustering algorithm and its kernel variant, kernel WFR. This algorithm computes resemblance scores between neighboring data instances, and after thresholding these scores, a resemblance graph is constructed. The connected components of this graph define the resulting clusters. Simulations on benchmark datasets demonstrate that WFR is an effective nonlinear clustering algorithm that does not require prior knowledge of the number of clusters or assumptions about their shapes.

Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Space

Self-supervised learning (SSL) has emerged as a powerful paradigm for representation learning by optimizing geometric objectives--such as invariance to augmentations, variance preservation, and feature decorrelation--without requiring labels. However, most existing methods operate in Euclidean space, limiting their ability to capture nonlinear dependencies and geometric structures. In this work, we propose Kernel VICReg, a novel self-supervised learning framework that lifts the VICReg objective into a Reproducing Kernel Hilbert Space (RKHS). By kernelizing each term of the loss-variance, invariance, and covariance--we obtain a general formulation that operates on double-centered kernel matrices and Hilbert-Schmidt norms, enabling nonlinear feature learning without explicit mappings. We demonstrate that Kernel VICReg not only avoids representational collapse but also improves performance on tasks with complex or small-scale data. Empirical evaluations across MNIST, CIFAR-10, STL-10, TinyImageNet, and ImageNet100 show consistent gains over Euclidean VICReg, with particularly strong improvements on datasets where nonlinear structures are prominent. UMAP visualizations further confirm that kernel-based embeddings exhibit better isometry and class separation. Our results suggest that kernelizing SSL objectives is a promising direction for bridging classical kernel methods with modern representation learning.

An Optimal Cascade Feature-Level Spatiotemporal Fusion Strategy for Anomaly Detection in CAN Bus

Autonomous vehicles represent a revolutionary advancement driven by the integration of artificial intelligence within intelligent transportation systems. However, they remain vulnerable due to the absence of robust security mechanisms in the Controller Area Network (CAN) bus. In order to mitigate the security issue, many machine learning models and strategies have been proposed, which primarily focus on a subset of dominant patterns of anomalies and lack rigorous evaluation in terms of reliability and robustness. Therefore, to address the limitations of previous works and mitigate the security vulnerability in CAN bus, the current study develops a model based on the intrinsic nature of the problem to cover all dominant patterns of anomalies. To achieve this, a cascade feature-level fusion strategy optimized by a two-parameter genetic algorithm is proposed to combine temporal and spatial information. Subsequently, the model is evaluated using a paired t-test to ensure reliability and robustness. Finally, a comprehensive comparative analysis conducted on two widely used datasets advocates that the proposed model outperforms other models and achieves superior accuracy and F1-score, demonstrating the best performance among all models presented to date.

Self-Supervised Learning Using Nonlinear Dependence

Self-supervised learning has gained significant attention in contemporary applications, particularly due to the scarcity of labeled data. While existing SSL methodologies primarily address feature variance and linear correlations, they often neglect the intricate relations between samples and the nonlinear dependencies inherent in complex data. In this paper, we introduce Correlation-Dependence Self-Supervised Learning (CDSSL), a novel framework that unifies and extends existing SSL paradigms by integrating both linear correlations and nonlinear dependencies, encapsulating sample-wise and feature-wise interactions. Our approach incorporates the Hilbert-Schmidt Independence Criterion (HSIC) to robustly capture nonlinear dependencies within a Reproducing Kernel Hilbert Space, enriching representation learning. Experimental evaluations on diverse benchmarks demonstrate the efficacy of CDSSL in improving representation quality.

Probabilistic Classification by Density Estimation Using Gaussian Mixture Model and Masked Autoregressive Flow

Density estimation, which estimates the distribution of data, is an important category of probabilistic machine learning. A family of density estimators is mixture models, such as Gaussian Mixture Model (GMM) by expectation maximization. Another family of density estimators is the generative models which generate data from input latent variables. One of the generative models is the Masked Autoregressive Flow (MAF) which makes use of normalizing flows and autoregressive networks. In this paper, we use the density estimators for classification, although they are often used for estimating the distribution of data. We model the likelihood of classes of data by density estimation, specifically using GMM and MAF. The proposed classifiers outperform simpler classifiers such as linear discriminant analysis which model the likelihood using only a single Gaussian distribution. This work opens the research door for proposing other probabilistic classifiers based on joint density estimation.

On Philomatics and Psychomatics for Combining Philosophy and Psychology with Mathematics

We propose the concepts of philomatics and psychomatics as hybrid combinations of philosophy and psychology with mathematics. We explain four motivations for this combination which are fulfilling the desire of analytical philosophy, proposing science of philosophy, justifying mathematical algorithms by philosophy, and abstraction in both philosophy and mathematics. We enumerate various examples for philomatics and psychomatics, some of which are explained in more depth. The first example is the analysis of relation between the context principle, semantic holism, and the usage theory of meaning with the attention mechanism in mathematics. The other example is on the relations of Plato's theory of forms in philosophy with the holographic principle in string theory, object-oriented programming, and machine learning. Finally, the relation between Wittgenstein's family resemblance and clustering in mathematics is explained. This paper opens the door of research for combining philosophy and psychology with mathematics.

Recurrent Neural Networks and Long Short-Term Memory Networks: Tutorial and Survey

This is a tutorial paper on Recurrent Neural Network (RNN), Long Short-Term Memory Network (LSTM), and their variants. We start with a dynamical system and backpropagation through time for RNN. Then, we discuss the problems of gradient vanishing and explosion in long-term dependencies. We explain close-to-identity weight matrix, long delays, leaky units, and echo state networks for solving this problem. Then, we introduce LSTM gates and cells, history and variants of LSTM, and Gated Recurrent Units (GRU). Finally, we introduce bidirectional RNN, bidirectional LSTM, and the Embeddings from Language Model (ELMo) network, for processing a sequence in both directions.

Gravitational Dimensionality Reduction Using Newtonian Gravity and Einstein's General Relativity

Due to the effectiveness of using machine learning in physics, it has been widely received increased attention in the literature. However, the notion of applying physics in machine learning has not been given much awareness to. This work is a hybrid of physics and machine learning where concepts of physics are used in machine learning. We propose the supervised Gravitational Dimensionality Reduction (GDR) algorithm where the data points of every class are moved to each other for reduction of intra-class variances and better separation of classes. For every data point, the other points are considered to be gravitational particles, such as stars, where the point is attracted to the points of its class by gravity. The data points are first projected onto a spacetime manifold using principal component analysis. We propose two variants of GDR -- one with the Newtonian gravity and one with the Einstein's general relativity. The former uses Newtonian gravity in a straight line between points but the latter moves data points along the geodesics of spacetime manifold. For GDR with relativity gravitation, we use both Schwarzschild and Minkowski metric tensors to cover both general relativity and special relativity. Our simulations show the effectiveness of GDR in discrimination of classes.