Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allow optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest are typically assumed to live in some (often unknown) low-dimensional manifold embedded in a high-dimensional ambient space. The result is a modelling mismatch since -- by construction -- the invertibility requirement implies high-dimensional support of the learned distribution. Injective flows, mappings from low-to high-dimensional spaces, aim to fix this discrepancy by learning distributions on manifolds, but the resulting volume-change term becomes more challenging to evaluate. Current approaches either avoid computing this term entirely using various heuristics, or assume the manifold is known beforehand and therefore are not widely applicable. Instead, we propose two methods to tractably calculate the gradient of this term with respect to the parameters of the model, relying on careful use of automatic differentiation and techniques from numerical linear algebra. Both approaches perform end-to-end nonlinear manifold learning and density estimation for data projected onto this manifold. We study the trade-offs between our proposed methods, empirically verify that we outperform approaches ignoring the volume-change term by more accurately learning manifolds and the corresponding distributions on them, and show promising results on out-of-distribution detection.

We present Primal, a deterministic feature mapping framework that harnesses the number-theoretic independence of prime square roots to construct robust, tunable vector representations. Diverging from standard stochastic projections (e.g., Random Fourier Features), our method exploits the Besicovitch property to create irrational frequency modulations that guarantee infinite non-repeating phase trajectories. We formalize two distinct algorithmic variants: (1) StaticPrime, a sequence generation method that produces temporal position encodings empirically approaching the theoretical Welch bound for quasi-orthogonality; and (2) DynamicPrime, a tunable projection layer for input-dependent feature mapping. A central novelty of the dynamic framework is its ability to unify two disparate mathematical utility classes through a single scaling parameter σ. In the low-frequency regime, the method acts as an isometric kernel map, effectively linearizing non-convex geometries (e.g., spirals) to enable high-fidelity signal reconstruction and compressive sensing. Conversely, the high-frequency regime induces chaotic phase wrapping, transforming the projection into a maximum-entropy one-way hash suitable for Hyperdimensional Computing and privacy-preserving Split Learning. Empirical evaluations demonstrate that our framework yields superior orthogonality retention and distribution tightness compared to normalized Gaussian baselines, establishing it as a computationally efficient, mathematically rigorous alternative to random matrix projections. The code is available at https://github.com/VladimerKhasia/primal

Large Language Models (LLMs) have shown immense potential in education, automating tasks like quiz generation and content summarization. However, generating effective presentation slides introduces unique challenges due to the complexity of multimodal content creation and the need for precise, domain-specific information. Existing LLM-based solutions often fail to produce reliable and informative outputs, limiting their educational value. To address these limitations, we introduce SlideBot - a modular, multi-agent slide generation framework that integrates LLMs with retrieval, structured planning, and code generation. SlideBot is organized around three pillars: informativeness, ensuring deep and contextually grounded content; reliability, achieved by incorporating external sources through retrieval; and practicality, which enables customization and iterative feedback through instructor collaboration. It incorporates evidence-based instructional design principles from Cognitive Load Theory (CLT) and the Cognitive Theory of Multimedia Learning (CTML), using structured planning to manage intrinsic load and consistent visual macros to reduce extraneous load and enhance dual-channel learning. Within the system, specialized agents collaboratively retrieve information, summarize content, generate figures, and format slides using LaTeX, aligning outputs with instructor preferences through interactive refinement. Evaluations from domain experts and students in AI and biomedical education show that SlideBot consistently enhances conceptual accuracy, clarity, and instructional value. These findings demonstrate SlideBot's potential to streamline slide preparation while ensuring accuracy, relevance, and adaptability in higher education.

The problem of identifying geometric structure in data is a cornerstone of (unsupervised) learning. As a result, Geometric Representation Learning has been widely applied across scientific and engineering domains. In this work, we investigate the use of Geometric Representation Learning for the data-driven discovery of system dynamics from spatial-temporal data. We propose to encode similarity structure in such data in a spatial-temporal proximity graph, to which we apply a range of classical and deep learning-based manifold learning approaches to learn reduced order dynamics. We observe that while manifold learning is generally capable of recovering reduced order dynamics, the quality of the learned representations varies substantially across different algorithms and hyperparameter choices. This is indicative of high sensitivity to the inherent geometric assumptions of the respective approaches and suggests a need for careful hyperparameter tuning, which can be expensive in practise. To overcome these challenges, we propose a framework for Automated Manifold Learning, which selects a manifold learning approach and corresponding hyperparameter choices based on representative subsamples of the input graph. We demonstrate that the proposed framework leads to performance gains both in scalability and in the learned representations' accuracy in capturing local and global geometric features of the underlying system dynamics.

Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allow optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest are typically assumed to live in some (often unknown) low-dimensional manifold embedded in a high-dimensional ambient space. The result is a modelling mismatch since -- by construction -- the invertibility requirement implies high-dimensional support of the learned distribution. Injective flows, mappings from low- to high-dimensional spaces, aim to fix this discrepancy by learning distributions on manifolds, but the resulting volume-change term becomes more challenging to evaluate. Current approaches either avoid computing this term entirely using various heuristics, or assume the manifold is known beforehand and therefore are not widely applicable.

Manifold clustering, with its exceptional ability to capture complex data structures, holds a pivotal position in cluster analysis. However, existing methods often focus only on finding the optimal combination between K-means and manifold learning, and overlooking the consistency between the data structure and labels. To address this issue, we deeply explore the relationship between K-means and manifold learning, and on this basis, fuse them to develop a new clustering framework. Specifically, the algorithm uses labels to guide the manifold structure and perform clustering on it, which ensures the consistency between the data structure and labels. Furthermore, in order to naturally maintain the class balance in the clustering process, we maximize the Schatten p-norm of labels, and provide a theoretical proof to support this. Additionally, our clustering framework is designed to be flexible and compatible with many types of distance functions, which facilitates efficient processing of nonlinear separable data. The experimental results of several databases confirm the superiority of our proposed model.

Laplacian-based methods are popular for dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.

Abstract--Traditional feature extraction and projection techniques, such as Principal Component Analysis, struggle to adequately represent X-Ray Transmission (XRT) Multi-Energy (ME) images, limiting the performance of neural networks in decision-making processes. To address this issue, we propose a method that approximates the dataset topology by constructing adjacency graphs using the Uniform Manifold Approximation and Projection. This technique not only preserves the global structure of the data but also enhances feature separability, leading to more accurate and robust classification results. Figure 1: Scheme of the experimental setting (left) view from aside. A top view of the detector is also given (right) to make I. Recent advances in Hyperspectral Images (HSI) analysis have primarily focused on reflection spectroscopy in the visible or near-infrared light domains.

In machine learning, Riemannian manifolds offer a useful abstraction for approximating commonly encountered, non-Euclidean empirical data distributions and optimization state spaces. While Euclidean machine learning algorithms have been adapted to Riemannian manifolds, these adaptations rely on computationally intensive differential-geometric primitives, such as exponential maps and parallel transports. Here, we present a framework for efficiently approximating differentialgeometric primitives on arbitrary manifolds via construction of an atlas graph representation, which leverages the canonical characterization of a manifold as a finite collection, or atlas, of overlapping coordinate charts. We first show the utility of this framework in a setting where the manifold is expressed in closed form, specifically, a runtime advantage, compared with state-of-the-art approaches, for first-order optimization over the Grassmann manifold. Moreover, using point cloud data for which a complex manifold structure was previously established, i.e., high-contrast image patches, we show that an atlas graph with the correct geometry can be directly learned from the point cloud. Finally, we demonstrate that learning an atlas graph enables downstream key machine learning tasks. In particular, we implement a Riemannian generalization of support vector machines that uses the learned atlas graph to approximate complex differential-geometric primitives, including Riemannian logarithms and vector transports. These settings suggest the potential of this framework for even more complex settings, where ambient dimension and noise levels may be much higher. Taken together, our results speed up and render more broadly applicable Riemannian optimization routines at the forefront of modern data science and machine learning.