Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

This paper introduces the Neural Differential Manifold (NDM), a novel neural network architecture that explicitly incorporates geometric structure into its fundamental design. Departing from conventional Euclidean parameter spaces, the NDM re-conceptualizes a neural network as a differentiable manifold where each layer functions as a local coordinate chart, and the network parameters directly parameterize a Riemannian metric tensor at every point. The architecture is organized into three synergistic layers: a Coordinate Layer implementing smooth chart transitions via invertible transformations inspired by normalizing flows, a Geometric Layer that dynamically generates the manifold's metric through auxiliary sub-networks, and an Evolution Layer that optimizes both task performance and geometric simplicity through a dual-objective loss function. This geometric regularization penalizes excessive curvature and volume distortion, providing intrinsic regularization that enhances generalization and robustness. The framework enables natural gradient descent optimization aligned with the learned manifold geometry and offers unprecedented interpretability by endowing internal representations with clear geometric meaning. We analyze the theoretical advantages of this approach, including its potential for more efficient optimization, enhanced continual learning, and applications in scientific discovery and controllable generative modeling. While significant computational challenges remain, the Neural Differential Manifold represents a fundamental shift towards geometrically structured, interpretable, and efficient deep learning systems.

Continuous attractors are critical for information processing in both biological and artificial neural systems, with implications for spatial navigation, memory, and deep learning optimization. However, existing research lacks a unified framework to analyze their properties across diverse dynamical systems, limiting cross-architectural generalizability. This study establishes a novel framework from the perspective of differential manifolds to investigate continuous attractors in artificial neural networks. It verifies compatibility with prior conclusions, elucidates links between continuous attractor phenomena and eigenvalues of the local Jacobian matrix, and demonstrates the universality of singular value stratification in common classification models and datasets. These findings suggest continuous attractors may be ubiquitous in general neural networks, highlighting the need for a general theory, with the proposed framework offering a promising foundation given the close mathematical connection between eigenvalues and singular values.

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.

In this paper, we focus on the intersections of a manifold's local structures to analyze the global structure of a manifold. We obtain local regions on data manifolds such as the latent space of StyleGAN2, using Mapper, a tool from topological data analysis. We impose gluing compatibility conditions on overlapping local regions, which guarantee that the local structures can be glued together to the global structure of a manifold. We propose a novel generative flow model called Atlas flow that uses compatibility to reattach the local regions. Our model shows that the generating processes perform well on synthetic dataset samples of well-known manifolds with noise. Furthermore, we investigate the style vector manifold of StyleGAN2 using our model.

Gaussian Processes (GPs) are highly expressive, probabilistic models. A major limitation is their computational complexity. Naively, exact GP inference requires $\mathcal{O}(N^3)$ computations with $N$ denoting the number of modeled points. Current approaches to overcome this limitation either rely on sparse, structured or stochastic representations of data or kernel respectively and usually involve nested optimizations to evaluate a GP. We present a new, generative method named Iterative Charted Refinement (ICR) to model GPs on nearly arbitrarily spaced points in $\mathcal{O}(N)$ time for decaying kernels without nested optimizations. ICR represents long- as well as short-range correlations by combining views of the modeled locations at varying resolutions with a user-provided coordinate chart. In our experiment with points whose spacings vary over two orders of magnitude, ICR's accuracy is comparable to state-of-the-art GP methods. ICR outperforms existing methods in terms of computational speed by one order of magnitude on the CPU and GPU and has already been successfully applied to model a GP with $122$ billion parameters.

We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional Euclidean space by learning a coordinate chart, but it requires that the entire manifold can be embedded in a single chart. This is impossible for manifolds with holes. In such cases, it is necessary to learn an atlas: a collection of charts that collectively cover the entire manifold. We begin with many small charts, and combine them in a bottom-up approach, where charts are only combined if doing so will not introduce problematic topological features. When it is no longer possible to combine any charts, each chart is individually embedded with standard manifold learning techniques, completing the construction of the atlas. We show the efficacy of our method by constructing atlases for challenging synthetic manifolds; learning human motion embeddings from motion capture data; and learning kinematic models of articulated objects.

Current methods focus on manifolds that are topologically Euclidean, enforce strong structural priors on the learned models or use operations that do not scale to high dimensions. In contrast, our model learns the local manifold topology piecewise by "gluing" it back together through a collection of learned coordinate charts. We demonstrate the efficiency of our approach on synthetic data of known manifolds, as well as higher dimensional manifolds of unknown topology, where we show better sample efficiency and competitive or superior performance against current state-of-the-art.

We introduce the Givens Transform, a novel transform between the space of orthonormal matrices and $\mathbb{R}^D$. The Givens Transform allows for the application of any general Bayesian inference algorithm to probabilistic models containing constrained unit-vectors or orthonormal matrix parameters. This includes a variety of matrix factorizations and dimensionality reduction models such as Probabilistic PCA (PPCA), Exponential Family PPCA (BXPCA), and Canonical Correlation Analysis (CCA). While previous Bayesian approaches to these models relied on separate sampling update rules for constrained and unconstrained parameters, the Givens Transform enables the treatment of unit-vectors and orthonormal matrices agnostically as unconstrained parameters. Thus any Bayesian inference algorithm can be used on these models without modification. This opens the door to not just sampling algorithms, but Variational Inference (VI) as well. We illustrate with several examples and supplied code, how the Givens Transform allows end-users to easily build complex models in their favorite Bayesian modeling framework such as Stan, Edward, or PyMC3, a task that was previously intractable due to technical constraints.