This work proposes a practical method for disentangling, i.e., Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER does not learn a global nonlinear embedding, instead, it learns a set of subspaces to assign to each point, where each subspace is the tangent space of one disentangled submanifold. Thus, GEOMANCER can be used to disentangle manifolds for which there may not be a global axis-aligned coordinate system. Experimental results on both synthetic data and Stanford 3D data are included in this paper.

This note offers a first bridge from machine learning to modern differential geometry. We show that the logits-to-probabilities step implemented by softmax can be modeled as a geometric interface: two potential-generated, conservative descriptions (from negative entropy and log-sum-exp) meet along a Legendrian "seam" on a contact screen (the probability simplex) inside a simple folded symplectic collar. Bias-shift invariance appears as Reeb flow on the screen, and the Fenchel-Young equality/KL gap provides a computable distance to the seam. We work out the two- and three-class cases to make the picture concrete and outline next steps for ML: compact logit models (projective or spherical), global invariants, and connections to information geometry where on-screen dynamics manifest as replicator flows.

This work proposes a practical method for disentangling, i.e., Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER does not learn a global nonlinear embedding, instead, it learns a set of subspaces to assign to each point, where each subspace is the tangent space of one disentangled submanifold. Thus, GEOMANCER can be used to disentangle manifolds for which there may not be a global axis-aligned coordinate system. Experimental results on both synthetic data and Stanford 3D data are included in this paper.

Nonlinear mean shift over riemannian manifolds.

This paper introduces Higher Gauge Flow Models, a novel class of Generative Flow Models. Building upon ordinary Gauge Flow Models (arXiv:2507.13414), these Higher Gauge Flow Models leverage an L$_{\infty}$-algebra, effectively extending the Lie Algebra. This expansion allows for the integration of the higher geometry and higher symmetries associated with higher groups into the framework of Generative Flow Models. Experimental evaluation on a Gaussian Mixture Model dataset revealed substantial performance improvements compared to traditional Flow Models.

This paper introduces Gauge Flow Models, a novel class of Generative Flow Models. These models incorporate a learnable Gauge Field within the Flow Ordinary Differential Equation (ODE). A comprehensive mathematical framework for these models, detailing their construction and properties, is provided. Experiments using Flow Matching on Gaussian Mixture Models demonstrate that Gauge Flow Models yields significantly better performance than traditional Flow Models of comparable or even larger size. Additionally, unpublished research indicates a potential for enhanced performance across a broader range of generative tasks.

The paper develops a mathematical framework for working with neural network representations in the context of finite differences and differential geometry. In this framework, data points going though layers have fixed coordinates but space is smoothly curved with each layer. The paper presents a very interesting framework for working with neural network representations, especially in the case of residual networks. Unfortunately, taking the limit as the number of layers goes to infinity does not make practical application very easy and somewhat limits the impact of this paper. The paper is not always completely clear.

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.

Principal curve is a well-known statistical method oriented in manifold learning using concepts from differential geometry. In this paper, we propose a novel metric-based principal curve (MPC) method that learns one-dimensional manifold of spatial data. Synthetic datasets Real applications using MNIST dataset show that our method can learn the one-dimensional manifold well in terms of the shape.