We propose Fiber Bundle Networks (FiberNet), a novel machine learning framework integrating differential geometry with machine learning. Unlike traditional deep neural networks relying on black-box function fitting, we reformulate classification as interpretable geometric optimization on fiber bundles, where categories form the base space and wavelet-transformed features lie in the fibers above each category. We introduce two innovations: (1) learnable Riemannian metrics identifying important frequency feature components, (2) variational prototype optimization through energy function minimization. Classification is performed via Voronoi tessellation under the learned Riemannian metric, where each prototype defines a decision region and test samples are assigned to the nearest prototype, providing clear geometric interpretability. This work demonstrates that the integration of fiber bundle with machine learning provides interpretability and efficiency, which are difficult to obtain simultaneously in conventional deep learning.

In this paper, we propose a framework for designing sliding mode controllers for a class of mechanical systems with symmetry, both unconstrained and constrained, that evolve on principal fiber bundles. Control laws are developed based on the reduced motion equations by exploring symmetries, leading to a sliding mode control strategy where the reaching stage is executed on the base space, and the sliding stage is performed on the structure group. Thus, design complexity is reduced, and difficult choices for coordinate representations when working with a particular Lie group are avoided. For this purpose, a sliding subgroup is constructed on the structure group based on a kinematic controller, and the sliding variable will converge to the identity of the state manifold upon reaching the sliding subgroup. A reaching law based on a general sliding vector field is then designed on the base space using the local form of the mechanical connection to drive the sliding variable to the sliding subgroup, and its time evolution is given according to the appropriate covariant derivative. Almost global asymptotic stability and local exponential stability are demonstrated using a Lyapunov analysis. We apply the results to a fully actuated system (a rigid spacecraft actuated by reaction wheels) and a subactuated nonholonomic system (unicycle mobile robot actuated by wheels), which is also simulated for illustration.

Neural Information Processing Systems http://nips.cc/

Anatomic tracer studies are critical for validating and improving diffusion MRI (dMRI) tractography. However, large-scale analysis of data from such studies is hampered by the labor-intensive process of annotating fiber bundles manually on histological slides. Existing automated methods often miss sparse bundles or require complex post-processing across consecutive sections, limiting their flexibility and generalizability. We present a streamlined, fully automated framework for fiber bundle segmentation in macaque tracer data, based on a U-Net architecture with large patch sizes, foreground aware sampling, and semi-supervised pre-training. Our approach eliminates common errors such as mislabeling terminals as bundles, improves detection of sparse bundles by over 20% and reduces the False Discovery Rate (FDR) by 40% compared to the state-of-the-art, all while enabling analysis of standalone slices. This new framework will facilitate the automated analysis of anatomic tracing data at a large scale, generating more ground-truth data that can be used to validate and optimize dMRI tractography methods.

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.

To fully understand the behavior of a large language model (LLM) requires our understanding of its input space. If this input space differs from our assumption, our understanding of and conclusions about the LLM is likely flawed, regardless of its architecture. Here, we elucidate the structure of the token embeddings, the input domain for LLMs, both empirically and theoretically. We present a generalized and statistically testable model where the neighborhood of each token splits into well-defined signal and noise dimensions. This model is based on a generalization of a manifold called a fiber bundle, so we denote our hypothesis test as the ``fiber bundle null.'' Failing to reject the null is uninformative, but rejecting it at a specific token indicates that token has a statistically significant local structure, and so is of interest to us. By running our test over several open-source LLMs, each with unique token embeddings, we find that the null is frequently rejected, and so the token subspace is provably not a fiber bundle and hence also not a manifold. As a consequence of our findings, when an LLM is presented with two semantically equivalent prompts, and if one prompt contains a token implicated by our test, that prompt will likely exhibit more output variability proportional to the local signal dimension of the token.

Hybrid systems are dynamical systems with continuous-time and discrete-time components in their dynamics. When hybrid systems are defined on a principal bundle we are able to define two classes of impacts for the discrete-time transition of the dynamics: interior impacts and exterior impacts. In this paper we define hybrid systems on principal bundles, study the underlying geometry on the switching surface where impacts occur and we find conditions for which both exterior and interior impacts are preserved by the mechanical connection induced in the principal bundle.

Vision-based tactile sensors have recently become popular due to their combination of low cost, very high spatial resolution, and ease of integration using widely available miniature cameras. The associated field of view and focal length, however, are difficult to package in a human-sized finger. In this paper we employ optical fiber bundles to achieve a form factor that, at 15 mm diameter, is smaller than an average human fingertip. The electronics and camera are also located remotely, further reducing package size. The sensor achieves a spatial resolution of 0.22 mm and a minimum force resolution 5 mN for normal and shear contact forces. With these attributes, the DIGIT Pinki sensor is suitable for applications such as robotic and teleoperated digital palpation. We demonstrate its utility for palpation of the prostate gland and show that it can achieve clinically relevant discrimination of prostate stiffness for phantom and ex vivo tissue.

High-dimensional motion planning problems can often be solved significantly faster by using multilevel abstractions. While there are various ways to formally capture multilevel abstractions, we formulate them in terms of fiber bundles. Fiber bundles essentially describe lower-dimensional projections of the state space using local product spaces, which allows us to concisely describe and derive novel algorithms in terms of bundle restrictions and bundle sections. Given such a structure and a corresponding admissible constraint function, we develop highly efficient and asymptotically-optimal sampling-based motion planning methods for high-dimensional state spaces. Those methods exploit the structure of fiber bundles through the use of bundle primitives. Those primitives are used to create novel bundle planners, the rapidly-exploring quotient-space trees (QRRT*), and the quotient-space roadmap planner (QMP*). Both planners are shown to be probabilistically complete and almost-surely asymptotically optimal. To evaluate our bundle planners, we compare them against classical sampling-based planners on benchmarks of four low-dimensional scenarios, and eight high-dimensional scenarios, ranging from 21 to 100 degrees of freedom, including multiple robots and nonholonomic constraints. Our findings show improvements up to 2 to 6 orders of magnitude and underline the efficiency of multilevel motion planners and the benefit of exploiting multilevel abstractions using the terminology of fiber bundles.

We present StreamNet, an autoencoder architecture for the analysis of the highly heterogeneous geometry of large collections of white matter streamlines. This proposed framework takes advantage of geometry-preserving properties of the Wasserstein-1 metric in order to achieve direct encoding and reconstruction of entire bundles of streamlines. We show that the model not only accurately captures the distributive structures of streamlines in the population, but is also able to achieve superior reconstruction performance between real and synthetic streamlines. Experimental model performance is evaluated on white matter streamlines resulting from T1-weighted diffusion imaging of 40 healthy controls using recent state of the art bundle comparison metric that measures fiber-shape similarities.