Full (exact) conformal set vs. split or cross-validated conformal set Non-connectedness of the conformal prediction set. This was initially suggested in [18, Remark 1]. We follow the actual practice in the literature [14, Remark 5]. We did not observe violations. We will also summarize the proposed algorithm in a direct pseudo-code.

Dimensionality reduction (DR) and deep neural networks (DNNs) are two important aspects in data analysis. In data analysis and deep learning, the datasets are often high-dimensional and exhibit some complicated topological structures due to various backgrounds from science to engineering [1,2,4-7]. Traditional approaches to data analysis and visualization, in particular on images recognition, often fail in the high-dimensional setting, and a common practice is to perform dimensionality reduction [2, 6, 11] in order to make data analysis tractable and economic, and the DNNs is a powerful tool in dealing with non-linear dimensionality reduction problems. It has now been recognized that practical datasets often consists of features of low intrinsic dimensions with some nontrivial topological structures [1,2,6], and the geometric structure of datasets heavily affect the architecture of the deep neural networks. Nonetheless, how and to what extent the geometric (topological) structure of datasets is connected with the architecture of a deep neural network remains unclear and is an active research area of deep learning in recent years. 1

Neural network training is typically viewed as gradient descent on a loss surface. We propose a fundamentally different perspective: learning is a structure-preserving transformation (a functor L) between the space of network parameters (Param) and the space of learned representations (Rep). This categorical framework reveals that different training runs producing similar test performance often belong to the same homotopy class (continuous deformation family) of optimization paths. We show experimentally that networks converging via homotopic trajectories generalize within 0.5% accuracy of each other, while non-homotopic paths differ by over 3%. The theory provides practical tools: persistent homology identifies stable minima predictive of generalization (R^2 = 0.82 correlation), pullback constructions formalize transfer learning, and 2-categorical structures explain when different optimization algorithms yield functionally equivalent models. These categorical invariants offer both theoretical insight into why deep learning works and concrete algorithmic principles for training more robust networks.

Besides per-pixel accuracy, topological correctness is also crucial for the segmentation of images with fine-scale structures, e.g., satellite images and biomedical images.

Natural language is replete with superficially different statements, such as ``Charles Darwin wrote" and ``Charles Darwin is the author of", which carry the same meaning. Large language models (LLMs) should generate the same next-token probabilities in such cases, but usually do not. Empirical workarounds have been explored, such as using k-NN estimates of sentence similarity to produce smoothed estimates. In this paper, we tackle this problem more abstractly, introducing a categorical homotopy framework for LLMs. We introduce an LLM Markov category to represent probability distributions in language generated by an LLM, where the probability of a sentence, such as ``Charles Darwin wrote" is defined by an arrow in a Markov category. However, this approach runs into difficulties as language is full of equivalent rephrases, and each generates a non-isomorphic arrow in the LLM Markov category. To address this fundamental problem, we use categorical homotopy techniques to capture ``weak equivalences" in an LLM Markov category. We present a detailed overview of application of categorical homotopy to LLMs, from higher algebraic K-theory to model categories, building on powerful theoretical results developed over the past half a century.

Energy-based models (EBMs) are a powerful class of probabilistic generative models due to their flexibility and interpretability. However, relationships between potential flows and explicit EBMs remain underexplored, while contrastive divergence training via implicit Markov chain Monte Carlo (MCMC) sampling is often unstable and expensive in high-dimensional settings. In this paper, we propose Variational Potential Flow Bayes (VPFB), a new energy-based generative framework that eliminates the need for implicit MCMC sampling and does not rely on auxiliary networks or cooperative training. VPFB learns an energy-parameterized potential flow by constructing a flow-driven density homotopy that is matched to the data distribution through a variational loss minimizing the Kullback-Leibler divergence between the flow-driven and marginal homotopies. This principled formulation enables robust and efficient generative modeling while preserving the interpretability of EBMs. Experimental results on image generation, interpolation, out-of-distribution detection, and compositional generation confirm the effectiveness of VPFB, showing that our method performs competitively with existing approaches in terms of sample quality and versatility across diverse generative modeling tasks. 1 1 Introduction

Narrow passage path planning is a prevalent problem from industrial to household sites, often facing difficulties in finding feasible paths or requiring excessive computational resources. Given that deep penetration into the environment can cause optimization failure, we propose a framework to ensure feasibility throughout the process using a series of subproblems tailored for narrow passage problem. We begin by decomposing the environment into convex objects and initializing collision constraints with a subset of these objects. By continuously interpolating the collision constraints through the process of sequentially introducing remaining objects, our proposed framework generates subproblems that guide the optimization toward solving the narrow passage problem. Several examples are presented to demonstrate how the proposed framework addresses narrow passage path planning problems.

The kinematic/robotic community is not only interested in measuring the closeness of a given robot configuration to its next singular one but also in a geometric meaningful index evaluating how far the robot design is away from being architecturally singular. Such an architecture singularity distance, which can be used by engineers as a criterion within the design process, is presented for a certain class of parallel manipulators of Stewart-Gough type; namely so-called linear pentapods. Geometrically the architecture singular designs are well-understood and can be subclassified into several cases, which allows to solve the optimization problem of computing the closest architecture singular design to a given linear pentapod with algorithms from numerical algebraic geometry.

In highly interactive driving scenarios, the actions of one agent greatly influences those of its neighbors. Planning safe motions for autonomous vehicles in such interactive environments, therefore, requires reasoning about the impact of the ego's intended motion plan on nearby agents' behavior. Deep-learning-based models have recently achieved great success in trajectory prediction and many models in the literature allow for ego-conditioned prediction. However, leveraging ego-conditioned prediction remains challenging in downstream planning due to the complex nature of neural networks, limiting the planner structure to simple ones, e.g., sampling-based planner. Despite their ability to generate fine-grained high-quality motion plans, it is difficult for gradient-based planning algorithms, such as model predictive control (MPC), to leverage ego-conditioned prediction due to their iterative nature and need for gradient. We present Interactive Joint Planning (IJP) that bridges MPC with learned prediction models in a computationally scalable manner to provide us the best of both the worlds. In particular, IJP jointly optimizes over the behavior of the ego and the surrounding agents and leverages deep-learned prediction models as prediction priors that the join trajectory optimization tries to stay close to. Furthermore, by leveraging homotopy classes, our joint optimizer searches over diverse motion plans to avoid getting stuck at local minima. Closed-loop simulation result shows that IJP significantly outperforms the baselines that are either without joint optimization or running sampling-based planning.