Unbalanced optimal transport (UOT) extends classical optimal transport to measures with different total masses, but statistical guarantees for Monge-type estimation remain limited. We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural population target is not a map alone, but a transport-growth pair. Consequently, we develop two estimators for the transport-growth pairs under several setups: an optimal transport plan-based estimator for a general case, and a kernel-based estimator for a case with smooth densities. We also show that an error of the estimator achieves the minimax optimal rate by deriving a matching lower bound of the minimax risk. Our main technical contribution is a value-based stability reduction that converts perturbations of the UOT objective into transport and growth risks through a UOT gap condition. These results provide a statistical foundation for Monge-type estimation in unbalanced optimal transport.

In Appendix A, we introduce and recall necessary notations for the supplementary material. Regarding Sinkhorn algorithm, uk,vk are the updates of thek-th iteration. The main idea for deriving this bound comes from the geometric convergence rate (i.e. First, we represent the above difference by other quantities that are straightforward to bound. Thus, it has an unique optimal solution which could be directly calculated as Xi =B(ui,vi;Ci).

Reasoning based on Large Language Models (LLMs) has garnered increasing attention due to outstanding performance of these models in mathematical and complex logical tasks. Beginning with the Chain-of-Thought (CoT) prompting technique, numerous reasoning methods have emerged that decompose problems into smaller, sequential steps (or thoughts). However, existing reasoning models focus on conventional problem-solving and do not necessarily generate creative solutions by ``creative reasoning''. In domains where the solution space is expansive and conventional solutions are suboptimal, such as drug discovery or business strategization, creative reasoning to discover innovative solutions is crucial. To address this gap, first we introduce a computational framework for creative reasoning inspired by established cognitive science principles. With this framework, we propose three core creative reasoning paradigms, namely, \textit{combinational}, \textit{exploratory}, and \textit{transformative} reasoning, where each offers specific directions for systematic exploration of the universe of thoughts to generate creative solutions. Next, to materialize this framework using LLMs, we introduce the \textit{Universe of Thoughts} (or \textit{UoT}, for short), a novel set of methods to implement the aforementioned three creative processes. Finally, we introduce three novel tasks that necessitate creative problem-solving, along with an evaluation benchmark to assess creativity from three orthogonal perspectives: feasibility as constraint, and utility and novelty as metrics. With a comparative analysis against the state-of-the-art (SOTA) reasoning techniques as well as representative commercial models with reasoning capability, we show that UoT demonstrates superior performance in creative reasoning.

In the face of uncertainty, the ability to seek information is of fundamental importance.

With advancements in sensing systems including those in medical imaging like MRI and CT scanners, the advent of the Internet of Things, and cheap abundant data storage, the volume of data collection has grown rapidly. Oftentimes, datasets in these fields contain spatial, temporal, and contextual features, and thus datasets have grown not only in volume, but also in dimensionality and complexity. High-dimensionality can lead to challenges for access, analysis, and interpretation. Without additional assumptions on structure within the data, there is typically some form of the curse of dimensionality that appears in a given task-oriented pipeline. For this reason, many works have explored when and how low-dimensional structures appear in high-dimensional data, and have considered how to detect and utilize such structure effectively for learning [1, 2].

This file is organized as follows: Section 2 presents the pseudo-code of UOT A approach.

Our work reveals a structured shortcoming of the existing mainstream self-supervised learning methods.