arXiv subjects

Leon Liu

Publications and source records attributed to Leon Liu.

4 recordsLinked to original sources

arXiv2025

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

math.QA↗

arXiv2025

Dreamland: Controllable World Creation with Simulator and Generative Models

Large-scale video generative models can synthesize diverse and realistic visual content for dynamic world creation, but they often lack element-wise controllability, hindering their use in editing scenes and training embodied AI agents. We propose Dreamland, a hybrid world generation framework combining the granular control of a physics-based simulator and the photorealistic content output of large-scale pretrained generative models. In particular, we design a layered world abstraction that encodes both pixel-level and object-level semantics and geometry as an intermediate representation to bridge the simulator and the generative model. This approach enhances controllability, minimizes adaptation cost through early alignment with real-world distributions, and supports off-the-shelf use of existing and future pretrained generative models. We further construct a D3Sim dataset to facilitate the training and evaluation of hybrid generation pipelines. Experiments demonstrate that Dreamland outperforms existing baselines with 50.8% improved image quality, 17.9% stronger controllability, and has great potential to enhance embodied agent training. Code and data will be made available.

cs.CV↗

arXiv2025

Cube: A Roblox View of 3D Intelligence

Foundation models trained on vast amounts of data have demonstrated remarkable reasoning and generation capabilities in the domains of text, images, audio and video. Our goal at Roblox is to build such a foundation model for 3D intelligence, a model that can support developers in producing all aspects of a Roblox experience, from generating 3D objects and scenes to rigging characters for animation to producing programmatic scripts describing object behaviors. We discuss three key design requirements for such a 3D foundation model and then present our first step towards building such a model. We expect that 3D geometric shapes will be a core data type and describe our solution for 3D shape tokenizer. We show how our tokenization scheme can be used in applications for text-to-shape generation, shape-to-text generation and text-to-scene generation. We demonstrate how these applications can collaborate with existing large language models (LLMs) to perform scene analysis and reasoning. We conclude with a discussion outlining our path to building a fully unified foundation model for 3D intelligence.

cs.CV↗

arXiv2022

Strongly nonexpansive mappings revisited: uniform monotonicity and operator splitting

The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of the class of uniformly monotone operators and their corresponding reflected resolvents. We show that the latter is closely related to the class of the strongly nonexpansive operators introduced by Bruck and Reich. Connections to duality via inverse operators are systematically studied. We provide applications to Douglas-Rachford and Peaceman-Rachford methods. Examples that illustrate and tighten our results are presented.

math.OC↗