Supplementary Material for "T ensor Completion Made Practical" Run Jennrich's algorithm (see Section F.2.1) to decompose T Here we give an outline of the proof of Theorem 3.2. This is our main contribution. A robust analysis of Jennrich's algorithm implies that we can then estimate the rank one See Section F and Section G for details. C.1 Basic Facts We use the following notation: The following claim gives us a simple relation for this. C.4 Concentration Inequalities Claim C.8. Say we have real numbers γ x In particular we will prove Theorem B.1.

We propose a new General Game Playing (GGP) system called Regular Games (RG). The main goal of RG is to be both computationally efficient and convenient for game design. The system consists of several languages. The core component is a low-level language that defines the rules by a finite automaton. It is minimal with only a few mechanisms, which makes it easy for automatic processing (by agents, analysis, optimization, etc.). The language is universal for the class of all finite turn-based games with imperfect information. Higher-level languages are introduced for game design (by humans or Procedural Content Generation), which are eventually translated to a low-level language. RG generates faster forward models than the current state of the art, beating other GGP systems (Regular Boardgames, Ludii) in terms of efficiency. Additionally, RG's ecosystem includes an editor with LSP, automaton visualization, benchmarking tools, and a debugger of game description transformations.

The pretraining-finetuning paradigm has powered major advances in domains such as natural language processing and computer vision, with representative examples including masked language modeling and next-token prediction. In molecular representation learning, however, pretraining tasks remain largely restricted to node-level denois-ing, which effectively captures local atomic environments but is often insufficient for encoding the global molecular structure critical to graph-level property prediction tasks such as energy estimation and molecular regression. To address this gap, we introduce Geo-Recon, a graph-level pretraining framework that shifts the focus from individual atoms to the molecule as an integrated whole. GeoRe-con formulates a graph-level reconstruction task: during pretraining, the model is trained to produce an informative graph representation that guides geometry reconstruction while inducing smoother and more transferable latent spaces. This encourages the learning of coherent, global structural features beyond isolated atomic details. Without relying on external supervision, GeoRecon achieves generally improves over backbones baselines on multiple molecular benchmarks including QM9, MD17, MD22, and 3BPA, demonstrating the effectiveness of graph-level reconstruction for holistic and geometry-aware molecular embeddings.

Multimodal Large Language Models (MLLMs) struggle with precise reasoning for structured visuals like charts and diagrams, as pixel-based perception lacks a mechanism for verification. To address this, we propose to leverage derendering -- the process of reverse-engineering visuals into executable code -- as a new modality for verifiable visual reasoning. Specifically, we propose RECODE, an agentic framework that first generates multiple candidate programs to reproduce the input image. It then uses a critic to select the most faithful reconstruction and iteratively refines the code. This process not only transforms an ambiguous perceptual task into a verifiable, symbolic problem, but also enables precise calculations and logical inferences later on. On various visual reasoning benchmarks such as CharXiv, ChartQA, and Geometry3K, RECODE significantly outperforms methods that do not leverage code or only use code for drawing auxiliary lines or cropping. Our work demonstrates that grounding visual perception in executable code provides a new path toward more accurate and verifiable multimodal reasoning.

We first investigate how many dimensions of the latent space are important to image synthesis.

Language Models (LMs) are increasingly being used for code generation, but ensuring the correctness of generated programs remains a significant challenge. Although imperfect code may be acceptable during software development with human oversight, domains such as video games and robotics require one-shot correctness for runtime-critical components. W e present a constrained decoding algorithm for generating semantically correct programs that incorporates a context-sensitive parser, which, at each step, outputs a regular expression that satisfies a critical non-extensible property to guide the generation of the next token sequence that can continue to a correct program. T o build such a context-sensitive parser, we propose a framework of a dynamic tree of parsers (T oP) during parsing, where each parser corresponds to a modular context-free grammar enriched with contextual information such as variable scopes and type constraints, with tree branches representing ambiguity in the future code segment. W e demonstrate our approach through sLua, a strongly typed variant of Lua, showing that our method can generate semantically correct programs conforming to any prescribed scripting API. W e further show that, with careful design, our semantic guarantees extend to runtime correctness, as validated in the application of generating game mechanics for a roguelike video game.

Supplementary Material for "T ensor Completion Made Practical" Run Jennrich's algorithm (see Section F.2.1) to decompose T Here we give an outline of the proof of Theorem 3.2. This is our main contribution. A robust analysis of Jennrich's algorithm implies that we can then estimate the rank one See Section F and Section G for details. C.1 Basic Facts We use the following notation: The following claim gives us a simple relation for this. C.4 Concentration Inequalities Claim C.8. Say we have real numbers γ x In particular we will prove Theorem B.1.

Molecular docking is a crucial aspect of drug discovery, as it predicts the binding interactions between small-molecule ligands and protein pockets. However, current multi-task learning models for docking often show inferior performance in ligand docking compared to protein pocket docking. This disparity arises largely due to the distinct structural complexities of ligands and proteins. To address this issue, we propose a novel game-theoretic framework that models the protein-ligand interaction as a two-player game called the Docking Game, with the ligand docking module acting as the ligand player and the protein pocket docking module as the protein player. To solve this game, we develop a novel Loop Self-Play (LoopPlay) algorithm, which alternately trains these players through a two-level loop. In the outer loop, the players exchange predicted poses, allowing each to incorporate the other's structural predictions, which fosters mutual adaptation over multiple iterations. In the inner loop, each player dynamically refines its predictions by incorporating its own predicted ligand or pocket poses back into its model. We theoretically show the convergence of LoopPlay, ensuring stable optimization. Extensive experiments conducted on public benchmark datasets demonstrate that LoopPlay achieves approximately a 10\% improvement in predicting accurate binding modes compared to previous state-of-the-art methods. This highlights its potential to enhance the accuracy of molecular docking in drug discovery.

The Okada model is a widely used analytical solution for displacements and strains caused by a point or rectangular dislocation source in a 3D elastic half-space. We present OkadaTorch, a PyTorch implementation of the Okada model, where the entire code is differentiable; gradients with respect to input can be easily computed using automatic differentiation (AD). Our work consists of two components: a direct translation of the original Okada model into PyTorch, and a convenient wrapper interface for efficiently computing gradients and Hessians with respect to either observation station coordinates or fault parameters. This differentiable framework is well suited for fault parameter inversion, including gradient-based optimization, Bayesian inference, and integration with scientific machine learning (SciML) models. Our code is available here: https://github.com/msomeya1/OkadaTorch