This is the kind of metric we take for granted in this work.

Generative AI, and specifically GPT, can produce a remarkable solution to a mechanical engineering analysis problem - but also, on occasion, a flawed solution. For example, an elementary mechanics problem is solved flawlessly in one GPT instance and incorrectly in a subsequent GPT instance, with a success probability of only 85%. This unreliability renders "out-of-the-box" GPT unsuitable for deployment in education or engineering practice. We introduce an "N-Plus-1" GPT Agency for Initial (Low-Cost) Analysis of mechanical engineering Problem Statements. Agency first launches N instantiations of Agent Solve to yield N independent Proposed Problem Solution Realizations; Agency then invokes Agent Compare to summarize and compare the N Proposed Problem Solution Realizations and to provide a Recommended Problem Solution. We argue from Condorcet's Jury Theorem that, for a Problem Statement characterized by per-Solve success probability greater than 1/2 (and N sufficiently large), the Predominant (Agent Compare) Proposed Problem Solution will, with high probability, correspond to a Correct Proposed Problem Solution. Furthermore, Agent Compare can also incorporate aspects of Secondary (Agent Compare) Proposed Problem Solutions, in particular when the latter represent alternative Problem Statement interpretations - different Mathematical Models - or alternative Mathematical Solution Procedures. Comparisons to Grok Heavy, a commercial multi-agent model, show similarities in design and performance, but also important differences in emphasis: our Agency focuses on transparency and pedagogical value.

Foundation models exhibit remarkable generalization across diverse tasks, largely driven by the characteristics of their training data. Recent data-centric methods like pruning and compression aim to optimize training but offer limited theoretical insight into how data properties affect generalization, especially the data characteristics in sample scaling. Traditional perspectives further constrain progress by focusing predominantly on data quantity and training efficiency, often overlooking structural aspects of data quality. In this study, we introduce SCAR, a principled scheme for characterizing the intrinsic structural properties of datasets across four key measures: Scale, Coverage, Authenticity, and Richness. Unlike prior data-centric measures, SCAR captures stable characteristics that remain invariant under dataset scaling, providing a robust and general foundation for data understanding. Leveraging these structural properties, we introduce Foundation Data-a minimal subset that preserves the generalization behavior of the full dataset without requiring model-specific retraining. We model single-modality tasks as step functions and estimate the distribution of the foundation data size to capture step-wise generalization bias across modalities in the target multi-modal dataset. Finally, we develop a SCAR-guided data completion strategy based on this generalization bias, which enables efficient, modality-aware expansion of modality-specific characteristics in multimodal datasets. Experiments across diverse multi-modal datasets and model architectures validate the effectiveness of SCAR in predicting data utility and guiding data acquisition. Code is available at https://github.com/McAloma/SCAR.

Real-world datasets often contain missing or corrupted values. Completing multidimensional tensor-structured data with missing entries is essential for numerous applications. Smoothness-constrained low-rank factorization models have shown superior performance with reduced computational costs. While effective at capturing global and long-range correlations, these models struggle to reproduce short-scale, high-frequency variations in the data. In this paper, we introduce the Generalized Least Squares Kernelized Tensor Factorization (GLSKF) framework for tensor completion. GLSKF integrates smoothness-constrained low-rank factorization with a locally correlated residual process; the resulting additive structure can effectively characterize both global dependencies and local variations. In particular, we define the covariance norm to enforce the smoothness of factor matrices in the global low-rank factorization, and use structured covariance/kernel functions to model the local processes. For model estimation, we develop an alternating least squares (ALS) procedure with closed-form solutions for each subproblem. To efficiently handle missing data, GLSKF utilizes projection matrices that preserve the Kronecker structure of covariances, facilitating fast computations through conjugate gradient (CG) and preconditioned conjugate gradient (PCG) algorithms. The proposed framework is evaluated on four real-world datasets across diverse tasks: traffic speed imputation, color image inpainting, video completion, and MRI image reconstruction. Experimental results confirm that GLSKF delivers superior effectiveness and scalability, establishing it as a robust solution for multidimensional tensor completion.

In drug discovery, in vitro and in vivo experiments reveal biochemical activities related to the efficacy and toxicity of compounds. The experimental data accumulate into massive, ever-evolving, and sparse datasets. Quantitative Structure-Activity Relationship (QSAR) models, which predict biochemical activities using only the structural information of compounds, face challenges in integrating the evolving experimental data as studies progress. We develop QSAR-Complete (QComp), a data completion framework to address this issue. Based on pre-existing QSAR models, QComp utilizes the correlation inherent in experimental data to enhance prediction accuracy across various tasks. Moreover, QComp emerges as a promising tool for guiding the optimal sequence of experiments by quantifying the reduction in statistical uncertainty for specific endpoints, thereby aiding in rational decision-making throughout the drug discovery process.

Boltzmann machines are able to learn highly complex, multimodal, structured and multiscale real-world data distributions. Parameters of the model are usually learned by minimizing the Kullback-Leibler (KL) divergence from training samples to the learned model. We propose in this work a novel approach for Boltzmann machine training which assumes that a meaningful metric between observations is known. This metric between observations can then be used to define the Wasserstein distance between the distribution induced by the Boltzmann machine on the one hand, and that given by the training sample on the other hand. We derive a gradient of that distance with respect to the model parameters. Minimization of this new objective leads to generative models with different statistical properties. We demonstrate their practical potential on data completion and denoising, for which the metric between observations plays a crucial role.

Fluid data completion is a research problem with high potential benefit for both experimental and computational fluid dynamics. An effective fluid data completion method reduces the required number of sensors in a fluid dynamics experiment, and allows a coarser and more adaptive mesh for a Computational Fluid Dynamics (CFD) simulation. However, the ill-posed nature of the fluid data completion problem makes it prohibitively difficult to obtain a theoretical solution and presents high numerical uncertainty and instability for a data-driven approach (e.g., a neural network model). To address these challenges, we leverage recent advancements in computer vision, employing the vector quantization technique to map both complete and incomplete fluid data spaces onto discrete-valued lower-dimensional representations via a two-stage learning procedure. We demonstrated the effectiveness of our approach on Kolmogorov flow data (Reynolds number: 1000) occluded by masks of different size and arrangement. Experimental results show that our proposed model consistently outperforms benchmark models under different occlusion settings in terms of point-wise reconstruction accuracy as well as turbulent energy spectrum and vorticity distribution.

Graph representation learning (GRL) on attribute-missing graphs, which is a common yet challenging problem, has recently attracted considerable attention. We observe that existing literature: 1) isolates the learning of attribute and structure embedding thus fails to take full advantages of the two types of information; 2) imposes too strict distribution assumption on the latent space variables, leading to less discriminative feature representations. In this paper, based on the idea of introducing intimate information interaction between the two information sources, we propose our Siamese Attribute-missing Graph Auto-encoder (SAGA). Specifically, three strategies have been conducted. First, we entangle the attribute embedding and structure embedding by introducing a siamese network structure to share the parameters learned by both processes, which allows the network training to benefit from more abundant and diverse information. Second, we introduce a K-nearest neighbor (KNN) and structural constraint enhanced learning mechanism to improve the quality of latent features of the missing attributes by filtering unreliable connections. Third, we manually mask the connections on multiple adjacent matrices and force the structural information embedding sub-network to recover the true adjacent matrix, thus enforcing the resulting network to be able to selectively exploit more high-order discriminative features for data completion. Extensive experiments on six benchmark datasets demonstrate the superiority of our SAGA against the state-of-the-art methods.

Neural autoregressive models are explicit density estimators that achieve state-of-the-art likelihoods for generative modeling. The D-dimensional data distribution is factorized into an autoregressive product of one-dimensional conditional distributions according to the chain rule. Data completion is a more involved task than data generation: the model must infer missing variables for any partially observed input vector. Previous work introduced an order-agnostic training procedure for data completion with autoregressive models. Missing variables in any partially observed input vector can be imputed efficiently by choosing an ordering where observed dimensions precede unobserved ones and by computing the autoregressive product in this order. In this paper, we provide evidence that the order-agnostic (OA) training procedure is suboptimal for data completion. We propose an alternative procedure (OA++) that reaches better performance in fewer computations. It can handle all data completion queries while training fewer one-dimensional conditional distributions than the OA procedure. In addition, these one-dimensional conditional distributions are trained proportionally to their expected usage at inference time, reducing overfitting. Finally, our OA++ procedure can exploit prior knowledge about the distribution of inference completion queries, as opposed to OA. We support these claims with quantitative experiments on standard datasets used to evaluate autoregressive generative models.

Boltzmann machines are able to learn highly complex, multimodal, structured and multiscale real-world data distributions. Parameters of the model are usually learned by minimizing the Kullback-Leibler (KL) divergence from training samples to the learned model. We propose in this work a novel approach for Boltzmann machine training which assumes that a meaningful metric between observations is known. This metric between observations can then be used to define the Wasserstein distance between the distribution induced by the Boltzmann machine on the one hand, and that given by the training sample on the other hand. We derive a gradient of that distance with respect to the model parameters. Minimization of this new objective leads to generative models with different statistical properties. We demonstrate their practical potential on data completion and denoising, for which the metric between observations plays a crucial role.