The first three rows present the model performance when trained on a single dataset.

We introduce Chi-Geometry - a library that generates graph data for testing and benchmarking GNNs' ability to predict chirality. Chi-Geometry generates synthetic graph samples with (i) user-specified geometric and topological traits to isolate certain types of samples and (ii) randomized node positions and species to minimize extraneous correlations. Each generated graph contains exactly one chiral center labeled either R or S, while all other nodes are labeled N/A (non-chiral). The generated samples are then combined into a cohesive dataset that can be used to assess a GNN's ability to predict chirality as a node classification task. Chi-Geometry allows more interpretable and less confounding benchmarking of GNNs for prediction of chirality in the graph samples which can guide the design of new GNN architectures with improved predictive performance. We illustrate Chi-Geometry's efficacy by using it to generate synthetic datasets for benchmarking various state-of-the-art (SOTA) GNN architectures. The conclusions of these benchmarking results guided our design of two new GNN architectures. The first GNN architecture established all-to-all connections in the graph to accurately predict chirality across all challenging configurations where previously tested SOTA models failed, but at a computational cost (both for training and inference) that grows quadratically with the number of graph nodes. The second GNN architecture avoids all-to-all connections by introducing a virtual node in the original graph structure of the data, which restores the linear scaling of training and inference computational cost with respect to the number of nodes in the graph, while still ensuring competitive accuracy in detecting chirality with respect to SOTA GNN architectures.

Hopfield networks are associative memory (AM) systems, designed for storing and retrieving patterns as local minima of an energy landscape. In the classical Hopfield model, an interesting phenomenon occurs when the amount of training data reaches its critical memory load $- spurious\,\,states$, or unintended stable points, emerge at the end of the retrieval dynamics, leading to incorrect recall. In this work, we examine diffusion models, commonly used in generative modeling, from the perspective of AMs. The training phase of diffusion model is conceptualized as memory encoding (training data is stored in the memory). The generation phase is viewed as an attempt of memory retrieval. In the small data regime the diffusion model exhibits a strong memorization phase, where the network creates distinct basins of attraction around each sample in the training set, akin to the Hopfield model below the critical memory load. In the large data regime, a different phase appears where an increase in the size of the training set fosters the creation of new attractor states that correspond to manifolds of the generated samples. Spurious states appear at the boundary of this transition and correspond to emergent attractor states, which are absent in the training set, but, at the same time, have distinct basins of attraction around them. Our findings provide: a novel perspective on the memorization-generalization phenomenon in diffusion models via the lens of AMs, theoretical prediction of existence of spurious states, empirical validation of this prediction in commonly-used diffusion models.

An interpretable comparison of generative models requires the identification of sample types produced more frequently by each of the involved models. While several quantitative scores have been proposed in the literature to rank different generative models, such score-based evaluations do not reveal the nuanced differences between the generative models in capturing various sample types. In this work, we attempt to solve a differential clustering problem to detect sample types expressed differently by two generative models. To solve the differential clustering problem, we propose a method called Fourier-based Identification of Novel Clusters (FINC) to identify modes produced by a generative model with a higher frequency in comparison to a reference distribution. FINC provides a scalable stochastic algorithm based on random Fourier features to estimate the eigenspace of kernel covariance matrices of two generative models and utilize the principal eigendirections to detect the sample types present more dominantly in each model. We demonstrate the application of the FINC method to large-scale computer vision datasets and generative model frameworks. Our numerical results suggest the scalability of the developed Fourier-based method in highlighting the sample types produced with different frequencies by widely-used generative models. Code is available at \url{https://github.com/buyeah1109/FINC}

Statistical samples, in order to be representative, have to be drawn from a population in a random and unbiased way. Nevertheless, it is common practice in the field of model-based diagnosis to make estimations from (biased) best-first samples. One example is the computation of a few most probable possible fault explanations for a defective system and the use of these to assess which aspect of the system, if measured, would bring the highest information gain. In this work, we scrutinize whether these statistically not well-founded conventions, that both diagnosis researchers and practitioners have adhered to for decades, are indeed reasonable. To this end, we empirically analyze various sampling methods that generate fault explanations. We study the representativeness of the produced samples in terms of their estimations about fault explanations and how well they guide diagnostic decisions, and we investigate the impact of sample size, the optimal trade-off between sampling efficiency and effectivity, and how approximate sampling techniques compare to exact ones.