Identifying low-dimensional sufficient structures in nonlinear sufficient dimension reduction (SDR) has long been a fundamental yet challenging problem. Most existing methods lack theoretical guarantees of exhaustiveness in identifying lower dimensional structures, either at the population level or at the sample level. We tackle this issue by proposing a new method, generative sufficient dimension reduction (GenSDR), which leverages modern generative models. We show that GenSDR is able to fully recover the information contained in the central $σ$-field at both the population and sample levels. In particular, at the sample level, we establish a consistency property for the GenSDR estimator from the perspective of conditional distributions, capitalizing on the distributional learning capabilities of deep generative models. Moreover, by incorporating an ensemble technique, we extend GenSDR to accommodate scenarios with non-Euclidean responses, thereby substantially broadening its applicability. Extensive numerical results demonstrate the outstanding empirical performance of GenSDR and highlight its strong potential for addressing a wide range of complex, real-world tasks.

Various neural network architectures rely on pooling operators to aggregate information coming from different sources. It is often implicitly assumed in such contexts that vectors encode epistemic states, i.e. that vectors capture the evidence that has been obtained about some properties of interest, and that pooling these vectors yields a vector that combines this evidence. We study, for a number of standard pooling operators, under what conditions they are compatible with this idea, which we call the epistemic pooling principle. While we find that all the considered pooling operators can satisfy the epistemic pooling principle, this only holds when embeddings are sufficiently high-dimensional and, for most pooling operators, when the embeddings satisfy particular constraints (e.g. having non-negative coordinates). We furthermore show that these constraints have important implications on how the embeddings can be used in practice. In particular, we find that when the epistemic pooling principle is satisfied, in most cases it is impossible to verify the satisfaction of propositional formulas using linear scoring functions, with two exceptions: (i) max-pooling with embeddings that are upper-bounded and (ii) Hadamard pooling with non-negative embeddings. This finding helps to clarify, among others, why Graph Neural Networks sometimes under-perform in reasoning tasks. Finally, we also study an extension of the epistemic pooling principle to weighted epistemic states, which are important in the context of non-monotonic reasoning, where max-pooling emerges as the most suitable operator.

It is an established assumption that pattern-based models are good at precision, while learning based models are better at recall. But is that really the case? I argue that there are two kinds of recall: d-recall, reflecting diversity, and e-recall, reflecting exhaustiveness. I demonstrate through experiments that while neural methods are indeed significantly better at d-recall, it is sometimes the case that pattern-based methods are still substantially better at e-recall. Ideal methods should aim for both kinds, and this ideal should in turn be reflected in our evaluations.

Virtual screening, including molecular docking, plays an essential role in drug discovery. Many traditional and machine-learning based methods are available to fulfil the docking task. The traditional docking methods are normally extensively time-consuming, and their performance in blind docking remains to be improved. Although the runtime of docking based on machine learning is significantly decreased, their accuracy is still limited. In this study, we take the advantage of both traditional and machine-learning based methods, and present a method Deep Site and Docking Pose (DSDP) to improve the performance of blind docking. For the traditional blind docking, the entire protein is covered by a cube, and the initial positions of ligands are randomly generated in the cube. In contract, DSDP can predict the binding site of proteins and provide an accurate searching space and initial positions for the further conformational sampling. The docking task of DSDP makes use of the score function and a similar but modified searching strategy of AutoDock Vina, accelerated by implementation in GPUs. We systematically compare its performance with the state-of-the-art methods, including Autodock Vina, GNINA, QuickVina, SMINA, and DiffDock. DSDP reaches a 29.8% top-1 success rate (RMSD < 2 {\AA}) on an unbiased and challenging test dataset with 1.2 s wall-clock computational time per system. Its performances on DUD-E dataset and the time-split PDBBind dataset used in EquiBind, TankBind, and DiffDock are also effective, presenting a 57.2% and 41.8% top-1 success rate with 0.8 s and 1.0 s per system, respectively.