Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization

Matrix factorization techniques, especially Nonnegative Matrix Factorization (NMF), have been widely used for dimensionality reduction and interpretable data representation. However, existing NMF-based methods are inherently single-scale and fail to capture the evolution of connectivity structures across resolutions. In this work, we propose persistent nonnegative matrix factorization (pNMF), a scale-parameterized family of NMF problems, that produces a sequence of persistence-aligned embeddings rather than a single one. By leveraging persistent homology, we identify a canonical minimal sufficient scale set at which the underlying connectivity undergoes qualitative changes. These canonical scales induce a sequence of graph Laplacians, leading to a coupled NMF formulation with scale-wise geometric regularization and explicit cross-scale consistency constraint. We analyze the structural properties of the embeddings along the scale parameter and establish bounds on their increments between consecutive scales. The resulting model defines a nontrivial solution path across scales, rather than a single factorization, which poses new computational challenges. We develop a sequential alternating optimization algorithm with guaranteed convergence. Numerical experiments on synthetic and single-cell RNA sequencing datasets demonstrate the effectiveness of the proposed approach in multi-scale low-rank embeddings.

Soft causal learning for generalized molecule property prediction: An environment perspective

Learning on molecule graphs has become an increasingly important topic in AI for science, which takes full advantage of AI to facilitate scientific discovery. Existing solutions on modeling molecules utilize Graph Neural Networks (GNNs) to achieve representations but they mostly fail to adapt models to out-of-distribution (OOD) samples. Although recent advances on OOD-oriented graph learning have discovered the invariant rationale on graphs, they still ignore three important issues, i.e., 1) the expanding atom patterns regarding environments on graphs lead to failures of invariant rationale based models, 2) the associations between discovered molecular subgraphs and corresponding properties are complex where causal substructures cannot fully interpret the labels. 3) the interactions between environments and invariances can influence with each other thus are challenging to be modeled. To this end, we propose a soft causal learning framework, to tackle the unresolved OOD challenge in molecular science, from the perspective of fully modeling the molecule environments and bypassing the invariant subgraphs. Specifically, we first incorporate chemistry theories into our graph growth generator to imitate expaned environments, and then devise an GIB-based objective to disentangle environment from whole graphs and finally introduce a cross-attention based soft causal interaction, which allows dynamic interactions between environments and invariances. We perform experiments on seven datasets by imitating different kinds of OOD generalization scenarios. Extensive comparison, ablation experiments as well as visualized case studies demonstrate well generalization ability of our proposal.