MOTIVATION: Structure-based drug design (SBDD) has advanced with deep generative models, but bridging the gap between continuous atomic coordinates and discrete atom types remains a challenge. Current approaches, such as diffusion and flow matching models, often fail to unify these heterogeneous modalities, relying on separate strategies or ill-fitting Euclidean metrics for discrete variables. This lack of a consistent framework limits generative models' ability to capture the geometric and chemical structure of protein-ligand complexes. RESULTS: We present MolPIF, a parameter interpolation flow mechanism designed to unify the generation of continuous and discrete molecular variables. Unlike traditional flow models that operate in sample space, MolPIF interpolates between distributions in the parameter space, theoretically recovering Wasserstein-2 optimal transport for continuous coordinates and establishing Fisher-Rao geodesics for discrete atom types. We further incorporate a geometry-enhanced learning strategy to improve the capture of atomic contexts. Extensive evaluations on the CrossDocked2020 dataset demonstrate that MolPIF outperforms baselines in binding affinity, chemical validity, geometric fidelity, and chemical space coverage. Additionally, MolPIF exhibits versatility in lead optimization and offers flexible prior distribution selection (such as Laplace), establishing a robust paradigm for SBDD. AVAILABILITY AND IMPLEMENTATION: Source code is freely available at https://github.com/BLEACH366/MolPIF.

Solving least squares (LS) problems is a major topic in many applications. With recent data explosion, traditional approach is no longer suitable while working with large datasets, instead, randomized algorithms become popular in addressing this issue. In this article we propose a new randomized algorithm - sparse subsampled randomized Hadamard transform (SpSRHT) for solving overdetermined least squares problems. Its unique block structure connects two most commonly used randomized algorithms subsampled randomized Hadamard transform (SRHT) and sparse subspace embedding (SpEmb) and creates a general framework which contains them as special cases. We have shown theoretically that SpSRHT with different parameters reaches the relative-error bound with sketch size ranging from the sketch size required by SpEmb to SRHT. The new algorithm closes the gap between SRHT and SpEmb which provides the possibility of balancing accuracy and efficiency demonstrated in them. This advantage of SpSRHT is also well illustrated in our numerical experiments.

Recently a deterministic method, frequent directions (FD) is proposed to solve the high dimensional low rank approximation problem. It works well in practice, but experiences high computational cost. In this paper, we establish a fast frequent directions algorithm for the low rank approximation problem, which implants a randomized algorithm, sparse subspace embedding (SpEmb) in FD. This new algorithm makes use of FD's natural block structure and sends more information through SpEmb to each block in FD. We prove that our new algorithm produces a good low rank approximation with a sketch of size linear on the rank approximated. Its effectiveness and efficiency are demonstrated by the experimental results on both synthetic and real world datasets, as well as applications in network analysis.