PathFinder: Joint Decompositions of Linked Multimodal Datasets

Low-rank matrix decompositions can uncover patterns and structure in data and have a number of different applications across many disciplines. Extensions to "joint" low-rank decompositions have been proposed to link datasets from different modalities. While these methods enable the discovery of common patterns across modalities, they require that all the multimodal data share one or more dimensions. We propose a new analysis method, PathFinder, that enables co-analysis of datasets that do not necessarily all share a dimension. The key insight is that as long as pairs or subgroups of matrices do share some dimension, and that there are one or more paths that link across the data matrices, a global joint decomposition can be sought out. This enables the joint estimation of common patterns across different modalities, species, or scales, where a one-to-one mapping across all data along some dimension is not necessarily available. We show that PathFinder is a general umbrella under which many matrix decomposition methods fall as special cases. It can be used to discover common patterns across disparate datasets and to make predictions for missing data or modalities.

Dependency-Guided Code Generation: Structured Matrix Decomposition and Consistency-Guided Refinement

The increasing complexity of modern software systems has made automated code generation a fundamental task in software engineering. However, existing approaches often fail to adequately capture the intricate, multi-level dependencies among code entities, leading to generated code that is logically incomplete or difficult to integrate into real-world systems. To address this limitation, we propose a dependency-aware code generation framework that explicitly models interactions among code entities through a graph-based representation. We decompose dependencies into two complementary components: a quantized matrix that captures strong, explicit relations, and a sparse low-rank factorization that models weaker, implicit interactions. The decomposition is efficiently learned via an alternating optimization procedure. During code generation, the learned dependency structure is incorporated as a constraint, ensuring both semantic coherence and structural consistency of the generated code. Furthermore, we introduce a sparse triplet representation for strong dependencies, significantly improving storage efficiency and computational scalability. Extensive experiments demonstrate that our approach consistently produces code with superior semantic alignment and structural fidelity compared to existing methods.

Hankel and Toeplitz Rank-1 Decomposition of Arbitrary Matrices with Applications to Signal Direction-of-Arrival Estimation

We consider the problems of computing the optimal rank-$1$ Hankel and Toeplitz-structured approximation of arbitrary matrices under $L_2$ and $L_1$-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the $L_2$ and $L_1$ norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.