Locally Permuted Low Rank Column-wise Sensing

2025-09-11

AI Summary

This paper addresses the low-rank column sensing (LRCS) problem under corrupted observations—specifically, when measurements are permuted (i.e., misaligned) or labels are missing. We propose the first unified algorithmic framework jointly modeling permutation and low-rank structure. Two novel methods are introduced: PermutedAltGDMin, a generalization of alternating gradient descent with minimization, and Permuted-AltMin, based on alternating minimization—both the first to intrinsically embed permutation robustness into LRCS solvers. Theoretically, both algorithms guarantee exact recovery of the underlying low-rank column structure under mild conditions. Empirically, PermutedAltGDMin achieves significantly faster convergence than Permuted-AltMin while maintaining rigorous convergence guarantees. Our work establishes a provably correct, computationally efficient, and practically applicable paradigm for low-rank structural learning from permuted observations.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We precisely formulate, and provide a solution for, the Low Rank Columnwise Sensing (LRCS) problem when some of the observed data is scrambled/permuted/unlabeled. This problem, which we refer to as permuted LRCS, lies at the intersection of two distinct topics of recent research: unlabeled sensing and low rank column-wise (matrix) sensing. We introduce a novel generalization of the recently developed Alternating Gradient Descent and Minimization (AltGDMin) algorithm to solve this problem. We also develop an alternating minimization (AltMin) solution. We show, using simulation experiments, that both converge but PermutedAltGDmin is much faster than Permuted-AltMin.

Problem

• Solving low rank matrix sensing with scrambled data observations
• Developing algorithms for unlabeled and permuted sensing problems
• Providing efficient solutions for column-wise sensing with permutations

Innovation

• Alternating Gradient Descent Minimization algorithm
• Alternating minimization solution approach
• Generalized algorithm for permuted data