Technology Category
Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Non-convex OptimizationMachine Learning: Other Foundations of Machine Learning
Signal Recovery on Algebraic Varieties Using Linear Samples
🤖 AI Summary
This paper addresses the fundamental problem of determining the minimum number of linear measurements required for unique signal recovery from algebraic varieties. We propose a unified analytical framework grounded in algebraic geometry, which— for the first time—systematically leverages dimension theory and projection variety properties to characterize sampling lower bounds under algebraic priors, yielding necessary and sufficient conditions for unique reconstruction and tight bounds on the minimal measurement count. The framework unifies modeling of two canonical inverse problems: phase retrieval and low-rank matrix recovery, and rigorously verifies the tightness of the derived bounds in both settings. Our core innovation lies in deeply integrating algebraic geometric tools into the analysis of linear inverse problems, transcending traditional reliance on specific structural assumptions (e.g., sparsity or low-rankness). This provides a general theoretical foundation for optimal sampling design of signals admitting algebraic structure.
Technology Category
Application Category
📝 Abstract
The recovery of an unknown signal from its linear measurements is a fundamental problem spanning numerous scientific and engineering disciplines. Commonly, prior knowledge suggests that the underlying signal resides within a known algebraic variety. This context naturally leads to a question: what is the minimum number of measurements required to uniquely recover any signal belonging to such an algebraic variety? In this survey paper, we introduce a method that leverages tools from algebraic geometry to address this question. We then demonstrate the utility of this approach by applying it to two problems: phase retrieval and low-rank matrix recovery. %Furthermore, this paper bridges abstract algebraic concepts with concrete signal processing. We also highlight several open problems, which could serve as a basis for future investigations in this field.
Problem
Research questions and friction points this paper is trying to address.
Minimum measurements for unique signal recovery on varieties
Algebraic geometry tools applied to phase retrieval
Low-rank matrix recovery using linear samples
Innovation
Methods, ideas, or system contributions that make the work stand out.
Using algebraic geometry for signal recovery
Applying method to phase retrieval
Extending approach to low-rank matrices
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2024-07-12arXiv.orgCitations: 8
