Signal Recovery on Algebraic Varieties Using Linear Samples

Zhiqiang Xu

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arXiv · 2506.17572

Signal Recovery on Algebraic Varieties Using Linear Samples

Zhiqiang Xu

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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. We also highlight several open problems, which could serve as a basis for future investigations in this field.

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Zhiqiang Xu. 2025-06-21. Signal Recovery on Algebraic Varieties Using Linear Samples. https://arxiv.org/abs/2506.17572

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