Algebraic compressed sensing

TL;DR

This work reframes compressed sensing for polynomial-defined models, replacing RIP-based guarantees with a primarily algebraic approach that leverages Noether normalization and genericity. By showing that, for generic measurement maps, the intrinsic model dimension $d$ governs recoverability ($s\ge d$) and identifiability ($s\ge d+1$), the authors obtain near-minimal sample complexity and localized well-posedness. They develop a robust mathematical toolkit from algebraic geometry to characterize existence, finite fibers, and local inverses, and derive a computable condition-number framework for inverse stability. Numerical experiments on low-rank tensor completion illustrate the theory and demonstrate the practical behavior of condition numbers and identifiability under coordinate projections. Overall, the paper provides a principled, geometry-driven path to understanding and solving algebraic compressed sensing problems with minimal measurements and computable stability guarantees.

Abstract

We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ powerful techniques from algebraic geometry to study well-posedness of sufficiently general compressed sensing problems, including existence, local recoverability, global uniqueness, and local smoothness. Our main results are summarized in thirteen questions and answers in algebraic compressed sensing. Most of our answers concerning the minimum number of required measurements for existence, recoverability, and uniqueness of algebraic compressed sensing problems are optimal and depend only on the dimension of the model.

Figures (1)

• Figure 1: Schematic description of the geometry. On the right is the model $\mathcal{X}$. On the left is the set of measurements $\mu(\mathcal{X})$ taken from $\mathcal{X}$ using the measurement map $\mu$. In the picture we have $\mu(x_1) = y$. We are concerned with the question whether $x_1$ is the only point in $\mathcal{X}$ mapping to $y$, or whether there exists another point $x_2\in \mathcal{X}$ with $\mu(x_2)=y$.

Theorems & Definitions (42)

• Example 1: Classic compressed sensing
• Example 2: Matrix completion
• Example 3: Tensor completion
• Example 4: Moment varieties
• Theorem 1.1
• Proposition 2.1
• Example 5: Random maps
• Example 6
• Example 7: Rank-$1$ matrix recovery
• Remark 3.1