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.

翻译：我们引入了代数压缩感知问题的广泛子类，其中结构化信号通过多项式进行显式或隐式建模。例如，这包括低秩矩阵和张量恢复。我们采用代数几何中的强大技术来研究足够一般的压缩感知问题的适定性，包括存在性、局部可恢复性、全局唯一性和局部光滑性。我们的主要结果总结为代数压缩感知中的十三个问题与解答。我们关于代数压缩感知问题存在性、可恢复性和唯一性所需最少测量数量的解答大多数是最优的，且仅取决于模型的维度。