The q-th order spectrum is a polynomial of degree q in the entries of a signal $x\in {\mathbb{C}}^N$, which is invariant under circular shifts of the signal. For $q\ge 3$, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum ($q=3$) and the trispectrum ($q=4$), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the q-th order spectrum is $N^{q-1}$, far exceeding the dimension of x, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be uniquely characterized up to symmetries, from only $N+1$ linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.

所述q个级光谱是次多项式q中的信号的条目$X\in {\mathbb{C}}^N$，它在信号的循环移位下是不变的。为了$q\ge 3$，该多项式唯一地确定信号，直至循环移位，称为高阶频谱。高阶光谱，特别是双谱（$q=3$) 和三谱 ($q=4$)，在各种统计信号处理和成像应用中发挥突出作用，例如相位检索和单粒子重建。然而，q阶谱的维数是$N^{q-1}$，远远超过x的维度，导致计算负载和存储需求增加。在这项工作中，我们表明没有必要存储和处理完整的高阶频谱：信号可以被唯一地表征到对称性，仅从$N+1$其高阶光谱的线性测量。证明依赖于代数几何的工具，并得到数值实验的证实。