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The $q$-th order spectrum is a polynomial of degree $q$ in the entries of asignal $x\in\mathbb{C}^N$, which is invariant under circular shifts of thesignal. For $q\geq 3$, this polynomial determines the signal uniquely, up to acircular 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 aprominent role in various statistical signal processing and imagingapplications, such as phase retrieval and single-particle reconstruction.However, the dimension of the $q$-th order spectrum is $N^{q-1}$, far exceedingthe dimension of $x$, leading to increased computational load and storagerequirements. In this work, we show that it is unnecessary to store and processthe full high-order spectra: a signal can be characterized uniquely, up tosymmetries, from only $N+1$ linear measurements of its high-order spectra. Theproof relies on tools from algebraic geometry and is corroborated by numericalexperiments.

Signal recovery from a few linear measurements of its high-order spectra