  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 $q\geq 3$, 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 characterized uniquely, 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.
