  Rank-one projections (ROP) of matrices and quadratic random sketching of
signals support several data processing and machine learning methods, as well
as recent imaging applications, such as phase retrieval or optical processing
units. In this paper, we demonstrate how signal estimation can be operated
directly through such quadratic sketches--equivalent to the ROPs of the "lifted
signal" obtained as its outer product with itself--without explicitly
reconstructing that signal. Our analysis relies on showing that, up to a minor
debiasing trick, the ROP measurement operator satisfies a generalised sign
product embedding (SPE) property. In a nutshell, the SPE shows that the scalar
product of a signal sketch with the "sign" of the sketch of a given pattern
approximates the square of the projection of that signal on this pattern. This
thus amounts to an insertion (an "inception") of a ROP model inside a ROP
sketch. The effectiveness of our approach is evaluated in several synthetic
experiments.
