  Lensless illumination single-pixel imaging with a multicore fiber (MCF) is a
computational imaging technique that enables potential endoscopic observations
of biological samples at cellular scale. In this work, we show that this
technique is tantamount to collecting multiple symmetric rank-one projections
(SROP) of a Hermitian \emph{interferometric} matrix -- a matrix encoding the
spectral content of the sample image. In this model, each SROP is induced by
the complex \emph{sketching} vector shaping the incident light wavefront with a
spatial light modulator (SLM), while the projected interferometric matrix
collects up to $O(Q^2)$ image frequencies for a $Q$-core MCF. While this scheme
subsumes previous sensing modalities, such as raster scanning (RS) imaging with
beamformed illumination, we demonstrate that collecting the measurements of $M$
random SLM configurations -- and thus acquiring $M$ SROPs -- allows us to
estimate an image of interest if $M$ and $Q$ scale linearly (up to log factors)
with the image sparsity level, hence requiring much fewer observations than RS
imaging or a complete Nyquist sampling of the $Q \times Q$ interferometric
matrix. This demonstration is achieved both theoretically, with a specific
restricted isometry analysis of the sensing scheme, and with extensive Monte
Carlo experiments. Experimental results made on an actual MCF system finally
demonstrate the effectiveness of this imaging procedure on a benchmark image.
