  Compressive learning is an emerging approach to drastically reduce the memory
footprint of large-scale learning, by first summarizing a large dataset into a
low-dimensional sketch vector, and then decoding from this sketch the latent
information needed for learning. In light of recent progress on information
preservation guarantees for sketches based on random features, a major
objective is to design easy-to-tune algorithms (called decoders) to robustly
and efficiently extract this information. To address the underlying non-convex
optimization problems, various heuristics have been proposed. In the case of
compressive clustering, the standard heuristic is CL-OMPR, a variant of sliding
Frank-Wolfe. Yet, CL-OMPR is hard to tune, and the examination of its
robustness was overlooked. In this work, we undertake a scrutinized examination
of CL-OMPR to circumvent its limitations. In particular, we show how this
algorithm can fail to recover the clusters even in advantageous scenarios. To
gain insight, we show how the deficiencies of this algorithm can be attributed
to optimization difficulties related to the structure of a correlation function
appearing at core steps of the algorithm. To address these limitations, we
propose an alternative decoder offering substantial improvements over CL-OMPR.
Its design is notably inspired from the mean shift algorithm, a classic
approach to detect the local maxima of kernel density estimators. The proposed
algorithm can extract clustering information from a sketch of the MNIST dataset
that is 10 times smaller than previously.
