  The compressive learning framework reduces the computational cost of training
on large-scale datasets. In a sketching phase, the data is first compressed to
a lightweight sketch vector, obtained by mapping the data samples through a
well-chosen feature map, and averaging those contributions. In a learning
phase, the desired model parameters are then extracted from this sketch by
solving an optimization problem, which also involves a feature map. When the
feature map is identical during the sketching and learning phases, formal
statistical guarantees (excess risk bounds) have been proven.
  However, the desirable properties of the feature map are different during
sketching and learning (e.g. quantized outputs, and differentiability,
respectively). We thus study the relaxation where this map is allowed to be
different for each phase. First, we prove that the existing guarantees carry
over to this asymmetric scheme, up to a controlled error term, provided some
Limited Projected Distortion (LPD) property holds. We then instantiate this
framework to the setting of quantized sketches, by proving that the LPD indeed
holds for binary sketch contributions. Finally, we further validate the
approach with numerical simulations, including a large-scale application in
audio event classification.
