This is interesting !
Compressive classification and the rare eclipse problem by Afonso S. Bandeira, Dustin G. Mixon, Benjamin Recht
This paper addresses the fundamental question of when convex sets remain disjoint after random projection. We provide an analysis using ideas from high-dimensional convex geometry. For ellipsoids, we provide a bound in terms of the distance between these ellipsoids and simple functions of their polynomial coefficients. As an application, this theorem provides bounds for compressive classification of convex sets. Rather than assuming that the data to be classified is sparse, our results show that the data can be acquired via very few measurements yet will remain linearly separable. We demonstrate the feasibility of this approach in the context of hyperspectral imaging.
from the conclusion:
It is also interesting how similar random projection and PCA perform. Note that the PCA method has an unfair advantage since it is data-adaptive, meaning that it uses the training data to select the projection, and in practical applications for which the sensing process is expensive, one might be interested in projecting in a non-adaptive way, thereby allowing for less sensing. Our simulations suggest that the expense is unnecessary, as a random projection will provide almost identical performance. As indicated in the previous subsection, random projection is also better understood as a means to maintain linear separability, and so there seems to be little bene t in choosing PCA over random projection (at least for this sort of classi cation task).
As such, it would be interesting to extend the results of this paper to more general classes of random projections, in particular, random projections which can be implemented with a hyperspectral imager (say).
Maybe, we could try random projections as instantiated by Nature.
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