Utility (entropy) of Multispace Random Projection

Prove that the Multispace Random Projection (MRP) transformation y = (1/√n) R x used for post-processing embeddings satisfies the Utility (Entropy sufficiency) property of the ideal primitive; specifically, establish that the computational HILL entropy of the protected embedding Y conditioned on the stored projection matrix R is at least the min-entropy of the input embedding distribution X (up to negligible loss), thereby showing that MRP does not create excessive collisions and largely preserves input entropy.

Background

The paper defines an ideal post-processing primitive with three properties: correctness (noise tolerance), security (fuzzy one-wayness), and utility (entropy sufficiency). The utility property requires that the entropy of the protected output Y, conditioned on the helper data S, be comparable to the min-entropy of the input embeddings X.

Multispace Random Projection (MRP) projects input embeddings x to a lower-dimensional space using a random Gaussian matrix R and stores both the projected vector y and R. While the Johnson–Lindenstrauss lemma supports correctness for distance preservation, the paper does not provide a proof of the utility (entropy) property for MRP.

The authors explicitly conjecture that MRP also satisfies the utility property, intuitively arguing that random projections largely transfer input entropy to the output without causing many collisions. Establishing this formally would clarify MRP’s suitability as an ideal primitive with respect to entropy.