Isometric sketching of any set via the Restricted Isometry Property

In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be computed in log-linear time, providing efficient dimensionality reduction of general sets. In particular, we show that using such matrices any set from high dimensions can be embedded into lower dimensions with near optimal distortion. We obtain our results by connecting dimensionality reduction of any set to dimensionality reduction of sparse vectors via a chaining argument.

💡 Research Summary

**
The paper “Isometric sketching of any set via the Restricted Isometry Property” establishes that structured random matrices, when combined with a random sign pattern, can serve as efficient substitutes for dense Gaussian matrices in the task of dimensionality reduction. The authors begin by recalling the classical Johnson‑Lindenstrauss (JL) lemma and Gordon’s “escape through the mesh” theorem, which guarantee that a random Gaussian matrix (A\in\mathbb{R}^{m\times n}) preserves Euclidean norms of all points in a set (T\subset\mathbb{R}^n) up to a factor ((1\pm\delta)) provided the embedding dimension satisfies (m = O(\omega(T)^2/\delta^2)). While these results are optimal in terms of the trade‑off between distortion (\delta) and the Gaussian width (\omega(T)), the computational cost of multiplying a dense Gaussian matrix by a vector is (O(mn)), which is prohibitive for large‑scale applications.

To overcome this bottleneck, the authors introduce a multiresolution version of the Restricted Isometry Property (MRIP). Standard RIP requires that a matrix (A) approximately preserves the Euclidean norm of all (s)-sparse vectors with a single distortion level (\delta). MRIP strengthens this by demanding that for every scale (\ell = 0,1,\dots, L) (with (L = \lceil \log_2 n\rceil)) the matrix simultaneously satisfies RIP with sparsity (s_\ell = 2^\ell s) and distortion (\delta_\ell = 2^\ell \delta). In other words, the matrix must behave like an RIP matrix at all dyadic sparsity levels, culminating in a uniform norm preservation for all vectors at the highest scale.

The central technical contribution (Theorem 3.1) shows that if a matrix (H) satisfies MRIP with parameters ((s,\tilde\delta)) and we multiply its columns by an independent random sign diagonal matrix (D) (i.e., (A = HD)), then for any set (T) we have with high probability \