Explicit construction of large restricted isometries beyond known barriers
Explicit restricted isometries with $k=m^{2/3-o(1)}$
Information Theory
Summary
The paper tackles a math problem about special matrices called restricted isometries, which are useful in data science and signal processing. The authors build specific matrices that work well even when the size and other parameters grow very large, surpassing previous limitations. They use a mix of graph theory and geometry ideas to do this, showing that a known supposed limit for certain matrix properties can be pushed further. Their approach challenges earlier beliefs about the difficulty of constructing these matrices explicitly.
What this means in practice
- •For signal processing engineers: Design measurement systems enabling more efficient recovery of sparse signals by using explicitly constructed matrices with improved restricted isometry bounds.
- •For coding theory developers: Develop error-correcting codes leveraging explicit matrix constructions with strong isometry properties to improve decoding reliability.
A theory result. No direct application yet.
Authors
Christian Kümmerle, Dustin G. Mixon
Abstract
We construct an explicit family of real $m\times n$ restricted isometries with $m\to\infty$, $m/n\to 0$, $δ_k\to 0$, and $k=m^{2/3-o(1)}$. We combine Gershgorin's circle theorem with a flat RIP estimate to reduce the problem to choosing a large vertex set in a strongly regular graph with the property that every small subset spans few edges. We select this vertex set using an incidence geometric result due to Zahl. The proof only uses the entrywise absolute value of the Gram matrix despite the conventional wisdom that such proof techniques exhibit a barrier of $m^{1/2}$. We prove that this barrier is actually $m^{2/3}$, and we show that our particular construction exhibits the same barrier.