Restricted isometry properties, a.k.a. uniform uncertainty principles (CaTa05, CaRT06), mutual incoherence (DoET06), irrepresentability conditions (ZhYu06)…

Recoverability conditions, as seen in sparse regression, sparse basis dictionaries, function approximation, compressed sensing etc. Uncertainty principles for a sparse world.

Terry Tao mentions the various related conditions for the compressed sensing problem, and which types of random matrices satisfy them. The casual lecture notes on uniform uncertainty look nice.

## Restricted Isometry

The compressed sensing formulation.

The restricted isometry constant of a matrix \(A\), is the smallest constant \(\delta_s\) \((1-\delta_s(A))\|x\|_2^2\leq \|Ax\|_2^2\leq (1+\delta_s(A))\|x\|_2^2\) for all \(s\)-sparse \(x\). That is, the measurement matrix does not change the norm of sparse signals “too much”, and in particular, does not null them when \(\delta_s \lt 1.\)

## Irrepresentability

TBD.

## Incoherence

TBD.

## Refs

- BDDW08
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- CaTa05
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- SoMe17
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- Zhang, C.-H. (2010) Nearly unbiased variable selection under minimax concave penalty.
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