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The Restricted Isometry Property

Module by: Ronald DeVore

We say that an n×Nn×Nn times N matrix ΦΦΦ has the restricted isometry property (RIP) for kkk if for each T⊆{1,…,N}T⊆{1,…,N}T subseteq { lbrace {1 , dotslow , N} rbrace} such that #T≤k#T≤kitalic "#" T <= k , ΦTΦTΦ_T (the matrix formed by choosing the columns of ΦΦΦ whose indices are in TTT ) has the property
( 1 − δ k ) ∥ x T ∥ ℓ 2 ≤ ∥ Φ T ( x ) ∥ ℓ 2 ≤ ( 1 + δ k ) ∥ x T ∥ ℓ 2 ( 1 − δ k ) ∥ x T ∥ ℓ 2 ≤ ∥ Φ T ( x ) ∥ ℓ 2 ≤ ( 1 + δ k ) ∥ x T ∥ ℓ 2 (RIP)
where 0<δk<10<δk<10 <δ_k <1 . This useful definition is by Candes and Tao. The idea is that the embedding of a kkk -dimensional space in MMM -dimensional space almost preserves norm – like an isometry. Another way of looking at it is to consider the matrix ΦTtΦTΦTtΦTΦ_T^t Φ_T , of size k×kk×kk times k . This matrix is symmetric, positive definite, and it’s eigen-values are between 1−δk1−δk1 - δ_k and 1+δk1+δk1 +δ_k .
I prefer the following modified condition (dubbed the MIRP), which is more convenient for mathematical analysis:
( c 1 ) − 1 ∥ x T ∥ ℓ 2 ≤ ∥ Φ T ( x ) ∥ ℓ 2 ≤ c 1 ∥ x T ∥ ℓ 2 ( c 1 ) − 1 ∥ x T ∥ ℓ 2 ≤ ∥ Φ T ( x ) ∥ ℓ 2 ≤ c 1 ∥ x T ∥ ℓ 2 (MRIP)
We can now state the following theorem.
theorem 1 
If ΦΦΦ satisfies MRIP for 2k2k2 k then ∃Δ∃Δ exists Δ s.t. (Φ,Δ)(Φ,Δ) \( {Φ , Δ} \) is instance optimal for ℓ1Nℓ1Nℓ_1^N for KKK .
This shows that whenever we have a matrix ΦΦΦ satisfying the MRIP for 2k2k2 k then it will perform well on encoding vectors (at least in the sense of ℓ1Nℓ1Nℓ_1^N accuracy). The question is how can we construct measurement matrices with this property? We can construct ΦΦΦ using Gaussian entries and then normalizing the columns.
theorem 2 
∃∃ exists constant c>0c>0c >0 s.t. if k≤cnlog(N∕n)k≤cnlog(N∕n)k <= c n over {l { \( {N ∕ n} \)}} then with high probability ΦΦΦ satisfies RIP and MRIP for kkk .
Given NNN and nnn , the range of kkk in the above results reflects how accurately we can recover data. There is another constant c′c′c^′ that serves as a converse bound for Theorem 3. This converse can be derived using Gluskin widths.
remark 1 
The following generic problem is of great interest: Consider the class of matrices ℳ = { Φ M × N , Φ has some prescribed property(eg. Toeplitz, circulant, etc.) } ℳ = { Φ M × N , Φ has some prescribed property(eg. Toeplitz, circulant, etc.) } . What is the largest kkk for which such a matrix can have the MRIP.

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