# 3.3 The restricted isometry property

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We say that an $n×N$ matrix $\Phi$ has the restricted isometry property (RIP) for $k$ if for each $T\subseteq \left\{1,\dots ,N\right\}$ such that $\text{#}T\le k$ , ${\Phi }_{T}$ (the matrix formed by choosing the columns of $\Phi$ whose indices are in $T$ ) has the property

 $\begin{array}{c}\left(1-{\delta }_{k}\right){\parallel {x}_{T}\parallel }_{{\ell }_{2}}{\le }_{}{\parallel {\Phi }_{T}\left(x\right)\parallel }_{{\ell }_{2}}{\le }_{}\left(1+{\delta }_{k}\right){{\parallel {x}_{T}\parallel }_{{\ell }_{2}}}_{}\end{array}$ (RIP)

where $0<{\delta }_{k}<1$ . This useful definition is by Candes and Tao. The idea is that the embedding of a $k$ -dimensional space in $M$ -dimensional space almost preserves norm – like an isometry. Another way of looking at it is to consider the matrix ${\Phi }_{T}^{t}{\Phi }_{T}$ , of size $k×k$ . This matrix is symmetric, positive definite, and it’s eigen-values are between $1-{\delta }_{k}$ and $1+{\delta }_{k}$ .

I prefer the following modified condition (dubbed the MIRP), which is more convenient for mathematical analysis:

 ${\left({c}_{1}\right)}^{-1}{\parallel {x}_{T}\parallel }_{{\ell }_{2}}\le {\parallel {\mathrm{\Phi }}_{T}\left(x\right)\parallel }_{{\ell }_{2}}\le {c}_{1}{\parallel {x}_{T}\parallel }_{{\ell }_{2}}$ (MRIP)

We can now state the following theorem.

If $\Phi$ satisfies MRIP for $2k$ then $\exists \Delta$ s.t. $\left(\Phi ,\Delta \right)$ is instance optimal for ${\ell }_{1}^{N}$ for $K$ .

This shows that whenever we have a matrix $\Phi$ satisfying the MRIP for $2k$ then it will perform well on encoding vectors (at least in the sense of ${\ell }_{1}^{N}$ accuracy). The question is how can we construct measurement matrices with this property? We can construct $\Phi$ using Gaussian entries and then normalizing the columns.

$\exists$ constant $c>0$ s.t. if $k\le c\frac{n}{\mathrm{log}\left(N∕n\right)}$ then with high probability $\Phi$ satisfies RIP and MRIP for $k$ .

Given $N$ and $n$ , the range of $k$ in the above results reflects how accurately we can recover data. There is another constant ${c}^{\prime }$ that serves as a converse bound for Theorem 3. This converse can be derived using Gluskin widths.

The following generic problem is of great interest: Consider the class of matrices $ℳ=\left\{\Phi M×N,\mathrm{\Phi }\text{has some prescribed property(eg. Toeplitz, circulant, etc.)}\right\}$ . What is the largest $k$ for which such a matrix can have the MRIP.

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