# 2.2 Sparse approximation and ℓp spaces

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We now look at how well $f\in \mathbf{X}$ can be approximated by $n$ functions in the dictionary $\mathcal{D}$ .

We define the error of $n$ -term approximation of $f$ by the elements of the dictionary $\mathcal{D}$ as $\begin{array}{cc}\text{(1)}& \sigma {}_{n}\left(f\right){}_{\mathbf{X}}:=\sigma {}_{n}\left(f,\mathcal{D}\right){}_{X}:=\text{inf}{}_{s\in {\mathrm{\Sigma }}_{n}}\parallel f-s\parallel {}_{\mathbf{X}}.\end{array}$

We also define the class of $r$ -smooth signals in $\mathcal{D}$ as ,

with the corresponding norm ${\parallel f\parallel }_{{\mathcal{A}}^{r}}={sup}_{n=1,2,\mathrm{\dots }}{\mathcal{n}}^{r}{\sigma }_{n}{\left(f\right)}_{\mathbf{X}}$ .

In general, the larger $r$ is, the 'smoother' the function $s\in {\mathcal{A}}^{r}{\left(\mathcal{D}\right)}_{\mathbf{X}}$ . Note also that ${\mathcal{A}}^{r}\subseteq {\mathcal{A}}^{{r}^{\prime }}$ if $r>{r}^{\prime }$ . Given $f$ , let $r\left(f\right)=\text{sup}\left\{r:f\in {\mathcal{A}}^{r}\right\}$ be a measure of the "smoothness" of $f$ , i.e. a quantification of compressibility.

Let $\mathbf{X}=H$ , a Hilbert space

A Hilbert space is a complete inner product space with the norm induced by the innerproduct
such as $\mathbf{X}={L}_{2}\left(\mathbb{R}\right)$ , and assume $\mathcal{D}=B$ - an orthonormal basis on $\mathbf{X}$ ; i.e. if $B={\left\{{\varphi }_{i}\right\}}_{i}$ , then $〈{\varphi }_{i},{\varphi }_{j}〉={\delta }_{i,j}$ , where ${\delta }_{i,j}$ is the Kronecker delta. This also means that each $f\in \mathbf{X}$ has an expansion $f={\sum }_{j}{c}_{j}\left(f\right){\varphi }_{j}$ , where ${c}_{j}\left(f\right)=〈f,{\varphi }_{j}〉$ . We also have ${\parallel f\parallel }_{\mathbf{X}}^{2}={\sum }_{j=1}^{\mathrm{\infty }}{|{c}_{j}\left(f\right)|}^{2}$ .

Recall the definition of ${\mathrm{\ell }}_{p}$ spaces: let $\left({a}_{j}\right)\in \mathbb{R}$ ; then $\left({a}_{j}\right)\in {\mathrm{\ell }}_{p}$ if ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}<\infty$ with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={\left({\sum }_{j}{|{a}_{j}|}^{p}\right)}^{1/p}$ for $p<\mathrm{\infty }$ and ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={sup}_{j}|{a}_{j}|$ for $p=\mathrm{\infty }$ . We also recall that for ${L}_{p}$ spaces on compact sets, ${L}_{p}\subset {L}_{{p}^{\prime }}$ if $p>{p}^{\prime }$ . The opposite is true for ${\mathrm{\ell }}_{p}$ spaces: ${\mathrm{\ell }}_{p}\subset {\mathrm{\ell }}_{{p}^{\prime }}$ if $p<{p}^{\prime }$ . Hence, the smaller the value of $p$ is, the “smaller” ${\mathrm{\ell }}_{p}$ is.

Does there exist a sequence $\left({a}_{j}\right)$ with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{1}}={\sum }_{j}|{a}_{j}|<\mathrm{\infty }$ but with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={\left({\sum }_{j}{|{a}_{j}|}^{p}\right)}^{\frac{1}{p}}=\mathrm{\infty }$ for all $0 ? Consider the sequence ${a}_{n}=\frac{1}{n{\left(\mathrm{log}n\right)}^{1+\delta }}$ . We see that $\left({a}_{n}\right)\in {\mathrm{\ell }}_{1}$ but ${\parallel \left({a}_{n}\right)\parallel }_{{\mathrm{\ell }}_{p}}=\mathrm{\infty }$ for all $0 .

A sequence $\left({a}_{n}\right)$ is in ${\mathrm{\ell }}_{p}$ if the sorted magnitudes of the ${a}_{n}$ decay faster than ${n}^{\mathrm{-}\frac{1}{p}}$ .

Define ${a}_{n}^{*}$ as the element of the sequence $\left({a}_{n}\right)$ with the ${n}^{\text{th}}$ largest magnitude, and denote $\left({a}_{n}^{*}\right)$ as the decreasing rearrangement of $\left({a}_{n}\right)$ . It is easy to show that $k{\left({a}_{k}^{*}\right)}^{p}\le {\sum }_{n}{\left({a}_{n}\right)}^{p}$ for all $k$ ; also, if $\left({a}_{n}\right)\in {\mathrm{\ell }}_{p}$ , then ${a}_{k}^{*}\le {\parallel \left({a}_{n}\right)\parallel }_{{\mathrm{\ell }}_{p}}{k}^{-\frac{1}{p}}$ .

A sequence $\left({a}_{n}\right)$ is in weak ${\mathrm{\ell }}_{p}$ , denoted $\left({a}_{n}\right)\in w{\mathrm{\ell }}_{p}$ , if ${a}_{k}^{*}\le M{k}^{-\frac{1}{p}}$ . We also define the quasinorm

A quasinorm is has the properties of a norm except thatthe triangle inequality is replaced by the condition $\parallel x+y\parallel \le {C}_{0}\left[\parallel x\parallel +\parallel y\parallel \right]$ for some absolute constant ${C}_{0}$ .
${\parallel \left({a}_{n}\right)\parallel }_{w{\mathrm{\ell }}_{p}}$ as the smallest $M>0$ such that ${a}_{k}^{*}\le M{k}^{-\frac{1}{p}}$ for each $k$ .

The sequence ${a}_{n}=\frac{1}{n}$ is in weak ${\mathrm{\ell }}_{1}$ but not in ${\mathrm{\ell }}_{1}$ .

For $p$ , ${p}^{\mathrm{\prime }}$ such that ${p}^{\mathrm{\prime }}\mathrm{>}p$ , we have ${\mathrm{\ell }}_{p}\mathrm{\subset }w{\mathrm{\ell }}_{p}\mathrm{\subset }{\mathrm{\ell }}_{{p}^{\mathrm{\prime }}}$ .

Let $\mathcal{D}\mathrm{=}B$ be an orthonormal basis for the Hilbert space $\mathbf{X}\mathrm{=}H$ . For $f\mathrm{\in }\mathbf{X}$ with representation in $B\mathrm{=}\left[{\varphi }_{1},{\varphi }_{2},\mathrm{\dots }\right]$ as $f\mathrm{=}{\mathrm{\sum }}_{n}{c}_{n}\left(f\right){\varphi }_{n}$ , we have $f\mathrm{\in }{\mathrm{A}}^{r}{\left(B\right)}_{\mathbf{X}}$ if and only if the sequence $\left({c}_{n}\left(f\right)\right)\mathrm{\in }w{\mathrm{\ell }}_{\tau }$ , with $\frac{1}{\tau }\mathrm{=}r\mathrm{+}\frac{1}{2}$ . Moreover, there exist ${C}_{0},{C}_{0}^{\mathrm{\prime }}\mathrm{\in }\mathbb{R}$ such that ${C}_{0}^{\mathrm{\prime }}{\parallel \left({c}_{n}\left(f\right)\right)\parallel }_{w{\mathrm{\ell }}_{\tau }}\mathrm{\le }{\parallel f\parallel }_{{\mathrm{A}}^{r}}\mathrm{\le }{C}_{0}{\parallel \left({c}_{n}\left(f\right)\right)\parallel }_{w{\mathrm{\ell }}_{\tau }}$ .

Let $r=\frac{1}{2}$ . $f\in {\mathcal{A}}^{\frac{1}{2}}$ if and only if $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , i.e. if ${c}_{n}^{*}\left(f\right)\le M{n}^{-1}=\frac{M}{n}$ .

We prove the converse statement; the forward statement proof is left to the reader. We would like to show thatif $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , then $f\in {\mathcal{A}}^{r}$ , with $r=\frac{1}{\tau }-\frac{1}{2}$ . The best $n$ -term approximation of $f$ in $B$ is of the form $s={\sum }_{k\in \mathrm{\Lambda }}{a}_{k}{\varphi }_{k},\mathrm{♯}\mathrm{\Lambda }\le n$ . Therefore, we have:

where $M:=\parallel \left(c{}_{n}\left(f\right)\parallel {}_{w{\mathrm{\ell }}_{p}}$ .

We prove the converse statement; the forward statement proof is left to the reader. We would like to show thatif $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , then $f\in {\mathcal{A}}^{r}$ , with $r=\frac{1}{\tau }-\frac{1}{2}$ . The best $n$ -term approximation of $f$ in $B$ is of the form $s={\sum }_{k\in \mathrm{\Lambda }}{a}_{k}{\varphi }_{k},\mathrm{♯}\mathrm{\Lambda }\le n$ . Therefore, we have:

where we define $C=\frac{{\lambda }_{0}}{r}$ . Using this result in the earlier statement, we get $\begin{array}{cc}\text{(5)}& \begin{array}{ccc}\hfill \sum _{k=n+1}^{\mathrm{\infty }}{|{c}_{k}^{*}\left(f\right)|}^{2}& \hfill \le \hfill & C{M}^{2}{n}^{-2r}\le {M}^{2}z{n}^{-r};\hfill \end{array}\end{array}$

this implies by definition that $\left({c}_{k}\left(f\right)\right)\in {\mathcal{A}}^{r}$ .

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