Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? (2006)

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Suppose we are given a vector &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$f$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; in a class &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;${\cal F} \subset{\BBR}^N$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$f$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; to be able to recover &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$f$&lt;/tex&gt; &lt;/formula&gt;&lt;/emphasis&gt; to within precision &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$\epsilon$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; in the Euclidean &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$(\ell_2)$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$f$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$n$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;th largest entry of the vector &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$\vert f\vert$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; (or of its coefficients in a fixed basis) obeys &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$\vert f\vert _{(n)} \le R \cdot n^{-1/p}$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;, where &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$R &gt; 0$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; and &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$p &gt; 0$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;. Suppose that we take measurements &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$y_k = \langle f, X_k\rangle, k = 1, \ldots, K$&lt;/tex&gt; &lt;/formula&gt;&lt;/emphasis&gt;, where the &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$X_k$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; are &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$N$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;-dimensional Gaussian vectors with independent standard normal entries. Then for each &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$f$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; obeying the decay estimate above for some &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$0 &lt; p &lt; 1$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; and with overwhelming probability, our reconstruction &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$f^\sharp$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt;, defined as the solution to the constraints &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$y_k = \langle f^\sharp, X_k \rangle$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; with minimal &lt;emphasis&gt;&lt;formula formulatype="inline"&gt; &lt;tex&gt;$\ell_1$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; norm, obeys &lt;emphasis&gt; &lt;formula formulatype="display"&gt;&lt;tex&gt;$$ \Vert f - f^\sharp\Vert _{\ell_2} \le C_p \cdot R \cdot (K/\log N)^{-r}, \quad r = 1/p - 1/2. $$&lt;/tex&gt; &lt;/formula&gt;&lt;/emphasis&gt;There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$K$&lt;/tex&gt;&lt;/formula&gt;&lt;/emphasis&gt; measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of &lt;emphasis&gt;&lt;formula formulatype="inline"&gt;&lt;tex&gt;$f$&lt;/tex&gt; &lt;/formula&gt;&lt;/emphasis&gt;. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed. </para>