UC Davis

The theory of Compressed Sensing asserts that an unknown signal $x\in\mathbb{R}^p$ can be accurately recovered from an underdetermined set of $n$ linear measurements with $n\ll p$, provided that $x$ is sufficiently sparse. However, in applications, the degree of sparsity $\|x\|_0$ is typically unknown, and the problem of directly estimating $\|x\|_0$ has been a longstanding gap between theory and practice. A closely related issue is that $\|x\|_0$ is a highly idealized measure of sparsity, and for real signals with entries not exactly equal to 0, the value $\|x\|_0=p$ is not a useful description of compressibility. In our previous conference paper that examined these problems, Lopes 2013, we considered an alternative measure of "soft" sparsity, $\|x\|_1^2/\|x\|_2^2$, and designed a procedure to estimate $\|x\|_1^2/\|x\|_2^2$ that does not rely on sparsity assumptions. The present work offers a new deconvolution-based method for estimating unknown sparsity, which has wider applicability and sharper theoretical guarantees. Whereas our earlier work was limited to estimating the quantity $\|x\|_1^2/\|x\|_2^2$, the current paper introduces a family of entropy-based sparsity measures $s_q(x):=\big(\frac{\|x\|_q}{\|x\|_1}\big)^{\frac{q}{1-q}}$ parameterized by $q\in[0,\infty]$. Two other main advantages of the new approach are that it handles measurement noise with infinite variance, and that it yields confidence intervals for $s_q(x)$ with asymptotically exact coverage probability (whereas our previous intervals were conservative). In addition to confidence intervals, we also analyze several other aspects of our proposed estimator $\hat{s}_q(x)$ and show that randomized measurements are an essential aspect of our procedure.