Estimating Unknown Sparsity in Compressed Sensing

Within the framework of compressed sensing, many theoretical guarantees for signal reconstruction require that the number of linear measurements $n$ exceed the sparsity ||x||_0 of the unknown signal x\in\R^p. However, if the sparsity ||x||_0 is unknown, the choice of $n$ remains problematic. This paper considers the problem of estimating the unknown degree of sparsity of $x$ with only a small number of linear measurements. Although we show that estimation of ||x||_0 is generally intractable in this framework, we consider an alternative measure of sparsity s(x):=\frac{\|x\|_1^2}{\|x\|_2^2}, which is a sharp lower bound on ||x||_0, and is more amenable to estimation. When $x$ is a non-negative vector, we propose a computationally efficient estimator \hat{s}(x), and use non-asymptotic methods to bound the relative error of \hat{s}(x) in terms of a finite number of measurements. Remarkably, the quality of estimation is \emph{dimension-free}, which ensures that \hat{s}(x) is well-suited to the high-dimensional regime where n<<p. These results also extend naturally to the problem of using linear measurements to estimate the rank of a positive semi-definite matrix, or the sparsity of a non-negative matrix. Finally, we show that if no structural assumption (such as non-negativity) is made on the signal $x$, then the quantity s(x) cannot generally be estimated when n<<p.