Recently, it was discovered that for a given function class F the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of F in the uniform norm. That analysis is based on deep results in discretization of the square norm of functions from finite dimensional subspaces. In this paper we show how very recent results on universal discretization of the square norm of functions from a collection of finite dimensional subspaces lead to an inequality between optimal sparse recovery in the square norm and best sparse approximations in the uniform norm with respect to appropriate dictionaries.

Recently, there was a big progress in studying sampling discretization of integral norms of finite dimensional subspaces and collections of such subspaces (universal discretization). It was established that sampling discretization results are useful in a number of applications. In particular, they turn out to be useful in sampling recovery. Typically, recent sampling discretization results provide existence of good points for discretization. The main goal of this paper is to show that in the problem of universal discretization the independent random points on a given domain that are identically distributed according to the given probabilistic measure provide good points with high probability. Also, we demonstrate that a simple greedy type algorithm based on good points for universal discretization provide good recovery in the square norm.