Nonparametric Estimation via Convex Programming

In the paper, we focus primarily on the problem of recovering a linear form g x of unknown “signal” x known to belong to a given convex compact set X ⊂ R from N independent realizations of a random variable ι taking values in a finite set, the distribution p of ι being affinely parameterized by x: p = Ax + b. With no additional assumptions on X and A, we develop minimax optimal, within an absolute constant factor, and computationally efficient estimation routine. We then apply this routine to recovering x itself in the Euclidean norm.