Error bounds for convex parameter estimation

We evaluate the accuracy of sparsity-based estimation methods inspired from compressed sensing. Typical estimation approaches consist of minimizing a non-convex cost function that exhibits local minima, and require excessive computational resources. A tractable alternative relies on a sparse representation of the observation vector using a large dictionary matrix and a convex cost function. This estimation approach converts the intractable high-dimensional non-convex problem into a simpler convex problem with reduced dimension. Unfortunately, the advantages come at the expense of increased estimation error. Therefore, an evaluation of the estimation error is of considerable interest. We consider the case of estimating a single parameter vector, and provide upper bounds on the achievable accuracy. The theoretical results are corroborated by simulations.