Monte Carlo wavelets: a randomized approach to frame discretization
This work addresses the discretization of wavelets for signal processing and analysis, but appears incremental as it builds on existing spectral calculus and Monte Carlo methods.
The authors tackled the problem of discretizing continuous wavelets on general domains by proposing Monte Carlo wavelets, a randomized approach based on stochastic discretization of integral operators, and established convergence with rates under regularity assumptions.
In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by spectral calculus. Then, we propose a stochastic discretization based on Monte Carlo estimates of integral operators. Using concentration of measure results, we establish the convergence of such a discretization and derive convergence rates under natural regularity assumptions.