Sparse harmonic transforms II: best s-term approximation guarantees for bounded orthonormal product bases in sublinear-time

In this paper we develop a sublinear-time compressive sensing algorithm for approximating functions of many variables which are compressible in given Bounded Orthonormal Product Basis (BOPB). The resulting is shown to both have an associated best s-term recovery guarantee the BOPB, and also work well numerically solving sparse approximation problems involving contained span fairly general sets as $$\sim 10^{230}$$ orthonormal basis functions. All code made publicly available. As part proof main new variants known CoSaMP proposed can utilize any sufficiently accurate support identification procedure satisfying Support Identification Property (SIP) order obtain strong guarantees. These then proven runtime error behavior largely determined by chosen method. theoretical results developing BOPB robust arbitrary additive errors. Using method create variant BOPB-compressible variables.