Tensor-structured sketching for constrained least squares

Constrained least squares problems arise in many applications. Their memory and computation costs are expensive in practice involving high-dimensional input data. We employ the so-called "sketching" strategy to project the least squares problem onto a space of a much lower "sketching dimension" via a random sketching matrix. The key idea of sketching is to reduce the dimension of the problem as much as possible while maintaining the approximation accuracy. Tensor structure is often present in the data matrices of least squares, including linearized inverse problems and tensor decompositions. In this work, we utilize a general class of row-wise tensorized sub-Gaussian matrices as sketching matrices in constrained optimizations for the sketching design's compatibility with tensor structures. We provide theoretical guarantees on the sketching dimension in terms of error criterion and probability failure rate. In the context of unconstrained linear regressions, we obtain an optimal estimate for the sketching dimension. For optimization problems with general constraint sets, we show that the sketching dimension depends on a statistical complexity that characterizes the geometry of the underlying problems. Our theories are demonstrated in a few concrete examples, including unconstrained linear regression and sparse recovery problems.