In recent years, tremendous progress has been made in high-dimensional inference problems by exploiting intrinsic low-dimensional structure. Sparsity is perhaps the simplest model for low-dimensional structure. It is based on the assumption that the object of interest can be represented as a linear combination of a small number of elementary functions, which are assumed to belong to a larger collection, or dictionary, of possible functions. Sparse recovery is the problem of determining which components are needed in the representation based on measurements of the object. Most theory and methods for sparse recovery are based on an assumption of non-adaptive measurements. This chapter investigates the advantages of sequential measurement schemes that adaptively focus sensing using information gathered throughout the measurement process. In particular, it is shown that adaptive sensing can be significantly more powerful when the measurements are contaminated with additive noise.

Introduction

High-dimensional inference problems cannot be accurately solved without enormous amounts of data or prior assumptions about the nature of the object to be inferred. Great progress has been made in recent years by exploiting intrinsic low-dimensional structure in high-dimensional objects. Sparsity is perhaps the simplest model for taking advantage of reduced dimensionality. It is based on the assumption that the object of interest can be represented as a linear combination of a small number of elementary functions. The specific functions needed in the representation are assumed to belong to a larger collection or dictionary of functions, but are otherwise unknown.