Model-based compressive sensing with Earth Mover’s Distance constraints

In compressive sensing, we want to recover a k-sparse signal x ∈ R from linear measurements of the form y = Φx, where Φ ∈ Rm×n describes the measurement process. Standard results in compressive sensing show that it is possible to exactly recover the signal x from only m = O(k log n k ) measurements for certain types of matrices. Model-based compressive sensing reduces the number of measurements even further by limiting the supports of x to a subset of the ( n k ) possible supports. Such a family of supports is called a structured sparsity model. In this thesis, we introduce a structured sparsity model for two-dimensional signals that have similar support in neighboring columns. We quantify the change in support between neighboring columns with the Earth Mover’s Distance (EMD), which measures both how many elements of the support change and how far the supported elements move. We prove that for a reasonable limit on the EMD between adjacent columns, we can recover signals in our model from only O(k log log k w ) measurements, where w is the width of the signal. This is an asymptotic improvement over the O(k log n k ) bound in standard compressive sensing. While developing the algorithmic tools for our proposed structured sparsity model, we also extend the model-based compressed sensing framework. In order to use a structured sparsity model in compressive sensing, we need a model projection algorithm that, given an arbitrary signal x, returns the best approximation in the model. We relax this constraint and develop a variant of IHT, an existing sparse recovery algorithm, that works with approximate model projection algorithms. Thesis Supervisor: Piotr Indyk Title: Professor of Computer Science and Engineering