Geometry and homotopy for l1 sparse representations

We explore the geometry of l1 sparse representations in both the noiseless (Basis Pursuit) and noisy (Basis Pursuit De-Noising) case using a homotopy method. We will see that the concept of the basis vertex c, which has unit inner product with active basis vectors, is a useful geometric concept, both for visualization and for algorithm construction. We derive an explicit homotopy continuation algorithm and find that this method has interesting parallels with the Polytope Faces Pursuit algorithm for the noiseless case. Numerical results confirm the operation of the algorithm.