We featured L1 Homotopy: A MATLAB Toolbox for Homotopy Algorithms in L1 Norm Minimization Problems before but it has now been updated to version 2.0 to deal with streaming measurements.
Here is the attendant paper: Sparse Recovery of Streaming Signals Using l_1-Homotopy by M. Salman Asif and Justin Romberg. The abstract reads:
Most of the existing methods for sparse signal recovery assume a static system: the unknown signal is a ﬁnite-length vector for which a ﬁxed set of linear measurements and a sparse representation basis are available and an `1-norm minimization program is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the unknown signal changes over time, and it is measured and reconstructed sequentially over small time intervals. A streaming framework for the reconstruction is particularly desired when dividing a streaming signal into disjoint blocks and processing each block independently is either infeasible or inefﬁcient.In this paper, we discuss two such streaming systems and a homotopy-based algorithm for quickly solving the associated weighted `1-norm minimization programs: 1) Recovery of a smooth, time-varying signal for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation. 2) Recovery of a sparse, time-varying signal that follows a linear dynamic model. For both the systems, we iteratively process measurements over a sliding interval and solve a weighted `1-norm minimization problem for estimating sparse coefﬁcients. Since we estimate overlapping portions of the streaming signal while adding and removing measurements, instead of solving a new `1 program from scratch at every iteration, we use an available signal estimate as a starting point in a homotopy formulation. Starting with a warm-start vector, our homotopy algorithm updates the solution in a small number of computationally inexpensive homotopy steps as the system changes. The homotopy algorithm presented in this paper is highly versatile as it can update the solution for the l_1 problem in a number of dynamical settings. We demonstrate with numerical experiments that our proposed streaming recovery framework outperforms the methods that represent and reconstruct a signal as independent, disjoint blocks, in terms of quality of reconstruction, and that our proposed homotopy-based updating scheme outperforms current state-of-the-art solvers in terms of the computation time and complexity.
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