An ADMM Algorithm for a Class of Total Variation Regularized Estimation Problems
This work provides a faster algorithm for specific convex optimization problems, which is incremental as it builds on existing ADMM methods for total variation regularization.
The paper tackles optimization problems with separable cost functions and total variation regularization, such as Fused Lasso and total variation denoising, by proposing an alternating augmented Lagrangian method that enables parallel computation and efficient updates. In a numerical example, their implementation achieved a speedup of around 10,000 times compared to the generic solver SDPT3.
We present an alternating augmented Lagrangian method for convex optimization problems where the cost function is the sum of two terms, one that is separable in the variable blocks, and a second that is separable in the difference between consecutive variable blocks. Examples of such problems include Fused Lasso estimation, total variation denoising, and multi-period portfolio optimization with transaction costs. In each iteration of our method, the first step involves separately optimizing over each variable block, which can be carried out in parallel. The second step is not separable in the variables, but can be carried out very efficiently. We apply the algorithm to segmentation of data based on changes inmean (l_1 mean filtering) or changes in variance (l_1 variance filtering). In a numerical example, we show that our implementation is around 10000 times faster compared with the generic optimization solver SDPT3.