Modular proximal optimization for multidimensional total-variation regularization
This work addresses computational bottlenecks in optimization for researchers and practitioners in signal processing and machine learning, offering incremental improvements through a modular design.
The paper tackles the problem of efficiently computing proximal operators for total-variation regularization, particularly for ℓ₁-norm TV, by introducing a new geometric analysis that connects to taut-string methods, resulting in solvers that surpass state-of-the-art methods in speed and performance across various applications like image denoising and deconvolution.
We study \emph{TV regularization}, a widely used technique for eliciting structured sparsity. In particular, we propose efficient algorithms for computing prox-operators for $\ell_p$-norm TV. The most important among these is $\ell_1$-norm TV, for whose prox-operator we present a new geometric analysis which unveils a hitherto unknown connection to taut-string methods. This connection turns out to be remarkably useful as it shows how our geometry guided implementation results in efficient weighted and unweighted 1D-TV solvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the backbone for building more complex (two or higher-dimensional) TV solvers within a modular proximal optimization approach. We review the literature for an array of methods exploiting this strategy, and illustrate the benefits of our modular design through extensive suite of experiments on (i) image denoising, (ii) image deconvolution, (iii) four variants of fused-lasso, and (iv) video denoising. To underscore our claims and permit easy reproducibility, we provide all the reviewed and our new TV solvers in an easy to use multi-threaded C++, Matlab and Python library.