  Clustering is a ubiquitous problem in data science and signal processing. In
many applications where we observe noisy signals, it is common practice to
first denoise the data, perhaps using wavelet denoising, and then to apply a
clustering algorithm. In this paper, we develop a sparse convex wavelet
clustering approach that simultaneously denoises and discovers groups. Our
approach utilizes convex fusion penalties to achieve agglomeration and
group-sparse penalties to denoise through sparsity in the wavelet domain. In
contrast to common practice which denoises then clusters, our method is a
unified, convex approach that performs both simultaneously. Our method yields
denoised (wavelet-sparse) cluster centroids that both improve interpretability
and data compression. We demonstrate our method on synthetic examples and in an
application to NMR spectroscopy.
