Simplicial Trend Filtering (Invited Paper)

Reconstructing simplicial signals, e.g., signals defined on nodes, edges, triangles, etc., of a network, from (partial) noisy observation is of interest in water/traffic flow estimation or currency exchange markets. Typically, this concerns solving a regularised problem w.r.t. the l2 norm of the divergence or the curl of the signal, i.e., the netflows at nodes and in triangles. Realworld simplicial signals are intrinsically divergence- or curl-free, which makes l2 regularizers inapplicable. To overcome this, we develop a simplicial trend filter (STF) by regularising the total divergence and the curl via their l1 norm. By tuning two scalars, the STF can reduce independently the divergence and curl much more than smooth filtering, leading to a better reconstructed signal. The SFT is a convex problem and can be solved by fast iterative algorithms. We apply the SFT to interpolation and denoising tasks in forex and music/artist transition recordings and show its superior performance to alternatives.