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
$\ell_{2}$
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
$\ell_{2}$
regularizers inapplicable. To overcome this, we develop a simplicial trend filter (STF) by regularising the total divergence and the curl via their
$\ell_{1}$
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.