Vector-Valued Graph Trend Filtering With Non-Convex Penalties

Elastic Graph Neural Networks

“…In the literature of graph total variation and graph trend filtering, the normalization step is often overlooked and the graph difference operator is directly used as in GTF (Wang et al, 2016;Varma et al, 2019). To achieve better numerical stability and handle diverse node degrees in real-world graphs, we propose to normalize each column of the incident matrix by the square root of node degrees for the 1 -based graph smoothing as follows 1 :…”

“…However, computing the estimators defined by ( 7) and ( 8) is much more challenging because of the nonsmoothness, and the two components, i.e., f (F) and g( ∆F), are non-separable as they are coupled by the graph difference operator ∆. In the literature, researchers have developed optimization algorithms for the graph trend filtering problem (4) such as Alternating Direction Method of Multipliers (ADMM) and Newton type algorithms (Wang et al, 2016;Varma et al, 2019). However, these algorithms require to solve the minimization of a nontrivial sub-problem in each single iteration, which incurs high computation complexity.…”

“…1 -based graph signal denoising has been explored in graph trend filtering (Wang et al, 2016;Varma et al, 2019) which tends to provide estimators with k-th order piecewise polynomials over graphs. Graph total variation has also been utilized in semi-supervised learning (Nie et al, 2011;Jung et al, 2016;Jung et al, 2019;Aviles-Rivero et al, 2019), spectral clustering (Bühler & Hein, 2009;Bresson et al, 2013b) and graph cut problems (Szlam & Bresson, 2010;Bresson et al, 2013a).…”

“…Theoretically, 1 -based methods tend to promote signal sparsity to trade for discontinuity (Rudin et al, 1992;Tibshirani et al, 2005;Sharpnack et al, 2012). Owning to these advantages, trend filtering (Tibshirani et al, 2014) and graph trend filter (Wang et al, 2016;Varma et al, 2019) demonstrate that 1 -based graph smoothing can adapt to inhomogenous level of smoothness of signals and yield estimators with k-th order piecewise polynomial functions, such as piecewise constant, linear and quadratic functions, depending on the order of the graph difference operator.…”

Compositional trend filtering

“…Recently, trend filtering has also been extended to many other contexts keeping the property that for a growing penalty parameter linear functions in the appropriate setting are selected; e.g. graphs [15], vector-valued graphs [13] and additive models [10].…”