8 research outputs found
Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space
This paper focuses on devising graph signal processing tools for the
treatment of data defined on the edges of a graph. We first show that
conventional tools from graph signal processing may not be suitable for the
analysis of such signals. More specifically, we discuss how the underlying
notion of a `smooth signal' inherited from (the typically considered variants
of) the graph Laplacian are not suitable when dealing with edge signals that
encode a notion of flow. To overcome this limitation we introduce a class of
filters based on the Edge-Laplacian, a special case of the Hodge-Laplacian for
simplicial complexes of order one. We demonstrate how this Edge-Laplacian leads
to low-pass filters that enforce (approximate) flow-conservation in the
processed signals. Moreover, we show how these new filters can be combined with
more classical Laplacian-based processing methods on the line-graph. Finally,
we illustrate the developed tools by denoising synthetic traffic flows on the
London street network.Comment: 5 pages, 2 figur
Signal Processing on Product Spaces
We establish a framework for signal processing on product spaces of
simplicial and cellular complexes. For simplicity, we focus on the product of
two complexes representing time and space, although our results generalize
naturally to products of simplicial complexes of arbitrary dimension. Our
framework leverages the structure of the eigenmodes of the Hodge Laplacian of
the product space to jointly filter along time and space. To this end, we
provide a decomposition theorem of the Hodge Laplacian of the product space,
which highlights how the product structure induces a decomposition of each
eigenmode into a spatial and temporal component. Finally, we apply our method
to real world data, specifically for interpolating trajectories of buoys in the
ocean from a limited set of observed trajectories
A Notion of Harmonic Clustering in Simplicial Complexes
We outline a novel clustering scheme for simplicial complexes that produces
clusters of simplices in a way that is sensitive to the homology of the
complex. The method is inspired by, and can be seen as a higher-dimensional
version of, graph spectral clustering. The algorithm involves only sparse
eigenproblems, and is therefore computationally efficient. We believe that it
has broad application as a way to extract features from simplicial complexes
that often arise in topological data analysis