Adjacency Spectral Embeddings of Correlation Networks

In many applications, weighted networks are constructed based on time series data. A time series is associated to each vertex, and edge weights are given by correlations between times series. This results in dependency among the edges, violating the assumptions of most common network models. Nonetheless, it is common to apply network embedding methods to networks built from correlation data. In this work, we show that this violation of assumptions is not critical. Provided that the time series under study are expressible in terms of a small number of orthogonal sequences, the adjacency spectral embedding provably recovers the true time series. That is, the adjacency spectral embedding applied to correlation networks serves as a denoising process, analogous to principal components analysis. In addition, we show that under suitable sparsity assumptions on the frequency domain, the embedding learned the adjacency spectral embedding recovers the Fourier coefficients of the true signals. This fact appears to be folklore in the signal processing community in the context of principle component analysis, but it is, to the best of our knowledge, new to the networks literature.

Networks

Embeddings

Correlation matrix

Spectral methods

Time series 

Section on Statistical Learning and Data Science