Graph connection Laplacian and random matrices with random blocks
This work provides foundational insights for data analysis using GCL, though it is incremental as it focuses on theoretical development for the null case rather than novel applications.
The authors tackled the problem of understanding the null case behavior of Graph Connection Laplacian (GCL) algorithms by developing a theory for the spectral distribution of random matrices with random block entries, including cases with significant dependence between blocks, and found that numerical simulations generally align well with their theoretical predictions.
Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the algorithms based on GCL. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries of fixed size. Part of the theory covers the case where there is significant dependence between the blocks. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.