## Geometric Insights into Spectral Clustering by Graph Laplacian Embeddings

We present new theoretical results for procedures identifying coarse structures in a given data set by means of appropriate spectral embeddings. We combine ideas from spectral geometry, metastability, optimal transport, and spectral analysis of weighted graph Laplacians to describe the embedding geometry. Our analysis focuses on 1) studying the embedding step of data clustering and 2) comparing the spectra of graph and continuum Laplacians, linking the original spectral clustering problem with a continuum counterpart.