components of ${\mathcal{M}}$ restricted to ${\mathcal{M}}_n$. The resulting Laplacian embedding map $F_n$ sends the original data set into a set of $N$ orthogonal vectors in ${\mathbb{R}}^N$. Such geometric structure of the embedded dataset allows one to readily identify the “true" coarse structure of the original dataset. The key question that we attempt to answer in this work is: what happens in more general settings where in particular ${\mathcal{M}}$ is not connected? We attempt to answer this question by:

1.  Relaxing the notion of $N$ connected components of ${\mathcal{M}}$, and replacing it with a notion of well-separated mixture model $\nu $. We highlight that our notion of well-separatedness only depends on the mixture model $\nu$ and not on extrinsic objects like a user chosen kernel in the construction of the graph $G$.

2.  Describing the geometry of the Laplacian embedding in terms of what we will call an *orthogonal cone structure*, a generalization of the orthogonality condition of the embedded dataset observed when ${\mathcal{M}}$ has $N$ connected components. We consider this notion for arbitrary probability measures and not just for point clouds, thus extending the notion introduced in [@BinYu].

The analysis that we present in this work is divided into two 