  In this work we study statistical properties of graph-based algorithms for
multi-manifold clustering (MMC). In MMC the goal is to retrieve the
multi-manifold structure underlying a given Euclidean data set when this one is
assumed to be obtained by sampling a distribution on a union of manifolds
$\mathcal{M} = \mathcal{M}_1 \cup\dots \cup \mathcal{M}_N$ that may intersect
with each other and that may have different dimensions. We investigate
sufficient conditions that similarity graphs on data sets must satisfy in order
for their corresponding graph Laplacians to capture the right geometric
information to solve the MMC problem. Precisely, we provide high probability
error bounds for the spectral approximation of a tensorized Laplacian on
$\mathcal{M}$ with a suitable graph Laplacian built from the observations; the
recovered tensorized Laplacian contains all geometric information of all the
individual underlying manifolds. We provide an example of a family of
similarity graphs, which we call annular proximity graphs with angle
constraints, satisfying these sufficient conditions. We contrast our family of
graphs with other constructions in the literature based on the alignment of
tangent planes. Extensive numerical experiments expand the insights that our
theory provides on the MMC problem.
