  In this paper, we consider a certain convolutional Laplacian for metric
measure spaces and investigate its potential for the statistical analysis of
complex objects. The spectrum of that Laplacian serves as a signature of the
space under consideration and the eigenvectors provide the principal directions
of the shape, its harmonics. These concepts are used to assess the similarity
of objects or understand their most important features in a principled way
which is illustrated in various examples. Adopting a statistical point of view,
we define a mean spectral measure and its empirical counterpart. The
corresponding limiting process of interest is derived and statistical
applications are discussed.
