  The canonical tensor model (CTM) is a tensor model in Hamilton formalism and
is studied as a model for gravity in both classical and quantum frameworks. Its
dynamical variables are a canonical conjugate pair of real symmetric
three-index tensors, and a question in this model was how to extract spacetime
pictures from the tensors. We give such an extraction procedure by using two
techniques widely known in data analysis. One is the tensor-rank (or CP, etc.)
decomposition, which is a certain generalization of the singular value
decomposition of a matrix and decomposes a tensor into a number of vectors. By
regarding the vectors as points forming a space, topological properties can be
extracted by using the other data analysis technique called persistent
homology, and geometries by virtual diffusion processes over points. Thus, time
evolutions of the tensors in the CTM can be interpreted as topological and
geometric evolutions of spaces. We have performed some initial investigations
of the classical equation of motion of the CTM in terms of these techniques for
a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy
spheres as spaces, and have obtained agreement with the general relativistic
system obtained previously in a formal continuum limit of the CTM. It is also
demonstrated by some concrete examples that the procedure is general for any
dimensions and topologies, showing the generality of the CTM.
