  We present a technique to coarse-grain quantum states in a finite-dimensional
Hilbert space. Our method is distinguished from other approaches by not relying
on structures such as a preferred factorization of Hilbert space or a preferred
set of operators (local or otherwise) in an associated algebra. Rather, we use
the data corresponding to a given set of states, either specified independently
or constructed from a single state evolving in time. Our technique is based on
principle component analysis (PCA), and the resulting coarse-grained quantum
states live in a lower dimensional Hilbert space whose basis is defined using
the underlying (isometric embedding) transformation of the set of fine-grained
states we wish to coarse-grain. Physically, the transformation can be
interpreted to be an "entanglement coarse-graining" scheme that retains most of
the global, useful entanglement structure of each state, while needing fewer
degrees of freedom for its reconstruction. This scheme could be useful for
efficiently describing collections of states whose number is much smaller than
the dimension of Hilbert space, or a single state evolving over time.
