  To analyze high-dimensional systems, many fields in science and engineering
rely on high-level descriptions, sometimes called "macrostates,"
"coarse-grainings," or "effective theories". Examples of such descriptions
include the thermodynamic properties of a large collection of point particles
undergoing reversible dynamics, the variables in a macroeconomic model
describing the individuals that participate in an economy, and the summary
state of a cell composed of a large set of biochemical networks.
  Often these high-level descriptions are constructed without considering the
ultimate reason for needing them in the first place. Here, we formalize and
quantify one such purpose: the need to predict observables of interest
concerning the high-dimensional system with as high accuracy as possible, while
minimizing the computational cost of doing so. The resulting State Space
Compression (SSC) framework provides a guide for how to solve for the {optimal}
high-level description of a given dynamical system, rather than constructing it
based on human intuition alone.
  In this preliminary report, we introduce SSC, and illustrate it with several
information-theoretic quantifications of "accuracy", all with different
implications for the optimal compression. We also discuss some other possible
applications of SSC beyond the goal of accurate prediction. These include SSC
as a measure of the complexity of a dynamical system, and as a way to quantify
information flow between the scales of a system.
