  Predictive rate-distortion analysis suffers from the curse of dimensionality:
clustering arbitrarily long pasts to retain information about arbitrarily long
futures requires resources that typically grow exponentially with length. The
challenge is compounded for infinite-order Markov processes, since conditioning
on finite sequences cannot capture all of their past dependencies. Spectral
arguments show that algorithms which cluster finite-length sequences fail
dramatically when the underlying process has long-range temporal correlations
and can fail even for processes generated by finite-memory hidden Markov
models. We circumvent the curse of dimensionality in rate-distortion analysis
of infinite-order processes by casting predictive rate-distortion objective
functions in terms of the forward- and reverse-time causal states of
computational mechanics. Examples demonstrate that the resulting causal
rate-distortion theory substantially improves current predictive
rate-distortion analyses.
