  Time series modeling is a well-established problem, which often requires that
methods (1) expressively represent complicated dependencies, (2) forecast long
horizons, and (3) efficiently train over long sequences. State-space models
(SSMs) are classical models for time series, and prior works combine SSMs with
deep learning layers for efficient sequence modeling. However, we find
fundamental limitations with these prior approaches, proving their SSM
representations cannot express autoregressive time series processes. We thus
introduce SpaceTime, a new state-space time series architecture that improves
all three criteria. For expressivity, we propose a new SSM parameterization
based on the companion matrix -- a canonical representation for discrete-time
processes -- which enables SpaceTime's SSM layers to learn desirable
autoregressive processes. For long horizon forecasting, we introduce a
"closed-loop" variation of the companion SSM, which enables SpaceTime to
predict many future time-steps by generating its own layer-wise inputs. For
efficient training and inference, we introduce an algorithm that reduces the
memory and compute of a forward pass with the companion matrix. With sequence
length $\ell$ and state-space size $d$, we go from $\tilde{O}(d \ell)$
na\"ively to $\tilde{O}(d + \ell)$. In experiments, our contributions lead to
state-of-the-art results on extensive and diverse benchmarks, with best or
second-best AUROC on 6 / 7 ECG and speech time series classification, and best
MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1)
fits AR($p$) processes that prior deep SSMs fail on, (2) forecasts notably more
accurately on longer horizons than prior state-of-the-art, and (3) speeds up
training on real-world ETTh1 data by 73% and 80% relative wall-clock time over
Transformers and LSTMs.
