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 $\tilde{O}(d \ell)$
naïvely 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.

提出了一种名为 SpaceTime 的新型状态空间时间序列体系结构，其通过一种新的 SSM 参数化基于伴随矩阵来提高表现力，并引入一个 “闭环” 变化的伴随 SSM，使其能够生成自己的逐层输入进行长期预测，同时通过算法实现前向传递的内存和计算效率提高，并在大量不同的基准测试中取得最先进的结果。

利用简单的离散状态空间有效建模时间序列

Effectively Modeling Time Series with Simple Discrete State Spaces

The paper is devoted to establishing some general exponential inequalities
for supermartingales. The inequalities improve or generalize many exponential
inequalities of Bennett, Freedman, de la Pe\~{n}a, Pinelis and van de Geer.
Moreover, our concentration inequalities also improve some known inequalities
for sums of independent random variables. Applications associated with linear
regressions, autoregressive processes and branching processes are provided. In
particular, an interesting application of {de la Pe\~{n}a's} inequality to
self-normalized deviations is also provided.

该论文旨在建立超鞅的一般指数不等式，提高或推广了许多人的指数不等式 Bennett、Freedman、de la Peña、Pinelis 和 van de Geer；此外，我们的集中不等式也提高了一些已知的独立随机变量和的不等式。提供了与线性回归、自回归过程和分支过程有关的应用；特别是对自标准化偏差的 de la Peña 不等式的有趣应用也提供了。