尺度等变生成预测：权重绑定的扩张卷积、小波散射输入和频谱一致性训练用于自相似时间序列
Scale-Equivariant Generative Forecasting: Weight-Tied Dilated Convolutions, Wavelet Scattering Inputs, and Spectral-Consistency Training for Self-Similar Time Series
AI总结：
本文提出了一种尺度等变生成预测方法，通过权重绑定的扩张卷积、小波散射输入和频谱一致性训练，用于自相似时间序列的生成，展示了在S&P 500日收益率上的优越性能。
摘要：
许多自然和工程时间序列--股票回报、气候异常、湍流速度、神经记录、分组网络流量--近似自相似：其时间跨度为T的分布与时间跨度为1的分布通过一个缩放指数H关联。标准深度生成序列模型（Transformer、扩张TCN、WaveNet家族）忽略了这一点。它们的感受野很宽，但内核参数在每个扩张级别独立存在，导致多尺度架构，而非尺度等变架构。我们有三个贡献。首先，我们为一维因果网络给出了离散尺度等变的精确定义，并证明了二进制扩张在边界效应范围内与任何内核权重在不同级别共享的扩张卷积堆栈相容。绑定内核将卷积参数预算减少L倍（L为深度），并强制自相似性作为归纳偏置。其次，我们将这种尺度等变WaveNet（SE-WaveNet）主干包裹在三个具有相同先验的组件中：一级Daubechies-4小波输入、Hurst-FiLM块暴露局部缩放指数、以及针对|f|^{-(2H+1)}幂律频谱的频谱一致性训练项。头部是条件归一化流，选择以保持等变性。第三，在30年S&P 500每日对数收益率上，SE-WaveNet样本在Allan方差前25个宇宙上重现经验缩放崩溃诊断（中位数C* = 0.020），而普通WaveNet在匹配容量下不（≥0.06）。NLL、KS校准和尾部能量距离与基线持平或优于基线，参数数量更少L倍。
英文摘要：
Many natural and engineered time series -- equity returns, climate anomalies, turbulent velocities, neural recordings, packet-level network traffic -- are approximately self-similar: their horizon-$T$ distribution is tied to the horizon-$1$ distribution by one scaling exponent $H$. Standard deep generative sequence models (transformers, dilated TCNs, the WaveNet family) ignore this. Their receptive fields are wide, but kernel parameters live independently at every dilation level, yielding a multi-scale architecture, not a scale-equivariant one. We make three contributions. First, we give a precise definition of discrete scale equivariance for 1D causal networks and prove that dyadic dilation commutes (up to boundary effects) with any dilated-convolution stack whose kernel weights are shared across levels. Tying the kernel shrinks the convolutional parameter budget by an $L$-fold factor (where $L$ is depth) and hard-wires self-similarity in as an inductive bias. Second, we wrap this Scale-Equivariant WaveNet (SE-WaveNet) backbone in three components that carry the same prior: a one-level Daubechies-4 wavelet input, a Hurst-FiLM block exposing the local scaling exponent, and a spectral-consistency training term targeting the $|f|^{-(2H+1)}$ power-law spectrum. The head is a conditional normalising flow, chosen to preserve equivariance. Third, on 30 years of S&P 500 daily log-returns, SE-WaveNet samples reproduce the empirical scaling-collapse diagnostic on the Allan-Variance top-25 universe (median $\mathcal{C}^\star = 0.020$), while a vanilla WaveNet at matched capacity does not ($\geq 0.06$). NLL, KS-calibration, and tail energy distance tie or beat the baseline, with $L\times$ fewer convolutional parameters.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
