Tempered-Stable Volatility Clock for Sequence Diffusion

1. Integration point: modify the forward-noise sampler of a 1D sequence diffusion model, trajectory diffusion model, or spectrogram diffusion model. For each training example, treat the sequence axis of length \(d\) as the volatility-clock axis. Do not change the U-Net or Transformer initially and do not provide \(A\) as a denoiser input; the model receives only \(x_t\), timestep \(t\), and the usual conditioning. 2. Pseudocode: estimate target excess kurtosis \(K\) and lag-one squared-return correlation \(r=\rho_{r^2}(1)\) from training sequences; set \(\varphi=3(K+2)r/K\), clip it to \([0,0.99]\), choose \(\alpha\in(0,1)\), set \(\theta=3(1-\alpha)/(K(1+\varphi))\), and set \(\delta=(1-\varphi)\theta^{1-\alpha}/\alpha\). For every minibatch item, burn in \(A_0\) and iterate A[i] = phi*A[i-1] + sample_TS(alpha,theta,delta) for \(i=1,...,d\); sample z ~ Normal(0,1); form eps=sqrt(A)*z; form x_t=sqrt(alpha_bar[t])*x0 + sqrt(1-alpha_bar[t])*eps; train the standard epsilon-prediction loss against eps. At generation time sample the same clock and use it in the reverse transition or score scaling. 3. Compute from the paper: the Laplace transform, cumulants, normalization of \(A\), and calibration identities. Estimate empirically: \(K\) and \(r\) from data, and validate that generated \(\operatorname{Var}(A)\), kurtosis, and squared-noise autocorrelation match. A production implementation needs a positive tempered-stable sampler; for the MVP use numerical inverse-CDF sampling from the specified Laplace transform or a reliable tempered-stable routine, and compare against a moment-matched Gamma-innovation ablation. 4. First experiment: train a small 1D temporal U-Net or Transformer diffusion model on synthetic stochastic-volatility sequences and on a financial-return benchmark, comparing ordinary iid Gaussian noise, iid per-coordinate variance mixtures, globally shared Student-t scale, and the proposed AR-tempered-stable clock at equal training steps and parameter count. Measure held-out likelihood or denoising loss, marginal excess kurtosis, lag-one through lag-20 squared-return autocorrelation, and calibration under a volatility-shock intervention. Success is lower denoising loss or better sample likelihood at equal FLOPs together with generated squared-noise autocorrelation that follows the predicted geometric decay \(\rho_{r^2}(h)=\varphi^hK/[3(K+2)]\), rather than the iid baseline's zero clustering.