# Tempered-Stable Volatility Clock for Sequence Diffusion

- ID: 1142
- Canonical URL: https://synthcore.org/idea/1142/tempered-stable-volatility-clock-for-sequence-diffusion
- API JSON: https://synthcore.org/api/idea/1142.json
- API Markdown: https://synthcore.org/api/idea/1142.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.19218](https://arxiv.org/abs/2607.19218)
- Category: sampling
- Solves: accuracy, sample-efficiency, generalization
- ML areas: diffusion, diffusion-sampling, training, world-model
- Math tags: stochastic-processes, probability, statistics, dynamical-systems
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Replace independent Gaussian diffusion noise across sequence positions with a positive, persistent variance chain and conditionally Gaussian perturbations. This gives the denoiser exposure to heavy tails and volatility clustering without requiring a more expressive neural architecture; keep the denoiser blind to the realized variance when the goal is for generated samples to retain this structure.

## Mathematical statement

The paper defines a tempered-stable innovation \(\eta\sim\mathrm{TS}(\alpha,\theta,\delta)\), with \(0<\alpha<1\), \(\theta>0\), and Laplace transform \(\mathbb{E}[e^{-u\eta}]=\exp(-\delta[(\theta+u)^\alpha-\theta^\alpha])\). Its cumulants are \(\kappa_n(\eta)=\delta\alpha\,\Gamma(n-\alpha)\theta^{\alpha-n}/\Gamma(1-\alpha)\). The variance clock is the stationary positive AR(1) process \(A_i=\varphi A_{i-1}+\eta_i\), where \(0\leq\varphi<1\), the \(\eta_i\) are iid, and \(\delta=(1-\varphi)\theta^{1-\alpha}/\alpha\), which ensures \(\mathbb{E}[A_i]=1\). For diffusion timestep \(t\), use \(x_t=\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\,\mathrm{diag}(A)^{1/2}z\), with \(z\sim\mathcal N(0,I)\). From the AR(1) recursion, \(\mathrm{Var}(A_i)=v_A=\kappa_2(\eta)/(1-\varphi^2)=(1-\alpha)/(\theta(1+\varphi))\). If \(\varepsilon_i=\sqrt{A_i}z_i\), its excess kurtosis is \(K=3v_A\), and its lag-one squared-noise autocorrelation is \(\rho_{r^2}(1)=\varphi v_A/(2+3v_A)=\varphi K/[3(K+2)]\), implying the feasibility bound \(\rho_{r^2}(1)<K/[3(K+2)]\). Thus one can estimate \(\varphi=3(K+2)\rho_{r^2}(1)/K\), then choose \(\alpha,\theta\) so that \(3(1-\alpha)/[\theta(1+\varphi)]=K\).

## Key formulas

- $$\mathbb{E}\,e^{-u\eta}=\exp\!\bigl(-\delta[(\theta+u)^{\alpha}-\theta^{\alpha}]\bigr),\qquad\kappa_{n}(\eta)=\delta\,\alpha\,\frac{\Gamma(n-\alpha)}{\Gamma(1-\alpha)}\,\theta^{\alpha-n}.$$
- $$A_{i}=\varphi A_{i-1}+\eta_{i},\qquad\eta_{i}\overset{\mathrm{iid}}{\sim}\mathrm{TS}(\alpha,\theta,\delta),\qquad\delta=\frac{(1-\varphi)\theta^{1-\alpha}}{\alpha}\quad\Rightarrow\quad\mathbb{E}A_{i}=1.$$
- $$x_t=\sqrt{\bar\alpha_t}\,x_0+\sqrt{1-\bar\alpha_t}\,\operatorname{diag}(A)^{1/2}z,\qquad z\sim\mathcal N(0,I).$$
- $$K=3\operatorname{Var}(A_i),\qquad \rho_{r^2}(1)=\frac{\varphi\operatorname{Var}(A_i)}{2+3\operatorname{Var}(A_i)}=\frac{\varphi K}{3(K+2)}<\frac{K}{3(K+2)}.$$

## Implementation notes

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.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a tempered-stable AR(1) volatility-clock sampler, forward-noise constructor, analytic verification harness, and blind toy denoiser comparison. The Laplace transform, stationary mean and variance, excess kurtosis, and geometric squared-noise autocorrelation matched theory across parameter sweeps. The denoiser showed a small favorable signal, but this is not evidence of a decisive modeling win.

### Mechanism check

- Verdict: Built a tempered-stable AR(1) volatility-clock sampler, forward-noise constructor, analytic verification harness, and blind toy denoiser comparison. The Laplace transform, stationary mean and variance, excess kurtosis, and geometric squared-noise autocorrelation matched theory across parameter sweeps. The denoiser showed a small favorable signal, but this is not evidence of a decisive modeling win.
- Confidence: 9/10
- Limitations: The denoiser test is a small synthetic blind noise-prediction task, not a full diffusion reverse-sampling, likelihood, financial-data, or multi-seed evaluation. Efficiency under difficult tempered-stable parameters and GPU acceleration were not tested.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.1657
- Paired wins: 2/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/453/bench_report.json)
- [bench_stage2.py](https://synthcore.org/code/453/bench_stage2.py)
- [denoiser_compare.py](https://synthcore.org/code/453/denoiser_compare.py)
- [denoiser_results.json](https://synthcore.org/code/453/denoiser_results.json)
- [report.md](https://synthcore.org/code/453/report.md)
- [report_bench_2026-08-31T123515.md](https://synthcore.org/code/453/report_bench_2026-08-31T123515.md)
- [results.json](https://synthcore.org/code/453/results.json)
- [volatility_clock.py](https://synthcore.org/code/453/volatility_clock.py)
- [Download all files as ZIP](https://synthcore.org/download/453)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.