Denoising Subordinated Probabilistic Models: Diffusion with a Tempered-Stable Volatility Clock, and What the Noise Mechanism Actually Controls
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摘要
Heavy-tailed extensions of denoising diffusion models replace the Gaussian noise by a Gaussian variance mixture εi = Ai zi: denoising Lévy probabilistic models (DLPM) take the mixing variables Ai i.i.d. across coordinates, Student-t EDM takes a single A shared by all coordinates. [page::0] Neither extreme has dynamics, yet temporal dependence of the noise amplitude— volatility clustering—is the defining stylized fact of the financial data these models target. [page::0] We introduce the Denoising Subordinated Probabilistic Model (DSPM), whose mixing vector is a stationary AR(1) chain driven by tempered-stable subordinator increments—the discretetime Barndorff-Nielsen–Shephard stochastic volatility process—running along the data axis. [page::0] Conditionally on the chain, the entire DDPM machinery survives verbatim, and we prove closed-form identities linking the chain parameters to the excess kurtosis and the squarednoise autocorrelation, yielding an exactly identified, analytically invertible calibration with a falsifiable feasibility bound ρr2(1) < K/(3(K + 2)). [page::0] DDPM, DLPM-type stable noise, and shared-scale Student-t-type noise are boundary cases of one memory parameter. [page::0] We then prove a result that delimits the entire design space: if the denoiser is conditioned on A (as sufficiency demands), the mixing law is a nuisance—in the exact-denoiser limit the generated distribution is invariant to it, and interventions on A have no effect. [page::0] Controlled experiments (identical denoiser, schedule, budget; only the law of A or the conditioning bit varies) confirm both halves: conditioned models match the data’s clustering regardless of the mixing law, and a designed 8 volatility shock moves the generated amplitude envelope by less than 13% against a naive prediction of +183%; blind models transmit the noise structure exactly as calibrated—lag-one squared-return autocorrelation 0.151 against the calibrated 0.161 with the predicted geometric decay, while the i.i.d. variant’s clustering collapses to 0.008. [page::0] The mixing law controls what structure the noise offers; the conditioning bit controls whether it transmits. [page::0] Finally we close the loop (v2): coupling the chain to the data through a variational volatility encoder (with a log-normal volatility prior, for closed-form KL) restores control—provided the objective retains the stochastic-volatility likelihood whose log-determinant term the simplified denoising loss discards; naive β-weighted couplings collapse or leak, and both failure modes are documented. [page::0] With it, the same 8 intervention scales the envelope by 3.07 (naive scaling: 2.83), the encoder tracks the true latent volatility of held-out paths (per-path correlation 0.57; 0.76 pooled), and the learned prior memory moves from its moment-calibrated initialization toward the true persistence. [page::0] The cost is substantial and stated plainly: the coupled model’s unconditional kurtosis falls below even the Gaussian baseline—fidelity and control are demonstrated separately, not yet jointly. [page::0]
