cs.LGAug 24, 2026

Change Detection in Probability Flow ODE: Online Testing in Diffusion Latent Spaces

Authors: Artem Kraevskiy, Artem Prokhorov

Organizations: National Research University Higher School of Economics Moscow, Russia · The University of Sydney Business School & CIREQ & CEBDA Sydney, Australia

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Abstract

A rapidly growing range of sequential data tasks, such as identifying trend reversals in financial markets, auto-segmenting video and audio recordings, detecting changes in movement direction from motion sensors cannot be fully addressed without detection of distributional shifts in time-ordered data. We consider a sequential change-point detection problem where the conditional density switches at an unknown time, yet neither the pre- nor post-change distribution admits a closed-form. Classical likelihood-ratio statistics are inapplicable in this settings. A conditional diffusion model, trained on pre-change-point data with a frozen context encoder, defines a deterministic bijection via the probability flow ODE. Pre-change observations are mapped onto standard Gaussian latent variables. Post-change observations, processed through the same frozen map, deviate from this reference. We employ the Maximum Mean Discrepancy as the test statistic, derive closed-form expressions for its components under the Gaussian null, and establish its asymptotic distribution as a degenerate U-statistic. Afterwards we apply an online detection procedure of Shiryaev--Roberts to the resulting statistic with exact threshold calibration. The method detects arbitrary distributional shifts, including covariance rotations and higher-order structural breaks, without parametric assumptions on either regime.

Figures & tables

Figure 1: Top: example draws from p0p_{0} and p1p_{1} for each pair. Middle: the rolling MMD^t2−μ0\widehat{\mathrm{MMD}}^{2}_{t}-\mu_{0} statistic, with its δ^2\hat{\delta}^{2} under H1H_{1} marked. Bottom: the Shiryaev–Roberts statistic log⁡(1+Rt)\log(1+R_{t}) , its calibrated threshold, the true changepoint τ\tau , and the declared alarm τ^\hat{\tau} .
Figure 2: MNIST digit 0→10\to 1 , real data. Top: conditional generation pθ(x∣history:t)p_{\theta}(x\mid\text{history}_{:t}) just before τ\tau , one step after, and 2525 steps after (four samples each, 2×22\times 2 ). Middle/bottom: the same rolling MMD 2 and Shiryaev–Roberts panels as Figure 1 .
ExperimentMethodnull dist.P(false alarm)\mathbb{P}(\text{false alarm})P(miss)\mathbb{P}(\text{miss})avg. delay
Gaussian mixture rotationOursexact1.9%\mathbf{1.9\%}9.7%9.7\%16.9\mathbf{16.9}
Aapprox.32.0%32.0\%13.8%13.8\%30.930.9
Bexact24.1%24.1\%16.3%16.3\%36.236.2
Casympt.67.0%67.0\%2.8%\mathbf{2.8}\%38.538.5
Four clusters →\to oneOursexact2.9%\mathbf{2.9\%}15.3%15.3\%29.3\mathbf{29.3}
Aapprox.44.3%44.3\%14.5%14.5\%40.640.6
Table 1: Operating characteristics over 1,0001{,}000 Monte Carlo trials per experiment ( τ\tau uniform after burn-in, shared PFA≤5%P_{\rm FA}\leq 5\% design budget). †\dagger Same encoder-denoiser checkpoint as the row above, only the DDIM encoding step at detection time replaced by EDICT ( Wallace et al., 2023 ) – no retraining involved.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Kernelk(x,y)k(x,y)CCBBw⋅MMD^2→dw\cdot\widehat{\mathrm{MMD}}^{2}\xrightarrow{d}Detectable shifts
Linearx⊤yx^{\top}y0000χ2(d)\chi^{2}(d) (exact, any ww )Mean only: E[Z]≠0\mathbb{E}[Z]\neq 0
Deg.- pp poly.(1+x⊤y/c)p(1+x^{\top}y/c)^{p}∑m=0⌊p/2⌋κm\displaystyle\sum_{m=0}^{\lfloor p/2\rfloor}\kappa_{m}∑m=0⌊p/2⌋τm(z)\displaystyle\sum_{m=0}^{\lfloor p/2\rfloor}\tau_{m}(z)∑l=1Kλl(Zl2−1)\sum_{l=1}^{K}\lambda_{l}(Z_{l}^{2}-1) , K=(p+dd)−1K=\binom{p+d}{d}-1Moments 1,…,p1,\ldots,p : mean, cov., skewness (p ⁣≥ ⁣3)(p\!\geq\!3)
RBFe−∥x−y∥2/2σ2e^{-\|x-y\|^{2}/2\sigma^{2}}(σ2σ2+2)d/2\left(\dfrac{\sigma^{2}}{\sigma^{2}+2}\right)^{d/2}(σ2σ2+1)d/2\left(\dfrac{\sigma^{2}}{\sigma^{2}+1}\right)^{d/2} ⋅ e−∥z∥2/2(σ2+1)\cdot\,e^{-\|z\|^{2}/2(\sigma^{2}+1)}∑l=1∞λl(Zl2−1)\sum_{l=1}^{\infty}\lambda_{l}(Z_{l}^{2}-1) , λl∝rl\lambda_{l}\propto r^{l}Any P ⁣≠ ⁣QP\!\neq\!Q
Laplacee−∥x−y∥/σe^{-\|x-y\|/\sigma}No closed formNo closed form∑l=1∞λl(Zl2−1)\sum_{l=1}^{\infty}\lambda_{l}(Z_{l}^{2}-1) , λl∼l−(d+1)/2\lambda_{l}\sim l^{-(d+1)/2}Any P ⁣≠ ⁣QP\!\neq\!Q
IMQ(c2 ⁣+ ⁣∥x ⁣− ⁣y∥2)−β(c^{2}\!+\!\|x\!-\!y\|^{2})^{-\beta}No closed formNo closed form∑l=1∞λl(Zl2−1)\sum_{l=1}^{\infty}\lambda_{l}(Z_{l}^{2}-1) , exp. decayAny P ⁣≠ ⁣QP\!\neq\!Q
Appendix
Table 2: MMD2\mathrm{MMD}^{2} against Q=N(0,I)Q=\mathcal{N}(0,I) for common kernels.
Experimentδ^2\hat{\delta}^{2}log⁡(1+A∗)\log(1+A^{*})τ\tauτ^\hat{\tau}delay
Gaussian mixture rotation0.2070.20723.823.81001001151151515
Four clusters →\to one0.0700.07043.443.41001001201202020
Blob →\to Ring0.2670.26725.625.61001001181181818
Appendix
Table 3: Pilot signal strength, calibrated threshold, and detection delay (raw time steps) for the three experiments, all at the shared PFA≤5%P_{\rm FA}\leq 5\% / 175175 -window budget.
 2D pairs (§ 4 )MNIST (§ 5 )
observation dim dd22256256 ( 16×1616\times 16 )
history encoderGRU, hidden 6464MLP embed ( 6464 ) →\to GRU, hidden 512512 †\dagger
denoiser conditioningFiLMconcatenation, CFG dropout †\dagger
denoiser width / # blocks128128 / 44256256 / 88 †\dagger
diffusion steps TT50050010001000
β\beta schedulelinear, 10−4 ⁣→ ⁣0.0210^{-4}\!\to\!0.02linear, 10−4 ⁣→ ⁣0.0210^{-4}\!\to\!0.02
Appendix
Table 4: Architecture and training hyperparameters for the two model families used throughout the paper. †\dagger : the MNIST column shows the corrected recipe used to produce Table 1 and the convergence study of Section 6 – hidden width 512512 , 88 denoiser blocks, learning rate 10−410^{-4} with gradient-norm clipping at 11 , no CFG dropout, 120,000120{,}000 steps, full 999999 -step DDIM budget. Section 5 ’s single-trajectory illustration (Figure 2 ) instead uses the original, smaller recipe disclosed there for tractability: hidden width 256256 , 44 denoiser blocks, learning rate 3×10−43\times 10^{-4} (unclipped), no CFG dropout, 6,0006{,}000 steps, and only 100100 DDIM steps; see the caveat stated there and the discussion in Section 6 for why the two recipes diverged and what the correction changed.
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