PDMD: Projected Distribution Matching Distillation for Video Diffusion Models

• Modern video diffusion transformers often require dozens of function evaluations, while step distillation compresses a many-step teacher into a few-step student.Source: §1 · p. 2
• At a fixed noisy query, the residual is an unbiased estimate of critic endpoint error; under high-dimensional assumptions, projection removes a constant error fraction while discarding only a vanishing fraction of useful signal.PDMD uses existing computations without extra data or a model pass.Source: §1 · pp. 2–3
• 83.73 VBench total on Wan2.1-T2V-1.3B is 1.03 points above matched DMD; MiniMax-H3 scores 83.17 VideoGen-Eval total, 0.41 above the strongest distilled baseline.Source: §1 · p. 3
• Experiments report more stable training, less oversaturation, more motion, and better audio; 2D experiments also improve energy distance and on-mode fraction.Source: §1 · p. 3
• On MiniMax-H3, PDMD leads all six audio metrics among 4-NFE models, and user studies prefer it to every 4-NFE baseline in visual, motion, and audio quality.Source: §1 · p. 3
• The paper connects DMD’s training instability and visual degradation to critic error in its score-difference update.Source: §1 · p. 3
• PDMD filters that error by projecting out the DMD update component parallel to the student–critic endpoint residual, an estimate of critic error.Source: §1 · p. 3

• DMD2 adds critic updates and discriminator supervision, while SiD, f-distill, and SGMD replace reverse-KL with alternative objectives requiring additional network evaluations (Xu et al., 2025).Source: §2 · p. 3
• Unlike these additions, PDMD keeps DMD’s score networks and critic training unchanged and alters only the student update computed from existing predictions.Source: §2 · p. 3
• Trajectory and hybrid approaches add consistency, flow-map, or other training components, often requiring another objective, rollout, or training stage (Song et al., 2023; Gu et al., 2026; Ge et al., 2026b; Nie et al., 2026; Kim et al., 2026).Source: §2 · pp. 3–4
• Rather than using paired prediction differences for image editing or projecting guidance components, PDMD projects the DMD update using the student–critic endpoint residual to suppress critic estimation error (Armandpour et al., 2023).Source: §2 · p. 4

3.1 PRELIMINARIES: DISTRIBUTION MATCHING DISTILLATION

• DMD trains a few-step student to match the teacher distribution using the difference between an online critic score and a frozen teacher score.Source: §3.1 · p. 4
• The student maps Gaussian input and condition to a clean endpoint, which is renoised before the online critic and frozen teacher are evaluated at the same query.Source: §3.1 · p. 4
• The score difference d is backpropagated through the student endpoint, and alternating student and critic optimization trains the student toward the teacher.Source: §3.1 · p. 4

3.2 PROJECTED DISTRIBUTION MATCHING DISTILLATION

• PDMD treats the student–critic endpoint residual as an estimate of critic error and projects the component of the DMD update parallel to that residual out of the update.Source: §3.2 · p. 4
• The algorithm replaces d with its residual-orthogonal component; when the residual is zero, the update remains unchanged.Source: §3.2 · p. 4
• The projection is the only change from DMD and requires no additional loss, network, model pass, data, or training stage.Source: §3.2 · p. 4

4.1 THE PROJECTION FILTERS THE CRITIC ERROR

• At a fixed noisy query, the student–critic endpoint residual is an unbiased estimate of critic endpoint error.The estimate follows because the student endpoint varies around the conditional-mean endpoint used by the population-optimal critic.Source: §4.1 · pp. 4–5
• Under high-dimensional assumptions, the projection removes a constant fraction of critic error while discarding only a vanishing fraction of ideal DMD signal.When conditional noise energy equals squared critic error, the conditional guarantee removes at least 50% of critic-error energy on average.Source: §4.1 · p. 5
• For error caused by critic lag, the residual remains the conditional mean of the error, and projection attenuates lag error in conditional mean square.In the two-mode model, strong lag can turn the critic’s restoring feedback into instability.Source: §4.1 · p. 5

4.2 2D VERIFICATION

• 28%: PDMD reduces mean squared error to the ideal update relative to DMD for σ ∈[0.15, 1.1], and only its projection improves both 2D distribution metrics over DMD.The comparison uses a ring of eight Gaussians with an exact teacher score; other tested projections remove similar update-norm shares without targeting critic error.Source: §4.2 · p. 6, Figure 4 · p. 6
• 3/20: PDMD collapses in three two-mode runs at one critic-to-student learning-rate setting, versus 14/20 for DMD; it typically recovers from early single-mode states.DMD collapse counts vary across two learning rates; its collapsed runs remain locked.Source: §4.2 · p. 6

5.1 EXPERIMENTAL SETUP

• PDMD is compared with matched four-NFE DMD† on Wan2.1, while the MiniMax-H3 study distills a 33B joint video–audio model to four NFE.For Wan2.1, PDMD differs from DMD† only by the projection, with training hyperparameters unchanged.Source: §5.1 · pp. 6–7
• The evaluation uses 944 AnyFlow-augmented VBench prompts for Wan2.1 and 387 VideoGen-Eval prompts for MiniMax-H3, with randomized side-by-side judgments from 20 annotators.Wan2.1 is evaluated at 480p with five seeds per prompt; MiniMax-H3 uses 544p, mixed aspect ratios, and a fixed seed.Source: §5.1 · p. 7

5.2 FOUR-STEP TEXT-TO-VIDEO GENERATION ON WAN2.1-T2V-1.3B

• 83.73: PDMD’s VBench total score on four-step Wan2.1 exceeds matched DMD† at 82.70, and annotators prefer it to every compared method on visual and motion quality.DMD† degrades after 1,500 iterations, while PDMD remains above 83.5 through 10,000; Figure 6 shows natural textures and less oversaturation.Source: §5.2 · p. 7, Figure 6 · p. 7

5.3 FOUR-STEP JOINT VIDEO–AUDIO GENERATION ON MINIMAX-H3-33B

• 83.17: PDMD’s VideoGen-Eval total on MiniMax-H3 is above every four-NFE baseline, and it leads the compared four-NFE models on all six audio metrics.PDMD comes within 0.04 of the 50-step teacher on production quality (PQ).Source: §5.3 · p. 7, Table 2 · p. 8

• PDMD shows fewer artifacts, more realistic textures, and better motion than four-NFE MiniMax-H3 baselines in the VideoGen-Eval examples.Each prompt compares an earlier and later frame from the same clip.Source: Figure 7 · p. 8
• The compared baselines show translucent ghosting and severely blurred animals, while PDMD keeps the animals sharp in the illustrated motion example.Source: Figure 8 · p. 8
• At 1,500 iterations, projecting out the update component parallel to the student–critic endpoint residual yields less saturation than the other tested directions in the shown example.Source: Figure 9 · p. 9
• From 500 to 3,000 iterations, PDMD holds its VideoGen-Eval total score while the other compared variants decline.Source: Figure 10 · p. 9
• 37.0 points: the 50-step H3 teacher remains preferred over PDMD overall, and by 35.4 points in visual quality, despite PDMD’s preference over every compared four-NFE model.Text-alignment judgments are mostly ties, at 64–80%.Source: §4 · p. 9

• From 500 to 3,000 iterations, PDMD maintains its total score while the other projection variants decline.At 1,500 iterations, only PDMD avoids saturation or high-frequency artifacts in the illustrated comparison.Source: §6 · p. 9
• Random projection removes almost no update norm and behaves like DMD, while the residual-parallel variant keeps only 11% and degrades fastest.PDMD removes a median 12% and remains stable; restoring half of the removed component also leads to degradation, but later than DMD.Source: §6 · p. 9
• Update shrinkage alone does not explain the stability gain: PDMD remains stable through 5,000 iterations at a 1.25× student learning rate, while low-rate DMD still oversaturates.At one critic update per student update, PDMD scores 82.90 total versus 82.37 for DMD.Source: §6 · p. 9

7 LIMITATIONS

• Single-step quality remains limited and may require an additional objective, such as a discriminator loss.Figure 11 is identified as illustrating the single-step limitation.Source: §7 · p. 9, Figure 11 · p. 9

8 CONCLUSION

• PDMD remains stable where DMD degrades, improves video scores, is preferred in user studies, and performs best on all audio metrics among compared 4-NFE models.The conclusion presents PDMD as a promising approach to stable few-step video distillation.Source: §8 · p. 10
• The study distills two released text-to-video models using existing prompts and public benchmarks, without real-video data, and does not evaluate preservation of base-model safety behavior.The distilled student inherits its teacher's capabilities and biases.Source: ETHICS STATEMENT · p. 10
• The authors will release code for video and 2D experiments, Wan2.1 student weights, and the MiniMax-H3 distillation LoRA.The supplement describes training settings, evaluation protocols, and proofs.Source: REPRODUCIBILITY STATEMENT · p. 10

A PROOFS

• For a fixed query, the ℓ2-optimal critic endpoint is the conditional mean, so its residual has zero conditional mean and covariance Σ.The conditional squared-error minimizer is unique.Source: A PROOFS · pp. 14–15

A.2 HIGH-DIMENSIONAL ERROR REMOVAL AND SIGNAL RETENTION

• Under the stated high-dimensional conditions and a/τ = Ω(1), PDMD removes a constant fraction of critic-error energy while removing a vanishing fraction of ideal-signal energy as effective dimension grows.The error alignment is structural because R = e − ξ contains the critic error explicitly; retaining the signal instead requires an incoherence assumption on the fixed ideal-signal direction.Source: §A.2 · pp. 16–18
• Unlike subtracting an unbiased error estimate, which injects its full conditional noise, projection needs no coefficient and asymptotically estimates the optimal fixed subtraction coefficient from the available vectors.The optimal fixed coefficient depends on unobservable a and τ; projection also never reverses or lengthens the DMD update and is invariant to nonzero rescaling of R.Source: §A.2 · p. 18
• A biased error estimate can remove more critic error when its mean-square deviation is smaller, while its bias adds at most a term proportional to ∥m∥²/(a + τ′) to signal removal.Under the stated concentration conditions, the error-removal ratio is a/(a + τ′) + O_P(d_eff^-1/2), so bias costs like an equal amount of variance.Source: §A.2 · p. 18

A.3 CRITIC LAG AND ALLOCATION STABILITY

• The scalar delayed-allocation model is asymptotically stable exactly when λτ < π/2, marginal at λτ = π/2, and unstable when λτ > π/2.The model assumes current-critic feedback restores matched allocation at rate λ, while the online critic responds to the student allocation τ time units earlier.Source: §A.3 · p. 19
• For a stale critic, the residual’s conditional mean is the lag-induced score error up to a scalar, and projection attenuates that error in conditional mean square.In the projected-feedback heuristic, the gain ρ yields the stability condition ρλτ < π/2; this is a heuristic because ρ varies with query and critic, and projection also affects the ideal update.Source: §A.3 · p. 20
• The projection’s effective gain falls toward zero when critic lag error dominates endpoint spread, then rises when the critic catches up and the residual becomes conditional noise.The passage reports that Table B3 is consistent with this interpretation and contrasts the state-dependent projection with a fixed reduction such as a random direction.Source: §A.3 · p. 21

B 2D EXPERIMENTS

• The 2D experiments use a variance-exploding diffusion toy in R² with an analytic teacher score and a learned online critic to test error removal, signal retention, and attenuation under critic lag.The first experiment estimates critic deviation from the population-optimal student score; the second measures mode allocation under critic lag.Source: B 2D EXPERIMENTS · p. 21
• The 2D experiments compare DMD with per-sample-projection PDMD under shared seeds, data, and schedules, measuring energy distance, modes covered, on-mode fraction, and removed fraction.Both networks are 3-layer MLPs of width 128; training uses Adam, learning rate 2 × 10−3, batch size 1,024, and 4,000 or 6,000 iterations across two experiments. Energy distance is zero iff the distributions match; modes covered counts components above 25% of their target on-mode share, on-mode fraction counts samples within 3σmode of a component center, and removed fraction is 1 −∥dkept∥/∥d∥.Source: B 2D EXPERIMENTS · p. 21

B.1 EXPERIMENT 1: ERROR FILTERING ON A MULTI-MODE TARGET

• The ring-of-eight experiment compares four rank-one deletions and unprojected DMD, with shared seeds, data, and schedule; similar step shrinkage tests whether direction matters beyond magnitude.The student–critic endpoint residual structurally contains critic error, whereas the random and teacher-residual directions do not.Source: §B.1 · pp. 21–22
• PDMD reaches energy distance 0.0080 and on-mode fraction 0.903, while DMD reaches 0.0273 and 0.838; only PDMD improves both metrics over DMD among tested deletions.The four deletions shrink the step by similar amounts, but random and teacher-residual deletions do not reproduce PDMD’s gain.Source: §B.1 · pp. 21–22, Figure B2 · p. 25
• In the error-carrying noise band, the student–critic endpoint residual removes more critic-error energy than ideal-signal energy, unlike the other tested directions.Its error-removal ratio exceeds the signal-removal ratio on average; the measured lower bound on error removal holds across snapshots and noise levels.Source: §B.1 · p. 22, Table B1 · p. 23, Figure B1 · p. 24

B.2 EXPERIMENT 2: TWO-MODE ALLOCATION UNDER CRITIC LAG

• With a critic learning rate 20× below the student’s, PDMD collapses in 3/20 seeds versus DMD’s 14/20 and has mean energy distance 0.386 versus 1.274.At the smaller step size, collapse occurs in 12/20 PDMD seeds versus 19/20 for DMD; PDMD trajectories recover toward balance after an early imbalanced transient.Source: §B.2 · p. 23, Table B3 · p. 24, Figure B3 · p. 26

C.1 WAN2.1-T2V-1.3B

• The Wan2.1 experiments train from Wan2.1-T2V-1.3B weights on 42K captions without real-video latents, and evaluate distilled models at 4 NFEs using 944 augmented VBench prompts.The user study compares PDMD against six baselines on a 95-prompt subset with 20 annotators.Source: §C.1 · pp. 24, 26, C EXPERIMENTAL DETAILS · p. 24

C.2 MINIMAX-H3-33B

• MiniMax-H3 training uses video and audio LoRA adapters, five critic updates per student update, and evaluation on 387 VideoGen-Eval prompts with a shared scoring protocol.The controlled experiments retain the 4-NFE recipe and change settings only where a subsection specifies; the evaluation renders one clip per prompt.Source: §C.2 · p. 27, WHERE THE PORTS DEPART FROM THE… · p. 28, D ADDITIONAL CONTROLLED EXPERIMENTS ON… · p. 29

D.1 PROJECTION DIRECTIONS OVER TRAINING

• At 1,500 iterations, all five control projections degrade in the illustrated prompts, while PDMD continues to follow its prompt through 3,000 iterations.The six variants differ only in the direction removed from the DMD update.Source: §D.1 · p. 29, Figure D1 · p. 30

D.2 RETAINED UPDATE NORM

• PDMD retains 88% of the update norm and remains stable, while the residual-parallel variant retains a median 11% and degrades fastest; random projection behaves like DMD.The results indicate that stability does not follow simply from the amount of update removed.Source: §D.2 · pp. 29, 31, Figure D2 · p. 31

D.3 RESTORING HALF OF THE REMOVED COMPONENT

• Restoring half of the residual-parallel component reproduces degradation: the run falls below the 4-step base-model score at 4,000 iterations and reaches 74.90 at 5,000.The fully projected PDMD run remains stable over the evaluated interval.Source: §D.3 · p. 31

D.4 A LARGER STUDENT STEP

• With a 1.25× larger student learning rate, PDMD shows no sustained decline over 5,000 iterations and ends at 83.18 total, while base-rate DMD falls to 70.07 by 2,500 iterations.The larger-step run’s ten checkpoints range from 82.82 to 83.48.Source: §D.4 · pp. 31–32, Figure D3 · p. 32

D.5 ONE CRITIC UPDATE PER STUDENT UPDATE ON MINIMAX-H3

• With one critic update per student update, MiniMax-H3 PDMD scores 82.90 total, above DMD with five critic updates and every 4-NFE baseline.This tests the projection without the five-update TTUR schedule used in the main MiniMax-H3 runs.Source: §D.5 · p. 34, Table D1 · p. 34

D.6 TWO-STEP GENERATION

• At 2 NFE, every PDMD checkpoint scores at least 80 total, while the unprojected run falls below 60; PDMD also leads on all six audio metrics at the reported checkpoints.The comparison changes only the projection, with shared setup and prompts.Source: §D.6 · p. 34, Table D2 · p. 34, Figure D5 · p. 35

E.1 THE REMOVED COMPONENT IN VIDEO SPACE

• In the decoded comparison, the PDMD target keeps the student endpoint’s subject, colors, and layout while showing less high-frequency speckle than the DMD target.The projection removes 5%–30% of video update norm across the 16 samples, with a median of 16%.Source: §E.1 · p. 40

E.2 SATURATION METRICS √

• PDMD has mean saturation 0.476 versus DMD’s 0.709 on Wan2.1, and 0.357 versus DMD’s 0.460 on MiniMax-H3.The paper cautions that lower saturation alone does not establish better perceptual quality.Source: §E.2 · p. 40, Table E1 · p. 40

E.3 MOTION COMPARISONS

• Across three motion examples, PDMD avoids baseline failures including near-static clips, misread rider motion, and bouquet ghosting.In prompt 895, PDMD shows the bat entering the frame, similar to the teacher; the figures use matched prompts and seeds across methods.Source: §E.3 · p. 42, Figure E2 · p. 43, Figure E3 · p. 44

E.4 ADDITIONAL SAMPLES

• The additional samples compare early and later Wan2.1 frames and two-frame MiniMax-H3 clips across prompts and models.The MiniMax-H3 figures cover eight prompts across two pages.Source: §E.4 · p. 46, Figure E5 · p. 46, Figure E6 · p. 48

F LIMITATIONS AND FUTURE WORK

• PDMD does not yet yield satisfactory single-step video generation; samples remain smeared and below the quality of four-step results.The authors leave possible use of an additional objective and the residual direction’s suitability for single-evaluation video generation to future work.Source: F LIMITATIONS AND FUTURE WORK · p. 50
• The theory guarantees conditional error removal, not convergence of coupled student–critic optimization, and depends on appreciable critic error and weak signal alignment with the residual.If useful signal aligns with the residual, or the critic is exact, the projection can still discard useful signal.Source: F LIMITATIONS AND FUTURE WORK · p. 50
• Video experiments cover two backbones, primarily at 4 NFE, and do not establish preservation of diversity across prompts and seeds.On MiniMax-H3, the 50-step teacher remains preferred in the user study despite PDMD’s higher aggregate video score, indicating a perceptual-quality gap and a limitation of automatic metrics.Source: F LIMITATIONS AND FUTURE WORK · p. 50