Discriminative Flow Matching: Beyond Time-Conditioning in Generative Restoration via Flow-State Representations

1 Introduction

• In restoration, samples at the same interpolation time may differ in degradation and proximity to the target manifold, so time may not unambiguously represent transport state.Unlike classically independent source and target distributions, restoration inputs contain the target signal, and that statistical dependence varies across samples and degradation conditions.Source: §1 · p. 1
• Time-unconditional CFM avoids conditioning on prescribed time but removes an explicit mechanism for adapting velocity to transport progress.Prior work explored this formulation partly to address trajectory-specific noise-scale learning and possible overfitting to the prescribed path (Zhang et al. 2026a,b).Source: §1 · p. 2
• The Discriminative Flow-State Hypothesis proposes that discriminative representations encode generative restoration state; DFM conditions the velocity field on those representations instead of explicit time.The representations are extracted from the iteratively enhanced sample to enable sample-adaptive restoration.Source: §1 · p. 2
• DFM improves restoration over conventional CFM and other baselines on speech enhancement and image denoising, while supporting adaptive inference with sample-dependent computational budgets.The introduction also states that discriminative representations capture restoration progress and can estimate the computational effort required for enhancement.Source: §1 · p. 2

2 Flow-State Theory

• CFM models transport from source p_0 to target p_1 along a probability path, commonly using linear interpolation x_t = (1 − t)x_0 + tx_1 to train a velocity network (Lee et al. 2025).The passage defines x_0 as the source observation at t = 0, x_1 as the target at t = 1, and x_t as an intermediate state.Source: §2 · p. 2
• As a probability path evolves toward the target distribution, the Wasserstein-2 distance provides a principled measure of generative progress by quantifying the remaining distance.Source: §2 · p. 2

2.1 Flow-State Properties

• Under OT-CFM with independent initialization such as a standard Gaussian, the trajectory interpolates x_0 and x_1 using the coupling that minimizes expected transport cost; Proposition 1 introduces properties of S(p_t) under this setup.The stated coupling attains the squared Wasserstein-2 distance; the supplied passage does not specify the proposition’s properties.Source: §2.1 · p. 2

2. Monotonicity: d

• In standard OT-CFM, the Flow-State simplifies to a linear scaling of the initial Wasserstein-2 distance over time, justifying t as its surrogate.The proof sketch derives an upper bound from the expected squared displacement and establishes equality by contradiction with optimal transport cost.Source: §2 · p. 3

2.2 Ambiguity of Time t in Restoration Tasks

• Restoration mixtures make t ambiguous: distinct mixture weights and time coordinates can yield identical states and marginal distributions, so t cannot unambiguously describe progress.The setup represents the initial state as x0 = αx1 + βν, with positive scaling parameters and additive noise.Source: §2.2 · p. 3
• The upper bound S(pt) ≤ |1 − t|(|α − 1| + |β|) decreases toward t = 1, while larger α reduces the bound by placing xt closer to x1.For the convex-combination case β = 1 − α, the stated bound is W2(pt, p1) ≤ 2|1 − t||α − 1|.Source: §2.2 · p. 3

2.3 Discriminative Flow-State Hypothesis

• Discriminative Flow-State Hypothesis: latents learned by discriminative restoration models encode the Flow-State governing generative transport and describe sample-specific progress, target proximity, and restoration complexity.Because the clean-data distribution is inaccessible at inference, the paper uses W2(pt, pd) as an approximate Flow-State; its deviation from W2(pt, p1) is bounded by the model’s mean squared reconstruction loss.Source: §2.3 · p. 3
• Under a Markov-chain assumption, the latent z retains at least as much target information as the discriminative estimate x̂1, jointly supporting its use to characterize transport progress.The paper notes that the assumption is strong but holds, for example, for an autoencoder without skip connections, and reports empirical validation of latent representations as predictors of observable Flow-State proxies.Source: §2.3 · pp. 3–4

3 Discriminative Flow Matching

• DFM conditions its velocity field on the pretrained encoder’s DFSR z = Eϕ(xζ), rather than explicit time t.The representation guides transport as it evolves during restoration.Source: §3 · p. 4
• The conditional path interpolates from x0 + σϵ at ζ = 0 to clean target x1 at ζ = 1, with linear mean and decreasing noise scale.Its target velocity is (x1 − x0) − σϵ.Source: §3 · p. 4
• Inference initializes at x0 + σϵ, uses Euler integration with uniform steps, and returns x(N) as the clean-data estimate.Because the velocity network does not depend on ζ, the evolving DFSR can guide adaptation of the evaluation budget or step sizes.Source: §3 · p. 4

4.1 Flow-State Preservation

• DFSR preserve Flow-State information: the latent predictor achieves the lowest MAE and RMSE across three targets, while both input types achieve iSNR Pearson correlations above 0.96.Latent representations also organize by degradation severity, with cleaner samples nearer the clean latent manifold; estimating ζ and gSNR remains challenging for both predictor types.Source: §4.1 · pp. 4–5, Table 1 · p. 5

4.2 Flow-State Monotonicity

• During DFM inference, signal- and latent-space Flow-States decrease monotonically (Spearman’s ρ = −1.000 for both), and their trajectories are rank-equivalent (ρ = 1.000).The latent trajectory has Pearson’s r = −0.973 and zero monotonicity violations, while the signal-space trajectory has r = −0.888 and a convergence floor of 0.112 ± 0.005.Source: §4.2 · p. 5, Figure 4 · p. 5

Dataset

• Evaluation uses the DNS Challenge synthetic non-reverberant test set of 150 utterances and the Real Recordings test set of 300 recordings.The training data comprise approximately 1,000 hours of noisy-clean pairs mixed at −10 to 30 dB SNR, with half additionally reverberated.Source: Dataset · p. 5

Implementation Details

• DFM conditions the NCSN++ backbone on frozen DCCRN-derived 256-dimensional DFSR embeddings, replacing the standard time embedding.Models use Adam with a 10−4 learning rate, batch size 8, and EMA decay 0.999; training runs for 800,000 steps and results average five runs.Source: Implementation Details · pp. 5–6

Experimental Results

• DFM achieves 19.63 dB SI-SDR and 2.96 PESQ on the synthetic test set, outperforming all CFM variants and improving 0.64 dB SI-SDR and 0.10 PESQ over baseline CFM.On real recordings, it obtains a P.808 MOS of 3.75 and best SIG (3.43), BAK (4.03), and OVRL (3.14) scores; it also consistently outperforms CFM, CFM+DL, and ARF under the data-prediction objective.Source: Table 2 · p. 6, Experimental Results · p. 6, Table 3 · p. 6

Complexity–Performance Trade-off

• DFM delivers superior speech-enhancement performance at competitive complexity, while adaptive inference reduces average cost to 6.27 GMACs per frame.The DCCRN encoder adds approximately 2.6M parameters, while reducing the conditioning dimension yields a 65.8M-parameter total model, comparable to FlowSE’s 65.6M.Source: Figure 6 · p. 7, Complexity–Performance Trade-off · p. 7

Adaptive Inference

• Adaptive ζ-based inference achieves 20.12 dB SI-SDR and 2.92 PESQ at 2.98 NFE, improving SI-SDR by 0.49 dB while reducing computation 40.4% versus Nmax = 5.Adaptive variants average 4.12, 3.32, and 2.98 NFE for gSNR, iSNR, and ζ predictors, respectively, while outperforming CFM baselines and surpassing fixed-budget DFM in SI-SDR at N = 5 with comparable PESQ.Source: Figure 5 · p. 6, Adaptive Inference · p. 7

4.4 Image Denoising

• On MNIST and Fashion-MNIST, DFM consistently outperforms CFM, class-conditioned CFM, and latent-conditioned CFM across degradation levels.DFM also supports adaptive inference for image denoising, allocating computation according to degradation while preserving restoration quality.Source: §4.4 · p. 7, Figure 7 · p. 7

• Across speech enhancement and image denoising, DFM improves restoration quality while enabling adaptive inference at reduced computational cost.Source: §5 · p. 7
• DFM replaces explicit time conditioning in CFM with discriminative Flow-State representations, grounded in the Discriminative Flow-State Hypothesis.Source: §5 · p. 7
• The conclusion presents discriminative conditioning as a powerful alternative to explicit time conditioning for CFM.Source: §5 · p. 8

A1.1 Proof of Proposition 1 (Properties of Flow-State for OT-CFM)

• For every t in [0, 1], Flow-State is bounded by 0 <= S(p_t) <= W_2(p_0,p_1).This is the boundedness property stated for the OT-CFM trajectory.Source: A1.1 Proof of Proposition 1… · p. 10
• Under the OT-CFM transport trajectory, S(p_t)=(1-t)W_2(p_0,p_1).The expression implies S(p_0)=W_2(p_0,p_1) and S(p_1)=0.Source: A1.1 Proof of Proposition 1… · p. 10

2. Monotonicity: d

• The proof establishes W2^2(p_t,p_1) = (1−t)^2 W2^2(p_0,p_1), since x_t−x_1=(1−t)(x_0−x_1) and it preserves the optimal coupling for t=0.Source: §2 · p. 10
• Differentiating the established equality yields Property (2).Source: §2 · p. 10
• Property (1) follows from the non-negativity of the Wasserstein metric and (1-t) ≤ 1 for t ∈ [0, 1].Source: §2 · p. 10

A1.2 Proof of Proposition 2: Ambiguity of Time Coordinate

• For the stated initial-state construction, a point on the interpolation admits distinct parameter sets and times t ≠ t′ in [0, 1).This is the ambiguity asserted by Proposition 2.Source: A1.2 Proof of Proposition 2: Ambiguity… · p. 10
• The proof constructs the alternative parameters by matching the coefficients of the clean signal and noise, with β′ > 0 and β′ ≠ β ensuring t′ ≠ t.The proof also gives a coefficient-matching relation for α′.Source: A1.2 Proof of Proposition 2: Ambiguity… · pp. 10–11
• A suitable α′ exists because the required bound is achievable for some t′ ∈ [0, 1), making α′ valid.The proof rules out the contrary assumption by noting that (1 − t)α + t is positive.Source: A1.2 Proof of Proposition 2: Ambiguity… · p. 11

A1.3 Proof of Proposition 3: Upper Bound of Flow-State

• Under E∥x_1∥² = E∥ν∥² = 1, the general bound’s coefficient is |α − 1| + |β| and its scale is |1 − t|.The proof obtains the bound by evaluating an arbitrary coupling and applying the triangle inequality and Cauchy–Schwarz.Source: A1.3 Proof of Proposition 3: Upper… · p. 11
• W2(p_t, p_1) ≤ 2|1 − t||α − 1| for the convex-combination case β = 1 − α, with α ∈ [0, 1].This follows by substituting β = 1 − α into the general upper bound.Source: A1.3 Proof of Proposition 3: Upper… · p. 11

A1.4 Proof of Proposition 4: Approximation of Flow-State

• Under the Markov-chain assumption xt → z → ˆx1, the latent z retains at least as much mutual information about clean x1 as the estimate ˆx1, while Flow-State approximation is tied to reconstruction error.The preservation proof assumes the decoder has no direct skip connection from xt; the approximation proof bounds the discrepancy between true and surrogate Wasserstein Flow-States using supervised reconstruction loss.Source: A1.4 Proof of Proposition 4:… · pp. 11–12, A1.5 Proof of Proposition 5:… · p. 12

A1.7 Flow-State Monotonicity

• Across eight SNR levels, relative kNN and MMD distances to clean latents decrease monotonically, with Spearman ρ = −0.9832 and no monotonicity violations.The analysis compares DCCRN latent representations of speech corrupted by rain and machinery noise against clean-speech latents.Source: Figure A8 · p. 13, Table A5 · p. 13, A1.7 Flow-State Monotonicity · p. 13

A1.9 Experiment with Different Discriminative Models

• DFM consistently outperforms time-conditioned FlowSE and standalone discriminative models on SI-SDR and PESQ; GCRN features yield 20.04 dB SI-SDR, while DCCRN features yield PESQ 2.96.The comparison holds across the evaluated discriminative representations, with similar trends on SCOREQ-based metrics.Source: Table A6 · p. 14, GCRN · pp. 14–15

A1.10 Impact of Input Features for Discriminative Models on DFM Performance

• DFM remains superior to CFM-related baselines across tested DCCRN inputs and STFT configurations; xζ-trained models reach SI-SDR 19.86 and PESQ 2.96.Using x0 also gives SI-SDR 19.63 and PESQ 2.96; matched STFT parameters help only minimally, and the results suggest no particular discriminative-model training pipeline is required. DFM projects features to 256 dimensions, interpolates them to the velocity network’s frame resolution, and adds them to residual-block features.Source: A1.8 Speech Enhancement · pp. 13–14, Table A7 · p. 15, A1.10 Impact of Input Features for… · p. 15

A1.11 Impact of Conditioning the Velocity Network with the Noisy Observed Signal

• Removing noisy-signal conditioning significantly degrades CFM variants, sometimes below the noisy input on reference metrics, although DNN metrics and listening indicate denoising and DFM remains superior.The reported explanation is energy mismatch: without the original noisy signal, the velocity network cannot faithfully preserve target-signal energy.Source: Table A8 · p. 15, A1.11 Impact of Conditioning the… · pp. 15–16

A1.12 Adaptive DFM Inference

• Adaptive inference uses ζ, iSNR, or gSNR as sample-difficulty estimates to choose an NFE budget and step sizes, with Nmax = 5, Trev = 1.0, and λ = 0.3 in this work.The method uses exponentially decaying weights to set thresholds and allocates the reverse interval according to difficulty.Source: A1.12 Adaptive DFM Inference · p. 16

Results

• For speech dereverberation, DFM outperforms or matches CFM, but SCOREQ gains are statistically negligible and may be limited by the discriminative DCCRN’s weaker Flow-State estimates.The experiment uses reverberation with T60 from 0.3 to 1 s and suggests that a better objective or non-mask-based discriminative model could improve results.Source: Table A9 · p. 17, Results · p. 17

A1.14 Image Denoising

• The image-denoising experiments train a classifier-based discriminative model and DFM/CFM baselines on MNIST and Fashion-MNIST; the lightweight encoder has 41.5k parameters.The classifier maps degraded 28 × 28 images to a latent representation and 10-class logits; no denoising outcome is reported in these passages.Source: A1.14 Image Denoising · p. 17, Image Classifier Architecture · p. 18

Image Classifier Architecture:

• The image classifier was trained independently on MNIST and Fashion-MNIST for 16 epochs with batch size 128, learning rate 10−3, weight decay 10−4, and Adam.These settings describe the classifier used in the image-denoising experiments.Source: Classifier Training · p. 18

Classifier Training

• The image classifier’s latent features showed semantic class structure and a heavily degraded cluster, while their MMD distance to class-wise clean features decreased as degradation scaling η decreased.The classifier achieved 75.7% MNIST and 68.9% Fashion-MNIST accuracy on generated degraded samples, supporting the proposed Flow-State hypothesis for image denoising.Source: Classifier Training · p. 18, Figure A10 · p. 19

Discriminative Flow Matching for Image Denoising

• For image denoising, DFM derives its conditioning representation as z=Cψ(xζ) from the image classifier and conditions the velocity field on the intermediate state and degraded image.The velocity network embeds the conditioning feature, adds it to the intermediate representation, then processes it through two residual convolutional blocks; DFM uses 232k parameters versus 255k for baseline CFM.Source: Discriminative Flow Matching for Image… · p. 19, Velocity Network Architecture · p. 19

Velocity Network Architecture

• DFM outperformed baseline methods across all degradation levels on MNIST and Fashion-MNIST, while adaptive inference matched fixed NFE=50 performance with a significantly lower average computational budget.Adaptive inference assigned more function evaluations to heavily degraded samples and fewer to less degraded samples.Source: Velocity Network Architecture · p. 19, Figure A12 · p. 20

A1.15 Statistical Analysis of the Results

• For SI-SDR, DFM improved over every competing model (p-value<0.001); for PESQ, superiority was significant (p-value<0.001) against all baselines except DCCRN, which was much inferior in SI-SDR.The analysis used paired Wilcoxon signed-rank tests; Figure A13 shows improvements with 95% confidence intervals.Source: A1.15 Statistical Analysis of the… · pp. 19–20, Table A10 · p. 21, Figure A13 · p. 21