VeRe-Flow: Clean-Guided Flow Matching for NR-BWE

• The paper introduces VeRe-Flow, a conditional flow matching framework that addresses noisy conditioning by coupling velocity contrastive regularization with representation alignment.
• At the velocity level, the method employs VeCoR to attract the predicted velocity toward clean trajectories while repelling noisy alternatives, directly resolving ambiguity in the flow.
• At the representation level, VeRe-Flow aligns internal hidden states with clean XEUS-derived SSL features, enhancing both spectral fidelity and perceptual quality.

Searching arXiv for the target paper and closely related work on flow matching, velocity contrastive regularization, and representation alignment. VeRe-Flow is a clean-guided conditional flow-matching framework for noise-robust bandwidth expansion (NR-BWE) that reconstructs a clean, wideband, high-fidelity speech signal from a noisy, bandwidth-limited input. Its defining premise is that ordinary flow matching becomes ambiguous under noisy conditioning because the model is supervised to move toward the clean target but is not explicitly penalized for drifting toward noisy yet semantically compatible directions. VeRe-Flow addresses that failure mode through multi-level clean supervision: Velocity Contrastive Regularization at the velocity-field level and Representation Alignment at the hidden-feature level. In the reported Valentini-Botinhao 8 kHz setting, it achieves the lowest LSD and the highest DNSMOS OVRL among all baselines, and the highest MOS among generative baselines (Koo et al., 28 Jun 2026).

1. Problem formulation and motivation

VeRe-Flow is defined for the NR-BWE setting, where the input speech is degraded in two ways simultaneously. First, narrowband or low-resolution speech lacks high-frequency content needed for naturalness and intelligibility. Second, environmental noise corrupts the observed signal and can dominate the conditioning signal. This makes NR-BWE harder than ordinary bandwidth expansion, which assumes clean narrowband input, and harder than ordinary speech enhancement, which removes noise but does not need to hallucinate missing spectral bands (Koo et al., 28 Jun 2026).

The method is motivated by a specific weakness of standard speech generation under noise. Flow matching has been effective for clean bandwidth expansion because it learns a deterministic transport from a simple prior to the target speech distribution with high perceptual quality and very few generation steps. In the noisy setting, however, the model must estimate a velocity field while being conditioned on corrupted speech. With the usual one-sided objective, supervision only attracts the predicted velocity toward the clean target; it does not explicitly repel the model from noisy directions that remain semantically consistent with the degraded input. The paper identifies this underdetermination of the velocity field as the central reason trajectories can move away from the clean speech manifold (Koo et al., 28 Jun 2026).

This yields the method’s two-level design. At the velocity level, VeRe-Flow explicitly prefers a clean trajectory over a noisy alternative. At the representation level, it pushes internal hidden states toward clean self-supervised speech representations. A plausible implication is that the method treats denoising and bandwidth completion not as separable subproblems, but as coupled constraints on both generative dynamics and intermediate representation geometry.

2. Conditional flow-matching formulation

The paper starts from conditional flow matching. Let $x \in \mathbb{R}^d$ denote a target sample from an unknown data distribution $q(x)$. Flow matching defines densities $p_t(x)$, $t \in [0,1]$, evolving from a simple prior $p_0$ to a target distribution $p_1 \approx q$, with dynamics

$$\frac{d}{dt}\phi_t(x)=v_t(\phi_t(x)), \qquad \phi_0(x)=x.$$

Direct marginal training is replaced by Conditional Flow Matching (CFM). Given a source sample $x_0 \sim p_0$ and target sample $x_1 \sim q(x)$, one defines an interpolation path $x_t=\phi_t(x_0,x_1)$ and trains a network to match the corresponding conditional velocity. The intended CFM objective is

$q(x)$0

For NR-BWE, VeRe-Flow departs from a data-dependent prior and instead uses a Gaussian prior,

$q(x)$1

with target

$q(x)$2

The noisy low-resolution speech is not folded into the prior. It is supplied explicitly as conditioning to the velocity network together with frame-level SSL features extracted from the noisy low-resolution input. The interpolation path is

$q(x)$3

where $q(x)$4 controls the minimum noise level. Differentiation gives the optimal conditional velocity

$q(x)$5

which is constant in $q(x)$6. The learned network therefore approximates

$q(x)$7

At inference, sampling begins from $q(x)$8, and the learned ODE is integrated while conditioned on the noisy input. The paper evaluates Euler and midpoint solvers and reports the best performance with Euler and $q(x)$9 function evaluations, i.e. $p_t(x)$0 (Koo et al., 28 Jun 2026).

3. Velocity-level clean guidance: Velocity Contrastive Regularization

The most distinctive component is Velocity Contrastive Regularization, abbreviated VeCoR in the paper. Its purpose is to remove ambiguity in velocity learning by introducing both a positive and a negative target in velocity space. Using the same Gaussian source $p_t(x)$1, the method defines

$p_t(x)$2

where $p_t(x)$3 is a mel-spectrogram from a semantically consistent noise-perturbed high-resolution speech signal paired with the same clean target. The VeCoR objective is

$p_t(x)$4

This is not an InfoNCE-style objective. It is a distance-contrastive objective: the predicted velocity is attracted to the clean target velocity and repelled from the noisy one. The first term is exactly the clean flow-matching loss, while the second term provides explicit negative supervision. The paper states that $p_t(x)$5 subsumes clean flow-matching supervision because its attractive term is the clean CFM objective (Koo et al., 28 Jun 2026).

The role of VeCoR is easiest to state geometrically. If a predicted velocity could support both a clean and a noisy continuation, ordinary CFM leaves that tie unresolved. VeCoR breaks the tie by pushing the learned field away from the noisy direction. A plausible implication is that the method treats noise robustness not as a property of the conditioning encoder alone, but as a property that should be enforced directly in the learned vector field.

Level	Mechanism	Function
Velocity	VeCoR	attract clean velocity, repel noisy velocity
Representation	RA	align hidden states with clean XEUS features
Conditioning	noisy low-resolution mel + XEUS	provide explicit noisy-acoustic and SSL guidance

4. Representation-level supervision and architecture

Representation Alignment (RA) complements VeCoR by acting inside the network rather than on the output velocity. The paper’s concern is that, even if the final velocity is supervised, hidden states under noisy conditioning may still drift away from the clean speech manifold. To counter this, the model extracts the hidden state

$p_t(x)$6

from the output of the first transformer layer, where $p_t(x)$7 is the number of frames and $p_t(x)$8 is the hidden size. This hidden state is projected into SSL space by a 3-layer MLP projection head $p_t(x)$9 with SiLU activations. The target is a frame-level clean representation from XEUS, a frozen SSL model pretrained with dereverberation and denoising objectives. The alignment loss is

$t \in [0,1]$0

where $t \in [0,1]$1 is cosine similarity. Minimizing negative cosine similarity maximizes similarity between the generator’s internal features and clean XEUS features (Koo et al., 28 Jun 2026).

The total training objective is

$t \in [0,1]$2

The reported formulation does not add adversarial, waveform reconstruction, or explicit spectrogram reconstruction losses beyond these terms. Functionally, the objective fits the clean conditional velocity, repels the model from a noisy velocity target, and aligns hidden representations with clean SSL embeddings (Koo et al., 28 Jun 2026).

Architecturally, VeRe-Flow uses 80-dimensional mel-spectrograms as both input and output representations. The backbone adds DiC-style Conv ResBlocks adapted to 1D speech representations. Each residual block has a GroupNorm–Activation–Conv1D structure with kernel size $t \in [0,1]$3, mid-block scale-and-shift conditioning projected from the time embedding, and a residual connection. These blocks are organized in a sandwich structure: a convolutional pre-stage, a central transformer stage, and a convolutional post-stage, with 4 stacked Conv ResBlocks in each convolutional stage. Conditioning uses the noisy low-resolution mel-spectrogram together with frame-level XEUS features, concatenated along the feature dimension and then projected into the model input space (Koo et al., 28 Jun 2026).

5. Training protocol, dataset, and empirical results

Speech is converted to 80-bin mel-spectrograms using a 20 ms hop size and a 1280-point window. XEUS features are extracted every 20 ms and aligned to the mel frames. Training runs for 400k iterations with batch size 16, using Adam and a cosine annealing learning-rate scheduler starting at $t \in [0,1]$4. During training, additive noise with random SNR uniformly sampled from $t \in [0,1]$5 is applied with probability $t \in [0,1]$6. The loss weights are

$t \in [0,1]$7

Generation produces a mel-spectrogram that is converted to waveform by BigVGAN. The best-performing flow configuration is a Gaussian prior, Euler solver, and $t \in [0,1]$8 (Koo et al., 28 Jun 2026).

Experiments use the Valentini-Botinhao noisy speech corpus. The original corpus is parallel clean/noisy data at 48 kHz with 28-speaker and 56-speaker subsets; the paper merges them for 84 training speakers. Evaluation uses the official test set with 2 unseen speakers and 20 noise conditions. Bandwidth limitation is created by a Chebyshev Type-I low-pass filter followed by downsampling. During training, filter order, ripple, and target sampling rate are randomized over 1–15 kHz. At test time, the filter is fixed to order 8 with 0.05 dB ripple, downsampled to 8 kHz, then reconstructed to 16 kHz and compared against 16 kHz ground truth (Koo et al., 28 Jun 2026).

The comparison includes non-generative baselines and two generative baselines retrained in the same NR-BWE setting for fairness. Metrics are LSD, DNSMOS SIG, BAK, and OVRL, plus MOS from human listening tests on Amazon Mechanical Turk using 40 test utterances and 11 raters per sample. On the noisy Valentini-Botinhao 8 kHz test setting, the reported generative results are: VeRe-Flow with LSD $t \in [0,1]$9, SIG $p_0$0, BAK $p_0$1, OVRL $p_0$2, and MOS $p_0$3; a prior flow baseline with LSD $p_0$4, SIG $p_0$5, BAK $p_0$6, OVRL $p_0$7, and MOS $p_0$8; and NU-Wave2 with LSD $p_0$9, SIG $p_1 \approx q$0, BAK $p_1 \approx q$1, OVRL $p_1 \approx q$2, and MOS $p_1 \approx q$3. Among all baselines, VeRe-Flow achieves the lowest LSD and the highest DNSMOS OVRL; among generative methods, it also achieves the highest MOS (Koo et al., 28 Jun 2026).

The ablations are technically informative. The best solver/prior setting is Gaussian prior plus Euler solver plus $p_1 \approx q$4, and increasing NFE to 6 does not improve results. In SSL ablations, XEUS is best, with LSD $p_1 \approx q$5 and OVRL $p_1 \approx q$6, compared with WavLM at LSD $p_1 \approx q$7, OVRL $p_1 \approx q$8, and wav2vec 2.0 at LSD $p_1 \approx q$9, OVRL $$\frac{d}{dt}\phi_t(x)=v_t(\phi_t(x)), \qquad \phi_0(x)=x.$$0. In component ablations, XEUS contributes most strongly to LSD, RA improves SIG and OVRL, and VeCoR improves BAK and OVRL, while the full model balances these gains (Koo et al., 28 Jun 2026).

6. Interpretation, related work, and limitations

The conceptual novelty of VeRe-Flow lies in how it reframes the noisy conditioning problem for flow matching. Rather than treating noise robustness as a preprocessing issue, it posits that the velocity field itself is underdetermined under one-sided supervision. The solution is a clean-guidance design with two distinct targets: one for trajectory dynamics and one for hidden-state semantics. VeCoR regulates where the generative path should and should not move; RA regulates how the model should internally encode speech while making that decision. This suggests a division of labor between dynamics-level regularization and representation-level regularization (Koo et al., 28 Jun 2026).

A useful clarification concerns nomenclature. The name “VeRe-Flow” is not unique in the broader flow-model literature. A different work, “Variational Rectified Flow Matching,” also associates “VeRe-Flow” with a latent-variable rectified-flow framework for modeling multi-modal velocity fields in image generation (Guo et al., 13 Feb 2025). That method addresses ambiguity in velocity directions through latent conditioning, whereas the speech-domain VeRe-Flow discussed here addresses ambiguity under noisy conditioning through velocity contrastive regularization and representation alignment. The shared label therefore masks substantially different technical objects.

The velocity-level component also sits within a broader trajectory of flow-model regularization. A later standalone work, “VeCoR - Velocity Contrastive Regularization for Flow Matching,” formulates an attract–repel objective for image generation and reports that such two-sided supervision improves stability and low-step sampling quality (Hong et al., 24 Nov 2025). A plausible implication is that the contrastive velocity mechanism introduced in speech NR-BWE is not an isolated heuristic, but part of a broader class of velocity-space regularizers.

The paper is comparatively restrained about limitations, but several constraints are explicit or implicit. Training requires paired clean and noisy high-resolution examples, and the velocity contrastive term depends on semantically matched noisy targets. The method also relies on a frozen SSL model, especially XEUS, whose denoising and dereverberation pretraining appears important to the reported performance. The paper points toward extending the method to other speech generation tasks, exploring stronger or more explicit contrastive formulations, and studying generalization under broader real-world degradation conditions (Koo et al., 28 Jun 2026).