Incremental Flow-Based Denoising Models

• Incremental flow-based denoising models are probabilistic generative frameworks that use sequential invertible transformations to iteratively reduce noise and impose structure.
• They leverage a unified path-space Kullback–Leibler objective with neural surrogates to construct backward generators that ensure universality and precise phase-wise control.
• Their phase-wise training, marked by early drift, intermediate transport, and late refinement stages, enables effective denoising across domains like image, video, and non-Euclidean spaces.

Incremental flow-based denoising models constitute a class of probabilistic generative frameworks designed to approximate complex data distributions via a sequence of invertible transformations (“flows”) that iteratively reduce noise and introduce structure. These models operate within the broader landscape of diffusion, score-based, and normalizing flow models, but are distinguished by their incremental, phase-wise approach to denoising which enables superior universality, theoretical guarantees, and adaptability to diverse stochastic forward processes.

1. Mathematical Foundations and Unified Frameworks

Incremental flow-based denoising models start from a forward Markov process—often a continuous-time diffusion, but potentially a discrete or Lévy-type process—defined over a state space $E \subset \mathbb{R}^d$. Let $(x_t)_{0 \leq t \leq T}$ denote the forward process with generator $\mathcal{L}_t$ and strictly positive, smooth marginals $p_t$. Under regularity, Feller, and density assumptions, the time-reversed process $(\bar{x}_t = x_{T-t})$ is again Markovian with an explicitly computable generator derived via a generalized Doob $h$-transform:

$$\overleftarrow{\mathcal{L}}_{T-t} f(x) = p_t^{-1} \mathcal{L}_t^*[p_t f](x) - p_t^{-1} f(x) \mathcal{L}_t^* p_t(x),$$

where $\mathcal{L}_t^*$ is the $L^2$ adjoint. In practice, the unknown true density $p_t$ is replaced by a neural-surreogate $\phi_t$, and optimization proceeds by minimizing a unified path-space Kullback–Leibler objective:

$$\mathfrak{L}[\widehat{\mathcal{L}}] = \mathbb{E}_{x_t \sim p_t} \left[ \int_0^T \left( \mathcal{L}_t \phi_t \, \phi_t^{-1} + \mathcal{L}_t \log \phi_t \right)(x_t) \, dt \right].$$

In continuous diffusion, this framework subsumes score-matching, while for pure-jump or discrete processes, the objective adapts to suitable forms involving transition rates and estimated score ratios (Ren et al., 2 Apr 2025). The backward generator in each case is incrementally constructed, yielding a stage-wise roadmap for sample generation.

2. Necessity and Universality of Incremental Generation

A rigorous approximation-theoretic analysis has demonstrated that a single-step (i.e., non-incremental) flow—even with arbitrary depth, width, and Lipschitz nonlinearity—is fundamentally non-universal for denoising generative modelling. Formally, the class of all time-1 flows of autonomous ODEs is meagre in the group of orientation-preserving homeomorphisms $Homeo^+([0,1]^d)$. This impossibility arises from the dynamical constraints of autonomous flows, notably their inability to model non-fixed periodic attractors, which can arise generically in invertible maps required by denoising pipelines (Rouhvarzi et al., 13 Nov 2025).

By contrast, every orientation-preserving Lipschitz homeomorphism $\varphi \in Homeo^+([0,1]^d)$ can be uniformly approximated to error $O(n^{-1/d})$ by a composition of at most $K_d$ incremental flows, where $K_d$ depends only on the input dimension $d$. Under additional $C^s$ smoothness, dimension-free rates $O((NL)^{-2s/d})$ can be attained, and the number of required flows remains bounded (and often small in practice for $d \leq 10$). Key architectural constraints are invertibility and well-controlled Lipschitz constants, commonly enforced using spectral normalization.

For practical denoising maps $f : [0,1]^d \to \mathbb{R}^D$, a canonical incremental “lifting” construction directly embeds $f$ into flows on $[0,1]^{d+1}$ via $V_f(x, y) = (0, f(x))$ and post-composes with a projection, ensuring injectivity and universality in $C([0,1]^d; \mathbb{R}^D)$ (Rouhvarzi et al., 13 Nov 2025).

3. Incremental Denoising as Phase-Wise Flow Matching

From the denoising perspective, flow-matching generative models are trained using objectives that interpolate between clean data $x_1$ and noise $x_0$, yielding time-indexed mixtures $x_t = (1-t)x_0 + t x_1$. The standard flow-matching loss is expressed as

$$\mathcal{L}_{FM}(\theta) = \mathbb{E}_{t, x_0, x_1}\left[ \left\| v_\theta(x_t, t) - (x_1 - x_0) \right\|^2 \right]$$

with minimizer $v^*(x_t, t) = \mathbb{E}[x_1 - x_0 \mid x_t, t]$ (Gagneux et al., 28 Oct 2025). The optimal MMSE denoiser at time $t$ is

$D^*_t(x) = x + (1 - t) v^*(x, t),$

and, crucially, all standard denoising objectives (classical, unweighted, and flow-matching with appropriate weighting) converge to $D^*_t(x)$ in the infinite-capacity limit. However, finite network capacity and phase-wise empirical behavior necessitate explicit analysis of weighting, scheduling, and parametrization to optimize denoising performance across all noise levels.

A key empirical finding is the presence of distinct dynamical phases:

• Early “drift” (mean attraction, low $t$)
• Intermediate “transport/content formation” (high Lipschitz, global transformation, $t \approx [\tau, 0.8]$)
• Late “denoising refinement” (local correction, small receptive field, $t\approx 1$)

Performance diagnostics such as PSNR(t), FID, and Jacobian norm provide insights into architectural and training choices, and controlled perturbations reveal phase sensitivities—for example, drift-type errors early have little effect on FID, whereas noise-type errors late dramatically increase FID (Gagneux et al., 28 Oct 2025).

4. Unified Training Algorithms and Practical Instantiations

Across frameworks (flow-matching, denoising Markov models, SFBD Flow), training proceeds via incremental, phase-wise score matching or generator matching. The following pseudocode (paraphrased from (Ren et al., 2 Apr 2025, Gagneux et al., 28 Oct 2025, Lu et al., 3 Jun 2025)) captures the core steps:

Training:

1. Sample clean data $x_0$ and noise level $t$ (or multi-scale step $k$).
2. Generate noised examples $x_t$ by interpolation or forward process simulation.
3. Compute surrogate (e.g., $D_\theta$ or $\phi_t$).
4. Minimize phase-weighted score-matching or flow-matching loss over batches.

Sampling:

1. Initialize at noise (or simple prior).
2. Simulate incrementally through $L$ time/discretization steps, using learned backward generator or denoiser.
3. Return clean sample after last incremental step.

The SFBD Flow algorithm (Lu et al., 3 Jun 2025) demonstrates that alternating projection-based denoising frameworks can be unified as continuous functional gradient flows, enabling end-to-end optimization with direct consistency constraints and without explicit alternation or retraining.

5. Model Variants: Image, Video, and Non-Euclidean Domains

Incremental flow-based denoising has been instantiated for a range of domains:

• Joint Image and Noise Models (FINO): Exploit normalizing flows to decouple image and noise in latent space via invertible block compositions, variable swapping, and correlation constraints. Coarse-to-fine intermediate denoisers can be implemented by zeroing out noise channels at partial flow depth, yielding progressively refined reconstructions (Guo et al., 2021).
• Video Denoising: Incorporate multi-scale, flow-refined guidance and bidirectional mutual feature propagation, as in the DFR+FMDP architecture, yielding robustness to very high noise regimes (e.g., PSNR 35.08 dB on DAVIS at $\sigma=50$). Multi-scale flow alignment and mutual fusion are essential for temporal consistency and state-of-the-art performance on synthetic and real noise benchmarks (Cao et al., 2022).
• General Markov/Lévy Processes: The unified denoising Markov models framework enables extension to geometric Brownian motion, jump processes, and other forward stochastic dynamics beyond pure diffusions, subject to minimal regularity and density assumptions (Ren et al., 2 Apr 2025).

6. Design Principles, Limitations, and Theoretical Guarantees

Effective design of incremental flow-based denoisers involves several empirically and theoretically grounded practices:

• Weight Scheduling: Emphasize mid-phase denoising; FM weighting $w_t = (1-t)^{-2}$ is often empirically optimal.
• Residual Parametrization: Structures like $D(x, t) = x + (1-t) N_\theta(x, t)$ enforce correct behavior at endpoints and bias towards incremental corrections.
• Jacobian Control: High Lipschitz constants should be restricted to intermediate regime to accommodate global content formation, while early/late phases require stronger regularization.
• Explicit Phase Probing and Robustification: Controlled training and evaluation perturbations in phase-specific windows guide architectural and loss scheduling.
• Limitation to Incremental Flows: Single-step flows are provably non-universal on $Homeo^+([0,1]^d)$; structured compositions or staged flows are necessary for universal approximation and for controlling $W_1$ distance in measure pushforwards (Rouhvarzi et al., 13 Nov 2025).

End-to-end error is bounded for incremental models by

$$(p_0 \| q_T) \leq (p_T \| q_0) + \mathfrak{L}[\widehat{\mathcal{L}}] + O(T \Delta t^r)$$

where $q_0$ is the prior, $\mathfrak{L}$ is the training loss, and $\Delta t^r$ is the time discretization error (Ren et al., 2 Apr 2025). When the forward process is expansive and converges ($p_T \| q_0$ decays), model quality depends primarily on estimation and discretization errors.

7. Connections, Extensions, and Research Directions

Incremental flow-based denoising models unify previously disparate threads in probabilistic generative modeling, subsuming diffusion models (Song et al.), flow-matching (Lipman et al.), generator-matching (Holderrieth et al.), and consistency constraint frameworks (e.g., Consistent Diffusion, TweedieDiff) (Ren et al., 2 Apr 2025, Gagneux et al., 28 Oct 2025, Lu et al., 3 Jun 2025). They facilitate principled extensions to non-Euclidean spaces, handle complex noise processes (including jumps and multiplicative volatility), and support phase-aware diagnostics critical for modern high-fidelity generative modeling.

Future directions include exploiting the full generality of Lévy generators, refining phase-wise scheduling and parametrization, and designing hybrid models that blend the strengths of normalizing flows, score-matching, and Markovian generator matching under unifying theoretical frameworks. For all such efforts, the necessity of truly incremental, multi-phase denoising flows is now theoretically established and practically validated.