Inertial-Accelerated Deep Inverse Priors

• The paper introduces RISP, which integrates momentum and restart mechanisms into deep inverse prior optimization, achieving up to 5–10× faster convergence with improved PSNR.
• Inertial-accelerated deep inverse priors leverage continuous-time ODE interpretations and second-order dynamics to provide convergence guarantees without relying on prior convexity.
• Empirical evaluations across deblurring, MRI, and inverse scattering tasks demonstrate significant speedups—up to 24× faster than traditional RED methods—while preserving fine image details.

Searching arXiv for the cited papers to ground the article in the current record. Inertial-accelerated deep inverse priors are methods for imaging inverse problems that combine deep image priors with inertial dynamics in order to improve convergence while preserving reconstruction quality. In the recent literature, this designation covers at least two technically distinct constructions: reconstruction algorithms that insert momentum and restart into score-regularized optimization, exemplified by Restarted Inertia with Score-based Priors (RISP), and self-supervised Deep Inverse Prior training rules in which the network parameters follow second-order inertial dynamics with viscous and geometric Hessian-driven dampings (Renaud et al., 8 Oct 2025, Buskulic et al., 3 Jun 2025). A broader architectural context is provided by unrolled optimization with deep priors, where classical iterative algorithms are truncated and reinterpreted as networks alternating between physics-informed data steps and learned prior steps (Diamond et al., 2017).

1. Inverse-problem setting and the role of deep priors

A central formulation is the inverse problem of recovering a latent image from measurements taken under a known physical image formation model. In the optimization-based setting used by RISP, imaging inverse problems are cast as MAP estimation,

$$\hat{\bm{x}} \in \arg\min_{\bm{x} \in \mathbb{R}^d} \left\{ F(\bm{x}) := f(\bm{x}) + g(\bm{x}) \right\},$$

where $f$ is the differentiable data-fidelity term and $g$ is the regularizer or prior, often nonconvex (Renaud et al., 8 Oct 2025). In the self-supervised Deep Inverse Prior setting, the image is generated by a neural network $x = G(u,\theta)$, and the parameters are fit by minimizing

$$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$

with $A$ the forward operator and $y$ the measurements (Buskulic et al., 3 Jun 2025).

Deep priors enter these formulations in several ways. RISP, like RED, uses a learned image score function

$\mathsf{S}(\bm{x}) \approx \nabla \log p(\bm{x}),$

with the relation formalized as $\mathsf{S} = -\nabla g$, so that the score acts as the negative gradient of the regularizer (Renaud et al., 8 Oct 2025). In ODP, the prior term of a classical iterative method is replaced by a learned CNN, while the data step explicitly encodes the known image formation operator $A$ (Diamond et al., 2017). In Deep Inverse Prior methods, the prior is implicit: the network architecture and the training dynamics regularize the reconstruction even without explicit penalization, and early stopping is part of that implicit regularization mechanism (Buskulic et al., 3 Jun 2025).

This division is important. Some methods place inertia in the optimization over the image variable; others place inertia in the optimization over network parameters. A plausible implication is that “inertial-accelerated deep inverse priors” are best understood as a family of techniques rather than a single algorithmic template.

2. RISP: restarted inertia with score-based priors

RISP is a principled extension of Regularization by Denoising designed to address two limitations identified for existing methods: slow convergence due to generic iterative optimization, and lack of principled acceleration compatible with nonconvex, deep score-based priors (Renaud et al., 8 Oct 2025). It combines inertial acceleration inspired by the heavy-ball method and Nesterov’s acceleration, a restart mechanism to counteract instability in nonconvex settings, and score-based image priors that may be learned by deep neural networks.

The method has two principal variants. In RISP-GM, the iterate first undergoes an inertial step,

$f$0

followed by a gradient step with the score prior,

$f$1

In RISP-Prox, the same inertial step is followed by a proximal update,

$f$2

The restarting mechanism resets momentum and the iteration counter whenever

$f$3

Operationally, this means that when the accumulated movement exceeds a budget $f$4, one resets $f$5 and restarts the method (Renaud et al., 8 Oct 2025).

The score-based prior enters exactly as in RED, via a denoiser or direct score estimator. The paper states that $f$6 can be constructed from a denoiser, for example GS-DRUNet, trained on clean/noisy image pairs (Renaud et al., 8 Oct 2025). This preserves compatibility with deep score priors while modifying the optimization backbone. The conceptual contribution is therefore not merely the insertion of momentum, but the construction of a restart-controlled inertial scheme for composite, possibly nonconvex, score-regularized objectives.

3. Dynamical-systems interpretation

RISP is analyzed through an associated continuous-time dynamical system. Its inertial behavior corresponds to the ODE

$f$7

where $f$8 is related to the momentum parameter $f$9 (Renaud et al., 8 Oct 2025). The paper presents this as a connection to the heavy-ball ODE and invokes the “rolling ball” analogy over $g$0. Restarting periodically resets velocity or negligible-gradient trajectories, which is described as essential in nonconvex settings to avoid overshooting or getting stuck.

The Deep Inverse Prior work adopts a more elaborate second-order ODE in parameter space,

$g$1

where $g$2 is the viscous damping and $g$3 is the geometric, Hessian-driven damping (Buskulic et al., 3 Jun 2025). When $g$4, this reduces to the heavy-ball ODE; when $g$5, the paper states that one obtains a Dynamical Inertial Newton-like flow. The geometric damping term is introduced to reduce oscillations and align updates with the curvature of the loss landscape.

Across these formulations, inertia is not treated as a purely heuristic addition. It is embedded in an ODE perspective that links acceleration, damping, and stability. This suggests a common interpretation: inertial acceleration in deep inverse methods is governed as much by the control of oscillatory behavior as by raw momentum accumulation.

4. Convergence guarantees and recovery theory

For RED baselines, the stationary-point convergence rate reported for RED-GM and RED-Prox is

$g$6

after $g$7 iterations, which the paper identifies as consistent with nonconvex gradient descent limits (Renaud et al., 8 Oct 2025). Under assumptions including Lipschitz continuous gradients and Lipschitz Hessians for $g$8 and the regularizer implied by the score, RISP-GM and RISP-Prox attain the faster rate

$g$9

For RISP-GM, the explicit bound quoted in the paper is

$x = G(u,\theta)$0

where $x = G(u,\theta)$1 is an explicit constant, for example 82, $x = G(u,\theta)$2, $x = G(u,\theta)$3 is the sum of gradient Lipschitz constants, and $x = G(u,\theta)$4 is a combined Hessian Lipschitz constant (Renaud et al., 8 Oct 2025). The restarting step is identified as crucial: without it, inertial methods cannot guarantee this speedup in general nonconvex landscapes. The paper also emphasizes that the acceleration is obtained without requiring convexity of the image prior.

The Deep Inverse Prior analysis gives convergence and recovery guarantees in both continuous and discrete time. In the continuous setting, with proper damping parameters $x = G(u,\theta)$5, the loss satisfies

$x = G(u,\theta)$6

and the parameters satisfy

$x = G(u,\theta)$7

where $x = G(u,\theta)$8 is the Jacobian at initialization (Buskulic et al., 3 Jun 2025). The paper further gives a signal recovery guarantee,

$x = G(u,\theta)$9

and quantifies meaningful early stopping times to avoid overfitting noise. Standard gradient flow is stated to have a slower exponential rate, approximately $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$0, so the inertial ODE yields an acceleration that the paper describes as matching optimal results in convex optimization (Buskulic et al., 3 Jun 2025).

In discrete time, the ODE is discretized as

$$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$1

$$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$2

with step size chosen via backtracking for stability (Buskulic et al., 3 Jun 2025). Under the paper’s assumptions, the method converges linearly: $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$3

Taken together, these results locate acceleration in two complementary regimes: stationarity-rate improvement for nonconvex score-regularized reconstruction, and accelerated exponential or linear convergence with recovery guarantees for self-supervised network training.

5. Relation to unrolled optimization with deep priors

Unrolled Optimization with Deep Priors provides a broader framework for understanding how deep priors and optimization interact. Diamond et al. describe ODP as a principled framework for infusing knowledge of the image formation into deep networks that solve inverse problems in imaging, inspired by classical iterative methods (Diamond et al., 2017). The method truncates a classical iterative optimization algorithm to $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$4 iterations and interprets the resulting sequence as a network. Each stage applies a CNN prior step and then a data step $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$5 encoding the physical model through $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$6.

The framework explicitly accommodates algorithms such as proximal gradient, ISTA/FISTA, ADMM, and Chambolle-Pock. In that sense, it forms a natural architectural setting in which inertial operations can be represented as layers or sub-graphs. The data block states that FISTA can be unrolled analogously, including its momentum operation (Diamond et al., 2017). At the same time, the ablation studies reported in ODP found minimal gains from using dual variables or momentum-enhanced variants such as ADMM, LADMM, and FISTA for the relatively small number of iterations typical in unrolled architectures; more aggressive approximate inversion in the data step was reported to be more beneficial.

This contrast is instructive. In RISP, the number of iterations is large enough for convergence-rate analysis and restart control to be central. In ODP, the network depth is fixed and comparatively shallow, and the paper emphasizes the value of hard-coding $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$7 through physics-guided layers. A plausible implication is that inertial acceleration plays different roles in optimization-to-convergence algorithms and in finite-depth learned reconstructor architectures.

6. Empirical behavior, applications, and common points of confusion

RISP was evaluated on linear inverse problems including deblurring, inpainting, super-resolution, and MRI; on a nonlinear inverse problem, Rician noise removal; and on large-scale inverse scattering at $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$8 resolution (Renaud et al., 8 Oct 2025). The paper reports up to $$\mathcal{L}(\theta) = \frac{1}{2}\|A G(u,\theta) - y\|^2,$$9 faster convergence in terms of both gradient norm reduction and PSNR similarity to ground truth versus RED, with the same or better PSNR in fewer iterations. For Rician noise removal, the reported example is that RISP-Prox achieves a PSNR of $A$0 dB in $A$1 ms, whereas RED-GM needs $A$2 longer runtime for comparable results. For inverse scattering, RISP can speed up convergence by up to $A$3, and the paper reports that RISP recovers fine structural details in cells within $A$4 minutes while RED algorithms are still blurry even after $A$5 minutes (Renaud et al., 8 Oct 2025).

Task	Reported speedup over RED	Reported quality statement
Deblurring	$A$6 faster	Equal or higher PSNR, sharper features in less time
Inpainting	$A$7 faster	Cleaner images, better inpainted regions
MRI	$A$8 faster	High PSNR, better detail recovery
Nonlinear (Rician)	$A$9 faster	Higher PSNR and SSIM at lower cost; visually cleaner images
Large-scale (ODT)	$y$0 faster	Ultra-large images, fine features preserved dramatically sooner

The Deep Inverse Prior work reports that inertial training with properly tuned inertia and geometric damping converges to zero loss in significantly fewer iterations than gradient descent or Polyak heavy ball, both on toy and imaging problems (Buskulic et al., 3 Jun 2025). The ODP paper reports strong empirical performance on denoising, deblurring, and compressed sensing MRI, and notes that removing the data step degrades performance significantly for deblurring and MRI, which it interprets as evidence for the value of explicitly encoding $y$1 (Diamond et al., 2017).

Several recurrent misconceptions can be addressed directly from these results. First, momentum in inverse imaging is not automatically a principled acceleration device: prior attempts to add inertia or momentum in RED/PnP were reported to show empirical benefits but to lack theoretical accelerated guarantees, whereas RISP was designed specifically to close that gap (Renaud et al., 8 Oct 2025). Second, accelerated convergence does not necessarily require a convex prior: RISP’s $y$2 rate is stated to hold without requiring prior convexity, which is precisely what enables deep network priors. Third, momentum is not uniformly beneficial in every deep inverse architecture: in ODP, for the relatively small number of iterations typical in unrolled architectures, momentum-enhanced variants offered minimal gains (Diamond et al., 2017). Fourth, inertial training does not remove the need for early stopping in self-supervised priors: the Deep Inverse Prior analysis explicitly quantifies stopping times to avoid overfitting noise (Buskulic et al., 3 Jun 2025).

The resulting picture is technically differentiated rather than uniform. Inertia may accelerate score-regularized reconstruction, stabilize and speed self-supervised parameter dynamics, or be comparatively unimportant in shallow unrolled networks. What unifies these settings is the attempt to combine learned priors with algorithmic structure precise enough to support either end-to-end trainability, formal convergence guarantees, or both.