Implicit Denoising Trajectory

• Implicit denoising trajectory is a framework where denoising emerges from inherent model dynamics and regularization without a dedicated transformation.
• It is implemented via implicit neural representations, denoising diffusion models, and autoencoder regularization to incrementally refine noisy data toward high-density regions.
• Empirical studies show that this approach enhances accuracy in trajectory prediction, system identification, and safe control by reducing noise-related errors.

Implicit denoising trajectory refers to a set of learning dynamics, modeling approaches, and algorithmic frameworks in which the trajectory of states—be they neural network parameters, data points, or stochastic process realizations—evolves from a noisy or imprecise initial state toward a clean, structured, or high-density region, all via a process that is not explicitly a denoising mapping but rather emerges from the choice of implicit representations, regularization, or generative process. This concept underlies several methodologies in trajectory prediction, generative modeling, and data-driven system identification, uniting them under the paradigm where "denoising" is embedded in the optimization trajectory or the probabilistic generative reverse process.

1. Conceptual Foundations

The notion of an implicit denoising trajectory arises where denoising is not performed by a dedicated, explicit transformation but is instead achieved as an emergent property of the model dynamics or fitting process. For instance, when a neural network (be it an MLP for implicit neural representation or a deep generative diffusion model) is tasked with fitting noisy data or generating samples from noise, the trajectory traversed by its outputs during training or sampling moves progressively closer to clean, high-probability data regions.

Two key mechanisms instantiate this concept:

• Spectral bias and neural architecture: In implicit neural representations (INRs), low-frequency (smooth, clean) components of signals are fit before high-frequency (noise) components. This "fitting order" means that the model's output is initially a denoised estimate, and only later becomes contaminated by overfitting the noise if allowed to continue training (Kim et al., 2022).
• Reverse diffusion process: In denoising diffusion probabilistic models (DDPMs), the “denoising trajectory” is the sequence of reverse transitions that map a noisy sample to a clean trajectory sample, each step removing incremental noise under learned stochastic drift fields (Botteghi et al., 2023, Zhu et al., 2023, Liu et al., 2024, Wang et al., 2024).

The core property is that these trajectories, whether in parameter space or data space, are not predetermined denoising maps but are implicitly defined by the statistical structure encoded in the learning dynamics or generative process.

2. Model Classes and Mathematical Frameworks

Multiple classes of models operationalize implicit denoising trajectories:

2.1 Implicit Neural Representations

INRs model signals (e.g., trajectories, images) as continuous functions parameterized by neural networks and fit directly to (possibly noisy) observations. The optimization path $\theta(t)$ under gradient descent moves from an initial state (often under-parameterized low-frequency fit) to more complex regimes, with spectral bias ensuring denoising occurs early in training (Kim et al., 2022). With proper early stopping or explicit regularization (e.g., layer-wise decay), a denoised trajectory can be extracted without a separate mapping.

The RKTV-INR approach (Yao et al., 17 Sep 2025) enhances this framework by:

• Fitting a sinusoidal MLP $\chi_\theta(t)$ to noisy trajectory data.
• Constraining the solution via data fidelity, Runge-Kutta self-consistency, and a total-variation penalty on the derivative.
• Leveraging the continuous structure and autodiff to obtain accurate denoised states and derivatives for system identification.

2.2 Denoising Diffusion Probabilistic Models

Diffusion-based generative models construct data by running a Markov chain that iteratively removes noise, starting from a Gaussian random initialization (trajectory in noise space) and ending at a data-like sample (trajectory in data space). The reverse chain is parameterized by neural networks fitted via denoising score matching objectives.

For trajectory domains:

• DiffTraj: Generates GPS trajectories via spatial-temporal DDPM, where denoising trajectory refers to the sequence of intermediate latent states from noise to sample, governed by a learnt reverse process (Zhu et al., 2023).
• C2F-TP: Uses a two-stage process. The first stage samples a coarse multimodal distribution, and the second stage denoises coarse samples via diffusion; the refinement stage constitutes an implicit denoising trajectory in the predicted path distribution (Wang et al., 2024).
• IDM (Intention-aware diffusion): Decouples intention (endpoint) and action (full path) uncertainties, first denoising a low-dimensional intention and then a high-dimensional trajectory, both via diffusion chains (Liu et al., 2024).
• Control and Planning via DDPMs: Generates control trajectories in safety-critical settings, enforcing trajectory constraints and reward optimization by guiding the denoising (sampling) process online (Botteghi et al., 2023).

2.3 Trajectory Denoising with Autoencoders

A denoising autoencoder (DAE) can regularize trajectory optimization in model-based reinforcement learning by penalizing trajectories that stray far from the training distribution. The DAE is trained to denoise noisy samples; its reconstruction residual is then used as a penalty during optimization, inherently biasing the optimized control sequence toward high-density, plausible trajectories (Boney et al., 2019).

3. Algorithmic Realizations

3.1 Gradient Trajectories in INR

• The fitting path $\theta(t)$ of an INR can be leveraged to provide a denoised output at intermediate times.
• The optimal early stopping time is often estimated using a proxy, e.g., training until the model's output error matches estimated noise power (Kim et al., 2022).

3.2 Reverse Diffusion Processes

• Forward process: $$q(x_{1:T}|x_0) = \prod_{t} \mathcal{N}(x_t; \sqrt{\alpha_t}x_{t-1}, \beta_t I)$$.
• Reverse (denoising) process: $$p_\theta(x_{t-1}|x_t,c) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t|c), \sigma_t^2 I)$$, with $\mu_\theta$ defined via the predicted noise (Zhu et al., 2023, Wang et al., 2024).
• Step-wise denoising (for coarse-to-fine or intention-aware models) enables efficient implicit denoising of sampled trajectory candidates.

3.3 Implicit Regularization in Planning

• Online model-based control is regularized by augmenting the cost with the DAE residual: $$G_{reg}(\tau) = G(\tau) - \alpha \sum_{t}\|g_\phi(x_t) - x_t\|^2$$.
• The optimizer consequently traverses a trajectory in action/state space that avoids unfamiliar, low-density regions (Boney et al., 2019).

4. Applications and Empirical Evidence

Implicit denoising trajectory mechanisms have been validated empirically in the following contexts:

• Trajectory Prediction: Coarse-to-fine denoising (C2F-TP) and intention-action diffusion (IDM) improve Final Displacement Error (FDE) on NGSIM, highD, and SDD benchmarks, with ablation demonstrating that denoising stages yield up to $75\%$ reduction in error compared to coarse or non-denoised baselines (Wang et al., 2024, Liu et al., 2024).
• System Identification: RKTV-INR achieves $39.3\%$ lower state estimation error and up to $73\%$ reduction in model identification error on nonlinear and chaotic system benchmarks relative to previous denoising or vanilla INR methods (Yao et al., 17 Sep 2025).
• Safe Control: DDPM planners incorporating safety constraints and reward gradients via guided denoising yield trajectories that comply with safety sets while maintaining task performance, outperforming standard RL baselines in success rate and sample efficiency (Botteghi et al., 2023).
• Model-based RL: DAE-regularized trajectory optimization in continuous control (MuJoCo, TE4) accelerates early learning, maintains stability against model exploitation, and outperforms both pure planning and probabilistic model-based baselines (Boney et al., 2019).
• Score Matching and Diffusion: Joint study of implicit and denoising score matching shows that score trajectories and their Jacobian converge optimally under low-dimensional structure, supporting stable deterministic ODE-based diffusion samplers (Yakovlev et al., 30 Dec 2025).

5. Theoretical Insights and Statistical Guarantees

Implicit denoising trajectories are closely related to score matching (ISM/DSM):

• Under smooth generator and additive noise models, both ISM and DSM estimators adapt to the intrinsic dimension $d$ of the data, providing estimation errors decaying as $O(n^{-2\beta/(2\beta+d)})$ for both the score and the log-density Hessian (Yakovlev et al., 30 Dec 2025).
• In flow-based and diffusion models, accurate estimation of both the score and its Jacobian along the trajectory suffices for convergence of generative ODE samplers.

Spectral bias in neural networks provides a natural filter: low-frequency (signal) is fit first, high-frequency (noise) is fit only later—leveraged for practical early stopping or layer-regularized training (Kim et al., 2022).

6. Comparative Architectures and Design Principles

Key architectural and algorithmic principles deduced from recent works:

Model/Method	Implicit Denoising Mechanism	Distinctive Feature
INR (Yao et al., 17 Sep 2025)	Spectral bias, early stopping, RK/TV reg	Direct continuous-time representation
DDPM (Botteghi et al., 2023)	Reverse Markov chain via learned drift	Conditional/safety/value guides
C2F-TP (Wang et al., 2024)	Coarse-to-fine, sampled + denoised path	Two-stage latent diffusion
IDM (Liu et al., 2024)	Decoupled (intention, action) diffusion	Dimension-reduced, fast inference
DAE regularizer (Boney et al., 2019)	DAE residual as data-local gradient	Implicit density support regularization

Layer-wise regularization and context-conditioned denoising are effective for balancing expressivity and robustness—highlighted in empirical ablation studies (Kim et al., 2022, Wang et al., 2024, Liu et al., 2024).

7. Implications, Limitations, and Future Directions

Implicit denoising trajectory frameworks unify generative, predictive, and control models under the perspective that denoising dynamics can be achieved via model architecture, optimization path, or reverse stochastic process. This removes the need for explicit, handcrafted denoising mappings.

A plausible implication is that further integration of implicit denoising mechanisms (e.g., via architecture, regularization, or hierarchical stochastic processes) could improve robustness, inference speed, and uncertainty quantification across broader real-world domains.

However, limitations include sensitivity to early stopping (in INR regimes), the complexity of training multi-stage diffusion models, and the computational burden of high-dimensional diffusion unless dimensionality is explicitly reduced (as with decoupling intention and action in IDM (Liu et al., 2024)). The development of lightweight or adaptive-steps diffusion/denoising mechanisms remains an open direction.

Advances in score-matching theory now guarantee that, for structured data, implicit denoising estimators not only achieve minimax optimal rates but also permit consistent estimation of higher-order structural features (e.g., log-density Hessians)—crucial for stability of deterministic samplers and theoretical underpinning of practical denoising generative models (Yakovlev et al., 30 Dec 2025).

Overall, the concept of implicit denoising trajectory has become foundational in data-driven modeling, generative trajectory synthesis, and robust planning, manifesting in both learning-theoretic guarantees and state-of-the-art empirical performance in complex, noise-dominated domains.