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Scalable Gaussian Process Regression via Deterministic Trigonometric Features: Uniform Bounds for Safe Model Predictive Control

This paper introduces a scalable Deterministic Trigonometric Feature Gaussian Process (DTF-GP) framework that enables high-probability uniform uncertainty bounds for safe learning-based Model Predictive Control, achieving the safety guarantees of full Gaussian processes with significantly improved computational efficiency in large-data regimes.

Original authors: Julius Jagdt, Johanna Menn, Sebastian Trimpe, Melanie N. Zeilinger, Anna Scampicchio

Published 2026-08-18

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CC BY 4.0

Original authors: Julius Jagdt, Johanna Menn, Sebastian Trimpe, Melanie N. Zeilinger, Anna Scampicchio

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a robot arm assembling delicate electronics or a self-driving car navigating a busy city street. For these machines to operate safely, they must constantly predict what will happen next. They rely on mathematical models—simplified versions of reality—to guess how their movements will change their position. But real life is messy. Friction, wind, and wear and tear mean these models are never perfect. If a robot relies on a flawed guess, it might push too hard, break a part, or crash. To stay safe, engineers use a strategy called Model Predictive Control. This method constantly recalculates the best path forward, checking every step against strict safety rules. However, this safety net only works if the robot knows how much it doesn't know. It needs to understand the limits of its own predictions.

For years, scientists have used a powerful statistical tool called a Gaussian process to fill in the gaps of these imperfect models. Think of a Gaussian process as a very cautious, highly intelligent mapmaker. Instead of just drawing a single line to show where a robot will go, it draws a wide, fuzzy band that represents all the possible places the robot could end up. The wider the band, the more uncertain the robot is. This uncertainty is crucial for safety: if the band gets too wide, the robot knows to slow down or stop. The problem is that as the robot collects more and more data to learn from, this mapmaking tool becomes incredibly slow. It is like trying to solve a massive puzzle where every new piece requires you to re-solve the entire picture from scratch. For large datasets, the computer simply cannot keep up, forcing engineers to choose between having a safe, slow system or a fast, unsafe one.

A team of researchers from universities in Germany, Switzerland, and Sweden has found a way to break this deadlock. They developed a new method that keeps the safety guarantees of the slow, perfect mapmaker but runs as fast as a simple, quick sketch. Their approach, which they call a deterministic trigonometric feature Gaussian process, replaces the heavy, complex calculations with a clever approximation using waves. Instead of trying to calculate every possible outcome at once, the new method breaks the problem down into a fixed set of wave patterns. By selecting these patterns in a specific, non-random way, the researchers proved that the resulting map is just as reliable as the original, slow version. They showed that this new tool can provide the same high-confidence safety warnings that a full system would, but without the crushing computational cost.

The researchers tested their idea on a simulated inverted pendulum, a classic control problem where a robot must balance a pole on a moving cart. In their experiments, they compared their new method against the traditional, slow approach. They found that both methods explored the environment with equal skill, learning the same amount about the system's hidden errors. The new method was just as good at finding the safest, most efficient paths. However, the difference in speed was dramatic. As the amount of data grew, the traditional method slowed down significantly, struggling to keep up with the volume of information. The new method, by contrast, maintained a steady pace regardless of how much data it processed. It handled large datasets with ease, proving that it is possible to have both high safety and high speed.

This work matters because it removes a major barrier to using advanced learning systems in the real world. Previously, engineers had to limit the amount of data their robots could learn from to keep the system running in real-time. Now, they can feed the system vast amounts of data to make it smarter and safer, without worrying that the computer will freeze. The researchers demonstrated that their new framework can be integrated into safety-critical systems, ensuring that the robot never takes a risk it cannot calculate. By turning a slow, heavy calculation into a fast, efficient one, they have opened the door for more robust and reliable autonomous systems that can learn continuously while staying safely within their limits.

Technical Summary: Scalable Gaussian Process Regression via Deterministic Trigonometric Features

Problem Statement
Learning-based Model Predictive Control (MPC) utilizing Gaussian Processes (GPs) offers a robust approach for controlling dynamical systems with unknown dynamics and model mismatch. A critical requirement for ensuring safety in these stochastic MPC frameworks is the availability of uncertainty bounds that hold uniformly over the entire state–input domain. While such uniform bounds have been derived for exact (full) GP regression, the computational cost of exact inference scales cubically with the number of data points ( $O(N^3)$ ). This poor scalability renders full GP regression impractical for large-data regimes. Existing scalable GP approximations—such as inducing-variable methods, subset-of-data approaches, and randomized feature methods (e.g., Random Fourier Features)—fail to provide the necessary uniform uncertainty bounds required for safety-critical control. Consequently, there is a gap between the need for scalable inference and the requirement for rigorous, uniform safety guarantees.

Methodology
The authors propose the Deterministic Trigonometric Feature Gaussian Process (DTF-GP), a scalable framework that reduces GP regression to Bayesian linear regression in a finite-dimensional feature space while retaining the ability to derive uniform uncertainty bounds.

Finite-Dimensional Kernel Approximation:
The method approximates stationary kernels using a finite set of deterministic trigonometric features. Leveraging Bochner's theorem, the stationary kernel is represented as a continuous superposition of trigonometric basis functions weighted by the spectral density. The DTF-GP discretizes this integral by selecting frequencies from a fixed, uniformly spaced grid.
- Deterministic Selection: Unlike Random Fourier Features, which sample frequencies stochastically, the DTF-GP selects frequencies deterministically.
- Spectral Parametrization: For the squared-exponential (RBF) kernel, the spectral weights are parameterized to mirror the exponential decay of the true spectral density.
- Frequency Truncation: Frequencies are retained within an ellipsoidal region in the frequency space, determined by a truncation radius. This ensures that the most significant spectral components are preserved while discarding negligible mass.
Uniform Uncertainty Bound Derivation:
The core theoretical contribution is the derivation of a high-probability uniform uncertainty bound for the DTF-GP.
- The bound decomposes the total error into a noise term (handled via self-normalized concentration inequalities) and an approximation term.
- The approximation term is analyzed using an orthogonal projection operator that maps the unknown function from the infinite-dimensional Reproducing Kernel Hilbert Space (RKHS) induced by the true kernel to the finite-dimensional RKHS induced by the DTF kernel.
- The bound explicitly accounts for the projection error (the difference between the true function and its projection onto the finite feature space).
- For the RBF kernel, the authors derive a closed-form upper bound for this projection error, which depends on the RKHS norm of the function, the spectral decay parameters, and the truncation radius.
Integration into Safe MPC:
The DTF-GP is integrated into a SafeMPC framework. The derived uniform uncertainty bound is used to propagate uncertainty sets over a prediction horizon. By ensuring that the true model error lies within these bounds with high probability, the controller can enforce state and input constraints, guaranteeing safe exploration.

Key Contributions

Formalization of DTF-GP: The paper formalizes a scalable GP framework using deterministic trigonometric features selected from a uniform frequency grid, reducing inference to Bayesian linear regression.
Uniform Uncertainty Bounds: The authors derive a high-probability uniform uncertainty bound for the DTF-GP. Crucially, they provide an explicit, closed-form projection error bound for the RBF kernel case, ensuring the resulting confidence bound relies on the same assumptions as full GP bounds (RKHS norm, sub-Gaussian noise, and posterior quantities).
Safe Learning-Based Control: The framework is successfully integrated into a learning-based MPC scheme, enabling safe exploration with high-probability safety guarantees in large-data regimes where full GPs are computationally infeasible.

Results
The proposed approach was evaluated on a nonlinear inverted pendulum control problem.

Exploration Performance: In dynamic exploration scenarios, the DTF-GP achieved cumulative mutual information (a measure of exploration efficiency) comparable to that of a full GP across various dataset sizes. This indicates that the finite-dimensional approximation does not significantly degrade the ability to learn the system dynamics when the truncation radius is chosen to meet a specific projection error target.
Computational Efficiency: The DTF-GP demonstrated a significant computational advantage in large-data regimes. While the computation time for full GP scales with the number of data points ( $O(N^2)$ $O (N^{2})$ for updates, $O(N^3)$ $O (N^{3})$ for likelihood evaluation), the DTF-GP scales with the number of features ( $M$ $M$ ), which is independent of the dataset size once the truncation is set.
- In static exploration experiments, the DTF-GP showed almost no dependence on the number of initial samples, whereas the full GP's runtime increased significantly.
- The DTF-GP maintained exploration performance comparable to the full GP while offering reduced computational costs, particularly beneficial for noise-rich or high-confidence-level settings requiring large datasets.

Significance and Claims
The paper claims to close the gap between scalable GP regression and the rigorous safety requirements of learning-based MPC. By providing a scalable framework that admits the derivation of uniform uncertainty bounds, the DTF-GP enables the deployment of safe learning-based controllers in scenarios with large datasets where exact GP inference is impractical. The authors emphasize that their method retains the theoretical safety guarantees of full GPs (high-probability uniform bounds) while offering the computational tractability of finite-dimensional approximations. The work is presented as a step toward making safe, data-driven control feasible for complex systems with significant model mismatch and large data availability.

Technical Summary: Scalable Gaussian Process Regression via Deterministic Trigonometric Features

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