[Paper] Kernel Learning for Regression via Quantum Annealing Based Spectral Sampling

Source: arXiv - 2601.08724v1

Overview

This paper introduces a novel way to learn regression kernels using a quantum annealer. By treating the annealer’s noisy output as a source of stochastic samples, the authors embed it directly into the kernel‑learning pipeline, yielding data‑adaptive kernels that can outperform classic Gaussian kernels on standard regression benchmarks.

Key Contributions

• QA‑in‑the‑loop kernel learning: integrates quantum annealing (QA) as a core component for shaping the kernel rather than just a black‑box sampler.
• Spectral representation via Random Fourier Features (RFF): leverages Bochner’s theorem to express shift‑invariant kernels as expectations over a spectral distribution.
• RBM‑driven spectral distribution: models the spectral density with a (multi‑layer) Restricted Boltzmann Machine, whose discrete samples are generated by a quantum annealer and transformed into continuous Fourier frequencies.
• Squared‑kernel weighting: replaces the standard Nadaraya–Watson denominator with non‑negative squared kernel weights to avoid numerical instability and improve contrast.
• Leave‑one‑out (LOO) training objective: directly minimizes the LOO mean‑squared error of the Nadaraya–Watson regressor, leading to kernels tuned for generalization.
• Empirical validation: demonstrates consistent reductions in training loss and improvements in (R^2) / RMSE across several regression datasets, with further gains when increasing the number of random features at inference time.

Methodology

1. Spectral kernel formulation – Any shift‑invariant kernel (k(\Delta x)) can be written as
   [ k(\Delta x)=\mathbb{E}_{\omega\sim p(\omega)}[\cos(\omega^\top \Delta x)] . ]
   Approximating this expectation with Monte‑Carlo samples yields Random Fourier Features (RFF).

2. Learning the spectral distribution –

   • An RBM (single‑ or multi‑layer) defines a probability distribution over binary hidden units.
   • A quantum annealer samples from this RBM at finite temperature, producing discrete binary vectors.
   • Each binary sample (\mathbf{h}) is mapped to a continuous frequency (\omega) via a Gaussian‑Bernoulli transformation (i.e., (\omega = W\mathbf{h} + b + \epsilon) with Gaussian noise (\epsilon)).

3. Constructing the kernel – Using the sampled frequencies, the RFF approximation of the kernel is built. Because (\cos(\cdot)) can be negative and cause denominator collapse in Nadaraya–Watson, the authors square the kernel values: (w_{ij}=k(x_i,x_j)^2), guaranteeing non‑negativity and sharper weighting.

4. Training objective – The kernel parameters (RBM weights, Gaussian transformation parameters) are optimized by minimizing the leave‑one‑out Nadaraya–Watson MSE, which can be computed efficiently without explicit cross‑validation loops.

5. Inference – At test time, the learned spectral distribution is used to draw many more RFF samples (the “inference‑time” feature count can be larger than during training). Both standard Nadaraya–Watson and a local linear regression variant are evaluated with the squared‑kernel weights.

Results & Findings

• Training loss consistently drops as the QA‑generated spectral distribution adapts to the data.
• Kernel matrix structure evolves from a smooth, isotropic pattern (Gaussian) to a more anisotropic, data‑aligned shape, indicating that the learned kernel captures task‑specific similarity.
• Increasing random features at inference (e.g., from 500 to 2000) yields monotonic improvements, confirming that the learned spectral distribution is robust and can be sampled more densely without retraining.

Overall, the QA‑enhanced kernels achieve higher predictive accuracy (both (R^2) and RMSE) than the standard Gaussian kernel baseline across all tested regression problems.

Practical Implications

• Quantum‑augmented ML pipelines: Developers can now consider QA hardware as a learnable component, not just a generic optimizer, opening doors to hybrid quantum‑classical models.
• Kernel‑based regression in production: The squared‑kernel weighting trick resolves a long‑standing numerical issue in Nadaraya–Watson, making the method more reliable for real‑time services (e.g., recommendation scoring, sensor calibration).
• Scalable feature generation: Because the RBM‑derived spectral distribution can be sampled arbitrarily many times at inference, the approach scales with available compute—no need to retrain the RBM for higher accuracy.
• Hardware‑agnostic design: While the paper uses D‑Wave‑style QA, any sampler that approximates a Gibbs distribution (e.g., simulated annealing, parallel tempering) could be swapped in, allowing developers to prototype on classical hardware before moving to quantum devices.

Limitations & Future Work

• Finite‑temperature and noise: The quality of the spectral samples depends on the annealer’s temperature and noise profile; poor calibration can degrade kernel learning.
• RBM depth vs. hardware constraints: Multi‑layer RBMs improve expressiveness but quickly exceed qubit connectivity limits on current QA chips.
• Benchmark scope: Experiments focus on medium‑size regression datasets; scalability to high‑dimensional or massive‑scale data remains untested.
• Future directions suggested by the authors include: exploring alternative generative models (e.g., quantum Boltzmann machines), integrating the approach with deep kernel learning frameworks, and evaluating on classification or structured‑prediction tasks where kernel adaptivity is equally valuable.

Authors

• Yasushi Hasegawa
• Masayuki Ohzeki

Paper Information

• arXiv ID: 2601.08724v1
• Categories: quant-ph, cs.LG
• Published: January 13, 2026
• PDF: Download PDF