Quantum-Assisted Support Vector Regression

TL;DR

This work introduces a quantum-assisted support vector regression (SVR) framework that uses simulated and hybrid quantum-classical annealing to train SVR models for facial-landmark detection (FLD) on small datasets. By formulating the SVR training as a QUBO and solving it with D-Wave's Leap Hybrid Solver (and simulated annealing as a baseline), the authors demonstrate QA-SVR alongside SA-SVR and SKL-SVR in a 2L single-output regression FLD setup with Gaussian kernels. Across LFW, LFPW, and BioID datasets, QA-SVR achieves comparable accuracy to classical approaches and, in some cases, lower variance due to ensemble-like averaging, while offering substantial speedups over pure SA, though not outperforming classic gradient-based SVR in raw training time. The study serves as a proof-of-concept for applying quantum-assisted SVR to real-world supervised learning tasks with limited data and outlines practical considerations for future quantum-enhanced regression. Potential improvements include larger feature sets, refined annealing hyperparameters, and experiments on larger quantum hardware to probe true quantum advantages.

Abstract

A popular machine-learning model for regression tasks, including stock-market prediction, weather forecasting and real-estate pricing, is the classical support vector regression (SVR). However, a practically realisable quantum SVR remains to be formulated. We devise annealing-based algorithms, namely simulated and quantum-classical hybrid, for training two SVR models and compare their empirical performances against the SVR implementation of Python's scikit-learn package for facial-landmark detection (FLD), a particular use case for SVR. Our method is to derive a quadratic-unconstrained-binary formulation for the optimisation problem used for training a SVR model and solve this problem using annealing. Using D-Wave's hybrid solver, we construct a quantum-assisted SVR model, thereby demonstrating a slight advantage over classical models regarding FLD accuracy. Furthermore, we observe that annealing-based SVR models predict landmarks with lower variances compared to the SVR models trained by gradient-based methods. Our work is a proof-of-concept example for applying quantum-assisted SVR to a supervised-learning task with a small training dataset.

Figures (8)

• Figure 1: An example image from the LFW database HRBL07. Using data from Ref. SWT13web, we show the five landmarks with manually determined positions, labels 1--5, and the face box as a red rectangle.
• Figure 2: Performance comparison between simulated annealing and quantum annealing for a non-linear regression problem. The true values are generated by adding random noise to a sinusoidal function. We fix $\varepsilon=0.1$, $B=4$, $B_\text{f}=0$, $\lambda=1$ and $\eta=0.125$. Quantum annealing is run on D-Wave's Advantage_system4.1 with $0.5~\mu$s annealing time and 1000 repetitions, whereas simulated annealing uses 1000 sweeps and 1000 repetitions (\ref{['appendix:workflow']}). RMSE is the root-mean-squared error between true and predicted values.
• Figure 3: Predictions from QA-landmark model: actual landmark positions SWT13web (in blue) and predicted landmark positions (in green) for a LFW test image HRBL07.
• Figure 4: Performance comparison between classical and quantum-classical hybrid optimisation techniques. (a) For each model, namely SKL-landmark, SA-landmark and QA-landmark, and each landmark, we show as barplots the average (in %) of five MNDEs and the average (in %) of five FRs, which are obtained from the 5-fold cross validation on 125 LFW images. (b) For each pair of optimisers ($X$ vs $Y$) we plot data points ($O_X,O_Y$), where $O_X$ and $O_Y$ are the objective values obtained by the optimisers $X$ and $Y$, respectively. There are 1000 data points corresponding to 20 different SVR models for each landmark coordinate and each fold. The black diagonal line represents $O_X=O_Y$.
• Figure 5: Performance comparison between SKL-landmark, SA-landmark and QA-landmark. The training dataset for (a) and (b) is the training set obtained in the $5^\text{th}$ fold of our 5-fold cross validation. For each landmark and the aggregate of all five landmarks, we plot the MNDE (in %) and FR (in %) for the test dataset comprising 164 LFPW images in (a) and for the test dataset comprising 1341 BioID images in (b).