Exploiting Hankel-Toeplitz Structures for Fast Computation of Kernel Precision Matrices (2408.02346v1)

Abstract: The Hilbert-space Gaussian Process (HGP) approach offers a hyperparameter-independent basis function approximation for speeding up Gaussian Process (GP) inference by projecting the GP onto M basis functions. These properties result in a favorable data-independent $\mathcal{O}(M^3)$ computational complexity during hyperparameter optimization but require a dominating one-time precomputation of the precision matrix costing $\mathcal{O}(NM^2)$ operations. In this paper, we lower this dominating computational complexity to $\mathcal{O}(NM)$ with no additional approximations. We can do this because we realize that the precision matrix can be split into a sum of Hankel-Toeplitz matrices, each having $\mathcal{O}(M)$ unique entries. Based on this realization we propose computing only these unique entries at $\mathcal{O}(NM)$ costs. Further, we develop two theorems that prescribe sufficient conditions for the complexity reduction to hold generally for a wide range of other approximate GP models, such as the Variational Fourier Feature (VFF) approach. The two theorems do this with no assumptions on the data and no additional approximations of the GP models themselves. Thus, our contribution provides a pure speed-up of several existing, widely used, GP approximations, without further approximations.

• The paper reduces precomputation from O(NM²) to O(NM) by exploiting Hankel–Toeplitz properties in Gaussian Process models.
• It leverages two theorems to universally decompose kernel precision matrices, ensuring robust application across various GP approximations.
• Experimental results validate significant computational gains in hyperparameter optimization without compromising model integrity.

Exploiting Hankel--Toeplitz Structures for Fast Computation of Kernel Precision Matrices

In the field of Gaussian Process (GP) modeling, improving computational efficiency remains a pivotal challenge. The paper "Exploiting Hankel--Toeplitz Structures for Fast Computation of Kernel Precision Matrices" by Frida Viset et al. addresses this issue by introducing a method that significantly reduces the computational complexity associated with hyperparameter optimization in GP inference, without introducing additional approximations or compromising the model's integrity.

Overview of Hyperparameter-Independent GP Approximation

The traditional Hyperparameter-Independent Basis Function Approximation (HGP) for accelerating GP inference leverages a projection of the GP onto $M$ basis functions. This results in a GP approximation with a computational complexity of $\mathcal{O}(M^3)$ for hyperparameter optimization. While this approach simplifies GP inference, it requires a one-time precomputation of the precision matrix, incurring an $\mathcal{O}(NM^2)$ cost. This precomputation stage becomes a bottleneck, particularly for large-scale datasets.

Reduction of Computational Complexity

The authors propose a novel method to reduce this precomputation complexity from $\mathcal{O}(NM^2)$ to $\mathcal{O}(NM)$. This improvement is achieved by recognizing that the precision matrix can be decomposed into a sum of Hankel--Toeplitz matrices, each characterized by only $\mathcal{O}(M)$ unique entries. Consequently, by focusing computational efforts only on these unique elements, the overall computational burden is significantly alleviated.

Theoretical Contributions

Two pivotal theorems underpin the robustness of this approach. These theorems establish sufficient conditions under which the outlined complexity reduction can be generally applied to a broad spectrum of approximate GP models, including the Variational Fourier Features (VFF) approach:

1. Condition for Hankel–Toeplitz Matrix Decomposition: The first theorem provides a framework to ensure that the precision matrix of various GP approximations can indeed be expressed as a sum of Hankel–Toeplitz matrices.
2. Universality of Computational Reduction: The second theorem affirms that this decomposition, and consequently the computational reduction, holds true under general conditions that require no assumptions on the underlying data and no further approximations of the GP models.

Experimental Validation

The experimental results substantiate the theoretical claims by demonstrating the computational gains on multiple datasets and scenarios. The reduction in computational complexity not only accelerates GP inference but does so while maintaining high fidelity in the results. This pure speed-up can be particularly advantageous for real-time applications and large-scale data scenarios where computational resources are a limiting factor.

Practical and Theoretical Implications

From a practical standpoint, the reductions in computational complexity proposed in this paper have significant implications for deploying GP models in resource-constrained environments. The approach allows for more efficient hyperparameter optimization, making it feasible to apply GP models to larger datasets than previously possible.

Theoretically, the paper extends the frontier of kernel methods by exploiting matrix structures inherent in the problem. The two theorems presented provide a robust foundation that could inspire further research into exploiting other matrix structures within machine learning algorithms to achieve similar computational benefits.

Future Developments

Looking forward, the principles elucidated in this paper could be extended to other areas within machine learning where large-scale kernel matrices pose computational challenges. There is potential for integrating these methods into standard GP libraries, enhancing accessibility and usability for practitioners. Further exploration might also consider the parallelization of these computations, pushing the boundaries of what is achievable with GP models in both speed and scale.

Conclusion

This paper offers a significant contribution to the computational efficiency of GP inference by leveraging the structural properties of Hankel–Toeplitz matrices. By reducing the precomputation complexity from $\mathcal{O}(NM^2)$ to $\mathcal{O}(NM)$, the authors provide an approach that benefits a wide range of approximate GP models without introducing additional approximations. This work not only advances the state-of-the-art in hyperparameter-independent GP approximations but also opens avenues for future research focused on exploiting matrix structures for computational gains.