Harnessing Multilevel Circulant Matrices for Generalizable Spectral Kernel Learning

Area of Science:

• Machine Learning
• Kernel Methods
• Signal Processing

Background:

• Kernel methods are crucial for data analysis, embedding distributions into reproducing kernel Hilbert spaces (RKHS).
• Learning generalizable kernels for high-dimensional, heterogeneous data with distribution shifts remains a significant challenge.
• Existing methods struggle with temporal or regional distribution shifts in complex datasets.

Purpose of the Study:

• To introduce SpectraMancer, a novel approach for learning generalizable kernels directly in the Fourier spectral domain.
• To address the limitations of current kernel learning methods in handling complex data and distribution shifts.
• To enable efficient and robust kernel learning for diverse applications.

Main Methods:

• SpectraMancer utilizes multilevel circulant matrices to embed shift-invariant kernel candidates into a common multilevel order.
• It employs randomized multilevel circulant matrices for fixed Fourier diagonalization, enabling elementwise Fast Fourier Transform (FFT) operations.
• The method incorporates kernel double centering, Frobenius normalization for scale invariance, and antithetic phase pairing to reduce spectral variance.

Main Results:

• SpectraMancer demonstrates the first kernel-learning approach leveraging randomized multilevel circulant matrices for joint diagonalization.
• The proposed spectral risk proxy (SRP) optimizes bandwidth weighting efficiently without repeated inner solves.
• Experimental results confirm improved spectrum-aware kernel selection and predictive performance on diverse benchmarks.

Conclusions:

• SpectraMancer offers a powerful new framework for generalizable kernel learning in the Fourier spectral domain.
• The method effectively handles high-dimensional, heterogeneous data and distribution shifts.
• SpectraMancer advances kernel methods by providing a scalable and efficient solution for complex machine learning tasks.