Matérn Kernel
A kernel function in Gaussian processes whose parameter ν controls the smoothness of sample paths. Smaller values of ν allow the function to vary more rapidly and are well suited to capturing local fluctuations and noise.
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The Matérn kernel controls the smoothness of sample paths through the parameter ν, and a smaller ν allows the function to vary faster and is good at capturing local fluctuations and noise; the Linear kernel assumes that the objective function is approximately linear and can model global trends and offsets.
After combining the Matérn kernel with the Linear kernel (usually by addition), the kernel function possesses both local adaptability and global trend capturing capability, showing robustness in static, drifting, and reactive environments.
Mild conditions usually refer to standard assumptions such as the kernel function being sufficiently smooth and its eigenvalues decaying at a polynomial rate, covering commonly used kernels such as the Gaussian kernel and the Matérn kernel.
The squared exponential kernel corresponds to the class of infinitely smooth functions, while the Matérn-ν kernel corresponds to the class of ν-times differentiable functions, with smaller ν yielding rougher function classes.
For the squared exponential kernel, γ_T = O((log T)^(d+1)), growing extremely slowly; for the Matérn-ν kernel, γ_T = Θ(T^(d/(2ν+d))), growing polynomially with T.
The paper proves that for the Matérn-ν kernel with parameter ν∈(0,2), the γ-exponential kernel with parameter γ∈(0,2), and certain piecewise polynomial kernels, the minimax optimal rate is precisely determined to be Θ(√(Tγ_T)), with upper and lower bounds matching within constant factors.
All Facts (6)
The Matérn kernel controls the smoothness of sample paths through the parameter ν, and a smaller ν allows the function to vary faster and is good at capturing local fluctuations and noise; the Linear kernel assumes that the objective function is approximately linear and can model global trends and offsets.
50%UnverifiedAfter combining the Matérn kernel with the Linear kernel (usually by addition), the kernel function possesses both local adaptability and global trend capturing capability, showing robustness in static, drifting, and reactive environments.
50%UnverifiedMild conditions usually refer to standard assumptions such as the kernel function being sufficiently smooth and its eigenvalues decaying at a polynomial rate, covering commonly used kernels such as the Gaussian kernel and the Matérn kernel.
50%UnverifiedThe squared exponential kernel corresponds to the class of infinitely smooth functions, while the Matérn-ν kernel corresponds to the class of ν-times differentiable functions, with smaller ν yielding rougher function classes.
50%UnverifiedFor the squared exponential kernel, γ_T = O((log T)^(d+1)), growing extremely slowly; for the Matérn-ν kernel, γ_T = Θ(T^(d/(2ν+d))), growing polynomially with T.
50%UnverifiedThe paper proves that for the Matérn-ν kernel with parameter ν∈(0,2), the γ-exponential kernel with parameter γ∈(0,2), and certain piecewise polynomial kernels, the minimax optimal rate is precisely determined to be Θ(√(Tγ_T)), with upper and lower bounds matching within constant factors.
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