Source code for gpytorch.kernels.piecewise_polynomial_kernel

import torch

from .kernel import Kernel


[docs]class PiecewisePolynomialKernel(Kernel): r""" Computes a covariance matrix based on the Piecewise Polynomial kernel between inputs :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`: .. math:: \begin{align} r &= \left\Vert x1 - x2 \right\Vert \\ j &= \lfloor \frac{D}{2} \rfloor + q +1 \\ K_{\text{ppD, 0}}(\mathbf{x_1}, \mathbf{x_2}) &= (1-r)^j_+ , \\ K_{\text{ppD, 1}}(\mathbf{x_1}, \mathbf{x_2}) &= (1-r)^{j+1}_+ ((j + 1)r + 1), \\ K_{\text{ppD, 2}}(\mathbf{x_1}, \mathbf{x_2}) &= (1-r)^{j+2}_+ ((1 + (j+2)r + \frac{j^2 + 4j + 3}{3}r^2), \\ K_{\text{ppD, 3}}(\mathbf{x_1}, \mathbf{x_2}) &= (1-r)^{j+3}_+ (1 + (j+3)r + \frac{6j^2 + 36j + 45}{15}r^2 + \frac{j^3 + 9j^2 + 23j +15}{15}r^3) \\ \end{align} where :math:`K_{\text{ppD, q}}` is positive semidefinite in :math:`\mathbb{R}^{D}` and :math:`q` is the smoothness coefficient. See `Rasmussen and Williams (2006)`_ Equation 4.21. .. note:: This kernel does not have an `outputscale` parameter. To add a scaling parameter, decorate this kernel with a :class:`gpytorch.kernels.ScaleKernel`. :param int q: (default= 2) The smoothness parameter. :type q: int (0, 1, 2 or 3) :param ard_num_dims: (Default: `None`) Set this if you want a separate lengthscale for each input dimension. It should be `d` if :attr:`x1` is a `... x n x d` matrix. :type ard_num_dims: int, optional :param batch_shape: (Default: `None`) Set this if you want a separate lengthscale for each batch of input data. It should be `torch.Size([b1, b2])` for a `b1 x b2 x n x m` kernel output. :type batch_shape: torch.Size, optional :param active_dims: (Default: `None`) Set this if you want to compute the covariance of only a few input dimensions. The ints corresponds to the indices of the dimensions. :type active_dims: Tuple(int) :param lengthscale_prior: (Default: `None`) Set this if you want to apply a prior to the lengthscale parameter. :type lengthscale_prior: ~gpytorch.priors.Prior, optional :param lengthscale_constraint: (Default: `Positive`) Set this if you want to apply a constraint to the lengthscale parameter. :type lengthscale_constraint: ~gpytorch.constraints.Positive, optional :param eps: (Default: 1e-6) The minimum value that the lengthscale can take (prevents divide by zero errors). :type eps: float, optional :var torch.Tensor lengthscale: The lengthscale parameter. Size/shape of parameter depends on the :attr:`ard_num_dims` and :attr:`batch_shape` arguments. .. _Rasmussen and Williams (2006): http://www.gaussianprocess.org/gpml/ Example: >>> x = torch.randn(10, 5) >>> # Non-batch option >>> covar_module = gpytorch.kernels.ScaleKernel( gpytorch.kernels.PiecewisePolynomialKernel(q = 2)) >>> # Non-batch: ARD (different lengthscale for each input dimension) >>> covar_module = gpytorch.kernels.ScaleKernel( gpytorch.kernels.PiecewisePolynomialKernel(q = 2, ard_num_dims=5) ) >>> covar = covar_module(x) # Output: LazyTensor of size (10 x 10) >>> batch_x = torch.randn(2, 10, 5) >>> # Batch: different lengthscale for each batch >>> covar_module = gpytorch.kernels.ScaleKernel( gpytorch.kernels.PiecewisePolynomialKernel(q = 2, batch_shape=torch.Size([2])) ) >>> covar = covar_module(batch_x) # Output: LazyTensor of size (2 x 10 x 10) """ has_lengthscale = True def __init__(self, q=2, **kwargs): super(PiecewisePolynomialKernel, self).__init__(**kwargs) if q not in {0, 1, 2, 3}: raise ValueError("q expected to be 0, 1, 2 or 3") self.q = q def fmax(self, r, j, q): return torch.max(torch.tensor(0.0), 1 - r).pow(j + q) def get_cov(self, r, j, q): if q == 0: return 1 if q == 1: return (j + 1) * r + 1 if q == 2: return 1 + (j + 2) * r + ((j ** 2 + 4 * j + 3) / 3.0) * r ** 2 if q == 3: return ( 1 + (j + 3) * r + ((6 * j ** 2 + 36 * j + 45) / 15.0) * r ** 2 + ((j ** 3 + 9 * j ** 2 + 23 * j + 15) / 15.0) * r ** 3 ) def forward(self, x1, x2, last_dim_is_batch=False, diag=False, **params): x1_ = x1.div(self.lengthscale) x2_ = x2.div(self.lengthscale) if last_dim_is_batch is True: D = x1.shape[1] else: D = x1.shape[-1] j = torch.floor(torch.tensor(D / 2.0)) + self.q + 1 if last_dim_is_batch and diag: r = self.covar_dist(x1_, x2_, last_dim_is_batch=True, diag=True) elif diag: r = self.covar_dist(x1_, x2_, diag=True) elif last_dim_is_batch: r = self.covar_dist(x1_, x2_, last_dim_is_batch=True) else: r = self.covar_dist(x1_, x2_) cov_matrix = self.fmax(r, j, self.q) * self.get_cov(r, j, self.q) return cov_matrix