Class ArcCosine

gpflow/kernels/misc.py:27–200

The Arc-cosine family of kernels which mimics the computation in neural networks. The order parameter specifies the assumed activation function. The Multi Layer Perceptron (MLP) kernel is closely related to the ArcCosine kernel of order 0. The key reference is :cite:t:`NIPS2009

class ArcCosine(Kernel):
    """
    The Arc-cosine family of kernels which mimics the computation in neural
    networks. The order parameter specifies the assumed activation function.
    The Multi Layer Perceptron (MLP) kernel is closely related to the ArcCosine
    kernel of order 0.

    The key reference is :cite:t:`NIPS2009_3628`.
    """

    implemented_orders = {0, 1, 2}

    @check_shapes(
        "variance: []",
        "weight_variances: [broadcast n_active_dims]",
        "bias_variance: []",
    )
    def __init__(
        self,
        order: int = 0,
        variance: TensorType = 1.0,
        weight_variances: TensorType = 1.0,
        bias_variance: TensorType = 1.0,
        *,
        active_dims: Optional[ActiveDims] = None,
        name: Optional[str] = None,
    ) -> None:
        """
        :param order: specifies the activation function of the neural network
          the function is a rectified monomial of the chosen order
        :param variance: the (initial) value for the variance parameter
        :param weight_variances: the (initial) value for the weight_variances parameter,
            to induce ARD behaviour this must be initialised as an array the same
            length as the the number of active dimensions e.g. [1., 1., 1.]
        :param bias_variance: the (initial) value for the bias_variance parameter
            defaults to 1.0
        :param active_dims: a slice or list specifying which columns of X are used
        """
        super().__init__(active_dims=active_dims, name=name)

        if order not in self.implemented_orders:
            raise ValueError("Requested kernel order is not implemented.")
        self.order = order

        self.variance = Parameter(variance, transform=positive())
        self.bias_variance = Parameter(bias_variance, transform=positive())
        self.weight_variances = Parameter(weight_variances, transform=positive())
        self._validate_ard_active_dims(self.weight_variances)

    @property
    def ard(self) -> bool:
        """
        Whether ARD behaviour is active.
        """
        ndims: int = self.weight_variances.shape.ndims
        return ndims > 0

    @check_shapes(