GPy/GPy/kern/_src/mlp.py

# Copyright (c) 2013, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)

from .kern import Kern
from ...core.parameterization import Param
from ...core.parameterization.transformations import Logexp
import numpy as np
four_over_tau = 2./np.pi

class MLP(Kern):
    """

    Multi layer perceptron kernel (also known as arc sine kernel or neural network kernel)

    .. math::

          k(x,y) = \\sigma^{2}\\frac{2}{\\pi }  \\text{asin} \\left ( \\frac{ \\sigma_w^2 x^\\top y+\\sigma_b^2}{\\sqrt{\\sigma_w^2x^\\top x + \\sigma_b^2 + 1}\\sqrt{\\sigma_w^2 y^\\top y \\sigma_b^2 +1}} \\right )


    :param input_dim: the number of input dimensions
    :type input_dim: int
    :param variance: the variance :math:`\sigma^2`
    :type variance: float
    :param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\sigma^2_w`
    :type weight_variance: array or list of the appropriate size (or float if there is only one weight variance parameter)
    :param bias_variance: the variance of the prior over bias parameters :math:`\sigma^2_b`
    :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter \sigma^2_w), otherwise there is one weight variance parameter per dimension.
    :type ARD: Boolean
    :rtype: Kernpart object


    """

    def __init__(self, input_dim, variance=1., weight_variance=1., bias_variance=100., active_dims=None, name='mlp'):
        super(MLP, self).__init__(input_dim, active_dims, name)
        self.variance = Param('variance', variance, Logexp())
        self.weight_variance = Param('weight_variance', weight_variance, Logexp())
        self.bias_variance = Param('bias_variance', bias_variance, Logexp())
        self.link_parameters(self.variance, self.weight_variance, self.bias_variance)


    def K(self, X, X2=None):
        self._K_computations(X, X2)
        return self.variance*self._K_dvar

    def Kdiag(self, X):
        """Compute the diagonal of the covariance matrix for X."""
        self._K_diag_computations(X)
        return self.variance*self._K_diag_dvar

    def update_gradients_full(self, dL_dK, X, X2=None):
        """Derivative of the covariance with respect to the parameters."""
        self._K_computations(X, X2)
        self.variance.gradient = np.sum(self._K_dvar*dL_dK)

        denom3 = self._K_denom**3
        base = four_over_tau*self.variance/np.sqrt(1-self._K_asin_arg*self._K_asin_arg)
        base_cov_grad = base*dL_dK

        if X2 is None:
            vec = np.diag(self._K_inner_prod)
            self.weight_variance.gradient = ((self._K_inner_prod/self._K_denom
                           -.5*self._K_numer/denom3
                           *(np.outer((self.weight_variance*vec+self.bias_variance+1.), vec)
                             +np.outer(vec,(self.weight_variance*vec+self.bias_variance+1.))))*base_cov_grad).sum()
            self.bias_variance.gradient = ((1./self._K_denom
                           -.5*self._K_numer/denom3
                           *((vec[None, :]+vec[:, None])*self.weight_variance
                           +2.*self.bias_variance + 2.))*base_cov_grad).sum()
        else:
            vec1 = (X*X).sum(1)
            vec2 = (X2*X2).sum(1)
            self.weight_variance.gradient = ((self._K_inner_prod/self._K_denom
                           -.5*self._K_numer/denom3
                           *(np.outer((self.weight_variance*vec1+self.bias_variance+1.), vec2) + np.outer(vec1, self.weight_variance*vec2 + self.bias_variance+1.)))*base_cov_grad).sum()
            self.bias_variance.gradient = ((1./self._K_denom
                           -.5*self._K_numer/denom3
                           *((vec1[:, None]+vec2[None, :])*self.weight_variance
                             + 2*self.bias_variance + 2.))*base_cov_grad).sum()

    def update_gradients_diag(self, dL_dKdiag, X):
        self._K_diag_computations(X)
        self.variance.gradient = np.sum(self._K_diag_dvar*dL_dKdiag)
        
        base = four_over_tau*self.variance/np.sqrt(1-self._K_diag_asin_arg*self._K_diag_asin_arg)
        base_cov_grad = base*dL_dKdiag/np.square(self._K_diag_denom)
        
        self.weight_variance.gradient = (base_cov_grad*np.square(X).sum(axis=1)).sum()
        self.bias_variance.gradient = base_cov_grad.sum()

    def gradients_X(self, dL_dK, X, X2):
        """Derivative of the covariance matrix with respect to X"""
        self._K_computations(X, X2)
        arg = self._K_asin_arg
        numer = self._K_numer
        denom = self._K_denom
        denom3 = denom*denom*denom
        if X2 is not None:
            vec2 = (X2*X2).sum(1)*self.weight_variance+self.bias_variance + 1.
            return four_over_tau*self.weight_variance*self.variance*((X2[None, :, :]/denom[:, :, None] - vec2[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)
        else:
            vec = (X*X).sum(1)*self.weight_variance+self.bias_variance + 1.
            return 2*four_over_tau*self.weight_variance*self.variance*((X[None, :, :]/denom[:, :, None] - vec[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)

    def gradients_X_diag(self, dL_dKdiag, X):
        """Gradient of diagonal of covariance with respect to X"""
        self._K_diag_computations(X)
        arg = self._K_diag_asin_arg
        denom = self._K_diag_denom
        #numer = self._K_diag_numer
        return four_over_tau*2.*self.weight_variance*self.variance*X*(1./denom*(1. - arg)*dL_dKdiag/(np.sqrt(1-arg*arg)))[:, None]


    def _K_computations(self, X, X2):
        """Pre-computations for the covariance matrix (used for computing the covariance and its gradients."""
        if X2 is None:
            self._K_inner_prod = np.dot(X,X.T)
            self._K_numer = self._K_inner_prod*self.weight_variance + self.bias_variance
            vec = np.diag(self._K_numer) + 1.
            self._K_denom = np.sqrt(np.outer(vec,vec))
        else:
            self._K_inner_prod = np.dot(X,X2.T)
            self._K_numer = self._K_inner_prod*self.weight_variance + self.bias_variance
            vec1 = (X*X).sum(1)*self.weight_variance + self.bias_variance + 1.
            vec2 = (X2*X2).sum(1)*self.weight_variance + self.bias_variance + 1.
            self._K_denom = np.sqrt(np.outer(vec1,vec2))
        self._K_asin_arg = self._K_numer/self._K_denom
        self._K_dvar = four_over_tau*np.arcsin(self._K_asin_arg)

    def _K_diag_computations(self, X):
        """Pre-computations concerning the diagonal terms (used for computation of diagonal and its gradients)."""
        self._K_diag_numer = (X*X).sum(1)*self.weight_variance + self.bias_variance
        self._K_diag_denom = self._K_diag_numer+1.
        self._K_diag_asin_arg = self._K_diag_numer/self._K_diag_denom
        self._K_diag_dvar = four_over_tau*np.arcsin(self._K_diag_asin_arg)