gpflow.kullback_leiblers#

Functions#

gpflow.kullback_leiblers.gauss_kl#

gpflow.kullback_leiblers.gauss_kl(q_mu, q_sqrt, K=None, *, K_cholesky=None)[source]#

Compute the KL divergence KL[q || p] between:

q(x) = N(q_mu, q_sqrt^2)

and:

p(x) = N(0, K)    if K is not None
p(x) = N(0, I)    if K is None

We assume L multiple independent distributions, given by the columns of q_mu and the first or last dimension of q_sqrt. Returns the sum of the divergences.

q_mu is a matrix ([M, L]), each column contains a mean.

q_sqrt can be a 3D tensor ([L, M, M]), each matrix within is a lower triangular square-root matrix of the covariance of q.
q_sqrt can be a matrix ([M, L]), each column represents the diagonal of a square-root matrix of the covariance of q.

K is the covariance of p (positive-definite matrix). The K matrix can be passed either directly as K, or as its Cholesky factor, K_cholesky. In either case, it can be a single matrix [M, M], in which case the sum of the L KL divergences is computed by broadcasting, or L different covariances [L, M, M].

Note: if no K matrix is given (both K and K_cholesky are None), gauss_kl computes the KL divergence from p(x) = N(0, I) instead.

Parameters:

q_mu (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) –
- q_mu has shape [M, L].
q_sqrt (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) –
- q_sqrt has shape [M_L_or_L_M_M…].
K (Union[ndarray[Any, Any], Tensor, Variable, Parameter, None]) –
- K has shape [broadcast L, M, M].
K_cholesky (Union[ndarray[Any, Any], Tensor, Variable, Parameter, None]) –
- K_cholesky has shape [broadcast L, M, M].

Return type:

Tensor

Returns:

return has shape [].