normalkl 0.1.2

A PyTorch package for computing KL divergences between normal distributions.

normalkl: KL divergences between normal distributions, without the stress

A PyTorch package for computing KL divergences between multivariate normal distributions.

Fully unit tested, so you don't have to worry about making sign errors.

How to use?

The function kl can be used to compute regular KL divergence between two normal distributions.

from normalkl import kl

mean1 = torch.tensor([4.0, 5.0])
covariance1 = torch.tensor([[1.0, 1.0], [2.0, 4.0]])
mean2 = torch.tensor([1.0, 2.0])
scalarvar2 = torch.tensor([3.0])

kl_div = kl(mean1, 'covmat', covariance1, mean2, 'scalarvar', scalarvar2)
print(kl_div)

Installation

Package available in pip:

pip install normalkl

Auto KL

The auto_kl computes KL divergence with prior variance automatically chosen such that the KL is minimized.

from normalkl import auto_kl
kl_div2 = auto_kl(mean1, 'cov', covariance1, mean2)

We can also separately compute the optimal prior variance using optimal_covariance,

from normalkl import kl, optimal_covariance, auto_kl

optimal_scalarvar = optimal_covariance(mean1, 'cov', cov1, mean2, 'scalarvar')
kl_div1 = kl(mean1, 'cov', covariance1, mean2, 'scalarvar', optimal_scalarvar)

kl_div2 = auto_kl(mean1, 'cov', covariance1, mean2)

print(kl_div1 == kl_div2) # True

Covariance types

For the first distribution (e.g. variational posterior), we support regular normal and Cholesky Kronecker Covariance (= matrix normal) parameterizations. For the second distribution (e.g. prior distribution), the following full, Kronecker and isotropic priors are supported.

Covariance Type	Abbreviation	Mathematical Formula	Expected Type (Shape)	cov1	cov2
Full Covariance Matrix	covmat	$\Sigma$	Full matrix (PSD), shape: $(D, D)$	✓	✓
Full Precision Matrix	precmat	$\Sigma^{-1}$	Full matrix (PSD), shape: $(D, D)$		✓
Diagonal Covariance Matrix (Vector)	diagvar	$\text{diag}(\mathbf{d})$, where $\mathbf{d}$ is a vector	Vector, shape: $(D,)$		✓
Diagonal Precision Matrix (Vector)	diagprec	$\text{diag}(\mathbf{d})^{-1}$, where $\mathbf{d}$ is a vector	Vector, shape: $(D,)$		✓
Scalar Variance	scalarvar	$\sigma^2 \mathbf{I}$, where $\sigma^2$ is a scalar	Scalar, shape: $(1,)$		✓
Scalar Precision	scalarprec	$\tau^{-1} \mathbf{I}$, where $\tau^{-1}$ is a scalar	Scalar, shape: $(1,)$		✓
Identity Matrix	identity	$\mathbf{I}$, the identity matrix	Flag or Boolean, shape: $(D, D)$		✓
Cholesky of Covariance Matrix	cholcov	$\mathbf{L}$, where $\Sigma = \mathbf{L}\mathbf{L}^\top$	Lower triangular matrix, shape: $(D, D)$		✓
Cholesky of Precision Matrix	cholprec	$\mathbf{L}$, where $\Sigma^{-1} = $\mathbf{L}\mathbf{L}^\top$	Lower triangular matrix, shape: $(D, D)$		✓
Kronecker-Factored Covariance Matrix	kroncov	$\Sigma = \mathbf{A} \otimes \mathbf{B}$	Pair of matrices, shapes: $(D_1, D_1)$, $(D_2, D_2)$		✓
Kronecker-Factored Precision Matrix	kronprec	$\Sigma^{-1} = \mathbf{A} \otimes \mathbf{B}$	Pair of matrices, shapes: $(D_1, D_1)$, $(D_2, D_2)$		✓
Cholesky Kronecker Covariance	cholkroncov	$\Sigma = (\mathbf{L}_A \mathbf{L}_A^\top \otimes \mathbf{L}_B \mathbf{L}_B^\top)$	Pair of lower triangular matrices, shapes: $(D_1, D_1)$, $(D_2, D_2)$	✓	✓
Cholesky Kronecker Precision	cholkronprec	$\Sigma^{-1} = (\mathbf{L}_A \mathbf{L}_A^\top \otimes \mathbf{L}_B \mathbf{L}_B^\top)$	Pair of lower triangular matrices, shapes: $(D_1, D_1)$, $(D_2, D_2)$		✓
Diagonal row+column variances	diagcovrowcol	$\text{diag}(\text{vec}(\mathbf{b} \mathbf{a}^T))$	Pair of vectors, shapes: $(D_1,)$, $(D_2,)$		✓
Diagonal row variances	diagcovrow	$\text{diag}(\text{vec}(\mathbf{1} \mathbf{a}^T))$	Vector, shape: $(D_1,)$		✓
Diagonal column variances	diagcovcol	$\text{diag}(\text{vec}(\mathbf{b} \mathbf{1}^T))$	Vector, shape: $(D_2,)$		✓

Optimal prior variances

Analytically optimal cov2 variances are available for the following types:

Type of cov1 \ Type of optimized cov2	covmat	precmat	diagvar	diagprec	scalarvar	scalarprec	identity	cholcov	cholprec	kroncov	kronprec	cholkroncov	cholkronprec	diagcovrowcol	diagcovrow	diagcovcol
covmat	✓	✓	✓	✓	✓	✓
precmat	✓	✓
diagvar
diagprec
scalarvar
scalarprec
identity
cholcov
cholprec
kroncov
kronprec
cholkroncov	✓	✓	✓	✓	✓	✓
cholkronprec
diagcovrowcol
diagcovrow
diagcovcol