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Lower bound on Kullback-Leibler (KL) divergence from Nguyen at al.

The lower bound was introduced by Nguyen, Wainwright, Jordan (NWJ) in [Nguyen et al. (2010)][1], and also known as f-GAN KL [(Nowozin et al., 2016)][2] and MINE-f [(Belghazi et al., 2018)][3].

I_NWJ = E_p(x,y)[f(x, y)] - 1/e * E_p(y)[Z(y)],

where f(x, y) is a critic function that scores pairs of samples (x, y), and Z(y) is the corresponding partition function:

Z(y) = E_p(x)[ exp(f(x, y)) ].


X, Y are samples from a joint Gaussian distribution, with correlation 0.8 and both of dimension 1.

batch_size, rho, dim = 10000, 0.8, 1
y, eps = tf.split(
    value=tf.random.normal(shape=(2 * batch_size, dim), seed=7),
    num_or_size_splits=2, axis=0)
mean, conditional_stddev = rho * y, tf.sqrt(1. - tf.square(rho))
x = mean + conditional_stddev * eps

# Scores/unnormalized likelihood of pairs of samples `x[i], y[j]`
# (For NWJ lower bound, the optimal critic is of the form `f(x, y) = 1 +
# log(p(x | y) / p(x))` [(Poole et al., 2018)][4]. )
conditional_dist = tfd.MultivariateNormalDiag(
    mean, scale_identity_multiplier=conditional_stddev)
conditional_scores = conditional_dist.log_prob(y[:, tf.newaxis, :])
marginal_dist = tfd.MultivariateNormalDiag(tf.zeros(dim), tf.ones(dim))
marginal_scores = marginal_dist.log_prob(y)[:, tf.newaxis]
scores = 1 + conditional_scores - marginal_scores

# Mask for joint samples in score tensor
# (The `scores` has its shape [x_batch_size, y_batch_size], i.e.
# `scores[i, j] = f(x[i], y[j]) = log p(x[i] | y[j])`.)
joint_sample_mask = tf.eye(batch_size, dtype=bool)

# Lower bound on KL divergence between p(x,y) and p(x)p(y),
# i.e. the mutual information between `X` and `Y`.
    logu=scores, joint_sample_mask=joint_sample_mask)

logu float-like Tensor of size [batch_size_1, batch_size_2] representing critic scores (scores) for pairs of points (x, y) with logu[i, j] = f(x[i], y[j]).
joint_sample_mask bool-like Tensor of the same size as logu masking the positive samples by True, i.e. samples from joint distribution p(x, y). Default value: None. By default, an identity matrix is constructed as the mask.
validate_args Python bool, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
name Python str name prefixed to Ops created by this function. Default value: None (i.e., 'lower_bound_nguyen_wainwright_jordan').

lower_bound float-like scalar for lower bound on KL divergence between joint and marginal distrbutions.


[1]: XuanLong Nguyen, Martin J. Wainwright, Michael I. Jordan. Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization. IEEE Transactions on Information Theory, 56(11):5847-5861, 2010. https://arxiv.org/abs/0809.0853 [2]: Sebastian Nowozin, Botond Cseke, Ryota Tomioka. f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. In Conference on Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.00709 [3]: Mohamed Ishmael Belghazi, et al. MINE: Mutual Information Neural Estimation. In International Conference on Machine Learning, 2018. https://arxiv.org/abs/1801.04062 [4]: Ben Poole, Sherjil Ozair, Aaron van den Oord, Alexander A. Alemi, George Tucker. On Variational Bounds of Mutual Information. In International Conference on Machine Learning, 2019. https://arxiv.org/abs/1905.06922