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Lower bound on Kullback-Leibler (KL) divergence from Nguyen at al.
tfp.vi.mutual_information.lower_bound_nguyen_wainwright_jordan( logu, joint_sample_mask=None, validate_args=False, name=None )
The lower bound was introduced by Nguyen, Wainwright, Jordan (NWJ) in
[Nguyen et al. (2010)], and also known as
f-GAN KL [(Nowozin et al.,
MINE-f [(Belghazi et al., 2018)].
I_NWJ = E_p(x,y)[f(x, y)] - 1/e * E_p(y)[Z(y)],
f(x, y) is a critic function that scores pairs of samples
Z(y) is the corresponding partition function:
Z(y) = E_p(x)[ exp(f(x, y)) ].
Y are samples from a joint Gaussian distribution, with correlation
0.8 and both of dimension
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)]. ) 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`. lower_bound_nguyen_wainwright_jordan( logu=scores, joint_sample_mask=joint_sample_mask)
: 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 : 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 : Mohamed Ishmael Belghazi, et al. MINE: Mutual Information Neural Estimation. In International Conference on Machine Learning, 2018. https://arxiv.org/abs/1801.04062 : 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