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InfoNCE lower bound on mutual information.

InfoNCE lower bound is proposed in [van den Oord et al. (2018)][1] based on noise contrastive estimation (NCE).

I(X; Y) >= 1/K sum(i=1:K, log( p_joint[i] / p_marginal[i])),

where the numerator and the denominator are, respectively,

p_joint[i] = p(x[i] | y[i]) = exp( f(x[i], y[i]) ),
p_marginal[i] = 1/K sum(j=1:K, p(x[i] | y[j]) )
              = 1/K sum(j=1:K, exp( f(x[i], y[j]) ) ),

and (x[i], y[i]), i=1:K are samples from joint distribution p(x, y). Pairs of points (x, y) are scored using a critic function f.


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

# Conditional distribution of p(x|y)
conditional_dist = tfd.MultivariateNormalDiag(
    mean, scale_identity_multiplier=conditional_stddev)

# Scores/unnormalized likelihood of pairs of samples `x[i], y[j]`
# (The scores has its shape [x_batch_size, distibution_batch_size]
# as the `lower_bound_info_nce` requires `scores[i, j] = f(x[i], y[j])
# = log p(x[i] | y[j])`.)
scores = conditional_dist.log_prob(x[:, tf.newaxis, :])

# Mask for joint samples
joint_sample_mask = tf.eye(batch_size, dtype=bool)

# InfoNCE lower bound on mutual information
lower_bound_info_nce(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_info_nce').

lower_bound float-like scalar for lower bound on mutual information.


[1]: Aaron van den Oord, Yazhe Li, Oriol Vinyals. Representation Learning with Contrastive Predictive Coding. arXiv preprint arXiv:1807.03748, 2018. https://arxiv.org/abs/1807.03748