The Arithmetic-Geometric Csiszar-function in log-space

A Csiszar-function is a member of F = { f:R_+ to R : f convex }.

vi_arithmetic_geometric(logu, self_normalized = FALSE, name = NULL)

Arguments

logu	float-like Tensor representing log(u) from above.
self_normalized	logical indicating whether f'(u=1)=0. When f'(u=1)=0 the implied Csiszar f-Divergence remains non-negative even when p, q are unnormalized measures.
name	name prefixed to Ops created by this function.

Value

arithmetic_geometric_of_u: float-like Tensor of the Csiszar-function evaluated at u = exp(logu).

Details

When self_normalized = True the Arithmetic-Geometric Csiszar-function is:

f(u) = (1 + u) log( (1 + u) / sqrt(u) ) - (1 + u) log(2)

When self_normalized = False the (1 + u) log(2) term is omitted.

Observe that as an f-Divergence, this Csiszar-function implies:

D_f[p, q] = KL[m, p] + KL[m, q]
m(x) = 0.5 p(x) + 0.5 q(x)

In a sense, this divergence is the "reverse" of the Jensen-Shannon f-Divergence. This Csiszar-function induces a symmetric f-Divergence, i.e., D_f[p, q] = D_f[q, p].

Warning: when self_normalized = Truethis function makes non-log-space calculations and may therefore be numerically unstable for|logu| >> 0`.

See also

Other vi-functions: vi_amari_alpha(), vi_chi_square(), vi_csiszar_vimco(), vi_dual_csiszar_function(), vi_fit_surrogate_posterior(), vi_jeffreys(), vi_jensen_shannon(), vi_kl_forward(), vi_kl_reverse(), vi_log1p_abs(), vi_modified_gan(), vi_monte_carlo_variational_loss(), vi_pearson(), vi_squared_hellinger(), vi_symmetrized_csiszar_function()