The Amari-alpha Csiszar-function in log-space

Source: R/vi-functions.R

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

vi_amari_alpha(logu, alpha = 1, self_normalized = FALSE, name = NULL)

Arguments

logu

float-like Tensor representing log(u) from above.
alpha

float-like scalar.
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

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

Details

When self_normalized = TRUE, the Amari-alpha Csiszar-function is:

f(u) = { -log(u) + (u - 1)},     alpha = 0
       { u log(u) - (u - 1)},    alpha = 1
       { ((u^alpha - 1) - alpha (u - 1) / (alpha (alpha - 1))},    otherwise

When self_normalized = FALSE the (u - 1) terms are omitted.

Warning: when alpha != 0 and/or self_normalized = True this function makes non-log-space calculations and may therefore be numerically unstable for |logu| >> 0.

References

  • A. Cichocki and S. Amari. "Families of Alpha-Beta-and GammaDivergences: Flexible and Robust Measures of Similarities." Entropy, vol. 12, no. 6, pp. 1532-1568, 2010.

See also

Other vi-functions: vi_arithmetic_geometric(), 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()