Y = g(X) = exp([X 0]) / sum(exp([X 0]))To implement softmax as a bijection, the forward transformation appends a value to the input and the inverse removes this coordinate. The appended coordinate represents a pivot, e.g., softmax(x) = exp(x-c) / sum(exp(x-c)) where c is the implicit last coordinate.
tfb_softmax_centered(validate_args = FALSE, name = "softmax_centered")a bijector instance.
Logical, default FALSE. Whether to validate input with asserts. If validate_args is FALSE, and the inputs are invalid, correct behavior is not guaranteed.
name prefixed to Ops created by this class.
At first blush it may seem like the Invariance of domain theorem implies this implementation is not a bijection. However, the appended dimension makes the (forward) image non-open and the theorem does not directly apply.
For usage examples see tfb_forward(), tfb_inverse(), tfb_inverse_log_det_jacobian().
Other bijectors:
tfb_absolute_value(),
tfb_affine_linear_operator(),
tfb_affine_scalar(),
tfb_affine(),
tfb_ascending(),
tfb_batch_normalization(),
tfb_blockwise(),
tfb_chain(),
tfb_cholesky_outer_product(),
tfb_cholesky_to_inv_cholesky(),
tfb_correlation_cholesky(),
tfb_cumsum(),
tfb_discrete_cosine_transform(),
tfb_expm1(),
tfb_exp(),
tfb_ffjord(),
tfb_fill_scale_tri_l(),
tfb_fill_triangular(),
tfb_glow(),
tfb_gompertz_cdf(),
tfb_gumbel_cdf(),
tfb_gumbel(),
tfb_identity(),
tfb_inline(),
tfb_invert(),
tfb_iterated_sigmoid_centered(),
tfb_kumaraswamy_cdf(),
tfb_kumaraswamy(),
tfb_lambert_w_tail(),
tfb_masked_autoregressive_default_template(),
tfb_masked_autoregressive_flow(),
tfb_masked_dense(),
tfb_matrix_inverse_tri_l(),
tfb_matvec_lu(),
tfb_normal_cdf(),
tfb_ordered(),
tfb_pad(),
tfb_permute(),
tfb_power_transform(),
tfb_rational_quadratic_spline(),
tfb_rayleigh_cdf(),
tfb_real_nvp_default_template(),
tfb_real_nvp(),
tfb_reciprocal(),
tfb_reshape(),
tfb_scale_matvec_diag(),
tfb_scale_matvec_linear_operator(),
tfb_scale_matvec_lu(),
tfb_scale_matvec_tri_l(),
tfb_scale_tri_l(),
tfb_scale(),
tfb_shifted_gompertz_cdf(),
tfb_shift(),
tfb_sigmoid(),
tfb_sinh_arcsinh(),
tfb_sinh(),
tfb_softplus(),
tfb_softsign(),
tfb_split(),
tfb_square(),
tfb_tanh(),
tfb_transform_diagonal(),
tfb_transpose(),
tfb_weibull_cdf(),
tfb_weibull()