The surjectivity of g as a map from the set of n x n positive-diagonal
lower-triangular matrices to the set of SPD matrices follows immediately from
executing the Cholesky factorization algorithm on an SPD matrix A to produce a
positive-diagonal lower-triangular matrix L such that A = L @ L.T.
To prove the injectivity of g, suppose that L_1 and L_2 are lower-triangular
with positive diagonals and satisfy A = L_1 @ L_1.T = L_2 @ L_2.T. Then
inv(L_1) @ A @ inv(L_1).T = [inv(L_1) @ L_2] @ [inv(L_1) @ L_2].T = I.
Setting L_3 := inv(L_1) @ L_2, that L_3 is a positive-diagonal
lower-triangular matrix follows from inv(L_1) being positive-diagonal
lower-triangular (which follows from the diagonal of a triangular matrix being
its spectrum), and that the product of two positive-diagonal lower-triangular
matrices is another positive-diagonal lower-triangular matrix.
A simple inductive argument (proceeding one column of L_3 at a time) shows
that, if I = L_3 @ L_3.T, with L_3 being lower-triangular with positive-
diagonal, then L_3 = I. Thus, L_1 = L_2, proving injectivity of g.
Tensor. The input to the "forward" Jacobian determinant evaluation.
event_ndims
Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.forward_min_event_ndims. The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event,
i.e. it has shape x.shape.ndims - event_ndims dimensions.
name
The name to give this op.
Returns
Tensor, if this bijector is injective.
If not injective this is not implemented.
Raises
TypeError
if self.dtype is specified and y.dtype is not
self.dtype.
NotImplementedError
if neither _forward_log_det_jacobian
nor {_inverse, _inverse_log_det_jacobian} are implemented, or
this is a non-injective bijector.
Note that forward_log_det_jacobian is the negative of this function,
evaluated at g^{-1}(y).
Args
y
Tensor. The input to the "inverse" Jacobian determinant evaluation.
event_ndims
Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.inverse_min_event_ndims. The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event,
i.e. it has shape y.shape.ndims - event_ndims dimensions.
name
The name to give this op.
Returns
Tensor, if this bijector is injective.
If not injective, returns the tuple of local log det
Jacobians, log(det(Dg_i^{-1}(y))), where g_i is the restriction
of g to the ith partition Di.
Raises
TypeError
if self.dtype is specified and y.dtype is not
self.dtype.