Type CholeskyOuterProduct
Namespace tensorflow.contrib.distributions.bijectors
Parent Bijector
Interfaces ICholeskyOuterProduct
Compute `g(X) = X @ X.T`; X is lower-triangular, positive-diagonal matrix. Note: the upper-triangular part of X is ignored (whether or not its zero). 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. #### Examples
Show Example
bijector.CholeskyOuterProduct().forward(x=[[1., 0], [2, 1]])
# Result: [[1., 2], [2, 5]], i.e., x @ x.T bijector.CholeskyOuterProduct().inverse(y=[[1., 2], [2, 5]])
# Result: [[1., 0], [2, 1]], i.e., cholesky(y).