View source on GitHub |
Concatenates chol @ chol.T
with additional rows and columns.
tfp.substrates.numpy.math.cholesky_concat(
chol, cols, name=None
)
This operation is conceptually identical to:
def cholesky_concat_slow(chol, cols): # cols shaped (n + m) x m = z x m
mat = tf.matmul(chol, chol, adjoint_b=True) # shape of n x n
# Concat columns.
mat = tf.concat([mat, cols[..., :tf.shape(mat)[-2], :]], axis=-1) # n x z
# Concat rows.
mat = tf.concat([mat, tf.linalg.matrix_transpose(cols)], axis=-2) # z x z
return tf.linalg.cholesky(mat)
but whereas cholesky_concat_slow
would cost O(z**3)
work,
cholesky_concat
only costs O(z**2 + m**3)
work.
The resulting (implicit) matrix must be symmetric and positive definite.
Thus, the bottom right m x m
must be self-adjoint, and we do not require a
separate rows
argument (which can be inferred from conj(cols.T)
).
Returns | |
---|---|
chol_concat
|
The Cholesky decomposition of:
|