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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)
).
Args | |
---|---|
chol
|
Cholesky decomposition of mat = chol @ chol.T .
|
cols
|
The new columns whose first n rows we would like concatenated to the
right of mat = chol @ chol.T , and whose conjugate transpose we would
like concatenated to the bottom of concat(mat, cols[:n,:]) . A Tensor
with final dims (n+m, m) . The first n rows are the top right rectangle
(their conjugate transpose forms the bottom left), and the bottom m x m
is self-adjoint.
|
name
|
Optional name for this op. |
Returns | |
---|---|
chol_concat
|
The Cholesky decomposition of:
|