# Copyright 2021 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License
"""A JIT-compatible library for QDWH-based polar decomposition.
QDWH is short for QR-based dynamically weighted Halley iteration. The Halley
iteration implemented through QR decmopositions does not require matrix
inversion. This is desirable for multicore and heterogeneous computing systems.
Reference: Nakatsukasa, Yuji, Zhaojun Bai, and François Gygi.
"Optimizing Halley's iteration for computing the matrix polar decomposition."
SIAM Journal on Matrix Analysis and Applications 31, no. 5 (2010): 2700-2720.
https://epubs.siam.org/doi/abs/10.1137/090774999
"""
import functools
import jax
from jax import core
import jax.numpy as jnp
from jax._src.lax import linalg as lax_linalg
def _use_qr(u, params):
"""Uses QR decomposition."""
a, b, c = params
m, n = u.shape
y = jnp.concatenate([jnp.sqrt(c) * u, jnp.eye(n)])
q, _ = lax_linalg.qr(y, full_matrices=False)
q1 = q[:m, :]
q2 = (q[m:, :]).T.conj()
e = b / c
u = (e * u + (a - e) / jnp.sqrt(c) * jnp.einsum('ij,jk->ik', q1, q2))
return u
def _use_cholesky(u, params):
"""Uses Cholesky decomposition."""
a, b, c = params
_, n = u.shape
x = c * u.T.conj() @ u + jnp.eye(n)
# `y` is lower triangular.
y = lax_linalg.cholesky(x, symmetrize_input=False)
z = lax_linalg.triangular_solve(
y, u.T, left_side=True, lower=True, conjugate_a=True).conj()
z = lax_linalg.triangular_solve(y, z, left_side=True, lower=True,
transpose_a=True, conjugate_a=True).T.conj()
e = b / c
u = e * u + (a - e) * z
return u
@functools.partial(jax.jit, static_argnums=(1, 2, 3))
def _qdwh(x, is_symmetric, max_iterations):
"""QR-based dynamically weighted Halley iteration for polar decomposition."""
# Estimates `alpha` and `beta = alpha * l`, where `alpha` is an estimate of
# norm(x, 2) such that `alpha >= norm(x, 2)` and `beta` is a lower bound for
# the smallest singular value of x.
eps = jnp.finfo(x.dtype).eps
alpha = jnp.sqrt(jnp.linalg.norm(x, ord=1) * jnp.linalg.norm(x, ord=jnp.inf))
l = eps
u = x / alpha
# Iteration tolerances.
tol_l = 10.0 * eps / 2.0
tol_norm = jnp.cbrt(tol_l)
def cond_fun(state):
_, _, _, is_unconverged, is_not_max_iteration = state
return jnp.logical_and(is_unconverged, is_not_max_iteration)
def body_fun(state):
u, l, iter_idx, _, _ = state
u_prev = u
# Computes parameters.
l2 = l**2
dd = jnp.cbrt(4.0 * (1.0 / l2 - 1.0) / l2)
sqd = jnp.sqrt(1.0 + dd)
a = (sqd + jnp.sqrt(8.0 - 4.0 * dd + 8.0 * (2.0 - l2) / (l2 * sqd)) / 2)
a = jnp.real(a)
b = (a - 1.0)**2 / 4.0
c = a + b - 1.0
# Updates l.
l = l * (a + b * l2) / (1.0 + c * l2)
# Uses QR or Cholesky decomposition.
def true_fn(u):
return _use_qr(u, params=(a, b, c))
def false_fn(u):
return _use_cholesky(u, params=(a, b, c))
u = jax.lax.cond(c > 100, true_fn, false_fn, operand=(u))
if is_symmetric:
u = (u + u.T.conj()) / 2.0
# Checks convergence.
iterating_l = jnp.abs(1.0 - l) > tol_l
iterating_u = jnp.linalg.norm((u-u_prev)) > tol_norm
is_unconverged = jnp.logical_or(iterating_l, iterating_u)
is_not_max_iteration = iter_idx < max_iterations
return u, l, iter_idx + 1, is_unconverged, is_not_max_iteration
iter_idx = 1
is_unconverged = True
is_not_max_iteration = True
u, _, num_iters, is_unconverged, _ = jax.lax.while_loop(
cond_fun=cond_fun, body_fun=body_fun,
init_val=(u, l, iter_idx, is_unconverged, is_not_max_iteration))
# Applies Newton-Schulz refinement for better accuracy.
u = 1.5 * u - 0.5 * u @ (u.T.conj() @ u)
h = u.T.conj() @ x
h = (h + h.T.conj()) / 2.0
# Converged within the maximum number of iterations.
is_converged = jnp.logical_not(is_unconverged)
return u, h, num_iters - 1, is_converged
# TODO: Add pivoting.
[docs]def qdwh(x, is_symmetric, max_iterations=10):
"""QR-based dynamically weighted Halley iteration for polar decomposition.
Args:
x: A full-rank matrix of shape `m x n` with `m >= n`.
is_symmetric: True if `x` is symmetric.
max_iterations: The predefined maximum number of iterations.
Returns:
A four-tuple of (u, h, num_iters, is_converged) containing the
polar decomposition of `x = u * h`, the number of iterations to compute `u`,
and `is_converged`, whose value is `True` when the convergence is achieved
within the maximum number of iterations.
"""
m, n = x.shape
if m < n:
raise ValueError('The input matrix of shape m x n must have m >= n.')
max_iterations = core.concrete_or_error(
int, max_iterations, 'The `max_iterations` argument must be statically '
'specified to use `qdwh` within JAX transformations.')
is_symmetric = core.concrete_or_error(
bool, is_symmetric, 'The `is_symmetric` argument must be statically '
'specified to use `qdwh` within JAX transformations.')
if is_symmetric:
eps = jnp.finfo(x.dtype).eps
tol = 50.0 * eps
relative_diff = jnp.linalg.norm(x - x.T.conj()) / jnp.linalg.norm(x)
if relative_diff > tol:
raise ValueError('The input `x` is NOT symmetric because '
'`norm(x-x.H) / norm(x)` is {}, which is greater than '
'the tolerance {}.'.format(relative_diff, tol))
with jax.default_matmul_precision('float32'):
u, h, num_iters, is_converged = _qdwh(x, is_symmetric, max_iterations)
return u, h, num_iters, is_converged
