240 lines
7.8 KiB
Python
240 lines
7.8 KiB
Python
# Copyright 2021 The JAX Authors.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# https://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License
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"""A JIT-compatible library for QDWH-based polar decomposition.
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QDWH is short for QR-based dynamically weighted Halley iteration. The Halley
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iteration implemented through QR decmopositions does not require matrix
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inversion. This is desirable for multicore and heterogeneous computing systems.
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Reference: Nakatsukasa, Yuji, Zhaojun Bai, and François Gygi.
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"Optimizing Halley's iteration for computing the matrix polar decomposition."
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SIAM Journal on Matrix Analysis and Applications 31, no. 5 (2010): 2700-2720.
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https://epubs.siam.org/doi/abs/10.1137/090774999
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"""
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import functools
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from typing import Optional, Tuple
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import jax
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import jax.numpy as jnp
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from jax import lax
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from jax._src import core
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from jax._src.lax import linalg as lax_linalg
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# Helpers for working with padded shapes
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def _mask(x, dims, alternative=0):
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"""Masks `x` up to the dynamic shape `dims`.
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Replaces values outside those dimensions with `alternative`. `alternative` is
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broadcast with `x`.
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"""
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assert jnp.ndim(x) == len(dims)
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mask = None
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for i, d in enumerate(dims):
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if d is not None:
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mask_dim_i = lax.broadcasted_iota(jnp.int32, x.shape, i) < d
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mask = mask_dim_i if mask is None else (mask & mask_dim_i)
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return x if mask is None else jnp.where(mask, x, alternative)
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def _pad_in_dim(x, low=0, high=0, interior=0, fill_value=0, axis=0):
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pads = [(0, 0, 0)] * x.ndim
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pads[axis] = (low, high, interior)
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return lax.pad(x, jnp.array(fill_value, x.dtype), pads)
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def _dynamic_concat(a, b, m, axis=0):
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"Concatenates padded arrays `a` and `b` where the true size of `a` is `m`."
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if m is None:
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return jnp.concatenate([a, b], axis=axis)
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return lax.dynamic_update_slice_in_dim(
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_pad_in_dim(a, high=b.shape[axis], axis=axis), b, m, axis)
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def _use_qr(u, m, n, params):
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"""QDWH iteration using QR decomposition.
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Args:
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u: a matrix, with static (padded) shape M x N.
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m, n: the dynamic shape of the matrix, where m <= M and n <= N.
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params: the QDWH parameters.
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"""
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a, b, c = params
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M, N = u.shape
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y = _dynamic_concat(jnp.sqrt(c) * u, jnp.eye(N, dtype=jnp.dtype(u)), m)
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q, _ = lax_linalg.qr(y, full_matrices=False)
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# q1 = q[:m, :]
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q1 = _mask(lax.slice(q, (0, 0), (M, N)), (m, n))
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# q2 = (q[m:, :]).T.conj()
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q2 = lax.dynamic_slice_in_dim(q, m, N, axis=0)
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q2 = _mask(q2, (n, n)).T.conj()
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e = b / c
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u = (e * u + (a - e) / jnp.sqrt(c) * jnp.einsum('ij,jk->ik', q1, q2))
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return u
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def _use_cholesky(u, m, n, params):
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"""QDWH iteration using Cholesky decomposition.
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Args:
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u: a matrix, with static (padded) shape M x N
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m, n: the dynamic shape of the matrix, where m <= M and n <= N.
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params: the QDWH parameters.
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"""
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a, b, c = params
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_, N = u.shape
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x = c * (u.T.conj() @ u) + jnp.eye(N, dtype=jnp.dtype(u))
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# Pads the lower-right corner with the identity matrix to prevent the Cholesky
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# decomposition from failing due to the matrix not being PSD if padded with
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# zeros.
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x = _mask(x, (n, n), jnp.eye(N, dtype=x.dtype))
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# `y` is lower triangular.
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y = lax_linalg.cholesky(x, symmetrize_input=False)
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z = lax_linalg.triangular_solve(
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y, u.T, left_side=True, lower=True, conjugate_a=True).conj()
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z = lax_linalg.triangular_solve(y, z, left_side=True, lower=True,
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transpose_a=True, conjugate_a=True).T.conj()
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e = b / c
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u = e * u + (a - e) * z
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return u
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def _qdwh(x, m, n, is_hermitian, max_iterations, eps):
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"""QR-based dynamically weighted Halley iteration for polar decomposition."""
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# Estimates `alpha` and `beta = alpha * l`, where `alpha` is an estimate of
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# norm(x, 2) such that `alpha >= norm(x, 2)` and `beta` is a lower bound for
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# the smallest singular value of x.
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if eps is None:
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eps = float(jnp.finfo(x.dtype).eps)
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alpha = (jnp.sqrt(jnp.linalg.norm(x, ord=1)) *
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jnp.sqrt(jnp.linalg.norm(x, ord=jnp.inf))).astype(x.dtype)
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l = eps
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u = x / alpha
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# Iteration tolerances.
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tol_l = 10.0 * eps / 2.0
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tol_norm = jnp.cbrt(tol_l)
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def cond_fun(state):
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_, _, _, is_unconverged, is_not_max_iteration = state
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return jnp.logical_and(is_unconverged, is_not_max_iteration)
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def body_fun(state):
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u, l, iter_idx, _, _ = state
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u_prev = u
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# Computes parameters.
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l2 = l**2
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dd = jnp.cbrt(4.0 * (1.0 / l2 - 1.0) / l2)
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sqd = jnp.sqrt(1.0 + dd)
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a = (sqd + jnp.sqrt(8.0 - 4.0 * dd + 8.0 * (2.0 - l2) / (l2 * sqd)) / 2)
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a = jnp.real(a)
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b = (a - 1.0)**2 / 4.0
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c = a + b - 1.0
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# Updates l.
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l = l * (a + b * l2) / (1.0 + c * l2)
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# Uses QR or Cholesky decomposition.
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def true_fn(u):
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return _use_qr(u, m, n, params=(a, b, c))
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def false_fn(u):
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return _use_cholesky(u, m, n, params=(a, b, c))
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u = jax.lax.cond(c > 100, true_fn, false_fn, operand=(u))
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if is_hermitian:
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u = (u + u.T.conj()) / 2.0
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# Checks convergence.
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iterating_l = jnp.abs(1.0 - l) > tol_l
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iterating_u = jnp.linalg.norm(u-u_prev) > tol_norm
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is_unconverged = jnp.logical_or(iterating_l, iterating_u)
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is_not_max_iteration = iter_idx < max_iterations
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return u, l, iter_idx + 1, is_unconverged, is_not_max_iteration
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iter_idx = 1
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is_unconverged = True
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is_not_max_iteration = True
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u, _, num_iters, is_unconverged, _ = jax.lax.while_loop(
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cond_fun=cond_fun, body_fun=body_fun,
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init_val=(u, l, iter_idx, is_unconverged, is_not_max_iteration))
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# Applies Newton-Schulz refinement for better accuracy.
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u = 1.5 * u - 0.5 * u @ (u.T.conj() @ u)
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h = u.T.conj() @ x
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h = (h + h.T.conj()) / 2.0
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# Converged within the maximum number of iterations.
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is_converged = jnp.logical_not(is_unconverged)
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return u, h, num_iters - 1, is_converged
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# TODO: Add pivoting.
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@functools.partial(jax.jit, static_argnames=('is_hermitian',))
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def qdwh(x, *, is_hermitian=False, max_iterations=None, eps=None,
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dynamic_shape: Optional[Tuple[int, int]] = None):
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"""QR-based dynamically weighted Halley iteration for polar decomposition.
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Args:
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x: A full-rank matrix, with shape `M x N`. The matrix may be
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padded up to that size from a smaller true shape (``dynamic_shape``).
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is_hermitian: True if `x` is Hermitian. Default to `False`.
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eps: The final result will satisfy
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``|x_k - x_k-1| < |x_k| * (4*eps)**(1/3)`` where `x_k` is the iterate.
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max_iterations: Iterations will terminate after this many steps even if the
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above is unsatisfied.
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dynamic_shape: the unpadded shape as an ``(m, n)`` tuple; optional.
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Returns:
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A four-tuple of (u, h, num_iters, is_converged) containing the
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polar decomposition of `x = u * h`, the number of iterations to compute `u`,
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and `is_converged`, whose value is `True` when the convergence is achieved
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within the maximum number of iterations.
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"""
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is_hermitian = core.concrete_or_error(
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bool, is_hermitian, 'The `is_hermitian` argument must be statically '
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'specified to use `qdwh` within JAX transformations.')
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if max_iterations is None:
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max_iterations = 10
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M, N = x.shape
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if M < N:
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raise ValueError('The input matrix of shape M x N must have M >= N.')
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if dynamic_shape is not None:
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m, n = dynamic_shape
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x = _mask(x, (m, n))
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else:
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m, n = M, N
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with jax.default_matmul_precision('float32'):
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u, h, num_iters, is_converged = _qdwh(x, m, n, is_hermitian, max_iterations,
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eps)
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return u, h, num_iters, is_converged
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