# mypy: allow-untyped-defs
"""Locally Optimal Block Preconditioned Conjugate Gradient methods.
"""
# Author: Pearu Peterson
# Created: February 2020
from typing import Dict, Optional, Tuple
import torch
from torch import Tensor
from . import _linalg_utils as _utils
from .overrides import handle_torch_function, has_torch_function
__all__ = ["lobpcg"]
def _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U):
# compute F, such that F_ij = (d_j - d_i)^{-1} for i != j, F_ii = 0
F = D.unsqueeze(-2) - D.unsqueeze(-1)
F.diagonal(dim1=-2, dim2=-1).fill_(float("inf"))
F.pow_(-1)
# A.grad = U (D.grad + (U^T U.grad * F)) U^T
Ut = U.mT.contiguous()
res = torch.matmul(
U, torch.matmul(torch.diag_embed(D_grad) + torch.matmul(Ut, U_grad) * F, Ut)
)
return res
def _polynomial_coefficients_given_roots(roots):
"""
Given the `roots` of a polynomial, find the polynomial's coefficients.
If roots = (r_1, ..., r_n), then the method returns
coefficients (a_0, a_1, ..., a_n (== 1)) so that
p(x) = (x - r_1) * ... * (x - r_n)
= x^n + a_{n-1} * x^{n-1} + ... a_1 * x_1 + a_0
Note: for better performance requires writing a low-level kernel
"""
poly_order = roots.shape[-1]
poly_coeffs_shape = list(roots.shape)
# we assume p(x) = x^n + a_{n-1} * x^{n-1} + ... + a_1 * x + a_0,
# so poly_coeffs = {a_0, ..., a_n, a_{n+1}(== 1)},
# but we insert one extra coefficient to enable better vectorization below
poly_coeffs_shape[-1] += 2
poly_coeffs = roots.new_zeros(poly_coeffs_shape)
poly_coeffs[..., 0] = 1
poly_coeffs[..., -1] = 1
# perform the Horner's rule
for i in range(1, poly_order + 1):
# note that it is computationally hard to compute backward for this method,
# because then given the coefficients it would require finding the roots and/or
# calculating the sensitivity based on the Vieta's theorem.
# So the code below tries to circumvent the explicit root finding by series
# of operations on memory copies imitating the Horner's method.
# The memory copies are required to construct nodes in the computational graph
# by exploting the explicit (not in-place, separate node for each step)
# recursion of the Horner's method.
# Needs more memory, O(... * k^2), but with only O(... * k^2) complexity.
poly_coeffs_new = poly_coeffs.clone() if roots.requires_grad else poly_coeffs
out = poly_coeffs_new.narrow(-1, poly_order - i, i + 1)
out -= roots.narrow(-1, i - 1, 1) * poly_coeffs.narrow(
-1, poly_order - i + 1, i + 1
)
poly_coeffs = poly_coeffs_new
return poly_coeffs.narrow(-1, 1, poly_order + 1)
def _polynomial_value(poly, x, zero_power, transition):
"""
A generic method for computing poly(x) using the Horner's rule.
Args:
poly (Tensor): the (possibly batched) 1D Tensor representing
polynomial coefficients such that
poly[..., i] = (a_{i_0}, ..., a{i_n} (==1)), and
poly(x) = poly[..., 0] * zero_power + ... + poly[..., n] * x^n
x (Tensor): the value (possible batched) to evalate the polynomial `poly` at.
zero_power (Tensor): the representation of `x^0`. It is application-specific.
transition (Callable): the function that accepts some intermediate result `int_val`,
the `x` and a specific polynomial coefficient
`poly[..., k]` for some iteration `k`.
It basically performs one iteration of the Horner's rule
defined as `x * int_val + poly[..., k] * zero_power`.
Note that `zero_power` is not a parameter,
because the step `+ poly[..., k] * zero_power` depends on `x`,
whether it is a vector, a matrix, or something else, so this
functionality is delegated to the user.
"""
res = zero_power.clone()
for k in range(poly.size(-1) - 2, -1, -1):
res = transition(res, x, poly[..., k])
return res
def _matrix_polynomial_value(poly, x, zero_power=None):
"""
Evaluates `poly(x)` for the (batched) matrix input `x`.
Check out `_polynomial_value` function for more details.
"""
# matrix-aware Horner's rule iteration
def transition(curr_poly_val, x, poly_coeff):
res = x.matmul(curr_poly_val)
res.diagonal(dim1=-2, dim2=-1).add_(poly_coeff.unsqueeze(-1))
return res
if zero_power is None:
zero_power = torch.eye(
x.size(-1), x.size(-1), dtype=x.dtype, device=x.device
).view(*([1] * len(list(x.shape[:-2]))), x.size(-1), x.size(-1))
return _polynomial_value(poly, x, zero_power, transition)
def _vector_polynomial_value(poly, x, zero_power=None):
"""
Evaluates `poly(x)` for the (batched) vector input `x`.
Check out `_polynomial_value` function for more details.
"""
# vector-aware Horner's rule iteration
def transition(curr_poly_val, x, poly_coeff):
res = torch.addcmul(poly_coeff.unsqueeze(-1), x, curr_poly_val)
return res
if zero_power is None:
zero_power = x.new_ones(1).expand(x.shape)
return _polynomial_value(poly, x, zero_power, transition)
def _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest):
# compute a projection operator onto an orthogonal subspace spanned by the
# columns of U defined as (I - UU^T)
Ut = U.mT.contiguous()
proj_U_ortho = -U.matmul(Ut)
proj_U_ortho.diagonal(dim1=-2, dim2=-1).add_(1)
# compute U_ortho, a basis for the orthogonal complement to the span(U),
# by projecting a random [..., m, m - k] matrix onto the subspace spanned
# by the columns of U.
#
# fix generator for determinism
gen = torch.Generator(A.device)
# orthogonal complement to the span(U)
U_ortho = proj_U_ortho.matmul(
torch.randn(
(*A.shape[:-1], A.size(-1) - D.size(-1)),
dtype=A.dtype,
device=A.device,
generator=gen,
)
)
U_ortho_t = U_ortho.mT.contiguous()
# compute the coefficients of the characteristic polynomial of the tensor D.
# Note that D is diagonal, so the diagonal elements are exactly the roots
# of the characteristic polynomial.
chr_poly_D = _polynomial_coefficients_given_roots(D)
# the code belows finds the explicit solution to the Sylvester equation
# U_ortho^T A U_ortho dX - dX D = -U_ortho^T A U
# and incorporates it into the whole gradient stored in the `res` variable.
#
# Equivalent to the following naive implementation:
# res = A.new_zeros(A.shape)
# p_res = A.new_zeros(*A.shape[:-1], D.size(-1))
# for k in range(1, chr_poly_D.size(-1)):
# p_res.zero_()
# for i in range(0, k):
# p_res += (A.matrix_power(k - 1 - i) @ U_grad) * D.pow(i).unsqueeze(-2)
# res -= chr_poly_D[k] * (U_ortho @ poly_D_at_A.inverse() @ U_ortho_t @ p_res @ U.t())
#
# Note that dX is a differential, so the gradient contribution comes from the backward sensitivity
# Tr(f(U_grad, D_grad, A, U, D)^T dX) = Tr(g(U_grad, A, U, D)^T dA) for some functions f and g,
# and we need to compute g(U_grad, A, U, D)
#
# The naive implementation is based on the paper
# Hu, Qingxi, and Daizhan Cheng.
# "The polynomial solution to the Sylvester matrix equation."
# Applied mathematics letters 19.9 (2006): 859-864.
#
# We can modify the computation of `p_res` from above in a more efficient way
# p_res = U_grad * (chr_poly_D[1] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k)).unsqueeze(-2)
# + A U_grad * (chr_poly_D[2] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k - 1)).unsqueeze(-2)
# + ...
# + A.matrix_power(k - 1) U_grad * chr_poly_D[k]
# Note that this saves us from redundant matrix products with A (elimination of matrix_power)
U_grad_projected = U_grad
series_acc = U_grad_projected.new_zeros(U_grad_projected.shape)
for k in range(1, chr_poly_D.size(-1)):
poly_D = _vector_polynomial_value(chr_poly_D[..., k:], D)
series_acc += U_grad_projected * poly_D.unsqueeze(-2)
U_grad_projected = A.matmul(U_grad_projected)
# compute chr_poly_D(A) which essentially is:
#
# chr_poly_D_at_A = A.new_zeros(A.shape)
# for k in range(chr_poly_D.size(-1)):
# chr_poly_D_at_A += chr_poly_D[k] * A.matrix_power(k)
#
# Note, however, for better performance we use the Horner's rule
chr_poly_D_at_A = _matrix_polynomial_value(chr_poly_D, A)
# compute the action of `chr_poly_D_at_A` restricted to U_ortho_t
chr_poly_D_at_A_to_U_ortho = torch.matmul(
U_ortho_t, torch.matmul(chr_poly_D_at_A, U_ortho)
)
# we need to invert 'chr_poly_D_at_A_to_U_ortho`, for that we compute its
# Cholesky decomposition and then use `torch.cholesky_solve` for better stability.
# Cholesky decomposition requires the input to be positive-definite.
# Note that `chr_poly_D_at_A_to_U_ortho` is positive-definite if
# 1. `largest` == False, or
# 2. `largest` == True and `k` is even
# under the assumption that `A` has distinct eigenvalues.
#
# check if `chr_poly_D_at_A_to_U_ortho` is positive-definite or negative-definite
chr_poly_D_at_A_to_U_ortho_sign = -1 if (largest and (k % 2 == 1)) else +1
chr_poly_D_at_A_to_U_ortho_L = torch.linalg.cholesky(
chr_poly_D_at_A_to_U_ortho_sign * chr_poly_D_at_A_to_U_ortho
)
# compute the gradient part in span(U)
res = _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
# incorporate the Sylvester equation solution into the full gradient
# it resides in span(U_ortho)
res -= U_ortho.matmul(
chr_poly_D_at_A_to_U_ortho_sign
* torch.cholesky_solve(
U_ortho_t.matmul(series_acc), chr_poly_D_at_A_to_U_ortho_L
)
).matmul(Ut)
return res
def _symeig_backward(D_grad, U_grad, A, D, U, largest):
# if `U` is square, then the columns of `U` is a complete eigenspace
if U.size(-1) == U.size(-2):
return _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
else:
return _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest)
class LOBPCGAutogradFunction(torch.autograd.Function):
@staticmethod
def forward( # type: ignore[override]
ctx,
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
# makes sure that input is contiguous for efficiency.
# Note: autograd does not support dense gradients for sparse input yet.
A = A.contiguous() if (not A.is_sparse) else A
if B is not None:
B = B.contiguous() if (not B.is_sparse) else B
D, U = _lobpcg(
A,
k,
B,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
ctx.save_for_backward(A, B, D, U)
ctx.largest = largest
return D, U
@staticmethod
def backward(ctx, D_grad, U_grad):
A_grad = B_grad = None
grads = [None] * 14
A, B, D, U = ctx.saved_tensors
largest = ctx.largest
# lobpcg.backward has some limitations. Checks for unsupported input
if A.is_sparse or (B is not None and B.is_sparse and ctx.needs_input_grad[2]):
raise ValueError(
"lobpcg.backward does not support sparse input yet."
"Note that lobpcg.forward does though."
)
if (
A.dtype in (torch.complex64, torch.complex128)
or B is not None
and B.dtype in (torch.complex64, torch.complex128)
):
raise ValueError(
"lobpcg.backward does not support complex input yet."
"Note that lobpcg.forward does though."
)
if B is not None:
raise ValueError(
"lobpcg.backward does not support backward with B != I yet."
)
if largest is None:
largest = True
# symeig backward
if B is None:
A_grad = _symeig_backward(D_grad, U_grad, A, D, U, largest)
# A has index 0
grads[0] = A_grad
# B has index 2
grads[2] = B_grad
return tuple(grads)
def lobpcg(
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
"""Find the k largest (or smallest) eigenvalues and the corresponding
eigenvectors of a symmetric positive definite generalized
eigenvalue problem using matrix-free LOBPCG methods.
This function is a front-end to the following LOBPCG algorithms
selectable via `method` argument:
`method="basic"` - the LOBPCG method introduced by Andrew
Knyazev, see [Knyazev2001]. A less robust method, may fail when
Cholesky is applied to singular input.
`method="ortho"` - the LOBPCG method with orthogonal basis
selection [StathopoulosEtal2002]. A robust method.
Supported inputs are dense, sparse, and batches of dense matrices.
.. note:: In general, the basic method spends least time per
iteration. However, the robust methods converge much faster and
are more stable. So, the usage of the basic method is generally
not recommended but there exist cases where the usage of the
basic method may be preferred.
.. warning:: The backward method does not support sparse and complex inputs.
It works only when `B` is not provided (i.e. `B == None`).
We are actively working on extensions, and the details of
the algorithms are going to be published promptly.
.. warning:: While it is assumed that `A` is symmetric, `A.grad` is not.
To make sure that `A.grad` is symmetric, so that `A - t * A.grad` is symmetric
in first-order optimization routines, prior to running `lobpcg`
we do the following symmetrization map: `A -> (A + A.t()) / 2`.
The map is performed only when the `A` requires gradients.
Args:
A (Tensor): the input tensor of size :math:`(*, m, m)`
B (Tensor, optional): the input tensor of size :math:`(*, m,
m)`. When not specified, `B` is interpreted as
identity matrix.
X (tensor, optional): the input tensor of size :math:`(*, m, n)`
where `k <= n <= m`. When specified, it is used as
initial approximation of eigenvectors. X must be a
dense tensor.
iK (tensor, optional): the input tensor of size :math:`(*, m,
m)`. When specified, it will be used as preconditioner.
k (integer, optional): the number of requested
eigenpairs. Default is the number of :math:`X`
columns (when specified) or `1`.
n (integer, optional): if :math:`X` is not specified then `n`
specifies the size of the generated random
approximation of eigenvectors. Default value for `n`
is `k`. If :math:`X` is specified, the value of `n`
(when specified) must be the number of :math:`X`
columns.
tol (float, optional): residual tolerance for stopping
criterion. Default is `feps ** 0.5` where `feps` is
smallest non-zero floating-point number of the given
input tensor `A` data type.
largest (bool, optional): when True, solve the eigenproblem for
the largest eigenvalues. Otherwise, solve the
eigenproblem for smallest eigenvalues. Default is
`True`.
method (str, optional): select LOBPCG method. See the
description of the function above. Default is
"ortho".
niter (int, optional): maximum number of iterations. When
reached, the iteration process is hard-stopped and
the current approximation of eigenpairs is returned.
For infinite iteration but until convergence criteria
is met, use `-1`.
tracker (callable, optional) : a function for tracing the
iteration process. When specified, it is called at
each iteration step with LOBPCG instance as an
argument. The LOBPCG instance holds the full state of
the iteration process in the following attributes:
`iparams`, `fparams`, `bparams` - dictionaries of
integer, float, and boolean valued input
parameters, respectively
`ivars`, `fvars`, `bvars`, `tvars` - dictionaries
of integer, float, boolean, and Tensor valued
iteration variables, respectively.
`A`, `B`, `iK` - input Tensor arguments.
`E`, `X`, `S`, `R` - iteration Tensor variables.
For instance:
`ivars["istep"]` - the current iteration step
`X` - the current approximation of eigenvectors
`E` - the current approximation of eigenvalues
`R` - the current residual
`ivars["converged_count"]` - the current number of converged eigenpairs
`tvars["rerr"]` - the current state of convergence criteria
Note that when `tracker` stores Tensor objects from
the LOBPCG instance, it must make copies of these.
If `tracker` sets `bvars["force_stop"] = True`, the
iteration process will be hard-stopped.
ortho_iparams, ortho_fparams, ortho_bparams (dict, optional):
various parameters to LOBPCG algorithm when using
`method="ortho"`.
Returns:
E (Tensor): tensor of eigenvalues of size :math:`(*, k)`
X (Tensor): tensor of eigenvectors of size :math:`(*, m, k)`
References:
[Knyazev2001] Andrew V. Knyazev. (2001) Toward the Optimal
Preconditioned Eigensolver: Locally Optimal Block Preconditioned
Conjugate Gradient Method. SIAM J. Sci. Comput., 23(2),
517-541. (25 pages)
https://epubs.siam.org/doi/abs/10.1137/S1064827500366124
[StathopoulosEtal2002] Andreas Stathopoulos and Kesheng
Wu. (2002) A Block Orthogonalization Procedure with Constant
Synchronization Requirements. SIAM J. Sci. Comput., 23(6),
2165-2182. (18 pages)
https://epubs.siam.org/doi/10.1137/S1064827500370883
[DuerschEtal2018] Jed A. Duersch, Meiyue Shao, Chao Yang, Ming
Gu. (2018) A Robust and Efficient Implementation of LOBPCG.
SIAM J. Sci. Comput., 40(5), C655-C676. (22 pages)
https://epubs.siam.org/doi/abs/10.1137/17M1129830
"""
if not torch.jit.is_scripting():
tensor_ops = (A, B, X, iK)
if not set(map(type, tensor_ops)).issubset(
(torch.Tensor, type(None))
) and has_torch_function(tensor_ops):
return handle_torch_function(
lobpcg,
tensor_ops,
A,
k=k,
B=B,
X=X,
n=n,
iK=iK,
niter=niter,
tol=tol,
largest=largest,
method=method,
tracker=tracker,
ortho_iparams=ortho_iparams,
ortho_fparams=ortho_fparams,
ortho_bparams=ortho_bparams,
)
if not torch._jit_internal.is_scripting():
if A.requires_grad or (B is not None and B.requires_grad):
# While it is expected that `A` is symmetric,
# the `A_grad` might be not. Therefore we perform the trick below,
# so that `A_grad` becomes symmetric.
# The symmetrization is important for first-order optimization methods,
# so that (A - alpha * A_grad) is still a symmetric matrix.
# Same holds for `B`.
A_sym = (A + A.mT) / 2
B_sym = (B + B.mT) / 2 if (B is not None) else None
return LOBPCGAutogradFunction.apply(
A_sym,
k,
B_sym,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
else:
if A.requires_grad or (B is not None and B.requires_grad):
raise RuntimeError(
"Script and require grads is not supported atm."
"If you just want to do the forward, use .detach()"
"on A and B before calling into lobpcg"
)
return _lobpcg(
A,
k,
B,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
def _lobpcg(
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
# A must be square:
assert A.shape[-2] == A.shape[-1], A.shape
if B is not None:
# A and B must have the same shapes:
assert A.shape == B.shape, (A.shape, B.shape)
dtype = _utils.get_floating_dtype(A)
device = A.device
if tol is None:
feps = {torch.float32: 1.2e-07, torch.float64: 2.23e-16}[dtype]
tol = feps**0.5
m = A.shape[-1]
k = (1 if X is None else X.shape[-1]) if k is None else k
n = (k if n is None else n) if X is None else X.shape[-1]
if m < 3 * n:
raise ValueError(
f"LPBPCG algorithm is not applicable when the number of A rows (={m})"
f" is smaller than 3 x the number of requested eigenpairs (={n})"
)
method = "ortho" if method is None else method
iparams = {
"m": m,
"n": n,
"k": k,
"niter": 1000 if niter is None else niter,
}
fparams = {
"tol": tol,
}
bparams = {"largest": True if largest is None else largest}
if method == "ortho":
if ortho_iparams is not None:
iparams.update(ortho_iparams)
if ortho_fparams is not None:
fparams.update(ortho_fparams)
if ortho_bparams is not None:
bparams.update(ortho_bparams)
iparams["ortho_i_max"] = iparams.get("ortho_i_max", 3)
iparams["ortho_j_max"] = iparams.get("ortho_j_max", 3)
fparams["ortho_tol"] = fparams.get("ortho_tol", tol)
fparams["ortho_tol_drop"] = fparams.get("ortho_tol_drop", tol)
fparams["ortho_tol_replace"] = fparams.get("ortho_tol_replace", tol)
bparams["ortho_use_drop"] = bparams.get("ortho_use_drop", False)
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker # type: ignore[method-assign]
if len(A.shape) > 2:
N = int(torch.prod(torch.tensor(A.shape[:-2])))
bA = A.reshape((N,) + A.shape[-2:])
bB = B.reshape((N,) + A.shape[-2:]) if B is not None else None
bX = X.reshape((N,) + X.shape[-2:]) if X is not None else None
bE = torch.empty((N, k), dtype=dtype, device=device)
bXret = torch.empty((N, m, k), dtype=dtype, device=device)
for i in range(N):
A_ = bA[i]
B_ = bB[i] if bB is not None else None
X_ = (
torch.randn((m, n), dtype=dtype, device=device) if bX is None else bX[i]
)
assert len(X_.shape) == 2 and X_.shape == (m, n), (X_.shape, (m, n))
iparams["batch_index"] = i
worker = LOBPCG(A_, B_, X_, iK, iparams, fparams, bparams, method, tracker)
worker.run()
bE[i] = worker.E[:k]
bXret[i] = worker.X[:, :k]
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker_orig # type: ignore[method-assign]
return bE.reshape(A.shape[:-2] + (k,)), bXret.reshape(A.shape[:-2] + (m, k))
X = torch.randn((m, n), dtype=dtype, device=device) if X is None else X
assert len(X.shape) == 2 and X.shape == (m, n), (X.shape, (m, n))
worker = LOBPCG(A, B, X, iK, iparams, fparams, bparams, method, tracker)
worker.run()
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker_orig # type: ignore[method-assign]
return worker.E[:k], worker.X[:, :k]
class LOBPCG:
"""Worker class of LOBPCG methods."""
def __init__(
self,
A: Optional[Tensor],
B: Optional[Tensor],
X: Tensor,
iK: Optional[Tensor],
iparams: Dict[str, int],
fparams: Dict[str, float],
bparams: Dict[str, bool],
method: str,
tracker: None,
) -> None:
# constant parameters
self.A = A
self.B = B
self.iK = iK
self.iparams = iparams
self.fparams = fparams
self.bparams = bparams
self.method = method
self.tracker = tracker
m = iparams["m"]
n = iparams["n"]
# variable parameters
self.X = X
self.E = torch.zeros((n,), dtype=X.dtype, device=X.device)
self.R = torch.zeros((m, n), dtype=X.dtype, device=X.device)
self.S = torch.zeros((m, 3 * n), dtype=X.dtype, device=X.device)
self.tvars: Dict[str, Tensor] = {}
self.ivars: Dict[str, int] = {"istep": 0}
self.fvars: Dict[str, float] = {"_": 0.0}
self.bvars: Dict[str, bool] = {"_": False}
def __str__(self):
lines = ["LOPBCG:"]
lines += [f" iparams={self.iparams}"]
lines += [f" fparams={self.fparams}"]
lines += [f" bparams={self.bparams}"]
lines += [f" ivars={self.ivars}"]
lines += [f" fvars={self.fvars}"]
lines += [f" bvars={self.bvars}"]
lines += [f" tvars={self.tvars}"]
lines += [f" A={self.A}"]
lines += [f" B={self.B}"]
lines += [f" iK={self.iK}"]
lines += [f" X={self.X}"]
lines += [f" E={self.E}"]
r = ""
for line in lines:
r += line + "\n"
return r
def update(self):
"""Set and update iteration variables."""
if self.ivars["istep"] == 0:
X_norm = float(torch.norm(self.X))
iX_norm = X_norm**-1
A_norm = float(torch.norm(_utils.matmul(self.A, self.X))) * iX_norm
B_norm = float(torch.norm(_utils.matmul(self.B, self.X))) * iX_norm
self.fvars["X_norm"] = X_norm
self.fvars["A_norm"] = A_norm
self.fvars["B_norm"] = B_norm
self.ivars["iterations_left"] = self.iparams["niter"]
self.ivars["converged_count"] = 0
self.ivars["converged_end"] = 0
if self.method == "ortho":
self._update_ortho()
else:
self._update_basic()
self.ivars["iterations_left"] = self.ivars["iterations_left"] - 1
self.ivars["istep"] = self.ivars["istep"] + 1
def update_residual(self):
"""Update residual R from A, B, X, E."""
mm = _utils.matmul
self.R = mm(self.A, self.X) - mm(self.B, self.X) * self.E
def update_converged_count(self):
"""Determine the number of converged eigenpairs using backward stable
convergence criterion, see discussion in Sec 4.3 of [DuerschEtal2018].
Users may redefine this method for custom convergence criteria.
"""
# (...) -> int
prev_count = self.ivars["converged_count"]
tol = self.fparams["tol"]
A_norm = self.fvars["A_norm"]
B_norm = self.fvars["B_norm"]
E, X, R = self.E, self.X, self.R
rerr = (
torch.norm(R, 2, (0,))
* (torch.norm(X, 2, (0,)) * (A_norm + E[: X.shape[-1]] * B_norm)) ** -1
)
converged = rerr < tol
count = 0
for b in converged:
if not b:
# ignore convergence of following pairs to ensure
# strict ordering of eigenpairs
break
count += 1
assert (
count >= prev_count
), f"the number of converged eigenpairs (was {prev_count}, got {count}) cannot decrease"
self.ivars["converged_count"] = count
self.tvars["rerr"] = rerr
return count
def stop_iteration(self):
"""Return True to stop iterations.
Note that tracker (if defined) can force-stop iterations by
setting ``worker.bvars['force_stop'] = True``.
"""
return (
self.bvars.get("force_stop", False)
or self.ivars["iterations_left"] == 0
or self.ivars["converged_count"] >= self.iparams["k"]
)
def run(self):
"""Run LOBPCG iterations.
Use this method as a template for implementing LOBPCG
iteration scheme with custom tracker that is compatible with
TorchScript.
"""
self.update()
if not torch.jit.is_scripting() and self.tracker is not None:
self.call_tracker()
while not self.stop_iteration():
self.update()
if not torch.jit.is_scripting() and self.tracker is not None:
self.call_tracker()
@torch.jit.unused
def call_tracker(self):
"""Interface for tracking iteration process in Python mode.
Tracking the iteration process is disabled in TorchScript
mode. In fact, one should specify tracker=None when JIT
compiling functions using lobpcg.
"""
# do nothing when in TorchScript mode
pass
# Internal methods
def _update_basic(self):
"""
Update or initialize iteration variables when `method == "basic"`.
"""
mm = torch.matmul
ns = self.ivars["converged_end"]
nc = self.ivars["converged_count"]
n = self.iparams["n"]
largest = self.bparams["largest"]
if self.ivars["istep"] == 0:
Ri = self._get_rayleigh_ritz_transform(self.X)
M = _utils.qform(_utils.qform(self.A, self.X), Ri)
E, Z = _utils.symeig(M, largest)
self.X[:] = mm(self.X, mm(Ri, Z))
self.E[:] = E
np = 0
self.update_residual()
nc = self.update_converged_count()
self.S[..., :n] = self.X
W = _utils.matmul(self.iK, self.R)
self.ivars["converged_end"] = ns = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
else:
S_ = self.S[:, nc:ns]
Ri = self._get_rayleigh_ritz_transform(S_)
M = _utils.qform(_utils.qform(self.A, S_), Ri)
E_, Z = _utils.symeig(M, largest)
self.X[:, nc:] = mm(S_, mm(Ri, Z[:, : n - nc]))
self.E[nc:] = E_[: n - nc]
P = mm(S_, mm(Ri, Z[:, n : 2 * n - nc]))
np = P.shape[-1]
self.update_residual()
nc = self.update_converged_count()
self.S[..., :n] = self.X
self.S[:, n : n + np] = P
W = _utils.matmul(self.iK, self.R[:, nc:])
self.ivars["converged_end"] = ns = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
def _update_ortho(self):
"""
Update or initialize iteration variables when `method == "ortho"`.
"""
mm = torch.matmul
ns = self.ivars["converged_end"]
nc = self.ivars["converged_count"]
n = self.iparams["n"]
largest = self.bparams["largest"]
if self.ivars["istep"] == 0:
Ri = self._get_rayleigh_ritz_transform(self.X)
M = _utils.qform(_utils.qform(self.A, self.X), Ri)
E, Z = _utils.symeig(M, largest)
self.X = mm(self.X, mm(Ri, Z))
self.update_residual()
np = 0
nc = self.update_converged_count()
self.S[:, :n] = self.X
W = self._get_ortho(self.R, self.X)
ns = self.ivars["converged_end"] = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
else:
S_ = self.S[:, nc:ns]
# Rayleigh-Ritz procedure
E_, Z = _utils.symeig(_utils.qform(self.A, S_), largest)
# Update E, X, P
self.X[:, nc:] = mm(S_, Z[:, : n - nc])
self.E[nc:] = E_[: n - nc]
P = mm(
S_,
mm(
Z[:, n - nc :],
_utils.basis(Z[: n - nc, n - nc :].mT),
),
)
np = P.shape[-1]
# check convergence
self.update_residual()
nc = self.update_converged_count()
# update S
self.S[:, :n] = self.X
self.S[:, n : n + np] = P
W = self._get_ortho(self.R[:, nc:], self.S[:, : n + np])
ns = self.ivars["converged_end"] = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
def _get_rayleigh_ritz_transform(self, S):
"""Return a transformation matrix that is used in Rayleigh-Ritz
procedure for reducing a general eigenvalue problem :math:`(S^TAS)
C = (S^TBS) C E` to a standard eigenvalue problem :math: `(Ri^T
S^TAS Ri) Z = Z E` where `C = Ri Z`.
.. note:: In the original Rayleight-Ritz procedure in
[DuerschEtal2018], the problem is formulated as follows::
SAS = S^T A S
SBS = S^T B S
D = (<diagonal matrix of SBS>) ** -1/2
R^T R = Cholesky(D SBS D)
Ri = D R^-1
solve symeig problem Ri^T SAS Ri Z = Theta Z
C = Ri Z
To reduce the number of matrix products (denoted by empty
space between matrices), here we introduce element-wise
products (denoted by symbol `*`) so that the Rayleight-Ritz
procedure becomes::
SAS = S^T A S
SBS = S^T B S
d = (<diagonal of SBS>) ** -1/2 # this is 1-d column vector
dd = d d^T # this is 2-d matrix
R^T R = Cholesky(dd * SBS)
Ri = R^-1 * d # broadcasting
solve symeig problem Ri^T SAS Ri Z = Theta Z
C = Ri Z
where `dd` is 2-d matrix that replaces matrix products `D M
D` with one element-wise product `M * dd`; and `d` replaces
matrix product `D M` with element-wise product `M *
d`. Also, creating the diagonal matrix `D` is avoided.
Args:
S (Tensor): the matrix basis for the search subspace, size is
:math:`(m, n)`.
Returns:
Ri (tensor): upper-triangular transformation matrix of size
:math:`(n, n)`.
"""
B = self.B
mm = torch.matmul
SBS = _utils.qform(B, S)
d_row = SBS.diagonal(0, -2, -1) ** -0.5
d_col = d_row.reshape(d_row.shape[0], 1)
# TODO use torch.linalg.cholesky_solve once it is implemented
R = torch.linalg.cholesky((SBS * d_row) * d_col, upper=True)
return torch.linalg.solve_triangular(
R, d_row.diag_embed(), upper=True, left=False
)
def _get_svqb(
self, U: Tensor, drop: bool, tau: float # Tensor # bool # float
) -> Tensor:
"""Return B-orthonormal U.
.. note:: When `drop` is `False` then `svqb` is based on the
Algorithm 4 from [DuerschPhD2015] that is a slight
modification of the corresponding algorithm
introduced in [StathopolousWu2002].
Args:
U (Tensor) : initial approximation, size is (m, n)
drop (bool) : when True, drop columns that
contribution to the `span([U])` is small.
tau (float) : positive tolerance
Returns:
U (Tensor) : B-orthonormal columns (:math:`U^T B U = I`), size
is (m, n1), where `n1 = n` if `drop` is `False,
otherwise `n1 <= n`.
"""
if torch.numel(U) == 0:
return U
UBU = _utils.qform(self.B, U)
d = UBU.diagonal(0, -2, -1)
# Detect and drop exact zero columns from U. While the test
# `abs(d) == 0` is unlikely to be True for random data, it is
# possible to construct input data to lobpcg where it will be
# True leading to a failure (notice the `d ** -0.5` operation
# in the original algorithm). To prevent the failure, we drop
# the exact zero columns here and then continue with the
# original algorithm below.
nz = torch.where(abs(d) != 0.0)
assert len(nz) == 1, nz
if len(nz[0]) < len(d):
U = U[:, nz[0]]
if torch.numel(U) == 0:
return U
UBU = _utils.qform(self.B, U)
d = UBU.diagonal(0, -2, -1)
nz = torch.where(abs(d) != 0.0)
assert len(nz[0]) == len(d)
# The original algorithm 4 from [DuerschPhD2015].
d_col = (d**-0.5).reshape(d.shape[0], 1)
DUBUD = (UBU * d_col) * d_col.mT
E, Z = _utils.symeig(DUBUD)
t = tau * abs(E).max()
if drop:
keep = torch.where(E > t)
assert len(keep) == 1, keep
E = E[keep[0]]
Z = Z[:, keep[0]]
d_col = d_col[keep[0]]
else:
E[(torch.where(E < t))[0]] = t
return torch.matmul(U * d_col.mT, Z * E**-0.5)
def _get_ortho(self, U, V):
"""Return B-orthonormal U with columns are B-orthogonal to V.
.. note:: When `bparams["ortho_use_drop"] == False` then
`_get_ortho` is based on the Algorithm 3 from
[DuerschPhD2015] that is a slight modification of
the corresponding algorithm introduced in
[StathopolousWu2002]. Otherwise, the method
implements Algorithm 6 from [DuerschPhD2015]
.. note:: If all U columns are B-collinear to V then the
returned tensor U will be empty.
Args:
U (Tensor) : initial approximation, size is (m, n)
V (Tensor) : B-orthogonal external basis, size is (m, k)
Returns:
U (Tensor) : B-orthonormal columns (:math:`U^T B U = I`)
such that :math:`V^T B U=0`, size is (m, n1),
where `n1 = n` if `drop` is `False, otherwise
`n1 <= n`.
"""
mm = torch.matmul
mm_B = _utils.matmul
m = self.iparams["m"]
tau_ortho = self.fparams["ortho_tol"]
tau_drop = self.fparams["ortho_tol_drop"]
tau_replace = self.fparams["ortho_tol_replace"]
i_max = self.iparams["ortho_i_max"]
j_max = self.iparams["ortho_j_max"]
# when use_drop==True, enable dropping U columns that have
# small contribution to the `span([U, V])`.
use_drop = self.bparams["ortho_use_drop"]
# clean up variables from the previous call
for vkey in list(self.fvars.keys()):
if vkey.startswith("ortho_") and vkey.endswith("_rerr"):
self.fvars.pop(vkey)
self.ivars.pop("ortho_i", 0)
self.ivars.pop("ortho_j", 0)
BV_norm = torch.norm(mm_B(self.B, V))
BU = mm_B(self.B, U)
VBU = mm(V.mT, BU)
i = j = 0
stats = ""
for i in range(i_max):
U = U - mm(V, VBU)
drop = False
tau_svqb = tau_drop
for j in range(j_max):
if use_drop:
U = self._get_svqb(U, drop, tau_svqb)
drop = True
tau_svqb = tau_replace
else:
U = self._get_svqb(U, False, tau_replace)
if torch.numel(U) == 0:
# all initial U columns are B-collinear to V
self.ivars["ortho_i"] = i
self.ivars["ortho_j"] = j
return U
BU = mm_B(self.B, U)
UBU = mm(U.mT, BU)
U_norm = torch.norm(U)
BU_norm = torch.norm(BU)
R = UBU - torch.eye(UBU.shape[-1], device=UBU.device, dtype=UBU.dtype)
R_norm = torch.norm(R)
# https://github.com/pytorch/pytorch/issues/33810 workaround:
rerr = float(R_norm) * float(BU_norm * U_norm) ** -1
vkey = f"ortho_UBUmI_rerr[{i}, {j}]"
self.fvars[vkey] = rerr
if rerr < tau_ortho:
break
VBU = mm(V.mT, BU)
VBU_norm = torch.norm(VBU)
U_norm = torch.norm(U)
rerr = float(VBU_norm) * float(BV_norm * U_norm) ** -1
vkey = f"ortho_VBU_rerr[{i}]"
self.fvars[vkey] = rerr
if rerr < tau_ortho:
break
if m < U.shape[-1] + V.shape[-1]:
# TorchScript needs the class var to be assigned to a local to
# do optional type refinement
B = self.B
assert B is not None
raise ValueError(
"Overdetermined shape of U:"
f" #B-cols(={B.shape[-1]}) >= #U-cols(={U.shape[-1]}) + #V-cols(={V.shape[-1]}) must hold"
)
self.ivars["ortho_i"] = i
self.ivars["ortho_j"] = j
return U
# Calling tracker is separated from LOBPCG definitions because
# TorchScript does not support user-defined callback arguments:
LOBPCG_call_tracker_orig = LOBPCG.call_tracker
def LOBPCG_call_tracker(self):
self.tracker(self)