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1168 lines
43 KiB
1168 lines
43 KiB
"""Locally Optimal Block Preconditioned Conjugate Gradient methods.
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"""
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# Author: Pearu Peterson
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# Created: February 2020
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from typing import Dict, Optional, Tuple
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import torch
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from torch import Tensor
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from . import _linalg_utils as _utils
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from .overrides import handle_torch_function, has_torch_function
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__all__ = ["lobpcg"]
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def _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U):
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# compute F, such that F_ij = (d_j - d_i)^{-1} for i != j, F_ii = 0
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F = D.unsqueeze(-2) - D.unsqueeze(-1)
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F.diagonal(dim1=-2, dim2=-1).fill_(float("inf"))
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F.pow_(-1)
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# A.grad = U (D.grad + (U^T U.grad * F)) U^T
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Ut = U.mT.contiguous()
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res = torch.matmul(
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U, torch.matmul(torch.diag_embed(D_grad) + torch.matmul(Ut, U_grad) * F, Ut)
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)
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return res
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def _polynomial_coefficients_given_roots(roots):
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"""
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Given the `roots` of a polynomial, find the polynomial's coefficients.
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If roots = (r_1, ..., r_n), then the method returns
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coefficients (a_0, a_1, ..., a_n (== 1)) so that
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p(x) = (x - r_1) * ... * (x - r_n)
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= x^n + a_{n-1} * x^{n-1} + ... a_1 * x_1 + a_0
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Note: for better performance requires writing a low-level kernel
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"""
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poly_order = roots.shape[-1]
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poly_coeffs_shape = list(roots.shape)
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# we assume p(x) = x^n + a_{n-1} * x^{n-1} + ... + a_1 * x + a_0,
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# so poly_coeffs = {a_0, ..., a_n, a_{n+1}(== 1)},
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# but we insert one extra coefficient to enable better vectorization below
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poly_coeffs_shape[-1] += 2
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poly_coeffs = roots.new_zeros(poly_coeffs_shape)
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poly_coeffs[..., 0] = 1
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poly_coeffs[..., -1] = 1
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# perform the Horner's rule
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for i in range(1, poly_order + 1):
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# note that it is computationally hard to compute backward for this method,
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# because then given the coefficients it would require finding the roots and/or
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# calculating the sensitivity based on the Vieta's theorem.
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# So the code below tries to circumvent the explicit root finding by series
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# of operations on memory copies imitating the Horner's method.
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# The memory copies are required to construct nodes in the computational graph
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# by exploting the explicit (not in-place, separate node for each step)
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# recursion of the Horner's method.
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# Needs more memory, O(... * k^2), but with only O(... * k^2) complexity.
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poly_coeffs_new = poly_coeffs.clone() if roots.requires_grad else poly_coeffs
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out = poly_coeffs_new.narrow(-1, poly_order - i, i + 1)
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out -= roots.narrow(-1, i - 1, 1) * poly_coeffs.narrow(
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-1, poly_order - i + 1, i + 1
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)
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poly_coeffs = poly_coeffs_new
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return poly_coeffs.narrow(-1, 1, poly_order + 1)
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def _polynomial_value(poly, x, zero_power, transition):
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"""
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A generic method for computing poly(x) using the Horner's rule.
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Args:
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poly (Tensor): the (possibly batched) 1D Tensor representing
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polynomial coefficients such that
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poly[..., i] = (a_{i_0}, ..., a{i_n} (==1)), and
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poly(x) = poly[..., 0] * zero_power + ... + poly[..., n] * x^n
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x (Tensor): the value (possible batched) to evalate the polynomial `poly` at.
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zero_power (Tensor): the representation of `x^0`. It is application-specific.
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transition (Callable): the function that accepts some intermediate result `int_val`,
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the `x` and a specific polynomial coefficient
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`poly[..., k]` for some iteration `k`.
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It basically performs one iteration of the Horner's rule
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defined as `x * int_val + poly[..., k] * zero_power`.
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Note that `zero_power` is not a parameter,
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because the step `+ poly[..., k] * zero_power` depends on `x`,
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whether it is a vector, a matrix, or something else, so this
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functionality is delegated to the user.
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"""
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res = zero_power.clone()
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for k in range(poly.size(-1) - 2, -1, -1):
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res = transition(res, x, poly[..., k])
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return res
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def _matrix_polynomial_value(poly, x, zero_power=None):
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"""
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Evaluates `poly(x)` for the (batched) matrix input `x`.
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Check out `_polynomial_value` function for more details.
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"""
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# matrix-aware Horner's rule iteration
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def transition(curr_poly_val, x, poly_coeff):
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res = x.matmul(curr_poly_val)
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res.diagonal(dim1=-2, dim2=-1).add_(poly_coeff.unsqueeze(-1))
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return res
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if zero_power is None:
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zero_power = torch.eye(
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x.size(-1), x.size(-1), dtype=x.dtype, device=x.device
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).view(*([1] * len(list(x.shape[:-2]))), x.size(-1), x.size(-1))
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return _polynomial_value(poly, x, zero_power, transition)
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def _vector_polynomial_value(poly, x, zero_power=None):
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"""
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Evaluates `poly(x)` for the (batched) vector input `x`.
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Check out `_polynomial_value` function for more details.
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"""
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# vector-aware Horner's rule iteration
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def transition(curr_poly_val, x, poly_coeff):
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res = torch.addcmul(poly_coeff.unsqueeze(-1), x, curr_poly_val)
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return res
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if zero_power is None:
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zero_power = x.new_ones(1).expand(x.shape)
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return _polynomial_value(poly, x, zero_power, transition)
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def _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest):
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# compute a projection operator onto an orthogonal subspace spanned by the
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# columns of U defined as (I - UU^T)
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Ut = U.mT.contiguous()
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proj_U_ortho = -U.matmul(Ut)
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proj_U_ortho.diagonal(dim1=-2, dim2=-1).add_(1)
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# compute U_ortho, a basis for the orthogonal complement to the span(U),
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# by projecting a random [..., m, m - k] matrix onto the subspace spanned
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# by the columns of U.
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#
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# fix generator for determinism
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gen = torch.Generator(A.device)
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# orthogonal complement to the span(U)
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U_ortho = proj_U_ortho.matmul(
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torch.randn(
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(*A.shape[:-1], A.size(-1) - D.size(-1)),
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dtype=A.dtype,
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device=A.device,
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generator=gen,
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)
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)
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U_ortho_t = U_ortho.mT.contiguous()
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# compute the coefficients of the characteristic polynomial of the tensor D.
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# Note that D is diagonal, so the diagonal elements are exactly the roots
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# of the characteristic polynomial.
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chr_poly_D = _polynomial_coefficients_given_roots(D)
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# the code belows finds the explicit solution to the Sylvester equation
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# U_ortho^T A U_ortho dX - dX D = -U_ortho^T A U
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# and incorporates it into the whole gradient stored in the `res` variable.
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#
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# Equivalent to the following naive implementation:
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# res = A.new_zeros(A.shape)
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# p_res = A.new_zeros(*A.shape[:-1], D.size(-1))
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# for k in range(1, chr_poly_D.size(-1)):
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# p_res.zero_()
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# for i in range(0, k):
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# p_res += (A.matrix_power(k - 1 - i) @ U_grad) * D.pow(i).unsqueeze(-2)
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# res -= chr_poly_D[k] * (U_ortho @ poly_D_at_A.inverse() @ U_ortho_t @ p_res @ U.t())
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#
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# Note that dX is a differential, so the gradient contribution comes from the backward sensitivity
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# 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,
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# and we need to compute g(U_grad, A, U, D)
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#
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# The naive implementation is based on the paper
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# Hu, Qingxi, and Daizhan Cheng.
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# "The polynomial solution to the Sylvester matrix equation."
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# Applied mathematics letters 19.9 (2006): 859-864.
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#
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# We can modify the computation of `p_res` from above in a more efficient way
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# p_res = U_grad * (chr_poly_D[1] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k)).unsqueeze(-2)
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# + A U_grad * (chr_poly_D[2] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k - 1)).unsqueeze(-2)
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# + ...
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# + A.matrix_power(k - 1) U_grad * chr_poly_D[k]
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# Note that this saves us from redundant matrix products with A (elimination of matrix_power)
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U_grad_projected = U_grad
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series_acc = U_grad_projected.new_zeros(U_grad_projected.shape)
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for k in range(1, chr_poly_D.size(-1)):
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poly_D = _vector_polynomial_value(chr_poly_D[..., k:], D)
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series_acc += U_grad_projected * poly_D.unsqueeze(-2)
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U_grad_projected = A.matmul(U_grad_projected)
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# compute chr_poly_D(A) which essentially is:
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#
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# chr_poly_D_at_A = A.new_zeros(A.shape)
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# for k in range(chr_poly_D.size(-1)):
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# chr_poly_D_at_A += chr_poly_D[k] * A.matrix_power(k)
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#
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# Note, however, for better performance we use the Horner's rule
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chr_poly_D_at_A = _matrix_polynomial_value(chr_poly_D, A)
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# compute the action of `chr_poly_D_at_A` restricted to U_ortho_t
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chr_poly_D_at_A_to_U_ortho = torch.matmul(
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U_ortho_t, torch.matmul(chr_poly_D_at_A, U_ortho)
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)
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# we need to invert 'chr_poly_D_at_A_to_U_ortho`, for that we compute its
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# Cholesky decomposition and then use `torch.cholesky_solve` for better stability.
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# Cholesky decomposition requires the input to be positive-definite.
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# Note that `chr_poly_D_at_A_to_U_ortho` is positive-definite if
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# 1. `largest` == False, or
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# 2. `largest` == True and `k` is even
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# under the assumption that `A` has distinct eigenvalues.
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#
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# check if `chr_poly_D_at_A_to_U_ortho` is positive-definite or negative-definite
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chr_poly_D_at_A_to_U_ortho_sign = -1 if (largest and (k % 2 == 1)) else +1
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chr_poly_D_at_A_to_U_ortho_L = torch.linalg.cholesky(
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chr_poly_D_at_A_to_U_ortho_sign * chr_poly_D_at_A_to_U_ortho
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)
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# compute the gradient part in span(U)
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res = _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
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# incorporate the Sylvester equation solution into the full gradient
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# it resides in span(U_ortho)
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res -= U_ortho.matmul(
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chr_poly_D_at_A_to_U_ortho_sign
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* torch.cholesky_solve(
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U_ortho_t.matmul(series_acc), chr_poly_D_at_A_to_U_ortho_L
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)
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).matmul(Ut)
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return res
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def _symeig_backward(D_grad, U_grad, A, D, U, largest):
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# if `U` is square, then the columns of `U` is a complete eigenspace
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if U.size(-1) == U.size(-2):
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return _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
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else:
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return _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest)
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class LOBPCGAutogradFunction(torch.autograd.Function):
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@staticmethod
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def forward( # type: ignore[override]
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ctx,
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A: Tensor,
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k: Optional[int] = None,
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B: Optional[Tensor] = None,
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X: Optional[Tensor] = None,
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n: Optional[int] = None,
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iK: Optional[Tensor] = None,
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niter: Optional[int] = None,
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tol: Optional[float] = None,
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largest: Optional[bool] = None,
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method: Optional[str] = None,
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tracker: None = None,
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ortho_iparams: Optional[Dict[str, int]] = None,
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ortho_fparams: Optional[Dict[str, float]] = None,
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ortho_bparams: Optional[Dict[str, bool]] = None,
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) -> Tuple[Tensor, Tensor]:
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# makes sure that input is contiguous for efficiency.
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# Note: autograd does not support dense gradients for sparse input yet.
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A = A.contiguous() if (not A.is_sparse) else A
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if B is not None:
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B = B.contiguous() if (not B.is_sparse) else B
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D, U = _lobpcg(
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A,
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k,
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B,
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X,
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n,
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iK,
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niter,
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tol,
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largest,
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method,
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tracker,
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ortho_iparams,
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ortho_fparams,
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ortho_bparams,
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)
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ctx.save_for_backward(A, B, D, U)
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ctx.largest = largest
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return D, U
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@staticmethod
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def backward(ctx, D_grad, U_grad):
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A_grad = B_grad = None
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grads = [None] * 14
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A, B, D, U = ctx.saved_tensors
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largest = ctx.largest
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# lobpcg.backward has some limitations. Checks for unsupported input
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if A.is_sparse or (B is not None and B.is_sparse and ctx.needs_input_grad[2]):
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raise ValueError(
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"lobpcg.backward does not support sparse input yet."
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"Note that lobpcg.forward does though."
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)
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if (
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A.dtype in (torch.complex64, torch.complex128)
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or B is not None
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and B.dtype in (torch.complex64, torch.complex128)
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):
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raise ValueError(
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"lobpcg.backward does not support complex input yet."
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"Note that lobpcg.forward does though."
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)
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if B is not None:
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raise ValueError(
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"lobpcg.backward does not support backward with B != I yet."
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)
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if largest is None:
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largest = True
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# symeig backward
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if B is None:
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A_grad = _symeig_backward(D_grad, U_grad, A, D, U, largest)
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# A has index 0
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grads[0] = A_grad
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# B has index 2
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grads[2] = B_grad
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return tuple(grads)
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def lobpcg(
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A: Tensor,
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k: Optional[int] = None,
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B: Optional[Tensor] = None,
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X: Optional[Tensor] = None,
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n: Optional[int] = None,
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iK: Optional[Tensor] = None,
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niter: Optional[int] = None,
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tol: Optional[float] = None,
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largest: Optional[bool] = None,
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method: Optional[str] = None,
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tracker: None = None,
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ortho_iparams: Optional[Dict[str, int]] = None,
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ortho_fparams: Optional[Dict[str, float]] = None,
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ortho_bparams: Optional[Dict[str, bool]] = None,
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) -> Tuple[Tensor, Tensor]:
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"""Find the k largest (or smallest) eigenvalues and the corresponding
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eigenvectors of a symmetric positive definite generalized
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eigenvalue problem using matrix-free LOBPCG methods.
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This function is a front-end to the following LOBPCG algorithms
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selectable via `method` argument:
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`method="basic"` - the LOBPCG method introduced by Andrew
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Knyazev, see [Knyazev2001]. A less robust method, may fail when
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Cholesky is applied to singular input.
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`method="ortho"` - the LOBPCG method with orthogonal basis
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selection [StathopoulosEtal2002]. A robust method.
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Supported inputs are dense, sparse, and batches of dense matrices.
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.. note:: In general, the basic method spends least time per
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iteration. However, the robust methods converge much faster and
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are more stable. So, the usage of the basic method is generally
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not recommended but there exist cases where the usage of the
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basic method may be preferred.
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.. warning:: The backward method does not support sparse and complex inputs.
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It works only when `B` is not provided (i.e. `B == None`).
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We are actively working on extensions, and the details of
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the algorithms are going to be published promptly.
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.. warning:: While it is assumed that `A` is symmetric, `A.grad` is not.
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To make sure that `A.grad` is symmetric, so that `A - t * A.grad` is symmetric
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in first-order optimization routines, prior to running `lobpcg`
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we do the following symmetrization map: `A -> (A + A.t()) / 2`.
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The map is performed only when the `A` requires gradients.
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Args:
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A (Tensor): the input tensor of size :math:`(*, m, m)`
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B (Tensor, optional): the input tensor of size :math:`(*, m,
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m)`. When not specified, `B` is interpreted as
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identity matrix.
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X (tensor, optional): the input tensor of size :math:`(*, m, n)`
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where `k <= n <= m`. When specified, it is used as
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initial approximation of eigenvectors. X must be a
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dense tensor.
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iK (tensor, optional): the input tensor of size :math:`(*, m,
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m)`. When specified, it will be used as preconditioner.
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k (integer, optional): the number of requested
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eigenpairs. Default is the number of :math:`X`
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columns (when specified) or `1`.
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n (integer, optional): if :math:`X` is not specified then `n`
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specifies the size of the generated random
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approximation of eigenvectors. Default value for `n`
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is `k`. If :math:`X` is specified, the value of `n`
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(when specified) must be the number of :math:`X`
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columns.
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tol (float, optional): residual tolerance for stopping
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criterion. Default is `feps ** 0.5` where `feps` is
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smallest non-zero floating-point number of the given
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input tensor `A` data type.
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largest (bool, optional): when True, solve the eigenproblem for
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the largest eigenvalues. Otherwise, solve the
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eigenproblem for smallest eigenvalues. Default is
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`True`.
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method (str, optional): select LOBPCG method. See the
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description of the function above. Default is
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"ortho".
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niter (int, optional): maximum number of iterations. When
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reached, the iteration process is hard-stopped and
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the current approximation of eigenpairs is returned.
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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(_utils.transpose(Z[: n - nc, n - nc :])),
|
|
),
|
|
)
|
|
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) * _utils.transpose(d_col)
|
|
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 * _utils.transpose(d_col), 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(_utils.transpose(V), 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(_utils.transpose(U), 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(_utils.transpose(V), 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)
|