import torch
import config as cfg
from linalg.svd_gesdd import SVDGESDD
from linalg.svd_symeig import SVDSYMEIG
from linalg.svd_arnoldi import SVDSYMARNOLDI, SVDARNOLDI
from linalg.svd_rsvd import RSVD
[docs]def truncated_svd_gesdd(M, chi, abs_tol=1.0e-14, rel_tol=None, keep_multiplets=False, \
eps_multiplet=1.0e-12, verbosity=0):
r"""
:param M: matrix of dimensions :math:`N \times L`
:param chi: desired maximal rank :math:`\chi`
:param abs_tol: absolute tolerance on minimal singular value
:param rel_tol: relative tolerance on minimal singular value
:param keep_multiplets: truncate spectrum down to last complete multiplet
:param eps_multiplet: allowed splitting within multiplet
:param verbosity: logging verbosity
:type M: torch.tensor
:type chi: int
:type abs_tol: float
:type rel_tol: float
:type keep_multiplets: bool
:type eps_multiplet: float
:type verbosity: int
:return: leading :math:`\chi` left singular vectors U, right singular vectors V, and
singular values S
:rtype: torch.tensor, torch.tensor, torch.tensor
Returns leading :math:`\chi`-singular triples of a matrix M by computing the full
SVD :math:`M= USV^T`. Returned tensors have dimensions
.. math:: dim(U)=(N,\chi),\ dim(S)=(\chi,\chi),\ \textrm{and}\ dim(V)=(L,\chi)
"""
U, S, V = SVDGESDD.apply(M)
# estimate the chi_new
chi_new= chi
if keep_multiplets and chi<S.shape[0]:
# regularize by discarding small values
gaps=S[:chi+1].clone().detach()
# S[S < abs_tol]= 0.
gaps[gaps < abs_tol]= 0.
# compute gaps and normalize by larger sing. value. Introduce cutoff
# for handling vanishing values set to exact zero
gaps=(gaps[:chi]-S[1:chi+1])/(gaps[:chi]+1.0e-16)
gaps[gaps > 1.0]= 0.
if gaps[chi-1] < eps_multiplet:
# the chi is within the multiplet - find the largest chi_new < chi
# such that the complete multiplets are preserved
for i in range(chi-1,-1,-1):
if gaps[i] > eps_multiplet:
chi_new= i
break
St = S[:chi].clone()
St[chi_new+1:]=0.
Ut = U[:, :St.shape[0]].clone()
Ut[:, chi_new+1:]=0.
Vt = V[:, :St.shape[0]].clone()
Vt[:, chi_new+1:]=0.
return Ut, St, Vt
St = S[:min(chi,S.shape[0])]
Ut = U[:, :St.shape[0]]
Vt = V[:, :St.shape[0]]
return Ut, St, Vt
[docs]def truncated_svd_symeig(M, chi, abs_tol=1.0e-14, rel_tol=None, keep_multiplets=False, \
eps_multiplet=1.0e-12, verbosity=0):
r"""
:param M: square matrix of dimensions :math:`N \times N`
:param chi: desired maximal rank :math:`\chi`
:param abs_tol: absolute tolerance on minimal singular value
:param rel_tol: relative tolerance on minimal singular value
:param keep_multiplets: truncate spectrum down to last complete multiplet
:param eps_multiplet: allowed splitting within multiplet
:param verbosity: logging verbosity
:type M: torch.tensor
:type chi: int
:type abs_tol: float
:type rel_tol: float
:type keep_multiplets: bool
:type eps_multiplet: float
:type verbosity: int
:return: leading :math:`\chi` left singular vectors U, right singular vectors V, and
singular values S
:rtype: torch.tensor, torch.tensor, torch.tensor
Returns leading :math:`\chi`-singular triples of a matrix M, where M is a symmetric
matrix :math:`M=M^T`, by computing the full symmetric decomposition :math:`M= UDU^T`.
Returned tensors have dimensions
.. math:: dim(U)=(N,\chi),\ dim(S)=(\chi,\chi),\ \textrm{and}\ dim(V)=(N,\chi)
"""
U, S, V = SVDSYMEIG.apply(M)
# estimate the chi_new
chi_new= chi
if keep_multiplets and chi<S.shape[0]:
# regularize by discarding small values
gaps=S[:chi+1].clone().detach()
# S[S < abs_tol]= 0.
gaps[gaps < abs_tol]= 0.
# compute gaps and normalize by larger sing. value. Introduce cutoff
# for handling vanishing values set to exact zero
gaps=(gaps[:chi]-S[1:chi+1])/(gaps[:chi]+1.0e-16)
gaps[gaps > 1.0]= 0.
if gaps[chi-1] < eps_multiplet:
# the chi is within the multiplet - find the largest chi_new < chi
# such that the complete multiplets are preserved
for i in range(chi-1,-1,-1):
if gaps[i] > eps_multiplet:
chi_new= i
break
St = S[:chi].clone()
St[chi_new+1:]=0.
Ut = U[:, :St.shape[0]].clone()
Ut[:, chi_new+1:]=0.
Vt = V[:, :St.shape[0]].clone()
Vt[:, chi_new+1:]=0.
return Ut, St, Vt
St = S[:min(chi,S.shape[0])]
Ut = U[:, :St.shape[0]]
Vt = V[:, :St.shape[0]]
return Ut, St, Vt
[docs]def truncated_svd_symarnoldi(M, chi, abs_tol=1.0e-14, rel_tol=None, keep_multiplets=False, \
eps_multiplet=1.0e-12, verbosity=0):
r"""
:param M: square matrix of dimensions :math:`N \times N`
:param chi: desired maximal rank :math:`\chi`
:param abs_tol: absolute tolerance on minimal singular value
:param rel_tol: relative tolerance on minimal singular value
:param keep_multiplets: truncate spectrum down to last complete multiplet
:param eps_multiplet: allowed splitting within multiplet
:param verbosity: logging verbosity
:type M: torch.tensor
:type chi: int
:type abs_tol: float
:type rel_tol: float
:type keep_multiplets: bool
:type eps_multiplet: float
:type verbosity: int
:return: leading :math:`\chi` left singular vectors U, right singular vectors V, and
singular values S
:rtype: torch.tensor, torch.tensor, torch.tensor
**Note:** `depends on scipy`
Returns leading :math:`\chi`-singular triples of a matrix M, where M is a symmetric matrix :math:`M=M^T`,
by computing the partial symmetric decomposition :math:`M= UDU^T` up to rank :math:`\chi`.
Returned tensors have dimensions
.. math:: dim(U)=(N,\chi),\ dim(S)=(\chi,\chi),\ \textrm{and}\ dim(V)=(N,\chi)
"""
U, S, V = SVDSYMARNOLDI.apply(M, chi+int(keep_multiplets))
# estimate the chi_new
chi_new= chi
if keep_multiplets:
# regularize by discarding small values
gaps=S.clone().detach()
# S[S < abs_tol]= 0.
gaps[gaps < abs_tol]= 0.
# compute gaps and normalize by larger sing. value. Introduce cutoff
# for handling vanishing values set to exact zero
gaps=(gaps[:len(S)-1]-S[1:len(S)])/(gaps[:len(S)-1]+1.0e-16)
gaps[gaps > 1.0]= 0.
if gaps[chi-1] < eps_multiplet:
# the chi is within the multiplet - find the largest chi_new < chi
# such that the complete multiplets are preserved
for i in range(chi-1,-1,-1):
if gaps[i] > eps_multiplet:
chi_new= i
break
St = S[:chi].clone()
St[chi_new+1:]=0.
Ut = U[:, :St.shape[0]].clone()
Ut[:, chi_new+1:]=0.
Vt = V[:, :St.shape[0]].clone()
Vt[:, chi_new+1:]=0.
return Ut, St, Vt
return U, S, V
[docs]def truncated_svd_arnoldi(M, chi, abs_tol=1.0e-14, rel_tol=None, keep_multiplets=False, \
eps_multiplet=1.0e-12, verbosity=0):
r"""
:param M: square matrix of dimensions :math:`N \times N`
:param chi: desired maximal rank :math:`\chi`
:param abs_tol: absolute tolerance on minimal singular value
:param rel_tol: relative tolerance on minimal singular value
:param keep_multiplets: truncate spectrum down to last complete multiplet
:param eps_multiplet: allowed splitting within multiplet
:param verbosity: logging verbosity
:type M: torch.tensor
:type chi: int
:type abs_tol: float
:type rel_tol: float
:type keep_multiplets: bool
:type eps_multiplet: float
:type verbosity: int
:return: leading :math:`\chi` left singular vectors U, right singular vectors V, and
singular values S
:rtype: torch.tensor, torch.tensor, torch.tensor
**Note:** `depends on scipy`
Returns leading :math:`\chi`-singular triples of a matrix M,
by computing the partial symmetric decomposition of :math:`H=M^TM` as :math:`H= UDU^T`
up to rank :math:`\chi`. Returned tensors have dimensions
.. math:: dim(U)=(N,\chi),\ dim(S)=(\chi,\chi),\ \textrm{and}\ dim(V)=(N,\chi)
"""
U, S, V = SVDARNOLDI.apply(M, chi+int(keep_multiplets))
# estimate the chi_new
chi_new= chi
if keep_multiplets:
# regularize by discarding small values
gaps=S.clone().detach()
# S[S < abs_tol]= 0.
gaps[gaps < abs_tol]= 0.
# compute gaps and normalize by larger sing. value. Introduce cutoff
# for handling vanishing values set to exact zero
gaps=(gaps[:len(S)-1]-S[1:len(S)])/(gaps[:len(S)-1]+1.0e-16)
gaps[gaps > 1.0]= 0.
if gaps[chi-1] < eps_multiplet:
# the chi is within the multiplet - find the largest chi_new < chi
# such that the complete multiplets are preserved
for i in range(chi-1,-1,-1):
if gaps[i] > eps_multiplet:
chi_new= i
break
St = S[:chi].clone()
St[chi_new+1:]=0.
Ut = U[:, :St.shape[0]].clone()
Ut[:, chi_new+1:]=0.
Vt = V[:, :St.shape[0]].clone()
Vt[:, chi_new+1:]=0.
return Ut, St, Vt
return U, S, V
def truncated_svd_rsvd(M, chi, abs_tol=None, rel_tol=None):
return RSVD.apply(M, chi)