[NDTensors] [ITensors] Implement optional rank reduction for QR/RQ/QL/LQ decompositions #1099
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Try and reduce complexity prior to implementing qr with combiners.
Also demonstrate index collection that fails for block sparse QR.
...With help from Niklas Tausendpfundt
Heisenberg Hamiltonian
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## main #1099 +/- ##
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- Coverage 59.45% 53.92% -5.53%
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- Hits 4943 4464 -479
- Misses 3371 3814 +443
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| # of Q. X = R or L | ||
| # | ||
| function trim_rows( | ||
| Q::AbstractMatrix, X::AbstractMatrix, rr_cutoff::Float64; rr_verbose=false, kwargs... |
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I would just name these keyword arguments cutoff and verbose (i.e. drop the rr_).
Also, cutoff has a particular meaning in the context of SVD, where it is the sum of squares of the singular values.
I see here it is used as:
zeros = map((r) -> (maximum(abs.(X[r, 1:Xnc])) <= rr_cutoff), 1:Xnr)so is checking that the maximum value in the row is less than the cutoff value/tolerance. Is that a standard approach for reducing the rank?
Maybe we can look at https://github.com/JuliaLinearAlgebra/LowRankApprox.jl and see what conventions they use for performing the rank reduction and what they call the keyword arguments (I see rtol and atol being used, which is used elsewhere in Julia say may be good choices).
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Issue 1) Do we need a distinct name for the rank revailing cutoff? Right now I have to code set up so that the user can just call truncate!(InfiniteMPO) without first calling orthogonalize!(InfiniteMPO). The truncate function checks if the iMPO is already orthogonalized and if not is calls orthogonalize before truncating with svd. So with this setup we need distinct control of Rank Revealing QR cutoff and the svd cutoff. rtol would work for this.
Issue 2) How to decide if a row is effectively zero? Here is what Daniel Parker says:
I looked in his thesis and there is no more info on this topic.
I agree that testing rr_cutoff > sqrt(sum(X[r, 1:Xnc]^2)) is closer to what svd truncation does.
Issue #3: Hornets nest: Conventional rank reducting QR does not do what Daniel did. Conventional methods do column pivoting QR, which is akin to "I am not going to solve A=QR for you, instead I am going solve AP=QR, where P is a column permutation matrix" This puts all near zero rows of R into the lower right corner. It acutally orders the diagonals of R similar singular value ordering. Maybe Parker purposely avoided this because he knew it would jumble up the MPOs and destroy his regular form. Maybe column pivoting QR and doing A = QP^T PRP^T (P^T=transpose(P)) would get us back to where we are now, but I haven't tried it. We could insist the library needs to support conventional rank reducting QR with colunm pivoting, not Daniels version , but I think users want to solve A=QR, not AP=QR, in other words they don't want to be told "Sorry I am not solving your problem, I will only solve my choice of problem". Maybe there is an easy solution and I am just not seeing it.
Perhaps I should email Daniel and get his opinion on max(abs(row)) vs. rms(row), and if column pivoting QR was tested and rejected for some reason.
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Within the qx functions, there is no fundamental reason why it needs a distinct name from cutoff. If we choose cutoff, we can choose different names in the higher level functions like truncate!(::MPO) to distinguish between the qr truncation tolerence/cutoff and the svd truncation/cutoff, for example with a code pattern like:
function truncate!(::MPO; qr_cutoff, svd_cutoff)
svd(T1; cutoff=svd_cutoff)
qr(T2; cutoff=qr_cutoff)
endHowever, I am just considering different options since the qx tolerance/cutoff seems to have a different meaning from the svd tolerance/cutoff, and it may be nice to keep consistency with packages like LowRankApprox.jl (or use that package directly for rank reducing QR).
Pivoted QR is a pretty standard technique and I think it is reasonable to support that. For example, Julia has a pivoted QR:
julia> using LinearAlgebra
julia> A = randn(4, 4);
julia> Q, R, p = qr(A, ColumnNorm())
QRPivoted{Float64, Matrix{Float64}, Vector{Float64}, Vector{Int64}}
Q factor:
4×4 LinearAlgebra.QRPackedQ{Float64, Matrix{Float64}, Vector{Float64}}:
-0.312837 -0.842554 -0.43312 -0.0681411
0.856948 -0.445627 0.228859 0.121162
0.341897 0.29066 -0.853861 0.263714
-0.225565 -0.0838845 0.175932 0.954532
R factor:
4×4 Matrix{Float64}:
3.20644 -0.309247 -1.45869 1.8667
0.0 2.13545 0.101833 -0.14863
0.0 0.0 1.81113 -0.0989319
0.0 0.0 0.0 -0.0716154
permutation:
4-element Vector{Int64}:
1
3
4
2
julia> norm(Q * R[:, invperm(p)] - A)
5.902547983198874e-16
julia> R[:, invperm(p)]
4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.14863 2.13545 0.101833
0.0 -0.0989319 0.0 1.81113
0.0 -0.0716154 0.0 0.0which would give better rank reduction than non-pivoted QR:
julia> Q2, R2 = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:
4×4 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.312837 -0.899079 0.232979 0.198774
0.856948 -0.181496 0.48104 -0.0360516
0.341897 -0.116376 -0.599446 0.714302
-0.225565 0.381016 0.595807 0.670046
R factor:
4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.192372 1.64988 1.01009
0.0 0.0 -1.35575 1.06884
0.0 0.0 0.0 -1.062I think it would be better to use pivoting, if possible (i.e. it would be better in general, so we should have it anyway), since it is the standard way to do rank reducing QR. We can just document that if you use pivoting and/or rank reduction then R is no longer guaranteed to be upper triangular (at the ITensor level, it isn't anyway because of the combiner transformation).
I think fundamentally the question is, does the Parker et al. algorithm actually rely on the upper/lower triangular form, or can it be framed more generally that you can do any orthogonal decomposition that does some rank reduction and preserves the proper block structure of the MPO? I have a hard time believing that such a detail as the ordering of the states within the block should matter, since it just corresponds to a simple gauge transformation by a permutation matrix on the links of the MPO. Once you gauge to a center bond/site, the left and right orthogonality centers are combined in order to perform the SVD, which would destroy the upper/lower triangular structure you were trying to preserve anyway.
I believe @LHerviou said he found he could reformulate the algorithm in terms of any orthogonal decomposition of the MPO block, which makes intuitive sense to me based on other tensor network gauging algorithms I know of.
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My statement was just:
The QR/QL does not seem important to me. What seems to matter is:
- Q -> the left part is a unitary, similar to what happens in left normalized MPS (assuming we are going for left canonical for now)
- We preserve the block structure of the canonical form of the InfiniteMPO
1 b c
A d
1
using the convention of upper triangular iMPOs in their paper.
Depending on left or right sweep, this means, orthogonalizing b or d with respect to 1, and then doing a QR on (b, A) or (A, d).
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- rr_cutoff -> cutoff
- rr_verbose -> verbose
- Switch from row removal to column pivoting
Done
|
Good point about the cutoff argument. It did not occur to me that I could
use rr_cutoff for the MPO truncate function and then simply pass it as
cutoff to the qr system.
I only realized very recently that Parker's method of removing rows is
non-standard. I will investigate how to use column pivoting in ITensors.
Regarding its interaction with MPO compression, keep in mind that block
respecting orthogonalization works on V block
[image: image.png]
And to maintain regular form, you have to preserve those I's and zeros.
SVD compression is different because it works on the center block (where
the A is) and so by construction avoids jumbling up the perimeter where the
regular form is defined. I agree that one can easily gauge transform the
regular form into a non regular form, but it is still local, and should be
othongonalizable and compresable. I also suspect the dependence on
triangular forms may be overstated in the Parker paper. I will experiment
with column pivoting in the ortho algo to see what the issues really are,
and then discuss with Loic if required.
…On Tue, Mar 28, 2023 at 1:37 PM Matt Fishman ***@***.***> wrote:
***@***.**** commented on this pull request.
------------------------------
In NDTensors/src/linearalgebra/linearalgebra.jl
<#1099 (comment)>:
> @@ -366,11 +366,58 @@ function LinearAlgebra.eigen(
V = complex(tensor(Dense(vec(VM)), Vinds))
return D, V, spec
end
+#
+# Trim out zero rows of R/X within tolerance rr_cutoff. Also trim the corresponding columns
+# of Q. X = R or L
+#
+function trim_rows(
+ Q::AbstractMatrix, X::AbstractMatrix, rr_cutoff::Float64; rr_verbose=false, kwargs...
Within the qx functions, there is no fundamental reason why it needs a
distinct name from cutoff. If we choose cutoff, we can choose different
names in the higher level functions like truncate!(::MPO) to distinguish
between the qr truncation tolerence/cutoff and the svd truncation/cutoff.
However, I am just considering different options since the qx
tolerance/cutoff seems to have a different meaning from the svd
tolerance/cutoff, and it may be nice to keep consistency with packages like
LowRankApprox.jl (or use that package directly for rank reducing QR).
Pivoted QR is a pretty standard technique and I think it is reasonable to
support that. For example, Julia has a pivoted QR:
julia> using LinearAlgebra
julia> A = randn(4, 4);
julia> Q, R, p = qr(A, ColumnNorm())
QRPivoted{Float64, Matrix{Float64}, Vector{Float64}, Vector{Int64}}
Q factor:4×4 LinearAlgebra.QRPackedQ{Float64, Matrix{Float64}, Vector{Float64}}:
-0.312837 -0.842554 -0.43312 -0.0681411
0.856948 -0.445627 0.228859 0.121162
0.341897 0.29066 -0.853861 0.263714
-0.225565 -0.0838845 0.175932 0.954532
R factor:4×4 Matrix{Float64}:
3.20644 -0.309247 -1.45869 1.8667
0.0 2.13545 0.101833 -0.14863
0.0 0.0 1.81113 -0.0989319
0.0 0.0 0.0 -0.0716154
permutation:4-element Vector{Int64}:
1
3
4
2
julia> norm(Q * R[:, invperm(p)] - A)5.902547983198874e-16
julia> R[:, invperm(p)]4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.14863 2.13545 0.101833
0.0 -0.0989319 0.0 1.81113
0.0 -0.0716154 0.0 0.0
which would give better rank reduction than non-pivoted QR:
julia> Q2, R2 = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:4×4 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.312837 -0.899079 0.232979 0.198774
0.856948 -0.181496 0.48104 -0.0360516
0.341897 -0.116376 -0.599446 0.714302
-0.225565 0.381016 0.595807 0.670046
R factor:4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.192372 1.64988 1.01009
0.0 0.0 -1.35575 1.06884
0.0 0.0 0.0 -1.062
I think it would be better to use pivoting, if possible (i.e. it would be
better in general, so we should have it anyway), since it is the standard
way to do rank reducing QR. We can either tell people that if you use rank
reduction then R is no longer guaranteed to be upper triangular (at the
ITensor level, it isn't anyway because of the combiner transformation).
I think fundamentally the question is, does the Parker et al. algorithm
actually rely on the upper/lower triangular form, or can it be framed more
generally that you can do any orthogonal decomposition that does some rank
reduction and preserves the proper block structure of the MPO? I have a
hard time believing that such a detail as the ordering of the states
*within* the block should matter, since it just corresponds to a simple
gauge transformation by a permutation matrix on the links of the MPO. Once
you gauge to a center bond/site, the left and right orthogonality centers
are combined in order to perform the SVD, which would destroy the
upper/lower triangular structure you were trying to preserve anyway.
I believe @LHerviou <https://github.com/LHerviou> said he found he could
reformulate the algorithm in terms of any orthogonal decomposition of the
MPO block, which makes intuitive sense to me based on other tensor network
gauging algorithm I know of.
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It seems LAPACK only supports column pivoting for QR (and therefore
indirectly for LQ), but not for QL/RQ. A table of supported orthogonal
factorizations is here: https://netlib.org/lapack/lug/node44.html
The MatrixFactorization.jl library also does not support pivoting ql, which is not
a surprise since LAPACK does not.
…On Tue, Mar 28, 2023 at 1:37 PM Matt Fishman ***@***.***> wrote:
***@***.**** commented on this pull request.
------------------------------
In NDTensors/src/linearalgebra/linearalgebra.jl
<#1099 (comment)>:
> @@ -366,11 +366,58 @@ function LinearAlgebra.eigen(
V = complex(tensor(Dense(vec(VM)), Vinds))
return D, V, spec
end
+#
+# Trim out zero rows of R/X within tolerance rr_cutoff. Also trim the corresponding columns
+# of Q. X = R or L
+#
+function trim_rows(
+ Q::AbstractMatrix, X::AbstractMatrix, rr_cutoff::Float64; rr_verbose=false, kwargs...
Within the qx functions, there is no fundamental reason why it needs a
distinct name from cutoff. If we choose cutoff, we can choose different
names in the higher level functions like truncate!(::MPO) to distinguish
between the qr truncation tolerence/cutoff and the svd truncation/cutoff.
However, I am just considering different options since the qx
tolerance/cutoff seems to have a different meaning from the svd
tolerance/cutoff, and it may be nice to keep consistency with packages like
LowRankApprox.jl (or use that package directly for rank reducing QR).
Pivoted QR is a pretty standard technique and I think it is reasonable to
support that. For example, Julia has a pivoted QR:
julia> using LinearAlgebra
julia> A = randn(4, 4);
julia> Q, R, p = qr(A, ColumnNorm())
QRPivoted{Float64, Matrix{Float64}, Vector{Float64}, Vector{Int64}}
Q factor:4×4 LinearAlgebra.QRPackedQ{Float64, Matrix{Float64}, Vector{Float64}}:
-0.312837 -0.842554 -0.43312 -0.0681411
0.856948 -0.445627 0.228859 0.121162
0.341897 0.29066 -0.853861 0.263714
-0.225565 -0.0838845 0.175932 0.954532
R factor:4×4 Matrix{Float64}:
3.20644 -0.309247 -1.45869 1.8667
0.0 2.13545 0.101833 -0.14863
0.0 0.0 1.81113 -0.0989319
0.0 0.0 0.0 -0.0716154
permutation:4-element Vector{Int64}:
1
3
4
2
julia> norm(Q * R[:, invperm(p)] - A)5.902547983198874e-16
julia> R[:, invperm(p)]4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.14863 2.13545 0.101833
0.0 -0.0989319 0.0 1.81113
0.0 -0.0716154 0.0 0.0
which would give better rank reduction than non-pivoted QR:
julia> Q2, R2 = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:4×4 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.312837 -0.899079 0.232979 0.198774
0.856948 -0.181496 0.48104 -0.0360516
0.341897 -0.116376 -0.599446 0.714302
-0.225565 0.381016 0.595807 0.670046
R factor:4×4 Matrix{Float64}:
3.20644 1.8667 -0.309247 -1.45869
0.0 -0.192372 1.64988 1.01009
0.0 0.0 -1.35575 1.06884
0.0 0.0 0.0 -1.062
I think it would be better to use pivoting, if possible (i.e. it would be
better in general, so we should have it anyway), since it is the standard
way to do rank reducing QR. We can either tell people that if you use rank
reduction then R is no longer guaranteed to be upper triangular (at the
ITensor level, it isn't anyway because of the combiner transformation).
I think fundamentally the question is, does the Parker et al. algorithm
actually rely on the upper/lower triangular form, or can it be framed more
generally that you can do any orthogonal decomposition that does some rank
reduction and preserves the proper block structure of the MPO? I have a
hard time believing that such a detail as the ordering of the states
*within* the block should matter, since it just corresponds to a simple
gauge transformation by a permutation matrix on the links of the MPO. Once
you gauge to a center bond/site, the left and right orthogonality centers
are combined in order to perform the SVD, which would destroy the
upper/lower triangular structure you were trying to preserve anyway.
I believe @LHerviou <https://github.com/LHerviou> said he found he could
reformulate the algorithm in terms of any orthogonal decomposition of the
MPO block, which makes intuitive sense to me based on other tensor network
gauging algorithm I know of.
—
Reply to this email directly, view it on GitHub
<#1099 (comment)>,
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If we determine that we don't care about the upper or lower triangular forms of the blocks (which presupposes that we can use pivoting for rank reduction in the first place for MPO compression), do we care about the |
|
Update from my end: I emailed Daniel Parker and he was glad to hear the ITensors team implementing his methods, and simultaneously a little skeptical that we can use pivoted QR. On Saturday I did some experiments with pivoted QR in place of QL and it worked for about 90% of my unit test cases, and failed on the more complex Hubbard model. So will I try and get the Hubbard working. I suspect you guys are right and I just need to understand the details well enough to reduce it to practice in the code. |
Take out rr prefixes
Factorize likes to pass cutoff=nothing down in the qr ?
Also add RowNorm for lq
RowNorm() for lq atol/rtol instead of cutoff block_rtol for future block sparse relative cutoff support
|
The interface update using atol/rtol instead of cutoff is done. |
Description
Additional functionality at the NDTensors level to strip out zero rows of R and the corresponding columns of Q, after QR decompoisiton. Same for LQ decomposition. The feature is controlled buy a new kwargs variable cutoff which is intended to act in a similar manner to cutoff in svd. Default is cutoff=-1.0 which is interpreted as no rank reduction. Typically for Float64, users will set 1e-15 < cutoff<1e-13 (depending on matrix size) to eliminate rows that are mostly likely zero to with roundoff error. This will leave Q*R invarient to within that precision level. The code uses column pivoting and removes bottom n rows in R norm(R_nn) < cutoff. When enabled ( cutoff>=0.0 ), rank reduction support for block sparse matrices will occur automaticlly on a block by block basis.
Rank reduction RQ/QL decomposition is not supported because lapack does not support column pivoting for these functions.
If practical and applicable, please include a minimal demonstration of the previous behavior and new behavior below.
Minimal demonstration of previous behavior
Minimal demonstration of new behavior
How Has This Been Tested?
See additional unit tests in:
NDTensors/test/linearalgebra.jl
test/base/test_decomp.jl testset Rank revealing QR/LQ decomp on MPS dense near line 389
Checklist:
using JuliaFormatter; format(".")in the base directory of the repository (~/.julia/dev/ITensors) to format your code according to our style guidelines.I plan to make a separate documentation PR for all of the QR/QL/LQ/RQ work, and corresponding updates to factorize().