Efficient Tensor Decomposition and Its Application Naoki KAWASHIMA - - PowerPoint PPT Presentation
TNSAA2018-2019 Kobe Dec.3-Dec.6, 2018 Efficient Tensor Decomposition and Its Application Naoki KAWASHIMA (ISSP) Dec. 3, 2018 Occam's Razor " Pluralitas non est ponenda sine necessitate. " (We should not make more
TNSAA2018-2019 （Kobe Dec.3-Dec.6, 2018）
William of Occam "Pluralitas non est ponenda sine necessitate." (We should not make more assumptions, if not necessary.)
from Wikipedia
Be stingy with model parameters!
( )
= = =
=
1 1 2 1 , , , 1
1 2 1 2
, , , Cont
S S N S S S S
N N
S S S T T
physical index virtual index
1
S
2
S
3
S
N
S
8
T
6
T
5
T
7
T
4
T
3
T
2
T
1
T
Parametrized by only O(N) tensors.
Traditional model O(1) TN model O(N) Exact model O(eN)
i
PEPS satisfies the area law by definition.
Majority of low-T condensed matter physics problems satisfy the "Area Law"
Gu, Levin, Wen: PRB 78 (2008); Schuch, et al: PRL 98 (2007)
𝑈 = 𝑉𝑇𝑊 = 𝑉 መ 𝑇 𝑊 = 𝑉 መ 𝑇 መ 𝑇 𝑊 = 𝑈
1𝑈2
𝑦 𝑦 𝑦 𝑦 𝑦
𝑈
1
𝑈
2
= ≈
𝑉 መ 𝑇 𝑊
𝑈 𝑉 𝑇 𝑊
=
𝑈
1
𝑈2
Singular Value Decomposition (SVD) with Truncation
𝑦2 𝑦2 𝑦2 𝑦
𝑦 𝑦 𝑦 𝑦
𝑈
𝑦 𝑦 𝑦 𝑦 𝑦2
𝑈
1
𝑈
2
= ≈
2D Potts model (L<=1,048,576) HOTRG calculation with χ～50
polynomial time calculation of Tc
arXiv:1806.10275
Finite Size Scaling 1st order nature
confirmed See Morita's talk
Tensor Network Calculation of ab-initio model for Na2IrO3
ab-initio model for Na2IrO3 Experimental
(zigzag state) is reproduced.
Hyunyong LEE (ISSP)
H.Y. Lee and NK: PRB 97, 205123 (2018) E dE/dΦ M Q
Optimization condition for u, v and w RG transformation: converges faster when D increased can get rid of local entanglement
Evenbly and Vidal: Phys. Rev. Lett. 115, 180405 (2015)
It also appears as the fixed point tensor of the TRG procedure in the disordered phase.
The fate of a local entanglement loop
Suppose each tensor is a CDL
Focus on a plaquette
The fate of a local entanglement loop
The 1st SVD The entanglement loop is deformed.
The fate of a local entanglement loop
The network after contraction of small squares. The green loop is still there.
The fate of a local entanglement loop
The 2nd SVD
The fate of a local entanglement loop
Some of the ent. loops have been
each generation the influence of the original ent. loop remains. The expressive capacity of the network is wasted. After the 2nd SVD The green loop still survives.
The fate of a local entanglement loop
By pinching the "information path", we can split the remaining loop, and remove them at the next contraction. It is essential that this line is thin.
Evenbly and Vidal: Phys. Rev. Lett. 115, 180405 (2015)
data set = 100 object image sets 1 object image set = 72 images 1 image = 128 x 128 dots 1 dot = 3 colors Zhao, Cichocki ら arXiv:1606.05535 Columbia Object Image Libraries (COIL)-100 dataset
xyci
T
1, ,128 1, ,128 1, ,3 1, ,7200 x y c i = = = = ... x-coordinate ... y-coordinate ... color ... image ID T
Z4 Z1 Z2 Z3
x y c i
COIL100 2D image classification task 128 x 128 x 3 x 7200 bits "open chain" "ring" Ring decomposition shows better performance.
maximum bond dim. average bond dim. tolerated error score （%) (large training) score （%) (small training)
Zhao, Cichocki ら arXiv:1606.05535 KNN classifier (K=1) applied to the image specifier core (Z4).
... but can we do that easily?
Z1 Z2 Z3 Z4
(1) random initial tensors Zi (2) for i=1,2,3,4, update Zi by 2
T
Z1 Z2 Z3 Z4
－
min
Zi (3) repeat until the error converges However, ALS is trapped by local loops.
sALS ... The initial condition obtained by sequential (open chain) SVD
ALS on CDL is either unstable or stuck with a local minimum. (At least partially, due to the local entanglement loops.)
H.-Y. Lee and N.K. arXiv:1807.03862
U
If we knew U, V, W and x, y, z explicitly, we can find Z1, Z2, Z3 of the TRD very easily. ... but how do we know them?
H.-Y. Lee and N.K. arXiv:1807.03862
( )
, , i pq i I p q
= 𝑓𝑗𝜚
When the given tensor T is a CDL, i.e., it must have the following form:
T
U V W
... then, we can find U, V and W by HOSVD
H.-Y. Lee and N.K. arXiv:1807.03862
( )
, , i pq i I p q
=
, I p q
... injection from (p,q) to i such as e.g.,
( ) ( ) ( ) ( )
0,0 0,1 1 1,0 2 1,1 3 I I I I = = = =
H.-Y. Lee and N.K. arXiv:1807.03862
t T
U V W
t t =
If T is expressed as a core tensor t and unitaries U, V, and W, where any matrix slices of t are mutually orthogonal, such an expression is unique up to the permutation within each index and the phase factors.
x mutual orthogonality
HOSVD
CDL satisfies mutual orthogonality → U,V,W can be obtained by HOSVD
H.-Y. Lee and N.K. arXiv:1807.03862
4th 4th with Random Noise By working directly with the inner structure, we can avoid the difficulty of the local minima in the optimization.
H.-Y. Lee and N.K. arXiv:1807.03862
4 16 16 16 16 4 4 4
H.-Y. Lee and N.K. arXiv:1807.03862
Quantum Information Condensed Matter Theory Information Processing
TNS
Optical Lattice
TN
Renormalization Group Ring Decomp.
mutual information Entanglement
MERA
■ CDL-like structure typical in TN-based RG often cause serious difficulty. ■ Index-splitting based on HOSVD may be useful in
■ TN representation makes it possible to handle extremely large systems, frustrated systems, etc.