Third-Order Tensor Decompositions and Their...

Third-Order Tensor Decompositions and TheirApplication in Quantum Chemistry

Tyler Ueltschi April 17, 2014

April 17, 2014

Background3rd-Order Tensor Decompositions

Application to Quantum ChemistryReferences

Table of Contents

1 Background

2 3rd-Order Tensor DecompositionsModal OperationsHigher Order SVD (HOSVD)CANDECOMP/PARAFAC Decomposition

3 Application to Quantum ChemistryThe ProblemA Rotation MatrixRotation by CP Decomposition

4 References

Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum Chemistry



3rd-Order Tensor

Definition: 3rd-Order Tensor

An array of n ×m matrices




3rd-Order Tensor

3rd-Order Tensor Definition

Fibers:

a

aFrom Bader and Kolda 2009




3rd-Order Tensor

3rd-Order Tensor Definition

Fibers

Slices:

a

aFrom Bader and Kolda 2009




Modal OperationsHigher Order SVD (HOSVD)CANDECOMP/PARAFAC Decomposition

Table of Contents

1 Background



4 References





Modal Operations

Modal Operations take Tensors to Matrices

Example: Modal Unfolding

A1 =

1 2 3 45 6 7 89 10 11 12

A2 =

13 14 15 1617 18 19 2021 22 23 24





Modal Operations



A1 =

1 2 3 45 6 7 89 10 11 12

A2 =

13 14 15 1617 18 19 2021 22 23 24

A(1) =

1 2 3 4 13 14 15 165 6 7 8 17 18 19 209 10 11 12 21 22 23 24





Modal Operations



A1 =

1 2 3 45 6 7 89 10 11 12

A2 =

13 14 15 1617 18 19 2021 22 23 24

A(2) =

1 5 9 13 17 212 6 10 14 18 223 7 11 15 19 234 8 12 16 20 24





Modal Operations



A1 =

1 2 3 45 6 7 89 10 11 12

A2 =

13 14 15 1617 18 19 2021 22 23 24

A(3) =

[1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24

]





Modal Operations


Modal Unfolding Example

Definition: Modal Product

The modal product, denoted ×k , of a 3rd-order tensorA ∈ Rn1×n2×n3 and a matrix U ∈ RJ×nk , where J is any integer, isthe product of modal unfolding A(k) with U. Such that

B = UA(k) = A×k U





Modal Product


Modal Unfolding Example

Modal Product A×1 U = UA(1)

Example: Modal Product

=

1 −1 11 1 −1−1 1 1

1 2 3 4 13 14 15 165 6 7 8 17 18 19 209 10 11 12 21 22 23 24

=

5 6 7 8 17 18 19 20−3 −2 −1 0 9 10 11 1213 14 15 16 25 26 27 28





Higher Order SVD

Definition: HOSVD

Suppose A is a 3rd-order tensor and A ∈ Rn1×n2×n3 . Then thereexists a Higher Order SVD such that

UTk A(k) = ΣkV

Tk (1 ≤ k ≤ d)

where Uk and Vk are unitary matrices and the matrix Σk containsthe singular values of A(k) on the diagonal, [Σk ]ij where i = j , andis zero elsewhere.





Higher Order SVD Definition

Example: 3rd-Order SVD

UT1 A(1) = A(1) → A

UT2 A(2) = Â(2) →

Â

UT3

Â(3) = S(3) → S







S1 =

−69.627 0.0914 −1.1× 10−14 3.1× 10−16

−0.033 −1.0453 2.2× 10−15 −7.0× 10−16

7.5× 10−15 1.9× 10−15 −4.9× 10−16 −2.6× 10−16

S2 =

0.0201 2.212 −2.8× 10−15 8.3× 10−16

−6.723 −0.935 −4.2× 10−16 9.8× 10−16

5.2× 10−15 −3.9× 10−16 3.2× 10−16 8.8× 10−16







U1S(1) = S(1) → S

U2S(2) = ˆS(2) →ˆS

U3ˆS(3) = A(3) → A







A1 =

[1.0 2.05.0 6.0

]

A2 =

[13.0 14.017.0 18.0

]S = A×1 U

T1 ×2 U

T2 ×3 U

T3

A = S ×1 U1 ×2 U2 ×3 U3







A1 =

[1.0 2.05.0 6.0

]

A2 =

[13.0 14.017.0 18.0

]

S = A×1 UT1 ×2 U

T2 ×3 U

T3

A = S ×1 U1 ×2 U2 ×3 U3







A1 =

[1.0 2.05.0 6.0

]

A2 =

[13.0 14.017.0 18.0

]S = A×1 U

T1 ×2 U

T2 ×3 U

T3

A = S ×1 U1 ×2 U2 ×3 U3





CP Decomposition

Definition: Rank of a Tensor

The rank of a tensor A is the smallest number of rank 1 tensorsthat sum to A.





CP Decomposition

Definition: CP Decomposition

A CP decomposition of a 3rd-order tensor, A, is defined as a sumof vector outer products, denoted ◦, that equal or approximatelyequal A. For R = rank(A)

A =R∑

r=1

ar ◦ br ◦ cr

and for R < rank(A)

A ≈R∑

r=1

ar ◦ br ◦ cr





CP Decomposition

Example: CP Decomposition

A1 =

[1 00 1

]A2 =

[0 1−1 0

]

The rank decomposition over R is A = [[A,B,C]], where

A =

[1 0 10 1 −1

]B =

[1 0 10 1 1

]C =

[1 1 0−1 1 1

]but over C

A =1√2

[1 1−i i

]B =

1√2

[1 1i −i

]C =

[1 1i −i

]





CP Decomposition


A1 =

[1 00 1

]A2 =

[0 1−1 0

]The rank decomposition over R is A = [[A,B,C]], where

A =

[1 0 10 1 −1

]B =

[1 0 10 1 1

]C =

[1 1 0−1 1 1

]

but over C

A =1√2

[1 1−i i

]B =

1√2

[1 1i −i

]C =

[1 1i −i

]





CP Decomposition


A1 =

[1 00 1

]A2 =

[0 1−1 0

]The rank decomposition over R is A = [[A,B,C]], where

A =

[1 0 10 1 −1

]B =

[1 0 10 1 1

]C =

[1 1 0−1 1 1

]but over C

A =1√2

[1 1−i i

]B =

1√2

[1 1i −i

]C =

[1 1i −i

]




The ProblemA Rotation MatrixRotation by CP Decomposition

Table of Contents

1 Background



4 References





The Problem

The Problem

We have a 3× 3× 3 hyperpolarizability tensor and need to rotateit about 3 axes in space and there is currently no known 3rd-orderrotation tensor.





The Problem

The Problem


For matrices and vectors we have rotation matrices that will rotateour matrix/vector around 3 axes:

R =

cos(φ) cos(ψ)− cos(θ) sin(φ) sin(ψ) − cos(θ) cos(ψ) sin(φ)− cos(φ) sin(ψ) sin(θ) sin(φ)cos(ψ) sin(φ) + cos(θ) cos(φ) sin(ψ) cos(θ) cos(φ) cos(ψ)− sin(φ) sin(ψ) − cos(φ) sin(θ)

sin(θ) sin(ψ) cos(ψ) sin(θ) cos(θ)





The Problem

The Problem


For matrices and vectors we have rotation matrices that will rotateour matrix/vector around 3 axes:





The Problem

The Problem


Rotation by CP Decomposition

X → Xrot

X =3∑

j=1

(aj) ◦ (bj) ◦ (cj)

Xrot =3∑

j=1

(Raj) ◦ (Rbj) ◦ (Rcj)





The Problem

Rotation by CP Decomposition

X → Xrot

X =3∑

j=1

(aj) ◦ (bj) ◦ (cj)

Xrot =3∑

j=1

(Raj) ◦ (Rbj) ◦ (Rcj)

= [Ra1|Ra2|Ra3]� [Rb1|Rb2|Rb3]� [Rc1|Rc2|Rc3]

= R[a1|a2|a3]� R[b1|b2|b3]� R[c1|c2|c3]

= RA� RB� RC




Table of Contents

1 Background



4 References




The End

1 G.H. Golub and C.F. Van Loan, Matrix Computations, (TheJohns Hopkins University Press 2013).

2 Hongfei Wang et al, “Quantitative spectral and orientationalanalysis in surface sum frequency generation vibrationalspectroscopy (SFG-VS)”, International Reviews in PhysicalChemistry (2005), (24) no. 2, 191-256.

3 Kolda, T.G. and Bader, B.W., “Tensor Decompositions andApplications”, SIAM Review (2009), (51) no. 3, 455-500.

4 Carroll, J.Douglas and Chang, Jih-Jie, “Analysis of individualdifferences in multidimensional scaling via an n-waygeneralization of Eckart-Young decomposition”,Psychometrika (1970), (35) no. 3, 283-319.




The End

1 P. Paatero, “A weighted non-negative least squares algorithmfor three-way PARAFAC factor analysis”, Chemometrics andIntelligent Laboratory Systems (1997), (38) no. 2, 223-242.

2 N. K. M. Faber and R. Bro, and P. K. Hopke, “Recentdevelopments in CANDECOMP/PARAFAC algorithms: Acritical review”, Chemometrics and Intelligent LaboratorySystems (2003), (65), 119-137.

3 J. B. Kruskal, “Rank, decomposition, and uniqueness for3-way and N-way arrays, in Multiway Data Analysis”, 7-18.


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