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Third-Order Tensor Decompositions and Their Application in Quantum Chemistry Tyler Ueltschi April 17, 2014 April 17, 2014 Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Table of Contents Background 1

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Third-Order Tensor Decompositions and Their Application in Quantum Chemistry

Tyler Ueltschi April 17, 2014 April 17, 2014

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

Table of Contents

1

Background

2

3rd-Order Tensor Decompositions Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

3

Application to Quantum Chemistry The Problem A Rotation Matrix Rotation by CP Decomposition

4

References

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

Table of Contents

1

Background

2

3rd-Order Tensor Decompositions Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

3

Application to Quantum Chemistry The Problem A Rotation Matrix Rotation by CP Decomposition

4

References

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

3rd-Order Tensor

Definition: 3rd-Order Tensor An array of n × m matrices

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

3rd-Order Tensor

3rd-Order Tensor Definition Fibers:

a

aFrom Bader and Kolda 2009 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

3rd-Order Tensor

3rd-Order Tensor Definition Fibers Slices:

a

aFrom Bader and Kolda 2009 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Table of Contents

1

Background

2

3rd-Order Tensor Decompositions Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

3

Application to Quantum Chemistry The Problem A Rotation Matrix Rotation by CP Decomposition

4

References

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Operations

Modal Operations take Tensors to Matrices Example: Modal Unfolding A1 =   1 2 3 4 5 6 7 8 9 10 11 12   A2 =   13 14 15 16 17 18 19 20 21 22 23 24  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Operations

Modal Operations take Tensors to Matrices Example: Modal Unfolding A1 =   1 2 3 4 5 6 7 8 9 10 11 12   A2 =   13 14 15 16 17 18 19 20 21 22 23 24   A(1) =   1 2 3 4 13 14 15 16 5 6 7 8 17 18 19 20 9 10 11 12 21 22 23 24  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Operations

Modal Operations take Tensors to Matrices Example: Modal Unfolding A1 =   1 2 3 4 5 6 7 8 9 10 11 12   A2 =   13 14 15 16 17 18 19 20 21 22 23 24   A(2) =     1 5 9 13 17 21 2 6 10 14 18 22 3 7 11 15 19 23 4 8 12 16 20 24    

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Operations

Modal Operations take Tensors to Matrices Example: Modal Unfolding A1 =   1 2 3 4 5 6 7 8 9 10 11 12   A2 =   13 14 15 16 17 18 19 20 21 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

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Operations

Modal Operations take Tensors to Matrices Modal Unfolding Example Definition: Modal Product The modal product, denoted ×k, of a 3rd-order tensor A ∈ Rn1×n2×n3 and a matrix U ∈ RJ×nk, where J is any integer, is the product of modal unfolding A(k) with U. Such that B = UA(k) = A ×k U

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Product

Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A ×1 U = UA(1) Example: Modal Product

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Product

Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A ×1 U = UA(1) Example: Modal Product =   1 −1 1 1 1 −1 −1 1 1     1 2 3 4 13 14 15 16 5 6 7 8 17 18 19 20 9 10 11 12 21 22 23 24  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Modal Product

Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A ×1 U = UA(1) Example: Modal Product =   1 −1 1 1 1 −1 −1 1 1     1 2 3 4 13 14 15 16 5 6 7 8 17 18 19 20 9 10 11 12 21 22 23 24   =   5 6 7 8 17 18 19 20 −3 −2 −1 9 10 11 12 13 14 15 16 25 26 27 28  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD

Definition: HOSVD Suppose A is a 3rd-order tensor and A ∈ Rn1×n2×n3. Then there exists a Higher Order SVD such that UT

k A(k) = ΣkVT k

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

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD UT

1 A(1) = ˆ

A(1) → ˆ A UT

2 ˆ

A(2) = ˆ ˆ A(2) → ˆ ˆ A UT

3 ˆ

ˆ A(3) = S(3) → S

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD 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  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD 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  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD ˆ U1S(1) = ˆ S(1) → ˆ S ˆ U2 ˆ S(2) = ˆ ˆ S(2) → ˆ ˆ S ˆ U3 ˆ ˆ S(3) = A(3) → A

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD A1 = 1.0 2.0 5.0 6.0

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD A1 = 1.0 2.0 5.0 6.0

  • A2 =

13.0 14.0 17.0 18.0

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD A1 = 1.0 2.0 5.0 6.0

  • A2 =

13.0 14.0 17.0 18.0

  • S = A ×1 UT

1 ×2 UT 2 ×3 UT 3

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

Higher Order SVD Definition Example: 3rd-Order SVD A1 = 1.0 2.0 5.0 6.0

  • A2 =

13.0 14.0 17.0 18.0

  • S = A ×1 UT

1 ×2 UT 2 ×3 UT 3

A = S ×1 U1 ×2 U2 ×3 U3

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

CP Decomposition

Definition: Rank of a Tensor The rank of a tensor A is the smallest number of rank 1 tensors that sum to A.

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

CP Decomposition

Definition: CP Decomposition A CP decomposition of a 3rd-order tensor, A, is defined as a sum

  • f vector outer products, denoted ◦, that equal or approximately

equal A. For R = rank(A) A =

R

  • r=1

ar ◦ br ◦ cr and for R < rank(A) A ≈

R

  • r=1

ar ◦ br ◦ cr

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

CP Decomposition

Example: CP Decomposition A1 = 1 1

  • A2 =

1 −1

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

CP Decomposition

Example: CP Decomposition A1 = 1 1

  • A2 =

1 −1

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

A = 1 1 1 −1

  • B =

1 1 1 1

  • C =

1 1 −1 1 1

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

CP Decomposition

Example: CP Decomposition A1 = 1 1

  • A2 =

1 −1

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

A = 1 1 1 −1

  • B =

1 1 1 1

  • C =

1 1 −1 1 1

  • but over C

A = 1 √ 2 1 1 −i i

  • B =

1 √ 2 1 1 i −i

  • C =

1 1 i −i

  • Tyler Ueltschi April 17, 2014

Third-Order Tensor Decompositions and Their Application in Quantum

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

Table of Contents

1

Background

2

3rd-Order Tensor Decompositions Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

3

Application to Quantum Chemistry The Problem A Rotation Matrix Rotation by CP Decomposition

4

References

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor.

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor. For matrices and vectors we have rotation matrices that will rotate

  • ur 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)  

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor. For matrices and vectors we have rotation matrices that will rotate

  • ur matrix/vector around 3 axes:

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor. Rotation by CP Decomposition X → Xrot

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor. Rotation by CP Decomposition X → Xrot X =

3

  • j=1

(aj) ◦ (bj) ◦ (cj)

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

The Problem We have a 3 × 3 × 3 hyperpolarizability tensor and need to rotate it about 3 axes in space and there is currently no known 3rd-order rotation tensor. Rotation by CP Decomposition X → Xrot X =

3

  • j=1

(aj) ◦ (bj) ◦ (cj) Xrot =

3

  • j=1

(Raj) ◦ (Rbj) ◦ (Rcj)

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

The Problem

Rotation by CP Decomposition X → Xrot X =

3

  • j=1

(aj) ◦ (bj) ◦ (cj) Xrot =

3

  • j=1

(Raj) ◦ (Rbj) ◦ (Rcj)

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

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]

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

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]

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References The Problem A Rotation Matrix Rotation by CP Decomposition

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

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

Table of Contents

1

Background

2

3rd-Order Tensor Decompositions Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition

3

Application to Quantum Chemistry The Problem A Rotation Matrix Rotation by CP Decomposition

4

References

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

The End

1 G.H. Golub and C.F. Van Loan, Matrix Computations, (The

Johns Hopkins University Press 2013).

2 Hongfei Wang et al, “Quantitative spectral and orientational

analysis in surface sum frequency generation vibrational spectroscopy (SFG-VS)”, International Reviews in Physical Chemistry (2005), (24) no. 2, 191-256.

3 Kolda, T.G. and Bader, B.W., “Tensor Decompositions and

Applications”, SIAM Review (2009), (51) no. 3, 455-500.

4 Carroll, J.Douglas and Chang, Jih-Jie, “Analysis of individual

differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition”, Psychometrika (1970), (35) no. 3, 283-319.

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

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Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References

The End

1 P. Paatero, “A weighted non-negative least squares algorithm

for three-way PARAFAC factor analysis”, Chemometrics and Intelligent Laboratory Systems (1997), (38) no. 2, 223-242.

2 N. K. M. Faber and R. Bro, and P. K. Hopke, “Recent

developments in CANDECOMP/PARAFAC algorithms: A critical review”, Chemometrics and Intelligent Laboratory Systems (2003), (65), 119-137.

3 J. B. Kruskal, “Rank, decomposition, and uniqueness for

3-way and N-way arrays, in Multiway Data Analysis”, 7-18.

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