Third-Order Tensor Decompositions and Their Application in Quantum - PowerPoint PPT Presentation

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 3rd-Order Tensor Decompositions 2 Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition Application to Quantum Chemistry 3 The Problem A Rotation Matrix Rotation by CP Decomposition References 4 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References Table of Contents Background 1 3rd-Order Tensor Decompositions 2 Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition Application to Quantum Chemistry 3 The Problem A Rotation Matrix Rotation by CP Decomposition References 4 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

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

Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References 3rd-Order Tensor 3rd-Order Tensor Definition Fibers: a a From Bader and Kolda 2009 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background 3rd-Order Tensor Decompositions Application to Quantum Chemistry References 3rd-Order Tensor 3rd-Order Tensor Definition Fibers Slices: a a From Bader and Kolda 2009 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Table of Contents Background 1 3rd-Order Tensor Decompositions 2 Modal Operations Higher Order SVD (HOSVD) CANDECOMP/PARAFAC Decomposition Application to Quantum Chemistry 3 The Problem A Rotation Matrix Rotation by CP Decomposition References 4 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Operations Modal Operations take Tensors to Matrices Example: Modal Unfolding     1 2 3 4 13 14 15 16 A 1 = 5 6 7 8 A 2 = 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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Operations Modal Operations take Tensors to Matrices Example: Modal Unfolding  1 2 3 4   13 14 15 16  A 1 = 5 6 7 8 A 2 = 17 18 19 20     9 10 11 12 21 22 23 24  1 2 3 4 13 14 15 16  A (1) = 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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Operations Modal Operations take Tensors to Matrices Example: Modal Unfolding     1 2 3 4 13 14 15 16 A 1 = 5 6 7 8 A 2 = 17 18 19 20     9 10 11 12 21 22 23 24   1 5 9 13 17 21 2 6 10 14 18 22   A (2) =   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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Operations Modal Operations take Tensors to Matrices Example: Modal Unfolding     1 2 3 4 13 14 15 16 A 1 = 5 6 7 8 A 2 = 17 18 19 20     9 10 11 12 21 22 23 24 � 1 � 2 3 4 5 6 7 8 9 10 11 12 A (3) = 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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References 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 ∈ R n 1 × n 2 × n 3 and a matrix U ∈ R J × n k , where J is any integer, is the product of modal unfolding A ( k ) with U . Such that B = U A ( k ) = A × k U Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Product Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A × 1 U = U A (1) Example: Modal Product Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Product Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A × 1 U = U A (1) Example: Modal Product     1 − 1 1 1 2 3 4 13 14 15 16 = 1 1 − 1 5 6 7 8 17 18 19 20     − 1 1 1 9 10 11 12 21 22 23 24 Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Modal Product Modal Operations take Tensors to Matrices Modal Unfolding Example Modal Product A × 1 U = U A (1) Example: Modal Product     1 − 1 1 1 2 3 4 13 14 15 16 = 1 1 − 1 5 6 7 8 17 18 19 20     − 1 1 1 9 10 11 12 21 22 23 24  5 6 7 8 17 18 19 20  = − 3 − 2 − 1 0 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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Higher Order SVD Definition: HOSVD Suppose A is a 3rd-order tensor and A ∈ R n 1 × n 2 × n 3 . Then there exists a Higher Order SVD such that U T k A ( k ) = Σ k V T (1 ≤ k ≤ d ) k where U k and V k are unitary matrices and the matrix Σ k contains the singular values of A ( k ) on the diagonal , [Σ k ] ij where i = j , and is zero elsewhere. Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Higher Order SVD Definition Example: 3rd-Order SVD U T 1 A (1) = ˆ A (1) → ˆ A A (2) = ˆ A (2) → ˆ U T 2 ˆ ˆ ˆ A 3 ˆ ˆ U T A (3) = S (3) → S Tyler Ueltschi April 17, 2014 Third-Order Tensor Decompositions and Their Application in Quantum

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Higher Order SVD Definition Example: 3rd-Order SVD − 1 . 1 × 10 − 14 3 . 1 × 10 − 16   − 69 . 627 0 . 0914 2 . 2 × 10 − 15 − 7 . 0 × 10 − 16 S 1 = − 0 . 033 − 1 . 0453   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

Background Modal Operations 3rd-Order Tensor Decompositions Higher Order SVD (HOSVD) Application to Quantum Chemistry CANDECOMP/PARAFAC Decomposition References Higher Order SVD Definition Example: 3rd-Order SVD − 1 . 1 × 10 − 14 3 . 1 × 10 − 16   − 69 . 627 0 . 0914 2 . 2 × 10 − 15 − 7 . 0 × 10 − 16 S 1 = − 0 . 033 − 1 . 0453   7 . 5 × 10 − 15 1 . 9 × 10 − 15 − 4 . 9 × 10 − 16 − 2 . 6 × 10 − 16 − 2 . 8 × 10 − 15 8 . 3 × 10 − 16   0 . 0201 2 . 212 − 4 . 2 × 10 − 16 9 . 8 × 10 − 16 S 2 = − 6 . 723 − 0 . 935   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