A t-SVD-based Nuclear Norm with Imaging Applications Ning Hao 2 Misha E. Kilmer 2 Oguz Semerci 1 Eric Miller 2 Shuchin Aeron 2 Gregory Ely 2 Zemin Zhang 2 1 Schlumberger-Doll Research 2 Tufts University Kilmer and Hao’s work supported by NSF-DMS 0914957 Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 1 / 28
Notation 1 A ( i,j,k ) = element of A in row i , column j , tube k 1 Graphics thanks to K. Braman Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 2 / 28
Notation 1 A ( i,j,k ) = element of A in row i , column j , tube k ← A 4 , 7 , 1 1 Graphics thanks to K. Braman Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 2 / 28
Notation 1 A ( i,j,k ) = element of A in row i , column j , tube k ← A 4 , 7 , 1 ← A : , 3 , 1 1 Graphics thanks to K. Braman Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 2 / 28
Notation 1 A ( i,j,k ) = element of A in row i , column j , tube k ← A 4 , 7 , 1 ← A : , 3 , 1 ← A : , : , 3 1 Graphics thanks to K. Braman Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 2 / 28
Motivation The application drives the choice of factorization (e.g. CP, Tucker) and the constraints. Today we are concerned with imaging applications, orientation dependent. Talk builds on: Closed multiplication operation between two tensors, factorizations reminiscent of matrix factorizations [K., Martin, Perrone, 2008; K., Martin 2010; Martin et al, 2012]. View of Third order tensors as operators on matrices, [K., Braman, Hoover, Hao, 2013] Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 3 / 28
Toward Defining Tensor-Tensor Multiplication For A ∈ R m × p × n , let A i = A : , : ,i . A 1 A 2 unfold A 3 ∈ R mn × p − → . . . A n Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 4 / 28
Toward Defining Tensor-Tensor Multiplication For A ∈ R m × p × n , let A i = A : , : ,i . A 1 A 2 unfold A 3 ∈ R mn × p − → . . . A n A 1 A 2 fold ∈ R m × p × n A 3 − → . . . A n Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 4 / 28
Block Circulant Matrix The block circulant matrix generated by unfold ( A ) is · · · A 1 A n A 3 A 2 A 2 A 1 A n · · · · · · ... ... ... circ ( A ) = A 3 A 2 . ... ... ... ... . . A n · · · · · · A 2 A 1 Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 5 / 28
Block Circulants A block circulant can be block-diagonalized by a (normalized) DFT in the 2nd dimension: ˆ 0 · · · 0 A 1 ˆ 0 A 2 0 · · · ( F ⊗ I )circ ( A ) ( F ∗ ⊗ I ) = ... 0 · · · 0 ˆ 0 · · · 0 A n Conveniently, an FFT along tube fibers of A gives � A . Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 6 / 28
Tensor - Tensor Multiplication [K., Martin, Perrone ‘08]: For A ∈ R m × p × n and B ∈ R p × q × n , define the t-product � � A ∗ B ≡ fold circ ( A ) · unfold ( B ) . Result is m × q × n . Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 7 / 28
Tensor - Tensor Multiplication [K., Martin, Perrone ‘08]: For A ∈ R m × p × n and B ∈ R p × q × n , define the t-product � � A ∗ B ≡ fold circ ( A ) · unfold ( B ) . Result is m × q × n . Example: A ∈ R m × p × 3 and B ∈ R p × q × 3 , A 1 A 3 A 2 B 1 . A ∗ B = fold A 2 A 1 A 3 B 2 A 3 A 2 A 1 B 3 Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 7 / 28
Tensor - Tensor Multiplication [K., Martin, Perrone ‘08]: For A ∈ R m × p × n and B ∈ R p × q × n , define the t-product � � A ∗ B ≡ fold circ ( A ) · unfold ( B ) . Result is m × q × n . Example: A ∈ R m × p × 3 and B ∈ R p × q × 3 , A 1 A 3 A 2 B 1 . A ∗ B = fold A 2 A 1 A 3 B 2 A 3 A 2 A 1 B 3 This tensor-tensor multiplication generalizes to higher-order tensors through recursion - see Martin et al, 2012. Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 7 / 28
More Definitions Definition A 1 × 1 × n tensor is called a tubal scalar. The t-product between tubal scalars is commutative ⇒ the t-product resembles matrix-matrix product with scalar mult replaced by t-product mult among tubal scalars. Definition The ℓ × ℓ × n identity tensor I is the tensor whose frontal slice is the ℓ × ℓ identity matrix, and whose other frontal slices are all zeros. Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 8 / 28
Transpose Definition If A is ℓ × m × n , then A T is the m × ℓ × n tensor obtained by transposing each of the frontal slices and then reversing the order of transposed frontal slices 2 through n . Example If A ∈ R ℓ × m × 4 A T 1 A T A T = fold 4 A T 3 A T 2 Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 9 / 28
Orthogonality Definition U ∈ R m × m × n is orthogonal if U T ∗ U = I = U ∗ U T . Can show Frobenius norm invariance: � U ∗ A � F = � A � F . Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 10 / 28
The t-SVD Theorem (K. and Martin, 2011) Let A ∈ R ℓ × m × n . Then A can be factored as A = U ∗ S ∗ V T where U , V are orthogonal ℓ × ℓ × n and m × m × n , and S is a ℓ × m × n f -diagonal tensor. Also, B = U 1: k, 1: k, : ∗ S 1: k, 1: k, : ∗ V T 1: k, 1: k, : satisfies M = { B = X ∗ Y , X ∈ R ℓ × k × n , Y ∈ R k × m × B = arg min M � A − B � F , Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 11 / 28
t-SVD Example Let A be 2 × 2 × 2 . � ˆ � A 1 0 ( F ⊗ I )circ ( A ) ( F ∗ ⊗ I ) = ˆ 0 A 2 � � σ (1) ˆ 0 1 � ˆ � � ˆ � � ˆ � σ (1) 0 ˆ 0 0 0 A 1 U 1 V ∗ 2 � � 1 = ˆ ˆ ˆ σ (2) 0 A 2 0 U 2 0 V ∗ ˆ 0 2 1 σ (2) 0 ˆ 2 The U , S , V T are formed by putting the hat matrices as frontal slices, ifft along tubes. � � σ (1) ˆ 1 e.g. s 1 = oriented into the screen. σ (2) ˆ 1 Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 12 / 28
Multi-rank Definition (K.,Braman,Hoover,Hao, 2013) Let A ∈ R ℓ × m × n . The multi-rank of A is length n vector consisting ( i ) , which must be symmetric about the of the ranks of all the � A “middle”. Example A ∈ R 2 × 2 × 4 , multi-rank possible: [ i, j, k, j ] T , 1 ≤ i, j, k ≤ 2 . Example A ∈ R 5 × 4 × 3 , multi-rank possible: [ i, j, j ] T , 1 ≤ i ≤ 4 , 1 ≤ j ≤ 4 . Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 13 / 28
Tensor Nuclear Norm If A is an ℓ × m , ℓ ≥ m matrix with singular values σ i , the nuclear norm � A � ⊛ = � m i =1 σ i . However, in the t-SVD, we have singular tubes (the entries of which need not be positive), which sum to a singular tube! The entries in the j th singular tube are the inverse Fourier coefficients of the length- n vector of the j th singular values of � A : , : ,i , i = 1 ..n . Definition For A ∈ R ℓ × m × n , our tensor nuclear norm is � A � ⊛ = � min( ℓ,m ) �√ nF s i � 1 = � min( ℓ,m ) � n j =1 � S i,i,j . (Same as the i =1 i =1 matrix nuclear norm of circ ( A ) ). Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 14 / 28
Tensor Nuclear Norm Theorem (Semerci,Hao,Kilmer,Miller) The tensor nuclear norm is a valid norm. Since the t-SVD extends to higher-order tensors [Martin et al, 2012], the norm does, as well. Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 15 / 28
TNN in Regularization and Optimization Yes, the t-SVD is orientation dependent, as is the norm. There are applications where this is particularly useful! Collection of “structurally similar” m × n images Video frames (3D, 4D= color); completing missing data Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 16 / 28
Multi-energy XRay CT k energy bins. µ ( r , E k ) → X k ∈ R N 1 × N 2 x k = vec ( X k ) ( φ, t ) space into N m source/det pairs Then A ∈ R N m × N p where [ A ] ij represents the length of that segment of ray i passing through pixel j . Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 17 / 28
Multi-energy XRay CT 0.8 cotton wax 0.6 nylon ethanol soap plexiglass 0.4 rubber 0.2 0.02 0.04 0.06 0.08 Energy (Mev) The log-liklihood function to be optimized, assuming Poisson noise L k ( x k ) = � D − 1 / 2 ( Ax k − m k ) � 2 k 2 where D k is diagonal, m k is log of scaled projection data. Misha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 18 / 28
Recommend
More recommend
