Misha E. Kilmer - A t-SVD-based Nuclear Norm with …nylon ethanol soap plexiglass rubber The...

A t-SVD-based Nuclear Norm withImaging Applications

Oguz Semerci1 Ning Hao2 Misha E. Kilmer 2 Eric Miller2

Shuchin Aeron2 Gregory Ely2 Zemin Zhang2

1Schlumberger-Doll Research

2Tufts 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

← A4,7,1

← A:,3,1

← A:,:,3

1Graphics thanks to K. BramanMisha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 2 / 28

Notation

1


← A4,7,1

← A:,3,1

← A:,:,3


Notation

1


← A4,7,1

← A:,3,1

← A:,:,3


Notation

1


← A4,7,1

← A:,3,1

← A:,:,3


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]


Toward Defining Tensor-TensorMultiplication

For A ∈ Rm×p×n, let Ai = A:,:,i.

unfold−→

A1

A2

A3...An

∈ Rmn×p


Toward Defining Tensor-TensorMultiplication

For A ∈ Rm×p×n, let Ai = A:,:,i.

unfold−→

A1

A2

A3...An

∈ Rmn×p

A1

A2

A3...An

fold−→ ∈ Rm×p×n


Block Circulant Matrix

The block circulant matrix generated by unfold (A) is

circ (A) =

A1 An · · · A3 A2

A2 A1 An · · · · · ·A3 A2

. . . . . . . . ....

. . . . . . . . . . . .

An · · · · · · A2 A1


Block Circulants

A block circulant can be block-diagonalized by a (normalized) DFT inthe 2nd dimension:

(F ⊗ I)circ (A) (F ∗ ⊗ I) =

A1 0 · · · 0

0 A2 0 · · ·0 · · · . . . 0

0 · · · 0 An

Conveniently, an FFT along tube fibers of A gives A.


Tensor - Tensor Multiplication

[K., Martin, Perrone ‘08]: For A ∈ Rm×p×n and B ∈ Rp×q×n, definethe t-product

A ∗B ≡ fold(

circ (A) · unfold (B)).

Result is m× q × n.




A ∗B ≡ fold(



Example: A ∈ Rm×p×3 and B ∈ Rp×q×3,

A ∗B = fold

A1 A3 A2

A2 A1 A3

A3 A2 A1

B1

B2

B3

.




A ∗B ≡ fold(



Example: A ∈ Rm×p×3 and B ∈ Rp×q×3,

A ∗B = fold

A1 A3 A2

A2 A1 A3

A3 A2 A1

B1

B2

B3

.

This tensor-tensor multiplication generalizes to higher-order tensorsthrough 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 betweentubal scalars is commutative ⇒ the t-product resemblesmatrix-matrix product with scalar mult replaced by t-product multamong tubal scalars.

Definition

The `× `× n identity tensor I is the tensor whose frontal slice isthe `× ` identity matrix, and whose other frontal slices are all zeros.


TransposeDefinition

If A is `×m× n, then AT is the m× `× n tensor obtained bytransposing each of the frontal slices and then reversing the order oftransposed frontal slices 2 through n.

Example

If A ∈ R`×m×4

AT = fold

AT1AT4AT3AT2


Orthogonality

Definition

U ∈ Rm×m×n is orthogonal if UT ∗U = I = U ∗UT.

Can show Frobenius norm invariance: ‖U ∗A‖F = ‖A‖F .


The t-SVD

Theorem (K. and Martin, 2011)

Let A ∈ R`×m×n. Then A can be factored as

A = U ∗ S ∗VT

where U,V are orthogonal `× `× n and m×m× n, and S is a`×m× n f -diagonal tensor. Also, B = U1:k,1:k,: ∗ S1:k,1:k,: ∗VT

1:k,1:k,:

satisfies

B = arg minM‖A−B‖F , M = B = X∗Y,X ∈ R`×k×n,Y ∈ Rk×m×n.


t-SVD Example

Let A be 2× 2× 2.

(F ⊗ I)circ (A) (F ∗ ⊗ I) =

[A1 0

0 A2

]

[A1 0

0 A2

]=

[U1 0

0 U2

][σ(1)1 0

0 σ(1)2

][σ(2)1 0

0 σ(2)2

][V ∗1 0

0 V ∗2

]

The U,S,VT are formed by putting the hat matrices as frontal slices,ifft along tubes.

e.g. s1 =

[σ(1)1

σ(2)1

]oriented into the screen.


Multi-rank

Definition (K.,Braman,Hoover,Hao, 2013)

Let A ∈ R`×m×n. The multi-rank of A is length n vector consisting

of the ranks of all the A(i)

, which must be symmetric about the“middle”.

Example

A ∈ R2×2×4, multi-rank possible: [i, j, k, j]T , 1 ≤ i, j, k ≤ 2.

Example

A ∈ R5×4×3, multi-rank possible: [i, j, j]T , 1 ≤ i ≤ 4, 1 ≤ j ≤ 4.


Tensor Nuclear Norm

If A is an `×m, ` ≥ m matrix with singular values σi, the nuclearnorm ‖A‖~ =

∑mi=1 σi.

However, in the t-SVD, we have singular tubes (the entries of whichneed not be positive), which sum to a singular tube!

The entries in the jth singular tube are the inverse Fourier coefficientsof the length-n vector of the jth singular values of A:,:,i, i = 1..n.

Definition

For A ∈ R`×m×n, our tensor nuclear norm is‖A‖~ =

∑min(`,m)i=1 ‖

√nF si‖1 =

∑min(`,m)i=1

∑nj=1 Si,i,j. (Same as the

matrix nuclear norm of circ (A)).


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.


TNN in Regularization and Optimization

Yes, the t-SVD is orientation dependent, as is the norm. There areapplications where this is particularly useful!

Collection of “structurally similar” m× n images

Video frames (3D, 4D= color); completing missing data


Multi-energy XRay CT

k energy bins.

µ(r, Ek)→ Xk ∈ RN1×N2

xk = vec(Xk)

(φ, t) space into Nm source/det pairs

Then A ∈ RNm×Np where [A]ij represents the length of thatsegment of ray i passing through pixel j.


Multi-energy XRay CT

0.02 0.04 0.06 0.08

0.2

0.4

0.6

0.8

Energy (Mev)

µ (1

/cm

)

cottonwaxnylonethanolsoapplexiglassrubber

The log-liklihood function to be optimized, assuming Poisson noise

Lk(xk) = ‖D−1/2k (Axk −mk)‖22

where Dk is diagonal, mk is log of scaled projection data.


Regularized Problem

Let Xs.t.X:,:,k = Xk; recall xk = vec(Xk).

minX

(

N3∑k=1

Lk(xk) + αkR(xk)) + γ‖Z‖∗, sbj to Z = X

where R() denotes possible additional regularization.

Optimized via an Alternating Direction Method of Multipliers(ADMM) (see Boyd et al 2011, Bertsekas 1999)Let Lη(X,Z,Y) denote the augmented Lagrangian. The updates are

Xn+1 := argminX

Lη (X,Zn,Yn) ,

Zn+1 := argminZ

Lη(Xn+1,Z,Yn

)Yn+1 := Yn + η(Xn+1 −Zn+1)

First decouples, 2nd is computed via t-SVD “shrinkage”


t-SVD Shrinkage

Zn+1 := argminZ

Lη(Xn+1,Z,Yn

)

Compute ( 1ηYn + Xn+1) = U ∗ S ∗VT. Recall S contains Σk for

frontal slices of transformed tensor.

Zn+1 := U ∗ ρ(S) ∗VT, where ρ(S) takes the non-zeros in S andreplaces them with the difference between them and rho, if thatdifference is greater than 0, and 0 otherwise. ρ depends on theparameter in the problem.Shown that Zn+1 := U ∗ (S ∗D) ∗VT


Numerical Results

0

0.2

0.4

0.6

0.8

1

0

0.05

0.1

0.15

0.2

0.25

25 keV 85 keVMisha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 21 / 28

Numerical Results

FBP, TNN, TV, TNN+TV

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Figure: Reconstruction results for 25 keV.


Numerical Results

0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

Figure: Reconstruction results for 85 keV.


Tensor Completion

Given unknown tensor M of size n1 × n2 × n3, given a subset ofentries Mijk : (i, j, k) ∈ Ω where Ω is an indicator tensor of sizen1 × n2 × n3. Recover the entire M:

min ‖X‖~subject to PΩ(X) = PΩ(M)

The (i, j, k)th component of PΩ(X) is equal to Mijk if (i, j, k) ∈ Ωand zero otherwise.

Similar to the previous problem, this can be solved by ADMM, with 3update steps, one which decouples, one that is ashrinkage/thresholding step.


Numerical Results

TNN minimization, Low Rank Tensor Completion (LRTC) [Liu, et al,2013] based on tensor-n-rank [Gandy, et al, 2011], and the nuclearnorm minimization on the vectorized video data [Cai, et al, 2010].MERL2 video, Basketball video

2with thanks to Dr. Amit AgrawalMisha E. Kilmer (Tufts University) Tensor Nuclear Norm June 2013 25 / 28

Numerical Results


Numerical Results

Basketball video data of size 144× 256× 3× 80.


Conclusions

Introduced the notion of a tensor nuclear norm around theconcept of the t-SVD for tensors

Discussed in 3rd order case, but generalizes to higher order

The tensor nuclear norm is useful in imaging applications wherewe can exploit certain features that are orientation dependent

Efficiency in implementations (parallelism); exploit complexconjugacy in the Fourier domain

A different t-SVD and associated factorizations and norm basedon fast trig-transform [Kernfeld, et al, 2013]


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