Diagonalization of Tensors with Circulant Structure

Transcription

1 Diagonalization of Tensors with Circulant Structure Mansoor Rezgi and Lars Eldén Linköping University Post Print N.B.: When citing this work, cite the original article. Original Publication: Mansoor Rezgi and Lars Eldén, Diagonalization of Tensors with Circulant Structure,, Linear Algebra and its Applications, (4),, Copyright: Elsevier Science B.V., Amsterdam. Postprint available at: Linköping University Electronic Press

2 Diagonalization of Tensors with Circulant Structure Mansoor Rezghi Lars Eldén Abstract The concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors. It is shown a tensor of arbitrary order, which is circulant with respect to two particular modes, can be diagonalized in those modes by discrete Fourier transforms. This property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure. Tensors with circulant structure occur in models for image blurring with periodic boundary conditions. It is shown that the new framework can be applied to such problems. Introduction Circulant matrices occur in many applications. For instance, they are used as models of the blurring process in digital image processing [4, p.8]. They also occur as preconditioners in the solution of linear systems with Toeplitz structure, see e.g. [6, 7, 7, ]. Circulant matrices are particularly useful since they are diagonalized by the Fourier matrix [8, Chapter.], which means that one can solve a linear system of equations with a circulant matrix of dimension n in O(n log n) operations. In this paper we generalize the concepts of diagonal and circulant matrices to tensors of arbitrary order. We show that a tensor can be circulant in different subsets of modes and that it can be transformed to diagonal form in the corresponding modes. Thus a tensor that is circulant with respect to the modes ( dimensions ) {l, k} is transformed to {l, k}-diagonal form by multiplication by Fourier matrices in the corresponding modes. This diagonalization can be used in fast contractive products of tensors and also for solving tensor equations, e.g. arising in image deblurring (restoration). In order to motivate further the development of the theory for tensors with circulant structure, we briefly discuss the application to image blurring. Matrices with circulant structure occur in connection with spatially invariant blurring models, where periodic boundary conditions are assumed, see e.g. [6, Chapter 4], and as preconditioners [] for problems with Toeplitz structure. There the images are treated as vectors, and the blurring model gives rise to a block circulant matrix with circulant blocks (BCCB), which can be diagonalized by a two-dimensional discrete Fourier transform [8, Chapter.8]. Department of Mathematics, Tarbiat Modares University, P.O. Box 4-7, Tehran, Iran (Rezghi@modares.ac.ir). This work has been done during a visit at Linköping University, Sweden. Department of Mathematics, Linköping University, SE-8 8 Linköping, Sweden(Laeld@mai.liu.se).

3 Next assume that we model a three-dimensional blurring process with periodic boundary conditions, or use a circulant type preconditioner for a three-dimensional problem with Toeplitz structure. It is straightforward to show that a generalization of the -D approach will lead to a matrix with doubly nested circulant structure. In addition, data and unknowns, which in the application represent volumes, are treated as vectors. We will show, see Section 6, that alternatively, and more naturally, the problem can be modeled by using tensor notation and techniques, and it will be shown that the blurring process is modeled by a order-6 tensor with circulant structure that operates on a volume, giving a volume. It appears that some of the results of this paper are known, partially or implicitly, in the numerical image processing community. For instance, the MATLAB codes in [6, Chapter 4.] can be thought of as tensor implementations of operations with BCCB matrices. Thus we do not claim that the results of this paper are new in essence, or they that they lead to more efficient algorithms. However, we believe that the consistent treatment in terms of a tensor framework is novel, and that the advantage of the tensor framework is that it is straightforward to generalize it to tensors of arbitrary order. In fact, in this paper we define the concepts and prove the results for the general case. The paper is organized as follows. In Section we define some tensor concepts that will be needed. Tensors with diagonal structure are defined in Section. We introduce tensors with circulant structure in Section 4. In Section we demonstrate that tensors with circulant structure are diagonalized by discrete Fourier transforms. The application to image blurring models is briefly described in Section 6. To our knowledge, tensors with diagonal structure were first introduced in []. The concept of totally diagonal tensors introduced in Section is used in the low-rank approximation by a tensor using the Candecomp/Parafac model, see e.g. []. A fast algorithm for computing multilinear SVD of special Toeplitz and Hankel tensors is discussed in []. In [] a tensor framework is introduced for analyzing preconditioners for linear equations with Toeplitz structure. Tensor Concepts. Notation and preliminaries. Tensors will be denoted by calligraphic letters, e.g A, B, matrices by capital roman letters and vectors by small roman letters. In order not to burden the presentation with too much detail, we sometimes will not explicitly mention the dimensions of matrices and tensors, and assume that they are such that the operations are well-defined. We will try to make our presentation easy to read by illustrating the concepts in terms of small examples and figures, mostly for order- tensors. For convenience we also introduce some concepts in terms of order- tensors. In such cases the generalization to order-n tensors is obvious. Let A denote a tensor in R I I I. The different dimensions of the tensor are referred to as modes. We will use both standard subscripts and MATLAB-like notation: a particular tensor element will be denoted in two equivalent ways: A(i, j, k) = a ijk. We will refer to subtensors in the following way. A subtensor obtained by fixing one of the indices is called a slice, e.g., A(i,:, :).

4 Such a slice is usually considered as a order- tensor. However, in an assignment we assume that the singleton mode is squeezed. For example A(, :, :) is in R I I, but when we define B = A(, :, :), we let B R I I, i.e, we identify R I I with R I I in the assignment *. A fibre is a subtensor, where all indices but one are fixed, An N-dimensional multi-index ī is defined The notation A(i,:, k). ī = (i,...,i N ). (.) ī k = (i,...,i k, i k+,...,i N ), (.) is used for a multi-index where the k th mode is missing. We define the order-n Kronecker delta as {, if i = = i δ i...i N = N ;, otherwise. The elementwise product of tensors X R I K L and Y R I K L is defined R I K L Z = X. Y, z ikl = x ikl y ikl. In the same way elementwise division is defined as R I K L Z = X./Y, z ikl = x ikl /y ikl. These elementwise operations can also be defined for vectors and matrices.. Tensor-Matrix Multiplication We define mode-p multiplication of a tensor by a matrix as follows. For concreteness we first let p =. The mode- product of a tensor A R J K L by a matrix W R M J is defined J R M K L B = (W) A, b mkl = w mj a jkl. This means that all column vectors (mode- fibres) in the order- tensor are multiplied by the matrix W. Similarly, mode- multiplication by a matrix X means that all row vectors (mode- fibres) are multiplied by the matrix X. Mode- multiplication is analogous. In the case when tensor-matrix multiplication is performed in all modes in the same formula, we omit the subscripts and write j= (X, Y, Z) A, (.) where the mode of each multiplication is understood from the order in which the matrices are given. * It may seem that the property of a vector being a column or a row may be lost in such a transformation. However, the notions of column and row vectors is not essential as long as one keeps track of the ordering of the modes.

5 The notation (.) was suggested by Lim []. An alternative notation was earlier given in [9]. Our (W) p A is the same as A p W in that system. It is convenient to introduce a separate notation for multiplication by a transposed matrix V R J M : R M K L C =. Matricization ( V T) A = A (V ), c mkl = J a jkl v jm. (.4) A tensor can be matricized * in many different ways. We use the convention introduced in [] (which differs from that in [, 4]). Let r = [r,, r L ] be the modes of A mapped to the rows and c = [c,, c M ] be the modes of A mapped to the columns. The matricization is denoted A (r;c) R J K, where J = i= j= L M I ri, and K = I ci. (.) For a given order-n tensor A, the element A(i,...,i N ) is mapped to A (r;c) (j, k) where.4 Contractions j = + k = + L l= M m= [ (irl l+ [ (icm m+ l ) l = m ) I rl l + ] m = I cm m + ] i=, (.6). (.7) Let A and B be order- tensors of the same dimensions. The inner product is defined e = A, B = λ,µ,ν a λµν b λµν. The inner product can be considered as a special case of the contracted product of two tensors, cf. [8, Chapter ], which is a tensor (outer) product followed by a contraction along specified modes. The contracted product can be defined also for tensors of different numbers of modes and contractions can be made along any two conforming modes. For example, with a order-4 tensor A and matrices (order- tensors) F and G we could have, A, F,4;, = G, a jkµν f µν = g jk, (.8) where the subscripts, 4 and, indicate the contracting modes of the two arguments. Obviously (.8) defines a linear system of equations. We also need the following relations in the paper. Proposition.. Let A R I I N, and X R I I k with k < N. Then the following relations hold. * Alternatively, unfolded [9] or flattened []. µ,ν 4

6 a) For every matrix V with conforming dimensions and j k, (V ) j A, X = A, (V T) X. :k;:k j :k;:k b) For every matrix V with conforming dimensions and j N k, (V ) j A, X :k;:k = (V ) k+j A, X :k;:k. Proof. The results immediately follow from the definitions of contractive and matrix-tensor product. Tensors With Diagonal Structure A starting point for our definitions and derivations will be to consider the concept of totally diagonal tensors. Straightforward generalization of a diagonal matrix is the following. Definition.. A tensor A R I I N is called totally diagonal if a i...i N can be nonzero only if i = i = = i N. Figure : A order- totally diagonal tensor. Note that we allow a diagonal element to be zero. Figure shows an order- totally diagonal tensor. In [] totally diagonal tensors are called maximally diagonal. Obviously a totally diagonal order- tensor is a diagonal matrix. The definition of a totally diagonal tensor is not general enough for our purposes. We also need to define tensors that are partially diagonal, i.e., diagonal in two or more modes. For example consider an order- tensor A, such that for every k, A(i, j, k) = if i j. (.) This tensor is diagonal with respect to the first and second modes. Figure illustrates all possible order- partially diagonal tensors. We now give a general definition. Definition.. Let < t N be a natural number, and let {s,...,s t } be a subset of {,...,N}. A tensor A R I I N is called {s,...,s t }-diagonal, if a i...i N can be nonzero only if i s = i s = = i st. By this definition a totally diagonal order-n tensor is {,...,N}-diagonal. Although it is not strictly required that a tensor be square with respect to diagonal modes, in this

7 mode- mode- mode- mode- mode- mode- mode- mode- mode- Figure : Order- partially diagonal tensors, from left to right: {, }, {, } and {, }. Note the convention for ordering the modes that we use in the illustrations. paper we make this assumption, i.e. that the tensor dimensions are the same in the modes for which it is diagonal. It is straightforward to show that matrix multiplication of an {s,...,s t }-diagonal tensor in the modes that are not diagonal, does not affect the {s,...,s t }-diagonality, i.e., the result of such a multiplication is also diagonal in the same modes. Proposition.. Let A R I I N be {s,...,s t }-diagonal. Then for every matrix X and k / {s,...,s t }, the tensor B = (X) k A is still {s,...,s t }-diagonal. Next we define tensors that are diagonal with respect to disjoint subsets of the modes. Definition.4. Let S = {s,...,s t } and Q = {q,...,q t } be two disjoint subsets of {,...,N}. A R I I N is called S, Q-diagonal if a i...i N can be nonzero only if i s = = i st and i q = = i qt. Figure illustrates an order-4 tensor A R n n, which is {, }, {, 4}-diagonal. Thus if i i or i i 4 then a i i i i 4 =. The matricization A (,;,4) R n n of A(:, :, :, ) A(:, :, :, ) A(:, :, :, ) Figure : The tensor A R n n is {, }, {, 4}-diagonal. A is a diagonal matrix. In general, the matricization of partially diagonal tensors, gives rise to multilevel block diagonal matrices. For example, the matricization A (;,) of the {, }-diagonal tensor in Figure, is a block matrix with diagonal blocks. Figure 4 shows different matricizations of a {, }-diagonal order-6 tensor A R. Sometimes a diagonal matrix A is represented by its diagonal elements as A = diag(d), where d is a vector. Thus we can write a diagonal matrix as a ij = δ ij d i, 6

8 nz = nz = nz = 48 Figure 4: The matricizations A (,,;,4,6), A (,,;4,6,), and A (,,;4,,6) of a{, }-diagonal A R. where δ ij is the Kronecker delta. In the same way we can represent diagonal tensors in different modes by using their diagonal elements only. For example a totally diagonal tensor A R I I N can be written as a i...i N = δ i...i N d i where d denotes the diagonal elements of A. Similarly, the {, }-diagonal tensor in (.), can be written A(i, j, k) = δ ij D(i, k), and so where D(i, k) = A(i, i, k). A(i, j,:) = δ ij D(i,:), Proposition.. A tensor A R I I N D R I I k I k+...i N, such that is {l, k}-diagonal if and only if there is a where multi-indices ī and ī k are defined in (.) and (.). A(ī) = δ il i k D(ī k ), (.) Proof. The first part is trivial. For the converse let A be {l, k}-diagonal. Then defining D as D(i,...,i l,...,i k, i k+,...,i N ) = A(i,...,i l,...,i k, i l, i k+,, i N ) completes the proof. The proposition shows that a tensor is {l, k}-diagonal if its k th mode exists only via the Kronecker delta. In the following example we show that tensors with diagonal structure occur naturally in the numerical solution of a self-adjoint Sylvester equation. Example.6. Consider a Sylvester equation V = AZ +ZB. If the matrices A and B are symmetric, then, by using their eigenvalue decompositions, one can transform the equation to the form Y = SX + XT, (.) where T = diag(t,...,t n ) and S = diag(s,...,s m ) are the diagonal matrices of eigenvalues. This is actually a special case of the Bartels-Stewart algorithm for solving the 7

9 Sylvester equation, see e.g. [, Chapter 7.6.]. If we set D(i, j) = s i + t j then (.) can be written as tensor-matrix equation Y = Ω, X,;,, (.4) where Ω is a {, }, {, 4}-diagonal tensor with diagonal elements D, i.e, Now it is easy to see that (.4) is equal to and thus the solution X, can be written Ω(i, j, k, l) = δ ik δ jl D(i, j). Y = D. X, X = Y./D, where. and./ are elementwise product and division. 4 Tensors With Circulant Structure We first consider a few properties of circulant matrices and then define tensors with circulant structure. 4. Circulant Matrices A matrix A = (a ij ) i,j=,...,n is said to be circulant if a ij = a i j, if i j i j ( mod n), (4.) i.e, A is a matrix of the form a a n... a. a a an. a n... a a Circulant matrices have special structure and properties. Every column (row) of A is down (right) cyclic shift of the column left(row above) of it, so if we define C = , (4.) and a and b T are the first column and row of A respectively, then A(:, j) = C j a, j =,...,n, (4.) A(i,:) = (C i b) T, i =,...,n. (4.4) 8

10 This means that a circulant matrix is completely determined by its first column or row. Furthermore it is well known [8, Chapter.] that C has a diagonal form as C = F ΩF, Ω = diag(, w, w,...,w n ), (4.) where F = n... w w... w n w w 4... w (n ) w n w (n )... w (n )(n ), w = exp( πi/n), is the Fourier matrix and F denotes the conjugate and transpose of F. By using (4.) it is easy to prove that any circulant matrix can be diagonalized by the Fourier matrix [8, Chapter.]. Proposition 4.. Let A R n n be a circulant matrix. Then A is diagonalized by the Fourier matrix F as, A = F Λ F, Λ = diag( nfa), (4.6) A = F Λ F, Λ = diag( nfb), (4.7) where Λ and Λ are conjugate, Λ = Λ. (4.8) For completeness we give the proof here. Proof. From (4.) A can be expressed in terms of powers of the matrix C, A = (a, Ca,...,C n a). (4.9) Using the eigenvalue decomposition (4.) we can write A = F (ā,ωā,...,ω n ā), where ā = Fa, i.e., ā ā... ā A = F ā ā ω... ā ω n ā n ā n ω n... ā n ω (n )(n ) nā... = F nā F.... nān Thus Similarly, by using (4.4), so by the above discussions A = F diag( nā)f. A T = [b, Cb,...,C n b], A T = F diag( nfb)f, 9

11 and A = FΛ F, Λ = diag( nfb). Since A is real, we have A = A, and by this the second statement is proved. Multiplication by F is the same as a discrete Fourier transform, which is usually implemented using the Fast Fourier Transform (FFT). In our comments on algorithms we will use the notation fft(a) = nfa and ifft(a) = n F a for the FFT and its inverse. Of course, both operations can be performed in O(n log n) operations, see e.g. []. If then by (4.6) and, by (4.7), Ax = y, y = ifft(fft(x). fft(a)), x = ifft(fft(y)./fft(a)), y = fft(ifft(x). fft(b)), x = fft(ifft(y)./fft(b)). It follows that matrix-vector multiplication and solving linear system with a circulant coefficient matrix can be performed using (4.6) and (4.7) in O(n log n) operations. 4. Tensors Circulant With Respect to Two Modes From Section 4. we see that the key property of a circulant matrix that allows it to be diagonalized using the discrete Fourier transform, is that its columns ( rows) can be written in terms of powers of the shift matrix C, see equation (4.), times the first column ( row). In this subsection we will consider tensors, whose slices are circulant with respect to a pair of modes. Then, naturally, it follows that the tensor can be expressed in terms of powers of C, which, in turn, makes it possible to diagonalize the tensor using the discrete Fourier transform. Consider an order- tensor A R n n n, where for every k, the A(:, :, k) slices are circulant, i.e, A(i, j, k) = A(i, j, k) if i j i j ( mod n), or equivalently, A(i, j,:) = A(i, j, :) if i j i j ( mod n). Thus A is circulant with respect to the first and second modes, and we define A to be {, }-circulant. Definition 4.. A R I I N is called {l, k}-circulant, if I l = I k = n, and A(:,...,:, i l,...,i k, :,...,:) = A(:,...,:, i l,...,i k, :,...,:), if i l i k i l i k ( mod n).

12 Using (4.) every column of the A(:, :, k) can be constructed from A(:,, k) as follows: For every j =,...,n A(:, j, k) = C j A(:,, k), so the corresponding holds also for the slices, A(:, j,:) = ( C j ) A(:,, :). (4.) This shows that every A(:, j,:) slice, is obtained by j cyclic shifts in the mode- direction of A(:,, :), see Figure. Considering shifts of slices, it is straightforward to A(:, 4, ) = C A(:,, ) A(:, 4, :) = ( C ) A(:,, :) Figure : Cyclic shifts of columns and slices. obtain the following relations, the general version of (4.). Lemma 4.. If A R I I N is {l, k}-circulant, then for every i k I k we have and, for every i l I l A(:,...,:,...,i k,...,:) = ( C i k ) A(:,...,:,...,,...,:), (4.) l A(:,...,i l,...,:,...,:) = ( C i l ) A(:,...,,...,:,...,:), (4.) k where is in the k th and l th mode of A in the first and second equations, respectively. Proof. For simplicity, and without loss of generality we assume that l =, k =. For fixed i, i 4,...,i N, each slice A(:, :, i, i 4,...,i N ) is circulant. The lemma now follows immediately from (4.) and (4.4). Note the analogy between the above result and Proposition. for a {l, k}-diagonal tensor: The k th mode exists only via the multiplication by the shift matrix C. Example 4.4. Let A R 4 4 be the {, }-circulant A(:, :, ) = 4, A(:, :, ) = 7 8, A(:, :, ) = So, for every k =,,, A(:, :, k) is circulant and by (4.) A(:,, :) = (C) A(:,, :) = 7 = 4 8, 9 7

13 i.e, A(:,, :) is a cyclic shift of A(:,, :) in the mode-. In the same way by (4.) A(, :, :) = (C) A(, :, :) = 7 = 9, A(, :, :) is a cyclic shift of A(, :, :) in the mode-. These are generalizations of column and row shifts in circulant matrices. Now consider an order-4 tensor A that is {, }-circulant and shown in Figure 6. Every A(:, :, :, ) A(:, :, :, ) Figure 6: {, }-circulant tensor A R. slice A(:, j,:, :) for j =,, is shown in Figure 7. Here every slice A(:, j,:, :) is obtained by a cyclic shift in the mode- direction on previous slice A(:, j, :, :) and so by j cyclic shifts in the mode- direction on A(:,, :, :). This confirms the result of Lemma 4. and so A(:, j,:, :) = ( C j ) A(:,, :, :). A(:,, :, :) A(:,, :, :) A(:,, :, :) Figure 7: A(:, j,:, :) slices of A for j =,,. If A is {l, k}-circulant, then tensor-matrix or contractive multiplication of the tensor in the modes other than l and k, do not destroy the circulant property. Proposition 4.. Let A R I I N be {l, k}-circulant. Then for every matrix X and s l, k, the tensor B = (X) s A is still {l, k}-circulant.

14 Proof. The proof is a direct result of the definition of a circulant tensor and tensor-matrix multiplication. 4. Tensors with Circulant Structure: Disjoint Sets of Modes Next we study the case when A is circulant in disjoint subsets of modes. This type of tensor occurs in image restoration, as we will see in Section 6. The following lemma shows how a tensor can be written in terms of powers of the shift matrix C. Lemma 4.6. Let A R I I N be circulant in two disjoint subset of modes {l, k} and {p, q}. Then for every i k, i q, A(:,...,i k,...,i q,...,:) = ( C ik, C iq ) A(:,...,,...,,...,:). (4.) l,p The s are in modes k and q of A. Proof. Without loss of generality, suppose that l =, k = and p =, q = 4. For every i, i 4 consider A(:, :, i, i 4, :,...,:), since A is {, 4}-circulant A(:, :, i, i 4, :,...,:) = ( C i 4 ) A(:, :, i,, :,...,:). But A(:, :, i,, :,...,:) is {, }-circulant, so By these two equations This proves the lemma. A(:, :, i,, :,...,:) = ( C i ) A(:, :,,, :,...,:). A(:, :, i, i 4, :,...,:) = ( C i, C i 4 ) A(:, :,,, :,...,:). (4.4), This lemma shows that A(:,...,i k,...,i q,...,:) is obtained by performing i k and i q cyclic shifts in the l and p modes respectively on A(:,...,,...,,...,:). Example 4.7. Let A R be the {, }, {, 4}-circulant tensor shown in Figure 8. Here by Lemma 4.6, all elements of A can be determined by shifts on A(:, :,, ). A(:, :, :, ) A(:, :, :, ) Figure 8: {, }, {, 4}-circulant tensor A. For example A(:, :,, ) is obtained after cyclic shifts on A(:, :,, ) in the mode- direction, A(:,:,,) = ( C ) A(:,:,,) 7 = 7 =.

15 In the same way, A(:, :,, ) is obtained by cyclic shift on A(:, :,, ) in the mode- direction, A(:,:,,) = (C) A(:,:,,) ( ) = 7 = 7. A(:, :,, ) is obtained after and cyclic shifts on A(:, :,, ) in mode- and mode-, respectively. A(:,:,,) = ( C,C ) A(:,:,,), ( ) 7 = 7 = The A (,;,4) matricization of this tensor is a block circulant matrix with circulant blocks (BCCB), 7 7 A (,;,4) 7 = Figure 9 shows an example, where A is {, }, {, 4}-circulant. A(:, :, :, ) A(:, :, :, ) Figure 9: {, }, {, 4}-circulant tensor A. Lemma 4.6 can be generalized for cases when A is circulant with respect to several different disjoint subsets of modes. The following special case occurs in image blurring models. Corollary 4.8. Let A R I I N be such that for every i =,...,N, A is {i, i + N}- circulant. Then for every i N+,...,i N, A(:,...,:, i N+,...,i N ) = ( C in+,...,c i N ) A(:,...,:,,...,) (4.),...,N Proof. The proof is straightforward by induction and using Lemma

16 4.4 Tensors with Circulant Structure: Coinciding Modes We now introduce the situation where two or more modes are circulant with respect to the same mode. For simplicity consider an order- tensor A R n n n that is {, }, {, }- circulant. So by Lemma 4. for every j and k These equations show that A(:, j,:) = ( C j ) A(:,, :), ( A(:, :, k) = C k ) A(:, :, ). A(:, j, k) = ( C j ) (C A(:,, k) = j C k ) A(:,, ), i.e, A can be constructed by A(:,, ). This proves the following lemma. Lemma 4.9. Let A R n n n be {, }, {, }-circulant. Then for every j, k ( A(:, j, k) = C j+k ) A(:,, ). (4.6) Now its natural to investigate relations between mode- and mode- when A is {, }, {, }- circulant. Proposition 4.. Let A R n n n be {, }, {, }-circulant. Then mode- and mode- have the following relations a) A(:, j, k) = A(:, j, k ) if j + k j + k ( mod n). (4.7) b) For every j and for every k, A(:, j,:) = A(:,, :) (C j ), (4.8) ( A(:, :, k) = A(:, :, ) C k ). (4.9) Proof. Since j + k = j + k + pn where p, is an integer, we get by (4.6), ( ) A(:, j, k) = C j +k +pn A(:,, ) ( ) = C j +k A(:,, ) = A(:, j, k ). This proves the first statement. Then, for fixed i, by (4.7) T T a ij a i,j, a ij a i,j,. =., a ij,n a i,j,n a ijn a i,j, i.e., A(i, j,:) = A(i, j, :)C. By continuing this process A(i, j,:) = A(i,, :)C (j ), so ( A(:, j,:) = A(:,, :) C (j )). This proves (4.8). In a similar way (4.9) can also be proved.

17 By (4.7) for every i, A(i,:, :) is symmetric. These results can be written for a tensor A R I I N, which is circulant in arbitrary modes{l, k} and {l, q}. Example 4.. Let A R be the {, }, {, }-circulant that is shown in Figure. By Lemma 4.9, for every j and k, A(:, j, k) can be constructed by cyclic shifts on fiber A(:,, ). For instance Figure : Order- {, }, {, }-circulant tensor A. A(:,,) = ( C ) A(:,,) = =. By writing A(i,:, :) for every i, It is easy to see the relations between mode- and mode-. A(, :, :) = ( ), A(, :, :) = ( ), A(, :, :) = ( ) Here, it can be seen that every column of these slices are up cyclic shift of the column left. {l, k}-diagonalization of a {l, k}-circulant Tensor In this section we will show that, if a tensor is circulant in some modes then by using the Fourier transform this tensor can be diagonalized in the corresponding modes. For instance, let A R be the {, }-circulant tensor given in Figure i -.9i +.9i -.+.8i Figure : {, }-circulant tensor A (left) and {, }-diagonal tensor Ω = (F, F ), A (right). By (4.6) for every k, F A(:, :, k)f is {, }-diagonal with diagonal elements 6

18 So as shown in Figure, ( nf ) A(:,, k). Ω = (F, F ), A is a {, }-diagonal tensor, where holds the diagonal elements of Ω, i.e., In this particular example ( ) D = F A(:,, :) = F and for example, D = ( nf ) A(:,, :), (.) Ω(i, j,:) = δ ij D(i,:). (.) Ω(,, :) = D(, :) 7 = = (..8i.9i ). 6..8i.9i. +.8i +.9i, This shows that A = (F, F), Ω where Ω and D are defined as (.) and (.), respectively. Now the following theorem shows that every {l, k}-circulant tensor can be diagonalized in the {l, k} modes. Theorem.. Let A R I I N be {l, k}-circulant. Then A satisfies A = (F, F) l,k Ω, where Ω is a {l, k}-diagonal tensor with diagonal elements D = ( nf ) A(:,...,, :,...,:); l here is in the k th mode of A. In particular. Ω(ī) = δ il i k D(ī k ), where the multi-indices ī and ī k are defined in (.), (.). Proof. For simplicity and without loss of generality we assume that l =, k =. For every fixed i...i N, by (4.6) (F, F ), A(:, :, i,...,i N ), is {, }-diagonal with diagonal elements nf A(:,, i,...,i N ). If we define D = ( nf ) A(:,, :,...,:), 7

19 and for every i,...,i N set Ω(:, :, i,...,i N ) = (F, F ), A(:, :, i,...,i N ), then Ω = (F, F ), A is {, }-diagonal, and its diagonal elements are D, i.e, Ω(ī) = δ i i D(ī ). This diagonalization can be used in fast matrix-tensor/contractive products.. Diagonalization of a Tensor in Disjoint Circulant Modes In this subsection we discuss the diagonalization of tensors that are circulant in different disjoint subsects of modes. First consider the tensor A in Example 4.7, which is {, }, {, 4}-circulant. Since A is {, 4}-circulant, (F, F ),4 A is {, 4}-diagonal. But (F, F ),4 A is still {, }-circulant. Thus ( ) Ω = (F, F ), (F, F ),4 A = (F, F, F, F ) :4 A is {, }-diagonal. But we know that Ω also preserves the {, 4}-diagonality of (F, F ),4 A, i.e., Ω is {, }, {, 4}-diagonal tensor. Figure confirms this result and shows that diagonal elements Ω are Ω(:, :, :, ) Ω(:, :, :, ) 4.-.i.+.i i -.-9.i Figure : {,},{,4}-diagonal tensor Ω. ( ) D = F, F A(:, :,, ), = ( F) 7 ( 4 9 F) =..i. + 9.i,. +.i. 9.i 8

20 and For example Ω(i, i, i, i 4 ) = δ i i δ i i 4 D(i, i ). Ω(,,, ) = D(, ) =. 9.i. In general if a tensor is circulant in different disjoint modes, then it can be diagonalized in the corresponding modes. Theorem.. Let A R I I N be circulant in two disjoint subset of modes {l, k}and {p, q}. Then A satisfies A = (F, F, F, F) l,p,k,q Ω where Ω is a {l, k}, {p, q}-diagonal tensor with diagonal elements D = ( nf, mf ) A(:,...,,...,,...,:); l,p here we denote n = I l = I k, m = I p = I q, and the s are in the k th and q th modes of A. Further, Ω(ī) = δ il i k δ ipi q D(ī k,q ), where ī is defined in (.) and ī k,q = (i,...,i k, i k+,...,i q, i q+,...,i N ). Proof. Without loss of generality we suppose that A is {, },{, 4}-circulant, i.e., l =, k =, p =, q = 4, and N=4. Since A is {, 4}-circulant we have where Ω is {, 4}-diagonal, A = (F, F),4 Ω (.) Ω(i, i, i, i 4 ) = δ i i 4 D(i, i, i ), (.4) and D = ( mf ) A(:, :, :, ). (.) By Proposition 4. and Theorem., D is {, }-circulant and satisfying D = (F, F), Ω, (.6) where Ω, is a {, }-diagonal tensor Ω(i, i, i ) = δ i i D(i, i ), (.7) and D = ( nf ) D(:, :, ). (.8) From (.), D(:, :, ) = ( mf) A(:, :,, ). So substituting D(:, :, ) in (.8), gives D = ( nf, mf ) A(:, :,, ). (.9), If we define Ω = (F, F ), Ω, (.) 9

21 then Ω(:, i, :, i 4 ) = (F, F ), Ω(:, i, :, i 4 ) = δ i i 4 (F, F ), D(:, i, :) = δ i i 4 Ω(:, i, :), where the last two equations come from (.4) and (.6), respectively. So by this equation and (.7) Ω(i, i, i, i 4 ) = δ i i 4 Ω(i, i, i ) = δ i i δ i i 4 D(i, i ). This shows that Ω is a {, }, {, 4}-diagonal tensor with diagonal elements D. Then by putting Ω from (.) in (.) and the theorem is proved. A = (F, F, F, F) :4 Ω, (.) Now we consider a special situation that is a generalization of the D algorithm in [6, Chapter 4.], see also Section 6. Corollary.. Let A R I I N be such that for every i =,...,N, A is {i, i + N}- circulant. Then A can be diagonalized as A = (F,...,F, F,...,F),...,N,N+,...,N Ω, (.) where Ω is a {, N + },...,{N, N}-diagonal tensor, with diagonal elements ( D = I F,..., ) I N F A(:,...,:,,,...,). For every i,...,i N,...,N N Ω(i,...,i N ) = ( δ isi s+n )D(i,...,i N ) Proof. The proof is straightforward by induction and using Theorem.. s= By a straightforward generalization of the procedure described in Section 6.. one can see that a tensor of the structure mentioned in Corollary. occurs in N-dimensional image blurring with periodic boundary conditions. We next show that a linear equation involving such a tensor can be solved cheaply. Corollary.4. Let A be a tensor satisfying the conditions of Corollary., and let X R I I N. The linear system of equations is equivalent to Y = A, X :N;:N (.) Y = D. X, (.4) where Y = (F,...,F ) :N Y, X = (F,...,F ) :N X and ( D = I F,..., ) I N F A(:,...,:,,...,) :N

22 Proof. By (.) the linear system (.) can be written Y = (F,...,F, F,...,F) :N Ω, X :N;:N ) = (F,...,F) :N ( (F,...,F ) :N Ω, X :N;:N ) = (F,...,F) :N ( Ω,(F,...,F ) :N X :N;:N, where the last two equations are obtained using Lemma.. By multiplying the result from modes to N by F, we get (F,...,F ) :N Y = Ω,(F,...,F ) :N X :N;:N. Now if we define Y = (F,...,F ) :N Y and X = (F,...,F ) :N X then Y = Ω, X :N;:N. Since Ω is {i, i + N}-diagonal for every i =,...,N, it is straightforward to show that this equation equal to Y = D. X, and thus can be solved by element-wise division, provided that all elements of D are non-zero. The solution X is then obtained by Fourier transforms. Let fftn(x) denote the N-dimensional Fourier transform, ( I F,..., I N F) :N X, and let ifftn(x) denote the inverse transform, ( ) F,..., F I IN By (.4), if we set P = A(:,...,:,,...,), then If all elements of D are nonzero, we have and the solution of (.) can be written as :N X. Y = fftn(fftn(p).*ifftn(x)). X = Y./D X = fftn(ifftn(y)./fftn(p)).. Diagonalization of a Tensor with Coinciding Circulant Modes Consider the {, } and {, }-circulant A R shown in Example 4.. We compute Ω as Ω = (F, F, F ) : A. Figure shows that Ω is {,, }-diagonal tensor. On the other hand.4 D = (F) A(:,, ) =.6.i,.6 +.i contains the diagonal elements of Ω. This confirms the following theorem.

23 i -.6+.i Figure : {,, }-diagonal tensor Ω = (F, F, F ) : A, that A is {, }, {, }-circulant. Theorem.. Let A R I I N be {, } and {, }-circulant, then A can be diagonalized as A = (F, F, F) : Ω, where Ω is {,, }-diagonal tensor and its diagonal elements are D = (nf) A(:,,, :,...,:) so, for every i, i, i Ω(ī) = δ iii D(ī, ) Proof. A is {, }-circulant, so Ω = (F, F ), A (.) which Ω is {, }-diagonal and Ω(i, i, i, i 4,...,i N ) = δ i i D(i, i, i 4,...,i N ) (.6) D = ( nf ) A(:, :,, :,...,:). (.7) Define Ω = (F ) Ω and D = (F ) D then by (.),(.6) and (.7) Ω = (F, F, F ) : A (.8) Ω(i, i, i, i 4,...,i N ) = δ i i D(i, i, i 4,...,i N ) (.9) D = (F, F ), na(:, :,, :,...,:). (.) By (.), D is {, }-diagonal because na(:, :,, :,...,:) is {, }-circulant. So diagonal elements of D are in D = ( nf ) na(:, :,, :,...,:) = (nf) A(:, :,, :,...,:) (.) and D(i, i, i 4,...,i N ) = δ i i D(i, i 4,...,i N ). Now by this equation and (.9) Ω(i, i, i, i 4,...,i N ) = δ i i D(i, i, i 4,...,i N ) = δ i i δ i i D(i, i 4,...,i N ) = δ i i i D(i, i 4,...,i N ) shows that Ω is {,, }-diagonal and its diagonal elements are in D = (nf) A(:,,, :,...,:). By this fact and (.8) the proof can be finished.

24 Corollary.6. Let A R I I N be {, i}-circulant for i =,,...,N. Then A can be diagonalized A = (F, F,...,F) :N Ω, (.) where Ω is totally diagonal with diagonal elements ( ) d = n (N )/ F A(:,,,...,), and Ω(i,...,i N ) = δ i...i N d(i ). Proof. By induction and using Theorem. the proof is straightforward. 6 Application in Image Blurring Models We consider the image deblurring problem with space invariant point spread function. The mathematical model is the following convolution equation p(s t)x(t)dt = y(s), Ω where the kernel p is called the point spread function (PSF), which often in applications has compact support. In the discrete version, pixels of the blurred image are obtained from a weighted sum of the corresponding pixel and its neighbors in the true image, where elements of the PSF array act as weights. In particular, in the one dimensional case, if the vectors y, x and p, are the blurred image, true image and PSF array, respectively, then discrete convolution can be summarized [6, Chapter 4]: Let p = Jp, where J is the reversal matrix, i.e, it reverses the ordering of elements of p. For computing the i th pixel of blurred image y, put the center in the rotated PSF array p on the i th pixel of true image x, and compute the contractive multiplication of the corresponding arrays. Convolution in higher dimensions is analogous, but the rotation with J must be done in all modes. For example in three dimensions, where the PSF array is P R n n n, the rotated PSF array is P = (J, J, J) : P. In this process, the blurred image is not only affected by the corresponding finite size true image, but it also depends on values of pixels on the boundaries of the true image. In order to apply the PSF at a point close to the boundary, we must impose boundary conditions, i.e. we must continue the image artificially outside its boundary, e.g. by using zero, periodic, reflective and anti-reflective boundary conditions [6, 9, ]. In this brief description we ignore the ill-posed nature of image deblurring. 6. Periodic Boundary Conditions One of the most common ways of imposing boundary conditions is to continue the image periodically outside the domain. The most important advantage of this type of boundary condition is that the linear system has circulant structure, which makes it possible to solve it using FFT. We now consider, in some detail, the -D and -D cases. Then we see that the -D case and higher are simple generalizations in the sense that one only increases the number of modes of the corresponding tensors. The algebra of solving a linear system with this circulant structure is the same independently of the number of modes.

25 6.. -D Consider one dimensional image blurring with periodic boundary condition. Let p R n be the PSF array with center located at the l th entry. Then the blurred image y R n is x x x x x x x x x p p p y = p x + p x + p x Figure 4: One dimensional discrete convolution with periodic boundary conditions when n= and the center is p. obtained from the true image x R n as y i = C l+i Jp, x, i =,...,n. For simplicity let ˆp = C l Jp, so y i = C i ˆp, x, i =,...,n. This can be written as matrix-vector equation y = Ax, (6.) where b T i = (C i ˆp) T is the i th row of A. Thus, A is a circulant matrix, and by using (4.7), the linear system of equations (6.) can be solved in O(n log n) operations: In MATLAB ˆp is computed as 6.. -D x=fft(ifft(y)./fft(ˆp)). ˆp = circshift(p(n:-:),l). The process in the two dimension is analogous. Assume that all boundaries are periodic, and let P R n n be the PSF array, where p l,k is its center. Put the center p l,k in the rotated PSF array over x, and compute the contracted product of X and P in that position, giving y. Then by moving P i steps down and j steps to the right and computing the contractive product of P and X one obtains the y ij pixel of the blurred image Y. This procedure can be written y ij = C i PC (j )T, X,;,, where P = C l JPJC (k)t. We can write this as a linear tensor-matrix transformation Y = A, X :;:, A(:, :, i, j) = C i PC (j )T. (6.) 4

26 p p p p p p x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x p p p x Figure : Two dimensional convolution with periodic boundary conditions and center p,. From Lemma 4.6 we see that A is a {, } and {, 4}-circulant tensor, so it can be diagonalized by Fourier matrices. Since A(:, :,, ) = P, by Corollary.4 we have where Y = F Y F, X = F XF and Now the image X can be computed by 6.. -D Y = P. X, P = ( nf, nf ) A(:, :,, ), = nf PF = fft( P). X = fft(ifft(y)./fft( P)). The three and higher dimensional cases are now handled simply by increasing the number of modes. Let X R n n n and Y R n n n be the true and blurred image, respectively, and let P be the PSF array with center at P(l, l, l ). The rotated PSF is given by ( ) P = C l J, C l J, C l J P. : Now the relation between the true and blurred image can be written as a tensor-tensor linear system ( Y = A, X :;:, A(:, :, :, i, j, k) = C i, C j, C k ) P. (6.) : where A is a {, 4},{, } and {, 6}-circulant tensor. By Corollary.4 this linear system is equivalent to Y = P. X,

27 where Y = (F, F, F ) : Y, X = (F, F, F ) : X, and P = ( nf, nf, nf ) A(:, :, :,,, ) : = ( nf, nf, nf ) : P. So it is straightforward to show that (6.) is solved by 7 Conclusions Y=fftn(fftn( P).*ifft(X)), (6.4) X=fftn(ifft(Y)./fftn( P)). (6.) In this paper we introduce a framework for handling tensors with diagonal and circulant structure. We show that every tensor that is circulant with respect to a pair of modes can be diagonalized, by the discrete Fourier transform, with respect to those modes. This means that the linear systems with circulant structure, which occur for instance in image deblurring in N dimensions, can be solved efficiently, using N-dimensional Fourier transforms. This is of course well-known. On the other hand, the derivation of these properties of the linear systems, has been based on complicated mappings of tensor data and tensor operators on vectors and matrices. In our framework no such mappings are needed, and the blurring process can be described using notation that is natural in the application. In addition, the generalization to higher dimensions is straightforward in the new framework. The tensor framework can be used also in connection with preconditioners with circulant structure. We are presently studying how other problems involving structured matrices can be generalized to tensors in a similar way. 8 Acknowledgement We are indebted to two anonymous referees, whose suggestions lead to improvements of the paper. References [] O. Aberth. The transformation of tensors to diagonal form. SIAM J. Appl. Math., :47, 967. [] R. Badeau and R. Boyer. Fast multilinear singular value decomposition for structured tensors. SIAM J. Matrix Anal. Appl, ():8, 8. [] B. Bader and T. Kolda. Algorithm 86: MATLAB tensor classes for fast algorithm prototyping. ACM Transactions on Mathematical Software, :6 6, 6. [4] B. W. Bader and T. G. Kolda. Efficient MATLAB computations with sparse and factored tensors. SIAM Journal on Scientific Computing, ():, 7. [] S. S. Capizzano. A note on antireflective boundary conditiones and fast deblurring models. SIAM J. Sci. Comput., (4):7,. 6

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