Numerical tensor methods and their applications

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1 Numerical tensor methods and their applications 8 May 2013

2 All lectures 4 lectures, 2 May, 08:00-10:00: Introduction: ideas, matrix results, history. 7 May, 08:00-10:00: Novel tensor formats (TT, HT, QTT). 8 May, 08:00-10:00: Advanced tensor methods (eigenproblems, linear systems). 14 May, 08:00-10:00: Advanced topics, recent results and open problems.

3 Brief recap of Lecture 2 Tensor Train format Arithmetics Rounding QTT-format (idea of tensorization) Cross approximation

4 Plan of lecture 2 QTT-format: explicit representations of functions QTT-format: explicit representation of operators Classification theory QTT-Fourier transform QTT-convolution Linear systems Eigenvalue problems

5 QTT-format (publications-first) [1] S. V. Dolgov, B. N. Khoromskij, and D. V. Savostyanov. Superfast Fourier transform using QTT approximation. J. Fourier Anal. Appl., 18(5): , [2] V. Kazeev, B. N. Khoromskij, and E. E. Tyrtyshnikov. Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity. Technical Report 36, MPI MIS, Leipzig, [3] V. A. Kazeev and B. N. Khoromskij. Low-rank explicit QTT representation of the Laplace operator and its inverse. SIAM J. Matrix Anal. Appl., 33(3): , [4] B. N. Khoromskij. O(d log n) Quantics approximation of N d tensors in high-dimensional numerical modeling. Constr. Appr., 34(2): , [5] I. V. Oseledets. Approximation of 2 d 2 d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl., 31(4): , 2010.

6 QTT-format We have a vector v of values of a function f on a uniform grid with 2 d points: v(i) = f (x i ), x i = a + ih, h = (b a)/(n 1).

7 QTT-format We have a vector v of values of a function f on a uniform grid with 2 d points: v(i) = f (x i ), x i = a + ih, h = (b a)/(n 1). Reshaping into a tensor: i (i 0, i 1,..., i d 1 ). i = i 1 + 2i 2 + 4i d 1 i d V (i 1,..., i d ) = v(i).

8 QTT-format Finally, we have: V (i 1,..., i d ) = f (t t d ), t k = a d + 2k i k h

9 Exponential function f (x) = exp λx, Then f (t t d ) = exp(λt 1 )... exp(λt d ), it has rank 1!

10 Linear function f (x) = x f (t t d ) = t t d

11 Linear function(full steps) t 1 + t 2 + t 3 + t 4 =

12 Linear function(full steps) t 1 + t 2 + t 3 + t 4 = ( t 1 1 ) ( 1 t 2 + t 3 + t 4 )

13 Linear function(full steps) t 1 + t 2 + t 3 + t 4 = ( t 1 1 ) ( ) 1 t 2 + t 3 + t 4 = ( t 1 1 ) ( ) ( ) t 2 1 t 3 + t 4 =

14 Linear function(full steps) t 1 + t 2 + t 3 + t 4 = ( t 1 1 ) ( ) 1 t 2 + t 3 + t 4 = ( t 1 1 ) ( ) ( ) = t 2 1 t 3 + t 4 = ( t 1 1 ) ( ) ( ( ) t 2 1 t 3 1) t 4 =

15 Sine function Similar representation can be obtained for a sine function: f (x) = sin λx f (t t d ) = sin(t t d ) = = ( ) ( ) ( ) ( ) sin t 2 cos t 2 sin td 1 cos t sin t 1 cos t 1... d 1 cos xd cos t 2 sin t 2 cos t d 1 sin t d 1 sin x d The rank is still 2!

16 General result Theorem Let f be such that (O. Const. Approx., 2013) f (x + y) = r u α (x)v α (y) α=1 then the QTT-ranks are bounded by r Interesting example: rational functions

17 TT-format for matrices What about matrices?

18 TT-format for matrices What about matrices? Solution - a vector x associated with a d-tensor X (i 1,..., i d ) Linear operators, acting on such tensors, can be indexed as A(i 1,..., i d ; j 1,..., j d ). Terminology: d-level matrix

19 Two-level matrix: illustration Even in 2d it is interesting: A, B are n n, Kronecker product a 11 B a 12 B a 13 B a 14 B C = A B = a 21 B a 22 B a 23 B a 24 B a 31 B a 32 B a 33 B a 34 B a 41 B a 42 B a 43 B a 44 B

20 Two-level matrix: illustration In the index form: C = A B C(i 1, i 2 ; j 1 j 2 ) = A(i 1, j 1 )B(i 2, j 2 )

21 Two-level matrix: illustration In the index form: C = A B C(i 1, i 2 ; j 1 j 2 ) = A(i 1, j 1 )B(i 2, j 2 ) Exactly rank-1 decomposition under permutation

22 Back to d-dimensions Let A be a d-level matrix: A(i 1,..., i d ; j 1,..., j d ) (say, d-dimensional Laplace operator)

23 Back to d-dimensions Let A be a d-level matrix: A(i 1,..., i d ; j 1,..., j d ) (say, d-dimensional Laplace operator) No low TT-ranks if considered as a 2d-array!

24 Back to d-dimensions Right way: permute indices B(i 1 j 1 ; i 2 j 2 ;... i d j d ) = A(i 1,..., i d ; j 1,..., j d ) A(i 1,..., i d ; j 1,..., j d ) = A 1 (i 1, j 1 )A 2 (i 2, j 2 )... A d (i d, j d )

25 Matrix-by-vector product A(i 1,..., i d ; j 1,... j d ) = A 1 (i 1, j 1 )... A d (i d, j d ) X (j 1,..., j d ) = X 1 (j 1 )... X d (j d ) Y (I ) = J A(I, J)X (J) Exercise: Find a formula

26 QTT-matrix Matrices in the QTT-format: a ij = 1 i j + 0.5, i, j = 1,..., 2d. Let us see what are the ranks. (Demo)

27 Laplace operator Consider an operator d 2 dx with Dirichlet boundary 2 conditions, discretized using the simplest finite-difference scheme: (Illustration for n = 4) = (Demo)

28 Laplace operator V. Kazeev: QTT-ranks of the Laplace operator are bounded by 3

29 Laplace operator: general results We will need a special matrix-by-matrix product: [ ] [ ] A 11 A 12 B 11 B 12 = A 21 A 22 B 21 B 22 [ ] A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 = A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 Doing Kronecker product for the blocks!

30 Basic QTT-blocks ( ) ( ) ( ) I =, J =, P =, ( ) ( ) I 2 =, I 1 =, ( ) 1 1 E =, F = 1 1 ( ) 1 1, 1 1 ( ) ( ) K =, L =,

31 QTT representation of 1D-Laplace ] (d) DD [I = J I J J J J J (d 2) 2I J J J J

32 QTT representation in the Toolbox In the TT-Toolbox, it is defined via the function tt_qlaplace_dd

33 Wavelets and tensor trains QTT can be applied to matrices: A(i, j) A(i 1,..., i d, j 1,..., j d ) A(i 1, j 1, i 2, j 2,..., i d, j d )

Wavelets and tensor trains Smooth function and/or special function: 512

34 Wavelets and tensor trains Smooth function and/or special function: Hilbert image : a ij = 1.0/(i j + 0.5) 540 bytes (QTT) vs 8 KB (JPEG)

35 Wavelets and tensor trains QTT compression of simple images Good: Triangle BAD: Circle 50 bytes (QTT) vs 8 KB (JPEG) 70 KB (QTT) vs 8 KB (JPEG)

36 Wavelets and tensor trains Wavelet tensor train: One step of TT-SVD is equivalent to: U A = v 11 v v 21 v v 31 v v 41 v 42...

37 Problems with a circle Wavelet: First (dominant) rows compress further, others are sparse QTT: Leave only rows of large norms (large singular values)

38 Problems with a circle That is why it is bad: r n, mem = O(n 2 )

39 Idea with the circle Leaving sparse singular vectors a novel digital data compression technique (to be combined with others)! Wavelet decomposition with adaptive number of moments!

40 Image compression New: 50 bytes (WTT) vs 8 KB (JPEG)

41 Image compression latex Lena 7 KB(WTT), PSNR KB (WTT), PSNR KB (WTT), PSNR 38.79

42 Fourier transform Consider now an important operation: the Fourier transform (FFT) Example of a matrix with large QTT-ranks! (We can test) y = Fx, F = w kl, w = exp 2πi n Question: Given x in the QTT-format, can we compute y?

43 FFT matrix and tensor networks Let us do quantization again: k = k 1 + 2k d 1 k d l = l 1 + 2l d 1 l d w kl = pq G pq(k p, l q ) G pq (k p, l q ) = w 2p 1 k p 2 q 1 l q Rank-1 two-dimensional tensor network!

44 FFT matrix and tensor networks(2) k 1, l 1 k 1, l 2 k 1, l 3 k 1, l 4 k 2, l 1 k 2, l 2 k 2, l 3 k 2, l 4 k 3, l 1 k 3, l 2 k 3, l 3 k 3, l 4 k 4, l 1 k 4, l 2 k 4, l 3 k 4, l 4

45 FFT of a vector ( ) l 1,...,l d pq G pq(k p, l q ) x 1 (l 1 )... x d (l d ) It can be rewritten as a Hadamard product of d rank-2 tensors! Hint: l k takes only two values Complexity: O(d 2 r 3 log n)

46 Convolution The convolution operation is defined as c i = j a i jb j. How to do it in the QTT-format?

47 Convolution Idea: Write convolution as c i = jk E ijka k b j E ijk = δ k i+j. Write down the tensor E: E ijk = E(i 1,..., i d, j 1,..., j d, k 1,..., k d+1 ). Permute dimensions: (i 1, j 1, k 1, i 2, j 2, k 2,...). Find an explicit representation

48 Convolution: results Summary of the results (Hackbusch and Kazeev, Khoromskij, Tyrtyshnikov) Toeplitz matrix generated by QTT-vectors of rank r has rank 2r Convolution of two vectors of rank r has rank 2r Multidimensional Toeplitz matrices have the same rank bound, but two QTT-matvecs required

49 Solving eigenvalue problems and linear systems Now, let us go to more advanced problems: Ax = λx Ax = f, with x = X (j 1,..., j d ), f = F (i 1,..., i d ), A(i 1,..., i d ; j 1,..., j d )

50 Tensor-structured solvers Path 1: Do iterative methods with truncation (Krylov, preconditioning, multigrid, etc.) Path 2: Use the information about the structure of the solution

51 Using the information about the solution How to use the information about the solution?

52 Using the information about the solution How to use the information about the solution? Formulate as an optimization problem!

53 Formulation as an optimization problem Ax = λx, A = A, We can minimize the Rayleigh quotient: (Ax, x) (x, x) min, Minimize not over the whole space, but over the set of structured tensors! Nonconvex optimization problem Have to guess the ranks

54 Idea of alternating iterations The idea of alternating iterations is simple: Fix all except X k (i k ). The local problem reduces to the linear eigenvalue problem Guess the rank!

55 Idea of DMRG Here comes the wonderful idea of DMRG (Density Matrix Renormalization Group, S. White, 1993) Generalization of the Wilson renormalization group (=ALS) Optimize not over one core, but over a pair of cores, X k and X k+1!

56 Idea of DMRG X (i 1, i 2, i 3, i 4 ) = X 1 (i 1 )X 2 (i 2 )X 3 (i 3 )X 4 (i 4 ) = = W 12 (i 1, i 2 )X 3 (i 3 )X 4 (i 4 ). Solve for W 12 Split back by the SVD: W 12 (i 1, i 2 ) = X 1 (i 1 )X 2 (i 2 ). The rank is determined adaptively!

57 DMRG and QTT The DMRG method creates modes of size n 2. Very good for spin systems (n = 2 or n = 4) Very good for the QTT (n = 2).

58 DMRG and QTT: papers Holtz, S., Rohwedder, T., Schneider, R., The alternating linear scheme for tensor optimization in the tensor train format (idea) S.V. Dolgov,, Solution of linear systems and matrix inversion in the TT-format (working code)

59 Difficulties in the DMRG S.V. Dolgov, Solution of linear systems and matrix inversion in the TT-format SVD-based truncation: L 2 -norm approximation of x, but the equation can be differential How to avoid local minima Fast solution of local systems Appplication to matrix inversion

60 Large mode sizes What if the mode size is large? Basically, the question is now how to increase the k-th rank. Answer: Using (projected) residuals of the Krylov methods!

61 Large mode sizes The k-th ALS-step: The x is in the special linear subspace: x = (U I V )φ, Analogously to the rounding procedure U and V are structured-orthogonal matrix and φ is r k 1 n k r k. The local problem is then Û AÛφ = Ûf. Can enrich the basis with the (low-rank) approximation of the two-block residual

62 AMEN: further details S.V. Dolgov and D.V. Savostyanov, 2013: Alternating minimal energy methods for linear systems in higher dimensions. Part I/II For more details: convergence estimates, algorithmic details and so on. Still A LOT to be done on algorithms...

63 Solving linear systems and eigenvalue problems (Demo) Solving the high-dimensional Poisson equation in the QTT-format: u = f.

64 Other important problems Solving block eigenvalue problems (minimizing block Rayleigh quotient) Solving nonstationary problems dy dt = Ay, how to rewrite as a minimization problem

65 Next lecture The plan for the next (and the last!) Applications, new results, open problems

Numerical tensor methods and their applications

Numerical tensor methods and their applications 14 May 2013 All lectures 4 lectures, 2 May, 08:00-10:00: Introduction: ideas, matrix results, history. 7 May, 08:00-10:00: Novel tensor formats (TT, HT,