NEW TENSOR DECOMPOSITIONS IN NUMERICAL ANALYSIS AND DATA PROCESSING

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1 NEW TENSOR DECOMPOSITIONS IN NUMERICAL ANALYSIS AND DATA PROCESSING Institute of Numerical Mathematics of Russian Academy of Sciences 11 October 2012

2 COLLABORATION MOSCOW: I.Oseledets, D.Savostyanov S.Dolgov, V.Kazeev, O.Lebedeva, A.Setukha, S.Stavtsev, D.Zheltkov S.Goreinov, N.Zamarashkin LEIPZIG: W.Hackbusch, B.Khoromskij, R.Schneider H.-J.Flad, V.Khoromskaia, M.Espig, L.Grasedyck

3 TENSORS IN 20TH CENTURY used chiefly as desriptive tools: physics differential geometry multiplication tables in algebras applied data management chemometrics sociometrics signal/image processing many others

4 WHAT IS TENSOR Tensor = d-linear form = d-dimensional array: A = [a i1 i 2...i d ] Tensor A possesses: dimensionality (order) d = number of indices (dimensions, modes, axes, directions, ways) size n 1... n d (number of points at each dimension)

5 EXAMPLES OF PROMINENT THEORIES FOR TENSORS IN 20th CENTURY Kruskal s theorem (1977) on essential uniqueness of canonical tensor decomposition introduced by Hitchcock (1927); canonical tensor decompositions as a base for Strassen s method of matrix multiplication of complexity less than n 3 (1969); interrelations between tensors (especially symmetric) and polynomials as a topic in algebraic geometry.

6 BEGIN WITH 2 2 MATRICES The column-by-row rule for 2 2 matrices yields 8 mults: [ ] [ ] a11 a 12 b11 b 12 = a 21 a 22 b 21 b 22 [ a11 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

7 DISCOVERY BY STRASSEN Only 7 mults is enough! IMPORTANT: for block 2 2 matrices these are 7 mults of blocks: α 1 = (a 11 + a 22 )(b 11 + b 22 ) α 2 = (a 21 + a 22 )b 11 α 3 = a 11 (b 12 b 22 ) α 4 = a 22 (b 21 b 11 ) α 5 = (a 11 + a 12 )b 22 α 6 = (a 21 a 11 )(b 11 + b 12 ) α 7 = (a 12 a 22 )(b 21 + b 22 ) c 11 = α 1 + α 4 α 5 + α 7 c 12 = α 3 + α 5 c 21 = α 2 + α 4 c 22 = α 1 + α 3 α 2 + α 6

8 HOW A TENSOR ARISES AND HELPS [ ] [ ] [ ] c1 c 2 a1 a = 2 b1 b 2 c 3 c 4 a 3 a 4 b 3 b 4 c k = h ijk = R α=1 w kα R u i α v jα w kα α=1 n 2 u iα a i i=1 n 2 n 2 c k = h ijk a i b j i=1 j=1 n 2 v jα b j j=1 Now only R mults of blocks! If n = 2 then R = 7 (Strassen, 1969). Recursion O(n log 2 7 ) scalar mults for any n.

9 GENERAL CASE BY RECURSION Two matrices of order n = 2 d can be multiplied with 7 d = n log 2 7 scalar multiplications and 7n log 2 7 scalar additions/subtrations. n = 2 d n/2 n/2 n/2 n/2 n/2 n/2 n/2

10 TENSORS IN 21ST CENTURY: NUMERICAL METHODS WITH TENSORIZATION OF DATA We consider typical problems of numerical analysis (matrix computations, interpolation, optimization) under the assumption that the input, output and all intermediate data are represented by tensors with many dimensions (tens, hundreds, even thousands). Of course, it assumes a very special structure of data. But we have it in really many problems!

11 THE CURSE OF DIMENSIONALITY The main problem is that using arrays as means to introduce tensors in many dimensions is infeasible: if d = 300 and n = 2, then such an array contains entries

12 NEW REPRESENTATION FORMATS Canonical polyadic and Tucker decompositions are of limited use for our purposes (by different reasons). New decompositions: TT (Tensor Train) HT (Hierarchical Tucker)

13 REDUCTION OF DIMENSIONALITY i 1 i 2 i 3 i 4 i 5 i 6 i 1 i 2 i 3 i 4 i 5 i 6 i 1 i 2 i 3 i 4 i 5 i 6 i 3 i 4 i 5 i 6

14 SCHEME FOR TT i 1 i 2 i 3 i 4 i 5 i 6 i 1 i 2 α i 3 i 4 i 5 i 6 α i 1 β i 2 αβ i 3 i 4 γ i 5 i 6 αγ i 3 δ i 4 γδ i 5 αη i 6 γη

15 SCHEME FOR HT i 1 i 2 i 3 i 4 i 5 i 6 i 1 i 2 α i 3 i 4 i 5 i 6 α i 1 β i 2 αβ i 3 i 4 γ i 5 i 6 αγ i 2 φ αβφ i 3 δ i 4 γδ i 5 i 6 ξ γηξ i 4 ψ γδψ i 5 ζ i 6 ξζ i 6 ν ξζν

16 THE BLESSING OF DIMENSIONALITY TT and HT provide new representation formats for d-tensors + algorithms with complexity linear in d. Let the amount of data be N. In numerical analysis, complexity O(N) is usually considered as a dream. With ultimate tensorization we go beyond the dream: since d log N, we may obtain complexity O(log N).

17 BASIC TT ALGORITHMS TT rounding. Like the rounding of machine numbers. COMLEXITY = O(dnr 3 ). ERROR d 1 BEST ERROR. TT interpolation. A tensor train is constructed from sufficiently few elements of the tensor, the number of them is O(dnr 2 ). TT quantization and wavelets. Low-dimensional high-dimensional algebraic wavelet tranbsforms (WTT). In matrix problems the complexity may drop from O(N) down to O(log N).

18 SUMMATION AGREEMENT Omit the symbol of summation. Assume summation if the index in a product of quantities with indices is repeated at least twice. Equations hold for all values of other indices.

19 SKELETON DECOMPOSITION A = UV = r u 1α... [ ] v 1α... v nα u mα α=1 According to the summation agreement, a(i, j) = u(i, α)v(j, α)

20 CANONICAL AND TUCKER CANONICAL DECOMPOSITION a(i 1... i d ) = u 1 (i 1 α)... u d (i d α) TUCKER DECOMPOSITION a(i 1... i d ) = g(α 1... α d )u 1 (i 1 α 1 )... u d (i d α d )

21 TENSOR TRAIN (TT) IN THREE DIMENSIONS a(i 1 ; i 2 i 3 ) = g 1 (i 1 ; α 1 )a 1 (α 1 ; i 2 i 3 ) a 1 (α 1 i 2 ; i 3 ) = g 2 (α 1 i 2 ; α 2 )g 3 (α 2 ; i 3 ) TENSOR TRAIN (TT) a(i 1 i 2 i 3 ) = g 1 (i 1 α 1 )g 2 (α 1 i 2 α 2 )g 3 (α 2 i 3 )

22 TENSOR TRAIN (TT) IN d DIMENSIONS a(i 1... i d ) = g 1 (i 1 α 1 )g 2 (α 1 i 2 α 2 )... g d 1 (α d 2 i d 1 α d 1 )g d (α d 1 i d ) a(i 1... i d ) = d g k (α k 1 i k α k ) k=1

23 KRONECKER REPRESENTATION OF TENSOR TRAINS A = G 1 α 1 G 2 α 1 α 2... G d 1 α d 2 α d 1 G d α d 1 A is of size (m 1... m d ) (n 1... n d ). G k α k 1 α k is of size m k n k.

24 ADVANTAGES OF TENSOR-TRAIN REPRESENTATION The tensor is determined through d tensor carriages g k (α k 1 i k α k ), each of size r k 1 n k r k. If the maximal size is r n r, then the number of representation parameters does not exceed dnr 2 n d.

25 TENSOR TRAIN PROVIDES STRUCTURED SKELETON DECOMPOSITIONS OF UNFOLDING MATRICES A k = a(i 1... i k ; i k+1... i d ) = u k (i 1... i k ; α k ) v k (α k ; i k+1... i d ) = U k V k u k (i 1... i k α k ) = g 1 (i 1 α 1 )... g k (α k 1 i k α k ) v k (α k i k+1... i d ) = g k+1 (α k i k+1 α k+1 )... g d (α k 1 i d )

26 TT RANKS ARE BOUNDED BY THE RANKS OF UNFOLDING MATRICES r k ranka k, A k = [a(i 1... i k ; i k+1... i d )] Equalities are always possible.

27 ORTHOGONAL TENSOR CARRIAGES A tensor carriage g(αiβ) is called row orthogonal if its first unfolding matrix g(α ; iβ) has orthonormal rows. A tensor carriage g(αiβ) is called column orthogonal if its second unfolding matrix g(αi ; β) has orthonormal columns.

28 ORTHOGONALIZATION OF TENSOR CARRIAGES tensor carriage g(αiβ) decomposition g(αiβ) = h(αα )q(α iβ) with q(α iβ) being row orthogonal. tensor carriage g(αiβ) decomposition g(αiβ) = q(αiβ )h(β β) with q(αiβ ) being column orthogonal.

29 PRODUCTS OF ORTHOGONAL TENSOR CARRIAGES A product of row (column) orthogonal tensor carriages p(α s, i s... i t, α t ) = t k=s+1 is also row (column) orthogonal. g k (α k 1 i k α k )

30 MAKING ALL CARRIAGES ORTHOGONAL Orthogonalize the columns of g 1 = q 1 h 1, then compute and orthogonalize h 1 g 2 = q 2 h 2. Thus, and after k steps g 1 g 2 = q 1 q 2 h 2 g 1... g k = q 1... q k h k. Similarly for the row orhogonalization, g k+1... g d = h k+1 z k+1... z d.

31 STRUCTURED ORTHOGONALIZATION TT decomposition a(i 1... i d ) = d g s (α s 1 i s α s ) s=1 column q k and row z k orthogonal carriages s. t. a(i ( 1... i k ; i k+1... i d ) = k ) ( q k (α s 1 i sα s) H k (α k, α k ) s=1 d s=k+1 ) z s (α s 1 i sα s ) q k and z k can be constructed in dnr 3 operations.

32 CONSEQUENCE: STRUCTURED SVD FOR ALL UNFOLDING MATRICES IN O(dnr 3 ) OPERATIONS It suffices to compute SVD for the matrices H k (α k α k ).

33 TENSOR APPROXIMATION VIA MATRIX APPROXIMATION We can approximate any fixed unfolding matrix using its structured SVD: a(i 1... i k ; i k+1... i d ) = a k + e k a k = U k (i 1... i k ; α k)σ k (α k)v k (α k ; i k+1... i d ) e k = e k (i 1... i k ; i k+1... i d )

34 ERROR ORTHOGONALITY U k (i 1... i k α k)e k (i 1... i k ; i k+1... i d ) = 0 e k (i 1... i k+1 ; i k+1... i d )V k (α ki k+1... i d ) = 0

35 COROLLARY OF ERROR ORTHOGONALITY Let a k be further approximated by a TT but so that u k or v k are kept. Then the further error, say e l, is orthogonal to e k. Hence, e k + e l 2 F = e k 2 F + e l 2 F

36 TENSOR-TRAIN ROUNDING Approximate successively A 1, A 2,..., A d 1 with the error bound ε. Then FINAL ERROR d 1 ε

37 TENSOR INTERPOLATION Interpolate an implicitly given tensor by a TT using only small part of its elements, of order dnr 2. Cross interpolation method for tensors is constructed as a generalization of the cross method for matrices (1995) and relies on the maximal volume principle from the matrix theory.

38 MAXIMAL VOLUME PRINCIPLE THEOREM (Goreinov, Tyrtyshnikov) Let [ ] A11 A A = 12, A 21 A 22 where A 11 is a r r block with maximal determinant in modulus (volume) among all r r blocks in A. Then the rank-r matrix ] A r = [ A11 A 21 A 1 11 [ A11 A 12 ] approximates A with the Chebyshev-norm error at most in (r + 1) 2 times larger than the error of best approximation of rank r.

39 BEST IS AN ENEMY OF GOOD Move a good submatrix M in A to the upper r r block. Use right-side multiplications by nonsingular matrices A = a r+1,1... a r+1,r a n1... a nr NECESSARY FOR MAXIMAL VOLUME: a ij 1, r + 1 i n, 1 j r

40 BEST IS AN ENEMY OF GOOD COROLLARY OF MAXIMAL VOLUME σ min (M) 1/ r(n r) + 1 ALGORITHM If a ij 1 + δ, then swap rows i and j. Make identity matrix in the first r rows by right-side multiplication. Quit if a ij < 1 + δ for all i, j. Otherwise repeat.

41 MATRIX CROSS ALGORITHM Given initial column indices j 1,..., j r. Find good row indices i 1,..., i r in these columns. Find good column indices in the rows i 1,..., i r. Proceed choosing good columns and rows until the skeleton cross approximations stabilize. E.E.Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Computing 64, no. 4 (2000),

42 CROSS TENSOR-TRAIN INTERPOLATION Let a 1 = a(i 1, i 2, i 3, i 4 ). Seek crosses in the unfolding matrices. On input: r initial columns in each. Select good rows. A 1 = [a(i 1 ; i 2, i 3, i 4 )], J 1 = {i (β 1) 2 i (β 1) 3 i (β 1) 4 } A 2 = [a(i 1, i 2 ; i 3, i 4 )], J 2 = {i (β 2) 3 i (β 2) 4 } A 3 = [a(i 1, i 2, i 3 ; i 4 )], J 3 = {i (β 3) 4 } rows matrix skeleton decomposition I 1 = {i (α 1) 1 } a 1 (i 1 ; i 2, i 3, i 4 ) a 1 = α 1 g 1 (i 1 ; α 1 ) a 2 (α 1 ; i 2, i 3, i 4 ) I 2 = {i (α 2) 1 i (α 2) 2 } a 2 (α 1, i 2 ; i 3, i 4 ) a 2 = α 2 g 2 (α 1, i 2 ; α 2 ) a 3 (α 2, i 3 ; i 4 ) I 3 = {i (α 3) 1 i (α 3) 2 i (α 3) 3 } a 3 (α 2, i 3 ; i 4 ) a 3 = α 3 g 3 (α 2, i 3 ; α 3 ) g 4 (α 3 ; i 4 ) Finally a = α 1,α 2,α 3,α 4 g 1 (i 1, α 1 ) g 2 (α 1, i 2, α 2 ) g 3 (α 2, i 3, α 3 ) g 4 (α 3, i 4 )

43 QUANTIZATION OF DIMENSIONS Increase the number of dimensions. E.g Extreme case is conversion of a vector of size N = 2 d to a d-tensor of size Using TT format with bounded TT ranks may reduce the complexity from O(N) to as little as O(log 2 N).

44 EXAMPLES OF QUANTIZATION f (x) is a function on [0, 1] a(i 1,..., i d ) = f (ih), i = i i i d 2 d The array of values of f is viewed as a tensor of size 2 2. EXAMPLE 1. f (x) = e x + e 2x + e 3x ttrank= 2.7 ERROR=1.5e-14 EXAMPLE 2. f (x) = 1 + x + x 2 + x 3 ttrank= 3.4 ERROR=2.4e-14 EXAMPLE 3. f (x) = 1/(x 0.1) ttrank= 10.1 ERROR=5.4e-14

45 THEOREMS If there is an ε-approximation with separated variables f (x + y) r u k (x)v k (y), k=1 r = r(ε), then a TT exists with error ε and TT-ranks r. If f (x) is a sum of r exponents, then an exact TT exists with the ranks r. For a polynomial of degree m an exact TT exists with the ranks r = m + 1. If f (x) = 1/(x δ) then r = log ε 1 + log δ 1.

46 ALGEBRAIC WAVELET FILTERS a(i 1... i d ) = u 1 (i 1 α 1 )a 1 (α 1 i 2... i d ) + e 1 u 1 (i 1 α 1 )u(i 1 α 1) = δ(α 1, α 1) a a 1 = u 1 a a 2 = u 2 a 1 a 3 = u 3 a 2...

47 TT QUADRATURE I (d) = sin(x 1 + x x d ) dx 1 dx 2... dx d = [0,1] d Im e i(x 1+x x d ) dx 1 dx 2... dx d = Im [0,1] d ( (e i ) d ) 1 n nodes in each dimension n d values in need! TT interpolation method uses only small part (n = 11) d I (d) Relative Error Timing e e e e e e e e i

48 QTT QUADRATURE 0 sinx x dx = π 2 Truncate the domain and use the rule of rectangles. Machine accuracy causes to use 2 77 values. The vector of values is treated as a tensor of size TT-ranks 12 for the machine precision. Less than 1 sec on notebook.

49 TT IN QUANTUM CHEMISTRY Really many dimensions are natural in quantum molecular dynamics: HΨ = ( V (R 1,..., R f ))Ψ = EΨ V is a Potential Energy Surface (PES) Calculation of V requires to solve Schredinger equation for a variety of coordinates of atoms R 1,..., R f. TT interpolation method uses only small part of values of V from which it produces a suitable TT approximation of PES.

50 TT IN QUANTUM CHEMISTRY Henon-Heiles PES: V (q 1,..., q f ) = 1 2 f f 1 qk 2 + λ k=1 k=1 ( qkq 2 k+1 1 ) 3 q3 k TT-ranks and timings (Oseledets-Khoromskij)

51 SPECTRUM IN THE WHOLE Use the evolution in time: Ψ t = ihψ, Ψ(0) = Ψ 0. Physical scheme reads Ψ(t) = e iht Ψ 0, then we find the autocorrelation function a(t) = (Ψ(t), Ψ 0 ) and its Fourier transform.

52 SPECTRUM IN THE WHOLE Henon-Heilse spectra for f = 2 and different TT-ranks.

53 SPECTRUM IN THE WHOLE Henon-Heiles spectra for f = 4 and f = 10.

54 TT FOR EQUATIONS WITH PARAMETERS Diffusion equation on [0, 1] 2. The diffusion coefficients are constant in each of p p square subdomains, i.e. p 2 parameters varing from 0.1 to points in each of parameters, space grid of size The solution for all values of parameters is approximated by TT with relative accuracy 10 5 : Number of parameters Storage 4 8 Mb Mb Mb

55 WTT FOR DATA COMPRESSION f (x) = sin(100x) A signal on uniform grid with the stepsize 1/2 d on 0 x 1 converts into a tensor of size with all TT-ranks = 2. The Dobechis transform gives much more nonzeros: storage for ε storage(wtt) storage(d4) storage(d8) filters sin(100x), n = 2 d, d = 20

56 WTT FOR COMPRESSION OF MATRICES WTT for vectorized matrices applies after reshaping: a(i 1... i d ; j 1... j d ) ã(i 1 j 1 ;... ; i d j d ). WTT compression with accuracy ε = 10 8 for the Cauchy-Hilbert matrix a ij = 1/(i j) for i j, a ii = 0. n = 2 d storage(wtt) storage(d4) storage(d8) storage(d20)

57 TT IN DISCRETE OPTIMIZATION Among all elements of a tensor given by TT find minimum or maximum. Discrete optimization problem is solved a an eigenvalue problem for diagonal matrices. Block minimization of Raleigh quotient in TT format, blocks of size 5, TT-ranks 5 (O.S.Lebedeva). Function Domain Size Iter. (Ax, x) (Ae i, e i ) e i x Exact max 3Q (1+0.1 x i +sin x i ) [1, 50] i=1 same [1, 50] Q (x + sin x i ) [1, 20] i=1 same [1, 20]

58 CONCLUSIONS AND PERSPECTIVES TT algorithms ( are efficient new instruments for compression of vectors and matrices. Storage and complexity depend on matrix size logarithmically. Free access to a current version of TT-library: There are some theorems with TT-rank estimates. Sharper and more general estimates are to be derived. Difficulty is in nonlinearity of TT decompositions.

59 CONCLUSIONS AND PERSPECTIVES TT interpolation methods provide new efficient methods for tabulation of functions of many variables, also those that are hard to evaluate. There are examples of application of TT methods for fast and accurate computation of multidimensional integrals. TT methods are successfully applied to image and signal processing and may compete with other known methods.

60 CONCLUSIONS AND PERSPECTIVES TT methods are a good base for numerical solution of multidimensional problems of quantum chemistry, quantum molecular dynamics, optimization in parameters, model reduction, multiparametric and stochastic differential equations.