The document discusses low-rank tensor methods for high-dimensional data analysis, detailing various techniques for efficient computation and approximation in tensor formats. It covers methodologies for evaluating specific parameters, finding maxima and minima, and computing level sets through tensor representations. The content also addresses the computational complexities of different tensor formats and their respective advantages and disadvantages.

1. Low-rank tensor methods for analysis of high
dimensional data
Alexander Litvinenko and Mike Espig
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http://sri-uq.kaust.edu.sa/
Extreme Computing Research Center, KAUST
Alexander Litvinenko and Mike Espig Low-rank tensor methods for analysis of high dimensional da

2. 4*
KAUST
I received very rich collaboration experience as a co-organizator of:
3 UQ workshops,
2 Scalable Hierarchical Algorithms for eXtreme Computing
(SHAXC) workshops
1 HPC Conference (www.hpcsaudi.org, 2017)

3. 4*
My previous work
After applying the stochastic Galerkin method, obtain:
Ku = f, where all ingredients are represented in a tensor format
Compute max{u}, var(u), level sets of u, sign(u)
[1] Efficient Analysis of High Dimensional Data in Tensor Formats,
Espig, Hackbusch, A.L., Matthies and Zander, 2012.
Research which ingredients influence on the tensor rank of K
[2] Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats,
W¨ahnert, Espig, Hackbusch, A.L., Matthies, 2013.
Approximate κ(x, ω), stochastic Galerkin operator K in Tensor
Train (TT) format, solve for u, postprocessing
[3] Polynomial Chaos Expansion of random coefficients and the solution of stochastic
partial differential equations in the Tensor Train format, Dolgov, Litvinenko, Khoromskij, Matthies, 2016.
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4. 4*
Typical quantities of interest
Keeping all input and intermediate data in a tensor
representation one wants to perform different tasks:
evaluation for specific parameters (ω1, . . . , ωM),
finding maxima and minima,
finding ‘level sets’ (needed for histogram and probability
density).
Example of level set: all elements of a high dimensional tensor
from the interval [0.7, 0.8].
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5. 4*
Canonical and Tucker tensor formats
Definition and Examples of tensors
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6. 4*
Canonical and Tucker tensor formats
[Pictures are taken from B. Khoromskij and A. Auer lecture course]
Storage: O(nd ) → O(dRn) and O(Rd + dRn).
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7. 4*
Definition of tensor of order d
Tensor of order d is a multidimensional array over a d-tuple
index set I = I1 × · · · × Id ,
A = [ai1...id
: i ∈ I ] ∈ RI
, I = {1, ..., n }, = 1, .., d.
A is an element of the linear space
Vn =
d
=1
V , V = RI
equipped with the Euclidean scalar product ·, · : Vn × Vn → R,
defined as
A, B :=
(i1...id )∈I
ai1...id
bi1...id
, for A, B ∈ Vn.
Let T := d
µ=1 Rnµ ,
RR(T ) := R
i=1
d
µ=1 viµ ∈ T : viµ ∈ Rnµ ,
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8. 4*
Examples of rank-1 and rank-2 tensors
Rank-1:
f(x1, ..., xd ) = exp(f1(x1) + ... + fd (xd )) = d
j=1 exp(fj(xj))
Rank-2: f(x1, ..., xd ) = sin( d
j=1 xj), since
2i · sin( d
j=1 xj) = ei d
j=1 xj
− e−i d
j=1 xj
Rank-d function f(x1, ..., xd ) = x1 + x2 + ... + xd can be
approximated by rank-2: with any prescribed accuracy:
f ≈
d
j=1(1 + εxj)
ε
−
d
j=1 1
ε
+ O(ε), as ε → 0
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9. 4*
Tensor and Matrices
Rank-1 tensor
A = u1 ⊗ u2 ⊗ ... ⊗ ud =:
d
µ=1
uµ
Ai1,...,id
= (u1)i1
· ... · (ud )id
Rank-1 tensor A = u ⊗ v, matrix A = uvT , A = vuT , u ∈ Rn,
v ∈ Rm,
Rank-k tensor A = k
i=1 ui ⊗ vi, matrix A = k
i=1 uivT
i .
Kronecker product of n × n and m × m matrices is a new block
matrix A ⊗ B ∈ Rnm×nm, whose ij-th block is [AijB].
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10. 4*
Computing QoI in low-rank tensor format
Now, we consider how to
find maxima in a high-dimensional tensor

11. 4*
Maximum norm and corresponding index
Let u = r
j=1
d
µ=1 ujµ ∈ Rr , compute
u ∞ := max
i:=(i1,...,id )∈I
|ui| = max
i:=(i1,...,id )∈I
r
j=1
d
µ=1
ujµ iµ
.
Computing u ∞ is equivalent to the following e.v. problem.
Let i∗
:= (i∗
1 , . . . , i∗
d ) ∈ I, #I = d
µ=1 nµ.
u ∞ = |ui∗ | =
r
j=1
d
µ=1
ujµ i∗
µ
and e(i∗
)
:=
d
µ=1
ei∗
µ
,
where ei∗
µ
∈ Rnµ the i∗
µ-th canonical vector in Rnµ (µ ∈ N≤d ).
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12. Then
u e(i∗
)
=


r
j=1
d
µ=1
ujµ




d
µ=1
ei∗
µ

 =
r
j=1
d
µ=1
ujµ ei∗
µ
=
r
j=1
d
µ=1
(ujµ)i∗
µ
ei∗
µ
=


r
j=1
d
µ=1
(ujµ)i∗
µ


ui∗ =
d
µ=1
e(i∗
µ) = ui∗ e(i∗
)
.
Thus, we obtained an “eigenvalue problem”:
u e(i∗
)
= ui∗ e(i∗
)
.
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13. 4*
Computing u ∞, u ∈ Rr by vector iteration
By defining the following diagonal matrix
D(u) :=
r
j=1
d
µ=1
diag (ujµ) µ µ∈N≤nµ
(1)
with representation rank r, obtain D(u)v = u v.
Now apply the well-known vector iteration method (with rank
truncation) to
D(u)e(i∗
)
= ui∗ e(i∗
)
,
obtain u ∞.
[Approximate iteration, Khoromskij, Hackbusch, Tyrtyshnikov 05],
and [Espig, Hackbusch 2010]
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14. 4*
How to compute the mean value in CP format
Let u = r
j=1
d
µ=1 ujµ ∈ Rr , then the mean value u can be
computed as a scalar product
u =


r
j=1
d
µ=1
ujµ

 ,


d
µ=1
1
nµ
˜1µ

 =
r
j=1
d
µ=1
ujµ, ˜1µ
nµ
=
(2)
=
r
j=1
d
µ=1
1
nµ
nµ
k=1
(ujµ)k , (3)
where ˜1µ := (1, . . . , 1)T ∈ Rnµ .
Numerical cost is O r · d
µ=1 nµ .
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15. 4*
How to compute the variance in CP format
Let u ∈ Rr and
˜u := u − u
d
µ=1
1
nµ
1 =
r+1
j=1
d
µ=1
˜ujµ ∈ Rr+1, (4)
then the variance var(u) of u can be computed as follows
var(u) =
˜u, ˜u
d
µ=1 nµ
=
1
d
µ=1 nµ


r+1
i=1
d
µ=1
˜uiµ

 ,


r+1
j=1
d
ν=1
˜ujν


=
r+1
i=1
r+1
j=1
d
µ=1
1
nµ
˜uiµ, ˜ujµ .
Numerical cost is O (r + 1)2 · d
µ=1 nµ .

16. 4*
Computing QoI in low-rank tensor format
Now, we consider how to
find ‘level sets’,
for instance, all entries of tensor u from interval [a, b].

17. 4*
Definitions of characteristic and sign functions
1. To compute level sets and frequencies we need
characteristic function.
2. To compute characteristic function we need sign function.
The characteristic χI(u) ∈ T of u ∈ T in I ⊂ R is for every multi-
index i ∈ I pointwise defined as
(χI(u))i :=
1, ui ∈ I,
0, ui /∈ I.
Furthermore, the sign(u) ∈ T is for all i ∈ I pointwise defined
by
(sign(u))i :=



1, ui > 0;
−1, ui < 0;
0, ui = 0.
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18. 4*
sign(u) is needed for computing χI(u)
Lemma
Let u ∈ T , a, b ∈ R, and 1 = d
µ=1
˜1µ, where
˜1µ := (1, . . . , 1)t ∈ Rnµ .
(i) If I = R<b, then we have χI(u) = 1
2 (1 + sign(b1 − u)).
(ii) If I = R>a, then we have χI(u) = 1
2(1 − sign(a1 − u)).
(iii) If I = (a, b), then we have
χI(u) = 1
2 (sign(b1 − u) − sign(a1 − u)).
Computing sign(u), u ∈ Rr , via hybrid Newton-Schulz iteration
with rank truncation after each iteration.
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19. 4*
Level Set, Frequency
Definition (Level Set, Frequency)
Let I ⊂ R and u ∈ T . The level set LI(u) ∈ T of u respect to I is
pointwise defined by
(LI(u))i :=
ui, ui ∈ I ;
0, ui /∈ I ,
for all i ∈ I.
The frequency FI(u) ∈ N of u respect to I is defined as
FI(u) := # supp χI(u).
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20. 4*
Computation of level sets and frequency
Proposition
Let I ⊂ R, u ∈ T , and χI(u) its characteristic. We have
LI(u) = χI(u) u
and rank(LI(u)) ≤ rank(χI(u)) rank(u).
The frequency FI(u) ∈ N of u respect to I is
FI(u) = χI(u), 1 ,
where 1 = d
µ=1
˜1µ, ˜1µ := (1, . . . , 1)T ∈ Rnµ .
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21. 4*
Numerical Experiments
2D L-shape domain, N = 557 dofs.
Total stochastic dimension is Mu = Mk + Mf = 20, there are
|J | = 231 PCE coefficients
u =
231
j=1
uj,0 ⊗
20
µ=1
ujµ ∈ R557
⊗
20
µ=1
R3
.
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22. 4*
Level sets
Now we compute level sets
sign(b u ∞1 − u)
for b ∈ {0.2, 0.4, 0.6, 0.8}.
Tensor u has 320 ∗ 557 ≈ 2 · 1012 entries ≈ 16 TB of
memory.
The computing time of one level set was 10 minutes.
Intermediate ranks of sign(b u ∞1 − u) and of rank(uk )
were less than 24.
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23. 4*
Example: Canonical rank d, whereas TT rank 2
d-Laplacian over uniform tensor grid. It is known to have the
Kronecker rank-d representation,
∆d = A⊗IN ⊗...⊗IN +IN ⊗A⊗...⊗IN +...+IN ⊗IN ⊗...⊗A ∈ RI⊗d ⊗I⊗d
(5)
with A = ∆1 = tridiag{−1, 2, −1} ∈ RN×N, and IN being the
N × N identity. Notice that for the canonical rank we have rank
kC(∆d ) = d, while TT-rank of ∆d is equal to 2 for any
dimension due to the explicit representation
∆d = (∆1 I) ×
I 0
∆1 I
× ... ×
I 0
∆1 I
×
I
∆1
(6)
where the rank product operation ”×” is defined as a regular
matrix product of the two corresponding core matrices, their
blocks being multiplied by means of tensor product. The similar
bound is true for the Tucker rank rankTuck (∆d ) = 2.

24. 4*
Advantages and disadvantages
Denote k - rank, d-dimension, n = # dofs in 1D:
1. CP: ill-posed approx. alg-m, O(dnk), hard to compute
approx.
2. Tucker: reliable arithmetic based on SVD, O(dnk + kd )
3. Hierarchical Tucker: based on SVD, storage O(dnk + dk3),
truncation O(dnk2 + dk4)
4. TT: based on SVD, O(dnk2) or O(dnk3), stable
5. Quantics-TT: O(nd ) → O(dlogq
n)