This document discusses tensor decomposition with Python. It begins by explaining what tensor decomposition and factorization are, and how they can be used to represent multi-dimensional datasets and perform dimensionality reduction. It then discusses matrix and tensor factorization methods like NMF, topic modeling, and CP/PARAFAC decomposition. The remainder of the document provides examples of tensor decomposition using Python tools and libraries, and discusses applications to analyzing temporal network and sensor data.

1. TENSOR DECOMPOSITION WITH PYTHON
LEARNING STRUCTURES FROM MULTIDIMENSIONAL DATA
ANDRÉ PANISSON
@apanisson
ISI Foundation, Torino & New York City

2. WHAT IS DATA DECOMPOSITION?
DECOMPOSITION == FACTORIZATION
Representation a dataset as a sum of (interpretable) parts
▸ Represent data as the combination of many components / factors
▸ Dimensionality reduction: each new dimension
represents a latent variable:
▸ text corpus => topics
▸ shopping behaviour => segments (user segmentation)
▸ social network => groups, communities
▸ psychology surveys => personality traits
▸ electronic medical records => health conditions
▸ chemical solutions => chemical ingredients

3. X
W
H

4. DATA DECOMPOSITION
▸ Decomposition of data represented in two dimensions:
MATRIX FACTORIZATION
▸ text: documents X terms
▸ surveys: subjects X questions
▸ electronic medical records: patients X diagnosis/drugs
▸ Decomposition of data represented in more dimensions:
TENSOR FACTORIZATION
▸ social networks: user X user (adjacency matrix) X time
▸ text: authors X terms X time
▸ spectroscopy:
solution sample X wavelength (emission) X wavelength (excitation)

5. WHY TENSOR FACTORIZATION + PYTHON?
▸ Matrix Factorization is already used in many fields
▸ Tensor Factorization is becoming very popular
for multiway data analysis
▸ TF is very useful to explore time-varying network data
▸ But still, the most used tool is Matlab
▸ There’s room for improvement in
the Python libraries for TF

6. MATRIX DECOMPOSITION

7. FACTOR ANALYSIS
Spearman ~1900
X≈WH
Xtests x subjects ≈ Wtests x intelligences Hintelligences x subjects
Spearman, 1927: The abilities of man.
≈
tests
subjects subjects
tests
Int.
Int.
X W
H

8. TOPIC MODELING / LATENT SEMANTIC ANALYSIS
Blei, David M. "Probabilistic topic models." Communications of the ACM 55.4 (2012): 77-84.
. , ,
. , ,
. . .
gene
dna
genetic
life
evolve
organism
brai n
neuron
nerve
data
number
computer
. , ,
Topics Documents
Topic proportions and
assignments
0.04
0.02
0.01
0.04
0.02
0.01
0.02
0.01
0.01
0.02
0.02
0.01
data
number
computer
. , ,
0.02
0.02
0.01

9. TOPIC MODELING / LATENT SEMANTIC ANALYSIS
X≈WH
Non-negative Matrix Factorization (NMF):
(~1970 Lawson, ~1995 Paatero, ~2000 Lee & Seung)
2005 Gaussier et al. "Relation between PLSA and NMF and implications."
arg min
W,H
kX WHk s. t. W, H 0
≈
documents
terms terms
documents
topic
topic
Sparse
Matrix! W
H

10. NON-NEGATIVE MATRIX FACTORIZATION (NMF)
NMF gives Part based representation
(Lee & Seung – Nature 1999)
NMF
=×
Original
PCA
×
=
NMF is similar to Spectral Clustering
(Ding et al. - SDM 2005)
arg min
W,H
kX WHk s. t. W, H 0
W W •
XHT
WHHT
H H •
WT
X
WTWH
NMF brings interpretation!

11. from sklearn import datasets, decomposition, utils
digits = datasets.fetch_mldata('MNIST original')
A = utils.shuffle(digits.data)
nmf = decomposition.NMF(n_components=20)
W = nmf.fit_transform(A)
H = nmf.components_
plt.rc("image", cmap="binary")
plt.figure(figsize=(8,4))
for i in range(20):
plt.subplot(2,5,i+1)
plt.imshow(H[i].reshape(28,28))
plt.xticks(())
plt.yticks(())
plt.tight_layout()

12. TENSORS AND TENSOR DECOMPOSITION

13. BEYOND MATRICES: HIGH DIMENSIONAL DATASETS
Cichocki et al. Nonnegative Matrix and Tensor Factorizations
Environmental analysis
▸ Measurement as a function of (Location, Time, Variable)
Sensory analysis
▸ Score as a function of (Wine sample, Judge, Attribute)
Process analysis
▸ Measurement as a function of (Batch, Variable, time)
Spectroscopy
▸ Intensity as a function of (Wavelength, Retention, Sample, Time,
Location, …)
…
MULTIWAY DATA ANALYSIS

14. DIGITAL TRACES FROM SENSORS AND IOT
USER
POSITION
TIME
…

15. TENSORS

16. WHAT IS A TENSOR?
A tensor is a multidimensional array
E.g., three-way tensor:
Mode-1
Mode-2
Mode-3
651a

18. FIBERS AND SLICES
Cichocki et al. Nonnegative Matrix and Tensor Factorizations
Column (Mode-1) Fibers Row (Mode-2) Fibers Tube (Mode-3) Fibers
Horizontal Slices Lateral Slices Frontal Slices
A[:, 4, 1] A[1, :, 4] A[1, 3, :]
A[1, :, :] A[:, :, 1]A[:, 1, :]

19. TENSOR UNFOLDINGS: MATRICIZATION AND VECTORIZATION
Matricization: convert a tensor to a matrix
Vectorization: convert a tensor to a vector

20. >>> T = np.arange(0, 24).reshape((3, 4, 2))
>>> T
array([[[ 0, 1],
[ 2, 3],
[ 4, 5],
[ 6, 7]],
[[ 8, 9],
[10, 11],
[12, 13],
[14, 15]],
[[16, 17],
[18, 19],
[20, 21],
[22, 23]]])
OK for dense tensors: use a combination
of transpose() and reshape()
Not simple for sparse datasets (e.g.: <authors, terms, time>)
for j in range(T.shape[1]):
for i in range(T.shape[2]):
print T[:, i, j]
[ 0 8 16]
[ 2 10 18]
[ 4 12 20]
[ 6 14 22]
[ 1 9 17]
[ 3 11 19]
[ 5 13 21]
[ 7 15 23]
# supposing the existence of unfold
>>> T.unfold(0)
array([[ 0, 2, 4, 6, 1, 3, 5, 7],
[ 8, 10, 12, 14, 9, 11, 13, 15],
[16, 18, 20, 22, 17, 19, 21, 23]])
>>> T.unfold(1)
array([[ 0, 8, 16, 1, 9, 17],
[ 2, 10, 18, 3, 11, 19],
[ 4, 12, 20, 5, 13, 21],
[ 6, 14, 22, 7, 15, 23]])
>>> T.unfold(2)
array([[ 0, 8, 16, 2, 10, 18, 4, 12, 20, 6, 14, 22],
[ 1, 9, 17, 3, 11, 19, 5, 13, 21, 7, 15, 23]])

21. RANK-1 TENSOR
The outer product of N vectors results in a rank-1 tensor
array([[[ 1., 2.],
[ 2., 4.],
[ 3., 6.],
[ 4., 8.]],
[[ 2., 4.],
[ 4., 8.],
[ 6., 12.],
[ 8., 16.]],
[[ 3., 6.],
[ 6., 12.],
[ 9., 18.],
[ 12., 24.]]])
a = np.array([1, 2, 3])
b = np.array([1, 2, 3, 4])
c = np.array([1, 2])
T = np.zeros((a.shape[0], b.shape[0], c.shape[0]))
for i in range(a.shape[0]):
for j in range(b.shape[0]):
for k in range(c.shape[0]):
T[i, j, k] = a[i] * b[j] * c[k]
T = a(1)
· · · a(N)
=
a
c
b
Ti,j,k = a
(1)
i a
(2)
j a
(3)
k

22. TENSOR RANK
▸ Every tensor can be written as a sum of rank-1 tensors
=
a1 aJ
c1 cJ
b1 bJ
+ +
▸ Tensor rank: smallest number of rank-1 tensors
that can generate it by summing up
X ⇡
RX
r=1
a(1)
r a(2)
r · · · a(N)
r ⌘ JA(1)
, A(2)
, · · · , A(N)
K
T ⇡
RX
r=1
ar br cr ⌘ JA, B, CK

23. array([[[ 61., 82.],
[ 74., 100.],
[ 87., 118.],
[ 100., 136.]],
[[ 77., 104.],
[ 94., 128.],
[ 111., 152.],
[ 128., 176.]],
[[ 93., 126.],
[ 114., 156.],
[ 135., 186.],
[ 156., 216.]]])
A = np.array([[1, 2, 3],
[4, 5, 6]]).T
B = np.array([[1, 2, 3, 4],
[5, 6, 7, 8]]).T
C = np.array([[1, 2],
[3, 4]]).T
T = np.zeros((A.shape[0], B.shape[0], C.shape[0]))
for i in range(A.shape[0]):
for j in range(B.shape[0]):
for k in range(C.shape[0]):
for r in range(A.shape[1]):
T[i, j, k] += A[i, r] * B[j, r] * C[k, r]
T = np.einsum('ir,jr,kr->ijk', A, B, C)
: Kruskal Tensorbr cr ⌘ JA, B, CK

24. TENSOR FACTORIZATION
▸ CANDECOMP/PARAFAC factorization (CP)
▸ extensions of SVD / PCA / NMF to tensors
NON-NEGATIVE TENSOR FACTORIZATION
▸ Decompose a non-negative tensor to
a sum of R non-negative rank-1 tensors
arg min
A,B,C
kT JA, B, CKk
with JA, B, CK ⌘
RX
r=1
ar br cr
subject to A 0, B 0, C 0

25. TENSOR FACTORIZATION: HOW TO
Alternating Least Squares(ALS):
Fix all but one factor matrix to which LS is applied
min
A 0
kT(1) A(C B)T
k
min
B 0
kT(2) B(C A)T
k
min
C 0
kT(3) C(B A)T
k
denotes the Khatri-Rao product, which is a
column-wise Kronecker product, i.e., C B = [c1 ⌦ b1, c2 ⌦ b2, . . . , cr ⌦ br]
T(1) = ˆA(ˆC ˆB)T
T(2) = ˆB(ˆC ˆA)T
T(3) = ˆC(ˆB ˆA)T
Unfolded Tensor
on the kth mode

26. F = [zeros(n, r), zeros(m, r), zeros(o, r)]
FF_init = np.rand((len(F), r, r))
def iter_solver(T, F, FF_init):
# Update each factor
for k in range(len(F)):
# Compute the inner-product matrix
FF = ones((r, r))
for i in range(k) + range(k+1, len(F)):
FF = FF * FF_init[i]
# unfolded tensor times Khatri-Rao product
XF = T.uttkrp(F, k)
F[k] = F[k]*XF/(F[k].dot(FF))
# F[k] = nnls(FF, XF.T).T
FF_init[k] = (F[k].T.dot(F[k]))
return F, FF_init
min
A 0
kT(1) A(C B)T
k
min
B 0
kT(2) B(C A)T
k
min
C 0
kT(3) C(B A)T
k
arg min
W,H
kX WHk s.
J. Kim and H. Park. Fast Nonnegative Tensor Factorization with an Active-set-like Method.
In High-Performance Scientific Computing: Algorithms and Applications, Springer, 2012, pp. 311-326.
W W •
XHT
WHHT
T(1)(C B)

27. HOW TO INTERPRET: USER X TERM X TIME
X is a 3-way tensor in which
xnmt is 1 if the term m was used by user n at interval t,
0 otherwise
ANxK
is the the association of each user n to a factor k
BMxK
is the association of each term m to a factor k
CTxK
shows the time activity of each factor
users
users
C
=
X
A
B
(N×M×T)
(T×K)
(N×K)
(M×K)
terms
tim
e
tim
e
terms
factors

28. http://www.datainterfaces.org/2013/06/twitter-topic-explorer/

29. TOOLS FOR TENSOR DECOMPOSITION

30. TOOLS FOR TENSOR FACTORIZATION

31. TOOLS: THE PYTHON WORLD
NumPy SciPy
Scikit-Tensor (under development):
github.com/mnick/scikit-tensor
NTF: gist.github.com/panisson/7719245

32. TENSOR DECOMPOSITION OF WEARABLE SENSOR DATA

34. direct proximity
sensing

35. primary
school
Lyon, France
primary school
231 students
10 teachers

36. TENSORS

37. 0 1 0
1 0 1
0 1 0
FROM TEMPORAL GRAPHS TO 3-WAY TENSORS

38. temporal network
tensorial
representation
tensor factorization
factors
communities temporal activity
factorization
quality
A,B C
tuning the complexity
of the model
nodes
communities
1B
5A
3B
5B
2B
2A
3A
4A
1A
4B
50
60
70
80
35
40
45
50
35
40
45
50
50
60
0
10
20
30
4040
0
5
10
15
20
25
30
50
60
70
80
35
40
45
50
40
45
50
50
60
0
10
20
30
4040
0
5
10
15
20
25
30
50
60
70
80
35
40
45
50
50 60
0
10
20
30
4040
0
5
10
15
20
25
30
structures in temporal networks
components
nodes
time
time interval
quality metrics
component

39. L. Gauvin et al., PLoS ONE 9(1), e86028 (2014)
1B
5A
3B
5B
2B
2A
3A
4A
1A
4B
TENSOR DECOMPOSITION OF SCHOOL NETWORK

40. https://github.com/panisson/ntf-school

41. Laetitia Gauvin Ciro Cattuto Anna Sapienza
.fit().predict()
( )

42. @apanisson
panisson@gmail.com
thank you