The document proposes using tensor decomposition on matrix products of multi-view data to perform unsupervised feature extraction. It applies this method to synthetic data containing embedded correlated variables and multi-omics data involving mRNA and miRNA. For the synthetic data, the method recovers the original orthogonal vectors after identifying correlations. For the multi-omics data, a few inter-correlated miRNAs and mRNAs are identified as statistically significant among many features.

1. Tensor decomposition­based unsupervised
feature extraction applied to matrix products
for multi­view data processing
Y­h. Taguchi
Department of Physics, Chuo University
Tokyo, Japan.
PLoS ONE 12(8): e0183933. PLoS ONE 12(8): e0183933.
DOI: 10.1371/journal.pone.0183933DOI: 10.1371/journal.pone.0183933

2. What's typical in Bioinformatics?What's typical in Bioinformatics?
Small samples（a few）, variables（=genes）are
huge（~104
）
→a typical “large p small n” problem
Difficult to apply usual statistical analyses
ex. small samples deep learning → ×
“large p small n” problem
→sparse modeling (lasso)variable selections ×
Approaches specific to bioinformatics are required

3. Purpose: multiview data analysis
persons
×
features
persons
features
persons
×
shoppings
shoppings
features：
A,B,D,M
persons：
β,δ,μ
shoppings：
1,3,4
persons

4. matrix tensor
×xij xil
xij ×xil
xijl
Tensor
decomposition
G
xik1
xjk2
xlk3
xijl=xij ×xil≒Σk1,k2,k3
Gk1,k2,k3
xik1
xjk2
xlk3
i:persons
j:features
l：shoppings

5. Demonstration using synthetic data set
50 50
1000
+20%ノイズ
50
100%noise
No correlationsNo correlations
＋＋
50
+20%ノイズ
50×1000
×1000
tensor
Tensor
decomposition

6. xik1
k1=1
1≦i 50≦
k1=2 k1=3
xjk2
k2=1
k2=2
xlk2
k3=1
k3=2
1≦j 1000≦ 1≦l 1000≦
persons
features shoppings

7. Advantages as multi­view data analysis toolsAdvantages as multi­view data analysis tools
・No weights required to integrate multiple views
・Complete unspervised learning
（no model buildings using pre­knowledge）
・smaller computational resources because of linearity
Disadvantages....
・tendency to require more memories
Solution：summing up Σi xij ×xil results in j×l matrix
that can be converted back （explains omitted）。

8. Feature extractionFeature extraction No real data separated well
Assume Gaussian
Detect outliers
Pi=P[ >∑k
(
xik
σ )
2
]
Benjamini­Hochberg
corrected P <0.01
P­values by χ2
dist
P(p)
1­p
0

9. Applications：multi­omics data
mRNA
sample1
sample2
sample3
sample4
sample5
miRNA
A group
B group
activeactive
expression interaction
xij ×xil i：161samples, j:13393mRNA, l:755miRNA,
(8 groups)

10. Selection of xik1
distinct between symptoms
k1=1 k1=2 k1=3 k1=4 k1=5
1≦k1 5 are symptom dependent≦
P­value

11. k2 k3 k1 G(k1,k2,k3)
1≦k1 k2 k3 5≦
k1 ：sample
k2 ：mRNA
k3 ：miRNA
1≦ k2 5≦Larger G
Smaller G
1≦ k3 2≦
xjk2
xlk3
assume Gaussian
Detect outliers
Benjamini­Hochberg
corrected P <0.01
P­values by χ2
dist
755miRNA中7miRNA
13393mRNA中427mRNA
（Biological validations omitted）

12. SummarySummary
・ As a feature selection in multi view data, after applying
tensor decomposition to a tensor generated by product of
matrices, I propose to select features associated with BH­
corrected P­values <0.01 computed by χ2
dist assumed for
a mode.
・ As for synthetic data set, apparently uncorrelated
variables embedded into noised are decomposed to original
orthogonal vectors after identifying correlated variables.
・As for muli omics data set, a few (a few %) inter­correlated
and biologically reasonable miRNAs and mRNAs are
identified among huge number of mRNAs and miRNAs

13. My presentation in GIW2017:
GIW 7 ­ RNA Bioinformatics
2nd
Nov. Morning (c.a. 10 AM)
at Adonis (1F)
Tensor decomposition­based unsupervised
feature extraction identified the universal
nature of sequence­non­specific off­target
regulation of mRNA mediated by microRNA
transfection