Top ranked 10 <i>G</i>(<i>ℓ</i>₁, <i>ℓ</i>₂, <i>ℓ</i>₃)s with larger absolute values among 1 ≤ <i>ℓ</i>₁, <i>ℓ</i>₂, <i>ℓ</i>₃ ≤ 10 when TD was applied to type I tensor generated from and (left) and and (right).

<p>Top ranked 10 <i>G</i>(<i>ℓ</i>₁, <i>ℓ</i>₂, <i>ℓ</i>₃)s with larger absolute values among 1 ≤ <i>ℓ</i>₁, <i>ℓ</i>₂, <i>ℓ</i>₃ ≤ 10 when TD was applied to type I tensor generated from and (left) and and (right).</p

Results of DIANA-mirath using seven miRNAs identified.

<p>Top 10 significant KEGG pathway was presented. gene: number of genes overlapped with miRNAs target genes, miRNA: number of overlapped miRNAs. Numbers both sides of “/” correspond to type I/type II tensors, respectively.</p

Top 10 significant overlap gene set in MSigDB with top ranked 400 (approx) mRNA probes identified by alternative methods SAM, limma, and RF, as well as 374 mRNA probes identified by HO GSVD.

<p>“BREAST_CANCER” was presented in bold to emphasize the overlap with breast cancer, whose counts are in parentheses at the right side of method names.</p

The results of TD applied to type II tensor generated from synthetic dataset (<i>M</i> = 50).

<p> (a) <i>ℓ</i>₃ = 1 (b) <i>ℓ</i>₃ = 2 (c) <i>ℓ</i>₃ = 3. (d) <i>ℓ</i>₃ = 1 (e) <i>ℓ</i>₃ = 2 (f) <i>ℓ</i>₃ = 3. (g): (a) vs (c), (h): (b) vs (d), <i>γ</i> = 0.97, <i>P</i> = 0, (i): (c) vs (f), <i>γ</i> = 0.97, <i>P</i> = 0. <i>γ</i>: Pearson correlation coefficients. P: associated <i>P</i>-values.</p

Hierarchical clustering of (x_mRNA) and (x_miRNA).

<p>When TD was applied to type II tensor (a) and <i>v</i>_{<i>ℓ</i>₃,<i>j</i>} (for mRNA, labelled as PC), and (for miRNA, labelled as PCM) when PCA was separately applied to miRNA and mRNA (b) (1 ≤ <i>ℓ</i>₃ ≤ 10). Distances were negative signed absolute values of Pearson correlation coefficients. Unweighted Pair Group Method with Arithmetic mean (UPGMA) was employed.</p

Tensor decomposition-based unsupervised feature extraction applied to matrix products for multi-view data processing

<div><p>In the current era of big data, the amount of data available is continuously increasing. Both the number and types of samples, or features, are on the rise. The mixing of distinct features often makes interpretation more difficult. However, separate analysis of individual types requires subsequent integration. A tensor is a useful framework to deal with distinct types of features in an integrated manner without mixing them. On the other hand, tensor data is not easy to obtain since it requires the measurements of huge numbers of combinations of distinct features; if there are <i>m</i> kinds of features, each of which has <i>N</i> dimensions, the number of measurements needed are as many as <i>N</i>^<i>m</i>, which is often too large to measure. In this paper, I propose a new method where a tensor is generated from individual features without combinatorial measurements, and the generated tensor was decomposed back to matrices, by which unsupervised feature extraction was performed. In order to demonstrate the usefulness of the proposed strategy, it was applied to synthetic data, as well as three omics datasets. It outperformed other matrix-based methodologies.</p></div

The results of TD applied to the type I tensor generated from a synthetic dataset (<i>M</i> = 50).

<p>(a) to (c) are orthogonal base functions: (a) constant, (b) linear, (c) half period sinusoidal. (d) and (e) base functions used for generating . (d) <i>k</i> = 1, (e) <i>k</i> = 2. (f) is the scatter plot of (d) and (e). (g) to (i) are the first, second, and third sample singular value vectors <i>x</i>_{<i>ℓ</i>₃,<i>j</i>} and <i>ℓ</i>₃ = 1, 2, 3, and are computed by applying TD to synthetic data.</p

The results of HO GSVD applied to the synthetic data.

<p>Red open circles are features with latent correlation (1 ≤ <i>j</i> ≤ <i>N</i>₀). The first (a) and the second (b) sample singular value vectors and the first vs the second feature singular value vectors of the first (c) and the second views (d).</p

Overlap between mRNAs identified (S1 Table) and MSigDB.

<p>Top 10 ranked gene sets are presented. Upper rows: type I, lower rows: type II tensors are considered in each gene set name, respectively. The word “BREAST_CANCER/_DUCTAL_CARCINOMA” was presented in bold face in order to emphasize the overlap with breast cancer related gene sets. K: The number of genes in each gene set, k: The number of genes overlapped.</p

The results of TD applied to type II tensor generated from vaccination.

<p>Sample singular value vectors, Black open circle: Red open circle: Green open circle: (a) <i>ℓ</i>₁ = <i>ℓ</i>₂ = <i>ℓ</i>₃ = 1 (b) <i>ℓ</i>₁ = <i>ℓ</i>₂ = <i>ℓ</i>₃ = 2 (c) Histogram of the correlation coefficients between sample singular value vectors and selected individual 104 mRNA probes expression profiles. (d) Boxplot of scaled and shifted selected individual 104 mRNA probe expression profiles. Black: P, Red:D, green:ND cell lines. <i>P</i>-values computed by categorical regression between P, D, and NP groups are below figures.</p