Fig 1.
The truncated HOSVD of a 3-mode tensor.
Part a shows the original tensor , and Part b displays the concept of a truncated HOSVD of
.
Fig 2.
Derivation of the optimal rank.
According to the relative error curve with different number of singular values used, the optimal rank of the Matrix 1 and Matrix 2 are 2 and 4, respectively.
Fig 3.
Block diagram of our co-clustering method based on hyperplane detection in singular vector spaces (HDSVS).
In the left-hand side, the flow for a truncated HOSVD is shown. The LGA module is presented in the middle. The ranking procedure based on a scoring function, for revealing significant co-clusters (δ-CLs) in a tensor, is listed in the right-hand side.
Fig 4.
Effects of noise and overlapping complexity on co-cluster identification in two synthetic tensors.
(a) Matching scores between true co-cluaters and the detected ones, with different SNRs. (b) Matching scores between two overlapping co-clusters and the true ones, with various overlapping degrees.
Fig 5.
Heat maps for gene×patient matrices at fixed time points.
(a) to (e), Scenarios for gene×patient heat maps corresponding to the baseline day, and 2 days, 1 month, 1 year and 2 years after the initiation of an IFN-β therapy.
Fig 6.
The linear patterns embedded in the 2D singular-vector space.
(a) to (c), The linear groups along the directions of first two singular vectors of U(n) (n = 1, 2, 3), respectively.
Fig 7.
Linear patterns embedded in the 3D singular-vector space of the gene×patient×time tensor.
(a) to (c), The scatter plots along the directions of first three singular vectors of U(n) (n = 1, 2, 3), respectively. (d) to (f), Linear or planar patterns of the 3D-points in (a) to (c).
Fig 8.
Heat maps (along each two modes) of the significant co-cluster CL121 in the gene×patient×time tensor.
(a) and (b), Heat maps of the gene×patient matrix at the baseline time and at time = 1 year. (c) to (h), Heat maps of the gene×time matrices for the 6 patients. (i) to (u), Heat maps of the patient×time matrices for the 13 genes.
Table 1.
Biological processes regulated by the 13 annotated genes in the significant co-cluster CL121 (identified in the gene×patient×time tensor).
Table 2.
Genetic pathways of the 13 annotated genes in the significant co-cluster CL121 (identified in the gene×patient×time tensor).
Fig 9.
Lineage tracing of C. elegans at the 200-cell stage.
Fig 10.
The linear patterns embedded in the 2D singular-vector space.
(a) to (c), The linear groups along the directions of first two singular vectors of U(n) (n = 1, 2, 3), respectively.
Fig 11.
Linear patterns embedded in the 3D singular-vector space for the lineage tracing data of C. elegans.
(a) to (c), The scatter plots along the directions of first three singular vectors of U(n) (n = 1, 2, 3), respectively. (d) to (f), Linear or planar patterns of the 3D-points in (a) to (c).
Table 3.
The linear groups in different modes of the gene×descendant×founder tensor (“*” denotes a founder cell).
Fig 12.
Heat maps of the descendant×founder matrices for the 42 genes involved in the significant co-cluster CL122 (identified in the gene×descendant×founder tensor).
Table 4.
Annotated functional categories revealed by the significant co-cluster CL122 (identified in the gene×descendant×founder tensor).
Fig 13.
Robustness test and performance comparison for HDSVS and a series of other biclustering methods (ISA, CC, FABIA, BSGP, SMR and BiMax).
(a) Signal-to-noise ratios (SNRs) vs. matching scores (MSs) for different biclustering methods, to search for constant biclusters. (b) SNR-MS curves for different biclustering methods, to search for constant-row/column biclusters. (c) SNR-MS curves for searching for additive biclusters. (d) SNR-MS curves for searching for multiplicative biclusters.
Fig 14.
GO enrichment significance tests.
Significance test results (under different thresholds) for HDSVS and existing biclustering methods (ISA, CC, FABIA, BSGP, SMR and BiMax), based on a 2D yeast gene expression tensor.