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Tensor GSVD of Patient- and Platform-Matched Tumor and Normal DNA Copy-Number Profiles Uncovers Chromosome Arm-Wide Patterns of Tumor-Exclusive Platform-Consistent Alterations Encoding for Cell Transformation and Predicting Ovarian Cancer Survival

Fig 1

Tensor generalized singular value decomposition (GSVD) of the patient- and platform-matched DNA copy-number profiles of the 6p+12p chromosome arms.

For each chromosome arm or combination of two chromosome arms, the structure of the tumor and normal discovery datasets (1 and 2) is that of two third-order tensors with one-to-one mappings between the column dimensions but different row dimensions. The patients, platforms, probes, and tissue types, each represent a degree of freedom. Unfolded into a single matrix, some of the degrees of freedom are lost and much of the information in the datasets might also be lost. We define a tensor GSVD that simultaneously separates the paired datasets into weighted sums of paired subtensors, i.e., combinations or outer products of three patterns each: Either one tumor-specific pattern of copy-number variation across the tumor probes, i.e., a tumor arraylet (a column basis vector of U1), or the corresponding normal-specific arraylet (a column basis vector of U2), combined with one pattern of variation across the patients, i.e., an x-probelet (a row basis vector of VxT), and one pattern across the platforms, i.e., a y-probelet (a row basis vector of VyT), which are identical for both the tumor and normal datasets (Equation 1). The tensor GSVD is depicted in a raster display, with relative copy-number gain (red), no change (black), and loss (green), explicitly showing the first through the 5th, and the 245th through the 249th 6p+12p x-probelets, both 6p+12p y-probelets, and the first through the 10th, and the 489th through the 498th 6p+12p tumor and normal arraylets. We prove that the significance of a subtensor in the tumor dataset relative to that of the corresponding subtensor in the normal dataset, i.e., the tensor GSVD angular distance, equals the row mode GSVD angular distance, i.e., the significance of the corresponding tumor arraylet in the tumor dataset relative to that of the normal arraylet in the normal dataset. The tensor GSVD angular distances for the 498 pairs of 6p+12p arraylets are depicted in a bar chart display, where the angular distance corresponding to the first pair of arraylets is ∼ π/4. For the 6p+12p combination of two chromosome arms, we find that the most significant subtensor in the tumor dataset (which corresponds to the coefficient of largest magnitude in ℛ1) is a combination of (i) the first y-probelet, which is approximately invariant across the platforms, (ii) the first x-probelet, which classifies the discovery set of patients into two groups of high and low coefficients, of significantly and robustly different prognoses, and (iii) the first, most tumor-exclusive tumor arraylet, which classifies the validation set of patients into two groups of high and low correlations of significantly different prognoses consistent with the x-probelet’s classification of the discovery set.

Fig 1