# Resolving the Sign Ambiguity in the Singular Value Decomposition and multi-way tensor models

Although the Singular Value Decomposition (SVD) and eigenvalue decomposition (EVD) are well-established and can be computed via state-of-the-art algorithms, it is not commonly mentioned that there is an intrinsic sign indeterminacy that can significantly impact the conclusions and interpretations drawn from their results. We provide a solution to the sign ambiguity problem by determining the sign of the singular vector from the sign of the inner product of the singular vector and the individual data vectors. The data vectors may have different orientation but it makes intuitive as well as practical sense to choose the direction in which the majority of the vectors point. This can be found by assessing the sign of the sum of the signed inner products.

Figure 1 Sixty-one 201-dimensional fluorescence emission spectra

For example, a set of fluorescence emission spectra each of dimension 201 is given for 61 different excitation wavelengths (Figure 1). These spectra represent three underlying spectral components and hence the three largest singular components should represent the systematic variation in the data.

Figure 2 Bootstrapped three first right singular vectors from Figure 1 before (top) and after (bottom) sign convention.

In an experiment, these three components are bootstrapped 100 times in order to be able to evaluate the uncertainty of the estimated components. The bootstrapping is done by sampling 61 rows with replacement 100 times, and the results are shown in the upper half of Figure 2. While the sign-flipping may be due to the bootstrapping, it is also likely to be due to the semi-random nature of the sign of the singular vectors. In the lower half of Figure 2, the result of applying the proposed sign convention is shown and as can been seen, all singular vectors can now be immediately compared because their signs do not change as long as they represent similar aspects of the data.

References:

R. Bro, E. Acar, and T. G. Kolda. Resolving the sign ambiguity in the singular value decomposition. J.Chemom. 22:135-140, 2008.

Resolving the Sign Ambiguity in the Singular Value Decomposition, Rasmus Bro, Evrim Acar and Tamara G. Kolda.

R. Bro, R. Leardi, and L. G. Johnsen. Solving the sign indeterminacy for multiway models. J.Chemom. 27:70-75, 2013