## 1 Introduction  
The singular value decomposition (SVD) is a mathematical result that allows the calculation of a low-rank approximation of any matrix and serves as the basis of many statistical methods used in data analysis [1]. The application areas are diverse, for example, SVD can be used in principal component analysis [2], the imputation of missing data [3], the graphical representation of multivariate data [4], the formulation of models to explain the interaction in two-way tables [5, 6] and non-parametric analysis of time series [7], to mention just a few.  
A problem inherent in the use of SVD is the determination of the appropriate rank of the approximation, i.e., the appropriate number of components to be retained, and a very convenient way to solve this problem uses the resampling technique known as cross-validation. Standard cross-validation consists of subdividing the study matrix into a certain number of groups, deleting each group in turn, evaluating the parameters of a chosen predictor from the remaining data and using the result to predict the deleted values [8].  
A numerical measure of agreement between predicted and actual values can then be computed for each possible rank, and the best rank to choose is the one providing the best calculated measure. Readers interested in classic references, alternative procedures, method comparisons and new developments of the subject can refer to the works of [PERSON] et al. [9], [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [12] and [PERSON] and [PERSON] [13].  
Among the wide variety of methods for conducting the cross-validation, one considered as classic and influential is the Eastment and Krzanowski [14] (EK) scheme proposed in 1982, and which currently (April 2022) has 452 citations in Google Scholar. The method depends on the SVD, so does not rest on any distributional or structural assumptions. Moreover, it provides exact calculations, so does not have the convergence problems that can occur with expectation maximization (EM) approximations [15].  
A frequent problem in data analysis is the presence of outliers [16] and the performance of the EK method can be affected by them because of its use of SVD, which is a least squares technique. To circumvent the problem, an option used by [PERSON] [17] was to compare the results obtained for the complete data with those for the reduced data, after eliminating the outliers. To the best of our knowledge, no specific study has been conducted to evaluate the effect of outliers on the EK method when applied to lower rank matrices; therefore, the aim of the current paper is to propose a methodology that combines robust singular value decomposition (rSVD) with the EK method and that does not require the elimination of outliers. This proposed methodology would therefore be applicable for any dataset that has some suspicion of contamination and the outliers do not correspond to errors in the collection or typing, as in that case the elimination of information is perfectly sensible.  
This article is organized as follows: the EK method is presented first, followed by the various proposed methods along with associated statistics that robustify the criterion for choosing the rank of a matrix. Subsequently, three rSVD's are presented that can be used as a way to reduce the effect of outliers, but without eliminating them. Then, a simulation study is described for evaluating possible cross-validation strategies for different levels of contamination using different resistant statistics. A study of a real dataset is also described to compare the proposed alternatives, and finally the results are presented together with a relevant discussion.