Determining the number of latent dimensions is a ubiquitous problem in machine learning. In this study, we introduce a novel method that relies on SVD to discover the number of latent dimensions. The general principle behind the method is to compare the curve of singular values of the SVD decomposition of a data set with the randomized data set curve. The inferred number of latent dimensions corresponds to the crossing point of the two curves. To evaluate our methodology, we compare it with competing methods such as Kaisers eigenvalue-greater-than-one rule (K1), Parallel Analysis (PA), Velicers MAP test (Minimum Average Partial). We also compare our method with the Silhouette Width (SW) technique which is used in different clustering methods to determine the optimal number of clusters.
The result on synthetic data shows that the Parallel Analysis and our method have similar results and more accurate than the other methods, and that our methods is slightly better result than the Parallel Analysis method for the sparse data sets.
The authors propose the use of bootstrapping the data (random sampling entries with replacement) to form surrogate data for which they can evaluate the singular value spectrum of the SVD of the matrix to the singular values of the bootstrapped data, thereby determining the number of latent dimensions in PCA by the point in which the singular values are no greater than the bootstrapped sampled values.  The procedure is contrasted to some existing methods for determining the number of latent components and found to perform similarly to another procedure based on bootstrapping correlation matrices, the PA procedure.
Pros:
Determining the number of components is an important problem that the authors here address.
Cons:
I find the paper poorly written and the methodology not sufficiently rooted in the existing literature. There are many approaches to determining the number of latent components in PCA that needs to be discussed and constrasted including:
Cross-validation:
http://scholar.google.dk/scholar_url?url=http%3A%2F%2Fwww.academia.edu%2Fdownload%2F43416804%2FGeneralizable_Patterns_in_Neuroimaging_H20160306-9605-1xf9c9h.pdf&hl=da&sa=T&oi=gga&ct=gga&cd=0&ei=rjkXWrzKKImMmAH-xo7gBw&scisig=AAGBfm2iRQhmI2EHEO7Cl6UZoRbfAxDRng&nossl=1&ws=1728x1023
Variational Bayesian PCA:
https://www.microsoft.com/en-us/research/publication/variational-principal-components/
Furthermore, the idea of bootstrapping for the SVD has been discussed in prior publications and the present work need to be related to these prior works. This includes:
Milan, Luis, and Joe Whittaker. “Application of the Parametric Bootstrap to Models That Incorporate a Singular Value Decomposition.” Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 44, no. 1, 1995, pp. 31–49. JSTOR, JSTOR, www.jstor.org/stable/2986193.
Fisher A, Caffo B, Schwartz B, Zipunnikov V. Fast, Exact Bootstrap Principal Component Analysis for p > 1 million. Journal of the American Statistical Association. 2016;111(514):846-860. doi:10.1080/01621459.2015.1062383.
Including the following package in R for performing bootstrapped SVD: https://cran.r-project.org/web/packages/bootSVD/bootSVD.pdf
The novelty of the present approach is therefore unclear given prior works on bootstrapping SVD/PCA.
Furthermore, for sparse data with missing entries there are specialized algorithms handling sparsity either using imputation or marginalization, which would be more principled to estimate the PCA parameters. 
Finaly, the performance appears almost identical with the PA procedure. In fact, it seems bootstrapping the correlation matrix has a very similar effect as the proposed bootstrapping procedure. Thus, it seems the proposed procedure which is very similar in spirit to PA does not have much benefit over this procedure.
Minor comments:
Explain what SW abbreviates when introduced first.
We will see that it PA a close relationship with BSVD-> We will see that PA is closely related to BSVD
more effective than SVD under certain conditions (?). – please provide reference instead of ?
But table 4 that shows -> But table 4 shows that
We can sum up with that the result seems ->To summarize, the result seems