I am reading a paper for robust, sparse PCA in which they propose a BIC-like criterion for selecting the appropriate value of the sparsity parameter $\lambda$. They define this as:


where $(\tilde{RV}, RV)$ refer to the total robust variance of the residuals matrix obtained from a sparse PCA and an unconstrained PCA.

An assumption of of PCA is that the data are jointly normally distributed. I often read that this assumption is largely ignorable when performing data exploration, and the comments usually make the point that PCA is not typically used for inference. But here it is. Focusing on the RV term in the equation, this seems to be the proxy for the likelihood function in a normal BIC. I understand the second term, which penalizes for model complexity. My question is that if my data is not jointly normally distributed, will this potentially invalidate inferences made from this information criterion (which is in model selection serves the role of a p-value: (Equivalence of AIC and p-values in model selection), and will these problems arise because the RV term is inappropriate? I notice that I get selected values of $\lambda$ which often seem too high to be intuitively reasonable, indicating that perhaps the RV term is smaller than it should be.


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