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Although it is often calculated differently, my intuitive understanding of PCA arises from its definition as the eigendecomposition of the sample covariance matrix. I have recently become aware of various popular methods for improving estimation of the covariance matrix (e.g. Donoho et al. 2013 Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model). Indeed, the Wikipedia page for Estimation of Covariance Matrix says:

Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample.

I frequently find myself performing PCA of data matrices where the number of samples (n) is comparable or less than the number of variables (p), which is precisely the case where the estimation of sample covariance matrix can be improved using shrinkage or other techniques. My goal in these situations is dimensionality reduction so as to find patterns in the data (e.g. in the famous Iris dataset, PCA reveals three differnt kinds of flowers). If I were to improve the estimation of the covariance matrix, would PCA then give a "better" understanding of the structure in the data (e.g. the groups would separate nicer)? Are there any examples where shrinkage is very useful in dimensionality reduction using PCA?

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The paper you cited (Donoho et al. 2013 Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model) is an impressive piece of work which I confess I did not really study. Nevertheless, I believe that it is easy to see that an answer to your question is negative: using any kind of shrinkage estimator of the covariance matrix will not improve your PCA results and, specifically, will not lead to "better understanding of the structure in the data".

In a nutshell, this is because shrinkage estimators only affect the eigenvalues of the sample covariance matrix and not the eigenvectors.

Let me quote the beginning of the abstract of Donoho et al.:

Since the seminal work of Stein (1956) it has been understood that the empirical covariance matrix can be improved by shrinkage of the empirical eigenvalues. In this paper, we consider a proportional-growth asymptotic framework with $n$ observations and $p_n$ variables having limit $p_n/n \to \gamma \in (0,1]$. We assume the population covariance matrix $\Sigma$ follows the popular spiked covariance model, in which several eigenvalues are significantly larger than all the others, which all equal $1$. Factoring the empirical covariance matrix $S$ as $S = V \Lambda V'$ with $V$ orthogonal and $\Lambda$ diagonal, we consider shrinkers of the form $\hat{\Sigma} = \eta(S) = V \eta(\Lambda) V'$ where $\eta(\Lambda)_{ii} = \eta(\Lambda_{ii})$ is a scalar nonlinearity that operates individually on the diagonal entries of $\Lambda$.

The abstract goes on to describe paper's contributions, but what is important for us here is that the sample covariance matrix $S$ and its shrinked version $\hat\Sigma$ have the same eigenvectors. Principal components are given by projections of the data onto these eigenvectors; so they will not be affected by the shrinkage.

The only thing that can get affected are the estimates of how much variance is explained by each PC because these are given by the eigenvalues. (And as @Aksakal wrote in the comments, this can affect the number of retained PCs.) But the PCs themselves will not change.

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    $\begingroup$ +1 Good point on rotation invariance of shrinkage estimators $\endgroup$ – Aksakal Dec 1 '16 at 20:45
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I think that shrinkage would not help in interpreting the data with PCA or reducing dimensionality of a given data set. The shrinkage will help to make your analysis robust, i.e. if you have to use the outcome of PCA on other data sets. When you estimate the covariance matrix of a small but high dimensional data set, the estimate becomes unstable, the estimation error is very high. So, if you apply what you have learned from this data set on other data sets, you may be up to an unpleasant surprise. Due to a sampling error you may see that your estimated covariance matrix doesn't match the new observations at all. So, shrinkage may help when there's some kind of a default or prior knowledge of the covariances, maybe theoretical asymptotthical limit etc.

On the other hand if this sample is all that you have to use the PCA analysis for then you're dealing essentially with the population. Hence, the sample estimate becomes the population parameter, and you're fine.

The example when a shrinkage works is in portfolio theory in finance. There are many strains of this beast, and some of them posit that the variables are highly correlated with common factors, while the residual correlatiuns between variables is small. This leads to a nice shrinkage target: a diagonal residual covariance matrix. It's not always that you have this kind of a case when you know to what to shrink your estimate though

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    $\begingroup$ So, it is kind of like preventing "overfitting," in a similar way as shrinkage of least squares when doing linear regression (e.g. LASSO) can make the fit more "robust" and more applicable to other datasets. Is this the case? @amoeba pointed out that the eigenvectors won't change for shrinkage--since that is the case, then what exactly would be "overfitting" in this context? Or am I missing the point? $\endgroup$ – user310374 Dec 1 '16 at 20:26
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    $\begingroup$ @amoeba is right that popular shrinkage estimators, such as Ledoit Wolf that is used a lot in finance, preserve the eigenvectors. This essentially means that your PCA coefficients will not change with shrinkage. What may change is which PCs you'll pick as your reduced dimensions. You usually pick the PCs based on the magnitude of the eigen values, which are going to change after shrinkage. So, you may end up with a different set of PCs as a result. $\endgroup$ – Aksakal Dec 1 '16 at 20:43
  • $\begingroup$ @Aksakal That's a good point (in the last comment above) about selecting PCs based on eigenvalues. I guess "different set" in this case means, more specifically, "different number [of leading PCs]" - I assume that shrinkage will lead to eigenvalues shrinking monotonically, i.e. will preserve their order (right?). $\endgroup$ – amoeba says Reinstate Monica Dec 1 '16 at 20:56
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    $\begingroup$ @amoeba, I'm not sure about preserving the order, never thought about this. That' why I used the word set. Ledoit Wolf's scheme will probably preserve the order $\endgroup$ – Aksakal Dec 1 '16 at 21:10

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