Why Classical PCA Can Distort Robust Optimization Inputs

Summary

The document discusses a reviewer’s concern that using principal component analysis to reduce the size of a robust optimization problem may introduce harmful distortions. The response suggests one possible interpretation: classical PCA’s asymptotic behavior depends on the data distribution, so substantial departures from normality can produce suboptimal or distorted estimates. Such distortions may matter when the reduced representation is used in an optimization problem that is meant to be robust.

As a possible remedy, the answer points to robust PCA and cites prior work, but it does not explain a particular robust PCA procedure or show how it changes an optimization result. The reviewer’s exact meaning is uncertain, and the response frames its explanation as a possibility rather than a demonstrated diagnosis. No dataset, experiment, or comparison is provided, so the claim should be treated as conceptual guidance rather than evidence that PCA will necessarily invalidate a given application.

Key ideas

• Classical PCA estimates can be sensitive to the distribution of the data.
• Large departures from normality may lead to suboptimal or distorted component estimates.
• Distortions in PCA inputs may affect a robust optimization model built from them.
• Robust PCA is suggested as an alternative, but no method or empirical comparison is detailed.

Tags

# The danger of using Principal Component Analysis (PCA) in Robust Optimization problems


# The danger of using Principal Component Analysis (PCA) in Robust Optimization problems












I have received a reviewer's comment on a paper which applies PCA to reduce the size of a problem and the application is in the robust optimization field. The reviewer implies that "In robust optimization, approximating the problem using PCA could be fatal because PCA creates uncontrollable distortions.". However no reference has been mentioned to address this point and my search for an answer didn't result in any reason for this comment. Would anyone please help me with a reason for this comment or introduce a book or paper that addresses this issue. Your help is greatly appreciated!

## Answer by Colin (score 2)

https://quant.stackexchange.com/a/23046

Possibly she is referring to the fact that classical PCA is not robust in the sense that its asymptotic properties depend on the distribution of the data. Large deviations from normality will result in sub-optimal estimates, or estimates that are distorted. If this is what she has in mind, then you can use robust PCA instead (cf. Candes et. al.)

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