I am trying to understand how low-rank approximation techniques such as PCA, factor analysis, total least squares, orthogonal regression, etc could be used in portfolio optimisation. Say I have a portfolio of n assets, I could break these down into 2-3 principal components (using PCA) achieving say 95% of the variance. Why would this be useful though? (Other than reduced time complexity of subsequent calculations). It would tell me that asset A is important in my portfolio since I am able to see the correlation between assets via the covariance/correlation matrix. I could also use it in fundamental analysis to determine which factors drive the price of a stock, by looking at the company's revenue, EBITDA, p/e, rev growth, etc. Despite this, I don't see how PCA or other low-rank approximation fundamentally improve portfolio optimisation beyond what the covariance matrix does. Are there any uses for example in forecasting of the stock price? Also, how do different low-rank approximation techniques differ in their approximation accuracy?
Thanks a lot.