In **Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors** [1], which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on pages 9-10, Section 6, Case study: security-based investment.

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> [...] We estimate every month the covariance matrix of the of the [sic] S&P stock returns $\Sigma\_{F}$ using a one-year rolling window of daily observations, and filtering the smallest eigenvalues to ensure positive definiteness.
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What is meant by "filtering the smallest eigenvalues"? What methods are there to transform the covariance matrix in such a way that the smallest eigenvalues (and corresponding eigenvectors) get removed, but the rest stay intact, and which one is used here in particular?

**Edit**: This SE question [2] is highly relevant to this one.

[1] <https://papers.ssrn.com/sol3/Papers.cfm?abstract_id=2276632>

[2] [What is the best way to "fix" a covariance matrix that is not positive semi-definite?](https://quant.stackexchange.com/questions/2074/what-is-the-best-way-to-fix-a-covariance-matrix-that-is-not-positive-semi-defi?rq=1)