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In Quantum-inspired classical algorithms for principal component analysis and supervised clustering, the PCA algorithm requires that the variances of the principal vectors differ by at least a constant fraction of the Frobenius norm squared ($\sigma_{i+1}^2 - \sigma_i^2 \ge \eta ||A||_{F}^2$) and that the variance is above a certain constant $\sigma^2$. Can the assumption that the principal vectors have different variances be dropped and the output changed to just the hypervolume created by all the (unordered) principal vectors with at least a certain variance, $\sigma^2$? And, if so, what is the modified algorithm's computational complexity? From the remark,

As we assume our eigenvalues have an $\eta ||A||_F^2$ gap, the precise eigenvector $|v_j\rangle$ sampled can be identified by the eigenvalue estimate. Then, by computing enough samples, we can learn all of the eigenvalues of at least $\sigma^2$ and get the corresponding states

It seems like this should be fine, but later Tang adds,

Note that we crucially use the assumptions in Problem 7 for our QML algorithm: without guarantee on the gap or that $\sigma_i \ge \sigma$, finding the top k singular vectors would be intractable, even with samples of $|v_i\rangle$’s.

leaves open the possibility it wouldn't work.

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