Timeline for Principal component analysis with boundedness constraints
Current License: CC BY-SA 4.0
6 events
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| Mar 4, 2022 at 15:48 | history | edited | Onur Oktay | CC BY-SA 4.0 |
added 395 characters in body
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| Feb 20, 2022 at 16:22 | comment | added | Onur Oktay | I'm also interested in decompositions where $\ell_\infty$ norm is replaced with $\ell_p$ norms for $p\notin\{1,2\}$. | |
| Feb 20, 2022 at 16:20 | comment | added | Onur Oktay | @NathanielJohnston thank you for your reply. I'm glad you brought the tensor norm on the table. If $X$, $Y$ are finite dimensional normed spaces, and $B_X$ & $B_Y$ & $B(X\hat{\otimes}_{\pi} Y)$ are the unit balls of the respective norms, then $B(X\hat{\otimes}_{\pi} Y)$ is the convex hull of $B_X\otimes B_Y$. Can we infer $r=mn$ or a multiple of $mn$? | |
| Feb 20, 2022 at 13:36 | comment | added | Nathaniel Johnston | A couple of remarks that fall short of an answer: (1) The value of the minimization problem that you've described is called the "projective tensor norm" of the $\ell^\infty$-norm (i.e., the norm given by $\|\mathbf{x}\|_{\infty} = \max_k\{|x_k|\}$). (2) The value of the minimum can be computed via semidefinite programming (in something like $O(m^3n^3)$ time). Decompositions themselves could be computed via the same techniques if we had a useful upper bound on $r$. | |
| Feb 20, 2022 at 12:46 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals, added tag
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| Feb 20, 2022 at 12:34 | history | asked | Onur Oktay | CC BY-SA 4.0 |