Timeline for Relationship between $L_1$ and $L_2$ distances of two Gaussian Mixture models
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 27, 2021 at 14:31 | comment | added | Ze-Nan Li | @leomonsaingeon Thanks for the hint. | |
| May 27, 2021 at 11:02 | comment | added | leo monsaingeon | Inside a big ball you can use Young/Cauchy-Schwarz to bound $L^2\lesssim L^1$. And outside of a big ball your mixtures behave essentially as the single Gaussian with the smallest $\sigma$, so you can compare $L^2$ and $L^1$ explicitly. Then you should take care of the details, optimizing the radius of the ball, taking into account the shifts $\mu_i=\mu_j$ etc, but I guess this should work | |
| May 27, 2021 at 10:03 | comment | added | Ze-Nan Li | @leomonsaingeon, thanks for the comment. Actually, I want to find an inequality relation between $L_1$ and $L_2$ distance, and how could I attain these upper and lower bounds? | |
| May 27, 2021 at 8:52 | comment | added | leo monsaingeon | What kind of relationship are you hoping for? an upper and lower bound should be attainable, but the constants would depend (plausibly, I guess) on the "diameter" $\max_{i,k} |\mu_k-\mu_i|$ as well as on the ratio $\frac{\min_j \sigma_j}{\max_j\sigma_j}$ (hopefully not on $K,N$). | |
| May 27, 2021 at 2:23 | history | asked | Ze-Nan Li | CC BY-SA 4.0 |