Timeline for 1-wasserstein distance v.s. total variation distance
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 19, 2020 at 21:57 | comment | added | dohmatob | It seems one can get another bound: $\|\mu_1 * \gamma - \mu_2 * \gamma\|_W \le (\underset{x,y,\;x \ne y}{\sup}\;\frac{\|\gamma_x - \gamma_y\|_W}{c(x,y)})\|\mu_1-\mu_2\|_W$, for any Polish space $(X,c)$ and transition kernel $\gamma$ thereupon. | |
| Jan 19, 2020 at 21:51 | comment | added | dohmatob | Lots of steps missing here. First note that $\delta_x * \gamma = \gamma_x$ for any transition kernel $\gamma$. Now, $$ \begin{split} \|\mu_1 * \gamma - \mu_2 * \gamma\|_{TV} &= \|(\mu_1-\mu_2)*\gamma\|_{TV} = \|\int ((\delta_x-\delta_y)dM(x,y)) * \gamma\|_{TV}\\ & = \|\int((\delta_x-\delta_y)* \gamma) dM(x,y) \|_{TV} = \|\int(\delta_x * \gamma -\delta_y * \gamma)dM(x,y) \|_{TV}\\ & = \|\int(\gamma_x-\gamma_y)dM(x,y)\|_{TV}\\ &\le \int \|\gamma_x-\gamma_y\|_{TV}dM(x,y) \le K_\gamma \int c(x,y) dM(x,y), \end{split} $$ where the last bust one inequality is the triangle inequality. | |
| Jan 25, 2015 at 20:05 | vote | accept | user58955 | ||
| Jan 25, 2015 at 15:49 | history | answered | R W | CC BY-SA 3.0 |