Semi-discrete optimal transport between mixture of Gaussians and their centers
$begingroup$
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=frac 1 k sum_{i=1}^k mathcal{N}(theta_i,sigma^2 I_d) $ be a $d$-dimensional Gaussian mixture and let $Q=frac 1 k sum_i delta_{theta_i}$ be a uniform discrete distribution on the set of centers ${theta_1,ldots,theta_k}$. For the case of $p=2$, my goal is to obtain the tightest bound as follows:
$$
W_2(P,Q) leq text{some function of } (sigma,d)
$$
My attempt: Let $X=Z+theta_D$ be a random variable where $Z sim mathcal{N}(0,sigma^2 I_d)$ and $Dsim mathsf{Unif}({1,ldots,k})$. Now define the random variable $Y=theta_D$. Clearly $X sim P$ and $Ysim Q$. Then using the definition of $W_2$, we obtain that
$$
W_2(P,Q) leq sqrt{mathbb{E}|X-Y|^2} = sqrt{mathbb{E}|Z|^2}=sigma sqrt{d}.
$$
So I am wondering if there are better bounds known for this specific case in the literature of optimal transport. Please point me to these references if it's already a well studied problem.
normal-distribution optimal-transport
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
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add a comment |
$begingroup$
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=frac 1 k sum_{i=1}^k mathcal{N}(theta_i,sigma^2 I_d) $ be a $d$-dimensional Gaussian mixture and let $Q=frac 1 k sum_i delta_{theta_i}$ be a uniform discrete distribution on the set of centers ${theta_1,ldots,theta_k}$. For the case of $p=2$, my goal is to obtain the tightest bound as follows:
$$
W_2(P,Q) leq text{some function of } (sigma,d)
$$
My attempt: Let $X=Z+theta_D$ be a random variable where $Z sim mathcal{N}(0,sigma^2 I_d)$ and $Dsim mathsf{Unif}({1,ldots,k})$. Now define the random variable $Y=theta_D$. Clearly $X sim P$ and $Ysim Q$. Then using the definition of $W_2$, we obtain that
$$
W_2(P,Q) leq sqrt{mathbb{E}|X-Y|^2} = sqrt{mathbb{E}|Z|^2}=sigma sqrt{d}.
$$
So I am wondering if there are better bounds known for this specific case in the literature of optimal transport. Please point me to these references if it's already a well studied problem.
normal-distribution optimal-transport
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
$endgroup$
add a comment |
$begingroup$
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=frac 1 k sum_{i=1}^k mathcal{N}(theta_i,sigma^2 I_d) $ be a $d$-dimensional Gaussian mixture and let $Q=frac 1 k sum_i delta_{theta_i}$ be a uniform discrete distribution on the set of centers ${theta_1,ldots,theta_k}$. For the case of $p=2$, my goal is to obtain the tightest bound as follows:
$$
W_2(P,Q) leq text{some function of } (sigma,d)
$$
My attempt: Let $X=Z+theta_D$ be a random variable where $Z sim mathcal{N}(0,sigma^2 I_d)$ and $Dsim mathsf{Unif}({1,ldots,k})$. Now define the random variable $Y=theta_D$. Clearly $X sim P$ and $Ysim Q$. Then using the definition of $W_2$, we obtain that
$$
W_2(P,Q) leq sqrt{mathbb{E}|X-Y|^2} = sqrt{mathbb{E}|Z|^2}=sigma sqrt{d}.
$$
So I am wondering if there are better bounds known for this specific case in the literature of optimal transport. Please point me to these references if it's already a well studied problem.
normal-distribution optimal-transport
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
$endgroup$
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=frac 1 k sum_{i=1}^k mathcal{N}(theta_i,sigma^2 I_d) $ be a $d$-dimensional Gaussian mixture and let $Q=frac 1 k sum_i delta_{theta_i}$ be a uniform discrete distribution on the set of centers ${theta_1,ldots,theta_k}$. For the case of $p=2$, my goal is to obtain the tightest bound as follows:
$$
W_2(P,Q) leq text{some function of } (sigma,d)
$$
My attempt: Let $X=Z+theta_D$ be a random variable where $Z sim mathcal{N}(0,sigma^2 I_d)$ and $Dsim mathsf{Unif}({1,ldots,k})$. Now define the random variable $Y=theta_D$. Clearly $X sim P$ and $Ysim Q$. Then using the definition of $W_2$, we obtain that
$$
W_2(P,Q) leq sqrt{mathbb{E}|X-Y|^2} = sqrt{mathbb{E}|Z|^2}=sigma sqrt{d}.
$$
So I am wondering if there are better bounds known for this specific case in the literature of optimal transport. Please point me to these references if it's already a well studied problem.
normal-distribution optimal-transport
normal-distribution optimal-transport
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
asked Dec 12 '18 at 23:07
pikachuchameleonpikachuchameleon
1,149623
1,149623
add a comment |
add a comment |
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