Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$

Question

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.

Question. What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ ?

Answer 1

Let $g \sim N(0, (1/d)I_d)$ be independent of $x$. Then $g_1 \overset{\mathcal L}{=} \|g\| x_1$, so $(x_1, \|g\|x_1)$ is a coupling between the marginal distribution of $x_1$ and $N(0, 1/d)$.

The norm $\|g\|$ is sharply concentrated around $1$, with fluctuations of order $1/\sqrt{d}$, so the Wasserstein distance is bounded above by $$\mathbb E|x_1 - \|g\|x_1| = (\mathbb E|1 - \|g\||) \mathbb E|x_1| = O(d^{-1}), $$ where we have used the fact that $\mathbb E|x_1|^2 = (1/d)\mathbb E\|x\|^2 = 1/d$ by isotropy of the distribution of $x$, so that $\mathbb E|x_1| \le \sqrt{\mathbb E|x_1|^2} = 1/\sqrt{d}$ thanks to Jensen's inequality.

Answer 2

Slightly weaker bound via a more robust method

Below I present an alternative method with slightly worse upper-bound than in the accepted answer, namely $d^{-1+o(1)}$ instead of $d^{-1}$.

The advantage of the new method is that it can be applied to (almost all) one-dimensional projections of arbitrary isotropic log-concave distributions (thanks to Klartag's CLT for convex bodies, namely Theorem 1.3 of this paper).

Disclaimer. It's possible I've made a mistake some. Thanks in advance for catching.

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Let $F$ be the marginal cdf of $\sqrt{d}x_1$ and let $F^c := 1 - F$ be its survival function. Let $\Phi$ be the standard gaudssian cdf.

We will make use of the following fact (see ref https://mathoverflow.net/a/315232/78539)

Fact 1 (Approximation). $Ko(x_1,N(0,1)) := \sup_{t \in \mathbb R}|F(t) - \Phi(t)| \le C/d$, for an absolute constant $C>0$ which doesn't depend on $d$.

We shall also need the following fact (which follows concentration of $\mathcal O(1)$-Lipschitz transformations of log-concave random vectors)

Fact 2 (Concentration). There exists an absolute constant $b>0$ such that $\max(F(-t),\Phi(-t),F^c(t),\Phi^c(t)) \le e^{-bt^2}$ for sufficiently large $t>0$.

Now, for any $T > 0$, one computes $$ W_1(\sqrt{d}x_1,N(0,1)) = \int_{-\infty}^\infty |F(t)-\Phi(t)|dt = A_1(T) + A_2(T) + A_3(T), $$ where the first is a a classical result (e.g see Proposition 2.17 of Santambrogio's OTAM), and the $A_k(T)$'s are defined by $$ \begin{split} A_1(T) &:= \int_{-\infty}^{-T}|F(t)-\Phi(t)|dt,\\ A_2(T) &:= \int_{-T}^T|F(t)-\Phi(t)|dt,\\ A_3(T) &:= \int_{T}^\infty|F(t)-\Phi(t)|dt=\int_{T}^\infty|F^c(t)-\Phi^c(t)|dt. \end{split} $$

Thanks to Fact 1, we know that $A_2(T) = \mathcal O(T/d)$. On the other hand, thanks to Fact 2, we have for sufficiently large $T>0$, $$ A_1(T) \le \int_{-\infty}^{-T} \max(F(t),\Phi(t))dt \le \int_{-\infty}^{-T} e^{bt^2}dt = \mathcal O(e^{-bT^2}). $$

By a symmetric argument, we also have $A_3(T) = \mathcal O(e^{-bT^2})$. Taking $T=\sqrt{(\log d)/b}$ then gives $A_2(T) = \mathcal O(\sqrt{\log d}/d)$ and $A_1(T),A_3(T) = \mathcal O(1/d)$. Thus $$ W_1(x_1,N(0,1/d)) = W_1(\sqrt{d} x_1,N(0,1)) = \mathcal O(\frac{\sqrt{\log d}}{d})=\mathcal O(d^{-1+o(1)}). $$

Extension to one-dimensional projections of isotropic log-concave distributions

Let $x$ be a random vector in $\mathbb R^d$ with an arbitrary isotropic log-concave density. Recall that by isotropy, we mean

• $\mathbb E [x] = 0$, and
• $\mathbb E |x^\top u|^2=1$ for any unit-vector $u \in \mathbb R^d$.

Now, for any unit-vector $u \in \mathbb R^d$, let $x_u := x^\top u$ be the projection of $x$ in the direction of $u$, and let $F_u$ be its cdf. Finally, let $\sigma_d$ be the uniform distribution on the unit-sphere in $\mathbb R^d$. We will prove the following

Theorem. For a proportion $1-e^{-\mathcal O(d^{0.99})}$ of unit-vectors $u$ in $\mathbb R^d$ (w.r.t the uniform distribution on the unit-sphere in $\mathbb R^d$), it holds that $$ W_1(x_u,N(0,1)) = \mathcal O((\log d)^{-1/2+o(1)})=o(1). $$

Thanks to Theorem 1.3 of Klartag's paper (referenced in the preamble), we know that for $\varepsilon_d = \left(\dfrac{\log \log d}{\log d}\right)^{1/2}=o(1)$,

Fact 1' (Approximation). There exists a measurable subset $\mathcal U$ of unit-vectors with $\sigma_d(\mathcal U) \ge 1 - e^{-\mathcal O(d^{0.99})}$ such that $$ Ko(x_u,N(0,1)) \le \mbox{TV}(x_u,N(0,1)) = \mathcal O(\varepsilon_d) $$ for all $u \in \mathcal U$.

Just as in Fact 2, we have the following property which is a direct consequence of the concentration of $\mathcal O(1)$-Lipschitz transformations of log-concave random vectors,

Fact 2' (Concentration). There exists an absolute constant $b>0$ such that $$ \max(F_u(-t),\Phi(-t),F_u^c(t),\Phi^c(t)) \le e^{-bt^2} $$ for all $u \in \mathcal U$ and for sufficiently large $t>0$.

Emulating the proof of the spherical case, but using the cutoff $T=\sqrt{\log (1/\varepsilon_d)/b}$ instead, we obtain $$ \sup_{u \in \mathcal U}W_1(x_u,N(0,1))) = \mathcal O(\varepsilon_d\sqrt{\log(1/\varepsilon_d)}) = \mathcal O((\log d)^{-1/2+o(1)}) = o(1). $$ Noting that $\sigma_d(\mathcal U)\ge 1 - e^{-\mathcal O(d^{0.99})}$, the proof is complete. $\quad\quad\quad\quad\quad\quad\Box$