  We consider the problem of embedding a finite set of points $\{x_1, \ldots,
x_n\} \in \mathbb{R}^d$ that satisfy $\ell_2^2$ triangle inequalities into
$\ell_1$, when the points are approximately low-dimensional. Goemans
(unpublished, appears in a work of [Magen and Moharammi, 2008]) showed that
such points residing in \emph{exactly} $d$ dimensions can be embedded into
$\ell_1$ with distortion at most $\sqrt{d}$. We prove the following robust
analogue of this statement: if there exists a $r$-dimensional subspace $\Pi$
such that the projections onto this subspace satisfy $\sum_{i,j \in [n]}\Vert
\Pi x_i - \Pi x_j \Vert _2^2 \geq \Omega(1) \sum_{i,j \in [n]}\Vert x_i - x_j
\Vert _2^2$, then there is an embedding of the points into $\ell_1$ with
$O(\sqrt{r})$ average distortion. A consequence of this result is that the
integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform
Sparsest Cut problem is $O(\sqrt{r})$ on graphs $G$ whose $r$-th smallest
normalized eigenvalue of the Laplacian satisfies $\lambda_r(G)/n \geq
\Omega(1)\Phi_{SDP} (G)$. Our result improves upon the previously known bound
of $O(r)$ on the average distortion, and the integrality gap of the
Goemans-Linial SDP under the same preconditions, proven in the previous works
of [Deshpande and Venkat, 2014] and [Deshpande, Harsha and Venkat, 2016].
