  Guruswami and Sinop give a $O(1/\delta)$ approximation guarantee for the
non-uniform Sparsest Cut problem by solving $O(r)$-level Lasserre semidefinite
constraints, provided that the generalized eigenvalues of the Laplacians of the
cost and demand graphs satisfy a certain spectral condition, namely,
$\lambda_{r+1} \geq \Phi^{*}/(1-\delta)$. Their key idea is a rounding
technique that first maps a vector-valued solution to $[0, 1]$ using
appropriately scaled projections onto Lasserre vectors. In this paper, we show
that similar projections and analysis can be obtained using only $\ell_{2}^{2}$
triangle inequality constraints. This results in a $O(r/\delta^{2})$
approximation guarantee for the non-uniform Sparsest Cut problem by adding only
$\ell_{2}^{2}$ triangle inequality constraints to the usual semidefinite
program, provided that the same spectral condition, $\lambda_{r+1} \geq
\Phi^{*}/(1-\delta)$, holds.
