We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSPs). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most eps worse than the relaxations value (normalized to lie in [0,1]), as long as r > kcdotrank_{geq theta}(Ins)/poly(e) ;, where k is the alphabet size of I, $theta=poly(e/k)$, and $rank_{geq theta}(Ins)$ denotes the number of eigenvalues larger than $theta$ in the normalized adjacency matrix of the constraint graph of $Ins$. In the case that $Ins$ is a uniquegames instance, the threshold $theta$ is only a polynomial in $e$, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for emph{every} instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khots Unique Games Conjecture. Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in some cases $r$ rounds of our program can be evaluated in time $2^{O(r)}poly(n)$.
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