By now, it is known that integrality gaps for the standard Unique Games SDP (see the paper of Khot and Vishnoi or Section 5.2 of this post) can be used to obtain integrality gaps for many other optimization problems, and often for very strong SDPs coming from various methods of SDP tightening; see, for instance, the paper of Raghavendra and Steurer.
Problematically, the Khot-Vishnoi gap is rather inefficient: To achieve the optimal gap for Unique Games with alphabet size 
The Walsh-Hadamard code
The Khot-Vishnoi construction is based on the Hadamard code.
(See Section 5.2 here for a complete description.) If we use 
where 
Of course, for two such sets 
In their construction, the variables are essentially all functions of the form 
![{\{u_S\}_{S \subseteq [k]}}](http://s0.wp.com/latex.php?latex=%7B%5C%7Bu_S%5C%7D_%7BS+%5Csubseteq+%5Bk%5D%7D%7D&bg=ffffff&fg=000000&s=0 "{\{u_S\}_{S \subseteq [k]}}")
A PSD lifting question
In an effort to improve this dependence, one could start with a much larger set of nearly orthogonal vectors, and then somehow lift them to a higher-dimensional space where they would become orthogonal. In order for the value of the SDP not to blow up, it would be necessary that this map has some kind of Lipschitz property. We are thus led to the following (possibly naïve) question.
Let 
There exists a map
such that
and whenever
satisfy
, we have
.
(Recall that 
One can show that
by randomly partitioning 
My question is simply: Is a better dependence on 
