Gaussian Projection Reduction

A randomized procedure to produce an approximate SDP solution with a desired low rank $d$. Again, let ${\mathbf{X}}^{\ast }$ be an optimal solution of SDP with rank $r>d$ and we factorize ${\mathbf{X}}^{\ast }$ as
$${\mathbf{X}}^{\ast }=({\mathbf{V}}^{\ast }{)}^T{\mathbf{V}}^{\ast },{\mathbf{V}}^{\ast }\in {\mathbb{E}}^{r\times n}$$
We then generate i.i.d Gaussian random variables ${\xi }_i^j$ with mean 0 and variance $1/d$, $i=1,\dots ,r;j=1,\dots ,d$, and we define
$$\hat{\mathbf{X}}=({\mathbf{V}}^{\ast }{)}^T[\sum _{j=1}^d{\xi }^j({\xi }^j{)}^T]{\mathbf{V}}^{\ast }\text{//rank is d}$$
Note that the rank of $\hat{\mathbf{X}}$ is $d$ and
$$\mathbb{E}(\hat{\mathbf{X}})=({\mathbf{V}}^{\ast }{)}^T\mathbf{I}{\mathbf{V}}^{\ast }={\mathbf{X}}^{\ast }\text{//approximate}$$
We can further show that $\hat{\mathbf{X}}$ would be a good rank-$d$ approximate SDP solution in many cases.

Randomized Binary Reduction

As QP optimization, we want to produce a vector $\mathbf{x}$ where each entry is either $-1$ or $1$. A procedure to achieve this is as follows. Let ${\mathbf{X}}^{\ast }$ be any optimal solution of SDP and we factorize ${\mathbf{X}}^{\ast }$ as
$${\mathbf{X}}^{\ast }=({\mathbf{V}}^{\ast }{)}^T{\mathbf{V}}^{\ast },{\mathbf{V}}^{\ast }\in {\mathbb{E}}^{n\times n}$$
Then, we generate a random $n$-dimensional vector $\xi$ where each entry is a i.i.d Gaussian random variable with mean 0 and variance 1. Define
$$\hat{\mathbf{x}}=sgn(({\mathbf{V}}^{\ast }{)}^T\xi )$$
It was proved by Sheppard:
$$\mathbb{E}[{\hat{x}}_i{\hat{x}}_j]=\frac{2}{\pi }\arcsin ({\mathbf{X}}_{ij}^{\ast }),i,j=1,2,\dots ,n$$
Consider the primal homogeneous binary quadratic maximization problem
z ∗ : = max ⁡ x T Q x s . t . x j = { 1 , − 1 } , j = 1 , … , n z^*:=\max \mathbf{x}^T\mathbf{Q}\mathbf{x}\\ s.t.\quad x_j=\{1, -1\}, j=1,\dots, n z∗:=maxxTQxs.t.xj​={1,−1},j=1,…,n
where we assume $\mathbf{Q}$ is positive semidefinite. Then, the SDP relaxation would be
$$\begin{gathered} z^{SDP}:=\max \mathbf{Q}\cdot \mathbf{X} \\ s.t.{\mathbf{I}}_j\cdot \mathbf{X}=1,j=1,\dots ,n \\ \mathbf{X}\in {\mathcal{S}}_{+}^n \end{gathered}$$
and let ${\mathbf{X}}^{\ast }$ be any optimal solution, from which we produced a random binary vector $\hat{\mathbf{x}}$. And, we evaluate the expected objective value
$$\mathbb{E}(\hat{\mathbf{x}}\mathbf{Q}\hat{\mathbf{x}})=\mathbb{E}(\mathbf{Q}\cdot \hat{\mathbf{x}}{\hat{\mathbf{x}}}^T)=\mathbf{Q}\cdot \mathbb{E}(\hat{\mathbf{x}}{\hat{\mathbf{x}}}^T)=\mathbf{Q}\cdot \frac{2}{\pi }\arcsin [{\mathbf{X}}^{\ast }]=\frac{2}{\pi }(\mathbf{Q}\cdot \arcsin [{\mathbf{X}}^{\ast }])$$
where $\arcsin [{\mathbf{X}}^{\ast }]\in {\mathcal{S}}^n$, and further
$$\arcsin [{\mathbf{X}}^{\ast }]-{\mathbf{X}}^{\ast }⪰0$$
from $\mathbf{Q}⪰0$, we have
$$\mathbf{Q}\cdot \arcsin [{\mathbf{X}}^{\ast }]\ge \mathbf{Q}\cdot {\mathbf{X}}^{\ast }=z^{SDP}\ge z^{\ast }$$