Let $X$ a $d \times T$ given matrix and $M$ a $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$ where $'$ denotes the transpose operator. We assume $n<T$. The question is : is it possible to estimate the scalar quantity : $$E=\mathbb{E}[||HX - X||^2]$$ as a function of $n, T, d$ and probably some quantities related to $X$ ?