Suppose $(x_1,\eta_1),\ldots,(x_n,\eta_n)$ are $n$ i.i.d points in $\mathbb R^{d+1}$ such that $\eta_1,\ldots,\eta_n$ are $\sigma$-subgaussian. Let $X \in \mathbb R^{n \times d}$ be the vertical stacking of the $x_i$'s and $\eta \in \mathbb R^n$ be the vertical stacking of the $\eta_i$'s


Are there any concentration inequalities which can be liveraged to bound the matrix $X^T\eta\eta^TX \in \mathbb R^{d \times d}$ ?


Naively, I'd guess that $X^T\eta\eta^TX \preceq \sigma^2X^TX + \text{"small thing"}$, with high probability.


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