Lipschitz constant of a Matrix Valued Function

Question

Consider the function $H(w) = \sum_{i=1}^n f(w^T x_i) x_i x_i^T $, where $w\in \mathbb{R}^d$, $\forall i$: $x_i \in \mathbb{R}^d$, and $f: \mathbb{R} \to [0,1]$. Further, know $|f'(y)| \leq 1$ for all $y\in \mathbb{R}$. Note, $d > n$ is possible.

What is the Lipschitz constant for this function, i.e. for which $M$ the following inequality is true?

$\|H(w) - H(w') \|_{op,2} \leq M \|w- w'\|_2$ for all $w,w'\in \mathbb{R}^d$

Answer 1

If you know that $|f'(x)| \le 1$, then you can quickly obtain the following estimate using the mean value theorem.

$H(w)-H(w') = \sum_i (f(x_i^T w)-f(x_i^Tw')) x_i x_i^T$, hence $\|H(w)-H(w')\| \le \sum_i |x_i^T (w-w')| \| x_i x_i^T\| $, and so $L=\sum_i \|x_i\|^3$ is one estimate.