Is this $(\Bbb R^{n \times n})^n \to \Bbb R$ function convex?

Question

Let $W := (W_1, W_2,\dots, W_n)$, where $W_i \in \Bbb R^{n \times n}$. Let $x$ be a constant vector. Is the following function convex?

$$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$

Answer 1

Function $W \mapsto f(W)$ is not convex when $n \ge 2$.

Take for example when $n=2$ and fix $x=1$.

If function $f(W_1,W_2)$ is convex then it must also be convex when restricted to a linear subspace.

Restricting to $W_1 = \begin{pmatrix} u & 0 \\ 0 & 0 \end{pmatrix}$ and $W_2 = \begin{pmatrix} v & 0 \\ 0 & 0 \end{pmatrix}$ then gives $f(W_1,W_2) = u^2 v^2$, which is not convex in $(u,v)$ (this can be checked by computing its Hessian matrix).