Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$. We are interested in this question in the number of points at which the function $L$ attains its minimum. (a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$. (b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$,
$$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$
and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$.
We are interested in this question in the number of points at which the function $L$ attains its minimum.
(a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$.
(b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.