Consider the generative model where,

\begin{align*} Z &\sim \mathrm{Bernoulli}\left( \frac{1}{2}\right) \\ Y &= \begin{cases} \pmb{\beta}_{1,*}^T \mathbf{X} + \eta &~~\text{given}~~ Z = 0\\ \pmb{\beta}_{2,*}^T \mathbf{X} + \eta &~~\text{given}~~ Z = 1 \end{cases} \end{align*} where $\mathbf{X} \in \mathbb{R}^d$ and is i.i.d standard normal per coordinate. Here, $\eta$ is $\mathcal{N}(0, \sigma^2)$ i.e a zero mean gaussian noise.

Now consider the population minimizers of the following loss, \begin{align*} \hat{\pmb{\beta}}_{1}, \hat{\pmb{\beta}}_{2} = \mathrm{argmin}_{\pmb{\beta}_{1}, \pmb{\beta}_{2}} \mathbb{E} \left[ min \left\{(Y - \pmb{\beta}_{1}^T \mathbf{X})^2, (Y - \pmb{\beta}_{2}^T \mathbf{X})^2 \right\}\right]. \end{align*}

Is it obvious that $\hat{\pmb{\beta}}_{1}, \hat{\pmb{\beta}}_{2} = \pmb{\beta}_{1,*}^T, \pmb{\beta}_{2,*}^T$ up to a permutation?



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