\label{eq:classif}$$ where $\| x \|_d = \sqrt{\sum_{j=1}^d x_j^2}$ denotes the Euclidean norm in $\mathbb{R}^d$; we also write $\langle x, y \rangle_d = \sum_{j=1}^d x_j y_j$ for the associated inner product.

[**Reinterpretation as a plug-in classifier**  ]{} We now explain why ${\widehat}\Phi_d$ can be reinterpreted as a plug-in classifier in a truncated space. Recall the expression  for the regression function $\eta$. It is thus natural to consider the ’truncated’ regression function $\eta_d$ by replacing $f$ and $g$ with their projections $\Pi_d(f) = \sum_{j=1}^d \theta_j \varphi_j$ and $\Pi_d(g) = \sum_{j=1}^d \mu_j \varphi_j$, [i.e.]{}, $$\begin{aligned}
\eta_d(X) & := \frac{\exp\left(\int_{0}^1 (\Pi_d(f)(s)-\Pi_d(g)(s)) dX_s - \frac{1}{2}\|\Pi_d(f)\|^2 + \frac{1}{2}\|\Pi_d(g)\|^2\right)}{1+\exp\left(\int_{0}^1 (\Pi_d(f)(s)-\Pi_d(g)(s)) dX_s - \frac{1}{2}\|\Pi_d(f)\|^2 + \frac{1}{2}\|\Pi_d(g)\|^2\right)} \label{eq:eta_d} \\
& = \frac{\exp\left( \langle {\boldsymbol{\theta}}_d - {\boldsymbol{\mu}}_d , {\mathbf{X}}_d \rangle_d -\frac{1}{2}\|{\boldsymbol{\theta}}_d\|_d^2 + \frac{1}{2}\|{\boldsymbol{\mu}}_d\|_d^2 \right)}{1+\exp\left( \langle {\boldsymbol{\theta}}_d - {\boldsymbol{\mu}}_d, {\mathbf{X}}_d \rangle_d -\frac{1}{2}\|{\boldsymbol{\theta}}_d\|_d^2 + \frac{1}{2}\|{\boldsymbol{\mu}}_d\|_d^2 \right)} \nonumber \,,\end{aligned}$$ where ${\boldsymbol{\theta}}_d := (\theta_j)_{1 {\leqslant}j {\leqslant}d}$, and ${\boldsymbol{\mu}}_d := (\mu_j)_{1 {\leqslant}j {\leqslant}d}$. We also define the associated oracle classifier (that still depends on the unknown functions $f$ and $g$): $$\label{eq:bayes_rule_d}
\Phi_d^{\star}(X) := 
\mathds{1}_{\left\lbrace \eta_d(X){\geqslant}\frac12 \right\rbrace} = \mathds{1}_{\left\lbrace 
\int_{0}^1 (\Pi_d(f)(s)-\Pi_d(g)(s)) dX_s {\geqslant}\frac{1}{2}\|\Pi_d(f)\|^2 - \frac{1}{2}\|\Pi_d(g)\|^2 \right\rbrace} \,.$$ As shown in Remark \[rmk:truncatedSpace\] (see Section \[sec:estimationerror\] below), $\eta_d$ and $\Phi_d^{\star}$ correspond to the regression function and the 