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Existence of optimal entropic weights for empirical modeling

Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
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Normalizing constants preserve metric entropy

Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation $$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...