Bernstein Bounded Implicit Representation

Use this as the final representation layer of a coordinate-based MLP or as a local decoder in a mesh-aware neural field. For a one-dimensional input, choose degree $n$ and compute $b_i^n(x)=\binom ni x^i(1-x)^{n-i}$; for triangular cells, map each point to barycentric coordinates $\mathbf b$ and compute $B^\alpha(x)=n!\mathbf b^\alpha/\alpha!$. Let the MLP output unconstrained logits $w_\alpha\in\mathbb R^q$ for every coefficient, then set $c_{\alpha,r}=\ell_r+(u_r-\ell_r)\sigma(w_{\alpha,r})$, where $r$ indexes output channels. Return $p_r(x)=\sum_{|\alpha|=n}c_{\alpha,r}B^\alpha(x)$. Pseudocode is: normalize the input into its cell; compute the MLP coefficient logits; apply the bounded sigmoid map; calculate all Bernstein basis weights; multiply and sum coefficients. Initialize logits using $w_\alpha=\operatorname{logit}((c^0_\alpha-\ell)/(u-\ell))$ when a useful initial field is known. The paper supplies the convex-hull guarantee; estimate approximation error empirically on dense validation points. Compare against an unconstrained MLP plus output clipping and a tanh output head on a 2D bounded-field reconstruction task, such as density or occupancy reconstruction. Measure maximum bound violation on a dense grid, interpolation error, boundary error, parameter count, and inference throughput. Success is exact zero violations between sampled points, lower error near sharp interfaces, and no meaningful slowdown relative to the baselines.