Adaptive hierarchical sparse-grid encoding

Integrate this at the coordinate-embedding stage of a coordinate MLP, implicit image representation, signed-distance network, or NeRF-like model. The input is a coordinate $x\in[0,1]^d$ and the output is a feature vector concatenated to the MLP input. Maintain anisotropic component grids indexed by level vectors $\boldsymbol\ell$, but allocate feature vectors only for active cells. For each active grid, interpolate its feature table at $x$ using multilinear interpolation and combine the results with the sparse-grid formula. Pseudocode: initialize all level vectors with $|\boldsymbol\ell|_1=n$; allocate their cells; for each minibatch compute $z_{\boldsymbol\ell}(x)$ by interpolation; return $z(x)=\sum_{i=0}^{d-1}(-1)^i\binom{d-1}{i}\sum_{|\boldsymbol\ell|_1=n+d-1-i}z_{\boldsymbol\ell}(x)$; train the MLP and active tables normally; every $K$ optimizer steps, evaluate a held-out coordinate set and compute either prediction residuals $r_\alpha=\|y_\alpha-\hat y(x_\alpha)\|_2$ or feature surpluses $s_\alpha=\|z_\alpha-\Pi_{\mathrm{coarse}}z_\alpha\|_2$; refine the top $p$ percent of cells with $s_\alpha>\tau$ by allocating their children. Initialize child features by coarse interpolation plus small Gaussian noise, and impose a 2:1 neighboring-level balance. The paper-derived components are the sparse level sets, alternating binomial weights, nested sparse/full-grid hierarchy, and surplus criterion; $K$, $p$, $\tau$, feature width, and child initialization noise are empirical choices. First test on 2D image fitting or signed-distance fitting, comparing against dense multiresolution grids and hash-grid encodings. Measure PSNR or geometric error, active parameter count, wall-clock training time, and peak memory at matched quality. Success is at least 2x fewer feature parameters at equal PSNR, or higher PSNR at equal memory, with refinement concentrated around edges and localized structures.