# Moment-Preserving Anisotropic Feature Tree - ID: 416 - Canonical URL: https://synthcore.org/idea/416/moment-preserving-anisotropic-feature-tree - API JSON: https://synthcore.org/api/idea/416.json - API Markdown: https://synthcore.org/api/idea/416.md - Verification status: mech_ok_no_baseline - Source: [arXiv:2607.04881](https://arxiv.org/abs/2607.04881) - Category: architecture - Solves: memory, accuracy, scalability - ML areas: mlp, embedding, world-model, memory - Math tags: harmonic-analysis, wavelet, geometry, linear-algebra - Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10 ## Idea description Replace a dense multiresolution voxel or hash-grid encoder with an omnitree-like anisotropic feature partition. Each cell stores a vector-valued scaling feature and its children are introduced only when local Haar detail energy is large; coarsening replaces children by their mean, so compression does not introduce an arbitrary offset. Splitting can be restricted to the coordinate whose one-dimensional detail coefficient is largest, allowing thin structures to receive resolution only in the relevant directions. ## Mathematical statement The paper represents a piecewise-constant function on leaf rectangles as f(x)=sum_v s^v phi^v(x), where v ranges over leaves, s^v is the value or feature attached to leaf v, and phi^v(x)=1 on rect(v) and 0 elsewhere. For a parent with 2^d equal children, the tensor-product Haar transform maps child values to one scaling coefficient and 2^d-1 detail coefficients. In two dimensions the transform is based on the Hadamard matrix H=[[1,1,1,1],[1,1,-1,-1],[1,-1,1,-1],[1,-1,-1,1]]; with normalized rows, the first output is the child mean and the other three outputs are horizontal, vertical, and diagonal details. For vector-valued features, apply the same transform independently to every feature channel. If c_i are the child feature vectors and m=2^d, the moment-preserving coarsening rule is c_parent=(1/m) sum_i c_i. Because the Haar basis is orthogonal, the squared error from dropping details is exactly proportional to E_detail=sum_{k != emptyset} ||w_k||_2^2. For anisotropic splitting, choose the coordinate with the largest directional detail norm, j*=argmax_j ||w_{j}||_2, and split only along that coordinate when E_detail exceeds a threshold. ## Key formulas - $$f(\mathbf{x})=\sum_{v:\,\vec b(v)=\vec 0}s^v\,\phi^v(\mathbf{x}),\qquad \phi^v(\mathbf{x})=\begin{cases}1,&\mathbf{x}\in\operatorname{rect}(v),\\0,&\text{else.}\end{cases}$$ - $$\begin{pmatrix}w_{\emptyset}\\w_{\{x\}}\\w_{\{y\}}\\w_{\{x,y\}}\end{pmatrix}=\frac{1}{4}\begin{pmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&1&-1\\1&-1&-1&1\end{pmatrix}\begin{pmatrix}c_{00}\\c_{01}\\c_{10}\\c_{11}\end{pmatrix},\qquad c_{ij}\in\mathbb{R}^{C}.$$ - $$c_{\mathrm{parent}}=w_{\emptyset}=\frac{1}{2^d}\sum_{i=1}^{2^d}c_i,\qquad E_{\mathrm{detail}}=\sum_{k\in\{0,1\}^{d}\setminus\{\mathbf{0}\}}\lVert w_k\rVert_2^2.$$ - $$j^*=\arg\max_{j\in\{1,\ldots,d\}}\lVert w_{\{j\}}\rVert_2,\qquad \text{split cell along coordinate }j^*\text{ if }E_{\mathrm{detail}}>\tau.$$ ## Implementation notes Integrate this at the spatial feature encoder, before the MLP or attention block that consumes coordinates. Start with a 2D coordinate MLP or a 3D NeRF-style voxel encoder: every active rectangular cell stores a C-dimensional feature vector and integer dyadic bounds, and the decoder is an unchanged small MLP receiving the selected cell feature together with the normalized coordinate. Maintain a priority queue keyed by detail energy. Pseudocode: initialize a root cell; for every candidate cell, accumulate or sample the feature vectors c_i in its possible child bins; compute the channelwise Haar outputs using the displayed matrix; calculate E_detail; if E_detail exceeds tau_split, create two children along j*=argmax_j ||w_{j}||_2 or all 2^d children; initialize child features by inverse Haar reconstruction; after every K optimizer steps, inspect sibling groups and merge them when E_detail