# Active-Contact Cone Projection - ID: 1895 - Canonical URL: https://synthcore.org/idea/1895/active-contact-cone-projection - API JSON: https://synthcore.org/api/idea/1895.json - API Markdown: https://synthcore.org/api/idea/1895.md - Verification status: unverified - Source: [arXiv:2608.06513](https://arxiv.org/abs/2608.06513) - Category: geometry - Solves: stability, accuracy, generalization - ML areas: embedding, optimizer, quantization, loss - Math tags: geometry, optimization, convex-analysis, linear-algebra - Ratings: usefulness 5/10; difficulty 5/10; novelty 4/10 ## Idea description Replace soft pairwise repulsion between learned prototypes or codebook vectors with an active-set feasibility layer based on the paper's first-order admissible cone. Pairs exactly at the minimum distance contribute linear half-space constraints to the update, while separated pairs do not unnecessarily restrict motion. This should reduce prototype collapse and make constrained embedding or quantization training less sensitive to penalty weights. ## Mathematical statement For disk centres ${\bf c}_i\in\mathbb{R}^2$, the feasible configuration space is defined by $\|{\bf c}_i-{\bf c}_j\|\ge 2$. The realised contact set is $E({\bf c})=\{\{i,j\}:\|{\bf c}_i-{\bf c}_j\|=2\}$. For each active pair, the contact normal is ${\bf u}_{ij}=({\bf c}_j-{f c}_i)/\|{\bf c}_j-{f c}_i\|$, and the paper's first-order admissible cone is ${\bf Adm}({\bf c})=\{\delta{\bf c}:\langle{\bf u}_{ij},\delta{\bf c}_j-\delta{\bf c}_i\rangle\ge0\ \text{for all }\{i,j\}\in E({\bf c})\}$. Positive inner product opens a contact, zero preserves it, and negative inner product violates first-order non-overlap. The rolling space replaces every active inequality by equality and is the subspace of motions preserving all realised contacts to first order. For a neural codebook with vectors $z_i\in\mathbb{R}^d$ and minimum distance $r$, use the same construction with active pairs $E(z)=\{(i,j):\|z_i-z_j\|\le r+\varepsilon\}$ and normals $u_{ij}=(z_j-z_i)/\|z_j-z_i\|$. Given the unconstrained gradient direction $g$, project it onto the cone by solving $v^*=\arg\min_v\frac12\|v-g\|_2^2$ subject to $u_{ij}^{\top}(v_j-v_i)\ge0$ for every active pair. ## Key formulas - $$\|{\bf c}_i- {\bf c}_j\|\geq 2,$$ - $$\langle{\bf u}_{ij},\delta{\bf c}_{j}-\delta{\bf c}_{i}\rangle\geq 0,\qquad {\bf u}_{ij}=\frac{{\bf c}_{j}-{f c}_{i}}{\|{\bf c}_{j}-{f c}_{i}\|},$$ - $$\text{\bf Adm}({\bf c})=\left\{\delta{\bf c}:\langle{\bf u}_{ij},\delta{\bf c}_{j}-\delta{\bf c}_{i}\rangle\geq0\ \text{for all }\{i,j\}\in E({\bf c})\right\},$$ - $$v^*=\operatorname*{arg\,min}_{v}\frac12\|v-g\|_2^2\quad\text{s.t.}\quad u_{ij}^{\top}(v_j-v_i)\geq0\ \forall(i,j)\in E(z),\qquad z\leftarrow z+\eta v^*.$$ ## Implementation notes Integrate this at the optimizer step for a learned prototype matrix $Z\in\mathbb{R}^{K\times d}$, such as VQ-VAE codebook vectors, class prototypes, retrieval anchors, or MoE expert embeddings. First compute the ordinary task gradient $g=-\nabla_Z L$ and flatten it into a vector with $Kd$ coordinates. Build the active set using pairwise distances: include $(i,j)$ when $\|z_i-z_j\|