Unverified 2026

Active-Contact Cone Projection

Implementation & benchmark of arXiv:2608.06513 — Intrinsic Geometry of Hard Disk Clusters

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Geometry Stability Accuracy Generalization Embedding Optimizer Quantization Loss Geometry Optimization Convex analysis Linear algebra

Source paper: Intrinsic Geometry of Hard Disk Clusters arXiv:2608.06513 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

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.

Formulas

$$\|{\bf c}_i- {\bf c}_j\|\geq 2,$$

$$\langle{\bf u}_{ij},\delta{\bf c}_{j}-\delta{\bf c}_{i}\rangle\geq 0,\qquad {\bf u}_{ij}=\frac{{\bf c}_{j}-{f c}_{i}}{\|{\bf c}_{j}-{f c}_{i}\|},$$

$$\text{\bf Adm}({\bf c})=\left\{\delta{\bf c}:\langle{\bf u}_{ij},\delta{\bf c}_{j}-\delta{\bf c}_{i}\rangle\geq0\ \text{for all }\{i,j\}\in E({\bf c})\right\},$$

$$v^*=\operatorname*{arg\,min}_{v}\frac12\|v-g\|_2^2\quad\text{s.t.}\quad u_{ij}^{\top}(v_j-v_i)\geq0\ \forall(i,j)\in E(z),\qquad z\leftarrow z+\eta v^*.$$

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.

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 $\|z_i-z_j\|<r+\varepsilon$, where $r$ is the desired minimum distance and $\varepsilon$ is a small active-band tolerance. For each active pair, calculate $u_{ij}=(z_j-z_i)/(\|z_j-z_i\|+10^{-8})$. Solve the Euclidean projection quadratic program $\min_v \frac12\|v-g\|^2$ subject to $u_{ij}^{\top}(v_j-v_i)\ge0$. A simple MVP uses projected-gradient iterations: initialize $v=g$, identify violated constraints, and subtract the minimum correction along the corresponding sparse constraint normal; alternatively use a batched QP solver for small $K$. Remove the mean update from $v$ if translation invariance is desired, and optionally separately remove the global rotational component. Apply $Z\leftarrow Z+\eta v^*$, then recompute the active set on the next optimizer step. The paper's mathematics supplies the contact-normal inequalities and active-set rule; the neural experiment must estimate the active set empirically from current distances. Start with a 512-vector VQ-VAE on CIFAR-10, comparing Adam plus a soft hinge penalty $\max(0,r-\|z_i-z_j\|)^2$ against Adam followed by cone projection. Measure codebook perplexity, nearest-neighbour distance violations, reconstruction loss, dead-code fraction, and wall-clock overhead. Success means fewer collapsed or dead prototypes and lower violation variance at equal reconstruction quality, or improved reconstruction at a fixed constraint budget.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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