# Persistent-rank token budget

- ID: 482
- Canonical URL: https://synthcore.org/idea/482/persistent-rank-token-budget
- API JSON: https://synthcore.org/api/idea/482.json
- API Markdown: https://synthcore.org/api/idea/482.md
- Verification status: unverified
- Source: [arXiv:2607.06524](https://arxiv.org/abs/2607.06524)
- Category: memory
- Solves: memory, accuracy, stability
- ML areas: graph-nn, pruning, embedding
- Math tags: topology, metric-geometry, linear-algebra, combinatorics
- Ratings: usefulness 5/10; difficulty 5/10; novelty 7/10

## Idea description

Add a topology-aware lower bound to point-cloud or graph token pruning: at each geometric scale, retain at least as many latent representatives as the persistent-homology rank between that scale and a larger scale. The method prevents the pruning module from collapsing independent connected components or cycles that remain persistent, while still allowing compression in topologically redundant regions.

## Mathematical statement

Lemma 3.1 states that for a simplicial filtration \(\mathcal{G}\), a finitely presented \(c\)-approximation \(\mathcal{F}\), homology degree \(i\geq 0\), scale \(r\geq 0\), and approximation factor \(c\geq 1\), one has \(|\mathcal{F}|\geq \dim H_i(\mathcal{F})_{cr}\geq \operatorname{rank}(H_i(\mathcal{G})_r\to H_i(\mathcal{G})_{c^2r})\). Here \(H_i(\mathcal{G})_r\) is the \(i\)-th homology vector space of the target filtration at scale \(r\), the arrow is the homology map induced by the inclusion \(\mathcal{G}_r\hookrightarrow \mathcal{G}_{c^2r}\), \(\operatorname{rank}\) counts homology classes surviving from \(r\) to \(c^2r\), and \(|\mathcal{F}|\) is the size of the finite presentation or approximation. For a point cloud \(X\) with metric \(\partial\), the target filtration is the Vietoris–Rips complex \(\mathcal{VR}(X)_r=\{\sigma\subset X:\sigma\neq\emptyset,\ \operatorname{diam}(\sigma)\leq 2r\}\). We adapt the theorem into a latent-slot constraint \(K_r\geq R_r\), where \(R_r=\operatorname{rank}(H_i(\mathcal{VR}(X))_r\to H_i(\mathcal{VR}(X))_{c^2r})\) and \(K_r\) is the number of retained point or token slots assigned to scale \(r\). This is a conservative architectural surrogate: the theorem lower-bounds filtration complexity, while the model enforces at least that many slots.

## Key formulas

- $$\mathcal{VR}(X)_{r}=\{\sigma\subset X\mid\sigma\neq\emptyset,\ \operatorname{diam}(\sigma)\leq 2r\},\qquad \operatorname{diam}(\sigma)=\max_{x,y\in\sigma}\partial(x,y).$$
- $$|\mathcal{F}|\geq\dim H_i(\mathcal{F})_{cr}\geq\operatorname{rank}\bigl(H_i(\mathcal{G})_{r}\to H_i(\mathcal{G})_{c^{2}r}\bigr).$$
- $$R_{i,r}=\operatorname{rank}\bigl(H_i(\mathcal{VR}(X))_{r}\to H_i(\mathcal{VR}(X))_{c^{2}r}\bigr),\qquad K_{i,r}\geq\max(K_{\min},R_{i,r}).$$
- $$\mathcal{L}_{\mathrm{topo}}=\lambda\sum_{i\in I}\sum_{r\in\mathcal{R}}\bigl[\max(0,R_{i,r}-K_{i,r})\bigr]^2.$$

## Implementation notes

Integrate this into the token-pruning or pooling stage of a point-cloud, molecular-graph, or geometric graph neural network. The input is a batch of coordinates or graph nodes \(X\), a metric \(\partial\), radii \(\mathcal{R}\), homology degrees \(I\) (start with \(i=0,1\)), and approximation factor \(c\). First construct sparse Vietoris–Rips complexes at \(r\) and \(c^2r\), using a k-nearest-neighbor graph and only edges, triangles, and optionally tetrahedra. Compute persistent ranks by finite-field boundary-matrix reduction: obtain bases for \(H_i(\mathcal{VR}(X)_r)\), map each cycle into the larger complex, reduce it modulo boundaries at \(c^2r\), and count linearly independent surviving images. During the pruning forward pass, score candidate tokens with the learned pruning head, sort by score, and retain at least \(K_{i,r}=\max(K_{\min},R_{i,r})\) tokens for every scale or group. Use a straight-through top-k mask for end-to-end training, or use the rank only to set a hard inference-time budget. Pseudocode: build sparse complexes; compute \(R_{i,r}\); assign budgets; score tokens; select highest-scoring tokens subject to all budgets; run the GNN; optionally add \(\mathcal{L}_{\mathrm{topo}}\). The exact mathematical quantity is the finite-field persistent rank; sparsification, field choice, radii, and token-to-scale assignment are empirical choices. Test first on ModelNet40 or ShapeNet with PointNet++ or Point Transformer and 4x token pruning. Compare uniform, learned-score, and rank-constrained pruning at equal FLOPs. Success means higher accuracy at the same retained-token count, fewer failures on disconnected or looped shapes, and errors concentrated less often on examples where the unconstrained pruner violates \(K_{i,r}\geq R_{i,r}\).

## Disclaimer

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