When and How to Canonize: A Generalization Perspective

Modern image encoders achieve high generalization by decoupling semantic meaning from resolution, an ability yet to be fully realized in the 3D domain. We investigate the failure of 3D point cloud encoders to achieve similar generalization and find that existing models are highly sensitive to sampling resolution and scale changes, leading to significant performance degradation. This sensitivity is a major bottleneck for real-world deployment in robotics, as it suggests models overfit to specific quantization densities and object scales rather than learning invariant semantic features. To mitigate this dependency, we propose Invaria, a point cloud encoder that achieves scale and density invariance through next-resolution prediction and receptive field calibration. While our objective is not the explicit generation of high-resolution point clouds, we find that this training objective encourages the model to learn robust, structural invariants. The resulting encoder achieves significant performance gains during resolution shifts while maintaining high efficiency through a compact model size and reduced token requirements. Specifically, on ScanNet, Invaria achieves a 56.0% higher mIoU at 3$\times$ lower resolution and a 20% improvement when the objects scale is reduced by a factor of 3. These gains are achieved with a 45% smaller model size and an average reduction of 40% in input tokens.

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; the closed-form weight rule $w_k^2 \propto (d_{k+1}^2 - d_k^2)/R_k^2$ maximizes the distortion slope in Mitra-Virk's affine certificate under $ν$-coherence. (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$, matched in the sample-starved regime $m \lesssim R/Δ$ by a Le Cam minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form ranker on a 64-descriptor chemical-graph pool (mean Spearman $ρ= +0.56$ across 11 benchmarks, positive on 10 of 11); the isotropic surrogate $Δ/\sqrt{\ell}$ admits a closed-form selection-consistency rate on the homogeneous protein/social pools. (iii) A training-time-decided certificate, with no per-prediction overhead, in three concrete radii (Pinelis, Gaussian plug-in, and variance-aware Pinelis-Bernstein). Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2; remaining gaps fall into two diagnosable regimes (descriptor blindness on NCI1/NCI109; pool-coverage limits elsewhere). The Pinelis-Bernstein radius fires on 8 of the 12 benchmarks; on MUTAG the empirical and population nearest-centroid rules agree on every one of 940 held-out test predictions, validating the certificate's mechanism.

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$, and farthest-point-sampling positions $2$-approximating the optimal $k$-center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ($91.3 \pm 1.0\%$, matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ($25\%$ on 4-class data).