Hyper^2 is proposed, a dual-space consistency framework that extends HyperbolicCD by reusing the identical arcosh(1+alpha d^2) functional form as a positional bias on the refinement attention (a hyperbolic distance encoding), paired with HyperbolicCD's hyperbolic Chamfer loss under a single shared curvature alpha.
Abstract
HyperbolicCD pioneered hyperbolic geometry for point cloud completion by replacing the Euclidean Chamfer distance with arcosh(1+alpha||x-y||^2), but the reported gains are modest (3-7% Chamfer reduction across SeedFormer, PointAttN and PMP-Net backbones on PCN and ShapeNet-55). We argue the bottleneck lies elsewhere: the loss is hyperbolic but the encoder it back-propagates through is Euclidean, so the position-dependent supervision of the loss is averaged away by the chain rule before it reaches the parameters. We call this a cross-geometry mismatch, and make it testable through two model-agnostic indicators, feature-loss correlation r_FL and effective gradient utilisation u_G. On an SVDFormer backbone trained with HyperbolicCD's loss alone we measure (r_FL, u_G) = (0.68, 39%). We propose Hyper^2, a dual-space consistency framework that extends HyperbolicCD by reusing the identical arcosh(1+alpha d^2) functional form as a positional bias on the refinement attention (a hyperbolic distance encoding), paired with HyperbolicCD's hyperbolic Chamfer loss under a single shared curvature alpha. Both operators are O(N log N) scalar non-linearities on Euclidean distances and together add only ~1.6% FLOPs over SVDFormer. Hyper^2 delivers -22.9% Chamfer on ShapeNet-55 over SVDFormer (well above the 13.2% linear sum of the -12.0% loss-only and -1.2% encoding-only single-space ablations) and -37.5% on the 21 unseen ShapeNet-34 categories. The two indicators remain essentially flat for any single-space configuration but jump together to (0.95, 87%) only when both encoder and loss are hyperbolic, supporting the claim that geometric consistency across encoder and loss, rather than either operator alone, is what enables hyperbolic supervision in point cloud completion. Code is available at https://github.com/Ethan-Zheng136/Hyper-2.
We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver). The Euclidean part establishes three core theorems: (1) circuit separation---IDA achieves exact retrieval with $\mathcal{O}(1)$ resources while softmax requires $\Omega((\log n)^2)$ width; (2) a Polyak--Lojasiewicz inequality with $\Omega(e^{\Delta^2/\sqrt{d}}/\Delta^2)$ stronger constant than softmax, implying linear convergence, $\mathcal{O}(\log n)$ Lipschitz scaling under a low-rank/clustering assumption, $\Theta(1)$ Hessian spread, and absence of spurious local minima; (3) a width-independent effective rank bound that limits noise memorization---softmax memorizes arbitrary labels when $d_h\ge n$, while IDA limits test error to $\mathcal{O}(\eta^2)$. The non-Euclidean extension then builds upon this prototype, replacing Euclidean distance with hyperbolic geodesic distance for storage and spherical geodesic distance for routing. The Riemann GeoResolver framework comprises ten integrated modules: four HIDA operators spanning $\Theta(n^2)$ to $\Theta(1)$ per token; Hyperbolic Curvature Compression (HCC) with provable error bounds; HyperGate with gradient lower-bound theorem; Spherical Inverse Distance Attention (SIDA) with sphere-analog PL inequalities; Dynamic Memory Genesis (DMG) with $\mathcal{O}(\log T)$ regret bounds; and Geodesic Sparse Routing (GSR) with quality and communication bounds. The Euclidean theorems are proved in full; the non-Euclidean extension theorems are proved with analogous arguments. This work establishes a theoretical arc: from Euclidean attention as a special case, to hyperbolic memory, to spherical retrieval.
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network localization, molecular conformation, and manifold learning. In this paper, we propose a Riemannian optimization framework for solving the EDMC problem by formulating it as a low-rank matrix completion task over the space of positive semi-definite Gram matrices. The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through nonnegativity and the triangle inequality, a structure inherited from classical multidimensional scaling. Under a Bernoulli sampling model for observed distances, we prove that Riemannian gradient descent on the manifold of rank-<inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> matrices locally converges linearly with high probability when the sampling probability satisfies <inline-formula> <tex-math notation="LaTeX">$p\geq {\mathcal {O}} (\nu ^{2} r^{2}\log (n)/n)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\nu $ </tex-math></inline-formula> is an EDMC-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies <inline-formula> <tex-math notation="LaTeX">$p \geq {\mathcal {O}} (\nu r^{3/2}\log ^{3/4}(n)/n^{1/4})$ </tex-math></inline-formula>. A key technical contribution of this work is the analysis of a symmetric linear operator arising from a dual basis expansion in the non-orthogonal basis, which requires analysis of a second order degenerate U-statistic to establish an optimal restricted isometry property in the presence of coupled terms. Empirical evaluations on synthetic data demonstrate that our algorithm achieves competitive performance relative to state-of-the-art methods. Moreover, we provide a geometric interpretation of matrix incoherence tailored to the EDMC setting and provide robustness guarantees for our method. Due to space constraints, the complete proofs of the convergence bounds and technical lemmas are provided in the extended pre-print 2508.00091, and the appendices containing said results are also available online at IEEE Xplore.
A completion theory for hyperbolic distance data is developed at the interface of matrix analysis, graph theory, and hyperbolic geometry. Krein's characterization of the metric space embeddability in Lobachevsky space leads to a natural anchoring procedure that transforms the indefinite data into a positive semidefinite kernel. In analogy with positive semidefinite and Euclidean distance matrix completion, chordality of the specification graph is shown to be the necessary and sufficient condition for local Lorentz-Gram data to admit global completion. Existence is complemented by explicit constructions. For trees, we obtain geodesic-rectification and product-distance completions; for chordal graphs, the latter extends to matrix-valued transfers along clique-trees. The resulting canonical completion is characterized by sparsity of its inverse and by a maximum-absolute-determinant principle. Its metric distortion exhibits a sharp dichotomy governed by clique separator size. Applications to exact recovery from sparse hyperbolic measurements and to hierarchical and phylogenetic data are developed.
Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.
We consider the Euclidean distance geometry problem (EDG): given a subset of the pairwise distances of an unknown cloud of $n$ points in $\mathbb{R}^\ell$, recover the point cloud up to rigid motions. When $n$ is large, a popular practical approach is to minimize a nonconvex quartic, known as the squared-stress or s-stress, over point clouds in $\mathbb{R}^k$, with $k$ potentially larger than $\ell$. It is a long-standing open problem to understand the optimization landscape of the s-stress when all pairwise distances are known (Malone and Trosset, 2000; Parhizkar, 2013). It was recently shown that the landscape is not benign when $k=\ell$, and it was conjectured that the landscape becomes benign as soon as $k\ge \ell+1$ (Song et al., 2025; Criscitiello et al., 2026). Here, we show that the complete-graph s-stress has a benign landscape whenever $k\ge 2(\ell+1)$, establishing the conjecture up to a factor of two. A key idea is to view second-order criticality as a containment of two ellipsoids; finding a descent direction then corresponds to finding a separating hyperplane that violates this containment. This dual perspective yields the stated landscape result, and also applies to any measurement operator whose inverse satisfies a simple frame condition.
Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing its Legendre dual, Fisher--Rao metric, Bregman divergence, and generalized Pythagorean theorem. We connect this framework to Craig--Sakamoto deformation, Wolfe duality, and, via Yoshizawa's embedding, Brockett--Bloch--Ratiu double-bracket flows, linking isospectral dynamics, Stiefel optimization, and component learning. The Bures--Wasserstein geometry provides a complementary gradient-flow structure. The singular theory emerges from boundary behavior: difference-of-convex deformations produce indefinite or degenerate Hessians while retaining pseudo-Hessian, dually flat, Legendre-self-dual structures. Newton flows exhibit finite-time collapse or {\L}ojasiewicz-controlled convergence near non-Morse critical sets. Fisher-metric degeneracies on the Birkhoff polytope and elliptic-curve moduli are resolved by explicit blow-ups, yielding a birationally invariant exponential decay law. We further derive a closed-form Kirillov Jacobian and introduce cross curvature as a spectral diagnostic of local escape rates, including a new Box--Cox interpolation. Reproducible numerical experiments support the closed-form results. Rather than claiming a completed theory, the paper provides foundations for singular information geometry centered on degenerate pencils, indefinite dual flatness, blow-up geometry, and {\L}ojasiewicz-type convergence.
Shintaro Yoshizawa· 0 citations
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