Nonlinear Geometrizability of State-Dependent Proto-Area in Approximate Holographic Codes

• URL: http://arxiv.org/abs/2606.22598v1
• Date: Sun, 21 Jun 2026 17:16:10 GMT
• Title: Nonlinear Geometrizability of State-Dependent Proto-Area in Approximate Holographic Codes
• Abstract summary: State-dependent proto-area data produced by approximate recovery need not be compatible with a single local bulk metric.<n>We derive exact finite-resolution criteria and, near the hyperbolic disk, necessary and sufficient conditions for a regular proto-area two-jet.<n>Hamiltonian-skewed codes realize both first-order nongeometry and a calibrated response whose first obstruction appears only at quadratic order.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: State-dependent proto-area data produced by approximate recovery need not be compatible with a single local bulk metric. Using recovery maps calibrated on the code channel and held fixed along a logical-state family, we derive exact finite-resolution criteria and, near the hyperbolic disk, necessary and sufficient conditions for a regular proto-area two-jet to arise from a metric two-jet on a time-reflection-symmetric asymptotically AdS$_3$ slice. Finite networks give a polyhedral realization problem with primal and dual certificates, stable reconstruction, and explicit witnesses of nongeometry. In the continuum, the geometric tangent space is the range of the rank-two geodesic X-ray transform. A metric-forced Jacobi equation determines the normal Hessian of the renormalized boundary-length image and yields a gauge-invariant quadratic obstruction. Under a split-regularity hypothesis, nearby geometric data form a local graph; the two-jet criterion itself is unconditional for regular data. Hamiltonian-skewed codes realize both first-order nongeometry and a response whose first obstruction appears only at quadratic order. The compatible metric perturbation is reconstructed modulo boundary-fixing diffeomorphisms.

Related papers

• Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm, with Application to Generative AI [0.7766379063545538]
  We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting parametric models into infinite-dimensional minimization non-asymptotic statistics.<n>We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable base manifold $mathcalB$, establishing a hard geometric containment of generative hallucinations.
  arXiv  Detail & Related papers  (2026-07-03T13:46:31Z)
• Fitting Unknown Number of Hyperplanes with Manifold Optimization [57.48093263119306]
  Fitting unknown number of linearplanes to data is a fundamental yet challenging problem in machine learning.<n>Existing approaches often struggle with optimum optima or lack geometric consistency.
  arXiv  Detail & Related papers  (2026-05-27T14:02:20Z)
• Bayesian Latent Space Models for Graphs Are Misspecified: Toward Robust Inference via Generalized Posteriors [0.0]
  Link-Sequential R-SafeBayes is a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization.<n> Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.
  arXiv  Detail & Related papers  (2026-05-18T12:53:43Z)
• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning [52.452902154360565]
  We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
  arXiv  Detail & Related papers  (2026-05-05T20:19:09Z)
• Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence [0.764671395172401]
  We develop a trajectory-restricted framework for linear convergence based on localized geometric regularity.<n>We show that classical convergence guarantees extend under these localized conditions.<n>Our work provides a geometric quantification for fast local convergence after active-set or manifold identification.
  arXiv  Detail & Related papers  (2026-04-18T17:02:06Z)
• On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry [0.0]
  In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-root parameter spaces, finite-sample behavior can deviate systematically from predictions.<n>We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a ((,g)) with Fisher-constrained-Rao metric.<n>We describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case.
  arXiv  Detail & Related papers  (2026-04-14T13:40:13Z)
• Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation [79.27003481818413]
  We introduce FlatVI, a training framework that regularises the latent manifold of discrete-likelihood variational autoencoders towards Euclidean geometry.<n>By encouraging straight lines in the latent space to approximate geodesics on the decoded single-cell manifold, FlatVI enhances compatibility with downstream approaches.
  arXiv  Detail & Related papers  (2025-07-15T23:08:14Z)
• Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent [25.846262685970164]
  This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization.<n>It reconstructs the point set configuration from partial Euclidean distance measurements.
  arXiv  Detail & Related papers  (2025-04-28T07:13:23Z)
• GMapLatent: Geometric Mapping in Latent Space [51.317738404571514]
  Cross-domain generative models based on encoder-decoder AI architectures have attracted much attention in generating realistic images.<n>We introduce a canonical latent space representation based on geometric mapping to align the cross-domain latent spaces in a rigorous and precise manner.<n>Experiments on gray-scale and color images validate the efficiency, efficacy and applicability of GMapLatent.
  arXiv  Detail & Related papers  (2025-03-30T12:02:36Z)
• Bridging Geometric States via Geometric Diffusion Bridge [79.60212414973002]
  We introduce the Geometric Diffusion Bridge (GDB), a novel generative modeling framework that accurately bridges initial and target geometric states. GDB employs an equivariant diffusion bridge derived by a modified version of Doob's $h$-transform for connecting geometric states. We show that GDB surpasses existing state-of-the-art approaches, opening up a new pathway for accurately bridging geometric states.
  arXiv  Detail & Related papers  (2024-10-31T17:59:53Z)
• Disentangled Representation Learning with the Gromov-Monge Gap [65.73194652234848]
  Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning.<n>We introduce a novel approach to disentangled representation learning based on quadratic optimal transport.<n>We demonstrate the effectiveness of our approach for quantifying disentanglement across four standard benchmarks.
  arXiv  Detail & Related papers  (2024-07-10T16:51:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.