Planar Curve Registration using Bayesian Inversion
- URL: http://arxiv.org/abs/2307.04909v1
- Date: Mon, 10 Jul 2023 21:26:43 GMT
- Title: Planar Curve Registration using Bayesian Inversion
- Abstract summary: We study parameterisation-independent closed curve matching as a Bayesian inverse problem.
The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space.
We adopt ensemble Kalman inversion using a negative Sobolev mismatch penalty to measure the discrepancy between the target and the ensemble mean shape.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We study parameterisation-independent closed planar curve matching as a
Bayesian inverse problem. The motion of the curve is modelled via a curve on
the diffeomorphism group acting on the ambient space, leading to a large
deformation diffeomorphic metric mapping (LDDMM) functional penalising the
kinetic energy of the deformation. We solve Hamilton's equations for the curve
matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite
element spaces for $2m^\text{th}$ order partial differential equations on
$\mathbb{R}^n$ simplicial grids when $m=n+1$, Mathematics of Computation 88
(316) (2019) 531-551] which provides mesh-independent Lipschitz constants for
the forward motion of the curve, and solve the inverse problem for the momentum
using Bayesian inversion. Since this element is not affine-equivalent we
provide a pullback theory which expedites the implementation and efficiency of
the forward map. We adopt ensemble Kalman inversion using a negative Sobolev
norm mismatch penalty to measure the discrepancy between the target and the
ensemble mean shape. We provide several numerical examples to validate the
approach.
Related papers
- Same Flow, Different Paths: Variance Reduction in Flow Matching [67.12978375116599]
We study the choice of this path from an optimization perspective.<n>Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD.
arXiv Detail & Related papers (2026-09-15T15:01:04Z) - Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics [0.0]
This work investigates the deformed-derivative formalism introduced by Borges.<n>It focuses on the relation between the linear operator $D_(q)$ and its nonlinear dual counterpart $D(q)$.<n>We show that this nonlinearity can be removed by a simultaneous transformation of the coordinate and of the wave function.
arXiv Detail & Related papers (2026-05-27T11:29:44Z) - Spherical Flows for Sampling Categorical Data [6.88204255655161]
We study the problem of learning generative models for discrete sequences in a continuous embedding space.<n>Experiments compare the vMF path against geodesic and Euclidean alternatives.<n>The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.
arXiv Detail & Related papers (2026-05-07T03:34:00Z) - Entropic Riemannian Neural Optimal Transport [41.69130102668252]
We propose a unified framework that combines intrinsic entropic OT and amortized out-of-sample evaluation.<n>Our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $mathbbS2$, $mathrmSO(3)$, $mathrmSPD(3)$, $mathrmSE(3)$, and $mathbbH2$.
arXiv Detail & Related papers (2026-05-05T19:43:03Z) - 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) - Axiomatic On-Manifold Shapley via Optimal Generative Flows [9.595059073171269]
Shapley-based attribution is critical for post-hoc XAI but suffers from off-manifold artifacts due to baselines.<n>We propose a formal theory of on-manifold Aumann-Shapley attributions driven by optimal generative flows.
arXiv Detail & Related papers (2026-03-05T12:05:20Z) - Klein--Gordon oscillator with linear--fractional deformed Casimirs in doubly special relativity [0.0]
We study the Klein--Gordon (KG) oscillator in a doubly special relativity (DSR) framework.<n>Timelike and lightlike deformations produce identical spectra characterized by a Planck-suppressed additive displacement.<n>We provide a compact $mathcalPT$-symmetric and pseudo-Hermitian formulation.
arXiv Detail & Related papers (2026-02-24T22:58:53Z) - Constructive conditional normalizing flows [0.8594140167290097]
We consider the problem of simultaneously approximating $$ and the pushforward $_#$ by means of the flow of a continuity equation.<n>We provide an explicit construction based on a polar-like decomposition of the Lagrange interpolant of $$.
arXiv Detail & Related papers (2026-02-09T12:52:47Z) - On the Superlinear Relationship between SGD Noise Covariance and Loss Landscape Curvature [1.6773271875801752]
Gradient Descent (SGD) introduces anisotropic noise that is correlated with the local curvature of the loss landscape, thereby biasing optimization toward flat minima.<n>We show that this assumption holds only under restrictive conditions that are typically violated in deep neural networks.<n>Experiments across datasets, architectures, and loss functions validate these bounds, providing a unified characterization of the noise-curvature relationship in deep learning.
arXiv Detail & Related papers (2026-02-05T12:35:13Z) - Robust Sublinear Convergence Rates for Iterative Bregman Projections [21.689846521201588]
We use entropic regularization to approximate linear programs whose constraints split into two (or more) tractable blocks.<n>We derive the flow-Sinkhorn algorithm for the Wasserstein-1 distance on graphs.
arXiv Detail & Related papers (2026-02-01T18:20:19Z) - Variational Inference with Mixtures of Isotropic Gaussians [8.365869192421865]
Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family.<n>We develop a variational framework and provide efficient algorithms suited for this family.
arXiv Detail & Related papers (2025-06-16T15:42:15Z) - Symmetry-Breaking Descent for Invariant Cost Functionals [0.0]
We study the problem of reducing a task cost functional $W(S)$, defined over Sobolev-class signals $S$, when the cost is invariant under a global symmetry group $G subset mathrmDiff(M)$.<n>We propose a variational method that exploits the symmetry structure to construct explicit, symmetry-breaking deformations of the input signal.
arXiv Detail & Related papers (2025-05-19T15:06:31Z) - Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space [49.1574468325115]
We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay.
We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace.
arXiv Detail & Related papers (2024-05-27T16:11:39Z) - Randomized Physics-Informed Machine Learning for Uncertainty
Quantification in High-Dimensional Inverse Problems [49.1574468325115]
We propose a physics-informed machine learning method for uncertainty quantification in high-dimensional inverse problems.
We show analytically and through comparison with Hamiltonian Monte Carlo that the rPICKLE posterior converges to the true posterior given by the Bayes rule.
arXiv Detail & Related papers (2023-12-11T07:33:16Z) - On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier [0.0]
General time-dependent PT-symmetric parametric oscillators for unbroken parity and time reversal regimes are studied theoretically.
We have demostrated the time variation of the Wigner distribution for the system consisting of two spatially separated ground state of the TD-parametric amplifier.
arXiv Detail & Related papers (2023-05-20T19:45:22Z) - One-dimensional pseudoharmonic oscillator: classical remarks and
quantum-information theory [0.0]
Motion of a potential that is a combination of positive quadratic and inverse quadratic functions of the position is considered.
The dependence on the particle energy and the factor $mathfraka$ describing a relative strength of its constituents is described.
arXiv Detail & Related papers (2023-04-13T11:50:51Z) - Adaptive Stochastic Variance Reduction for Non-convex Finite-Sum
Minimization [52.25843977506935]
We propose an adaptive variance method, called AdaSpider, for $L$-smooth, non-reduction functions with a finitesum structure.
In doing so, we are able to compute an $epsilon-stationary point with $tildeOleft + st/epsilon calls.
arXiv Detail & Related papers (2022-11-03T14:41:46Z) - Unified Fourier-based Kernel and Nonlinearity Design for Equivariant
Networks on Homogeneous Spaces [52.424621227687894]
We introduce a unified framework for group equivariant networks on homogeneous spaces.
We take advantage of the sparsity of Fourier coefficients of the lifted feature fields.
We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation.
arXiv Detail & Related papers (2022-06-16T17:59:01Z) - Feature Cross Search via Submodular Optimization [58.15569071608769]
We study feature cross search as a fundamental primitive in feature engineering.
We show that there exists a simple greedy $(1-1/e)$-approximation algorithm for this problem.
arXiv Detail & Related papers (2021-07-05T16:58:31Z) - Machine Learning and Variational Algorithms for Lattice Field Theory [1.198562319289569]
In lattice quantum field theory studies, parameters defining the lattice theory must be tuned toward criticality to access continuum physics.
We introduce an approach to "deform" Monte Carlo estimators based on contour deformations applied to the domain of the path integral.
We demonstrate that flow-based MCMC can mitigate critical slowing down and observifolds can exponentially reduce variance in proof-of-principle applications.
arXiv Detail & Related papers (2021-06-03T16:37:05Z) - Moreau-Yosida $f$-divergences [0.0]
Variational representations of $f$-divergences are central to many machine learning algorithms.
We generalize the so-called tight variational representation of $f$-divergences in the case of probability measures on compact metric spaces.
We provide an implementation of the variational formulas for the Kullback-Leibler, reverse Kullback-Leibler, $chi2$, reverse $chi2$, squared Hellinger, Jensen-Shannon, Jeffreys, triangular discrimination and total variation divergences.
arXiv Detail & Related papers (2021-02-26T11:46:10Z) - Estimating 2-Sinkhorn Divergence between Gaussian Processes from
Finite-Dimensional Marginals [4.416484585765028]
We study the convergence of estimating the 2-Sinkhorn divergence between emphGaussian processes (GPs) using their finite-dimensional marginal distributions.
We show almost sure convergence of the divergence when the marginals are sampled according to some base measure.
arXiv Detail & Related papers (2021-02-05T16:17:55Z) - Debiased Sinkhorn barycenters [110.79706180350507]
Entropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning.
We show how this bias is tightly linked to the reference measure that defines the entropy regularizer.
We propose debiased Wasserstein barycenters that preserve the best of both worlds: fast Sinkhorn-like iterations without entropy smoothing.
arXiv Detail & Related papers (2020-06-03T23:06:02Z)
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.