Search arXiv⌕ Search

SEARCH · Search arXiv

Results for “math.DG”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

600 recordsLinked to original sources
arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗
arXiv2026

The Discrete Harmonic Center of a Quadrilateral

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

math.DG↗
arXiv2026

Accelerate Vector Diffusion Maps by Landmarks

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

stat.ML↗
arXiv2026

Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements

Determining the minimal number of intensity measurements required for phase retrieval in $\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].

quant-ph↗
arXiv2026

Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses

Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.

math.NA↗
arXiv2026

Cultural Bias Without a Cultural Self:A Disassociation Study of LLM's Persona and Bias

Language models prompted with cultural personas increasingly stand in for human respondents in cross-cultural research. Their responses separate personas cleanly, and that separation is read as evidence of a cultural point of view. We show that the separation is real, that the point of view is not, and that one criterion tells them apart. A trait is structure internal to one respondent that survives a change of measurement frame; a bias needs only group-specific item means. To test for the first, we represent a single response set as an Item--Dimension matrix and treat its correlation matrix as a point on the manifold of symmetric positive definite matrices. In humans this carries what a trait should: it reproduces across test--retest sessions sharing no items, order or context ($r=0.77$, $N=89$); on public NEO-PI-R data it identifies individuals at up to $76\%$ against a $0.4\%$ chance level ($N=263$); and it predicts GPA ($R^2=0.281$, $p=0.003$) where BigFive aggregates from the same responses predict nothing ($R^2=0.018$). In four frontier LLMs it returns nothing. Persona structure is readable only while every instance shares one item order: give each its own order and separation falls from $94.7\%$ to chance, while realigning instances to \emph{any} shared random order restores it to $82$--$84\%$. Responses generated independently item by item, with no latent structure, reproduce the entire pattern. The cultural signal is a group template, not a property of any instance, and alignment regimes differ only in which stereotype survives on the surface.

stat.ML↗
arXiv2026

Denoising-Aware Temporal Point Cloud Completion for 3D Crop Architecture Recovery and Phenotypic Trait Extraction

High-throughput phenotyping depends on accurate 3D reconstruction of plants across growth stages, yet the development and evaluation of temporal completion methods are limited by the lack of datasets with complete geometric ground truth. To address this challenge, we introduce SynthCrop4D, a procedurally generated synthetic dataset of temporally evolving plant point clouds that provides controllable noise, occlusion, and complete plant geometry for benchmarking reconstruction methods. Using this dataset, we evaluate a two-stage pipeline that combines spatial denoising and temporal point cloud completion. First, a denoising module removes structural artifacts from raw laser-scanned point clouds. The resulting data are then processed by an Adaptive Temporal PoinTr model that reconstructs the current growth stage (t) using information from the previous stage (t-1), enabling recovery of regions missing due to self-occlusion. We evaluate the proposed framework on both SynthCrop4D and the real-world Pheno4D dataset (tomato and maize) under settings with and without denoising. Results show that denoising substantially improves reconstruction quality, with the best configuration achieving a Chamfer Distance of 0.0061 on SynthCrop4D (Temporal PoinTr + Mamba-DG) and an F-Score of 0.2080 on Pheno4D (Vanilla PoinTr + Mamba-DG). We further demonstrate the use of completed point clouds for phenotypic trait extraction, including plant height, canopy width, and convex hull volume, obtaining hull-volume MAEs of 0.021 on synthetic data and 0.343 on real data. Together, SynthCrop4D and the proposed pipeline provide a benchmark and methodology for temporal plant reconstruction and high-throughput crop phenotyping.

cs.CV↗
arXiv2026

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA↗
arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗
arXiv2026

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR↗
arXiv2026

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO↗
arXiv2026

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR↗
arXiv2026

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph↗
arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗
arXiv2026

Difference equations of average entropies

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

math-ph↗
arXiv2026

A Complete Characterization of Tensorizable $f$-divergences

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

cs.IT↗
arXiv2026

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn↗