A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.
problem How many values can a non-constant slice regular function of a quaternionic variable avoid?
method Investigates slice regular functions of quaternionic variables, extending the classical Picard theorem.
result A non-constant slice regular function of a quaternionic variable can avoid at most one value, similar to the classical Picard theorem.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
Study shows surfaces with many twistor lines can't be odd-degree.
problem Characterizing algebraic surfaces with infinitely many twistor lines.
method Utilized quaternionic slice regularity and normalization map theory.
result Constructive existence of even-degree surfaces with infinitely many twistor lines.
In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{ge…
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains Ω of R^4. When Ω is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which Ω is the complement of a parabola is st…
Smooth manifold structure on Möbius transformations of quaternionic ball identified.
problem Identifying the manifold structure of Möbius transformations of quaternionic unit ball.
method Realizing M(B) as a quotient of Sp(1,1) and using Lie group properties. result The manifold M(B) is diffeomorphic to R4imesS3. We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on S1×S7, we study their group of automorphisms and their deformations.
Regular sliceness implies once-stably decomposable sliceness in symplectizations.
problem Relationship between regular and decomposable Lagrangian cobordisms in symplectizations.
method Stabilization-free strategy and satellite operations.
result Regular sliceness implies once-stably decomposable sliceness.
Study slice-regular polynomial functions via twistor space group actions.
problem Characterize slice-regular functions and their polynomial subclasses.
method Employ the twistor construction and group actions of PGL(2,H). result Characterize slice-regular functions with planar twistor lifts and normal classes of polynomials.
Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
Study of zero-divisors in sedenions via determinant factorization.
problem Characterizing zero-divisors in the sedenion algebra.
method Factorization of determinant of left multiplication, reduction to quaternionic normal form, block computation.
result Quartic polynomial factorization of determinant, geometric model of zero-divisor locus.
Paper tackles regularization and sparsification for quaternion neural networks.
problem Regularizing and sparsifying quaternion neural networks for compactness and real-time applications.
method Developed targeted regularization strategies for quaternion neural networks, extending l1 and structured regularization.
result Tailored strategies significantly reduce the number of connections and neurons, resulting in smaller, more compact networks.
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Given a slice regular function f:Ω⊂H→H, with Ω∩R=∅, it is possible to lift it to a surface in the twistor space CP3 of S4≃H∪{∞} (see~\cite{gensalsto}). In this paper we show that the same result is true if one rem…
Sliced-regularized OT improves transport plan accuracy.
problem Optimal transport (OT) approximation accuracy.
method Sliced-regularized optimal transport (SROT) formulation.
result SROT yields more accurate approximations of exact OT than entropic OT.
In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate m-dimensional Delzant polytopes, we obtain manifolds of real dimension 4m, acted on by m copies of the group Sp(1) of unit quaternions. Th…
New autoencoder improves latent space learning by optimizing sliced Gromov-Wasserstein discrepancies.
problem Improving inner discrepancy between prior and posterior distributions in autoencoders.
method Proposed spherical sliced fused Gromov Wasserstein (SSFG) and variants (MSSFG, PSSFG) to find important directions.
result New autoencoders achieve favorable performance in latent manifold learning, image generation, and reconstruction.
The paper examines the topology of quaternionic toric actions on manifolds.
problem Understanding the global topology of manifolds with quaternionic toric actions.
method Established toric, differential, and tetraplectic foundations. Constructed spectral sequences for the orbit projection to describe cohomology and K-theory.
result Explicit descriptions of cohomology and K-theory for manifolds with quaternionic toric actions, extending complex toric topology.
The paper finds transformation formulas for quaternionic complex structures.
problem Quaternionic projective invariance of k-Cauchy-Fueter complex. method Explicit transformation formulae under mSL(n+1,H). result Quaternionic projectively invariant operator and defining density.
We prove that every projective special Kähler manifold with \emph{regular boundary behaviour} is complete and defines a family of complete quaternionic Kähler manifolds depending on a parameter c≥0. We also show that, irrespective of its boundary behaviour, every complete projective special Kähler manifold with \e…
The paper develops quaternionic toric geometry and classifies local actions.
problem Classifying local quaternionic torus actions on manifolds.
method Develops local Qn-actions, introduces invariants, and studies tetraplectic structures. result Classifies local quaternionic torus actions up to homeomorphism.
The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fibre…
A new method classifies color images using quaternion algebra.
problem Classifying color images with preserved intrinsic relationships.
method LSQMM model with quaternion nuclear norm regularization and ADMM algorithm.
result LSQMM outperforms state-of-the-art methods in classification accuracy and efficiency.
The study proves the regularity of inverse mean curvature flow in specific geometric settings.
problem Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds.
method Utilizing the behavior of Hawking masses, the study shows star-shaped slices after a long time.
result The weak solution of inverse mean curvature flow becomes regular over time.
This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve …
Paper explores properties of slice-matching operators for measure transfer.
problem Efficiently transferring measures in high dimensions.
method Examines an associated slice-matching operator with source, target measures and slicing directions.
result Establishes invariance, equivariance, Lipschitz continuity, and error bounds.
Extended elliptical slice sampling for infinite-dimensional spaces, proving reversibility.
problem Proving reversibility of elliptical slice sampling in infinite-dimensional spaces.
method Extended elliptical slice sampling to infinite-dimensional separable Hilbert spaces, providing an alternative proof of reversibility.
result The approach yields a positive semi-definite Markov operator, proving reversibility.
Gromov-Hausdorff convergence of time-slices of singular Ricci flows
problem Gromov-Hausdorff convergence of time-slices of singular Ricci flows
method Completion of singular Ricci flow with respect to a natural spacetime distance
result Gromov-Hausdorff convergence at the first singular time
Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.
problem Understanding positive scalar curvature metrics on manifolds with boundary.
method Minimal slicing via capillary hypersurfaces to prove rigidity statements.
result Proves rigidity statement in dimension 4 for specific geometric conditions.
Elliptical slice sampling converges geometrically, providing reliable sampling for Bayesian learning.
problem Sampling from posterior distributions in Bayesian learning.
method Elliptical slice sampling, geometric ergodicity.
result Elliptical slice sampling yields geometric convergence guarantees under weak regularity assumptions.
In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar cu…
A new distance measure balances projection exploration and informativeness.
problem Inefficient and incomplete projection sampling in existing sliced-Wasserstein distances.
method Proposes Distributional Sliced-Wasserstein (DSW) that optimally balances projection exploration and informativeness.
result DSW generalizes Max-SW and can be computed efficiently.
A new tree-sliced Wasserstein distance improves optimal transport computations.
problem Computational and statistical drawbacks in optimal transport.
method Introducing tree metrics and averaging Wasserstein distances using random tree metrics.
result Tree-sliced Wasserstein distance outperforms other methods on benchmarks.
Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent orbits inside sl_{2m}, and intersections of those with regular semisimple orbits. Th…
Study of quaternion-Kähler structures via harmonic flow on 8-manifolds.
problem Geometric flow of quaternion-Kähler structures on 8-manifolds.
method Formulated gradient Dirichlet flow, analyzed harmonicity, proved almost-monotonicity.
result Proved long-time existence and constructed harmonic solitons.
We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de-Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geome…
Quaternionic analysis proves minimum of Willmore functional on Riemann surfaces.
problem Finding minimum of Willmore functional on Riemann surfaces.
method Extending quaternionic analysis to weakly conformal maps and using Darboux transformation.
result Minimum of Willmore functional on Riemann surfaces is achieved by weakly conformal maps.
Recently, the principal component pursuit has received increasing attention in signal processing research ranging from source separation to video surveillance. So far, all existing formulations are real-valued and lack the concept of phase, which is inherent in inputs such as complex spectrograms or color images. Thus,…
In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.
New bounds improve neural network generalization through slicing.
problem Difficulty in evaluating mutual information in high dimensions for neural networks.
method Slicing the parameter space and using disintegrated mutual information and k-sliced mutual information.
result Slicing improves generalization and offers significant computational and statistical advantages.
In this paper we study generative modeling via autoencoders while using the elegant geometric properties of the optimal transport (OT) problem and the Wasserstein distances. We introduce Sliced-Wasserstein Autoencoders (SWAE), which are generative models that enable one to shape the distribution of the latent space int…
Study evaluates using multiple slices as input for CNNs in medical image segmentation.
problem Improving segmentation performance in medical images with limited computational resources.
method Compared pseudo-3D and 2D approaches using different CNN architectures and datasets.
result Multi-slice inputs did not significantly improve segmentation performance over 2D or 3D CNNs.
Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…
Study on confidence intervals for Sliced Wasserstein distance, with optimal risk bounds.
problem Statistical inference for the Sliced Wasserstein distance.
method Construct minimax confidence intervals with adaptive lengths.
result Minimax optimal confidence intervals for the Sliced Wasserstein distance.
We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…
The study finds conditions for quaternionic structures on symmetric spaces.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.