Domain Manifold Research Articles

A Schwarz lemma of harmonic maps into metric spaces

<p>We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.</p>

Cross-subject EEG emotion recognition using multi-source domain manifold feature selection

Recent researches on emotion recognition suggests that domain adaptation, a form of transfer learning, has the capability to solve the cross-subject problem in Affective brain-computer interface (aBCI) field. However, traditional domain adaptation methods perform single to single domain transfer or simply merge different source domains into a larger domain to realize the transfer of knowledge, resulting in negative transfer. In this study, a multi-source transfer learning framework was proposed to promote the performance of multi-source electroencephalogram (EEG) emotion recognition. The method first used the data distribution similarity ranking (DDSA) method to select the appropriate source domain for each target domain off-line, and reduced data drift between domains through manifold feature mapping on Grassmann manifold. Meanwhile, the minimum redundancy maximum correlation algorithm (mRMR) was employed to select more representative manifold features and minimized the conditional distribution and marginal distribution of the manifold features, and then learned the domain-invariant classifier by summarizing structural risk minimization (SRM). Finally, the weighted fusion criterion was applied to further improve recognition performance. We compared our method with several state-of-the-art domain adaptation techniques using the SEED and DEAP dataset. Results showed that, compared with the conventional MEDA algorithm, the recognition accuracy of our proposed algorithm on SEED and DEAP dataset were improved by 6.74% and 5.34%, respectively. Besides, compared with TCA, JDA, and other state-of-the-art algorithms, the performance of our proposed method was also improved with the best average accuracy of 86.59% on SEED and 64.40% on DEAP. Our results demonstrated that the proposed multi-source transfer learning framework is more effective and feasible than other state-of-the-art methods in recognizing different emotions by solving the cross-subject problem.

Achieving domain generalization for underwater object detection by domain mixup and contrastive learning

The performance of existing underwater object detection methods severely degrades when they face the domain shift caused by complicated underwater environments. Due to the limited domain diversity in collected data, deep detectors easily memorize a few seen domains, which leads to low generalization ability. There are two common ideas to improve the domain generalization performance. First, it can be inferred that the detector trained on as many domains as possible is domain-invariant. Second, their hidden features should be equivalent because the images with the same semantic content are in different domains. This paper further excavates these two ideas and proposes a domain generalization framework that learns how to generalize across domains from Domain Mixup and Contrastive Learning (DMCL). First, based on the formation of underwater images, an image in one kind of underwater environment is the linear transformation of another underwater environment. Therefore, a style transfer model, which outputs a linear transformation matrix instead of the whole image, is proposed to transform images from one source domain to another, enriching the domain diversity of the training data. Second, the Mixup operation interpolates different domains on the feature level, sampling new domains on the domain manifold. Third, a contrastive loss is selectively applied to features from different domains to force the model to learn domain-invariant features but retain the discriminative capacity. With our method, detectors will be robust to domain shift. Also, a domain generalization benchmark S-UODAC2020 for detection is set up to measure the performance of our method. Comprehensive experiments on S-UODAC2020 and two object recognition benchmarks (PACS and VLCS) demonstrate that the proposed method is able to learn domain-invariant representations and outperforms other domain generalization methods. The code is available inhttps://github.com/mousecpn/DMC-Domain-Generalization-for-Underwater-Object-Detection.git

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

Biharmonic maps and biharmonic submanifolds with small curvature integral

In this article, we study biharmonic maps and biharmonic submanifolds with small curvature integral. Let φ:(Mn,g)→(Nm,h) be a biharmonic map from a complete noncompact Riemannian manifold (Mn,g) into a Riemannian manifold (Nm,h) satisfying that the L2-norm of the tension field of the map is finite. If the domain manifold of the map satisfies a Sobolev inequality and the Ln2-norm of the sectional curvature on the image φ(M) is sufficiently small, then we are able to prove the harmonicity of the biharmonic map. It turns out that the fundamental tone of M is sufficiently large, then such a biharmonic map φ must be harmonic. In case where the map is an isometric immersion, we prove that if M satisfies a Sobolev inequality, then M must be minimal under the assumption that the Ln2-norm of the Ricci curvature on M is sufficiently small. Moreover it is shown that if the fundamental tone of a biharmonic submanifold is sufficiently big, then it is minimal.

Computing harmonic maps between Riemannian manifolds

AbstractIn our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.

Differentiating densities on smooth manifolds

Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. We highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.

A note on harmonic maps of statistical manifolds

We show an existence and uniqueness result for a class of maps from a flat statistical manifold into a Riemannian manifold in a given homotopy class when the target Riemannian manifold is of negative sectional curvature under a global topological non-triviality condition. We also show that due to dualistic structure of the domain manifold the result is still valid in dual coordinates.

Coupled local–global adaptation for multi-source transfer learning

This paper presents a novel unsupervised multi-source domain adaptation approach, named as coupled local–global adaptation (CLGA). At the global level, in order to maximize the adaptation ability, CLGA regards multiple domains as a unity, and jointly mitigates the gaps of both marginal and conditional distributions between source and target dataset. At the local level, with the intention of maximizing the discriminative ability, CLGA investigates the relationship among distinctive domains, and exploits both class and domain manifold structures embedded in data samples. We formulate both local and global adaptation in a concise optimization problem, and further derive an analytic solution for the objective function. Extensive evaluations verify that CLGA performs better than several existing methods not only in multi-source adaptation tasks but also in single source scenarios.

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.

Smooth long-time existence of Harmonic Ricci Flow on surfaces

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.

Adaptively refined multi-patch B-splines with enhanced smoothness

A spline space suitable for Isogeometric Analysis (IgA) on multi-patch domains is presented. Our construction is motivated by emerging requirements in isogeometric simulations. In particular, IgA spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution, even across interfaces of adjacent patches.Given a domain manifold M consisting of individual patches (isomorphic to the unit square or cube) that are glued together along interfaces, we present a construction of multi-patch B-splines defined on them. Their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch. The resulting multi-patch B-splines with enhanced smoothness (MPBES) possess the property of local linear independence and form a non-negative partition of unity. Moreover, their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties.Subsequently, adaptively refined MPBES are obtained by generalizing the construction of truncated hierarchical (TH) B-splines. More precisely, a nested sequence of spaces spanned by MPBES is considered, corresponding to steps of local enrichment. In addition, an inversely nested sequence of subdomains (which are submanifolds of M) is used to specify the local refinement level of functions in these spaces. Finally, truncated hierarchical MPBES are obtained by means of the selection and truncation mechanism of THB-splines. The desired properties of linear independence and convex partition of unity are maintained.The paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis.

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

We study the new geometric flow that was introduced in the paper [12] of Topping and the author that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper of Topping and the author [12] where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.

Schrödinger soliton from Lorentzian manifolds

In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension ( k − 1 ) n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.

On the homotopy class of maps with finite p-energy into non-positively curved manifolds

We prove that a map f : M → N with finite p-energy, p > 2, from a complete manifold \({\left(M,\left\langle ,\right\rangle \right)}\) into a non-positively curved, compact manifold N is homotopic to a constant, provided the negative part of the Ricci curvature of the domain manifold is small in a suitable spectral sense. The result relies on a Liouville-type theorem for finite q-energy, p-harmonic maps under spectral assumptions.

Geometry-aware domain decomposition for T-spline-based manifold modeling

This paper presents a new and effective method to construct manifold T-splines of complicated topology/geometry. The fundamental idea of our novel approach is the geometry-aware object segmentation, by which an arbitrarily complicated surface model can be decomposed into a group of disjoint components that comprise branches, handles, and base patches. Such a domain decomposition simplifies objects of arbitrary topological type into a family of genus-zero/one open surfaces, each of which can be conformally parameterized into a set of rectangles. In contrast to the conventional decomposition approaches, our method can guarantee that the cutting locus are consistent on the parametric domain. As a result, the resultant T-splines of decomposed components are automatically glued and have high-order continuity everywhere except at the extraordinary points. We show that the number of extraordinary points of the domain manifold is bounded by the number of segmented components. Furthermore, the entire mesh-to-spline data conversion pipeline can be implemented with full automation, and thus, has potential in shape modeling and reverse engineering applications of complicated real-world objects.

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincare inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.

Topology of Sobolev mappings IV

We will classify the path connected components of spaces of Sobolev maps between manifolds and study the strong and weak density of smooth maps in the spaces of Sobolev maps for the case the domain manifold has nonempty boundary and Dirichlet problems.

Solving variational problems and partial differential equations mapping into general target manifolds

A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(2) (2001) 759]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system where this embedding function is defined. In the case of variational problems, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we re-write all the equation’s geometric characteristics with respect to the embedding function. We then obtain a set of equations that, while defined on the whole Euclidean space, are intrinsic to the implicitly defined target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional hypersurfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is a hypersphere; probability maps, where the target manifold is a hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional manifolds.