Equivariant Embedding Research Articles

The G-invariant graph Laplacian part II: Diffusion maps

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

Equivariant embedding of finite-dimensional dynamical systems

AbstractWe prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

Multiplicative Higgs bundles and involutions

In this paper we generalize the theory of multiplicative G-Higgs bundles over a curve to pairs (G,θ), where G is a reductive algebraic group and θ is an involution of G. This generalization involves the notion of a multiplicative Higgs bundle taking values in a symmetric variety associated to θ, or in an equivariant embedding of it. We also study how these objects appear as fixed points of involutions of the moduli space of multiplicative G-Higgs bundles, induced by the involution θ.

Normal Completions of Toric Varieties Over Rank One Valuation Rings and Completions of Γ-Admissible Fans

Abstract We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma $ is discrete or divisible then no base-change is needed. We give explicit examples that show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma $-admissible fans admit $\Gamma $-admissible completions. In order to show this we prove a combinatorial analog of Noetherian reduction, which we believe will be of independent interest.

Embedding Higson compactification into a product of adelic solenoids

The Higson compactification of any simply connected proper geodesic metric space admits an embedding into a product of adelic solenoids that induces an isomorphism of 1-dimensional cohomology.The Higson compactification of the universal covering Y‾ of a finite complex Y supplied with a lifted from Y metric admits a π1(Y)-equivariant embedding into a product of adelic solenoids that induces an epimorphism of 1-dimensional cohomology.

Extrinsic Bayesian Optimization on Manifolds

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.

Force-field-enhanced neural network interactions: from local equivariant embedding to atom-in-molecule properties and long-range effects.

We introduce FENNIX (Force-Field-Enhanced Neural Network InteraXions), a hybrid approach between machine-learning and force-fields. We leverage state-of-the-art equivariant neural networks to predict local energy contributions and multiple atom-in-molecule properties that are then used as geometry-dependent parameters for physically-motivated energy terms which account for long-range electrostatics and dispersion. Using high-accuracy ab initio data (small organic molecules/dimers), we trained a first version of the model. Exhibiting accurate gas-phase energy predictions, FENNIX is transferable to the condensed phase. It is able to produce stable Molecular Dynamics simulations, including nuclear quantum effects, for water predicting accurate liquid properties. The extrapolating power of the hybrid physically-driven machine learning FENNIX approach is exemplified by computing: (i) the solvated alanine dipeptide free energy landscape; (ii) the reactive dissociation of small molecules.

Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

Equivariant embeddings of manifolds into Euclidean spaces

Suppose a finite group G acts on a manifold M. By a theorem of Mostow, also Palais, there is a G-equivariant embedding of M into the m-dimensional Euclidean space Rm for some m. We are interested in some explicit bounds of such m.First we provide an upper bound: there exists a G-equivariant embedding of M into Rd|G|+1, where |G| is the order of G and M embeds into Rd. Next we provide a lower bound for finite cyclic group action G: If there are l points having pairwise co-prime lengths of G-orbits greater than 1 and there is a G-equivariant embedding of M into Rm, then m≥2l.Some applications to surfaces are given.

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design.

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Szegő kernels and equivariant embedding theorems for CR manifolds

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an $\mathbb R$-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasakian manifolds. As applications of our results, we obtain Torus equivariant Kodaira and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant Kodaira embedding theorem for complex manifolds.

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $\Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincaré disk with $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces into a tube domain $\Omega^\prime \subset \Omega$ and derive a contradiction by means of the Poincaré–Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z \subset \Omega$. More precisely, if $\check{\Gamma} \subset \operatorname{Aut} (\Omega)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z / \check{\Gamma}$ is compact, we prove that $Z \subset \Omega$ is totally geodesic. In particular, letting $\Gamma \subset \operatorname{Aut} (\Omega)$ be a torsion-free cocompact lattice, and $\pi : \Omega \to \Omega / \Gamma =: X_\Gamma$ be the uniformization map, a subvariety $Y \subset X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $\pi-(Y)$ is an algebraic subset of $\Omega$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_\Gamma$ by means of bi-algebraicity without recourse to the celebrated monodromy result of André–Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of André–Deligne we exploit the existence theorem of Aubin and Yau on Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ and make use of Nadel’s semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

Equivariant completions of affine spaces

Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства $\mathbb{C}^n$ в полные алгебраические многообразия $X$, для которых действие векторной группы $\mathbb{G}_a^n$ на $\mathbb{C}^n$ параллельными переносами продолжается до действия $\mathbb{G}_a^n$ на $X$. В первой части работы мы подробно изучаем соответствие Хассетта-Чинкеля, описывающее эквивариантные вложения $\mathbb{C}^n$ в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические многообразия и в многообразия Фано определeнных типов. Библиография: 109 названий.

Smooth Nonprojective Equivariant Completions of Affine Space

In this paper we construct an equivariant embedding of the affine space $\mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n \geq 3$. The theory of toric varieties is used as the main tool for this construction. In the case of $n = 3$ we describe the orbit structure on the variety $X$.

The Minimal Dimension of a Sphere with an Equivariant Embedding of the Bouquet of g Circles is $$2g-1$$

To embed the bouquet of $g$ circles $B_g$ into the $n$-sphere $S^n$ so that its full symmetry group action extends to an orthogonal actions on $S^n$, the minimal $n$ is $2g-1$. This answers a question raised by B. Zimmermann.

Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal G rtimes S^1-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

Equivariant Kodaira Embedding for CR Manifolds with Circle Action

We consider a compact CR manifold with a transversal CR locally free circle action endowed with an S1-equivariant positive CR line bundle. We prove that a certain weighted Fourier–Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

On the deformation rigidity of smooth projective symmetric varieties with Picard number one

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.

Effective quasimorphisms on right-angled Artin groups

We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they “see” all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24. These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings.