Showing papers on "Equivariant map published in 2019"

General E(2)-Equivariant Steerable CNNs

Abstract: The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as drop in replacement for non-equivariant convolutions.

Gauge Equivariant Convolutional Networks and the Icosahedral CNN

Abstract: The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.

A General Theory of Equivariant CNNs on Homogeneous Spaces

Abstract: We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also answer a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? We show that such maps correspond one-to-one with generalized convolutions with an equivariant kernel, and characterize the space of such kernels.

Universal Invariant and Equivariant Graph Neural Networks

Abstract: Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the \emph{equivariant} case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Additionally, unlike many previous works that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size.

Gauge Equivariant Convolutional Networks and the Icosahedral CNN

Abstract: The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.

Universal Invariant and Equivariant Graph Neural Networks

Abstract: Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity and either an invariant or equivariant linear operator. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the equivariant case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Finally, unlike many previous settings that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size.

Equivariant Multi-View Networks

Abstract: Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, where view permutation invariance is achieved through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fashion, up to the very last layer. We further develop this idea to operate on smaller discrete homogeneous spaces of the rotation group, where a polar view representation is used to maintain equivariance with only a fraction of the number of input views. We set the new state of the art in several large scale 3D shape retrieval tasks, and show additional applications to panoramic scene classification.

General $E(2)$-Equivariant Steerable CNNs.

Abstract: The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of $E(2)$-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group $E(2)$ and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. $E(2)$-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in replacement for non-equivariant convolutions.

Construction of two-bubble solutions for energy-critical wave equations

Abstract: We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k > 2. The concentrating speed of the second bubble is exponential for the first two models and a power function in the last case.

Equivariant Hamiltonian Flows.

Abstract: This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while providing an equivariant representation of the data. We provide proof of principle demonstrations of how such flows can be learnt, as well as how the addition of symmetry invariance constraints can improve data efficiency and generalisation. Finally, we make connections to disentangled representation learning and show how this work relates to a recently proposed definition.

Universal approximations of permutation invariant/equivariant functions by deep neural networks

Abstract: In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine transformations are $G$-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.

Equivariant Flows: sampling configurations for multi-body systems with symmetric energies

Abstract: Flows are exact-likelihood generative neural networks that transform samples from a simple prior distribution to the samples of the probability distribution of interest. Boltzmann Generators (BG) combine flows and statistical mechanics to sample equilibrium states of strongly interacting many-body systems such as proteins with 1000 atoms. In order to scale and generalize these results, it is essential that the natural symmetries of the probability density - in physics defined by the invariances of the energy function - are built into the flow. Here we develop theoretical tools for constructing such equivariant flows and demonstrate that a BG that is equivariant with respect to rotations and particle permutations can generalize to sampling nontrivially new configurations where a nonequivariant BG cannot.

Quantum geometry of moduli spaces of local systems and representation theory

Abstract: Let G be a split semi-simple adjoint group, and S an oriented surface with punctures and special boundary points. We introduce a moduli space P(G,S) parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S)) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le. We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.

Splitting theorems for Poisson and related structures

Abstract: According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known e.g. for Lie algebroids, Dirac structures and generalized complex structures. In this paper, we develop a novel approach towards these results that leads to various generalizations, including their equivariant versions as well as their formulations in new contexts.

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Abstract: Motivic Chern classes are elements in the K-theory of an algebraic variety $X$ depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory flag manifolds $G/B$. The calculation is recursive starting from the class of a point, and using the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of $G$. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Feh{e}r, Rim{a}nyi and Weber. The Hecke algebra action on the K-theory of the dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the Poincar{e} dual version of the motivic Chern class to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations, and {holomorphy} properties, of the coefficients in the transition matrix between the standard and the Casselman's basis.

Probabilistic symmetries and invariant neural networks.

Abstract: Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures, in an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings. By considering group invariance from the perspective of probabilistic symmetry, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of probability distributions that are invariant or equivariant under the action of a compact group. Our representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We demonstrate that examples from the recent literature are special cases, and develop the details of the general program for exchangeable sequences and arrays.

Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras

Abstract: We introduce the shifted quantum affine algebras. They map homomorphically into the quantized K-theoretic Coulomb branches of \(3d\ {\mathcal N}=4\) SUSY quiver gauge theories. In type A, they are endowed with a coproduct, and they act on the equivariant K-theory of parabolic Laumon spaces. In type A1, they are closely related to the type A open relativistic quantum Toda system.

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Abstract: Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kahler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and Kahler parameters of $X$, respectively, with the Kahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rimanyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

$K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds

Abstract: Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on $\mathbb{C}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a vertex formalism. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi-Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the $K$-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov-Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of $K$-theoretic stable pair invariants of the local resolved conifold. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov-Okounkov-Osinenko.

Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

Abstract: We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-Tao and Vakil.

3d Mirror Symmetry and Elliptic Stable Envelopes

Abstract: We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters on one side with the Kahler parameters on the dual side.

Ring objects in the equivariant derived Satake category arising from Coulomb branches

Abstract: This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in arXiv:1601.03586 work for an arbitrary commutative ring object. The second purpose of this paper is to study Coulomb branches associated with star shaped quivers, which are expected to be conjectural Higgs branches of $3d$ Sicilian theories in type $A$ by arXiv:1007.0992.

Equivariant Transformer Networks.

Abstract: How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transformation groups in several parameters. On a real-world image classification task, ETs improve the sample efficiency of ResNet classifiers, achieving relative improvements in error rate of up to 15% in the limited data regime while increasing model parameter count by less than 1%.

Quaternion Equivariant Capsule Networks for 3D Point Clouds

Abstract: We present a 3D capsule module for processing point clouds that is equivariant to 3D rotations and translations, as well as invariant to permutations of the input points. The operator receives a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end transformation equivariance through a novel dynamic routing procedure on quaternions. Further, we theoretically connect dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving \emph{iterative re-weighted least squares} (IRLS) problems with provable convergence properties. It is shown that such group dynamic routing can be interpreted as robust IRLS rotation averaging on capsule votes, where information is routed based on the final inlier scores. Based on our operator, we build a capsule network that disentangles geometry from pose, paving the way for more informative descriptors and a structured latent space. Our architecture allows joint object classification and orientation estimation without explicit supervision of rotations. We validate our algorithm empirically on common benchmark datasets.

Equivariant Transformer Networks

Abstract: How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transformation groups in several parameters. On a real-world image classification task, ETs improve the sample efficiency of ResNet classifiers, achieving relative improvements in error rate of up to 15% in the limited data regime while increasing model parameter count by less than 1%.

Equivariant Multi-View Networks

Abstract: Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, achieving view permutation invariance through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fashion, up to the very last layer. We further develop this idea to operate on smaller discrete homogeneous spaces of the rotation group, where a polar view representation is used to maintain equivariance with only a fraction of the number of input views. We set the new state of the art in several large scale 3D shape retrieval tasks, and show additional applications to panoramic scene classification.

Cluster theory of the coherent Satake category

Abstract: We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of G. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized r-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding. We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case G = GL_n in detail and prove that the loop rotation equivariant coherent Satake category of GL_n is a monoidal categorification of an explicit quantum cluster algebra. More generally, we construct renormalized r-matrices in any monoidal category whose product is compatible with an auxiliary chiral category, and explain how the appearance of cluster algebras in 4d N=2 field theory may be understood from this point of view.

Rota-Net: Rotation Equivariant Network for Simultaneous Gland and Lumen Segmentation in Colon Histology Images

Abstract: Analysis of the shape of glands and their lumen in digitised images of Haematoxylin & Eosin stained colon histology slides can provide insight into the degree of malignancy. Segmenting each glandular component is an essential prerequisite step for subsequent automatic morphological analysis. Current automated segmentation approaches typically do not take into account the inherent rotational symmetry within histology images. We incorporate this rotational symmetry into an encoder-decoder based network by utilising group equivariant convolutions, specifically using the symmetry group of rotations by multiples of 90\(^\circ \). Our rotation equivariant network splits into two separate branches after the final up-sampling operation, where the output of a given branch achieves either gland or lumen segmentation. In addition, at the output of the gland branch, we use a multi-class strategy to assist with the separation of touching instances. We show that our proposed approach achieves the state-of-the-art performance on the GlaS challenge dataset.

Fast Computation of Tukey Trimmed Regions and Median in Dimension p > 2

Abstract: Given data in Rp , a Tukey κ-trimmed region is the set of all points that have at least Tukey depth κ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful to...