Convolution plays a crucial role in various applications in signal and image processing, analysis and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processing point cloud data. However, one of the major challenges is to define a proper way to "sweep" filters through the point cloud as a natural generalization of the planar convolution and to reflect the point cloud's geometry at the same time. In this paper, we consider generalizing convolution by adapting parallel transport on the point cloud. Inspired by a triangulated surface based method [Stefan C. Schonsheck, Bin Dong, and Rongjie Lai, arXiv:1805.07857.], we propose the Narrow-Band Parallel Transport Convolution (NPTC) using a specifically defined connection on a voxelized narrow-band approximation of point cloud data. With that, we further propose a deep convolutional neural network based on NPTC (called NPTC-net) for point cloud classification and segmentation. Comprehensive experiments show that the proposed NPTC-net achieves similar or better results than current state-of-the-art methods on point clouds classification and segmentation.
The newly proposed PTC @cite_15 defines convolution directly on the manifold, while uses tangent planes to transport kernels by a properly chosen parallel transport. PTC can be equivalently cast into the form of ) using the inverse exponential map, and implementation of the proposed parallel transported is realized through choosing specific vector fields @math guided by a Eikonal equation for transforming vectors along geodesic curves on manifolds.
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Deep generative models learn a mapping from a low-dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. Abstract Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation. The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with Log-Euclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low efficiency especially when the dimensionality of SPD matrix is high. To overcome this limitation, we propose a novel metric learning approach to work directly on logarithms of SPD matrices. Specifically, our method aims to learn a tangent map that can directly transform the matrix logarithms from the original tangent space to a new tangent space of more discriminability. Under the tangent map framework, the novel metric learning can then be formulated as an optimization problem of seeking a Mahalanobis-like matrix, which can take the advantage of traditional metric learning techniques. Extensive evaluations on several image set classification tasks demonstrate the effectiveness of our proposed metric learning method. A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair @math of points of M there exists a trajectory of S from m to @math . Here a trajectory of S is a curve which is an integral curve of some @math or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class @math , @math , or @math , there exists a completely controllable set S consisting of two vector fields of class @math . We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form A new architecture, denoted spatial pyramid matching on the semantic manifold (SPMSM), is proposed for scene recognition. SPMSM is based on a recent image representation on a semantic probability simplex, which is now augmented with a rough encoding of spatial information. A connection between the semantic simplex and a Riemmanian manifold is established, so as to equip the architecture with a similarity measure that respects the manifold structure of the semantic space. It is then argued that the closed-form geodesic distance between two manifold points is a natural measure of similarity between images. This leads to a conditionally positive definite kernel that can be used with any SVM classifier. An approximation of the geodesic distance reveals connections to the well-known Bhattacharyya kernel, and is explored to derive an explicit feature embedding for this kernel, by simple square-rooting. This enables a low-complexity SVM implementation, using a linear SVM on the embedded features. Several experiments are reported, comparing SPMSM to state-of-the-art recognition methods. SPMSM is shown to achieve the best recognition rates in the literature for two large datasets (MIT Indoor and SUN) and rates equivalent or superior to the state-of-the-art on a number of smaller datasets. In all cases, the resulting SVM also has much smaller dimensionality and requires much fewer support vectors than previous classifiers. This guarantees much smaller complexity and suggests improved generalization beyond the datasets considered. By using optimal mass transportation and a quantitative Holder inequality, we provide estimates for the Borell–Brascamp–Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell–Brascamp–Lieb inequalities (including Brunn–Minkowski and Prekopa–Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge–Ampere equation, we give a new proof of Dubuc's characterization of the equality in Borell–Brascamp–Lieb inequalities in the Euclidean setting. When the n-dimensional Riemannian manifold has Ricci curvature Ric(M) ≥ (n-1)k for some k ⋲ ℝ, it turns out that equality in the Borell–Brascamp–Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k. A precise characterization is provided for the equality in the Lott–Sturm–Villani-type distorted Brunn–Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented. Analyzing data representing multifarious trajectories is central to the many fields in Science and Engineering; for example, trajectories representing a tennis serve, a gymnast’s parallel bar routine, progression remission of disease and so on. We present a novel geometric algorithm for performing statistical analysis of trajectories with distinct number of samples representing longitudinal (or temporal) data. A key feature of our proposal is that unlike existing schemes, our model is deployable in regimes where each participant provides a different number of acquisitions (trajectories have different number of sample points or temporal span). To achieve this, we develop a novel method involving the parallel transport of the tangent vectors along each given trajectory to the starting point of the respective trajectories and then use the span of the matrix whose columns consist of these vectors, to construct a linear subspace in Rm. We then map these linear subspaces (possibly of distinct dimensions) of Rm on to a single high dimensional hypersphere. This enables computing group statistics over trajectories by instead performing statistics on the hypersphere (equipped with a simpler geometry). Given a point on the hypersphere representing a trajectory, we also provide a “reverse mapping” algorithm to uniquely (under certain assumptions) reconstruct the subspace that corresponds to this point. Finally, by using existing algorithms for recursive Fr´echet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few principal components. We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance τ(s), for fixed physical distance s, thus becomes (1 + τ(s) a)− a, with standard exponential decay recovered when a → ∞. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized transport model are stated in integral form for d = 1, 2, 3 spatial dimensions. A deterministic numerical solution is developed in d = 1 using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when a = ∞. Differential diffusion equations are not presently known when a < ∞, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in d = 2, a... This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation. We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out. This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called A-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of P-trajectories and A-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and A-trajectories. It shows that projective transformations map P-trajectories into P-trajectories. It presents Karmarkar's interpretation of A-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, x2 restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of A-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that P-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The A-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947 89 @math .25 per page We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative. A convenient way of analysing Riemannian manifolds is to embed them in Euclidean spaces, with the embedding typically obtained by flattening the manifold via tangent spaces. This general approach is not free of drawbacks. For example, only distances between points to the tangent pole are equal to true geodesic distances. This is restrictive and may lead to inaccurate modelling. Instead of using tangent spaces, we propose embedding into the Reproducing Kernel Hilbert Space by introducing a Riemannian pseudo kernel. We furthermore propose to recast a locality preserving projection technique from Euclidean spaces to Riemannian manifolds, in order to demonstrate the benefits of the embedding. Experiments on several visual classification tasks (gesture recognition, person re-identification and texture classification) show that in comparison to tangent-based processing and state-of-the-art methods (such as tensor canonical correlation analysis), the proposed approach obtains considerable improvements in discrimination accuracy. Interpolation between pairs of values, typically vectors, is a fundamental operation in many computer graphics applications. In some cases simple linear interpolation yields meaningful results without requiring domain knowledge. However, interpolation between pairs of distributions or pairs of functions often demands more care because features may exhibit translational motion between exemplars. This property is not captured by linear interpolation. This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending. The functions can be non-uniformly sampled, high-dimensional, and defined on non-Euclidean manifolds, e.g., spheres and tori. Our method decomposes distributions or functions into sums of radial basis functions (RBFs). We solve a mass transport problem to pair the RBFs and apply partial transport to obtain the interpolated function. We describe practical methods for computing the RBF decomposition and solving the transport problem. We demonstrate the interpolation approach on synthetic examples, BRDFs, color distributions, environment maps, stipple patterns, and value functions. Geometric methods for construction of codes in the Grassmann manifolds are presented. The methods follow the geometric approach to space-time coding for the non-coherent MIMO channel where the code design is interpreted as a packing problem on Grassmann manifolds. The differential structure of the Grassmann manifold provides parametrization with the tangent space at the identity element. Grassmann codes for the non-coherent channel are constructed by mapping suitable subsets of lattices from the tangent space to the Grassmann manifold via the exponential map. As examples, constructions from the rotated Gosset, Barnes-Wall and Leech lattice are presented. Due to the specifics of the mapping, some of the structure is preserved after the mapping to the manifold. The method is further improved by modifying the mapping from the tangent space to the manifold. Ideas for other constructions of Grassmann codes are also presented and discussed. Common human actions are instantly recognizable by people and increasingly machines need to understand this language if they are to engage smoothly with people. Here we introduce a new method for automated human action recognition. The proposed method represents videos as a tangent bundle on a Grassmann manifold. Videos are expressed as third order tensors and factorized to a set of tangent spaces. Tangent vectors are then computed between elements on a Grassmann manifold and exploited for action classification. In particular, logarithmic mapping is applied to map a point from the manifold to tangent vectors centered at a given element. The canonical metric is used to induce the intrinsic distance for a set of tangent spaces. Empirical results show that our method is effective on both uniform and non-uniform backgrounds for action classification. We achieve recognition rates of 91 on the Cambridge gesture dataset, 88 on the UCF sport dataset, and 97 on the KTH human action dataset. Additionally, our method does not require prior training. Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approximate the optimal transport plan by a pushforward of a reference distribution, and cast the optimal transport problem into a minimax problem. We then can solve OT problems efficiently using primal dual stochastic gradient-type algorithms. We also show that we can recover the density of the optimal transport plan using neural ordinary differential equations. Numerical experiments on both synthetic and real datasets illustrate that SPOT is robust and has favorable convergence behavior. SPOT also allows us to efficiently sample from the optimal transport plan, which benefits downstream applications such as domain adaptation. We present a general method of studying the transport process ( X(t) ) , t≥0, in the Euclidean space ℝ m , m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of ( X(t) ) are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of ( X(t) ) in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of ( X(t) ) in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given. In computer vision applications, features often lie on Riemannian manifolds with known geometry. Popular learning algorithms such as discriminant analysis, partial least squares, support vector machines, etc., are not directly applicable to such features due to the non-Euclidean nature of the underlying spaces. Hence, classification is often performed in an extrinsic manner by mapping the manifolds to Euclidean spaces using kernels. However, for kernel based approaches, poor choice of kernel often results in reduced performance. In this paper, we address the issue of kernel selection for the classification of features that lie on Riemannian manifolds using the kernel learning approach. We propose two criteria for jointly learning the kernel and the classifier using a single optimization problem. Specifically, for the SVM classifier, we formulate the problem of learning a good kernel-classifier combination as a convex optimization problem and solve it efficiently following the multiple kernel learning approach. Experimental results on image set-based classification and activity recognition clearly demonstrate the superiority of the proposed approach over existing methods for classification of manifold features.