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.
There are mainly two types of point-based convolution. The first type is to combine the information of points directly. These methods can be formulated as where @math is a neighborhood of @math and kernel @math takes different forms in different methods. PointNet @cite_1 is an early attempt to extract features on point cloud. PointNet is a network structure without convolution, or alternatively we can interpret the convolution defined by PointNet has the simplest kernel @math where @math is the Kronecker-Delta. Various later works attempt to improve PointNet by choosing different forms of the kernel @math . For example, PointNet++ @cite_23 introduces a max pooling among local points, i.e. choosing kernel @math as an indicator function: @math . Pointcnn @cite_19 chooses @math where @math and @math are trainable variables with @math . DGCNN @cite_7 proposes an "edge convolution" that can be viewed as fixing @math and @math MLP @math , where MLP means the Multi-Layer Perceptron.
@cite_19 @cite_1 @cite_7 @cite_23
2902302021 2560609797 2785053089 2963121255
We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a so-called Fourier integral operator (FIO) of the form @math at points given on a Cartesian grid. Here, @math is a frequency variable, @math is the Fourier transform of the input @math , @math is an amplitude, and @math is a phase function, which is typically as large as @math ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size @math by @math would require @math operations. This paper develops a new numerical algorithm which requires @math operations and as low as @math in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel @math into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology. Unlike on images, semantic learning on 3D point clouds using a deep network is challenging due to the naturally unordered data structure. Among existing works, PointNet has achieved promising results by directly learning on point sets. However, it does not take full advantage of a point's local neighborhood that contains fine-grained structural information which turns out to be helpful towards better semantic learning. In this regard, we present two new operations to improve PointNet with a more efficient exploitation of local structures. The first one focuses on local 3D geometric structures. In analogy to a convolution kernel for images, we define a point-set kernel as a set of learnable 3D points that jointly respond to a set of neighboring data points according to their geometric affinities measured by kernel correlation, adapted from a similar technique for point cloud registration. The second one exploits local high-dimensional feature structures by recursive feature aggregation on a nearest-neighbor-graph computed from 3D positions. Experiments show that our network can efficiently capture local information and robustly achieve better performances on major datasets. Our code is available at this http URL Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math Unlike on images, semantic learning on 3D point clouds using a deep network is challenging due to the naturally unordered data structure. Among existing works, PointNet has achieved promising results by directly learning on point sets. However, it does not take full advantage of a point's local neighborhood that contains fine-grained structural information which turns out to be helpful towards better semantic learning. In this regard, we present two new operations to improve PointNet with a more efficient exploitation of local structures. The first one focuses on local 3D geometric structures. In analogy to a convolution kernel for images, we define a point-set kernel as a set of learnable 3D points that jointly respond to a set of neighboring data points according to their geometric affinities measured by kernel correlation, adapted from a similar technique for point cloud registration. The second one exploits local high-dimensional feature structures by recursive feature aggregation on a nearest-neighbor-graph computed from 3D positions. Experiments show that our network can efficiently capture local information and robustly achieve better performances on major datasets. Our code is available at http: www.merl.com research license#KCNet We introduce a kernel approximation strategy that enables computation of the Gaussian process log marginal likelihood and all hyperparameter derivatives in @math time. Our GRIEF kernel consists of @math eigenfunctions found using a Nystrom approximation from a dense Cartesian product grid of inducing points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor products, computational complexity of the training procedure can be practically independent of the number of inducing points. This allows us to use arbitrarily many inducing points to achieve a globally accurate kernel approximation, even in high-dimensional problems. The fast likelihood evaluation enables type-I or II Bayesian inference on large-scale datasets. We benchmark our algorithms on real-world problems with up to two-million training points and @math inducing points. Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. We consider the problem of learning an unknown function @math on the @math -dimensional sphere with respect to the square loss, given i.i.d. samples @math where @math is a feature vector uniformly distributed on the sphere and @math . We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: (RF) The random feature model of Rahimi-Recht; (NT) The neural tangent kernel model of Jacot-Gabriel-Hongler. Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and hence enjoy universal approximation properties when the number of neurons @math diverges, for a fixed dimension @math . We prove that, if both @math and @math are large, the behavior of these models is instead remarkably simpler. If @math , then RF performs no better than linear regression with respect to the raw features @math , and NT performs no better than linear regression with respect to degree-one and two monomials in the @math . More generally, if @math then RF fits at most a degree- @math polynomial in the raw features, and NT fits at most a degree- @math polynomial. Low-rank approximation is a common tool used to accelerate kernel methods: the @math kernel matrix @math is approximated via a rank- @math matrix @math which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error @math -rank approximation to @math is at least as difficult as multiplying the input data matrix @math by an arbitrary matrix @math . Barring a breakthrough in fast matrix multiplication, when @math is not too large, this requires @math time where @math is the number of non-zeros in @math . This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16,MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to @math input sparsity runtimes. At the same time there is hope: we show for the first time that @math time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . The Euclidean @math -means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of @math points in Euclidean space @math , and the goal is to choose @math centers in @math so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general @math and a @math -approximation which runs in time @math . At the other extreme, the only known computational complexity result for this problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in @math can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all @math . In this paper we provide the first hardness of approximation for the Euclidean @math -means problem. Concretely, we show that there exists a constant @math such that it is NP-hard to approximate the @math -means objective to within a factor of @math . We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform @math , a known hard vertex cover instance, by taking a graph product with a suitably chosen graph @math , and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest. Let us consider the differential equation @math with an @math from @math to @math and suppose that there exists a transformation @math from @math to @math ( @math ) such that @math obeys a differential equation @math with some function @math ; then the first equation is said to be lumpable to the second by @math . Here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction. We provided a series of necessary and sufficient conditions for the existence of such functions @math and @math ; some of them are formulated in terms of @math and @math only. Beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point ... This paper investigates a zero-sum game played on a weighted connected graph @math between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree @math and the edge player chooses an edge @math . The payoff to the edge player is @math , defined as follows: If @math lies in the tree @math then @math ; if @math does not lie in the tree then @math , where @math is the weight of edge @math and @math is the weight of the unique cycle formed when edge @math is added to the tree @math . The main result is that the value of the game on any @math -vertex graph is bounded above by @math . It is conjectured that the value of the game is @math . The game arises in connection with the @math -server problem on a road network; i.e., a metric space that can be represented as a multigraph @math in which each edge @math represents a road of length @math . It is shown that, if the value of the game on @math is @math , then there is a randomized strategy that achieves a competitive ratio of @math against any oblivious adversary. Thus, on any @math -vertex road network, there is a randomized algorithm for the @math -server problem that is @math competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any @math -vertex weighted, connected multigraph, the algorithm constructs a spanning tree @math such that the average, over all edges @math , of @math is less than or equal to @math . This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph. Many real-world problems face the dilemma of choosing best @math out of @math options at a given time instant. This setup can be modelled as combinatorial bandit which chooses @math out of @math arms at each time, with an aim to achieve an efficient tradeoff between exploration and exploitation. This is the first work for combinatorial bandit where the reward received can be a non-linear function of the chosen @math arms. The direct use of multi-armed bandit requires choosing among @math -choose- @math options making the state space large. In this paper, we present a novel algorithm which is computationally efficient and the storage is linear in @math . The proposed algorithm is a divide-and-conquer based strategy, that we call CMAB-SM. Further, the proposed algorithm achieves a regret bound of @math for a time horizon @math , which is sub-linear in all parameters @math , @math , and @math . The evaluation results on different reward functions and arm distribution functions show significantly improved performance as compared to standard multi-armed bandit approach with @math choices. This paper presents a number of new methods for visual tracking using the output of an event-based asynchronous neuromorphic dynamic vision sensor. It allows the tracking of multiple visual features in real time, achieving an update rate of several hundred kilohertz on a standard desktop PC. The approach has been specially adapted to take advantage of the event-driven properties of these sensors by combining both spatial and temporal correlations of events in an asynchronous iterative framework. Various kernels, such as Gaussian, Gabor, combinations of Gabor functions, and arbitrary user-defined kernels, are used to track features from incoming events. The trackers described in this paper are capable of handling variations in position, scale, and orientation through the use of multiple pools of trackers. This approach avoids the @math operations per event associated with conventional kernel-based convolution operations with @math kernels. The tracking performance was evaluated experimentally for each type of kernel in order to demonstrate the robustness of the proposed solution. State-of-the-art semantic image segmentation methods are mostly based on training deep convolutional neural networks (CNNs). In this work, we proffer to improve semantic segmentation with the use of contextual information. In particular, we explore patch-patch' context and patch-background' context in deep CNNs. We formulate deep structured models by combining CNNs and Conditional Random Fields (CRFs) for learning the patch-patch context between image regions. Specifically, we formulate CNN-based pairwise potential functions to capture semantic correlations between neighboring patches. Efficient piecewise training of the proposed deep structured model is then applied in order to avoid repeated expensive CRF inference during the course of back propagation. For capturing the patch-background context, we show that a network design with traditional multi-scale image inputs and sliding pyramid pooling is very effective for improving performance. We perform comprehensive evaluation of the proposed method. We achieve new state-of-the-art performance on a number of challenging semantic segmentation datasets including @math , @math - @math , @math , @math - @math , @math - @math , @math - @math , and @math datasets. Particularly, we report an intersection-over-union score of @math on the @math - @math dataset. This paper explores the complexity of deep feedforward networks with linear pre-synaptic couplings and rectified linear activations. This is a contribution to the growing body of work contrasting the representational power of deep and shallow network architectures. In particular, we offer a framework for comparing deep and shallow models that belong to the family of piecewise linear functions based on computational geometry. We look at a deep rectifier multi-layer perceptron (MLP) with linear outputs units and compare it with a single layer version of the model. In the asymptotic regime, when the number of inputs stays constant, if the shallow model has @math hidden units and @math inputs, then the number of linear regions is @math . For a @math layer model with @math hidden units on each layer it is @math . The number @math grows faster than @math when @math tends to infinity or when @math tends to infinity and @math . Additionally, even when @math is small, if we restrict @math to be @math , we can show that a deep model has considerably more linear regions that a shallow one. We consider this as a first step towards understanding the complexity of these models and specifically towards providing suitable mathematical tools for future analysis. We study the aggregate group nearest neighbor searching for the MAX operator in the plane. For a set @math of @math points and a query set @math of @math points, the query asks for a point of @math whose maximum distance to the points in @math is minimized. We present data structures for answering such queries for both @math and @math distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the @math version, we build a data structure of O(n) size in @math time, such that each query can be answered in @math time. For the @math version, we build a data structure in @math time and @math space, such that each query can be answered in @math time, and alternatively, we build a data structure in @math time and space for any @math , such that each query can be answered in @math time. Further, we extend our result for the @math version to the top- @math queries where each query asks for the @math points of @math whose maximum distances to @math are the smallest for any @math with @math : We build a data structure of O(n) size in @math time, such that each top- @math query can be answered in @math time. Interpolators---estimators that achieve zero training error---have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum @math norm ( ridgeless') interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors @math are obtained by applying a linear transform to a vector of i.i.d. entries, @math (with @math ); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, @math (with @math , @math a matrix of i.i.d. entries, and @math an activation function acting componentwise on @math ). We recover---in a precise quantitative way---several phenomena that have been observed in large-scale neural networks and kernel machines, including the double descent' behavior of the prediction risk, and the potential benefits of overparametrization. A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the @math -dimensional setting, however, it requires the inversion of an @math covariance matrix, @math , as well as the evaluation of its determinant, @math . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form @math , where @math is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix @math is typically dense, causing standard direct methods for inversion and determinant evaluation to require @math work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix @math can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an @math algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant @math , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining @math . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.