Geometric Structure Of Data Research Articles

Nonlinear Endmember Identification for Hyperspectral Imagery via Hyperpath-Based Simplex Growing and Fuzzy Assessment

Nonlinear geometric manifold of hyperspectral data usually makes great trouble for accurate endmember extraction in literature. To address this issue, we propose a novel nonlinear endmember extraction algorithm by building a hypergraph and a fuzzy assessment strategy. The global change of nonlinear data manifold is first measured in a hypergraph whose hyperedges correspond to different local pixel subgroups. In contrast to edges in a simple graph, every hyperpath connected by multiple hyperedges instead of individual pixels effectively facilitates the determination of the simplex spanned by endmembers on complex data manifolds interfered by noises and outliers. Furthermore, in the hypergraph-based manifold system, a reliable fuzzy assessment mechanism for extracting the final endmembers is established by combining the classic simplex volume maximization rule with the inherent properties of hyperspectral data and hypergraph. In this procedure, the impact of outliers and the complex geometric structures of data can be further effectively reduced, leading to better unmixing results. Experimental results of various types of simulated data and real hyperspectral imagery demonstrate that the proposed algorithm (hypergraph and fuzzy-assessment-based nonlinear endmember extraction) can extract more accurate endmembers from complex data manifolds, and it is more robust to noises and outliers compared with state-of-the-art algorithms.

Graph Moving Object Segmentation

Moving Object Segmentation (MOS) is a fundamental task in computer vision. Due to undesirable variations in the background scene, MOS becomes very challenging for static and moving camera sequences. Several deep learning methods have been proposed for MOS with impressive performance. However, these methods show performance degradation in the presence of unseen videos; and usually, deep learning models require large amounts of data to avoid overfitting. Recently, graph learning has attracted significant attention in many computer vision applications since they provide tools to exploit the geometrical structure of data. In this work, concepts of graph signal processing are introduced for MOS. First, we propose a new algorithm that is composed of segmentation, background initialization, graph construction, unseen sampling, and a semi-supervised learning method inspired by the theory of recovery of graph signals. Second, theoretical developments are introduced, showing one bound for the sample complexity in semi-supervised learning, and two bounds for the condition number of the Sobolev norm. Our algorithm has the advantage of requiring less labeled data than deep learning methods while having competitive results on both static and moving camera videos. Our algorithm is also adapted for Video Object Segmentation (VOS) tasks and is evaluated on six publicly available datasets outperforming several state-of-the-art methods in challenging conditions.

Learning-Based Hole Detection in 3D Point Cloud Towards Hole Filling

Abstract In this paper, we propose a learning-based approach for automatic detection of hole boundary points in a 3D point cloud. Point cloud is an important geometric data structure used in 3D modelling. Data obtained from automatic acquisition techniques often result in geometric deficiencies such as holes in the 3D point cloud. For successful hole-filling to achieve better surface reconstruction, accurate detection of hole boundary points in the point cloud is necessary. Most of the existing methods use threshold values for different parameters which need to be set manually after analyzing the nature of point cloud. It becomes difficult to generalize the threshold values for a wide variety of point clouds. To overcome this limitation, we propose a deep learning framework to detect holes in the point cloud. The architecture directly consumes point cloud and considers the permutation invariance of points. Each point in the point cloud is classified as a hole boundary point or not. The detected hole boundary points are used for hole filling by fitting a surface and interpolating points on the surface.

Adaptive Hypergraph Embedded Semi-Supervised Multi-Label Image Annotation

Multilabel image annotation attracts a lot of research interest due to its practicability in multimedia and computer vision fields, while the need for a large amount of labeled training data to achieve promising performance makes it a challenging task. Fortunately, unlabeled and relevant data are widely available and these data can be used to serve the annotation task. To this end, we propose a novel adaptive hypergraph learning (AHL) method for multilabel image annotation in a semisupervised way, in which both the limited labeled data and abundant unlabeled data are utilized to facilitate the annotation performance. In detail, we seek a multilabel propagation scheme by learning a hypergraph which is used to preserve the local geometric structures of data in a high-order manner. Meanwhile, a feature projection is integrated into AHL to obtain a latent feature space where unlabeled instances can be effectively and robustly assigned with multiple labels. Experiments on six widely used image datasets are conducted to evaluate our model and the results demonstrate that the proposed AHL outperforms other state-of-the-art semisupervised methods.

Unsupervised feature selection via adaptive hypergraph regularized latent representation learning

Due to the rapid development of multimedia technology, a large number of unlabelled data with high dimensionality need to be processed. The high dimensionality of data not only increases the computation burden of computer hardware, but also hinders algorithms to obtain optimal performance. Unsupervised feature selection, which is regarded as a means of dimensionality reduction, has been widely recognized as an important and challenging pre-step for many machine learning and data mining tasks. However, we observe that there are at least two issues in previous unsupervised feature selection methods. Firstly, traditional unsupervised feature selection algorithms usually assume that the data instances are identically distributed and there is no dependency between them. However, the data instances are not only associated with high dimensional features but also inherently interconnected with each other. Secondly, the traditional similarity graph used in previous methods can only describe the pair-wise relations of data, but cannot capture the high-order relations, so that the complex structures implied in the data cannot be sufficiently exploited. In this work, we propose a robust unsupervised feature selection method which embeds the latent representation learning into feature selection. Instead of measuring the feature importances in original data space, the feature selection is carried out in the learned latent representation space which is more robust to noises. In order to capture the local manifold geometrical structure of original data in a high-order manner, a hypergraph is adaptively learned and embedded into the resultant model. An efficient alternating algorithm is developed to optimize the problem. Experimental results on eight benchmark data sets demonstrate the effectiveness of the proposed method.

Bidirectional Discrete Matrix Factorization Hashing for Image Search.

Unsupervised image hashing has recently gained significant momentum due to the scarcity of reliable supervision knowledge, such as class labels and pairwise relationship. Previous unsupervised methods heavily rely on constructing sufficiently large affinity matrix for exploring the geometric structure of data. Nevertheless, due to lack of adequately preserving the intrinsic information of original visual data, satisfactory performance can hardly be achieved. In this article, we propose a novel approach, called bidirectional discrete matrix factorization hashing (BDMFH), which alternates two mutually promoted processes of 1) learning binary codes from data and 2) recovering data from the binary codes. In particular, we design the inverse factorization model, which enforces the learned binary codes inheriting intrinsic structure from the original visual data. Moreover, we develop an efficient discrete optimization algorithm for the proposed BDMFH. Comprehensive experimental results on three large-scale benchmark datasets show that the proposed BDMFH not only significantly outperforms the state-of-the-arts but also provides the satisfactory computational efficiency.

Semi-Supervised Graph Regularized Deep NMF With Bi-Orthogonal Constraints for Data Representation.

Semi-supervised non-negative matrix factorization (NMF) exploits the strengths of NMF in effectively learning local information contained in data and is also able to achieve effective learning when only a small fraction of data is labeled. NMF is particularly useful for dimensionality reduction of high-dimensional data. However, the mapping between the low-dimensional representation, learned by semi-supervised NMF, and the original high-dimensional data contains complex hierarchical and structural information, which is hard to extract by using only single-layer clustering methods. Therefore, in this article, we propose a new deep learning method, called semi-supervised graph regularized deep NMF with bi-orthogonal constraints (SGDNMF). SGDNMF learns a representation from the hidden layers of a deep network for clustering, which contains varied and unknown attributes. Bi-orthogonal constraints on two factor matrices are introduced into our SGDNMF model, which can make the solution unique and improve clustering performance. This improves the effect of dimensionality reduction because it only requires a small fraction of data to be labeled. In addition, SGDNMF incorporates dual-hypergraph Laplacian regularization, which can reinforce high-order relationships in both data and feature spaces and fully retain the intrinsic geometric structure of the original data. This article presents the details of the SGDNMF algorithm, including the objective function and the iterative updating rules. Empirical experiments on four different data sets demonstrate state-of-the-art performance of SGDNMF in comparison with six other prominent algorithms.

A generalized least-squares approach regularized with graph embedding for dimensionality reduction

In current graph embedding methods, low dimensional projections are obtained by preserving either global geometrical structure of data or local geometrical structure of data. In this paper, the PCA (Principal Component Analysis) idea of minimizing least-squares reconstruction errors is regularized with graph embedding, to unify various local manifold embedding methods within a generalized framework to keep global and local low dimensional subspace. Different from the well-known PCA method, our proposed generalized least-squares approach considers data distributions together with an instance penalty in each data point. In this way, PCA is viewed as a special instance of our proposed generalized least squares framework for preserving global projections. Applying a regulation of graph embedding, we can obtain projection that preserves both intrinsic geometrical structure and global structure of data. From the experimental results on a variety of face and handwritten digit recognition, our proposed method has advantage of superior performances in keeping lower dimensional subspaces and higher classification results than state-of-the-art graph embedding methods.

Lossy asymptotic equipartition property for geometric networked data structures

This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation principles for properly defined empirical measures on CGRGs. As a motivation for this article, we apply our results to some data from Wireless Sensor Network for Monitoring Water Quality from a Lake.

Adaptive robust principal component analysis

Robust Principal Component Analysis (RPCA) is a powerful tool in machine learning and data mining problems. However, in many real-world applications, RPCA is unable to well encode the intrinsic geometric structure of data, thereby failing to obtain the lowest rank representation from the corrupted data. To cope with this problem, most existing methods impose the smooth manifold, which is artificially constructed by the original data. This reduces the flexibility of algorithms. Moreover, the graph, which is artificially constructed by the corrupted data, is inexact and does not characterize the true intrinsic structure of real data. To tackle this problem, we propose an adaptive RPCA (ARPCA) to recover the clean data from the high-dimensional corrupted data. Our proposed model is advantageous due to: (1) The graph is adaptively constructed upon the clean data such that the system is more flexible. (2) Our model simultaneously learns both clean data and similarity matrix that determines the construction of graph. (3) The clean data has the lowest-rank structure that enforces to correct the corruptions. Extensive experiments on several datasets illustrate the effectiveness of our model for clustering and low-rank recovery tasks.

The Vector Heat Method

This article describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian . As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method, we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations so that they can be used in the context of tangent vector field processing.

Two Approaches to Building Time-Windowed Geometric Data Structures

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal et al. (in: Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pp 240–254, 2015). In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al. ’s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for both the time-windowed 2D diameter decision problem and the time-windowed 2D convex hull area decision problem in $$O(n \log n)$$ time, improving Bokal et al. ’s $$O(n \log ^2 n)$$ and $$O(n \log n \log \log n)$$ solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first $$O(n\, \mathrm{polylog}\, n)$$ algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

Gene selection for microarray data classification via adaptive hypergraph embedded dictionary learning

Due to the rapid development of DNA microarray technology, a large number of microarray data come into being and classifying these data has been verified useful for cancer diagnosis, treatment and prevention. However, microarray data classification is still a challenging task since there are often a huge number of genes but a small number of samples in gene expression data. As a result, a computational method for reducing the dimension of microarray data is necessary. In this paper, we introduce a computational gene selection model for microarray data classification via adaptive hypergraph embedded dictionary learning (AHEDL). Specifically, a dictionary is learned from the feature space of original high dimensional microarray data, and this learned dictionary is used to represent original genes with a reconstruction coefficient matrix. Then we use a l2, 1-norm regularization to impose the row sparsity on the coefficient matrix for selecting discriminate genes. Meanwhile, in order to capture the localmanifold geometrical structure of original microarray data in a high-order manner, a hypergraph is adaptively learned and embedded into the model. An iterative updating algorithm is designed for solving the optimization problem. In order to validate the efficacy of the proposed model, we have conducted experiments on six publicly available microarray data sets and the results demonstrate that AHEDL outperforms other state-of-the-art methods in terms of microarray data classification. AbbreviationsUnlabelled TableAHEDLAdaptive Hypergraph Embedded Dictionary LearningADMMAlternating Direction method of MultipliersSVMSupport Vector MachineRFRandom Forestk-NNk-Nearest NeighborCVcross validationMSVM-RFMulticlass Support Vector Machine-Recursive Feature EliminationKernelPLSKernel Partial Least SquaresWLMGSWeight Local Modularity based Gene SelectionGRSL-GSGene Selection via Subspace Learning and Manifold RegularizationLNNFWLocal-Nearest-Neighbors-based Feature Weighting for Gene SelectionRLRRegularized Logistic RegressionACCaccuracySDstandard deviationsANOVAAnalysis of VarianceDFDegrees of FreedomSSSum-of-SquareMSMean Sum-of-SquareFF-valueSigstatistical significanceSRBCTSmall Round Blue Cell TumorsGCMGlobal Cancer MapCLL_SUB_111B-cell chronic lymphocytic leukemia

Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Given a finite range space $$\Sigma =(\mathsf {X},\mathcal {R})$$, with $$N= |\mathsf {X}| + |\mathcal {R}|$$, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of $$\Sigma $$. The first algorithm is a simpler variant of the Bronnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to $$O(N\mathrm {polylog}(N))$$ expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if $$\mathsf {X}$$ is a set of points and $$\mathcal {R}$$ a set of disks in $$\mathbb {R}^2$$.

Manifold regularized sparse representation of injected details for pansharpening

ABSTRACTPansharpening with sparse representation (SR) and details injection (ID) can both produce visually and quantitatively pleasing images. The former constructs pansharpened image by combining the dictionary and estimated sparse coefficients while the later by sharpening the multispectral bands through adding the proper spatial details from panchromatic (Pan) image. The combination of these two methods has been putting forward as the pansharpening method based on sparse representation of injected details (SR-D). Although SR-D has achieved better results both in visual and quantitative parts than many state-of-art methods, it ignores the intrinsic geometric structure connection between the multispectral image (MS) and the corresponding high-resolution MS image. In this paper, we propose a new pansharpening method, called manifold regularized sparse representation of injected details (MR-SR-D) by introducing a manifold regularization (MR) into the former SR-D model. The manifold regularization utilized a graph Laplacian to incorporate the locally geometrical structure of the multispectral data. Experimental results on the IKONOS, QuickBird and WorldView2 data sets show that the proposed method can achieve remarkable spectral and spatial quality on both reduced scale and full scale.

Graph Regularized Sparse Autoencoders with Nonnegativity Constraints

Unsupervised feature learning with deep networks has been widely studied in recent years. Among these networks, deep autoencoders have shown a decent performance in discovering hidden geometric structure of the original data. Both nonnegativity and graph constraints show the effectiveness in representing intrinsic structures in the high dimensional ambient space. This paper combines the nonnegativity and graph constraints to find the original geometrical information intrinsic to high dimensional data, keeping it in a dimensionality reduced space. In the experiments, we test the proposed networks on several standard image data sets. The results demonstrate that they outperform existing methods.

Signal enrichment with strain-level resolution in metagenomes using topological data analysis

BackgroundA metagenome is a collection of genomes, usually in a micro-environment, and sequencing a metagenomic sample en masse is a powerful means for investigating the community of the constituent microorganisms. One of the challenges is in distinguishing between similar organisms due to rampant multiple possible assignments of sequencing reads, resulting in false positive identifications. We map the problem to a topological data analysis (TDA) framework that extracts information from the geometric structure of data. Here the structure is defined by multi-way relationships between the sequencing reads using a reference database.ResultsBased primarily on the patterns of co-mapping of the reads to multiple organisms in the reference database, we use two models: one a subcomplex of a Barycentric subdivision complex and the other a Čech complex. The Barycentric subcomplex allows a natural mapping of the reads along with their coverage of organisms while the Čech complex takes simply the number of reads into account to map the problem to homology computation. Using simulated genome mixtures we show not just enrichment of signal but also microbe identification with strain-level resolution.ConclusionsIn particular, in the most refractory of cases where alternative algorithms that exploit unique reads (i.e., mapped to unique organisms) fail, we show that the TDA approach continues to show consistent performance. The Čech model that uses less information is equally effective, suggesting that even partial information when augmented with the appropriate structure is quite powerful.

Flexible unsupervised feature extraction for image classification

Dimensionality reduction is one of the fundamental and important topics in the fields of pattern recognition and machine learning. However, most existing dimensionality reduction methods aim to seek a projection matrix W such that the projection WTx is exactly equal to the true low-dimensional representation. In practice, this constraint is too rigid to well capture the geometric structure of data. To tackle this problem, we relax this constraint but use an elastic one on the projection with the aim to reveal the geometric structure of data. Based on this context, we propose an unsupervised dimensionality reduction model named flexible unsupervised feature extraction (FUFE) for image classification. Moreover, we theoretically prove that PCA and LPP, which are two of the most representative unsupervised dimensionality reduction models, are special cases of FUFE, and propose a non-iterative algorithm to solve it. Experiments on five real-world image databases show the effectiveness of the proposed model.

RGB-D action recognition based on discriminative common structure learning model

The emergence of low-cost depth cameras creates potential for RGB-D based human action recognition. However, most of the existing RGB-D based approaches simply concatenate original heterogeneous features without discovering the latent relations among different modalities. We propose a discriminative common structure learning (DCSL) model for human action recognition from RGB-D sequences. Specifically, we extract deep learning-based features and hand-crafted features from multimodal data (skeleton, depth, and RGB). In particular, we propose a deep architecture based on 3-D convolutional neural network to automatically extract deep spatiotemporal features from raw sequences. The proposed DCSL model utilizes a generalized version of collective matrix factorization to learn shared features among different modalities. To perform supervised learning and preserve intermodal similarity, we formulate a graph regularization term by considering both label information and similar geometric structure of multimodal data, which intends to improve the discriminative power of shared features. Moreover, we solve the objective function using an iterative optimization algorithm. Then, an improved collaborative representation classifier is employed to perform computationally efficient action recognition. Experimental results on four action datasets demonstrate the superior performance of the proposed method.

Spatial-spectral neighbour graph for dimensionality reduction of hyperspectral image classification

ABSTRACTRecently, graph embedding-based methods have drawn increasing attention for dimensionality reduction (DR) of hyperspectral image (HSI) classification. Graph construction is a critical step for those DR methods. Pairwise similarity graph is generally employed to reflect the geometric structure in the original data. However, it ignores the similarity of neighbouring pixels. In order to further improve the classification performance, both spectral and spatial-contextual information should be taken into account in HSI classification. In this paper, a novel spatial-spectral neighbour graph (SSNG) is proposed for DR of HSI classification, which consists of the following four steps. First, a superpixel-based segmentation algorithm is adopted to divide HSI into many superpixels. Second, a novel distance metric is utilized to reflect the similarity of two spectral pixels in each superpixel. In the third step, a spatial-spectral neighbour graph is constructed according to the above distance metric. At last, support vector machine with a composite kernel (SVM-CK) is adopted to classify the dimensionality-reduced HSI. Experimental results on three real hyperspectral datasets demonstrate that our method can achieve higher classification accuracy with relatively less consumed time than other graph embedding-based methods.