We consider the problem of localizing wireless devices in an ad-hoc network embedded in a d-dimensional Euclidean space. Obtaining a good estimation of where wireless devices are located is crucial in wireless network applications including environment monitoring, geographic routing and topology control. When the positions of the devices are unknown and only local distance information is given, we need to infer the positions from these local distance measurements. This problem is particularly challenging when we only have access to measurements that have limited accuracy and are incomplete. We consider the extreme case of this limitation on the available information, namely only the connectivity information is available, i.e., we only know whether a pair of nodes is within a fixed detection range of each other or not, and no information is known about how far apart they are. Further, to account for detection failures, we assume that even if a pair of devices is within the detection range, it fails to detect the presence of one another with some probability and this probability of failure depends on how far apart those devices are. Given this limited information, we investigate the performance of a centralized positioning algorithm MDS-MAP introduced by , and a distributed positioning algorithm, introduced by , called HOP-TERRAIN. In particular, for a network consisting of n devices positioned randomly, we provide a bound on the resulting error for both algorithms. We show that the error is bounded, decreasing at a rate that is proportional to R Rc, where Rc is the critical detection range when the resulting random network starts to be connected, and R is the detection range of each device.
The localization problem and its variants has attracted significant research interest in recent years. A general survey of the area and an overview of recent techniques can be found in @cite_3 and @cite_1 , respectively. The problem is also closely related to dimensionality reduction @cite_31 and manifold learning @cite_23 in which the objects data come from a high dimensional space, and the goal is to compute a low-dimensional, neighbourhood preserving embeddings.
@cite_23 @cite_31 @cite_1 @cite_3
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The problem of dimensionality reduction arises in many fields of information processing, including machine learning, data compression, scientific visualization, pattern recognition, and neural computation. Here we describe locally linear embedding (LLE), an unsupervised learning algorithm that computes low dimensional, neighborhood preserving embeddings of high dimensional data. The data, assumed to be sampled from an underlying manifold, are mapped into a single global coordinate system of lower dimensionality. The mapping is derived from the symmetries of locally linear reconstructions, and the actual computation of the embedding reduces to a sparse eigenvalue problem. Notably, the optimizations in LLE---though capable of generating highly nonlinear embeddings---are simple to implement, and they do not involve local minima. In this paper, we describe the implementation of the algorithm in detail and discuss several extensions that enhance its performance. We present results of the algorithm applied to data sampled from known manifolds, as well as to collections of images of faces, lips, and handwritten digits. These examples are used to provide extensive illustrations of the algorithm's performance---both successes and failures---and to relate the algorithm to previous and ongoing work in nonlinear dimensionality reduction. Previous efforts in hashing intend to preserve data variance or pairwise affinity, but neither is adequate in capturing the manifold structures hidden in most visual data. In this paper, we tackle this problem by reconstructing the locally linear structures of manifolds in the binary Hamming space, which can be learned by locality-sensitive sparse coding. We cast the problem as a joint minimization of reconstruction error and quantization loss, and show that, despite its NP-hardness, a local optimum can be obtained efficiently via alternative optimization. Our method distinguishes itself from existing methods in its remarkable ability to extract the nearest neighbors of the query from the same manifold, instead of from the ambient space. On extensive experiments on various image benchmarks, our results improve previous state-of-the-art by 28-74 typically, and 627 on the Yale face data. Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, that is, how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low- dimensional space. Experiments on synthetic data, as well as real-world images, demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings. We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in two-dimensional three-dimensional (2D 3D) Euclidean spaces and in higher-dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements. Over the past few years, some embedding methods have been proposed for feature extraction and dimensionality reduction in various machine learning and pattern classification tasks. Among the methods proposed are Neighborhood Preserving Embedding (NPE), Locality Preserving Projection (LPP) and Local Discriminant Embedding (LDE) which have been used in such applications as face recognition and image video retrieval. However, although the data in these applications are more naturally represented as higher-order tensors, the embedding methods can only work with vectorized data representations which may not capture well some useful information in the original data. Moreover, high-dimensional vectorized representations also suffer from the curse of dimensionality and the high computational demand. In this paper, we propose some novel tensor embedding methods which, unlike previous methods, take data directly in the form of tensors of arbitrary order as input. These methods allow the relationships between dimensions of a tensor representation to be efficiently characterized. Moreover, they also allow the intrinsic local geometric and topological properties of the manifold embedded in a tensor space to be naturally estimated. Furthermore, they do not suffer from the curse of dimensionality and the high computational demand. We demonstrate the effectiveness of the proposed tensor embedding methods on a face recognition application and compare them with some previous methods. Extensive experiments show that our methods are not only more effective but also more efficient. In this paper, we focus on tackling the problem of automatic accurate localization of detected objects in high-resolution remote sensing images. The two major problems for object localization in remote sensing images caused by the complex context information such images contain are achieving generalizability of the features used to describe objects and achieving accurate object locations. To address these challenges, we propose a new object localization framework, which can be divided into three processes: region proposal, classification, and accurate object localization process. First, a region proposal method is used to generate candidate regions with the aim of detecting all objects of interest within these images. Then, generic image features from a local image corresponding to each region proposal are extracted by a combination model of 2-D reduction convolutional neural networks (CNNs). Finally, to improve the location accuracy, we propose an unsupervised score-based bounding box regression (USB-BBR) algorithm, combined with a nonmaximum suppression algorithm to optimize the bounding boxes of regions that detected as objects. Experiments show that the dimension-reduction model performs better than the retrained and fine-tuned models and the detection precision of the combined CNN model is much higher than that of any single model. Also our proposed USB-BBR algorithm can more accurately locate objects within an image. Compared with traditional features extraction methods, such as elliptic Fourier transform-based histogram of oriented gradients and local binary pattern histogram Fourier, our proposed localization framework shows robustness when dealing with different complex backgrounds. Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text. How do we judge similarity? Our mental representations of the world are formed by processing large numbers of sensory in Different aspects of the curse of dimensionality are known to present serious challenges to various machine-learning methods and tasks. This paper explores a new aspect of the dimensionality curse, referred to as hubness, that affects the distribution of k-occurrences: the number of times a point appears among the k nearest neighbors of other points in a data set. Through theoretical and empirical analysis involving synthetic and real data sets we show that under commonly used assumptions this distribution becomes considerably skewed as dimensionality increases, causing the emergence of hubs, that is, points with very high k-occurrences which effectively represent "popular" nearest neighbors. We examine the origins of this phenomenon, showing that it is an inherent property of data distributions in high-dimensional vector space, discuss its interaction with dimensionality reduction, and explore its influence on a wide range of machine-learning tasks directly or indirectly based on measuring distances, belonging to supervised, semi-supervised, and unsupervised learning families. Local features have proven very useful for recognition. Manifold learning has proven to be a very powerful tool in data analysis. However, manifold learning application for images are mainly based on holistic vectorized representations of images. The challenging question that we address in this paper is how can we learn image manifolds from a punch of local features in a smooth way that captures the feature similarity and spatial arrangement variability between images. We introduce a novel framework for learning a manifold representation from collections of local features in images. We first show how we can learn a feature embedding representation that preserves both the local appearance similarity as well as the spatial structure of the features. We also show how we can embed features from a new image by introducing a solution for the out-of-sample that is suitable for this context. By solving these two problems and defining a proper distance measure in the feature embedding space, we can reach an image manifold embedding space. We address dimensionality estimation and nonlinear manifold inference starting from point inputs in high dimensional spaces using tensor voting. The proposed method operates locally in neighborhoods and does not involve any global computations. It is based on information propagation among neighboring points implemented as a voting process. Unlike other local approaches for manifold learning, the quantity propagated from one point to another is not a scalar, but is in the form of a tensor that provides considerably richer information. The accumulation of votes at each point provides a reliable estimate of local dimensionality, as well as of the orientation of a potential manifold going through the point. Reliable dimensionality estimation at the point level is a major advantage over competing methods. Moreover, the absence of global operations allows us to process significantly larger datasets. We demonstrate the effectiveness of our method on a variety of challenging datasets. Local features have been widely used in computer vision tasks, e.g., human action recognition, but it tends to be an extremely challenging task to deal with large-scale local features of high dimensionality with redundant information. In this paper, we propose a novel fully supervised local descriptor learning algorithm called discriminative embedding method based on the image-to-class distance (I2CDDE) to learn compact but highly discriminative local feature descriptors for more accurate and efficient action recognition. By leveraging the advantages of the I2C distance, the proposed I2CDDE incorporates class labels to enable fully supervised learning of local feature descriptors, which achieves highly discriminative but compact local descriptors. The objective of our I2CDDE is to minimize the I2C distances from samples to their corresponding classes while maximizing the I2C distances to the other classes in the low-dimensional space. To further improve the performance, we propose incorporating a manifold regularization based on the graph Laplacian into the objective function, which can enhance the smoothness of the embedding by extracting the local intrinsic geometrical structure. The proposed I2CDDE for the first time achieves fully supervised learning of local feature descriptors. It significantly improves the performance of I2C-based methods by increasing the discriminative ability of local features while greatly reducing the computational burden by dimensionality reduction to handle large-scale data. We apply the proposed I2CDDE algorithm to human action recognition on four widely used benchmark datasets. The results have shown that I2CDDE can significantly improve I2C-based classifiers and achieves state-of-the-art performance. Abstract This paper mainly focuses on dimensional reduction of fused dataset of holistic and geometrical face features vectors by solving singularity problem of linear discriminant analysis and maximizing the Fisher ratio in nonlinear subspace region with the preservation of local discriminative features. The combinational feature vector space is projected into low dimensional subspace using proposed Kernel Locality Preserving Symmetrical Weighted Fisher Discriminant Analysis (KLSWFDA) method. Matching score level fusion technique has been applied on projected subspace and combinational entire Gabor subspace is framed. Euclidean distance metric (L2) and support vector machine (SVM) classifier has been implemented to recognize and classify the expressions. Performance of proposed approach is evaluated and compared with state of art approaches. Experimental results on JAFFE, YALE and FD expression database demonstrate the effectiveness of the proposed approach. We consider the problem of positioning a cloud of points in the Euclidean space ℝ d , using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. It is also closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using local (or partial) metric information. Here we propose a reconstruction algorithm based on semidefinite programming. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm’s performance: in the noiseless case, we find a radius r 0 beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension d, and the average degree of the nodes in the graph. Low-dimensional embedding, manifold learning, clustering, classification, and anomaly detection are among the most important problems in machine learning. The existing methods usually consider the case when each instance has a fixed, finite-dimensional feature representation. Here we consider a different setting. We assume that each instance corresponds to a continuous probability distribution. These distributions are unknown, but we are given some i.i.d. samples from each distribution. Our goal is to estimate the distances between these distributions and use these distances to perform low-dimensional embedding, clustering classification, or anomaly detection for the distributions. We present estimation algorithms, describe how to apply them for machine learning tasks on distributions, and show empirical results on synthetic data, real word images, and astronomical data sets. We study the problem of discovering a manifold that best preserves information relevant to a nonlinear regression. Solving this problem involves extending and uniting two threads of research. On the one hand, the literature on sufficient dimension reduction has focused on methods for finding the best linear subspace for nonlinear regression; we extend this to manifolds. On the other hand, the literature on manifold learning has focused on unsupervised dimensionality reduction; we extend this to the supervised setting. Our approach to solving the problem involves combining the machinery of kernel dimension reduction with Laplacian eigenmaps. Specifically, we optimize cross-covariance operators in kernel feature spaces that are induced by the normalized graph Laplacian. The result is a highly flexible method in which no strong assumptions are made on the regression function or on the distribution of the covariates. We illustrate our methodology on the analysis of global temperature data and image manifolds. Robust visual localization under a wide range of viewing conditions is a fundamental problem in computer vision. Handling the difficult cases of this problem is not only very challenging but also of high practical relevance, e.g., in the context of life-long localization for augmented reality or autonomous robots. In this paper, we propose a novel approach based on a joint 3D geometric and semantic understanding of the world, enabling it to succeed under conditions where previous approaches failed. Our method leverages a novel generative model for descriptor learning, trained on semantic scene completion as an auxiliary task. The resulting 3D descriptors are robust to missing observations by encoding high-level 3D geometric and semantic information. Experiments on several challenging large-scale localization datasets demonstrate reliable localization under extreme viewpoint, illumination, and geometry changes. Reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high-dimensional data analysis. Fisher discriminant analysis (FDA) is a traditional technique for supervised dimensionality reduction, but it tends to give undesired results if samples in a class are multimodal. An unsupervised dimensionality reduction method called locality-preserving projection (LPP) can work well with multimodal data due to its locality preserving property. However, since LPP does not take the label information into account, it is not necessarily useful in supervised learning scenarios. In this paper, we propose a new linear supervised dimensionality reduction method called local Fisher discriminant analysis (LFDA), which effectively combines the ideas of FDA and LPP. LFDA has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem. We demonstrate the practical usefulness and high scalability of the LFDA method in data visualization and classification tasks through extensive simulation studies. We also show that LFDA can be extended to non-linear dimensionality reduction scenarios by applying the kernel trick. Robust and accurate visual localization across large appearance variations due to changes in time of day, seasons, or changes of the environment is a challenging problem which is of importance to application areas such as navigation of autonomous robots. Traditional feature-based methods often struggle in these conditions due to the significant number of erroneous matches between the image and the 3D model. In this paper, we present a method for scoring the individual correspondences by exploiting semantic information about the query image and the scene. In this way, erroneous correspondences tend to get a low semantic consistency score, whereas correct correspondences tend to get a high score. By incorporating this information in a standard localization pipeline, we show that the localization performance can be significantly improved compared to the state-of-the-art, as evaluated on two challenging long-term localization benchmarks. We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the spl lscr sub 1 -norm. A number of recent theoretical results on sparsifying properties of spl lscr sub 1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame spl acute r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization. This paper introduces self-taught object localization, a novel approach that leverages deep convolutional networks trained for whole-image recognition to localize objects in images without additional human supervision, i.e., without using any ground-truth bounding boxes for training. The key idea is to analyze the change in the recognition scores when artificially masking out different regions of the image. The masking out of a region that includes the object typically causes a significant drop in recognition score. This idea is embedded into an agglomerative clustering technique that generates self-taught localization hypotheses. Our object localization scheme outperforms existing proposal methods in both precision and recall for small number of subwindow proposals (e.g., on ILSVRC-2012 it produces a relative gain of 23.4 over the state-of-the-art for top-1 hypothesis). Furthermore, our experiments show that the annotations automatically-generated by our method can be used to train object detectors yielding recognition results remarkably close to those obtained by training on manually-annotated bounding boxes.