Multiple Manifolds Research Articles

Multiple Riemannian Kernel Hashing for Large-Scale Image Set Classification and Retrieval.

Conventional image set methods typically learn from small to medium-sized image set datasets. However, when applied to large-scale image set applications such as classification and retrieval, they face two primary challenges: 1) effectively modeling complex image sets; and 2) efficiently performing tasks. To address the above issues, we propose a novel Multiple Riemannian Kernel Hashing (MRKH) method that leverages the powerful capabilities of Riemannian manifold and Hashing on effective and efficient image set representation. MRKH considers multiple heterogeneous Riemannian manifolds to represent each image set. It introduces a multiple kernel learning framework designed to effectively combine statistics from multiple manifolds, and constructs kernels by selecting a small set of anchor points, enabling efficient scalability for large-scale applications. In addition, MRKH further exploits inter- and intra-modal semantic structure to enhance discrimination. Instead of employing continuous feature to represent each image set, MRKH suggests learning hash code for each image set, thereby achieving efficient computation and storage. We present an iterative algorithm with theoretical convergence guarantee to optimize MRKH, and the computational complexity is linear with the size of dataset. Extensive experiments on five image set benchmark datasets including three large-scale ones demonstrate the proposed method outperforms state-of-the-arts in accuracy and efficiency particularly in large-scale image set classification and retrieval.

Multimanifold Bistructured Low Rank Representation of hyperspectral images

Hyperspectral images (HSIs) are obtained by imaging surface materials using a spectral imager. In practical applications, the hyperspectral remote sensing images are inevitably contaminated by noise, leading to errors in image processing results. Gaussian noise, Poisson noise, and stripe noise are common in HSIs. This article presents an HSI denoising model based on the low rank representation of multi manifold double constructions. Introducing multiple manifolds into the field of hyperspectral image denoising for the first time, comprehensively obtaining complex information on hyperspectral surfaces. We extended the graph-regularized low-rank representation model (GLRR) by using maximum data variance to construct global information. GLRR obtains local information, and the parallel effects of nearest neighbor and maximum variance graphs take into account both global and local consistency in the denoising process. Our model is not only applicable to the case of a single manifold, but also to the case of multiple manifolds, especially in the case of complex ground target images with multiple manifold spectra. Use the LADMAP iterative algorithm to iteratively solve the proposed algorithm. Avoid introducing auxiliary variables to reduce runtime costs and algorithm complexity. Both on simulated and real datasets, the runtime is relatively fast. We experiments on both real and simulated datasets, as well as real datasets, to verify that our method has good denoising performance when Gaussian, Poisson, and stripe noise occur simultaneously.

Hyperspectral Image Dimensionality Reduction Algorithm Based on Spatial–Spectral Adaptive Multiple Manifolds

Hyperspectral images contain rich spatial–spectral information and have high dimensions, which can lead to challenges related to feature extraction for classification tasks, resulting in suboptimal performance. We propose a hyperspectral image dimensionality reduction algorithm based on spatial–spectral adaptive multiple manifolds to address the problem of small differences between features of dissimilar samples in the subspace caused by the uniform projection transformation in traditional dimensionality reduction methods. Firstly, to address spatial boundary mismatch problems caused by re-characterizing a pixel using pixels in a fixed area around it as its near neighbors in traditional algorithms, an adaptive weight representation method based on super-pixel segmentation is proposed, which enhances the similarity of similar samples and the dissimilarity of dissimilar samples. Secondly, to address the problem that a single manifold cannot completely characterize the near neighbor between samples of different categories, an adaptive multi-manifold representation method is proposed. The feature representation of the entire hyperspectral data in the low-dimensional subspace is obtained by adaptively fusing the intra- and inter-manifold maps constructed for each category of samples in the spatial and spectral dimensions. Experimental results on two public datasets show that the proposed method achieves better results when performing the hyperspectral image dimensionality reduction task.

Mixture Manifold Networks: A Computationally Efficient Baseline for Inverse Modeling

We propose and show the efficacy of a new method to address generic inverse problems. Inverse modeling is the task whereby one seeks to determine the hidden parameters of a natural system that produce a given set of observed measurements. Recent work has shown impressive results using deep learning, but we note that there is a trade-off between model performance and computational time. For some applications, the computational time at inference for the best performing inverse modeling method may be overly prohibitive to its use. In seeking a faster, high-performing model, we present a new method that leverages multiple manifolds as a mixture of backward (e.g., inverse) models in a forward-backward model architecture. These multiple backwards models all share a common forward model, and their training is mitigated by generating training examples from the forward model. The proposed method thus has two innovations: 1) the multiple Manifold Mixture Network (MMN) architecture, and 2) the training procedure involving augmenting backward model training data using the forward model. We demonstrate the advantages of our method by comparing to several baselines on four benchmark inverse problems, and we furthermore provide analysis to motivate its design.

Multi-view clustering for multiple manifold learning via concept factorization

Clustering of multi-view data has attracted great interest in machine learning. Multi-view data is based on complementary and consistent information due to the collection of numerous sources. However, the current multi-view clustering method does not focus on learning the manifold structure for consensus representation over the kernel space, the overfitting redundant features of diverse view data, and ignoring the correlation among these complementary multiple views. In this article, a novel multi-view clustering approach is introduced to tackle this issue, which deploys concept factorization to indulge the intrinsic data information. The intrinsic geometry of the data and consensus representation are preserved through the utilization of manifold learning. In addition, the correlation constraint is adopted for learning the common shared structure from diverse views, and the smooth regularization term is deployed to alleviate the overfitting of views. Finally, we fuse all individual terms to formulate the objective function and design a theoretically guaranteed optimization procedure to solve the model. Extensive experiments conducted on the benchmark datasets suggest that the proposed algorithm surpasses state-of-the-art approaches regarding clustering performance.

Training-Based Multiple Source Tracking Using Manifold-Learning and Recursive Expectation-Maximization

In this paper we propose a data-driven approach for multiple speaker tracking in reverberant enclosures. The speakers are uttering, possibly overlapping, speech signals while moving in the environment. The method comprises two stages. The first stage executes a single source localization using semi-supervised learning on multiple manifolds. The second stage, which is unsupervised, uses time-varying maximum likelihood estimation for tracking. The feature vectors, used by both stages, are the relative transfer functions (RTFs), which are known to be related to source positions. The number of sources is assumed to be known while the microphone positions are unknown. In the training stage, a large database of RTFs is given. A small percentage of the data is attributed with exact positions (namely, labelled data) and the rest is assumed to be unlabelled, i.e. the respective position is unknown. Then, a nonlinear, manifold-based, mapping function between the RTFs and the source positions is inferred. Applying this mapping function to all unlabelled RTFs constructs a dense grid of localized sources. In the test phase, this RTFs grid serves as the centroids for a (MoG) model. The MoG parameters are estimated by applying a recursive variant of the (EM) procedure that relies on the sparsity and intermittency of the speech signals. We present a comprehensive simulation study in various reverberation levels, including static and dynamic scenarios, for both two or three (partially) overlapping speakers. For the dynamic case we provide simulations with several speakers trajectories, including intersecting sources. The proposed scheme outperforms baseline methods that use a simpler propagation model in terms of localization accuracy and tracking capabilities.

Global Scattering Center Representation of Target Wide-Angle Single Reflection/Diffraction Mechanisms Based on the Multiple Manifold Concept

Many radar applications, such as system design and testing and target detection and recognition, need to compress and rapidly reconstruct the target-scattering characteristics data through some suitable sparse representations, while the scattering center (SC) model resulting from different scattering mechanisms is just a prospective candidate. For the target scattering characteristics data with single reflection/diffraction mechanisms, the multimanifold structures of wide-angle SCs were revealed in light of asymptotic high-frequency theory and ray theory of electromagnetic field. Then the multimanifold clustering and curve/surface fitting algorithms are introduced to construct the target global SC (GSC) model. Compared with simulation data of sphere-cone target, the RCS at elevation 90°, azimuth 0–180° can be accurately reconstructed by only 77 parameters, the compress rate and root mean square error are 0.66 and 1.17 dB respectively. Simulation results showed that the GSC model could greatly compress the wide-angle scattering data while ensuring a suitable reconstruction accuracy. The proposed multimanifold GSC representation is convenient to implement and can effectively replace the redundant original scattering characteristics data.

Spaks: Self-paced multiple kernel subspace clustering with feature smoothing regularization

Considerable attention to kernel subspace clustering has emerged in recent years due to its excellent performance in grouping data points from multiple manifolds. In particular, multiple kernel learning (MKL) is commonly used in kernel subspace clustering models to alleviate the difficulty of choosing a suitable kernel function, which brings out numerous MKL-based subspace clustering methods. Despite these successes, these MKL methods cannot guarantee that the feature representation produced by the learned kernel function is easy-to-divide, which is critical for the clustering task. Besides, the weighting strategies in these methods ignore the respective fitting ability of each base kernel, which may result in inappropriate weight assignment and inferior performance. To address these problems, this paper proposes a Self-Paced Smooth Multiple Kernel Subspace Clustering (Spaks) method. In Spaks, we first incorporate a feature smoothing regularization into the kernel subspace clustering model to facilitate the smoothness of feature representation on the affinity graph, thus ensuring that the data points between different subspaces are sufficiently separable. Then we implement the multiple kernel mechanism in the self-paced learning framework. In this way, the weights of base kernels depend on their corresponding single-kernel optimization objective function value, which indicates that a greater weight will be assigned to a kernel with a smaller objective. At last, we optimize the formulated model by an alternative iteration scheme, so that the optimal affinity matrix is obtained for clustering. Extensive experiments on seven benchmark data sets validate the superiority of the proposed Spaks method on the clustering task.

Multi-manifold discriminant local spline embedding

Manifold learning reveals the intrinsic low-dimensional manifold structure of high-dimensional data and has achieved great success in a wide spectrum of applications. However, traditional manifold learning methods assume that all the data lie on a common manifold, hence fail to capture the complicated geometry structure of the real-world data lying on multiple manifolds. This paper proposes a novel Multi-manifold Discriminant Local Spline Embedding (MDLSE) algorithm for high-dimensional classification, which considers a more realistic scenario where data of the same class lies on the same manifold. On the basis of this assumption, MDLSE seeks to reconstruct multiple manifolds for different classes of data in the embedding and separate them as apart as possible. In order to preserve the geometry structure of all the manifolds, MDLSE employs thin plate splines to align the local patches within each manifold compatibly in the global embedding. Meanwhile, to separate the different manifolds, MDLSE utilizes discriminative information to ensure the neighboring data from different manifolds to be mapped far from each other. Extensive experiments on both synthetic and real-world datasets demonstrate the superiority of MDLSE over the other representative manifold learning algorithms. The advantage of MDLSE is often more obvious on smaller size of training data and in lower embedding dimensions.

Characterizing interactions between cardiac shape and deformation by non-linear manifold learning.

In clinical routine, high-dimensional descriptors of the cardiac function such as shape and deformation are reduced to scalars (e.g. volumes or ejection fraction), which limit the characterization of complex diseases. Besides, these descriptors undergo interactions depending on disease, which may bias their computational analysis. In this paper, we aim at characterizing such interactions by unsupervised manifold learning. We propose to use a sparsified version of Multiple Manifold Learning to align the latent spaces encoding each descriptor and weighting the strength of the alignment depending on each pair of samples. While this framework was up to now only applied to link different datasets from the same manifold, we demonstrate its relevance to characterize the interactions between different but partially related descriptors of the cardiac function (shape and deformation). We benchmark our approach against linear and non-linear embedding strategies, among which the fusion of manifolds by Multiple Kernel Learning, the independent embedding of each descriptor by Diffusion Maps, and a strict alignment based on pairwise correspondences. We first evaluated the methods on a synthetic dataset from a 0D cardiac model where the interactions between descriptors are fully controlled. Then, we transfered them to a population of right ventricular meshes from 310 subjects (100 healthy and 210 patients with right ventricular disease) obtained from 3D echocardiography, where the link between shape and deformation is key for disease understanding. Our experiments underline the relevance of jointly considering shape and deformation descriptors, and that manifold alignment is preferable over fusion for our application. They also confirm at a finer scale the characteristic traits of the right ventricular diseases in our population.

Clustered Multiple Manifold Metric Learning for Hyperspectral Image Dimensionality Reduction and Classification

Dimensionality reduction (DR) technology is an important part of hyperspectral image (HSI) processing. The DR technology can effectively remove the redundant information in the HSIs and avoid the Hughes phenomenon, which is beneficial to image classification. Metric learning is widely used in DR technology, the goal of which is to achieve DR by maximizing the distance of between-class while minimizing the distance of within-class. However, traditional metric learning does not consider the intrinsic structure of data during the training process, resulting in insufficient information utilization. In order to solve the above problems, this article proposes clustered multiple manifold metric learning (CM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> L) by combining manifold learning with metric learning for DR and classification of HSI. The proposed CM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> L algorithm first divides the original data into multiple independent clusters and regards each sample point and its near point in the cluster as a whole, constructing it as a manifold. Then, CM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> L uses the manifold metric distance to replace the traditional metric distance. Finally, by making full use of the local information of HSI in such a way, the discrimination ability is enhanced. Intensive experimental results on three real HSI datasets validate the effectiveness of our proposed CM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> L algorithm.

Multi-View Clustering Based on Multiple Manifold Regularized Non-Negative Sparse Matrix Factorization

Clustering of multi-view data has got broad consideration of the researchers. Multi-view data is composed through different domain which shows the consistent and complementary behavior. The existing studies did not draw attention of over-fitting and sparsity among the diverse view, which is the considerable issue for getting the unique consensus knowledge from these complementary data. Herein article, a multi-view clustering approach is recommended to provide the consensus solution from the multi-view data. To accomplish this task, we exploit non-negative matrix factorized method to generate a cost function. Further, manifold learning model is used to build the graph through the nearest neighbor strategy, which is effective to save the geometrical design for data and feature matrix. Furthermore, the over-fitting problem, sparsity is handled through adaption of frobenious norm, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> -norm on basis and coefficient matrices. The whole formulation is done through the mathematical function, which is optimized through the iterative updating strategy to get the optimal solution. The computational experiment is carried on the available datasets to exhibits that the proposed strategy beats the current methodologies in terms of clustering execution.

LAAT: Locally Aligned Ant Technique for discovering multiple faint low dimensional structures of varying density

Dimensionality reduction and clustering are often used as preliminary steps for many complex machine learning tasks. The presence of noise and outliers can deteriorate the performance of such preprocessing and therefore impair the subsequent analysis tremendously. In manifold learning, several studies indicate solutions for removing background noise or noise close to the structure when the density is substantially higher than that exhibited by the noise. However, in many applications, including astronomical datasets, the density varies alongside manifolds that are buried in a noisy background. We propose a novel method to extract manifolds in the presence of noise based on the idea of Ant colony optimization. In contrast to the existing random walk solutions, our technique captures points that are locally aligned with major directions of the manifold. Moreover, we empirically show that the biologically inspired formulation of ant pheromone reinforces this behavior enabling it to recover multiple manifolds embedded in extremely noisy data clouds. The algorithm performance in comparison to state-of-the-art approaches for noise reduction in manifold detection and clustering is demonstrated, on several synthetic and real datasets, including an N-body simulation of a cosmological volume.

SSMD: Dimensionality Reduction and Classification of Hyperspectral Images Based on Spatial–Spectral Manifold Distance Metric Learning

Metric learning, which aims to obtain a metric matrix M such that samples from the same class are close to one another and samples of different classes are far from one another, is widely used in the field of hyperspectral dimensionality reduction (DR) and classification. Traditional metric learning is based on the Mahalanobis distance, which measures the similarity between samples via point-to-point distance, ignoring the structural features of the hyperspectral images (HSIs). To solve the above problem, we proposed clustered multiple manifold metric learning (CM3L), which obtains a manifold distance (MD), aimed at improving discrimination by introducing structural features of the HSIs and achieving good results. However, this manifold distance still has certain shortcomings in specific application situations. MD only considers the labeled data in the construction of the manifold and ignores the unlabeled data, resulting in the destruction of the manifold. Therefore, this article proposes a new spatial–spectral manifold distance (SSMD) to improve the performance of metric learning in hyperspectral DR and classification by maintaining the integrity of the constructed manifolds. The SSMD selects suitable neighboring points in the labeled and unlabeled data through the spectral–spatial information in order to participate in the construction of the manifold. Then, the distance between the manifolds is calculated to replace the traditional Mahalanobis distance. The results of seven sets of comparison experiments on three real HSI datasets demonstrate the effectiveness of SSMD in improving the classification results of HSIs.

&amp;lt;italic&amp;gt;t&amp;lt;/italic&amp;gt;-SNE for Complex Multi-Manifold High-Dimensional Data

<p indent=0mm>To solve the problem that the <italic>t</italic>-SNE method cannot distinguish multiple manifolds that intersect each other well, a visual dimensionality reduction method is proposed. Based on the <italic>t-</italic>SNE method, Euclidean metric and local PCA are considered when calculating high-dimensional probability to distinguish different manifolds. Then the <italic>t-</italic>SNE gradient solution method can be directly used to get the dimensionality reduction result. Finally, three generated data and two real data are used to test proposed method, and quantitatively evaluate the discrimination of different manifolds and the degree of neighborhood preservation within the manifold in the dimensionality reduction results. These results show that proposed method is more useful when processing multi-manifold data, and can keep the neighborhood structure of each manifold well.

Exploring the Latent Manifold of City Patterns

Understanding how cities evolve through time and how humans interact with their surroundings is a complex but essential task that is necessary for designing better urban environments. Recent developments in artificial intelligence can give researchers and city developers powerful tools, and through their usage, new insights can be gained on this issue. Discovering a high-level structure in a set of observations within a low-dimensional manifold is a common strategy used when applying machine learning techniques to tackle several problems while finding a projection from and onto the underlying data distribution. This so-called latent manifold can be used in many applications such as clustering, data visualization, sampling, density estimation, and unsupervised learning. Moreover, data of city patterns has some particularities, such as having superimposed or natural patterns that correspond to those of the depicted locations. In this research, multiple manifolds are explored and derived from city pattern images. A set of quantitative and qualitative tests are proposed to examine the quality of these manifolds. In addition, to demonstrate these tests, a novel specialized dataset of city patterns of multiple locations is created, with the dataset capturing a set of recognizable superimposed patterns.

A Novel Demodulation Method Based on Spectral Clustering for Phase-Modulated Signals Interrupted by the Plasma Sheath Channel

The phenomenon of radio blackout due to a plasma sheath during reentry has attracted much attention over the last several decades. However, radio blackout has long puzzled aerospace researchers and has not yet been completely resolved. Owing to the effects of the time-varying plasma sheath channel environment, the constellation of received signals exhibits multiple manifolds, making the traditional algorithms based on Euclidean distance unable to perform demodulation. In the current study, we proposed a novel demodulation method based on spectral clustering for phase-modulated signals interrupted by the plasma sheath channel. Experimental results revealed that as long as the receiving radio waves were above the floor of receiver’s dynamic range, the proposed algorithm effectively classified received signals after time-varying plasma. The parasitic modulation interference caused by time-varying plasma was restrained effectively through the novel algorithm. In contrast to the traditional communication method, the proposed method can be applied to telemetry, track and command, as well as communication between reentry vehicles and near-space hypersonic vehicles in future.

Adaptive Fusion of Heterogeneous Manifolds for Subspace Clustering.

Multiview clustering (MVC) has recently received great interest due to its pleasing efficacy in combining the abundant and complementary information to improve clustering performance, which overcomes the drawbacks of view limitation existed in the standard single-view clustering. However, the existing MVC methods are mostly designed for vectorial data from linear spaces and, thus, are not suitable for multiple dimensional data with intrinsic nonlinear manifold structures, e.g., videos or image sets. Some works have introduced manifolds' representation methods of data into MVC and obtained considerable improvements, but how to fuse multiple manifolds efficiently for clustering is still a challenging problem. Particularly, for heterogeneous manifolds, it is an entirely new problem. In this article, we propose to represent the complicated multiviews' data as heterogeneous manifolds and a fusion framework of heterogeneous manifolds for clustering. Different from the empirical weighting methods, an adaptive fusion strategy is designed to weight the importance of different manifolds in a data-driven manner. In addition, the low-rank representation is generalized onto the fused heterogeneous manifolds to explore the low-dimensional subspace structures embedded in data for clustering. We assessed the proposed method on several public data sets, including human action video, facial image, and traffic scenario video. The experimental results show that our method obviously outperforms a number of state-of-the-art clustering methods.

Multi-view image clustering based on sparse coding and manifold consensus

Multi-view clustering has received an increasing attention in many applications, where different views of objects can provide complementary information to each other. Existing approaches on multi-view clustering mainly focus on extending Non-negative Matrix Factorization (NMF) by enforcing the constraint over the coefficient matrices from different views in order to preserve their consensus. In this paper, we argue that it is more reasonable to utilize the high-level manifold consensus rather than the low-level coefficient matrix consensus (as conducted in state-of-the-art approaches) to better capture the underlying clustering structure of the data. For this purpose, we propose MMRSC (Multiple Manifold Regularized Sparse Coding), which aims to preserve the consensus over multiple manifold structures from different views. Experimental results on two publicly available real-world image datasets demonstrate that our proposed approach can significantly outperform the state-of-the-art approaches for the multi-view image clustering task. Moreover, we also conduct computational complexity analysis and the result shows that MMRSC can effective handle the multi-view clustering problem without increasing the computational cost as compared to GraphSC.

Exploiting Linear Manifold Features With Parts-Based Representation in Various Scenes

Image recognition in complex scenes is a big challenge in computer vision. Manifold learning has become one of the most popular tools in the application of data dimensionality reduction and image recognition due to its efficiency in retrieving the intrinsic geometric features of image data. In this paper, we propose a new manifold feature extracting model based on the nonnegative matrix factorization (NMF) for image clustering in various scenes. In this model, Pearson distance with multiple manifold regulation constraints are adopted as the objective function to derive NMF based learning algorithms for the feature capturing of high dimensional data. With a variable neighborhood size in the learning, the proposed model can learn the linear features and at the same time learn the local similarity of images in multi-scale neighborhoods of a graph space. For different settings of learning parameters $\lambda _{lx}$ and $\lambda _{sx}$ , tests show that the proposed algorithms can efficiently retrieve low dimensional structures of images. Test results on four different image datasets demonstrate that the algorithms can achieve the state of art performance on the clustering of images in different types of scene.