Manifold Embedding Research Articles

Overcoming graph topology imbalance for inductive and scalable semi-supervised learning

Graph-based semi-supervised learning (GSSL) has received much attention recently. Despite some progress made in this area by some recent methods, some limitations still need to be addressed. Namely, there are two main shortcomings. First, the graphs used are very often built in advance and independently of the task at hand, using a heuristic, and generally do not represent the true topology of the data. The second shortcoming is the ability of the model to handle a very large number of unlabeled samples. This can make the GSSL solution impractical from a computational resource perspective. In this paper, we propose the Weighted Simultaneous Graph Construction and Reduced Flexible Manifold Embedding (W-SGRFME) method, which is a scalable and inductive GSSL framework. The main contributions are as follows. First, we extend the concept of graph topology imbalance to large datasets. Second, we integrate the computed weights of the labeled samples into the unified semi-supervised model. The latter jointly estimates the labels of the unlabeled samples, the mapping of the feature space to the label space, and the graph matrix of the anchor graph. Moreover, the fusion of labels and features of anchors is used to adaptively construct the graph. Experimental results on three large datasets from semi-supervised learning show the effectiveness of the proposed scalable method. These datasets are NORB, RCV1, and Covtype. Experimental results on large datasets show the superiority of the proposed method over existing scalable models.

Expression Atlas update: insights from sequencing data at both bulk and single cell level.

Expression Atlas (www.ebi.ac.uk/gxa) and its newest counterpart the Single Cell Expression Atlas (www.ebi.ac.uk/gxa/sc) are EMBL-EBI's knowledgebases for gene and protein expression and localisation in bulk and at single cell level. These resources aim to allow users to investigate their expression in normal tissue (baseline) or in response to perturbations such as disease or changes to genotype (differential) across multiple species. Users are invited to search for genes or metadata terms across species or biological conditions in a standardised consistent interface. Alongside these data, new features in Single Cell Expression Atlas allow users to query metadata through our new cell type wheel search. At the experiment level data can be explored through two types of dimensionality reduction plots, t-distributed Stochastic Neighbor Embedding (tSNE) and Uniform Manifold Approximation and Projection (UMAP), overlaid with either clustering or metadata information to assist users' understanding. Data are also visualised as marker gene heatmaps identifying genes that help confer cluster identity. For some data, additional visualisations are available as interactive cell level anatomograms and cell type gene expression heatmaps.

Rolling bearing variable condition fault diagnosis method based on manifold embedding adaptive graph label propagation

Domain adaptation has shown good performance in fault diagnosis of bearing transferable working conditions. However, most of the existing cross-domain diagnosis models are carried out in linear space, which are subject to strong linear constraints and ignore the manifold correlation information between samples, which greatly limits the learning ability of the diagnosis model. Therefore, we propose a manifold embedding adaptive graph label propagation fault diagnosis method for rolling bearings under variable working conditions. The purpose is to reduce the cross-condition domain offset and mine the neglected nonlinear manifold information in the linear space to improve the adaptive ability of the diagnosis model. First, a similarity graph is constructed on the source domain of the known fault type and the target domain of the unknown information, and the cross-domain propagation of the label information is performed on the graph. Then, in this process, the clustering domain adaptation is performed simultaneously, the accuracy of label propagation is improved by reducing the cross-domain offset, and the graph is updated according to the new sample distribution. Finally, according to the known graph and target domain labels, the regression residuals of linear labels are introduced to relax the strong linear constraints in the original space, so that the linear label space approximates the manifold label space and obtains more accurate fault identification results. Experiments show that the new method, as an unsupervised domain adaptive method in transfer learning, has better fault recognition ability than similar transfer learning methods on Paderborn university datasets and own test rig datasets, and the convergence performance of the algorithm is excellent. The highest fault recognition accuracy can reach 100%.

Dimensionality reduction by t-Distribution adaptive manifold embedding

Dimensionality reduction by t-Distribution adaptive manifold embedding

Action Recognition via Adaptive Semi-Supervised Feature Analysis

This study presents a new semi-supervised action recognition method via adaptive feature analysis. We assume that action videos can be regarded as data points in embedding manifold subspace, and their matching problem can be quantified through a specific Grassmannian kernel function while integrating feature correlation exploration and data similarity measurement into a joint framework. By maximizing the intra-class compactness based on labeled data, our algorithm can learn multiple features and leverage unlabeled data to enhance recognition. We introduce the Grassmannian kernels and the Projected Barzilai–Borwein (PBB) method to train a subspace projection matrix as a classifier. Experiment results show our method has outperformed the compared approaches when a few labeled training samples are available.

On the well-posed variational principle in degenerate point particle systems using embeddings of the symplectic manifold

Abstract A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position variables of a coordinate system of a given system has to be fixed on the boundaries and that such systems are always Ostrogradski stable. For these systems, Frobenius integrability conditions are derived in explicit form. Relationships between integral constants indicated from the conditions and boundary conditions in a given coordinate system are also investigated by introducing three fundamental correspondences between Lagrange and Hamilton formulation. Based on these ingredients, we formulate problems that have to be resolved to realize the well-posedness in the degenerate systems. To resolve the problems, we compose a set of embeddings that extract a subspace holding the symplectic structure of the entire phase space in which the variational principle should be well-posed. Using these embeddings, we establish a methodology to set appropriate boundary conditions that the well-posed variational principle demands. Finally, we apply the methodology to examples and summarize this work as a three-step procedure such that one can use just by following it.

Toward a computational theory of manifold untangling: from global embedding to local flattening.

It has been hypothesized that the ventral stream processing for object recognition is based on a mechanism called cortically local subspace untangling. A mathematical abstraction of object recognition by the visual cortex is how to untangle the manifolds associated with different object categories. Such a manifold untangling problem is closely related to the celebrated kernel trick in metric space. In this paper, we conjecture that there is a more general solution to manifold untangling in the topological space without artificially defining any distance metric. Geometrically, we can either embed a manifold in a higher-dimensional space to promote selectivity or flatten a manifold to promote tolerance. General strategies of both global manifold embedding and local manifold flattening are presented and connected with existing work on the untangling of image, audio, and language data. We also discuss the implications of untangling the manifold into motor control and internal representations.

0424 Visualizing insomnia phenotypes using dimensionality reduction techniques

Abstract Introduction A large number of features can be extracted from a single hypnogram, such as stages durations, onsets, or transitions probabilities. Those numerous indicators can turn a collection of sleep records into a high dimension space. Dimensionality reduction techniques are then useful to reveal patterns in data. We used 3 dimensionality reduction techniques to visualize insomnia phenotypes from a dataset of insomnia and control sleep records: principal component analysis (PCA), t-distributed stochastic neighbor embedding (t-SNE) and uniform manifold approximation and projection (UMAP). Methods 519 sleep records have been included with the following diagnoses: 46 sleep onset insomnia, 83 sleep state misperception, 223 sleep maintenance insomnia, 117 sleep onset and maintenance insomnia and 50 controls (good sleep). We limited the feature extraction to the hypnogram as macrostructure constitute the primary source of information used for diagnosis in polysomnography. We used common macrostructure indicators such as wake after sleep onset (WASO), as well as more intricate features such as stages transitions probabilities. A first set of 54 features per hypnogram was computed. Those features were then projected in a 2 dimensional space using PCA, t-SNE and UMAP. Results Co-ranking matrix of the projections was computed for the 3 techniques (PCA: Kmax=103, Qlocal=0.38; tSNE: Kmax=20, Qlocal=0.44; UMAP: Kmax=160, Qlocal=0.44). UMAP was the technique that projected the hypnograms feature sets in the most meaningful way, by reflecting the individual diagnoses. Interestingly in the UMAP representation, group outliers, such as controls having nevertheless experienced nocturnal awakenings were projected next to similar insomnia subjects, which indicate a fine-grained capture of sleep quality at the individual level. Conclusion Dimensionality reduction and unsupervised techniques are promising methods when approaching high dimensionality spaces. A fine analysis of projected clusters could show subgroups in already known phenotypes. As literature showed robustness and applicability of those methods to larger datasets, dimensionality reduction of insomnia records could be improved by the addition of new features computed from hypnodensities generated by machine learning algorithms, or by spectral features extracted from polysomnographic signals. Support (if any) None

SENE: A novel manifold learning approach for distracted driving analysis with spatio-temporal and driver praxeological features

Although there are many studies conducted on distracted driving, the growing number of accidents on roads demands further serious attention. The majority of the distracted driving-related data in real life are unlabeled and higher dimensional, leading to complex analyses. There is a lack of existence of proper indices for understanding the perilousness due to distracted driving, which makes it very difficult to understand which road or neighborhood has a higher risk of accidents. Despite earlier studies have focused on either spatiotemporal or praxeological factors separately, they have not considered both factors together. Moreover, crisp rule extraction and interpretation are lacking in the literature. Therefore, to deal with such issues, we have proposed a new methodology which: (i) develops Schrodinger Eigenmap Neighborhood Embedding (SENE) manifold learning for dimensionality reduction, followed by K-means clustering, (ii) develops road perilousness index (RPI) and neighborhood PI (NPI) to explain dangerousness of roads or neighborhoods, (iii) uses both spatiotemporal and driver praxeological factors, and (iv) develops Tolerance Rough Set Approach (TRSA) for crisp rules generation and interpretation. Road accident data from the Nevada Department of Transportation is used as a case study. Besides, a total of four benchmark datasets from the University of California Irvine repository are also used for comparative study to prove the superiority of our proposed methodology over some state-of-the-art. Experimental results reveal that the proposed methodology outperforms others by providing the highest clustering accuracy with four clusters. Finally, a set of 16 crisp rules are extracted and interpreted from clusters using TRSA.

A Transformer Condition Identification Model under Crossing Measuring Points Sampling

Due to the difference in sampling data distribution among different measuring points, the common transformer condition identification method of vibration analysis combined with machine learning has decreased accuracy. This work proposes a non-deep transformer condition identification model based on domain adaptation to solve it. Firstly, multi-scale feature extraction of vibration data is fulfilled by variational mode decomposition and time-frequency analysis. Then, manifold embedding method is employed to adapt source and target domains, which are sampled from different measuring points of transformer. The experimental results manifest that the presented approach could improve the accuracy of the transformer condition identification to a certain extent.

Opening the black-box of Neighbor Embeddings with Hotelling’s [formula omitted] statistic and [formula omitted]-residuals

In contrast to classical techniques for exploratory analysis of high-dimensional data sets, such as principal component analysis (PCA), neighbor embedding (NE) techniques tend to better preserve the local structure/topology of high-dimensional data. However, the ability to preserve local structure comes at the expense of interpretability: techniques such as t-Distributed Stochastic Neighbor Embedding (t-SNE) or Uniform Manifold Approximation and Projection (UMAP) do not provide insights into which input or to what extent the variables underlie the topological (cluster) structure seen in the corresponding embedding. We here introduce X-MAP, a workflow that employs different “tricks” from the chemometrics field based on PCA, Q-residuals and Hotelling’s T2 contributions in combination with novel visualization approaches to derive local and global explanations of neighbor embeddings. We show how our approach is capable of identifying discriminatory features between groups of data points that remain unnoticed when exploring NEs using standard univariate or multivariate approaches.

A geometric framework for outlier detection in high‐dimensional data

AbstractOutlier or anomaly detection is an important task in data analysis. We discuss the problem from a geometrical perspective and provide a framework which exploits the metric structure of a data set. Our approach rests on the manifold assumption, that is, that the observed, nominally high‐dimensional data lie on a much lower dimensional manifold and that this intrinsic structure can be inferred with manifold learning methods. We show that exploiting this structure significantly improves the detection of outlying observations in high dimensional data. We also suggest a novel, mathematically precise and widely applicable distinction between distributional and structural outliers based on the geometry and topology of the data manifold that clarifies conceptual ambiguities prevalent throughout the literature. Our experiments focus on functional data as one class of structured high‐dimensional data, but the framework we propose is completely general and we include image and graph data applications. Our results show that the outlier structure of high‐dimensional and non‐tabular data can be detected and visualized using manifold learning methods and quantified using standard outlier scoring methods applied to the manifold embedding vectors.This article is categorized under: Technologies > Structure Discovery and Clustering Fundamental Concepts of Data and Knowledge > Data Concepts Technologies > Visualization

Open books and embeddings of smooth and contact manifolds

Abstract We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n − k . We then prove that every closed manifold M 2n+1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊2⌊3n/2⌋+3. We also prove that every closed manifold M 2n+1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ2n+3. Finally, we give various examples of contact open book embeddings of contact (2n + 1)-manifolds in the trivial supporting open book of the standard contact structure on 𝕊4n+1.

Multi-similarity semi-supervised manifold embedding for facial attractiveness scoring

Multi-similarity semi-supervised manifold embedding for facial attractiveness scoring

Incomplete multi-view clustering network via nonlinear manifold embedding and probability-induced loss

Incomplete multi-view clustering, which included missing data in different views, is more challenging than multi-view clustering. For the purpose of eliminating the negative influence of incomplete data, researchers have proposed a series of solutions. However, the present incomplete multi-view clustering methods still confront three major issues: (1) The interference of redundant features hinders these methods to learn the most discriminative features. (2) The importance role of local structure is not considered during clustering. (3) These methods fail to utilize data distribution information to guide models update to decrease the effects of outliers and noise. To address above issues, a novel deep clustering network which exerted on incomplete multi-view data was proposed in this paper. We combine multi-view autoencoders with nonlinear manifold embedding method UMAP to extract latent consistent features of incomplete multi-view data. In the clustering method, we introduce Gaussian Mixture Model (GMM) to fit the complex distribution of data and deal with the interference of outliers. In addition, we reasonably utilize the probability distribution information generated by GMM, using probability-induced loss function to integrate feature learning and clustering as a joint framework. In experiments conducted on multiple benchmark datasets, our method captures incomplete multi-view data features effectively and perform excellent.

Semi-supervised machinery health assessment framework via temporal broad learning system embedding manifold regularization with unlabeled data

In the era of industrial big data, data-driven methods based on deep learning have made a splash in the field of mechanical PHM. However, the performance of deep learning models is limited by the depth of networks while previous supervised learning methods can only be applied to a small amount of labeled data while most of data acquired in the field are unlabeled, resulting in overfitting and low generalization of models. To address the above issues, a semi-supervised framework via temporal broad learning system embedding manifold regularization (TBLSMR) is proposed for machinery health assessment with unlabeled data. Specifically, linear and nonlinear temporal feature information is first learned from labeled and unlabeled data by TBLS. Then the Laplacian matrix is introduced embedding manifold regularization and used to construct objective function. Meanwhile, a specific ridge regression algorithm is designed to easily calculate the weights of network. Finally, an end-to-end mapping between the feature information layer and the label layer is established. The effectiveness of the proposed method is verified by two case studies. The comparison results with traditional models show that TBLSMR obtains high prediction accuracy while achieving training consumption savings, making it a promising approach for machinery health assessment that meets industrial needs.

Simplicial Complex-Enhanced Manifold Embedding of Spatiotemporal Data for Structural Health Monitoring

Structural Health Monitoring requires the continuous assessment of a structure’s operational conditions, which involves the collection and analysis of a large amount of data in both spatial and temporal domains. Conventionally, both data-driven and physics-based models for structural damage detection have relied on handcrafted features, which are susceptible to the practitioner’s expertise and experience in feature selection. The limitations of handcrafted features stem from the potential for information loss during the extraction of high-dimensional spatiotemporal data collected from the sensing system. To address this challenge, this paper proposes a novel, automated structural damage detection technique called Simplicial Complex Enhanced Manifold Embedding (SCEME). The key innovation of SCEME is the reduction of dimensions in both the temporal and spatial domains for efficient and information-preserving feature extraction. This is achieved by constructing a simplicial complex for each signal and using the resulting topological invariants as key features in the temporal domain. Subsequently, curvature-enhanced topological manifold embedding is performed for spatial dimension reduction. The proposed methodology effectively represents both intra-series and inter-series correlations in the low-dimensional embeddings, making it useful for classification and visualization. Numerical simulations and two benchmark experimental datasets validate the high accuracy of the proposed method in classifying different damage scenarios and preserving useful information for structural identification. It is especially beneficial for structural damage detection using complex data with high spatial and temporal dimensions and large uncertainties in reality.

Comparison of manifold representations of sonar data

There are many challenges in active sonar target recognition due to the dynamic nature of the environment, unknown target geometries, and various clutter present within the ocean. These factors combine and entangle the target features within the received response. This research assumes the informative and discriminatory acoustic target features lie on a low dimensional manifold and can be extracted using statistical and machine learning techniques. Linear techniques, such as principal component analysis and linear discriminant analysis, are useful for dimension reduction but will typically not accurately capture the underlying non-linearity within a dataset. Non-linear manifold learning techniques, such as T-distributed stochastic neighbor embedding and uniform manifold approximation and projection, are relatively recent techniques that create low-dimensional embeddings in an unsupervised fashion and can capture the non-linearity. Previously, persistent braid features have been reported in real sonar data1. Our objective here is to do comparison between these different manifold representations for both simulated data with a known ground truth and experimental active sonar data. [1] A. Sen Gupta, B. Kubicek, A. Christensen, and I. Kirsteins, “Geometric feature representation in active sonar signal processing,” OCEANS 2022—Chennai, 2022, pp. 1–5. [Work supported by NDSEG 2021 and ONR Grant No. N00014-21-1-2420].

MEQA: Manifold embedding quality assessment via anisotropic scaling and Kolmogorov-Smirnov test

Manifold learning methods unfold the manifold structures and embed them in a lower-dimensional space. The quality of such an embedding should be measured both qualitatively and quantitatively. The proposed manifold embedding quality assessment (MEQA) method does so by taking into account of local and global structure preservation as both are important traits of an embedding. To measure the local structure preservation MEQA uses two transformations. Initially anisotropic scaling, rotation and translation are incorporated to measure the closeness between the original and the embedded data points. In the next stage, rigid transformation is incorporated to quantify the previous transformation which involved anisotropic scaling. For quantifying the global structure preservation, the Kolmogorov-Smirnov test is applied in a distributed manner over each dimension and averaged over them. To establish the superiority of MEQA we conducted several studies over standard synthetic and real-life datasets across separate existing feature extraction techniques.

Manifold Embeddings by Heat Kernels of Connection Laplacian

We show that any closed n-dimensional manifold (M, g) can be embedded by a map constructed using the heat kernels of the connection Laplacian as well as maps constructed using truncated heat kernel at a certain time t from a $$\delta $$ -net $$\{q_i\}_{i=1}^{N_0}$$ via a rescaling technique. Both the time t and $$N_0$$ are bounded in terms of the dimension, bounds on the Ricci curvature and its derivative, the injectivity radius, and the volume. Moreover, both maps can be made arbitrarily close to an isometry.