Manifold Representation Research Articles

Robust subspace clustering via two-way manifold representation

Robust subspace clustering via two-way manifold representation

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in \(\mathbb {R}^2 \) or \(\mathbb {R}^3 \) , such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

Shufflemono: Rethinking Lightweight Network for Self-Supervised Monocular Depth Estimation

Abstract Self-supervised monocular depth estimation has been widely applied in autonomous driving and automated guided vehicles. It offers the advantages of low cost and extended effective distance compared with alternative methods. However, like automated guided vehicles, devices with limited computing resources struggle to leverage state-of-the-art large model structures. In recent years, researchers have acknowledged this issue and endeavored to reduce model size. Model lightweight techniques aim to decrease the number of parameters while maintaining satisfactory performance. In this paper, to enhance the model’s performance in lightweight scenarios, a novel approach to encompassing three key aspects is proposed: (1) utilizing LeakyReLU to involve more neurons in manifold representation; (2) employing large convolution for improved recognition of edges in lightweight models; (3) applying channel grouping and shuffling to maximize the model efficiency. Experimental results demonstrate that our proposed method achieves satisfactory outcomes on KITTI and Make3D benchmarks while having only 1.6M trainable parameters, representing a reduction of 27% compared with the previous smallest model, Lite-Mono-tiny, in monocular depth estimation.

Join multiple Riemannian manifold representation and multi‐kernel non‐redundancy for image clustering

AbstractImage clustering has received significant attention due to the growing importance of image recognition. Researchers have explored Riemannian manifold clustering, which is capable of capturing the non‐linear shapes found in real‐world datasets. However, the complexity of image data poses substantial challenges for modelling and feature extraction. Traditional methods such as covariance matrices and linear subspace have shown promise in image modelling, and they are still in their early stages and suffer from certain limitations. However, these include the uncertainty of representing data using only one Riemannian manifold, limited feature extraction capacity of single kernel functions, and resulting incomplete data representation and redundancy. To overcome these limitations, the authors propose a novel approach called join multiple Riemannian manifold representation and multi‐kernel non‐redundancy for image clustering (MRMNR‐MKC). It combines covariance matrices with linear subspace to represent data and applies multiple kernel functions to map the non‐linear structural data into a reproducing kernel Hilbert space, enabling linear model analysis for image clustering. Additionally, the authors use matrix‐induced regularisation to improve the clustering kernel selection process by reducing redundancy and assigning lower weights to identical kernels. Finally, the authors also conducted numerous experiments to evaluate the performance of our approach, confirming its superiority to state‐of‐the‐art methods on three benchmark datasets.

Latent space representation method for structural acoustic assessments

When targeting structural acoustic objectives, engineering practitioners face epistemic uncertainties in the selection of optimal geometries and material distributions, particularly during early stages of the design process. Models built for simulating acoustic phenomena generally produce vector-valued output quantities of interest, such as autospectral density and frequency response functions. Given finite compute resources and time we seek computationally parsimonious ways to distill meaningful design information into actionable results from a limited set of model runs, and thus aim to use machine learning to perform model order reduction. Unlike time series data for which recurrent neural networks can learn from prior time steps to inform subsequent steps, frequency-dependent data demands a different machine learning paradigm. We thus evaluate the utility of autoencoders to represent structural acoustic results with a low dimensional latent space to enable such objectives as surrogate modeling for design optimization. We demonstrate the accuracy of autoencoder based methods of constructing a manifold representation for frequency dependent functions of varying modal density and damping, and discuss the predictive capability thereof.

Local Sample-Weighted Multiple Kernel Clustering With Consensus Discriminative Graph.

Multiple kernel clustering (MKC) is committed to achieving optimal information fusion from a set of base kernels. Constructing precise and local kernel matrices is proven to be of vital significance in applications since the unreliable distant-distance similarity estimation would degrade clustering performance. Although existing localized MKC algorithms exhibit improved performance compared with globally designed competitors, most of them widely adopt the KNN mechanism to localize kernel matrix by accounting for τ -nearest neighbors. However, such a coarse manner follows an unreasonable strategy that the ranking importance of different neighbors is equal, which is impractical in applications. To alleviate such problems, this article proposes a novel local sample-weighted MKC (LSWMKC) model. We first construct a consensus discriminative affinity graph in kernel space, revealing the latent local structures. Furthermore, an optimal neighborhood kernel for the learned affinity graph is output with naturally sparse property and clear block diagonal structure. Moreover, LSWMKC implicitly optimizes adaptive weights on different neighbors with corresponding samples. Experimental results demonstrate that our LSWMKC possesses better local manifold representation and outperforms existing kernel or graph-based clustering algorithms. The source code of LSWMKC can be publicly accessed from https://github.com/liliangnudt/LSWMKC.

Enhanced video clustering using multiple riemannian manifold-valued descriptors and audio-visual information

Videos inherently blend multiple modalities in real-world scenarios, primarily visual and auditory cues. When synergized, these cues foster enhanced data representations. Standard clustering techniques, primarily designed for managing vectorial data in Euclidean spaces, struggle to handle multidimensional data with nonlinear manifold structures, such as video or image sets. While recent subspace clustering methods using Riemannian manifold representation tackle this issue, they often sideline auditory information, overlooking the potential harmony between visual and auditory modalities. This paper presents an innovative approach that crafts multiple Riemannian manifold-valued descriptors to bridge this gap, encapsulating multimodal video information in a unified structure. We architect a single-modality Riemannian subspace clustering for individual modal data and extend it to a multi-modality framework, leveraging the interplay of audio-visual data. Detailed optimization and convergence analysis are also provided. The proposed approach significantly outperforms the existing state-of-the-art methods, improving accuracy by 4%, 1%, and 2% on UCF-101, UCF-sport, and AVE datasets, respectively.

Data-driven modeling of resolved and filtered thermo-diffusively unstable hydrogen–air flames

Hydrogen premixed flames, under technically relevant conditions, are affected by intrinsic instabilities due to hydrodynamic and thermodiffusive effects, which are not taken into account by traditional turbulent combustion models. In this work, a data-driven approach is proposed for lean, premixed, unstable, self-wrinkling, hydrogen flames, based on a tabulated lower-dimensional manifold representation, wherein two scalars are defined based on species mass fractions. The model leverages a fully-resolved, two-dimensional, multi-step chemistry DNS dataset for the development of the tables. Both a flame-resolved and a filtered formulation are presented, and the additional terms that appear in the latter are closed following the filtered tabulated chemistry formalism. Two sets of tables are generated, using as a data source a small-scale and a medium-scale DNS, respectively. The a-priori analysis demonstrated the capability of the model to reproduce the flame structure with limited errors for both formulations. The flame-resolved formulation is validated a-posteriori for both table sets, showing that the flame correctly transitions to a fully-developed, corrugated state, although with a systematic small underestimation of the consumption speed. Given the agreement between the results obtained with the two sets of tables, the filtered a-posteriori analysis is conducted only for the table originating from the small-scale DNS, considering three different filter sizes. A set of one-dimensional flames is first simulated to assess the grid requirements for the non-trivial filtered simulations. Then, both the small-scale and large-scale domains are simulated. For both, it is observed that the characteristic flame structures are fairly reproduced using the small filter size, whereas the medium and large filter sizes produce a significant underestimation of consumption speed, as they fail to generate sufficient flame area through self-wrinkling. Overall, this approach represents a starting point for the development of turbulent combustion models including multi-step chemistry and the intrinsic instabilities effects.

Dual subspace manifold learning based on GCN for intensity-invariant facial expression recognition

Facial expression recognition (FER) is one of the most important computer vision tasks for understanding human inner emotions. However, the poor generation ability of the FER model limits its applicability due to tremendous intraclass variation. Especially for expressions of varying intensities, the appearance differences among weak expressions are subtle, which makes FER tasks challenging. In response to these issues, this paper presents a dual subspace manifold learning method based on a graph convolutional network (GCN) for intensity-invariant FER tasks. Our method treats the target task as a node classification problem and learns the manifold representation using two subspace analysis methods: locality preserving projection (LPP) and peak-piloted locality preserving projection (PLPP). Inspired by the classic LPP, which maintains local similarity among data, this paper introduces a novel PLPP that maintains the locality between peak expressions and non-peak expressions to enhance the representation of weak expressions. This paper also reports two subspace fusion methods, one based on a weighted adjacency matrix and another on a self-attention mechanism, that combine the LPP and PLPP to further improve FER performance. The second method achieves a recognition accuracy of 93.83% on the CK+, 74.86% on the Oulu-CASIA and 75.37% on the MMI for weak expressions, outperforming state-of-the-art methods.

Spontaneity competes with intention to influence the coordination dynamics of interpersonal performance tendencies

Research has shown that spontaneous visual coupling supports frequency entrainment, phase attraction, and intermittent interpersonal coordination when co-actors are switched from a no-vision (NV) to vision (V) context. In two experiments, co-actors started in a NV context while producing the same or different amplitude movements. The same amplitude resulted in similar self-paced frequencies, while different amplitudes resulted in disparate frequencies. In experiment 1, co-actors were instructed to maintain amplitude while receiving no instructions to coordinate their actions. Frequency and phase entrainment was limited in the V context even when co-actors started the NV context with the same amplitude. In experiment 2, co-actors were instructed to maintain amplitude and intentionally coordinate together, but not at a specific pattern. Significant frequency modulations occurred to maintain amplitude as the co-actors sought to coordinate their actions. With the open-ended instructions, co-actors produced in-phase and anti-phase coordination along with intermittent performance exhibited by shifts between a variety of stable relative phase patterns. The proposed hypotheses and findings are discussed within the context of a shared manifold representation for joint action contexts, with the coordination dynamics expressed by the HKB model of relative phase serving to conceptualization the representations in the shared manifold.

Autoencoder-based conditional optimal transport generative adversarial network for medical image generation

Medical image generation has recently garnered significant interest among researchers. However, the primary generative models, such as Generative Adversarial Networks (GANs), often encounter challenges during training, including mode collapse. To address these issues, we proposed the AE-COT-GAN model (Autoencoder-based Conditional Optimal Transport Generative Adversarial Network) for the generation of medical images belonging to specific categories. The training process of our model comprises three fundamental components. The training process of our model encompasses three fundamental components. First, we employ an autoencoder model to obtain a low-dimensional manifold representation of real images. Second, we apply extended semi-discrete optimal transport to map Gaussian noise distribution to the latent space distribution and obtain corresponding labels effectively. This procedure leads to the generation of new latent codes with known labels. Finally, we integrate a GAN to train the decoder further to generate medical images. To evaluate the performance of the AE-COT-GAN model, we conducted experiments on two medical image datasets, namely DermaMNIST and BloodMNIST. The model’s performance was compared with state-of-the-art generative models. Results show that the AE-COT-GAN model had excellent performance in generating medical images. Moreover, it effectively addressed the common issues associated with traditional GANs.

Elastic deep autoencoder for text embedding clustering by an improved graph regularization

Text clustering is a task for grouping extracted information of the text in different clusters, which has many applications in recommender systems, sentiment analysis, and more. Deep learning-based methods have become increasingly popular due to their high accuracy in identifying nonlinear structures. They usually consist of two major parts: dimensionality reduction and clustering. Autoencoders are simple unsupervised neural networks used for better representation of low-dimensional data and have shown good performance in dealing with non-linear features. However, while they utilize the Frobenius norm to deal well with Gaussian noise, they are sensitive to outlier data and Laplacian noise. In this paper, a deep autoencoder with an adapted elastic loss for text embedding clustering (EDA-TEC) is proposed. The elastic loss is a combination of the Frobenius norm and L2,1-norm to consider both types of noises. Additionally, to maintain the high-dimensional data geometric structure, a modified graph regularization term based on the weighted cosine similarity measure is used. EDA-TEC also improves clustering results by considering the sparsity regularization of the manifold representation data. In this jointly end-to-end deep learning model, better representation and text clustering results are achieved with high accuracy on common datasets compared to existing methods.11https://github.com/safinal/text-embedding-clustering.

A Cyclic Differentiable Manifold Representation of Redundant Manipulator Kinematics

Abstract An inverse kinematic mapping for redundant serial manipulators is presented at the configuration level, for which periodic manipulator operational trajectories map into periodic input trajectories, i.e., for which serial manipulators are cyclic. The inverse kinematic mapping defines a differentiable manifold on which output and self-motion coordinates comprise operational coordinates that represent manipulator redundant degrees-of-freedom. The inverse kinematic mapping and differentiable manifold are defined in analytical form and a computational method for their evaluation is presented. Numerical examples are presented to illustrate the validity of the formulation on subsets of manipulator regular configuration space, or on the entire space.

Characterizing and understanding galaxies with two parameters

ABSTRACT We report the discovery of a 2D Galaxy Manifold within the multidimensional luminosity space of local galaxies. The multidimensional luminosity space is constructed using 11 bands that span from far ultraviolet to near-infrared for redshift < 0.1 galaxies observed with GALEX, SDSS, and UKIDSS. The manifold captures the diversity of observed galaxies in terms of stellar-dominated emissions and ties the correlations of various physical properties to the manifold. We find that two latent parameters are sufficient to express 93.2 per cent of the variance in the galaxy sample, suggesting that this Galaxy Manifold is one of the most efficient representations of galaxies. The transformation between the observed luminosities and the manifold parameters as an analytic mapping is provided. The manifold representation provides accurate (precision = 0.85) morphological classifications with a simple linear boundary, and galaxy properties can be estimated with minimal scatter (0.12 and 0.04 dex for star formation rate and stellar mass, respectively) by calibrating with the 2D manifold location. Under the assumption that the manifold expresses the possible parameter space of galaxies, the evolution on the manifold is considered. We find that constant and exponentially decreasing star formation histories form almost orthogonal modes of evolution on the manifold. Through these simple models, we understand that the two modes are closely related to gas content, which suggests the close relationship of the manifold to gas accretion. Lastly, the found manifold suggests a paradigm where galaxies are characterized by their mass/scale and specific SFR, which agrees with previous studies of dimensionality reduction.

Expert knowledge guided manifold representation learning for magnetic resonance imaging-based glioma grading

Radiomics and deep learning have shown high popularity in automatic glioma grading. Radiomics extracts handcrafted features that quantitatively describe the expert knowledge of gliomas, and deep learning is powerful in extracting numerous learned features that facilitate the final classification. However, the performance of existing methods can still be improved as their complementary strengths haven’t been sufficiently investigated and integrated. Furthermore, the final prediction at the testing phase normally needs lesion maps which rely too much on the segmentation accuracy and could be troublesome. To address these challenges, we propose an expert knowledge guided manifold representation learning (ENROL) framework for accurate glioma grading without the reliance on segmentation maps during testing. Low-dimensional manifolds of the handcrafted features and learned features are constructed to mine the implicit relationship between radiomics and deep learning, and therefore to dig mutual consent representation for the glioma grades. With a specially designed manifold discrepancy measurement, the grading model can exploit the MRI data and expert knowledge more effectively and get rid of the requirement of lesion segmentation maps at the testing phase. The proposed framework is flexible regarding deep learning architectures to be utilized. Three different architectures have been evaluated on both single-sequence and multi-sequence MRI data. Results show that our framework can generate promising glioma grading performance.

Loop Closure Detection With Bidirectional Manifold Representation Consensus

Loop closure detection (LCD) is an indispensable module in simultaneous localization and mapping. It is responsible to recognize pre-visited areas during the navigation of a robot, providing auxiliary information to revise pose estimation. Unlike most current methods which focus on seeking an appropriate representation of images, we propose a novel two-stage pipeline dominated by the estimation of spatial geometric relationship. Specifically, to avoid unnecessary memory costs, consecutive images are segmented into sequences as per the similarity of their global features. Then the sequence descriptor is incrementally inserted into hierarchical navigable small world for the construction of reference database, from which the most similar image for the query one is searched parallelly. To further identify whether the candidate pair is geometry-consistent, a feature matching method termed as bidirectional manifold representation consensus (BMRC) is proposed. It constructs local neighborhood structures of feature points via manifold representation, and formulates the matching problem into an optimization model, enabling linearithmic time complexity via a closed-form solution. Meanwhile, an accelerated version of it is introduced (BMRC*), which performs about 63% faster than BMRC in an image pair with 352 initial correspondences. Extensive experiments on nine publicly available datasets demonstrate that BMRC and BMRC* perform well in feature matching and the proposed pipeline has remarkable performance in the LCD task.

SynapseCLR: Uncovering features of synapses in primary visual cortex through contrastive representation learning

3D electron microscopy (EM) connectomics image volumes are surpassing 1mm3, providing information-dense, multi-scale visualizations of brain circuitry and necessitating scalable analysis techniques. We present SynapseCLR, a self-supervised contrastive learning method for 3D EM data, and use it to extract features of synapses from mouse visual cortex. SynapseCLR feature representations separate synapses by appearance and functionally important structural annotations. We demonstrate SynapseCLR's utility for valuable downstream tasks, including one-shot identification of defective synapse segmentations, dataset-wide similarity-based querying, and accurate imputation of annotations for unlabeled synapses, using manual annotation of only 0.2% of the dataset's synapses. In particular, excitatory versus inhibitory neuronal types can be assigned with >99.8% accuracy to individual synapses and highly truncated neurites, enabling neurite-enhanced connectomics analysis. Finally, we present a data-driven, unsupervised study of synaptic structural variation on the representation manifold, revealing its intrinsic axes of variation and showing that representations contain inhibitory subtype information.

Application of machine learning in low-order manifold representation of chemistry in turbulent flames

The Uniform Conditional State (UCS) and the Multidimensional Flamelet Manifold (MFM) models are methods for the tabulation of chemistry in simulations of turbulent flames. The high-dimensionality of the tables these models generate and many possible combinations of the values for the input variables necessitate the allocation of a considerable size of memory during CFD calculations. This issue becomes even more problematic when adding more conditioning variables to the model. In this study, two Artificial Intelligence (AI)-based approaches referred to as Decision Tree (DT) and Artificial Neural Network (ANN) are developed and tested to provide in situ chemistry representation. The goal is to predict four parameters (outputs) accurately with low memory demand and computational cost. The trained AI models are then employed for simulation of a turbulent premixed flame. Comparison of the results from the AI-based approaches to those from the conventional UCS model shows acceptable agreement. The memory and CPU requirements from the different approaches are compared. It is found that the ANN model reduces the size of the chemistry table by around 92%. Conversely, the DT-based model reduces the size of the chemistry model by only 40%. The CPU time for using the DT model during the CFD calculations was around 10% shorter than the conventional approach while it was 8% higher for the ANN model. It was concluded that, based on the particular applications, different AI-based methods can facilitate an efficient representation of the chemistry manifold.

Unsupervised multilayer fuzzy neural networks for image clustering

Currently, labelling a large number of images is still a very challenging task. To tackle the problem of unlabelled data, unsupervised learning has been proposed. Among many unsupervised learning algorithms, K-means is the most popular algorithm. However, in a low-dimensional space, fuzzy c-means, which is more robust and less sensitive to initialization, has several advantages over K-means clustering. On the other hand, stacked convolutional pooling structures and manifold representation play pivotal roles in image clustering. In this paper, we propose an unsupervised multilayer fuzzy neural network for image clustering that unifies fuzzy systems, multilayer convolutional structures and manifold representation. The main contributions are as follows. First, we extend fuzzy systems to unsupervised tasks by introducing manifold representation, which expands the applications of fuzzy systems. Next, we propose the idea of using only a small number of attributes to compute firing strengths. This is implemented to prevent the firing strengths from falling to zero. Finally, randomly generated convolutional weights are used to extract features, which is a good choice for data without labels. It is demonstrated on a wide range of image datasets that the proposed approach is competitive with existing fuzzy and nonfuzzy clustering algorithms.

Linear Manifold Modeling and Graph Estimation based on Multivariate Functional Data with Different Coarseness Scales

We develop a high-dimensional graphical modeling approach for functional data where the number of functions exceeds the available sample size. This is accomplished by proposing a sparse estimator for a concentration matrix when identifying linear manifolds. As such, the procedure extends the ideas of the manifold representation for functional data to high-dimensional settings where the number of functions is larger than the sample size. By working in a penalized setting it enriches the functional data framework by estimating sparse undirected graphs that show how functional nodes connect to other functional nodes. The procedure allows multiple coarseness scales to be present in the data and proposes a simultaneous estimation of several related graphs. Its performance is illustrated using a real-life fMRI dataset and with simulated data.