Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

R-MFE-TCN: A correlation prediction model between body surface and tumor during respiratory movement

BACKGROUND

2D CT image-guided radiofrequency ablation (RFA) is an exciting minimally invasive treatment that can destroy liver tumors without removing them. However, CT images can only provide limited static information, and the tumor will move with the patient's respiratory movement. Therefore, how to accurately locate tumors under free conditions is an urgent problem to be solved at present.

PURPOSE

The purpose of this study is to propose a respiratory correlation prediction model for mixed reality surgical assistance system, Riemannian and Multivariate Feature Enhanced Temporal Convolutional Network (R-MFE-TCN), and to achieve accurate respiratory correlation prediction.

METHODS

The model adopts a respiration-oriented Riemannian information enhancement strategy to expand the diversity of the dataset. A new Multivariate Feature Enhancement module (MFE) is proposed to retain respiratory data information, so that the network can fully explore the correlation of internal and external data information, the dual-channel is used to retain multivariate respiratory feature, and the Multi-headed Self-attention obtains respiratory peak-to-valley value periodic information. This information significantly improves the prediction performance of the network. At the same time, the PSO algorithm is used for hyperparameter optimization. In the experiment, a total of seven patients' internal and external respiratory motion trajectories were obtained from the dataset, and the first six patients were selected as the training set. The respiratory signal collection frequency was 21 Hz.

RESULTS

A large number of experiments on the dataset prove the good performance of this method, which improves the prediction accuracy while also having strong robustness. This method can reduce the delay deviation under long window prediction and achieve good performance. In the case of 400 ms, the average RMSE and MAE are 0.0453  and 0.0361 mm, respectively, which is better than other research methods.

CONCLUSION

The R-MFE-TCN can be extended to respiratory correlation prediction in different clinical situations, meeting the accuracy requirements for respiratory delay prediction in surgical assistance.

Joint Metric Learning-Based Class-Specific Representation for Image Set Classification

With the rapid advances in digital imaging and communication technologies, recently image set classification has attracted significant attention and has been widely used in many real-world scenarios. As an effective technology, the class-specific representation theory-based methods have demonstrated their superior performances. However, this type of methods either only uses one gallery set to measure the gallery-to-probe set distance or ignores the inner connection between different metrics, leading to the learned distance metric lacking robustness, and is sensitive to the size of image sets. In this article, we propose a novel joint metric learning-based class-specific representation framework (JMLC), which can jointly learn the related and unrelated metrics. By iteratively modeling probe set and related or unrelated gallery sets as affine hull, we reconstruct this hull sparsely or collaboratively over another image set. With the obtained representation coefficients, the combined metric between the query set and the gallery set can then be calculated. In addition, we also derive the kernel extension of JMLC and propose two new unrelated set constituting strategies. Specifically, kernelized JMLC (KJMLC) embeds the gallery sets and probe sets into the high-dimensional Hilbert space, and in the kernel space, the data become approximately linear separable. Extensive experiments on seven benchmark databases show the superiority of the proposed methods to the state-of-the-art image set classifiers.

Manifold Learning-Based Common Spatial Pattern for EEG Signal Classification

EEG signal classification using Riemannian manifolds has shown great potential. However, the huge computational cost associated with Riemannian metrics poses challenges for applying Riemannian methods, particularly in high-dimensional feature data. To address these, we propose an efficient ensemble method called MLCSP-TSE-MLP, which aims to reduce the computational cost while achieving superior performance. MLCSP of the ensemble utilizes a Riemannian graph embedding strategy to learn intrinsic low-dimensional sub-manifolds, enhancing discrimination. TSE uses the Euclidean mean as the reference point for tangent space mapping and reducing computational cost. Finally, the ensemble incorporates the MLP classifier to offer improved classification performance. Classification results conducted on three datasets demonstrate that MLCSP-TSE-MLP achieves significant superior performance compared to various competing methods. Notably, the MLCSP-TSE module achieves a remarkable increase in training speed and exhibits much lower test time compared to traditional Riemannian methods. Based on these results, we believe that the proposed MLCSP-TSE-MLP is a powerful tool for handling high-dimensional data and holds great potential for practical applications.

Learning to Optimize on Riemannian Manifolds

Many learning tasks are modeled as optimization problems with nonlinear constraints, such as principal component analysis and fitting a Gaussian mixture model. A popular way to solve such problems is resorting to Riemannian optimization algorithms, which yet heavily rely on both human involvement and expert knowledge about Riemannian manifolds. In this paper, we propose a Riemannian meta-optimization method to automatically learn a Riemannian optimizer. We parameterize the Riemannian optimizer by a novel recurrent network and utilize Riemannian operations to ensure that our method is faithful to the geometry of manifolds. The proposed method explores the distribution of the underlying data by minimizing the objective of updated parameters, and hence is capable of learning task-specific optimizations. We introduce a Riemannian implicit differentiation training scheme to achieve efficient training in terms of numerical stability and computational cost. Unlike conventional meta-optimization training schemes that need to differentiate through the whole optimization trajectory, our training scheme is only related to the final two optimization steps. In this way, our training scheme avoids the exploding gradient problem, and significantly reduces the computational load and memory footprint. We discuss experimental results across various constrained problems, including principal component analysis on Grassmann manifolds, face recognition, person re-identification, and texture image classification on Stiefel manifolds, clustering and similarity learning on symmetric positive definite manifolds, and few-shot learning on hyperbolic manifolds.

Functional connectivity learning via Siamese-based SPD matrix representation of brain imaging data

Measurement of brain functional connectivity has become a dominant approach to explore the interaction dynamics between brain regions of subjects under examination. Conventional functional connectivity measures largely originate from deterministic models on empirical analysis, usually demanding application-specific settings (e.g., Pearson's Correlation and Mutual Information). To bridge the technical gap, this study proposes a Siamese-based Symmetric Positive Definite (SPD) Matrix Representation framework (SiameseSPD-MR) to derive the functional connectivity of brain imaging data (BID) such as Electroencephalography (EEG), thus the alternative application-independent measure (in the form of SPD matrix) can be automatically learnt: (1) SiameseSPD-MR first exploits graph convolution to extract the representative features of BID with the adjacency matrix computed considering the anatomical structure; (2) Adaptive Gaussian kernel function then applies to obtain the functional connectivity representations from the deep features followed by SPD matrix transformation to address the intrinsic functional characteristics; and (3) Two-branch (Siamese) networks are combined via an element-wise product followed by a dense layer to derive the similarity between the pairwise inputs. Experimental results on two EEG datasets (autism spectrum disorder, emotion) indicate that (1) SiameseSPD-MR can capture more significant differences in functional connectivity between neural states than the state-of-the-art counterparts do, and these findings properly highlight the typical EEG characteristics of ASD subjects, and (2) the obtained functional connectivity representations conforming to the proposed measure can act as meaningful markers for brain network analysis and ASD discrimination.

U-SPDNet: An SPD manifold learning-based neural network for visual classification

With the development of neural networking techniques, several architectures for symmetric positive definite (SPD) matrix learning have recently been put forward in the computer vision and pattern recognition (CV&PR) community for mining fine-grained geometric features. However, the degradation of structural information during multi-stage feature transformation limits their capacity. To cope with this issue, this paper develops a U-shaped neural network on the SPD manifolds (U-SPDNet) for visual classification. The designed U-SPDNet contains two subsystems, one of which is a shrinking path (encoder) making up of a prevailing SPD manifold neural network (SPDNet (Huang and Van Gool, 2017)) for capturing compact representations from the input data. Another is a constructed symmetric expanding path (decoder) to upsample the encoded features, trained by a reconstruction error term. With this design, the degradation problem will be gradually alleviated during training. To enhance the representational capacity of U-SPDNet, we also append skip connections from encoder to decoder, realized by manifold-valued geometric operations, namely Riemannian barycenter and Riemannian optimization. On the MDSD, Virus, FPHA, and UAV-Human datasets, the accuracy achieved by our method is respectively 6.92%, 8.67%, 1.57%, and 1.08% higher than SPDNet, certifying its effectiveness.

Learning Log-Determinant Divergences for Positive Definite Matrices

Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the αβ-log-det divergence, which is a meta-divergence parametrized by scalars α and β, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian gradient descent schemes for optimizing our formulations efficiently, and show the usefulness of our method on eight standard computer vision tasks.

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Motor Imagery Classification via Kernel-Based Domain Adaptation on an SPD Manifold

Background: Recording the calibration data of a brain–computer interface is a laborious process and is an unpleasant experience for the subjects. Domain adaptation is an effective technology to remedy the shortage of target data by leveraging rich labeled data from the sources. However, most prior methods have needed to extract the features of the EEG signal first, which triggers another challenge in BCI classification, due to small sample sets or a lack of labels for the target. Methods: In this paper, we propose a novel domain adaptation framework, referred to as kernel-based Riemannian manifold domain adaptation (KMDA). KMDA circumvents the tedious feature extraction process by analyzing the covariance matrices of electroencephalogram (EEG) signals. Covariance matrices define a symmetric positive definite space (SPD) that can be described by Riemannian metrics. In KMDA, the covariance matrices are aligned in the Riemannian manifold, and then are mapped to a high dimensional space by a log-Euclidean metric Gaussian kernel, where subspace learning is performed by minimizing the conditional distribution distance between the sources and the target while preserving the target discriminative information. We also present an approach to convert the EEG trials into 2D frames (E-frames) to further lower the dimension of covariance descriptors. Results: Experiments on three EEG datasets demonstrated that KMDA outperforms several state-of-the-art domain adaptation methods in classification accuracy, with an average Kappa of 0.56 for BCI competition IV dataset IIa, 0.75 for BCI competition IV dataset IIIa, and an average accuracy of 81.56% for BCI competition III dataset IVa. Additionally, the overall accuracy was further improved by 5.28% with the E-frames. KMDA showed potential in addressing subject dependence and shortening the calibration time of motor imagery-based brain–computer interfaces.

A Framework for Short Video Recognition Based on Motion Estimation and Feature Curves on SPD Manifolds

Given the prosperity of video media such as TikTok and YouTube, the requirement of short video recognition is becoming more and more urgent. A significant feature of short video is that there are few switches of scenes in short video, and the target (e.g., the face of the key person in the short video) often runs through the short video. This paper presents a new short video recognition algorithm framework that transforms a short video into a family of feature curves on symmetric positive definite (SPD) manifold as the basis of recognition. Thus far, no similar algorithm has been reported. The results of experiments suggest that our method performs better on three changeling databases than seven other related algorithms published in the top issues.

SymNet: A Simple Symmetric Positive Definite Manifold Deep Learning Method for Image Set Classification

By representing each image set as a nonsingular covariance matrix on the symmetric positive definite (SPD) manifold, visual classification with image sets has attracted much attention. Despite the success made so far, the issue of large within-class variability of representations still remains a key challenge. Recently, several SPD matrix learning methods have been proposed to assuage this problem by directly constructing an embedding mapping from the original SPD manifold to a lower dimensional one. The advantage of this type of approach is that it cannot only implement discriminative feature selection but also preserve the Riemannian geometrical structure of the original data manifold. Inspired by this fact, we propose a simple SPD manifold deep learning network (SymNet) for image set classification in this article. Specifically, we first design SPD matrix mapping layers to map the input SPD matrices into new ones with lower dimensionality. Then, rectifying layers are devised to activate the input matrices for the purpose of forming a valid SPD manifold, chiefly to inject nonlinearity for SPD matrix learning with two nonlinear functions. Afterward, we introduce pooling layers to further compress the input SPD matrices, and the log-map layer is finally exploited to embed the resulting SPD matrices into the tangent space via log-Euclidean Riemannian computing, such that the Euclidean learning applies. For SymNet, the (2-D)²principal component analysis (PCA) technique is utilized to learn the multistage connection weights without requiring complicated computations, thus making it be built and trained easier. On the tail of SymNet, the kernel discriminant analysis (KDA) algorithm is coupled with the output vectorized feature representations to perform discriminative subspace learning. Extensive experiments and comparisons with state-of-the-art methods on six typical visual classification tasks demonstrate the feasibility and validity of the proposed SymNet.

Symmetric positive definite manifold learning and its application in fault diagnosis

Locally linear embedding (LLE) is an effective tool to extract the significant features from a dataset. However, most of the relevant existing algorithms assume that the original dataset resides on a Euclidean space, unfortunately nearly all the original data space is non-Euclidean. In addition, the original LLE does not use the discriminant information of the dataset, which will degrade its performance in feature extraction. To address these problems raised in the conventional LLE, we first employ the original dataset to construct a symmetric positive definite manifold, and then estimate the tangent space of this manifold. Furthermore, the local and global discriminant information are integrated into the LLE, and the improved LLE is operated in the tangent space to extract the important features. We introduce Iris dataset to analyze the capability of the proposed method to extract features. Finally, several experiments are performed on five machinery datasets, and experimental results indicate that our proposed method can extract the excellent low-dimensional representations of the original dataset. Compared with the state-of-the-art methods, the proposed algorithm shows a strong capability for fault diagnosis.

Common Spatial Pattern Reformulated for Regularizations in Brain-Computer Interfaces

Common spatial pattern (CSP) is one of the most successful feature extraction algorithms for brain-computer interfaces (BCIs). It aims to find spatial filters that maximize the projected variance ratio between the covariance matrices of the multichannel electroencephalography (EEG) signals corresponding to two mental tasks, which can be formulated as a generalized eigenvalue problem (GEP). However, it is challenging in principle to impose additional regularization onto the CSP to obtain structural solutions (e.g., sparse CSP) due to the intrinsic nonconvexity and invariance property of GEPs. This article reformulates the CSP as a constrained minimization problem and establishes the equivalence of the reformulated and the original CSPs. An efficient algorithm is proposed to solve this optimization problem by alternately performing singular value decomposition (SVD) and least squares. Under this new formulation, various regularization techniques for linear regression can then be easily implemented to regularize the CSPs for different learning paradigms, such as the sparse CSP, the transfer CSP, and the multisubject CSP. Evaluations on three BCI competition datasets show that the regularized CSP algorithms outperform other baselines, especially for the high-dimensional small training set. The extensive results validate the efficiency and effectiveness of the proposed CSP formulation in different learning contexts.

Spontaneous and deliberate modes of creativity: Multitask eigen-connectivity analysis captures latent cognitive modes during creative thinking

Despite substantial progress in the quest of demystifying the brain basis of creativity, several questions remain open. One such issue concerns the relationship between two latent cognitive modes during creative thinking, i.e., deliberate goal-directed cognition and spontaneous thought generation. Although an interplay between deliberate and spontaneous thinking is often implicated in the creativity literature (e.g., dual-process models), a bottom-up data-driven validation of the cognitive processes associated with creative thinking is still lacking. Here, we attempted to capture the latent modes of creative thinking by utilizing a data-driven approach on a novel continuous multitask paradigm (CMP) that widely sampled a hypothetical two-dimensional cognitive plane of deliberate and spontaneous thinking in a single fMRI session. The CMP consisted of eight task blocks ranging from undirected mind wandering to goal-directed working memory task, while also included two widely-used creativity tasks, i.e., alternate uses task (AUT) and remote association task (RAT). Using eigen-connectivity (EC) analysis on the multitask whole-brain functional connectivity (FC) patterns, we embedded the multitask FCs into a low-dimensional latent space. The first two latent components, as revealed by the EC analysis, broadly mapped onto the two cognitive modes of deliberate and spontaneous thinking, respectively. Further, in this low-dimensional space, both creativity tasks were located in the upper right corner of high deliberate and spontaneous thinking (creative cognitive space). Neuroanatomically, the creative cognitive space was represented by not only increased intra-network connectivity within executive control and default mode network, but also by higher coupling between the two canonical brain networks. Further, individual differences reflected in the low-dimensional connectivity embeddings were related to differences in deliberate and spontaneous thinking abilities. Altogether, using a continuous multitask paradigm and a data-driven approach, we provide initial empirical evidence for the contribution of both deliberate and spontaneous modes of cognition during creative thinking.

Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.

Heterogeneous Clutter Suppression for Airborne Radar STAP Based on Matrix Manifolds

Clutter suppression in heterogeneous environments is a serious challenge for airborne radar. To address this problem, a matrix-manifold-based clutter suppression method is proposed. First, the distributions of training data in heterogeneous environments are analyzed, while the received data are characterized on a Riemannian manifold of Hermitian positive definite matrices. It is indicated that the training data with different distributions with the same power are separated, whereas data with the same distribution are closer together. This implies that the underlying geometry of the data can be better revealed by manifolds than by Euclidean space. Based on these properties, homogeneous training data are selected by establishing a binary hypothesis test such that the negative effects of the use of heterogeneous samples are alleviated. Moreover, as exploiting a geometric metric on manifolds to reveal the underlying information of data, experimental results on both simulated and real data validate that the proposed method has a superior performance with small sample support.

A Robust Context-Based Deep Learning Approach for Highly Imbalanced Hyperspectral Classification

Hyperspectral imaging is an area of active research with many applications in remote sensing, mineral exploration, and environmental monitoring. Deep learning and, in particular, convolution-based approaches are the current state-of-the-art classification models. However, in the presence of noisy hyperspectral datasets, these deep convolutional neural networks underperform. In this paper, we proposed a feature augmentation approach to increase noise resistance in imbalanced hyperspectral classification. Our method calculates context-based features, and it uses a deep convolutional neuronet (DCN). We tested our proposed approach on the Pavia datasets and compared three models, DCN, PCA + DCN, and our context-based DCN, using the original datasets and the datasets plus noise. Our experimental results show that DCN and PCA + DCN perform well on the original datasets but not on the noisy datasets. Our robust context-based DCN was able to outperform others in the presence of noise and was able to maintain a comparable classification accuracy on clean hyperspectral images.

Probabilistic learning vector quantization on manifold of symmetric positive definite matrices

In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive definite matrices, which are inherently points that live on a curved Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, traditional Euclidean machine learning algorithms yield poor results on such data. In this paper, we generalize the probabilistic learning vector quantization algorithm for data points living on the manifold of symmetric positive definite matrices equipped with Riemannian natural metric (affine-invariant metric). By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent. Empirical investigations on synthetic data, image data , and motor imagery electroencephalogram (EEG) data demonstrate the superior performance of the proposed method.

Dimensionality Transcending: A Method for Merging BCI Datasets With Different Dimensionalities

OBJECTIVE

We present a transfer learning method for datasets with different dimensionalities, coming from different experimental setups but representing the same physical phenomena. We focus on the case where the data points are symmetric positive definite (SPD) matrices describing the statistical behavior of EEG-based brain computer interfaces (BCI).

METHOD

Our proposal uses a two-step procedure that transforms the data points so that they become matched in terms of dimensionality and statistical distribution. In the dimensionality matching step, we use isometric transformations to map each dataset into a common space without changing their geometric structures. The statistical matching is done using a domain adaptation technique adapted for the intrinsic geometry of the space where the datasets are defined.

RESULTS

We illustrate our proposal on time series obtained from BCI systems with different experimental setups (e.g., different number of electrodes, different placement of electrodes). The results show that the proposed method can be used to transfer discriminative information between BCI recordings that, in principle, would be incompatible.

CONCLUSION AND SIGNIFICANCE

Such findings pave the way to a new generation of BCI systems capable of reusing information and learning from several sources of data despite differences in their electrodes positioning.

Hierarchical Gaussian Descriptors with Application to Person Re-Identification

Describing the color and textural information of a person image is one of the most crucial aspects of person re-identification (re-id). Although a covariance descriptor has been successfully applied to person re-id, it loses the local structure of a region and mean information of pixel features, both of which tend to be the major discriminative information for person re-id. In this paper, we present novel meta-descriptors based on a hierarchical Gaussian distribution of pixel features, in which both mean and covariance information are included in patch and region level descriptions. More specifically, the region is modeled as a set of multiple Gaussian distributions, each of which represents the appearance of a local patch. The characteristics of the set of Gaussian distributions are again described by another Gaussian distribution. Because the space of Gaussian distribution is not a linear space, we embed the parameters of the distribution into a point of Symmetric Positive Definite (SPD) matrix manifold in both steps. We show, for the first time, that normalizing the scale of the SPD matrix enhances the hierarchical feature representation on this manifold. Additionally, we develop feature norm normalization methods with the ability to alleviate the biased trends that exist on the SPD matrix descriptors. The experimental results conducted on five public datasets indicate the effectiveness of the proposed descriptors and the two types of normalizations.

A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold

The symmetric positive definite (SPD) matrices, forming a Riemannian manifold, are commonly used as visual representations. The non-Euclidean geometry of the manifold often makes developing learning algorithms (e.g., classifiers) difficult and complicated. The concept of similarity-based learning has been shown to be effective to address various problems on SPD manifolds. This is mainly because the similarity-based algorithms are agnostic to the geometry and purely work based on the notion of similarities/distances. However, existing similarity-based models on SPD manifolds opt for holistic representations, ignoring characteristics of information captured by SPD matrices. To circumvent this limitation, we propose a novel SPD distance measure for the similarity-based algorithm. Specifically, we introduce the concept of point-to-set transformation, which enables us to learn multiple lower dimensional and discriminative SPD manifolds from a higher dimensional one. For lower dimensional SPD manifolds obtained by the point-to-set transformation, we propose a tailored set-to-set distance measure by making use of the family of alpha-beta divergences. We further propose to learn the point-to-set transformation and the set-to-set distance measure jointly, yielding a powerful similarity-based algorithm on SPD manifolds. Our thorough evaluations on several visual recognition tasks (e.g., action classification and face recognition) suggest that our algorithm comfortably outperforms various state-of-the-art algorithms.

Riemannian classifier enhances the accuracy of machine-learning-based diagnosis of PTSD using resting EEG

Recently, objective and automated methods for the diagnosis of post-traumatic stress disorder (PTSD) have attracted increasing attention. However, previous studies on machine-learning-based diagnosis of PTSD with resting-state electroencephalogram (EEG) have reported poor accuracies of as low as 60%. Here, a Riemannian geometry-based classifier, the Fisher geodesic minimum distance to the mean (FgMDM), was employed for PTSD classification for the first time. Eyes-closed resting-state EEG data of 39 healthy individuals and 42 PTSD patients were used for the analysis. EEG source activities in 148 cortical regions were parcellated based on the Destrieux atlas, and their covariances were evaluated for each individual. Thirty epochs of preprocessed EEG were employed to calculate source activities. In addition, the FgMDM approach was applied to each EEG source covariance to construct the classifier. For a comparison, linear discriminant analysis (LDA), support vector machine (SVM), and random forest (RF) classifiers employing source band powers and network features as feature candidates were also tested. The FgMDM classifier showed an average classification accuracy of 75.240.80%. In contrast, the maximum accuracies of LDA, SVM, and RF classifiers were 66.54 ± 2.99%, 61.11 ± 2.98%, and 60.99 ± 2.19%, respectively. Our study demonstrated that the diagnostic accuracy of PTSD with resting-state EEG could be significantly improved by employing the FgMDM framework, which is a type of Riemannian geometry-based classifier.

Spectral Convolution Feature-Based SPD Matrix Representation for Signal Detection Using a Deep Neural Network

Convolutional neural networks have powerful performances in many visual tasks because of their hierarchical structures and powerful feature extraction capabilities. SPD (symmetric positive definition) matrix is paid attention to in visual classification, because it has excellent ability to learn proper statistical representation and distinguish samples with different information. In this paper, a deep neural network signal detection method based on spectral convolution features is proposed. In this method, local features extracted from convolutional neural network are used to construct the SPD matrix, and a deep learning algorithm for the SPD matrix is used to detect target signals. Feature maps extracted by two kinds of convolutional neural network models are applied in this study. Based on this method, signal detection has become a binary classification problem of signals in samples. In order to prove the availability and superiority of this method, simulated and semi-physical simulated data sets are used. The results show that, under low SCR (signal-to-clutter ratio), compared with the spectral signal detection method based on the deep neural network, this method can obtain a gain of 0.5-2 dB on simulated data sets and semi-physical simulated data sets.

Matrix Information Geometry for Spectral-Based SPD Matrix Signal Detection with Dimensionality Reduction

In this paper, a novel signal detector based on matrix information geometric dimensionality reduction (DR) is proposed, which is inspired from spectrogram processing. By short time Fourier transform (STFT), the received data are represented as a 2-D high-precision spectrogram, from which we can well judge whether the signal exists. Previous similar studies extracted insufficient information from these spectrograms, resulting in unsatisfactory detection performance especially for complex signal detection task at low signal-noise-ratio (SNR). To this end, we use a global descriptor to extract abundant features, then exploit the advantages of matrix information geometry technique by constructing the high-dimensional features as symmetric positive definite (SPD) matrices. In this case, our task for signal detection becomes a binary classification problem lying on an SPD manifold. Promoting the discrimination of heterogeneous samples through information geometric DR technique that is dedicated to SPD manifold, our proposed detector achieves satisfactory signal detection performance in low SNR cases using the K distribution simulation and the real-life sea clutter data, which can be widely used in the field of signal detection.

Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network

The symmetric positive definite (SPD) matrix has attracted much attention in classification problems because of its remarkable performance, which is due to the underlying structure of the Riemannian manifold with non-negative curvature as well as the use of non-linear geometric metrics, which have a stronger ability to distinguish SPD matrices and reduce information loss compared to the Euclidean metric. In this paper, we propose a spectral-based SPD matrix signal detection method with deep learning that uses time-frequency spectra to construct SPD matrices and then exploits a deep SPD matrix learning network to detect the target signal. Using this approach, the signal detection problem is transformed into a binary classification problem on a manifold to judge whether the input sample has target signal or not. Two matrix models are applied, namely, an SPD matrix based on spectral covariance and an SPD matrix based on spectral transformation. A simulated-signal dataset and a semi-physical simulated-signal dataset are used to demonstrate that the spectral-based SPD matrix signal detection method with deep learning has a gain of 1.7–3.3 dB under appropriate conditions. The results show that our proposed method achieves better detection performances than its state-of-the-art spectral counterparts that use convolutional neural networks.

Analyzing Dynamical Brain Functional Connectivity as Trajectories on Space of Covariance Matrices

Human brain functional connectivity (FC) is often measured as the similarity of functional MRI responses across brain regions when a brain is either resting or performing a task. This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs). We use a recently developed metric on the space of SPDMs for quantifying differences across FC observations, and for clustering and classification of FC trajectories. To facilitate large scale and high-dimensional data analysis, we propose a novel, metric-based dimensionality reduction technique to reduce data from large SPDMs to small SPDMs. We illustrate this comprehensive framework using data from the Human Connectome Project (HCP) database for multiple subjects and tasks, with task classification rates that match or outperform state-of-the-art techniques.

A Riemannian Geometry Approach to Reduced and Discriminative Covariance Estimation in Brain Computer Interfaces

OBJECTIVE

Spatial covariance matrices are extensively employed as brain activity descriptors in brain computer interface (BCI) research that, typically, involve the whole array of sensors. Here, we introduce a methodological framework for delineating the subset of sensors, the covariance structure of which offers a reduced, but more powerful, representation of brain's coordination patterns that ultimately leads to reliable mind reading.

METHODS

Adopting a Riemannian geometry approach, we turn the problem of sensor selection as a maximization of a functional that is computed over the manifold of symmetric positive definite (SPD) matrices and encapsulates class separability in a way that facilitates the search among subsets of different size. The introduced optimization task, namely discriminative covariance reduction (DCR), lacks an analytical solution and is tackled via the cross-entropy optimization technique.

RESULTS

Based on two different EEG datasets and three distinct classification schemes, we demonstrate that the DCR approach provides a noteworthy gain in terms of accuracy (in some cases exceeding 20%) and a remarkable reduction in classification time (on average 82%). Additionally, results include the intriguing empirical finding that the pattern of selected sensors in the case of disabled persons depends on the type of disability.

CONCLUSION

The proposed DCR framework can speed up the classification time in BCI-systems operating on the SPD manifolds by simultaneously enhancing their reliability. This is achieved without sacrificing the neuroscientific interpretability endowed in the topographical arrangement of the selected sensors.

SIGNIFICANCE

Riemannian geometry is exploited for DCR in BCI systems, in a dimensionality-agnostic manner, guaranteeing improved performance.

Sensors Information Fusion System with Fault Detection Based on Multi-Manifold Regularization Neighborhood Preserving Embedding

Electrical drive systems play an increasingly important role in high-speed trains. The whole system is equipped with sensors that support complicated information fusion, which means the performance around this system ought to be monitored especially during incipient changes. In such situation, it is crucial to distinguish faulty state from observed normal state because of the dire consequences closed-loop faults might bring. In this research, an optimal neighborhood preserving embedding (NPE) method called multi-manifold regularization NPE (MMRNPE) is proposed to detect various faults in an electrical drive sensor information fusion system. By taking locality preserving embedding into account, the proposed methodology extends the united application of Euclidean distance of both designated points and paired points, which guarantees the access to both local and global sensor information. Meanwhile, this structure fuses several manifolds to extract their own features. In addition, parameters are allocated in diverse manifolds to seek an optimal combination of manifolds while entropy of information with parameters is also selected to avoid the overweight of single manifold. Moreover, an experimental test based on the platform was built to validate the MMRNPE approach and demonstrate the effectiveness of the fault detection. Results and observations show that the proposed MMRNPE offers a better fault detection representation in comparison with NPE.

Supervised Dimensionality Reduction on Grassmannian for Image Set Recognition

Modeling videos and image sets by linear subspaces has achieved great success in various visual recognition tasks. However, subspaces constructed from visual data are always notoriously embedded in a high-dimensional ambient space, which limits the applicability of existing techniques. This letter explores the possibility of proposing a geometry-aware framework for constructing lower-dimensional subspaces with maximum discriminative power from high-dimensional subspaces in the supervised scenario. In particular, we make use of Riemannian geometry and optimization techniques on matrix manifolds to learn an orthogonal projection, which shows that the learning process can be formulated as an unconstrained optimization problem on a Grassmann manifold. With this natural geometry, any metric on the Grassmann manifold can theoretically be used in our model. Experimental evaluations on several data sets show that our approach results in significantly higher accuracy than other state-of-the-art algorithms.

Robust Visual Tracking Based on Adaptive Convolutional Features and Offline Siamese Tracker

Robust and accurate visual tracking is one of the most challenging computer vision problems. Due to the inherent lack of training data, a robust approach for constructing a target appearance model is crucial. The existing spatially regularized discriminative correlation filter (SRDCF) method learns partial-target information or background information when experiencing rotation, out of view, and heavy occlusion. In order to reduce the computational complexity by creating a novel method to enhance tracking ability, we first introduce an adaptive dimensionality reduction technique to extract the features from the image, based on pre-trained VGG-Net. We then propose an adaptive model update to assign weights during an update procedure depending on the peak-to-sidelobe ratio. Finally, we combine the online SRDCF-based tracker with the offline Siamese tracker to accomplish long term tracking. Experimental results demonstrate that the proposed tracker has satisfactory performance in a wide range of challenging tracking scenarios.

Heteroscedastic Max-min Distance Analysis for Dimensionality Reduction

Max-min distance analysis (MMDA) performs dimensionality reduction by maximizing the minimum pairwise distance between classes in the latent subspace under the homoscedastic assumption, which can address the class separation problem caused by the Fisher criterion, but is incapable of tackling heteroscedastic data properly. In this paper, we propose two heteroscedastic MMDA (HMMDA) methods to employ the differences of class covariances. Whitened HMMDA (WHMMDA) extends MMDA by utilizing the Chernoff distance as the separability measure between classes in the whitened space. Orthogonal HMMDA (OHMMDA) incorporates the maximization of the minimal pairwise Chernoff distance and the minimization of class compactness into a trace quotient formulation with an orthogonal constraint of the transformation, which can be solved by bisection search. Two variants of OHMMDA further encode the margin information by using only neighboring samples to construct the intra-class and inter-class scatters. Experiments on several UCI datasets and two face databases demonstrate the effectiveness of the HMMDA methods.