Principal Angles Research Articles

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

Characterizing the microstructures of mammalian enamel by synchrotron phase contrast microCT

The enamel of mammalian teeth is a highly mineralized tissue that must endure a lifetime of cyclic contact and is inspiring the development of next-generation engineering materials. Attempts to implement enamel-inspired structures in synthetic materials have had limited success, largely due to the absence of a detailed understanding of its microstructure. The present work used synchrotron phase-contrast microCT imaging to evaluate the three-dimensional microstructure of enamel from four mammals including Lion, Gray Wolf, Snow Leopard, and Black Bear. Quantitative results of image analysis revealed that the decussation pattern of enamel consists of discrete diazone (D) and parazone (P) bands of rods organized with stacking arrangement of D+/P/D-/P in all mammals evaluated; the D+ and D- refer to distinct diazone bands with juxtaposed rod orientations from the reference plane. Furthermore, the rod orientations in the bands can be described in terms of two principal angles, defined here as the pitch and yaw. While the pitch angle increases from the outer enamel to a maximum (up to ≈ 40°) near the dentin enamel junction, minimal spatial variations are observed in yaw across the enamel thickness. There are clear differences in the decussation parameters of enamel across species that are interpreted here with respect to the structural demands placed on their teeth. The rod pitch and band width of enamel are identified as important design parameters and appear to be correlated with the bite force quotient of the four mammals evaluated. Statement of significanceThe multi-functionality of tooth enamel requires both hardness and resistance to fracture, properties that are generally mutually exclusive. Ubiquitous to all mammalian teeth, the enamel is expected to have undergone adaptations in microstructure to accommodate the differences in diet, body size and bite force across animals. For the first time, we compare the complex three-dimensional microstructure of enamel from teeth of multiple mammalian species using synchrotron micro-computed tomography. The findings provide new understanding of the “design” of mammalian enamel microstructures, as well as how specific parameters associated with the decussation of rods appear to be engineered to modulate its fracture resistance.

Anisotropic Quantum Hall Droplets

We study two-dimensional (2D) droplets of noninteracting electrons in a strong magnetic field, placed in a confining potential with arbitrary shape. Using semiclassical methods adapted to the lowest Landau level, we obtain near-Gaussian energy eigenstates that are localized on level curves of the potential and have a position-dependent height. This one-particle insight allows us to deduce explicit formulas for expectation values of local many-body observables, such as density and current, in the thermodynamic limit. In particular, correlations along the edge are long-ranged and inhomogeneous. As we show, this is consistent with the system’s universal low-energy description as a free 1D chiral conformal field theory of edge modes, known from earlier works in simple geometries. A delicate interplay between radial and angular dependencies of eigenfunctions ultimately ensures that the theory is homogeneous in terms of the canonical angle variable of the potential, despite its apparent inhomogeneity in terms of more naïve angular coordinates. Finally, we propose a scheme to measure the anisotropy by subjecting the droplet to microwave radiation; we compute the corresponding absorption rate and show that it depends on the droplet’s shape and the waves’ polarization. These results, both local and global, are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential. Published by the American Physical Society 2024

A closer look at some new lower bounds on the minimum singular value of a matrix

There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, P+Q, in relation to the eigenvalues of P and Q. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, λmin(P+Q), under the conditions that matrices P and Q are symmetric positive semi-definite and their sum P+Q is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices P and Q, which are denoted by R(P) and R(Q), respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on λmin(P+Q) is the distinction of the case where R(P) and R(Q) have no principal angles between 0 and π2. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace R(Q), which is employed to formulate the lower bounds for λmin(P+Q). At last, an example that showcases the potential for improving these two lower bounds is presented.

A novel unsupervised domain adaptation-based method for lithium-ion batteries state of health prognostic

Precise state-of-health prognostic is crucial for the safe and reliable operation of Lithium-ion batteries in energy storage systems. Nevertheless, most data-driven state-of-health prognostic methods are only effective when the distribution of laboratory training data and test data is consistent. This paper proposes an unsupervised domain adaptation method for individualized SOH estimation. More specifically, a convolutional neural network is used to automatically extract battery aging features from raw charging voltage curves. Besides, we define a geometrical distance between different domain feature subspaces based on the principal angles and learn deep transferable features by minimizing it. In this way, we can narrow the domain gap by using orthogonal bases of the feature spaces, while avoiding changes in feature scales. To maintain the geometrical properties of feature subspaces to the maximum extent possible, we further apply the bases mismatch penalization to penalize the matching of orthogonal bases of different importance. Our method is evaluated on three different datasets with different chemistries, profiles, and ambient temperatures to validate the effectiveness and generalizability. Our method outperforms several other domain adaptation approaches significantly. The results show that the proposed transfer learning-based method is widely applicable and can greatly shorten the duration of battery aging experiments.

Phasic and tonic muscle synergies are different in number, structure and sparseness

In the last two decades, muscle synergies analysis has been commonly used to assess the neurophysiological mechanisms underlying human motor control. Several synergy models and algorithms have been employed for processing the electromyographic (EMG) signal, and it has been shown that the coordination of motor control is characterized by the presence of phasic (movement-related) and tonic (anti-gravity and related to co-contraction) EMG components. Neural substrates indicate that phasic and tonic components have non-homogeneous origin; however, it is still unclear if these components are generated by the same set of synergies or by distinct synergies. This study aims at testing whether phasic and tonic components are generated by distinct phasic and tonic synergies or by the same set of synergies with phasic and tonic activation coefficients. The study also aims at characterizing the differences between the phasic and the tonic synergies. Using a comprehensive mapping of upper-limb point-to-point movements, synergies were extracted from phasic and tonic EMG signal separately, estimating the tonic components with a linear ramp model. The goodness of reconstruction (R2) as a function of the number of synergies was compared, and sets of synergies extracted from each dataset at three R2 threshold levels (0.80, 0.85, 0.90) were retained for further analysis. Then, shared, phasic-specific, and tonic-specific synergies were extracted from the two datasets concatenated. The dimensionality of the synergies shared between the phasic and the tonic datasets was estimated with a bootstrap procedure based on the evaluation of the distribution of principal angles between the subspaces spanned by phasic and tonic synergies due to noise. We found only few shared synergies, indicating that phasic and tonic synergies have in general different structures. To compare consistent differences in synergy composition, shared, phasic-specific, and tonic-specific synergies were clustered separately. Phasic-specific clusters were more numerous than tonic-specific ones, suggesting that they were more differentiated among subjects. The structure of phasic clusters and the higher sparseness indicated that phasic synergies capture specific muscle activation patterns related to the movement while tonic synergies show co-contraction of multiple muscles for joint stabilization and holding postures. These results suggest that in many scenarios phasic and tonic synergies should be extracted separately, especially when performing muscle synergy analysis in patients with abnormal tonic activity and for tuning devices with gravity support.

Slow feature subspace: A video representation based on slow feature analysis for action recognition

This paper proposes a new video representation for subspace-based action recognition. Traditional subspace-based methods represent a video as a subspace by applying principal component analysis (PCA) to its frames. However, this subspace might lead to an imprecise representation of actions, as PCA loses the temporal information of frames. Therefore, we introduce the slow feature subspace based on the slow feature analysis (SFA). SFA extracts a set of slow features of an input video by projecting the video frames onto the weight vectors that minimize the data variance over time. Motivated by these properties, several methods based on SFA were proposed for action recognition. However, they do not explicitly consider the distribution of the slow features, which represents essential action information. Therefore, our key idea is to capture this distribution through a low-dimensional subspace called slow feature subspace. Our subspace is generated by applying PCA to several weight vectors corresponding to the slowest components obtained by SFA. Our framework replaces the subspaces of several traditional mutual subspace methods with our slow feature subspace to improve their results in action recognition. This approach transforms the comparison between two videos into the comparison between two slow feature subspaces using the canonical angles between them, avoiding vector concatenation and data aggregation. The effectiveness of our framework is demonstrated through extensive experiments with various datasets. Our results show that our slow feature subspace can improve the traditional subspace-based methods and achieve competitive performance compared to different methods, including state-of-the-art neural networks.

Data-driven motor imagery EEG classifier using difference subspace method

Motor imagery (MI) electroencephalography (EEG) based brain-computer interface (BCI) systems are in great demand for many real-time applications both in the medical and consumer sectors. However, the processing of MI EEG data is challenging, and many of the approaches are complex and suffer from poor classification accuracy due to the unstability of EEG signals. To enhance the classification accuracy, a novel computationally efficient difference subspace method (DSM) of classification is proposed in this study. The proposed method is driven directly by the data without the necessity of extracting any handcrafted features. Subspaces are learned using principal component analysis (PCA). A search strategy based on overlapping criteria is used for selecting the subspace dimensions. The similarity between the two subspaces, measured based on the canonical angles, is used as the criteria to classify MI activity. The proposed method is implemented on BCI competition III and IV datasets, and the performance is evaluated on the basis of classification accuracy (%CA), Cohen’s kappa coefficient, F1-score etc., By retention of only the difference components between the two groups of data after removing the common components, the difference subspace method could do dominatingly well in classifying the left and right hand MI classification compared to the subspace method (SM) and many of state-of-the-art techniques. The results show that the DSM achieves an average classification accuracy and Cohen’s kappa coefficient for BCI competition III and IV datasets as 83.97% & 0.68 and 86.22% & 0.72, respectively.

Efficient Distribution Similarity Identification in Clustered Federated Learning via Principal Angles between Client Data Subspaces

Clustered federated learning (FL) has been shown to produce promising results by grouping clients into clusters. This is especially effective in scenarios where separate groups of clients have significant differences in the distributions of their local data. Existing clustered FL algorithms are essentially trying to group together clients with similar distributions so that clients in the same cluster can leverage each other's data to better perform federated learning. However, prior clustered FL algorithms attempt to learn these distribution similarities indirectly during training, which can be quite time consuming as many rounds of federated learning may be required until the formation of clusters is stabilized. In this paper, we propose a new approach to federated learning that directly aims to efficiently identify distribution similarities among clients by analyzing the principal angles between the client data subspaces. Each client applies a truncated singular value decomposition (SVD) step on its local data in a single-shot manner to derive a small set of principal vectors, which provides a signature that succinctly captures the main characteristics of the underlying distribution. This small set of principal vectors is provided to the server so that the server can directly identify distribution similarities among the clients to form clusters. This is achieved by comparing the similarities of the principal angles between the client data subspaces spanned by those principal vectors. The approach provides a simple, yet effective clustered FL framework that addresses a broad range of data heterogeneity issues beyond simpler forms of Non-IIDness like label skews. Our clustered FL approach also enables convergence guarantees for non-convex objectives.

Distance problems in the behavioral setting

Motivated by the distance to uncontrollability problem, we define a distance between finite-length linear time-invariant behaviors. The method proposed in this paper for computing the distance exploits the principal angles associated with structured matrices representing the systems.

Deep-learning-based biomarker of spinal cartilage endplate health using ultra-short echo time magnetic resonance imaging.

T2* relaxation times in the spinal cartilage endplate (CEP) measured using ultra-short echo time magnetic resonance imaging (UTE MRI) reflect aspects of biochemical composition that influence the CEP's permeability to nutrients. Deficits in CEP composition measured using T2* biomarkers from UTE MRI are associated with more severe intervertebral disc degeneration in patients with chronic low back pain (cLBP). The goal of this study was to develop an objective, accurate, and efficient deep-learning-based method for calculating biomarkers of CEP health using UTE images. Multi-echo UTE MRI of the lumbar spine was acquired from a prospectively enrolled cross-sectional and consecutive cohort of 83 subjects spanning a wide range of ages and cLBP-related conditions. CEPs from the L4-S1 levels were manually segmented on 6,972 UTE images and used to train neural networks utilizing the u-net architecture. CEP segmentations and mean CEP T2* values derived from manually- and model-generated segmentations were compared using Dice scores, sensitivity, specificity, Bland-Altman, and receiver-operator characteristic (ROC) analysis. Signal-to-noise (SNR) and contrast-to-noise (CNR) ratios were calculated and related to model performance. Compared with manual CEP segmentations, model-generated segmentations achieved sensitives of 0.80-0.91, specificities of 0.99, Dice scores of 0.77-0.85, area under the receiver-operating characteristic curve values of 0.99, and precision-recall (PR) AUC values of 0.56-0.77, depending on spinal level and sagittal image position. Mean CEP T2* values and principal CEP angles derived from the model-predicted segmentations had low bias in an unseen test dataset (T2* bias =0.33±2.37 ms, angle bias =0.36±2.65°). To simulate a hypothetical clinical scenario, the predicted segmentations were used to stratify CEPs into high, medium, and low T2* groups. Group predictions had diagnostic sensitivities of 0.77-0.86 and specificities of 0.86-0.95. Model performance was positively associated with image SNR and CNR. The trained deep learning models enable accurate, automated CEP segmentations and T2* biomarker computations that are statistically similar to those from manual segmentations. These models address limitations with inefficiency and subjectivity associated with manual methods. Such techniques could be used to elucidate the role of CEP composition in disc degeneration etiology and guide emerging therapies for cLBP.

Detecting Microstate Transition in Human Brain via Eigenspace of Spatiotemporal Graph

In this letter, a novel approach for detecting the transition of electroencephalography (EEG) microstates of the human brain has been proposed. We have considered the EEG electrodes as nodes of a graph and the correlation between the electrodes' signals as the edge weights. Then, using spectral analysis of the graph, a method has been proposed for detecting the transition of the microstates. The proposed method is comprised of two steps. First, a spatiotemporal graph is constructed using the correlation between the Laplacian of the spatial graph at consecutive time instants. Then, using the principal angles of the eigenspace of the graph, the transition of the EEG microstate has been detected. Experimental results on two publicly available datasets show that the proposed method performs more accurately than state of the art. On the first dataset for the 15 out of 20 subjects, the spatiotemporal graph approach improved the accuracy of detecting the transition time. On the second dataset, it missed only two transitions as opposed to the spatial graph, which failed to detect overall ten transitions across all the subjects.

Rigid-Body Attitude Control, Synchronization, and Bipartite Consensus Using Feedback Reshaping

Improved attitude control laws based on a feedback reshaping strategy using rotation matrices are proposed for attitude control of a single rigid body on and for multibody attitude synchronization on . For attitude control of a single body the proposed control law improves the closed-loop response for initial principal rotation angles near 180° while producing a globally continuous control torque. For the problem of multibody attitude synchronization, an alternative feedback reshaping strategy is proposed, which destabilizes nonconsensus equilibria and produces attitude synchronization with almost global stability and globally continuous control. Finally, a previously proposed bipartite consensus algorithm based on the concept of a structurally balanced graph with antagonistic interactions represented by negative edge weights is applied for the first time to rigid-body attitude consensus control with feedback reshaping in which the consensus attitudes and angular velocities are divided into two distinct groups. Simulations serve as illustrative examples of each of these problems and confirm the advantages of the proposed strategies. For multibody synchronization the focus is on ring graph topologies for which locally stable nonconsensus equilibria arise when using an existing attitude consensus control protocol in the literature.

Upper bounds and spectrum for approximation exponents for subspaces ofℝn

This paper uses W. M. Schmidt’s idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of ℝ n . Given two subspaces of ℝ n , A and B of respective dimensions d and e, with d+e⩽n, the proximity between A and B is measured by t=min(d,e) canonical angles 0⩽θ 1 ⩽⋯⩽θ t ⩽π/2; we set ψ j (A,B)=sinθ j . If B is a rational subspace, its complexity is measured by its height H(B)=covol(B∩ℤ n ). We denote by μ n (A|e) j the exponent of approximation defined as the upper bound (possibly equal to +∞) of the set of β>0 such that for infinitely many rational subspaces B of dimension e, the inequality ψ j (A,B)⩽H(B) -β holds. We are interested in the minimal value μ ˚ n (d|e) j taken by μ n (A|e) j when A ranges through the set of subspaces of dimension d of ℝ n such that for all rational subspaces B of dimension e one has dim(A∩B)<j. We show that if A is included in a rational subspace F of dimension k, its exponent in ℝ n is the same as its exponent in ℝ k via a rational isomorphism F→ℝ k . This allows us to deduce new upper bounds for μ ˚ n (d|e) j . We also study the values taken by μ n (A|e) e when A is a subspace of ℝ n satisfying dim(A∩B)<e for all rational subspaces B of dimension e.

Differences in Tridimensional Shoulder Kinematics between Asymptomatic Subjects and Subjects Suffering from Rotator Cuff Tears by Means of Inertial Sensors: A Cross-Sectional Study.

Background: The aim of this study was to analyze differences in three-dimensional shoulder kinematics between asymptomatic subjects and patients who were diagnosed with rotator cuff tears. Methods: This cross-sectional study recruited 13 symptomatic subjects and 14 asymptomatic subjects. Data were obtained from three inertial sensors placed on the humerus, scapula and sternum. Kinematic data from the glenohumeral, scapulothoracic and thoracohumeral joints were also calculated. The participants performed shoulder abductions and flexions. The principal angles of movements and resultant vectors in each axis were studied. Results: The glenohumeral joint showed differences in abduction (p = 0.001) and flexion (p = 0.000), while differences in the scapulothoracic joint were only significant during flexion (p = 0.001). The asymptomatic group showed higher velocity values in all sensors for both movements, with the differences being significant (p < 0.007). Acceleration differences were found in the scapula during abduction (p = 0.001) and flexion (p = 0.014), as well as in the sternum only during shoulder abduction (p = 0.022). Conclusion: The results showed kinematic differences between the patients and asymptomatic subjects in terms of the mobility, velocity and acceleration variables, with lower values for the patients.

Principal angles in pseudo-euclidean spaces of index 1

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space.

Applying muscle synergy analysis to forearm high-density electromyography of healthy people.

Muscle synergy is regarded as a motor control strategy deployed by the central nervous system (CNS). Clarifying the modulation of muscle synergies under different strength training modes is important for the rehabilitation of motor-impaired patients. To represent the subtle variation of neuromuscular activities from the smaller forearm muscles during wrist motion, we proposed to apply muscle synergy analysis to preprocessed high-density electromyographic data (HDEMG). Here, modulation of muscle synergies within and across the isometric and isotonic training modes for strengthening muscles across the wrist were investigated. Surface HDEMGs were recorded from healthy subjects (N = 10). Three different HDEMG electrode configurations were used for comparison and validation of the extracted muscle synergies. The cosine of principal angles (CPA) and the Euclidian distance (ED) between synergy vectors were used to evaluate the intra- and inter-mode similarity of muscle synergies. Then, how the activation coefficients modulate the excitation of specific synergy under each mode was examined by pattern recognition. Next, for a closer look at the mode-specific synergies and the synergies shared by the two training modes, k-means clustering was applied. We observed high similarity of muscle synergies across different tasks within each training mode, but decreased similarity of muscle synergies across different training modes. Both intra- and intermode similarity of muscle synergies were consistently robust to electrode configurations regardless of the similarity metric used. Overall, our findings suggest that applying muscle synergy analysis to HDEMG is feasible, and that the traditional muscle synergies defined by whole-muscle components may be broadened to include sub-muscle components represented by the HDEMG channels. This work may lead to an appropriate neuromuscular analysis method for motor function evaluation in clinical settings and provide valuable insights for the prescription of rehabilitation training therapies.

Eigenvalue bounds for saddle-point systems with singular leading blocks

We derive bounds on the eigenvalues of saddle-point matrices with singular leading blocks. The technique of proof is based on augmenting the singular leading block to replace it with a positive definite matrix. Our bounds depend on the principal angles between the ranges or kernels of the matrix blocks. Numerical experiments validate our analytical findings.

On the Sensitivity of the Canonical Angles of a Unitoid Matrix

On the Sensitivity of the Canonical Angles of a Unitoid Matrix

Grassmannian learning mutual subspace method for image set recognition

This paper addresses the problem of object recognition given a set of images as input (e.g., multiple camera sources and video frames). Convolutional neural network (CNN)-based frameworks do not exploit these sets effectively, processing a pattern as observed, not capturing the underlying feature distribution as it does not consider the variance of images in the set. To address this issue, we propose the Grassmannian learning mutual subspace method (G-LMSM), a NN layer embedded on top of CNNs that can process image sets more effectively and can be trained in an end-to-end manner. The image set is first represented by a low-dimensional input subspace and then this input subspace is matched with dictionary subspaces by a similarity of their canonical angles, an interpretable and easy to compute metric. The key idea of G-LMSM is that the dictionary subspaces are learned as points on the Grassmann manifold, optimized with Riemannian stochastic gradient descent. This learning is stable, efficient and theoretically well-grounded. We demonstrate the effectiveness of our proposed method on hand shape recognition, face identification, and facial emotion recognition.