Principal Angles Research Articles

Rapid Recognition of Tomato’s Disease Stages Based on the Kernel Mutual Subspace Method

Highlights A low time complexity and high accuracy recognition method of tomato’s disease stages was proposed.Small number of disease image samples can support a stable and high accuracy method with less computation time and less memory cost.Provided a solution to establish a real-time and low energy consumption disease recognize system.Abstract. Disease stages recognition of tomato is important for the timely diagnosis and prevention of tomato diseases. In this article, the color and texture features were extracted, and the kernel mutual subspace method (KMSM) was introduced to establish a rapid Tomato’s disease stage recognition method. Firstly, the color and textural features were extracted from tomato leaf and mapped to a high-dimensional space using a Gaussian kernel function. Secondly, a nonlinear disease feature subspace was established by applying principal component analysis (PCA) based on the mapped high-dimensional space. Finally, disease stages are recognized by calculating the canonical angles between the testing subspace and reference sub-spaces. To validate recognition rate of the proposed method, we conducted 10-fold cross-validation for experiments using the tomato disease sets contained in PlantVillage and artificial intelligence (AI) Challenger 2018 datasets. Also compared with the support vector machine (SVM) and VGG 16 methods, the results show that, accuracy of those three evaluated methods on the PlantVillage datasets are 99.34%, 89.2%, and 97%, respectively; and accuracy on the AI Challenger 2018 datasets are 98.66%, 71.14%, and 73.93%, respectively. Moreover, average training and recognition time of the proposed method is 0.1496 and 0.008 s, respectively, which is faster than SVM and VGG16 methods. In conclusion, the proposed method can be carried out in real-time recognition intelligent equipment which requires less memory space, less computation time, and with higher recognition rate, and low energy consumption. Keywords: Color and textural feature, High dimensional feature space, Kernel mutual subspace method, Tomato’s disease stages recognition,.

Subspace clustering using ensembles of K-subspaces

Abstract Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the $K$-subspace (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble $K$-subspaces (EKSSs), forms a co-association matrix whose $(i,j)$th entry is the number of times points $i$ and $j$ are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace.

Counting Collisions in an N-Billiard System Using Angles Between Collision Subspaces

The principal angles between binary collision subspaces in an N-billiard system in d-dimensional Euclidean space are computed. These angles are computed for equal masses and arbitrary masses. We then provide a bound on the number of collisions in the planar 3-billiard system problem. Comparison of this result with known billiard collision bounds in lower dimensions is discussed.

Performance of plastic concrete under true tri-axial compressive stress

Based on the tests among 45 group specimens of five mixes under true tri-axial compressive stress, the characteristics of strain–stress curve, the impact of the stress path and the failure criterion of the plastic concrete were illustrated. The results indicated that the stress - strain curves can be generally divided into four segments, the initial segment was equipped with significantly elastic characteristics, while the last segment indicated severely plastic deformation had occurred. Specimen failure pattern was similar to that of axial compression specimen when the levels of the second and third principal stress were smaller. Otherwise, specimen failure pattern looked like eight diagonal crack. Principal stresses, cohesive forces and angles were influenced dramatically by different stress paths. Based on 93 group test data, three formulas with four parameters, three parameters and two parameters are put forward to express the failure criterion of plastic concrete under true tri-axial compression.

On the effect of anisotropy on drained static liquefaction triggering

A number of recent tailings storage facility (TSF) failures has been indicated through post-failure investigations to have been triggered in tailings that were under drained conditions up to the initiation of instability. Such field-scale behaviour is consistent with many laboratory element studies on drained triggering of instability. However, essentially all available data on the drained initiation of instability has been carried out under conditions where the principal stress angle (α) is zero. This is despite the principal stress angle in zones of TSFs where instability is likely to be triggered being below-slope, where α is higher than zero. To assess the implications of anisotropy on the drained initiation of instability, the limited data available that enables assessment of the effects of α on instability in undrained testing is referenced. These data suggest that the likely range of principal stress angles below-slope areas of TSFs are such that anisotropy should be considered in estimating relevant triggering stress ratios. The need for more experimental data of drained triggering of instability at principal stress angles relevant to below-slope conditions is outlined.

Shallow Versus Deep Neural Networks in Gear Fault Diagnosis

Accurate gear defect detection in induction machine-based systems is a fundamental issue in several industrial applications. At this aim, shallow neural networks, i.e., architectures with only one hidden layer, have been used after a feature extraction step from vibration, torque, acoustic pressure and electrical signals. Their additional complexity is justified by their ability in extracting its own features and in the very high-test classification rates. These signals are here analyzed, both geometrically and topologically, in order to estimate the class manifolds and their reciprocal positioning. At this aim, the different states of the gears are studied by using linear (Pareto charts, biplots, principal angles) and nonlinear (curvilinear component analysis) techniques, while the class clusters are visualized by using the parallel coordinates. It is deduced that the class manifolds are compact and well separated. This result justifies the use of a shallow neural network, instead of a deep one, as already remarked in the literature, but with no theoretical justification. The experimental section confirms this assertion, and also compares the shallow neural network results with the other machine learning techniques used in the literature.

Differences in the principal strain angles during activities performed on natural hilly terrain versus engineered surfaces

BackgroundTibial stress fractures in military recruits occur beginning with the fourth week of training. In and ex vivo tibial strain experiments indicate that the repetitive mechanical loading during this time may not alone be sufficient to cause stress fracture. This has led to the hypothesis that the development of tibial stress fracture is mediated by the bone remodeling response to high repetitive strains. This study assesses the differences in the strain and angle of the principal strain during military field activities versus common civilian activities. MethodsIn vivo strain measurements were made from a rosette strain gauge bonded to the midshaft of the medial tibia. Measurements of principal strains and their angles were made while performing level and inclined walking and running on an asphalt surface, while fast walking up and down stairs, while performing a standing vertical jump and while zig-zag running up and down a 30° inclined dirt hill. FindingsThe angle of the principal strain varied little (5.40° to +2.74°) during activities performed on engineered surfaces. During zig-zag running on a dirt hill the strain levels were higher (maximum shear = 4290 με). At the pivot points of zig-zag running the angle of the principal strain varied between −115° to −123° downhill and between −32.8° to −51° uphill. InterpretationActivities that mimic those performed by infantry recruits on irregular hilly surfaces result in higher tibial strains and have more variation in principal strain angles than activities of ordinary civilian life performed on engineered surfaces.

Cost-Benefit Analysis of Moving-Target Defense in Power Grids

We study moving-target defense (MTD) that actively perturbs transmission line reactances to thwart stealthy false data injection (FDI) attacks against state estimation in a power grid. Prior work on this topic lacks an analysis of the relationship between MTD's effectiveness (in detecting FDI attacks) and the associated cost of the perturbations (incurred by the grid operator). To address the issue, we present formal design criteria to select MTD reactance perturbations that are truly effective. Based on a key optimal power flow (OPF) formulation, we find that the effective MTD may incur a non-trivial operational cost. We show that MTD's detection capability and the associated cost depend on the separation between the column spaces of measurement matrices before and after the MTD perturbation. We use the metric of smallest principal angles between the subspaces to characterize the separation. We show that different degrees of the separation provide a spectrum of tradeoffs between the MTD's detection capability and its cost. Furthermore, we present closed-form expressions in the case of a two-bus system to illustrate the tradeoffs. While our analysis is primarily based on a direct current (dc) power flow model, we show that the perturbations designed using this model are also effective in detecting FDI attacks against ac power flows. Similarly, the cost-benefit tradeoff still holds under the ac power model. Extensive simulations, using the MATPOWER simulator and benchmark IEEE bus systems, verify and illustrate the proposed design approach that for the first time addresses both key aspects of cost and effectiveness of the MTD.

Consistent Failure to Produce a Cognitive Load Effect in Visual Working Memory Using a Standard Dual-Task Procedure.

Working memory performance is impaired when an attention-demanding task is executed during memory retention. The cognitive load effect is the consistent finding that the size of the memory impairment is determined by the relative amount of time that the secondary processing task occupies attention during memory retention. Cognitive load has been proposed to be a Priority-A benchmark any model of working memory should be able to explain (Oberauer et al., 2018), in part because the effect appears to generalize across different experimental procedures and materials. Using a standard dual-task procedure, we detail four experiments using a visual working memory recall task, two requiring memory for low-level features and two requiring memory for canonical angles (up, down, left, right, etc.). In all four experiments, we failed to find a cognitive load effect, calling into question the generality of the cognitive load effect and whether it is driving forgetting in multitasking contexts.

On angles, projections and iterations

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of “consistency projections” in this context.

Nonasymptotic upper bounds for the reconstruction error of PCA

We analyse the reconstruction error of principal component analysis (PCA) and prove nonasymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. The bounds reveal that the excess risk differs significantly from usually considered subspace distances based on canonical angles. Our approach relies on the analysis of empirical spectral projectors combined with concentration inequalities for weighted empirical covariance operators and empirical eigenvalues.

When a maximal angle among cones is nonobtuse

Principal angles between linear subspaces have been studied for their application to statistics, numerical linear algebra, and other areas. In 2005, Iusem and Seeger defined critical angles within a single convex cone as an extension of antipodality in a compact set. Then, in 2016, Seeger and Sossa extended that notion to two cones. This was motivated in part by an application to regression analysis, but also allows their cone theory to encompass linear subspaces which are themselves convex cones. One obstacle to computing the maximal critical angle between cones is that, in general, the maximum will not occur at a pair of generators of the cones. We show that in the special case where the maximal angle between the cones is nonobtuse, it does suffice to check only the generators. This special case can be checked at essentially no extra cost, and we incorporate that information into an improved algorithm to find the maximal angle.

On the phases of a complex matrix

In this paper, we define the phases of a complex sectorial matrix to be its canonical angles, which are uniquely determined from a sectorial decomposition of the matrix. Properties of matrix phases are studied, including those of compressions, Schur complements, matrix products, and sums. In particular, by exploiting a notion known as the compound numerical range, we establish a majorization relation between the phases of the eigenvalues of AB and the phases of A and B. This is then applied to investigate the rank of I+AB with phase information of A and B, which plays a crucial role in feedback stability analysis. A pair of problems: banded sectorial matrix completion and decomposition is studied. The phases of the Kronecker and Hadamard products are also discussed.

Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition

Given a matrix $\bf A$ with numerical rank $k$ , the two-sided orthogonal decomposition (TSOD) computes a factorization ${\bf A} = {\bf UDV}^T$ , where ${\bf U}$ and ${\bf V}$ are orthogonal, and ${\bf D}$ is (upper/lower) triangular. TSOD is rank-revealing as the middle factor ${\bf D}$ reveals the rank of $\bf A$ . The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the $k$ approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm.

Perturbation theory for Hermitian quadratic eigenvalue problem – damped and simultaneously diagonalizable systems

The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems (λ2M+λD+K)x=0. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=[x1,…,xk] and X2=[xk+1,…,xn] are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X˜1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.

Subspace perspective on canonical correlation analysis: Dimension reduction and minimax rates

Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by the recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose two losses based on the principal angles between the model spaces spanned by the sample canonical variates and their population correspondents, respectively. We further characterize the non-asymptotic error bounds for the estimation risks under the proposed error metrics, which reveal how the performance of sample CCA depends adaptively on key quantities including the dimensions, the sample size, the condition number of the covariance matrices and particularly the population canonical correlation coefficients. The optimality of our uniform upper bounds is also justified by lower-bound analysis based on stringent and localized parameter spaces. To the best of our knowledge, for the first time our paper separates $p_{1}$ and $p_{2}$ for the first order term in the upper bounds without assuming the residual correlations are zeros. More significantly, our paper derives $(1-\lambda_{k}^{2})(1-\lambda_{k+1}^{2})/(\lambda_{k}-\lambda_{k+1})^{2}$ for the first time in the non-asymptotic CCA estimation convergence rates, which is essential to understand the behavior of CCA when the leading canonical correlation coefficients are close to $1$.

Principal angles and principal azimuths of a high-index transparent thin film on a low-index transparent substrate: Si on glass in the near infrared

Principal angles of incidence and associated principal azimuths, at which incident linearly polarized light becomes circularly polarized upon reflection, are determined for an IR-transparent high-index Si film (of the proper thickness) which is deposited on a low-index SiO2 (glass) substrate at the near-IR wavelength λ = 1.55 μm. The results are substantially different from those reported by the author [Appl. Opt. 57, 9529–9532 (2018)] for the SiO2–Si (film-substrate) system at the same wavelength. The intensity reflectances for incident p- and s-polarized light, and the linear-to-circular polarization conversion efficiency are also calculated as functions of the principal angle.

Provable Subspace Clustering: When LRR Meets SSC

An important problem in analyzing big data is subspace clustering, i.e., to represent a collection of points in a high-dimensional space via the union of low-dimensional subspaces. Sparse subspace clustering (SSC) and Low-rank representation (LRR) are the state-of-the-art methods for this task. These two methods are fundamentally similar in that both are based on convex optimization exploiting the intuition of “Self-Expressiveness”. The main difference is that the SSC minimizes the vector ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the representation matrix to induce sparsity while LRR minimizes the nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-rank sparse subspace clustering (LRSSC), by the combining SSC and LRR, and develop theoretical guarantees of the success of the algorithm. The results reveal interesting insights into the strengths and the weaknesses of SSC and LRR, and demonstrate how the LRSSC can take advantage of both methods in preserving the “Self-Expressiveness Property” and “Graph Connectivity” at the same time. A byproduct of our analysis is that it also expands the theoretical guarantee of SSC to handle cases when the subspaces have arbitrarily small canonical angles but are “nearly independent”.

Analysis of Deformation of Circular Roadway Considering Effects of Intermediate Principal Stress and Dilatancy

Based on the unified strength theory and non-associated flow rule, with the consideration of the effects of intermediate principal stress and dilation characteristics, the analytical expressions of surrounding rock stress, displacement and plastic zone radius were obtained. Then the influences of the intermediate principal stresses and dilation angles on the surrounding rock stress, displacement and the plastic zone radius were analyzed by engineering example. The analysis shows that surrounding rock stress distribution and plastic zone radius have close relations with the intermediate principal stress. With the increase of the intermediate principal stress coefficient, the ultimate strength of surrounding rock increased, but the plastic zone radius shows a decreasing trend. The displacement of plastic zone is not only related to the intermediate principal stress coefficient, but also to the dilation angle. The displacement of plastic zone increased with the increase of dilation angle, and the displacement of plastic zone decreased with the increase of the intermediate principal stress coefficient.

Evaluation of tissue deformation during radiofrequency and microwave ablation procedures: Influence of output energy delivery.

The purpose of this study was to quantitatively analyze tissue deformation during radiofrequency (RF) and microwave ablation for varying output energy levels. A total of 46 fiducial markers which were classified into outer, middle, and inner lines were positioned into a single plane around an RF or microwave ablation applicator in each ex vivo bovine liver sample (8cm×6cm×4cm, n=18). Radiofrequency (500kHz; ~35W average) or microwave (2.4GHz; 50-100W output, ~35-70W delivered) ablation was performed for 10min (n=4-6 each setting). CT images were acquired over the entire liver volume every 15s. Principle strain magnitude and direction were determined from fiducial marker displacement. Normal and shear strain were then calculated such that negative strain denoted contraction and positive strain denoted expansion. Temporal variations, the final magnitudes, and angles of the strain were compared across energy delivery settings, using one-way ANOVA with post hoc Tukey's tests. On average, tissue strain rates peak at around 1min and decayed exponentially over time. No evidence of tissue expansion was observed. The tissue strains from RF and 50W, 75W, and 100W microwave ablation at 10min were -8.5%, -38.9%, -54.4%, and -65.7%, respectively, from the inner region and -3.6%, -23.7%, -41.8%, and -44.3%, respectively, from the outer region. Negative strain magnitude was positively correlated to energy delivery in the inner region (Spearman's =-0.99). Microwaves at higher powers (75-100W) induced significantly more strain than at lower power (50W) or after RF ablation (P<0.01). Principal strain angles ranged from 0.8° to -8.1°, indicating that tissue deformed more in the direction transverse to the applicator than along the direction of the applicator. The influence of output energy on tissue deformation during RF and microwave ablation was analyzed. Microwave ablation created significantly greater contraction than RF ablation with similar energy delivery. During microwave ablation, more contraction was noted at higher power levels and in proximity to the antenna. Contraction primarily transverse to the antenna produces ablation zones that are more elongated than the original tissue volume.