Linear Subspace Research Articles

Linear subspaces in cubic hypersurfaces

Linear subspaces in cubic hypersurfaces

Whole-brain T2 mapping with radial sampling and retrospective motion correction at 3T.

Several barriers prevent the use of whole-brain T2 mapping in routine use despite increasing interest in this parameter. One of the main barriers is the long scan time resulting in patient discomfort and motion corrupted data. To address this challenge, a method for accurate whole-brain T2 mapping with a limited acquisition time and motion correction capabilities is investigated. A 3D radial multi-echo spin-echo sequence was implemented with optimized sampling trajectory enabling the estimation of intra-scan motion, subsequently used to correct the raw data. Motion corrected echo images are then reconstructed with linear subspace constrained reconstruction. Experiments were carried out on phantom and volunteers at 3T to evaluate the accuracy of the T2 estimation, the sensitivity to lesions and the efficiency of the correction on motion corrupted data. Whole-brain T2 mapping acquired in less than 7 min enabled the depiction of lesions in the white matter with longer T2. Data retrospectively corrupted with typical motion traces of pediatric patients highly benefited from the motion correction by reducing the error in T2 estimates within the lesions. All datasets acquired on seven volunteers, with deliberate motion, also showed that motion corrupted T2 maps could be improved with the retrospective motion correction both at the voxel level and the structure level. A whole-brain T2 mapping sequence with retrospective intra-scan motion correction and reasonable acquisition time is proposed. The method necessitates advanced iterative reconstruction strategies but no additional navigator, external device, or increased scan time is required.

Linear sections of determinantal varieties

Denote Mmn(k) to be the set of all m n matrices over k, and PMmn(k) to be its affine space. Let Dm,n,r PMmn(k) be the subvariety consisting of the projective classes of all m n matrices with rank less than or equal to r. We study the minimal integer k > 0 such that for any linear subspace L of dimension k in PMmn(k), Dm,n,r L = . We get some partial results about this problem. Specially, when m = n = r + 1, we solve this problem for k = R, C and Q.

A hybrid multilinear-linear subspace learning approach for enhanced person re-identification in camera networks

Several tasks in surveillance systems depend on camera networks, one of them is Person re-identification (PRe-ID) which has wide interest as a research topic in the computer vision field. The current techniques in this issue encounter many obstacles in handling the variability of appearance, extracting effective features, and capturing intricate data relationships, leading to suboptimal performance causing systems to have low performance. This paper presents a novel hybrid subspace learning approach for Person re-identification (PRe-ID), that leverages the strengths of both multilinear and linear techniques to significantly enhance the performance of PRe-ID. The proposed process begins by extracting distinctive effective features from pedestrian images using LOMO and GOG as shallow feature descriptors with CNN for deep features, followed by the application of two stages of subspace learning. In the first stage, we employ a multilinear subspace learning algorithm known as Tensor-based Quadratic Discriminant Analysis (TXQDA). TXQDA captures complex relationships and interactions among different features, resulting in a more discriminative feature representation. Subsequently, we employ a linear subspace learning technique called Side Information Linear Discriminant Analysis (SILD) to refine the subspace learned from TXQDA. SILD utilizes linear projections to enhance discriminability and preserve crucial inter-class variations, thereby improving the overall performance of PRe-ID. Ultimately, we implement logistic regression (LR) fusion at the score level to enhance the performance of our hybrid subspace learning approach, particularly within various tensor feature representation scenarios. Extensive experiments are conducted on three challenging datasets to evaluate the effectiveness of our hybrid subspace learning approach. The experimental results demonstrate that our method achieves superior performance compared to state-of-the-art techniques in the field of PRe-ID.

Characteristic value decomposition: A unifying paradigm for data-driven modal analysis

Data-driven modal analysis is an indispensable means to understand the dynamic behavior of engineered structures and natural systems. It effectively captures active dynamic behavior and embeds them in a set of identified linear modal subspaces, unveiling complex dynamic behavior via the superposition of simpler hierarchical spatiotemporal dynamics. Despite the existence of many methods, there is no unified perspective connecting these modal analysis techniques. This paper proposes a new modal decomposition method, the characteristic value decomposition (CVD), that provides a unifying paradigm for data-driven modal analysis. CVD identifies identical spatial modes and temporal coordinates from two measurement data matrices. The modal parameters are identified through an eigendecomposition to a quotient of auto-covariance and cross-covariance matrices of the two data matrices. This paper shows that several data-driven modal analysis methods, namely, the state-variable decomposition, the dynamic mode decomposition, the Ibrahim time domain method, and the eigensystem realization algorithm, are special cases of CVD when the two data matrices are appropriately selected. Moreover, the CVD identifies the modal parameters with higher accuracy and is robust to added noise compared to established methods, as demonstrated using numerical experiments. It is also shown that the CVD flexibly utilizes data from multiple experiments and undersampled data sets to unveil system dynamics.

Feedback equivalence of the chained mechanical system to the almost linear form and its use for the sustainable multi-step walking design

The main theoretical novelty of this paper is the state and feedback equivalence of the underactuated 4-degrees of freedom planar walking-like mechanical chain system with 3 actuators to its 8-dimensional almost linear form with 3 virtual inputs. Moreover, the only residual nonlinearity vanishes on the 4-dimensional linear subspace being forward invariant when 2 of 3 virtual inputs are set to be zero. Dynamics inside that subsystem is actually the chain of 4 integrators fed by the remaining single virtual input and it can be interpreted as a rich variety of synchronous movements of torsos and legs. In such a way, the seemingly abstract and purely theoretical result can be used to design the walking-like movement during the single-support phase. The impact effect during the impulsive-like double-support phase is then attenuated by further special trajectories tuning and finite-time stabilization technique which provides the sustainable multi-step walking design. Moreover, the target walking-like trajectory is attracted by nearby trajectories. This further justify the importance and usefulness of the mentioned state and feedback equivalence. Its viability is further demonstrated by the simulations of various scenarios of the walking-like movement and the respective torsos behaviors.

Dual contrastive learning for multi-view clustering

Self-representation subspace clustering, based on the self-representation property of linear subspaces, has demonstrated superior performance in subspace clustering. However, many existing methods learn self-representation in the embedding space, which fails to accurately capture the clustering structure of the data. Additionally, in the data embedding process, only the shared features among clusters are removed, and the discriminative features that distinguish different clusters cannot be effectively selected. To address these limitations, we propose a contrastive learning-based self-representation subspace clustering method named Multi-view Contrastive Subspace Clustering (MCSC). This method integrates feature selection and self-representation learning into a unified framework. It learns a shared self-representation coefficient in both the embedding space and the original space using contrastive learning, which allows for a more accurate description of the linear relationship between samples and a better depiction of the inter-sample structure simultaneously. Moreover, we employ the ℓ1,2-norm for feature selection, which eliminates shared redundancies while preserving important features specific to each cluster, making the samples more distinguishable. Extensive experiments on six datasets verify the validity of our proposed model.

Local reduced subspaces of subchannel‐inspired subdomains

AbstractIn this article, we study an update of the traditional subchannel approximation utilizing local reduced order bases. Through employing the symmetries and periodicity of a 7‐pin bundle, the global domain is decomposed into numerous repeating subdomains following several dividing strategies. We locally study the reduced basis generated by proper orthogonal decomposition. We analyze the similarities, assessing the truncation error and the distance between the linear subspaces spanned by the reduced bases. We focus on the first stage of building a reduced order model, the generation of the reduced subspace, which is usually not regarded in detail in our application problem. Our assessment related to flow blockage in liquid metal‐cooled nuclear reactors, a postulated high‐risk accident that results in potential fuel damage.

Buckling of steel-composite cylindrical pressure hulls with initial delamination

This study investigated the buckling behavior of steel–composite cylindrical pressure hulls with initial delamination. Hydrostatic tests were conducted on three groups of steel–composite cylindrical pressure hulls with different delamination parameters, and the study determined the relationships between the delamination parameters and buckling behavior of delaminated pressure hulls. Numerical analyses were conducted using linear subspace and nonlinear Riks methods, followed by a comprehensive comparison of the numerical and experimental results. Numerical and response surface analyses were conducted, and two sets of practical equations were proposed to evaluate the linear and nonlinear buckling loads of steel–composite cylindrical pressure hulls. The results indicated that initial delamination reduced the buckling loads by 50 %, with the ratio of delamination length to total length being the key factor determining the nonlinear buckling loads of delaminated steel–composite cylindrical pressure hulls.

Quantum algorithm for computing distances between subspaces

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given data collection. Computing distances between different subspaces is generally a computationally challenging problem with both theoretical and applicable consequences, as, for example, the results can be used to classify data from different categories. Fueled by the fast-growing development of quantum algorithms, we consider such problems in the quantum context and provide a quantum algorithm for estimating two kinds of distance: Grassmann distance and ellipsoid distance. Under appropriate assumptions and conditions, the speedup of our quantum algorithm is exponential with respect to both the dimension of the given data and the number of data points. Some extensions regarding estimating different kinds of distance are then discussed as a corollary of our main quantum algorithmic method.

Accelerated 3D multi-echo spin-echo sequence with a subspace constrained reconstruction for whole mouse brain mapping.

To accelerate whole-brain quantitative mapping in preclinical imaging setting. A three-dimensional (3D) multi-echo spin echo sequence was highly undersampled with a variable density Poisson distribution to reduce the acquisition time. Advanced iterative reconstruction based on linear subspace constraints was employed to recover high-quality raw images. Different subspaces, generated using exponential or extended-phase graph (EPG) simulations or from low-resolution calibration images, were compared. The subspace dimension was investigated in terms of precision. The method was validated on a phantom containing a wide range of and was then applied to monitor metastasis growth in the mouse brain at 4.7T. Image quality and estimation were assessed for 3 acceleration factors (6/8/10). The EPG-based dictionary gave robust estimations of a large range of . A subspace dimension of 6 was the best compromise between precision and image quality. Combining the subspace constrained reconstruction with a highly undersampled dataset enabled the acquisition of whole-brain maps, the detection and the monitoring of metastasis growth of less than 500 . Subspace-based reconstruction is suitable for 3D mapping. This method can be used to reach an acceleration factor up to 8, corresponding to an acquisition time of 25 min for an isotropic 3D acquisition of 156 m on the mouse brain, used here for monitoring metastases growth.

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

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

On the set of supercyclic operators

AbstractIn this article, we address a problem posed by Bayart regarding the existence of an infinite‐dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We fully resolve this problem by establishing the existence of the closed subspace. Furthermore, we prove that the set of supercyclic operators on contains, up to the null operator, an isometric copy of .

A joint learning framework for optimal feature extraction and multi-class SVM

In high-dimensional data classification, effectively extracting discriminative features while eliminating redundancy is crucial for enhancing the performances of classifiers, such as Support Vector Machine (SVM). However, previous studies have decoupled the process of feature extraction from the development of SVM, leading to suboptimal classification accuracy. To address this problem, we propose a novel joint learning framework that combines optimal feature extraction and multi-class SVM, incorporating a generalized regression form to learn a discriminative latent subspace. The projected data in this subspace are more likely to have a larger margin between different classes and align with the properties of the SVM classification mechanism, enhancing the overall classification performance. Three iterative algorithms were presented to obtain optimal solutions with guaranteed convergence, and theoretical analyses were also conducted to reveal their fundamental nature. The optimal linear projection subspace is equivalent to that obtained from Linear Discriminant Analysis (LDA) in some special cases. We conducted extensive experiments using diverse datasets to evaluate the performances of the proposed algorithms. Our algorithms achieved an accuracy improvement of up to 7.55% compared to other conventional methods.

On max-plus two-sided linear systems whose solution sets are min-plus linear

The max-plus algebra R∪{−∞} is defined in terms of a combination of the following two operations: addition a⊕b:=max⁡(a,b) and multiplication a⊗b:=a+b. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system A⊗x=B⊗x. We demonstrate that the minimum “min-plus” linear subspace containing the “max-plus” solution space can be computed by applying the alternating method, which is a well-known algorithm to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the “min-plus” and “max-plus” subspaces to be identical. The computational complexity of the algorithm presented in this study is pseudo-polynomial.

On Angles between Linear Subspaces in $$\mathbb R^4$$ and the Singularity

On Angles between Linear Subspaces in $$\mathbb R^4$$ and the Singularity

The L-fuzzy vector subspace degrees and its induced convex structure

The L-fuzzy vector subspace degrees and its induced convex structure

Structure discovery in PAC-learning by random projections

High dimensional learning is data-hungry in general; however, many natural data sources and real-world learning problems posses some hidden low-complexity structure that permit effective learning from relatively small sample sizes. We are interested in the general question of how to discover and exploit such hidden benign traits when problem-specific prior knowledge is insufficient. In this work, we address this question through random projection’s ability to expose structure. We study both compressive learning and high dimensional learning from this angle by introducing the notions of compressive distortion and compressive complexity. We give user-friendly PAC bounds in the agnostic setting that are formulated in terms of these quantities, and we show that our bounds can be tight when these quantities are small. We then instantiate these quantities in several examples of particular learning problems, demonstrating their ability to discover interpretable structural characteristics that make high dimensional instances of these problems solvable to good approximation in a random linear subspace. In the examples considered, these turn out to resemble some familiar benign traits such as the margin, the margin distribution, the intrinsic dimension, the spectral decay of the data covariance, or the norms of parameters—while our general notions of compressive distortion and compressive complexity serve to unify these, and may be used to discover benign structural traits for other PAC-learnable problems.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

A Borel linear subspace of R^\omega that cannot be covered by countably many closed Haar-meager sets

We prove that the countable product of lines contains a Haar-null Haar-meager Borel linear subspace $L$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma-Ger classes in the topological vector spaces ${\mathbb R}^n$ for $n\le \omega$.