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Learning-to-learn, a progressive speedup of learning while solving a series of similar problems, represents a core process of knowledge acquisition that draws attention in both neuroscience and artificial intelligence. To investigate its underlying brain mechanism, we trained a recurrent neural network model on arbitrary sensorimotor mappings known to depend on the prefrontal cortex. The network displayed an exponential time course of accelerated learning. The neural substrate of a schema emerges within a low-dimensional subspace of population activity; its reuse in new problems facilitates learning by limiting connection weight changes. Our work highlights the weight-driven modifications of the vector field, which determines the population trajectory of a recurrent network and behavior. Such plasticity is especially important for preserving and reusing the learnt schema in spite of undesirable changes of the vector field due to the transition to learning a new problem; the accumulated changes across problems account for the learning-to-learn dynamics.

Anomaly or outlier detection is consider as one of the vital application of data mining, which deals with anomalies or outliers. Anomalies are considered as data points that are dramatically different from the rest of the data points. In this survey, we comprehensively present anomaly detection algorithms in an organized manner. We begin this survey with the definition of anomaly, then provide essential elements of anomaly detection, such as different types of anomaly, different application domains, and evaluation measures. Such anomaly detection algorithms are categorized in seven categories based on their working mechanisms, which includes total of 52 algorithms. The categories are anomaly detection algorithms based on statistics, density, distance, clustering, isolation, ensemble and subspace. For each category, we provide the time complexity of each algorithm and their general advantages and disadvantages. In the end, we compared all discussed anomaly detection algorithms in detail.

In this paper, we devise a unitary noise subspace fitting method for monostatic multiple-input multiple-output radar, which can obtain high-resolution direction-of-arrival (DOA) estimation with non-orthogonal waveforms. Firstly, the cross-covariance matrix without spatially colored noise can be obtained by the spatial cross-correlation technique of two transmit subarrays. Secondly, unitary transformation is introduced to convert the singular value decomposition from the complex-valued matrix to the real-valued matrix. Finally, DOAs are achieved by solving the optimization problem. Some numerical simulation results confirm the effectiveness and superiority of the proposed approach, which can provide higher angular resolution, more accurate estimation performance and low computational cost.

[EN[ In this work, a new mathematical algorithm for sparse and orthogonal constrained biplots, called CenetBiplots, is proposed. Biplots provide a joint representation of observations and variables of a multidimensional matrix in the same reference system. In this subspace the relationships between them can be interpreted in terms of geometric elements. CenetBiplots projects a matrix onto a low-dimensional space generated simultaneously by sparse and orthogonal principal components. Sparsity is desired to select variables automatically, and orthogonality is necessary to keep the geometrical properties that ensure the biplots graphical interpretation. To this purpose, the present study focuses on two different objectives: 1) the extension of constrained singular value decomposition to incorporate an elastic net sparse constraint (CenetSVD), and 2) the implementation of CenetBiplots using CenetSVD. The usefulness of the proposed methodologies for analysing high-dimensional and low-dimensional matrices is shown. Our method is implemented in R software and available for download from https://github.com/ananieto/SparseCenetMA., Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was not supported by any grant., Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCLE

Multi-label transfer learning aiming to learn robust classifiers for the target domain by leveraging knowledge from a source domain has been received considerable attention recently. The core part of such research is similarity measurement. Nevertheless, the existing similarity measurement functions of probability distributions are still too simple to fully describe the similarity of probability distributions.. In order to address this problem, we propose a Multi-label Transfer Learning Via Latent Graph Alignment (G-MLTL). G-MLTL uses subspace learning to make the feature distribution of the target domain consistent with the source domain. At the same time, G-MLTL decomposes the label matrix to ensure data points sharing the same labels to have identical latent semantic representation in the new reconstruction space. The proposed G-MLTL also focuses on directly utilizing Latent Graph Alignment to guide the knowledge transfer process. Extensive experiments demonstrate that G-MLTL significantly outperforms the existing multi-label transfer learning methods. Especially when the number of labels is more than four, the mean Average Precision of G-MLTL is higher than the baseline algorithm by 2.1-10.5%.

Linear codes with good parameters have wide applications in secret sharing schemes, authentication codes, association schemes, consumer electronics and communications, etc. During the past four decades, constructions of linear codes with good parameters received much attention and many classes of such codes were presented. In this paper, we obtain a family of linear codes with good parameters over $$\mathbb {F}_p$$ by exploring further properties of constant dimension subspace codes, where p is a prime. The weight distribution of three classes of linear codes presented in this family is determined. Most notably, three classes of linear codes presented in this family are distance-optimal with respect to the Griesmer bound. Also, this paper presents a sufficient and necessary condition for this family of linear codes to have a $$\lambda $$ -dimensional hull. In addition, we show that our linear codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs.

We consider the subspace of the Schwartz space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in a multidimensional real space equipped with the topology defined by a countable family of norms constructed with the help of a family of convex separately radial functions in ℝn. We describe the strong dual of this subspace in terms of the Fourier–Laplace transform of functionals.

A method for recursive subspace identification of continuous-time systems based on generalized Poisson moment functionals is proposed. Most of the existing subspace identification methods have concentrated mainly on the time-invariant discrete-time systems. The results of subspace identification methods are confined to the discrete-time cases, due to the difference on the construction of Hankel matrices. In addition, the time-invariant identification algorithms are not suitable for online identification cases. In order to solve the problems above, the time derivatives of Hankel matrices can be evaluated by generalized Poisson moment functionals, which provides a simple linear mapping for identification algorithm without the amplification of stochastic noises. The size of the data matrices is fixed a priori to fade the influence of old data to the updated data, which is a key to reduce computational burden and storage cost of recursive algorithms. The efficiency of the presented method is provided by comparing simulation results.

Heterogeneous domain adaptation (HDA) aims to adapt a trained model on a source domain with different input feature space to an unlabeled target domain. In fact, HDA is a challenging issue, since there exists feature and distribution discrepancies across domains. In this paper, we propose a novel approach named as heterogeneous domain adaptation with statistical distribution alignment and progressive pseudo label selection (SDA-PPLS). SDA-PPLS learns two projection matrices for source and target domains to map them into a latent subspace to have a shared feature space. Moreover, to mitigate the distribution gap, SDA-PPLS aligns both first-order and second-order statistical information, simultaneously, to improve the target classification model performance. In addition, to discriminate instances into distinct classes, SDA-PPLS aligns the class conditional distributions by pseudo label refinement of target domain data. Finally, to prevent the propagation of inaccurate pseudo labels to the next iteration, a progressive technique is proposed to select instances with higher probability. Experimental results on several real-word datasets on image to image, text to text and text to image tasks with different feature representations, demonstrate that the proposed method outperforms other state-of-the-art HDA methods.

In this paper, we study the spectrum and the invariant subspace of compressed shifts on the Beurling type quotient module $${{\cal K}_\theta}$$ of Hardy space H2(ⅅ2). Suppose θ is a rational inner function with degree (1, 1). Firstly, we characterize the spectrum of $${S_{{z_1}}}$$ on $${{\cal K}_\theta}$$ ; secondly, we characterize the Agler type subspace of the compressed shift.

EEG signal classification is a key technology for EEG signal processing and identification systems. Dictionary learning has shown excellent performance due to its sparse representation and learning capability. Usually dictionary learning requires sufficient labeled EEG signals to build classification models, and assumes that the data distribution of training and test signals are the same. However, in new EEG signal domain, often only a small amount of signals are labeled, and more are not labeled. At the same time, data dynamicity, confounding factors and strong interclass similarity also seriously disrupt the performance of EEG signal classifier. To this end, a geometric preserving transfer discriminative dictionary learning method called GPTDDL is developed for cross-domain EEG signal classification. Through projected signals of different domains to the common subspace, a shared discriminative dictionary is obtained, which explores the geometric structure information by graph Laplacian regularization and discriminative information by principal component analysis regularization. Benefiting from the discriminative information transferred from source domain, the discriminability of the learned sparse coding of target domain is strengthened. GPTDDL integrates this idea into the framework of LC-KSVD, and learns the subspace and dictionary learning parameters in an iterative strategy. The experimental results on Bonn EEG signal dataset demonstrate the validity of the GPTDDL method.

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

It is shown that the set of finite multisets can be identified with the space of symmetric polynomials on a linear space of finite sequences whose completion in a suitable metric gives a topologization of the set of multisets as a subspace of the space of symmetric analytic functions on l1. As one of possible topologizations, we can mention the topology of Hilbert space. Moreover, we consider some algebraic operations on the set of multisets, in particular, the analogs of fractional mappings.

In through-wall radar imaging (TWRI), the presence of the wall greatly reduces the performance of target detection algorithms. The signal reflected from the wall is stronger than the signal reflected from the target and masks the target. The physical properties of the wall or the reflections from the back and side walls in the environment where the target is located make the problem even more difficult. Within the scope of this study, the non-negative matrix factorization (NMF)-based approaches that we proposed for clutter removal in ground penetrating radar systems were adapted to the TWR problem. Moreover, a new NMF-based method which provides a better modelling of the wall component using sparsity constraint is introduced. Comparison with traditional subspace-based methods such as principal component analysis, singular value decomposition and low rank and sparse method robust principal component analysis for an experimental dataset validates that sparsity-guided NMF-based methods provide the best results.

When using arbitrarily oriented subspace clustering algorithms one obtains a partitioning of a given data set and for each partition its individual subspace. Since clustering is an unsupervised machine learning task, we may not have “ground truth” labels at our disposal or do not wish to rely on them. What is needed in such cases are internal measure which permits a label-less analysis of the obtained subspace clustering. In this work, we propose methods for revising clusters obtained from arbitrarily oriented correlation clustering algorithms. Initial experiments conducted reveal improvements in the clustering results compared to the original clustering outcome. Our proposed approach is simple and can be applied as a post-processing step on arbitrarily oriented correlation clusterings.

Given a pseudocompact $$\varDelta $$ -space X, we establish that countable subsets of X must be scattered. This implies that pseudocompact $$\varDelta $$ -spaces of countable tightness are scattered. If a pseudocompact $$\varDelta $$ -space has the Souslin property, then it is separable and has a dense set of isolated points. It is shown that adding a countable subspace to a pseudocompact $$\varDelta $$ -space can destroy the $$\varDelta $$ -property. However, if X is countably compact and $$Y\subset X$$ is a $$\varDelta $$ -space for some $$Y\subset X$$ such that $$|X\backslash Y|\le \omega $$ , then X is a $$\varDelta $$ -space. We also show that monotonically normal $$\varDelta $$ -spaces must be hereditarily paracompact. Besides, if X is a subspace of an ordinal with its order topology, then X is hereditarily paracompact if and only if it has the $$\varDelta $$ -property.

In recent years, tensor data appear more frequently in machine learning. Tucker decomposition is a powerful tool for processing tensor data. However, from the perspective of dimensionality reduction, Tucker decomposition without any regularization is only a tensor version of principal component analysis (PCA) that learning a linear subspace to minimize the distance between the data and their projections on the subspace. However, many high-dimensional tensor data usually reside in low-dimensional submanifolds, rather than low-dimensional linear subspaces. It is necessary to fill the gap between submanifolds and linear subspaces to achieve superior performance. In this paper, we proposed a new dimensionality reduction algorithm of tensor data based on manifold regularized Tucker decomposition (called MRTD for short), in which manifold regularization is in addition to Tucker decomposition. Furthermore, unlike many other similar algorithms that ignore the matrices of Tucker decomposition by absorbing them into a whole matrix, in this paper, we proposed an iterative solution to MRTD that the matrices are calculated based on each other iteratively. Matrices in Tucker decomposition represent different dimensionality reduction operations for each dimension of tensor. Therefore, MRTD is a dimensionality reduction algorithm specially designed for tensor data, not for vector data. The experimental results of MRTD and 5 other related state-of-the-art algorithms on 6 commonly-used real-world datasets are presented, showing the better performance of MRTD.

How to design effective multi-view subspace clustering (MVSC) algorithms has recently become a research hotspot. In this paper, we propose a new MVSC algorithm, termed latent multi-view self-representation for clustering via the tensor nuclear norm (LMVS/TNN), which can seamlessly unify multi-view clustering and dimensionality reduction into a framework. Specifically, for each view data, LMVS/TNN learns the transformed data from the original space, which can maintain the original manifold structure, and each subspace representation matrix from the transformed latent space simultaneously. Furthermore, to use the high-order correlations and complementary information from multi-view data, LMVS/TNN constructs a third-order tensor by taking the representation matrix extracted from the transformed latent space as the frontal slice of the third-order tensor and the tensor is constrained by a new low-rank tensor constraint, i.e., the tensor nuclear norm (TNN). In addition, based on the augmented Lagrangian scheme, we develop an efficient procedure to solve LMVS/TNN. To verify the performance of LMVS/TNN, we conduct experiments on public datasets and find that LMVS/TNN outperforms some representative clustering algorithms.

In this paper, a general search direction of three-term conjugate gradient method which always satisfy sufficient descent condition is adopted. By solving a new accelerated subspace quadratic optimization problem which can take the advantages of the linear conjugate gradient method, a new conjugate parameter is obtained and a new truncation strategy is proposed to modify the conjugate parameter. Thus, a three-term conjugate gradient method for solving unconstrained optimization problem is proposed under the standard Wolfe line search technique. Under general assumption, the global convergence of the algorithm is discussed and obtained. Finally, the numerical performance for our method is reported via testing about 100 examples and comparing with other two algorithms. Numerical results show that the proposed method is promising.

This paper develops the analysis of discrete-time periodically time-varying linear systems over finite fields. It is shown that the conditions for the existence of Floquet Transform for periodic linear systems over reals (or complex) do not carry forward for this case over finite fields. The existence of Floquet Transform is shown to be equivalent to the existence of an Nth root of the monodromy matrix for the class of non-singular periodic linear systems. As the verification of existence and computation of the Nth root is a computationally hard problem, an independent analysis of the solutions of such systems is carried without the use of Floquet Transform. It is proved that all initial conditions of such systems lie either in a periodic orbit or a chain leading to a periodic orbit. A subspace of the state space is also identified, containing all initial conditions lying on a periodic orbit. Further, whenever the Floquet Transform exists, more concrete results on the orbit lengths are established depending on the coprimeness of the orbit length with the system period.

This paper establishes the algorithmic principle of alternating projections onto incoherent low-rank subspaces (APIS) as a unifying principle for designing robust regression algorithms that offer consistent model recovery even when a significant fraction of training points are corrupted by an adaptive adversary. APIS offers the first algorithm for robust non-parametric (kernel) regression with an explicit breakdown point that works for general PSD kernels under minimal assumptions. APIS also offers, as straightforward corollaries, robust algorithms for a much wider variety of well-studied settings, including robust linear regression, robust sparse recovery, and robust Fourier transforms. Algorithms offered by APIS enjoy formal guarantees that are frequently sharper than (especially in non-parametric settings) or competitive to existing results in these settings. They are also straightforward to implement and outperform existing algorithms in several experimental settings.

The subspace minimization conjugate gradient methods based on Barzilai–Borwein method (SMCG_BB) and regularization model (SMCG_PR), which were proposed by Liu and Liu (J Optim Theory Appl 180(3):879–906, 2019) and Zhao et al. (Numer Algorithm, 2020. https://doi.org/10.1007/s11075-020-01017-1), respectively, are very interesting and efficient for unconstrained optimization. In this paper, two accelerated subspace minimization conjugate gradient methods are proposed for unconstrained optimization. Motivated by the subspace minimization conjugate gradient methods and the finite termination of linear conjugate gradient methods, we derive an acceleration parameter by the quadratic interpolation function to improve the stepsize, and the modified stepsize may be more closer to the stepsize obtained by exact line search. Moreover, several specific acceleration criteria to enhance the efficiency of the algorithm are designed. Under standard conditions, the global convergence of the proposed methods can be guaranteed. Numerical results show that the proposed accelerated methods are superior to two excellent subspace minimization conjugate gradient methods SMCG_BB and SMCG_PR.

We examine the geometry of the subspace of homogeneous probability measures in terms of the 2-Wasserstein metric on the space of all probability measures on Hilbert spaces of functions. We show that, on appropriate Hilbert spaces, the geodesics joining homogeneous measures stay in the space of homogeneous measures and, as a result, the homogeneous measures themselves form a space of nonpositive curvature in the sense of A. D. Alexandrov.

Dimensionality reduction is vital in many fields, such as computer vision and pattern recognition. This paper proposes an unsupervised dimensionality reduction algorithm based on multi-local linear regression. The algorithm first divides the high-dimensional data into many localities. Under the criterion of local homeomorphism, the continuous dependency relationship of the high-dimensional data is maintained in each locality in the low-dimensional space. At the same time, due to the overlap of locality divisions, that is, each data may belong to multiple localities. Therefore, the algorithm performs a multi-local linear prediction on each target data point, to better capture the internal geometric structure of the data. Finally, to coordinate the predictions of the target data points by each locality, we require that the variance between the predictions of each locality to the same target point should be as small as possible. We perform experiments on synthetic and real datasets. Compared with the existing advanced algorithms, the experimental results show that the proposed algorithm has good feasibility.

In this paper, the subspace properties of trust-region methods are employed to develop a regularized Newton-type (RNT) algorithm for solving convex optimization problem. The proposed RNT algorithm is analyzed for quadratic convergence under the local error bound conditions and global convergence for unconstrained convex optimization problems which may have singular Hessian at the solutions. Afterwards numerical results are presented to show the efficiency and robustness of the proposed algorithm in producing an optimal solution for the given problem.

Properties of a secondary bent function construction that adds the indicator of an affine subspace of arbitrary dimension to a given bent function in n variables are obtained. Some results regarding normal and weakly normal bent functions are generalized. An upper bound for the number of generated bent functions is proven. This bound is attained if and only if the given bent function is quadratic. In certain cases, the addition of the indicator of an m-dimensional subspace, for different m, will not generate bent functions. Such examples are presented for any even n ≥ 10. It is proven that there exists an infinite family of Maiorana–McFarland bent functions such that the numbers of generated bent functions differ for the bent function and its dual function.

A channel prediction scheme with channel matrix doubling and temporal-spatial smoothing is proposed. With the assumption that the underlying channel model parameters are unchanged, a virtual doubled channel matrix is constructed to make full use of the intrinsic information contained in the CSI observations, then temporal-spatial smoothing technology is adopted to form the smoothed Hankel matrix, whose rows and columns are mutually space-correlated and time-correlated. After that, the multi-dimensional rotational invariance techniques algorithm is employed to estimate the channel model parameters, and finally time extrapolation is performed on the estimated channel model to predict the future channel state. By channel matrix doubling and temporal-spatial smoothing, the subspace estimation error can be significantly reduced and a more accurate channel model can be obtained, which leads to better channel prediction performance.

Unsupervised dimensionality reduction presents one of the greatest challenges facing the machine learning community. However, as for 2-dimensional time series data processing, conventional unsupervised dimensionality reduction methods usually focus on the similarity calculation techniques through 1-dimensional (1D) feature vectors, while ignoring the underlying structural and time clues. In fact, the structural information implies the spatial relationship between objects in an image and the time information holds the temporal correlation among images, both are crucial discriminative information for unsupervised dimensionality reduction. In view of this,we appeal to take structural and temporal clues into account in unsupervised dimensionality reduction. Specifically, we derive a similarity learning model, which projects the original data space to a compact feature subspace by jointly exploring the adaptive structural and temporal clues (2d-UDRAT). Besides, a practical iterative strategy is conceived to obtain the solution of the proposed model. To verify the efficiency of our method, we take video stream as an example and elaborately design a series of comparison experiments versus several 2D unsupervised dimensionality reduction methods. The experimental results indicate that our proposed is more effective than other state-of-the-art methods.

A basic problem of the constant dimension subspace coding is to determine the maximal possible size Aq(n,d,k) of a set of k-dimensional subspaces in $\mathbf {F}_{q}^{n}$ such that the subspace distance satisfies $\text {dis}(U,V) =2k-2 \dim (U \cap V) \geq d$ for any two different subspaces U and V in this set. We propose two constructions of constant dimension subspace codes that can insert flexibly into the generalized parallel linkage construction. In our constructions matrix blocks from small constant dimension codes and rank metric codes play important roles. Through a well-arranged combination for the matrix blocks, more than 120 new constant dimension subspace codes of distance 4, 6, 8 better than previously best known codes are constructed.

Identifying reduced-order models (ROMs) of nonlinear dynamical systems is difficult, especially when the system equation is unknown with only measurement data available. In such a case, not only a reduced subspace but also the associated dynamics need to be identified from data only, leading to a challenging data-driven ROM problem. In this study, we present a hierarchical deep learning approach to identify ROM from measurement only; it simultaneously identifies the nonlinear normal modal (NNM) subspace with a hierarchical order and the associated nonlinear modal dynamics. We conduct study to validate such an approach on both unforced and forced nonlinear dynamical systems, and find that the identified hierarchical NNMs-spanned subspace enables an efficient and effective dimensional truncation to achieve optimally lowest-dimensional ROM. We discuss in detail its performance and applicability.

In this work, an efficient direction of arrival (DOA) tracking method for coherently distributed sources is proposed. The central DOA and angle spread are estimated by the proposed method with a particle filter at each snapshot. The spectrum calculated by the distributed source parameter estimator (DSPE) is used as a pseudo-likelihood function for particle updating, which enables the particle filter to process sensor signals directly without estimating the source amplitude. The proposed method is compared with the total least square estimating signal parameter via rotational invariance technique and the fast implementation of a power iteration subspace updating (FAPI-TLS-ESPRIT). The proposed method is verified by Monte Carlo simulations, and simulation results show that the proposed method can achieve an excellent DOA tracking performance and outperforms the FAPI-TLS-ESPRIT method. In addition, the proposed method can simultaneously estimate the central DOA and angle spread.

PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to $1$ and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations., 4 Figures, 6 Tables. 19 pages, the current version has been improved further and accepted by {\em Numerical Algorithms}

Suppose that $$\mathbb {V}$$ is a n-dimensional vector space and $$\mathbb {W}$$ is its fixed k-dimensional subspace such that $$n-k\ge 1.$$ In the present article, we initiate the study of a new graph structure “subspace-based subspace sum graph” $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V}),$$ where the vertex set $${\mathcal {V}}({\mathcal {G}}_{\mathbb {W}}(\mathbb {V}))$$ is the collection of all subspaces $${\mathcal {W}}$$ of $$\mathbb {V}$$ such that $${\mathcal {W}}+\mathbb {W}\ne \mathbb {V}$$ and $${\mathcal {W}}\nsubseteq \mathbb {W},$$ i.e., $${\mathcal {V}}({\mathcal {G}}_{\mathbb {W}}(\mathbb {V}))=$$ $$\{{\mathcal {W}}\subseteq \mathbb {V}~|~{\mathcal {W}}+\mathbb {W}\ne \mathbb {V}, {\mathcal {W}}\nsubseteq \mathbb {W}\}$$ and any two distinct vertices $${\mathcal {W}}_{1}$$ and $${\mathcal {W}}_{1}$$ of $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V})$$ are adjacent if and only if $${\mathcal {W}}_{1}+{\mathcal {W}}_{2}+\mathbb {W}=\mathbb {V}.$$ The girth, diameter, domination number, clique number, chromatic number and maximal independent sets of $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V})$$ have been studied. It is proved that two subspace-based subspace sum graphs $${\mathcal {G}}_{\mathbb {W}_{1}}(\mathbb {V})$$ and $${\mathcal {G}}_{\mathbb {W}_{2}}(\mathbb {V})$$ are isomorphic if and only if $$\mathbb {W}_{1}$$ and $$\mathbb {W}_{2}$$ are isomorphic. Further, we show that $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V})$$ is weakly perfect and characterize all vector spaces $$\mathbb {V}$$ and subspaces $$\mathbb {W}$$ of $$\mathbb {V}$$ for which $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V})$$ is perfect. Finally, in case of finite field degree of a given vertex, order and size of $${\mathcal {G}}_{\mathbb {W}}(\mathbb {V})$$ have also been obtained.

In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Frechet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*.

System parameters identification of nonlinear bistable structures has attracted considerable interest because the performance enhancement of energy harvesting and vibration control is significantly dependent on the model parameter of nonlinear systems. Therefore, a two-stage subspace method is proposed to identify the critical parameters in the system equation of nonlinear bistable piezoelectric structures. The dynamic equation of nonlinear bistable piezoelectric structures is separated into an underlying linear electromechanical coupling equation and a nonlinear mechanical equation. At first, for the underlying linear electromechanical coupling equation, a force–displacement subspace is constructed to identify the linear mass, damping and stiffness. Meanwhile, a velocity–voltage subspace is created for the identification of the electromechanical coupling coefficient. Next, for the nonlinear mechanical equation, the nonlinear restoring force in bistable structures can be estimated by the extended nonlinear frequency response function. Numerical simulation on a magnetic coupled bistable piezoelectric structure is performed to investigate the influence of frequency-swept responses, the noise intensity and polynomial order on identification accuracy. Experimental measurement of a magnetic coupled asymmetric bistable piezoelectric beam is conducted under different excitation conditions. Experimental results demonstrate the effectiveness of the proposed identification method.

A method is presented in which the ground-state subspace is projected out of a Hamiltonian representation. As a result of this projection, an effective Hamiltonian is constructed where its ground-state coincides with an excited-state of the original problem. Thus, low-lying excited-state energies can be calculated using existing hybrid quantum classical techniques and variational algorithm(s) for determining ground-state. The method as formulated is shown to be fully valid for the $$\hbox{H}_2$$ and LiH molecules. The primary restriction with our technique is the number of terms required in the projection operator, therefore we explore arguments to reduce the number of terms and discuss applicability to different classes of Hamiltonians.

The standard machine learning tasks often assume that the training (source domain) and test (target domain) data follow the same distribution and feature space. However, many real-world applications suffer from the limited number of training labeled data and benefit from the related available labeled datasets to train the model. In this way, since there is the distribution difference across the source and target domains (i.e., domain shift problem), the learned classifier on the training set might perform poorly on the test set. To address the shift problem, domain adaptation provides variety of solutions to learn robust classifiers to deal with distribution mismatch across the source and target domains. In this paper, we put forward a novel domain adaptation approach, referred to as cross- and multiple-domains visual transfer learning via iterative Fischer linear discriminant analysis (CIDA) to tackle shift problem across domains. CIDA transfers the source and target domains into a shared low-dimensional Fischer linear discriminant analysis (FLDA)-based subspace in an unsupervised manner. CIDA benefits joint FLDA and domain adaptation criterions to reduce the distribution mismatch across the training and test sets. Moreover, CIDA employs an adaptive classifier to build a robust model against data drift across different domains. Also, CIDA generates the intermediate pseudotarget labels to utilize the target data in training process. In this way, CIDA refines the pseudolabels using an iterative manner to converge the model. Our extensive experiments illustrate that CIDA significantly outperforms the baseline machine learning and other state-of-the-art transfer learning methods on nine visual benchmark datasets under different difficulties.

Partial multi-label learning is of great significant interest due to accurate supervision is difficult to be obtained. Recently, multi-view learning has been developed to deal with partial multi-label learning tasks. Although few multi-view partial multi-label learning methods have been proposed, all of them are designed under the full-view assumption. However, due to the difficulties in multi-view data collection, some views may not contain complete information in real task. The appearance of missing views will affect the performance of traditional partial multi-label learning algorithms. To solve this problem, we propose a novel I ncomplete M ulti-V iew P artial M ulti-L abel learning (IMVPML) framework which makes use of incomplete multi-view feature representation and utilizes the low-rank and sparse decomposition scheme to remove the noisy labels. Specifically, we first learn a shared subspace across heterogenous incomplete views. Secondly, we utilize the low-rank and sparse decomposition scheme to obtain the ground-truth labels. Thirdly, we introduce a graph Laplacian regularization to constrain the ground-truth labels and impose orthogonality constraints on the correlations between subspace. Finally, a predictive model is learned by shared subspace and disambiguation labels. Enormous experimental results demonstrate that the proposed method can achieve competitive performance in solving the problem of incomplete multi-view partial multi-label learning.

Non-negative matrix factorization (NMF), as an explanatory feature extraction technology, has powerful dimensionality reduction and semantic representation capabilities. In recent years, it has attracted great attention in the process of dimensionality reduction analysis of real high-dimensional data. However, the classic NMF algorithm is an unsupervised method in terms of learning methods. In the calculation process, the spatial structure information in the original data is often ignored, resulting in poor clustering performance of the algorithm in the subspace. In order to overcome the above problems, this paper proposes a semi-supervised NMF algorithm called semi-supervised dual graph regularized NMF with biorthogonal constraints (SDGNMF-BO). In this algorithm, the semi-supervised NMF three-factor decomposition is based on the dual graph model of the data space and feature space of the original data, which can effectively improve the learning ability of the algorithm in the subspace, and the biorthogonal constraint conditions are added in the decomposition process and achieve better local representation, significantly reduce the inconsistency between the original matrix and the basic vector. In order to prove the superiority of the algorithm under fair conditions, compares the multi-directional clustering experiments of 4 real image data sets and 1 text data set, and uses 2 clustering evaluation indexes to prove that the algorithm is better than other comparison algorithms.

A basic problem about a constant dimension subspace code is to find its maximal possible size Aq(n, d, k). In this paper, we investigate constant dimension codes with parallel linkage construction and multilevel construction and obtain new lower bounds on Aq(18,6,9). By combining the Johnson type bound, we obtain new lower bounds on Aq(17,6,8). These lower bounds are larger than previously best known bounds in Heinlein et al. (2019).

Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup G is non-locally compact and G has a first-countable remainder, then $$\chi (G)\le \omega _{1}$$ , $$\omega (G)\le 2^{\omega }$$ and $$|bG|\le 2^{\omega _{1}}$$ . Moreover, it is proved that the property of paracompact p-space of a strongly topological gyrogroup G is equivalent with G having a Lindelof remainder in a compactification. By this result, we prove that if H is a dense subspace of a strongly topological gyrogroup G which is locally pseudocompact and not locally compact, then every remainder of H is pseudocompact. Furthermore, if a strongly topological gyrogroup G has countable pseudocharacter and G is non-metrizable, then all remainders of G are pseudocompact. These two results give partial answers to a question posed by Arhangel’ skiǐ and Choban, see (Topol Appl 157:789–799, 2010, Problem 5.1). Finally, it is shown that the Lindelof property of a non-locally compact strongly topological gyrogroup G is equivalent with having a remainder with subcountable type for some compactifications of G.

Subspace clustering is one of the efficient techniques for determining the clusters in different subsets of dimensions. Ideally, these techniques should find all possible non-redundant clusters in which the data point participates. Unfortunately, existing hard subspace clustering algorithms fail to satisfy this property. Additionally, with the increase in dimensions of data, classical subspace algorithms become inefficient. This work presents a new density-based subspace clustering algorithm (S_FAD) to overcome the drawbacks of classical algorithms. S_FAD is based on a bottom-up approach and finds subspace clusters of varied density using different parameters of the DBSCAN algorithm. The algorithm optimizes parameters of the DBCAN algorithm through a hybrid meta-heuristic algorithm and uses hashing concepts to discover all non-redundant subspace clusters. The efficacy of S_FAD is evaluated against various existing subspace clustering algorithms on artificial and real datasets in terms of F_Score and rand_index. Performance is assessed based on three parameters: average ranking, SRR ranking, and scalability on varied dimensions. Statistical analysis is performed through the Wilcoxon signed-rank test. Results reveal that S_FAD performs considerably better on the majority of the datasets and scales well up to 6400 dimensions on the actual dataset.

Domain adaption is to transform the source and target domain data into a certain space through a certain transformation, so that the probability distribution of the transformed data is as close as possible. The domain adaption algorithm based on Maximum Mean Difference (MMD) Maximization and Reproducing Kernel Hilbert Space (RKHS) subspace transformation is the current main algorithm for domain adaption, in which the RKHS subspace transformation is determined by MMD of the transformed source and target domain data. However, MMD has inherent defects in theory. The probability distributions of two different random variables will not change after subtracting their respective mean values, but their MMD becomes zero. A reasonable method should be that the MMD of the source and target domain data with the same label should be as small as possible after RKHS subspace transformation. However, the labels of target domain data are unknown and there is no way to model according to this criterion. In this paper, a domain adaption algorithm based on source dictionary regularized RKHS subspace learning is proposed, in which the source domain data are used as a dictionary, and the target domain data are approximated by the sparse coding of the dictionary. That is to say, in the process of RKHS subspace transformation, the target domain data are distributed around the mostly relevant source domain data. In this way, the proposed algorithm indirectly achieves the MMD of the source and target domain data with the same label after RKHS subspace transformation. So far there has been no similar work reported in the published academic papers. The experimental results presented in this paper show that the proposed algorithm outperforms 5 other state-of-the-art domain adaption algorithms on 5 commonly used datasets.

Sparse multinomial logistic regression has recently received widespread attention. It provides a useful tool for solving multi-classification problems in various fields, such as signal and image processing, machine learning and disease diagnosis. In this paper, we first study the group sparse multinomial logistic regression model and establish its optimality conditions. Based on the theoretical results of this model, we hence propose an efficient algorithm called the subspace quadratic regularization algorithm to compute a stationary point of a given problem. This algorithm enjoys excellent convergence properties, including the global convergence and locally quadratic convergence. Finally, our numerical results on standard benchmark data clearly demonstrate the superior performance of our proposed algorithm in terms of logistic loss value, sparsity recovery and computational time.

Complete temperature field estimation from limited local measurements is widely desired in many industrial and scientific applications of thermal engineering. Since the sensor configuration dominates the reconstruction performance, some progress has been made in designing sensor placement methods. But these approaches remain to be improved in terms of both accuracy and efficiency due to the lack of comprehensive schemes and efficient optimization algorithms. In this work, we develop a data-driven sensor placement framework for thermal field reconstruction. Specifically, we first tailor the low-dimensional model from the prior thermal maps to represent the high-dimensional temperature distribution states by virtue of proper orthogonal decomposition technique. Then, on such subspace, a recursive greedy algorithm with determinant maximization as the objective function is developed to optimize the sensor placement configuration. Furthermore, we find that the same sensor configuration can be yielded faster by the standard procedures of column-pivoted QR factorization, which allows concise software implementation with readily available function packages. When the sensor locations are determined, we advocate using the data-based closed-form estimator to minimize the reconstruction error. Real-time thermal monitoring on the multi-core processor is employed as the case to demonstrate the effectiveness of the proposed methods for thermal field reconstruction. Extensive evaluations are conducted on simulation or experimental datasets of three processors with different architectures. The results show that our method achieves state-of-the-art reconstruction performance while possessing the lowest computational complexity when compared with the existing methods.

Graph regularization has shown its effectiveness in multi-view subspace clustering methods. Many multi-view subspace clustering methods based on graph regularization build the adjacency matrix directly based on a simple similarity measure between data points for each view. However, these simply constructed graphs are sensitive to light corruptions and even generate misleading manifold. Considering this shortcoming, this paper presents a multi-view subspace clustering algorithm (CGMSC) with a well-defined locally consistent graph regularization. We formulate CGMSC by a two-stage procedure. In the first stage, an adaptive self-weighted multi-view local linear embedding (ASWMVLLE) method is proposed to build the locally consistent geometric relationship between instances. In the second stage, ASWMVLLE is introduced into CGMSC by defining a local graph regularization term about the consensus latent subspace representation, which can not only effectively keep the manifold structure of data, but also ensure the consistency across different views. Experiments on eight real-world datasets demonstrate that our method has good robustness and clustering performance.

High-throughput sequencing of single-cell gene expression reveals a complex mechanism of individual cell's heterogeneity in a population. An important purpose for analyzing single-cell RNA sequencing (scRNA-seq) data is to identify cell subtypes and functions by cell clustering. To deal with high levels of noise and cellular heterogeneity, we introduced a new single cell data analysis model called Adaptive Total-Variation Regularized Low-Rank Representation (ATV-LRR). In scRNA-seq data, ATV-LRR can reconstruct the low-rank subspace structure to learn the similarity of cells. The low-rank representation can not only segment multiple linear subspaces, but also extract important information. Moreover, adaptive total variation also can remove cell noise and preserve cell feature details by learning the gradient information of the data. At the same time, to analyze scRNA-seq data with unknown prior information, we introduced the maximum eigenvalue method into the ATV-LRR model to automatically identify cell populations. The final clustering results show that the ATV-LRR model can detect cell types more effectively and stably.

International audience; This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on integral representation formulas. For the parallel and sequential forms of the method appearing in the literature, we propose a general abstract formulation in a given Hilbert setting and interpret the procedure in terms of subspace corrections. We then prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. An alternative proof of convergence is provided in a case which does not fit into the previous framework. We finally present some numerical tests.

Large scale multi-dimensional time series can be found in many disciplines, including finance, econometrics, biomedical engineering, and industrial engineering systems. It has long been recognized that the time dependent components of the vector time series often reside in a subspace, leaving its complement independent over time. In this paper we develop a method for projecting the time series onto a low-dimensional time-series that is predictable, in the sense that an auto-regressive model achieves low prediction error. Our formulation and method follow ideas from principal component analysis, so we refer to the extracted low-dimensional time series as principal time series. In one special case we can compute the optimal projection exactly; in others, we give a heuristic method that seems to work well in practice. The effectiveness of the method is demonstrated on synthesized and real time series.

Land subsidence (LS) is significant problem that can lead to casualties, destruction of infrastructure, and socio-economic and environmental problems. In this study, we examine the Damghan Plain of Iran where LS poses a major obstacle to growth and management of the region. Dagging and random subspace (RSS) as meta- or ensemble-classifiers of a radial basis function neural network (RBFnn) were combined into two novel-ensemble intelligence approaches (Dagging-RBFnn and RSS-RBFnn) to predict and map the susceptibility of land units to subsidence. The goodness-of-fit (of training data) and prediction accuracy (of testing data) for the ensemble models were contrasted with the RBFnn, which is used as the benchmark for improvement. Details of 120 LS locations were examined and the data for twelve LS conditioning factors (LSCFs) were compiled. The LS points were randomly divided into four groups or folds, each comprised of 25 percent of the cases. The novel ensemble models were constructed using 75 percent (3 folds) and tested with the remaining 25 percent (onefold) in a four-fold cross-validation (CV) mechanism. Information-gain ratio and multicollinearity tests were used to select the LSCFs that would be used to estimate LS probabilities. The importance of each factor was calculated using a random forest (RF) model. The most important LSCFs were groundwater drawdown, land uses and land covers, elevation, and lithology. Twelve land subsidence susceptibility maps were generated using the k-fold CV approaches as each of the three models (RBFnn, Dagging-RBFnn and RSS-RBFnn) was applied to each of the four folds. The LS susceptibility models reveal a strong probability for LS on 15% to 24% of the plain. All of the maps generated achieved adequate levels of prediction accuracies and goodness-of-fits. The Dagging-RBFnn ensemble yielded the most robust maps, however. The ensemble of Dagging-RBFnn enhances the accuracy of modeling but the opposite condition was found for the RSS-RBFnn ensemble. It is evident that ensembles with meta classifiers might not always increase the accuracy of the base classifier. Overall, the southern part of the plain shows the highest LS risk. The results of this study suggests that groundwater withdrawal levels should be tracked and possibly restricted in regions with higher (extreme or moderate) probabilities of LS. This demonstrates that new approaches can support land use planning and decision making to minimize LS and improve sustainability.