Subspace Sampling Research Articles

Nonuniform Sampling in Lp-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform

In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the Lp-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a Lp-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations.

Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators

Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators

Random Subspace Sampling for Classification with Missing Data

Random Subspace Sampling for Classification with Missing Data

Gabor frames for rational functions

We study the frame properties of the Gabor systems G(g;α,β):={e2πiβmxg(x-αn)}m,n∈Z.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathfrak {G}}(g;\\alpha ,\\beta ):=\\{e^{2\\pi i \\beta m x}g(x-\\alpha n)\\}_{m,n\\in {\\mathbb {Z}}}. \\end{aligned}$$\\end{document}In particular, we prove that for Herglotz windows g such systems always form a frame for L^2({mathbb {R}}) if alpha ,beta >0, alpha beta le 1. For general rational windows gin L^2({mathbb {R}}) we prove that {mathfrak {G}}(g;alpha ,beta ) is a frame for L^2({mathbb {R}}) if 0<alpha ,beta , alpha beta <1, alpha beta not in {mathbb {Q}} and {hat{g}}(xi )ne 0, xi >0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L^2({mathbb {R}}).

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

Functional principal subspace sampling for large scale functional data analysis

Functional data analysis (FDA) methods have computational and theoretical appeals for some high dimensional data, but lack the scalability to modern large sample datasets. To tackle the challenge, we develop randomized algorithms for two important FDA methods: functional principal component analysis (FPCA) and functional linear regression (FLR) with scalar response. The two methods are connected as they both rely on the accurate estimation of functional principal subspace. The proposed algorithms draw subsamples from the large dataset at hand and apply FPCA or FLR over the subsamples to reduce the computational cost. To effectively preserve subspace information in the subsamples, we propose a functional principal subspace sampling probability, which removes the eigenvalue scale effect inside the functional principal subspace and properly weights the residual. Based on the operator perturbation analysis, we show the proposed probability has precise control over the first order error of the subspace projection operator and can be interpreted as an importance sampling for functional subspace estimation. Moreover, concentration bounds for the proposed algorithms are established to reflect the low intrinsic dimension nature of functional data in an infinite dimensional space. The effectiveness of the proposed algorithms is demonstrated upon synthetic and real datasets.

Average sampling in certain subspaces of Hilbert–Schmidt operators on $$L^2(\mathbb {R}^d)$$

The concept of translation of an operator allows to consider the analogue of shift-invariant subspaces in the class of Hilbert–Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of square integrable functions in the phase space \(\mathbb {R}^d\times \widehat{\mathbb {R}}^d\) and the Hilbert space of Hilbert–Schmidt operators on \(L^2(\mathbb {R}^d)\), which permits to take advantage of some well established sampling results.

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

Improved Initialization Method for Metaheuristic Algorithms: A Novel Search Space View

As an essential step of metaheuristic optimizers, initialization seriously affects the convergence speed and solution accuracy. The main motivation of the state-of-the-art initialization method is to generate a small initial population to cover the search space as much as possible uniformly. However, these approaches have suffered from the curse of dimensionality, high computational cost, and sensitivity to parameters, which ultimately reduce the algorithm’s convergence speed. In this paper, a new initialization technique named diagonal linear uniform initialization (DLU) is proposed, which follows a novel search view, i.e., adopting the diagonal subspace sampling instead of the whole space. By considering the algorithm’s update mechanism, the improved sampling method dramatically improves the convergence speed and solution accuracy of metaheuristic algorithms. Compared with the other eight widely used initialization strategies, the differential evolution (DE) algorithm with DLU obtains the best performance in search accuracy and convergence speed. In the extension experiments, results show that the DLU is still effective for three swarm-based algorithms: particle swarm optimization (PSO), cuckoo search (CS), and artificial bee colony (ABC). Especially for the multi-objective problem, the DLU still demonstrates its powerful performance compared with other strategies.

Hybrid Deep Transfer Network and Rotational Sample Subspace Ensemble Learning for Early Cancer Detection

Accurate histopathology cell image classification plays an important role in early cancer detection and diagnosis. Currently, Convolutional Neural Network is used to assist pathologists for histopathology image classification. In the paper, a Min mice model was applied to evaluate the capability of Convolutional Neural Network features for detecting early-stage carcinogenesis. However, due to the limited histopathology images of the mice model, it may cause overfitting for the classification. Hence, hybrid deep transfer network and rotational sample subspace ensemble learning is proposed for the histopathology image classification. First, deep features are obtained by deep transfer network based on regularized loss functions. Then, the rotational sample subspace sampling is applied to increase the diversity between training sets. Subsequently, subspace projection learning is introduced to achieve dimensionality reduction. Finally, the ensemble learning is used for histopathology image classification. The proposed method was tested on 126 histopathology images of the mouse model. The experimental results demonstrate that the proposed method has achieved a remarkable classification accuracy (99.39%, 99.74%, 100%). It has demonstrated that the proposed approach is promising for early cancer diagnosis.

Hybrid Deep Transfer Network and Rotational Sample Subspace Ensemble Learning for Early Cancer Detection

Accurate histopathology cell image classification plays an important role in early cancer detection and diagnosis. Currently, Convolutional Neural Network is used to assist pathologists for histopathology image classification. In the paper, a Min mice model was applied to evaluate the capability of Convolutional Neural Network features for detecting early-stage carcinogenesis. However, due to the limited histopathology images of the mice model, it may cause overfitting for the classification. Hence, hybrid deep transfer network and rotational sample subspace ensemble learning is proposed for the histopathology image classification. First, deep features are obtained by deep transfer network based on regularized loss functions. Then, the rotational sample subspace sampling is applied to increase the diversity between training sets. Subsequently, subspace projection learning is introduced to achieve dimensionality reduction. Finally, the ensemble learning is used for histopathology image classification. The proposed method was tested on 126 histopathology images of the mouse model. The experimental results demonstrate that the proposed method has achieved a remarkable classification accuracy (99.39%, 99.74%, 100%). It has demonstrated that the proposed approach is promising for early cancer diagnosis.

Development of an efficient global optimization method based on adaptive infilling for structure optimization

For problems with expensive black-box functions, the surrogate-based optimization (SBO) is more efficient than the conventional evolutionary algorithms in searching for the global optimum. However, the SBO converges much slower and shows imperfection in local exploitation, along with the increase of the scale of the design space, the number of the design variables, and the nonlinearity of the problems. This paper proposes an efficient global optimization method, which integrates an adaptive infilling by fuzzy clustering algorithm into an SBO process based on Kriging model. In each refinement cycle, a Kriging model is first built using samples in the current design space; then a fuzzy clustering algorithm is adopted to partition the design space into several subspaces considering inner features of the samples. Thus, new infilling samples are selected within each subspace by maximizing the expected improvement of the objective function and minimizing the surrogate prediction. Thereafter, the design space is updated by merging those subspaces, resulting in a diminishing design space during refinement. Furthermore, the parameters for the adaptive infilling procedure are studied to recommend reasonable settings for running optimizations. The proposed method is finally validated and assessed by eight analytical tests with bound constraints, and then employed in a beam optimization problem and a rocket interstage optimization problem under nonlinear constraints. The results indicate that the adaptive infilling behaves quite well in space exploration due to sampling in clustered subspaces, and possesses good performance in local exploitation as well because of space reduction.

Domain adaptation for object recognition using subspace sampling demons

Manually labeling data for training machine learning models is time-consuming and expensive. Therefore, it is often necessary to apply models built in one domain to a new domain. However, existing approaches do not evaluate the quality of intermediate features that are learned in the process of transferring from the source domain to the target domain, which results in the potential for sub-optimal features. Also, transfer learning models in existing work do not provide optimal results for a new domain. In this paper, we first propose a fast subspace sampling demons (SSD) method to learn intermediate subspace features from two domains and then evaluate the quality of the learned features. To show the applicability of our model, we test our model using a synthetic dataset as well as several benchmark datasets. Extensive experiments demonstrate significant improvements in classification accuracy over the state of the art.

Motion Planning by Sampling in Subspaces of Progressively Increasing Dimension

This paper introduces an enhancement to traditional sampling-based planners, resulting in efficiency increases for high-dimensional holonomic systems such as hyper-redundant manipulators, snake-like robots, and humanoids. Despite the performance advantages of modern sampling-based motion planners, solving high dimensional planning problems in near real-time remains a considerable challenge. The proposed enhancement to popular sampling-based planning algorithms is aimed at circumventing the exponential dependence on dimensionality, by progressively exploring lower dimensional volumes of the configuration space. Extensive experiments comparing the enhanced and traditional version of RRT, RRT-Connect, and Bidirectional T-RRT on both a planar hyper-redundant manipulator and the Baxter humanoid robot show significant acceleration, up to two orders of magnitude, on computing a solution. We also explore important implementation issues in the sampling process and discuss the limitations of this method.

Analogue beamforming solution for beam‐alignment problem

Analogue only beamforming generally refers to a structure of analogue phase shifters that have no control over the amplitude of the signals and are utilised in millimetre-wave massive multi-input multi-output systems to reduce the cost and power consumption. Conventional analogue only beamforming methods are based on beam-training and subspace sampling where the transmitter and receiver select the best beamformer and combiner from a list of predefined beam patterns to maximise the beamforming gain. In this study, first, the beam-alignment problem is formulated as a quadratic optimisation problem with quadratic constraints. Then, a practical solution for analogue only beamforming is proposed to maximise beamforming gain. The proposed method has no assumption on the array structure or isotropic antenna elements in the array. Also, a new method is proposed to design sub-codebooks to reduce the construction time of the codebook. Simulation results also corroborate the analytical results and performance of the proposed method.

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $${\mathcal {H}}$$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on $${\mathcal {H}}$$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

A Hierarchical Gamma Mixture Model-Based Method for Classification of High-Dimensional Data

Data classification is an important research topic in the field of data mining. With the rapid development in social media sites and IoT devices, data have grown tremendously in volume and complexity, which has resulted in a lot of large and complex high-dimensional data. Classifying such high-dimensional complex data with a large number of classes has been a great challenge for current state-of-the-art methods. This paper presents a novel, hierarchical, gamma mixture model-based unsupervised method for classifying high-dimensional data with a large number of classes. In this method, we first partition the features of the dataset into feature strata by using k-means. Then, a set of subspace data sets is generated from the feature strata by using the stratified subspace sampling method. After that, the GMM Tree algorithm is used to identify the number of clusters and initial clusters in each subspace dataset and passing these initial cluster centers to k-means to generate base subspace clustering results. Then, the subspace clustering result is integrated into an object cluster association (OCA) matrix by using the link-based method. The ensemble clustering result is generated from the OCA matrix by the k-means algorithm with the number of clusters identified by the GMM Tree algorithm. After producing the ensemble clustering result, the dominant class label is assigned to each cluster after computing the purity. A classification is made on the object by computing the distance between the new object and the center of each cluster in the classifier, and the class label of the cluster is assigned to the new object which has the shortest distance. A series of experiments were conducted on twelve synthetic and eight real-world data sets, with different numbers of classes, features, and objects. The experimental results have shown that the new method outperforms other state-of-the-art techniques to classify data in most of the data sets.

BAYESIAN FINITE ELEMENT MODEL UPDATING METHOD BASED ON MULTI-CHAIN DIFFERENTIAL EVOLUTION

To cope with the shortages of low sampling efficiency and difficult convergence of traditional Bayesian method under high-dimension parameters, a Bayesian based finite element model is proposed by the base of a multi-chain differential evolution algorithm. Based on the standard Markov chain Monte Carlo (MCMC) method, a differential evolution algorithm is introduced, and a random difference operation among multiple Markov chains is derived from the size and direction of an adaptive selection condition distribution to quickly approximate the target distribution. A subspace sampling algorithm which uses adaptive selection to select good parameter dimensions for sampling is introduced to improve sampling efficiency. Also, an anomaly Markov chain detection algorithm which detects and eliminates abnormities in Markov chains in the non-stationary period is introduced to improve the sampling efficiency in the stationary phase. The correction results of a simply-supported-beam model and a four-floor-frame-structure model show that the proposed method has a higher correction accuracy, better noise resistance, and better correction effects than that of DRAM algorithm under high order frequency and mode shapes, which provides a new way to improve the computational accuracy in uncertainty model correction.

On Cesáro Averages for Weighted Trees in the Random Forest

The random forest is a popular and effective classification method. It uses a combination of bootstrap resampling and subspace sampling to construct an ensemble of decision trees that are then averaged for a final prediction. In this paper, we propose a potential improvement on the random forest that can be thought of as applying a weight to each tree before averaging. The new method is motivated by the potential instability of averaging predictions of trees that may be of highly variable quality, and because of this, we replace the regular average with a Cesaro average. We provide both a theoretical analysis that gives exact conditions under which the new approach outperforms the traditional random forest, and numerical analysis that shows the new approach is competitive when training a classification model on numerous realistic data sets.

Leveraging the Restricted Isometry Property: Improved Low-Rank Subspace Decomposition for Hybrid Millimeter-Wave Systems

Communication at millimeter wave frequencies will be one of the essential new technologies in 5G. Acquiring an accurate channel estimate is the key to facilitate advanced millimeter wave hybrid multiple-input multiple-output (MIMO) precoding techniques. Millimeter wave MIMO channel estimation, however, suffers from a considerably increased channel use overhead. This happens due to the limited number of radio frequency (RF) chains that prevent the digital baseband from directly accessing the signal at each antenna. To address this issue, recent research has focused on adaptive closed-loop and two-way channel estimation techniques. In this paper, unlike the prior approaches, we study a non-adaptive, hence rather simple, open-loop millimeter wave MIMO channel estimation technique. We present a simple random design of channel subspace sampling signals and show that they obey the restricted isometry property (RIP) with high probability. We then formulate the channel estimation as a low-rank subspace decomposition problem and, based on the RIP, show that the proposed framework reveals resilience to a low signal-to-noise ratio. It is revealed that the required number of channel uses ensuring a bounded estimation error is linearly proportional to the degrees of freedom of the channel, whereas it converges to a constant value if the number of RF chains can grow proportionally to the channel dimension while keeping the channel rank fixed. In particular, we show that the tighter the RIP characterization the lower the channel estimation error is. We also devise an iterative technique that effectively finds a suboptimal but stationary solution to the formulated problem. The proposed technique is shown to have improved channel estimation accuracy with a low channel use overhead as compared to that of previous closed-loop and two-way adaptation techniques.