Dimensional Subspace Research Articles

Orthogonal Subspace Representation for Generative Adversarial Networks.

Disentanglement learning aims to separate explanatory factors of variation so that different attributes of the data can be well characterized and isolated, which promotes efficient inference for downstream tasks. Mainstream disentanglement approaches based on generative adversarial networks (GANs) learn interpretable data representation. However, most typical GAN-based works lack the discussion of the latent subspace, causing insufficient consideration of the variation of independent factors. Although some recent research analyzes the latent space on pretrained GANs for image editing, they do not emphasize learning representation directly from the subspace perspective. Appropriate subspace properties could facilitate corresponding feature representation learning to satisfy the independent variation requirements of the obtained explanatory factors, which is crucial for better disentanglement. In this work, we propose a unified framework for ensuring disentanglement, which fully investigates latent subspace learning (SL) in GAN. The novel GAN-based architecture explores orthogonal subspace representation (OSR) on vanilla GAN, named OSRGAN. To guide a subspace with strong correlation, less redundancy, and robust distinguishability, our OSR includes three stages, self-latent-aware, orthogonal subspace-aware, and structure representation-aware, respectively. First, the self-latent-aware stage promotes the latent subspace strongly correlated with the data space to discover interpretable factors, but with poor independence of variation. Second, the following orthogonal subspace-aware stage adaptively learns some 1-D linear subspace spanned by a set of orthogonal bases in the latent space. There is less redundancy between them, expressing the corresponding independence. Third, the structure representation-aware stage aligns the projection on the orthogonal subspace and the latent variables. Accordingly, feature representation in each linear subspace can be distinguishable, enhancing the independent expression of interpretable factors. In addition, we design an alternating optimization step, achieving a tradeoff training of OSRGAN on different properties. Despite it strictly constrains orthogonality, the loss weight coefficient of distinguishability induced by orthogonality could be adjusted and balanced with correlation constraint. To elucidate, this tradeoff training prevents our OSRGAN from overemphasizing any property and damaging the expressiveness of the feature representation. It takes into account both interpretable factors and their independent variation characteristics. Meanwhile, alternating optimization could keep the cost and efficiency of forward inference unchanged and will not burden the computational complexity. In theory, we clarify the significance of OSR, which brings better independence of factors, along with interpretability as correlation could converge to a high range faster. Moreover, through the convergence behavior analysis, including the objective functions under different constraints and the evaluation curve with iterations, our model demonstrates enhanced stability and definitely converges toward a higher peak for disentanglement. To depict the performance in downstream tasks, we compared the state-of-the-art GAN-based and even VAE-based approaches on different datasets. Our OSRGAN achieves higher disentanglement scores on FactorVAE, SAP, MIG, and VP metrics. All the experimental results illustrate that our novel GAN-based framework has considerable advantages on disentanglement.

A rank-updating technique for the Kronecker canonical form of singular pencils

For a linear time-invariant system x˙(t)=Ax(t)+Bu(t), the Kronecker canonical form (KCF) of the matrix pencil (sI−A|B) provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil sF−G using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.

Generalized bent functions with large dimension

By a fundamental result by Mesnager et al. in 2018, a generalized bent function (originally defined as a class of functions from an $ n $-dimensional vector space $ \mathbb{V}_n^{(p)} $ into a cyclic group), is a bent function $ g: \mathbb{V}_n^{(p)}\rightarrow \mathbb{F}_p $ with a partition $ \mathcal{P} $ of $ \mathbb{V}_n^{(p)} $, such that for every function $ C $ which is constant on the sets of $ \mathcal{P} $, the function $ g + C $ is bent. The set of these bent functions forms then an affine space of dimension $ |\mathcal{P}| \le p^{n/2} $. This characterization of generalized bent functions is much more comprehensive than any earlier description.In this article, we analyse some classes of bent functions under this perspective. As shown earlier, Maiorana-McFarland bent functions permit the largest possible partitions, giving rise to $ p^{n/2} $-dimensional affine bent function spaces. The reason behind is their characterization as the bent functions, which are affine restricted to the $ n/2 $-dimensional affine subspaces of a trivial cover of $ \mathbb{V}_n^{(p)} $. We will show that maximal possible partitions can also be obtained for other classes of (regular) bent functions. Most notably, these classes are described as bent functions which are affine restricted to non-trivial covers of $ \mathbb{V}_n^{(p)} $. We investigate (largest) partitions for other types of bent functions, including weakly regular and non-weakly respectively non-dual bent functions. We round off the article giving partitions for bent functions obtained from bent partitions, which includes the partial spread class, and partitions for Carlet's class $ \mathcal{C} $ and $ \mathcal{D} $.

Unsupervised Feature Selection via Controllable Adaptive Graph Learning and Discriminative Feature Learning.

Unsupervised feature selection is challenging in machine learning, pattern recognition, and data mining. The crucial difficulty is to learn a moderate subspace that preserves the intrinsic structure and to find uncorrelated or independent features simultaneously. The most common solution is first to project the original data into a lower dimensional space and then force them to preserve the similar intrinsic structure under linear uncorrelation constraint. However, there are three shortcomings. First, the final graph generated by the iterative learning process differs significantly from the initial graph in which the original intrinsic structure is embedded. Second, it requires prior knowledge about a moderate dimension of subspace. Third, it is inefficient when dealing with high-dimensional datasets. The first shortcoming, which is longstanding and undiscovered, makes the previous methods fail to achieve their expected results. The last two ones increase the difficulty of applying in different fields. Therefore, two unsupervised feature selection methods are proposed based on controllable adaptive graph learning and uncorrelated/independent feature learning (CAG-U and CAG-I) to address the abovementioned issues. In the proposed methods, the final graph that preserves intrinsic structure can be adaptively learned while the difference between the two graphs can be well controlled. Besides, relatively uncorrelated/independent features can be selected using a discrete projection matrix. The experimental results on 12 datasets in different fields show the superiority of CAG-U and CAG-I.

Widths and rigidity

We consider the Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions in $L_2$ is rigid, that is, it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$ for $p<2$ the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We present some sufficient conditions for rigidity. We prove that the independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$, the metric that corresponds to convergence in measure. In the case of $L_p$ for $1<p<2$ the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, for example, that the first $N$ trigonometric functions can be approximated by very low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$ for ${p<1}$. Bibliography: 34 titles.

New constant dimension subspace codes from multilevel linkage construction

<p style='text-indent:20px;'>One of the main problems in subspace coding is to determine the maximal possible size of a constant dimension subspace code with given parameters. In this paper, we introduce a family of new codes, named rank-restricted Ferrers diagram rank-metric codes, to give an improved construction of constant dimension subspace codes. Our method constructs many new CDCs with larger sizes than previously best-known codes.</p>

Self-adaptive subspace representation from a geometric intuition

In this paper, we address the unsupervised subspace learning task from a geometric intuition, which offers a direct understanding of the data representation. First, we consider all subspaces of the Euclidean space as a series of graded Grassmannian manifolds, and represent them by orbits of rotation group action. Then, we reformulate the unsupervised subspace learning by a least square problem with respect to rotation and projection operator. Second, we introduce a low-rank regularization to obtain a low-dimensional and robust subspace representation. Then, the model is translated into a minimization problem on the Grassmannian manifold. By dividing the model into two subproblems of solving the optimal rotation and subspace dimension, we design an alternating iteration strategy, where the locally geodesic structure of the rotation group and the unconstrained quadratic 0-1 programming are utilized. Finally, we apply the proposed method to the image classification problem on five benchmark datasets and compare it with nine state-of-the-art methods. Numerical results show that our proposed method has better feature representation ability and almost achieves the best performance in terms of classification accuracy and robustness.

Improved eigenspace-based method for robust adaptive beamforming with dimension search

The eigenspace-based technique, as a classic method for robust adaptive beamforming (RAB), can substantially mitigate the negative impact caused by model mismatch. However, the conventional eigenspace-based beamformer (CESB) undergoes severe performance degradation at low signal-to-noise ratio (SNR). This is because it firmly sets the dimension of the projection subspace (DOPS) to the source number. To alleviate this problem, an improved eigenspace-based approach is proposed in this communication, where the DOPS is auto-selected to adapt to different cases. We roughly analyze the rationality that the robustness of the eigenspace-based technique can be enhanced by simply adjusting the DOPS at low SNR. In addition, we concretely propose an automatic selection method for the DOPS derived from a minimum beamformer sensitivity estimation problem. The proposed method is advantageous in that it is parameter-free. Moreover, simulation experiments are performed to demonstrate the improvement of the CESB and its robustness against steering vector mismatch.

On the frame complex of symplectic spaces

For a symplectic space V of dimension 2n over Fq, we compute the eigenvalues of its orthogonality graph. This is the simple graph with vertices the 2-dimensional non-degenerate subspaces of V and edges between orthogonal vertices. As a consequence of Garland's method, we obtain vanishing results on the homology groups of the frame complex of V, which is the clique complex of this graph. We conclude that if n<q+3 then the poset of frames of size ≠0,n−1, which is homotopy equivalent to the frame complex, is Cohen-Macaulay over a field of characteristic 0. However, we also show that this poset is not Cohen-Macaulay if the dimension is big enough.

A note on varieties of weak CM-type

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of Hn(X;Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H⁎(X;Q) is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3×T2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.

Dimensionality-Driven Evolution of Electronic Structure and Transport Properties in Pressure-Induced Phases of Ca&lt;sub&gt;2&lt;/sub&gt;N Electride

We investigate how a change in dimensionality of interstitial electronic states in the Ca2N electride influences its electronic structure and transport properties. Employing the Maximally Localized Wannier Functions (MLWF) approach, we successfully describe the interstitial quasi-atomic states (ISQ) located in non-nuclear Wyckoff positions between Ca atoms. This allowed us to conclude that the electride subsystem is responsible for the formation of a band structure in the vicinity of the Fermi level in all Ca2N phases observed under pressure. Using the obtained MLWF basis, we calculate the electronic and thermal conductivity, along with the Seebeck coefficient, by solving the semi-classical Boltzmann transport equations. The results achieved permit the conclusion that the counterintuitive increase in resistance under pressure observed experimentally is attributed to enhanced localization of interstitial electronic states through electride subspace dimensionality transformations. We also established a substantial anisotropy in the transport properties within the 2D phase and found that the conductivity inside the plane of the electride layers is provided by electrons, while along the direction normal to the layers, holes become the majority carriers.

STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES

Previously, we have described the structure of the sets of Picard and Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros. It was proved that the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension not higher than p-1 of a p-dimensional complex Euclidean space. In addition, the sum of the dimensions of all these subspaces does not exceed p and any pairwise intersection of these subspaces contains only the zero vector. The set of Picard exceptional vectors has the same structure. For an integer curve of non-integer or zero order, the set of Borel exceptional vectors together with the zero vector is a single subspace of dimension no higher than p-1, which is not the case for the set of Picard exceptional vectors of an integer curve of non-integer order. The sufficiency of these results is confirmed to some extent by the corresponding examples. In this paper, we show that the above results on the structure of the set of Borel and Picard exceptional vectors for integer curves can be transferred to the case of analytic curves in the circle. We have also constructed an analytic curve in the circle, which confirms, to some extent, the sufficiency of the results. However, it was not possible to find out whether the corresponding result is transferred to the case of analytic curves of non-integer or zero order.

Krylov construction and complexity for driven quantum systems.

Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos and has recently been much studied for various time-independent systems. We initiate the study of K complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K complexity with the system coupling constant interpolating between the weak and strong coupling regimes. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.

Sparse robust adaptive unsupervised subspace learning for dimensionality reduction

This work is devoted to the investigation of dimension reduction problem. As an efficient dimension reduction method, much attention has been paid on unsupervised subspace learning since it does not rely on expensive labels. Firstly, we implant a robust estimator in the error term of objective function, this leads to that small coefficients can be automatically allocated to the abnormal points. Thus, our model is robust to noise and outliers. Posteriorly, the L2,r-norm (1≤r≤2) is used as a measure of error, then, the performance of the model can be improved by selecting the appropriate adaptive parameter r. Further, a L2,p-norm (0<p≤1) regularization term is added to the objective function, therefore the gained sparse subspace can further improve the efficiency and accuracy of the expression, as well as enhance the generalization ability of the model and reduce over-fitting. Moreover, an efficient algorithm with fast convergence speed is designed to solve the model. Finally, the experimental results on 14 datasets show that the subspace dimension obtained by our method is no more than 6. From the results of classification test, our algorithm has obvious advantages over the other similar six algorithms.

Green function based ultrasonic super resolution imaging for defect location and characterization

Green function is the important component for ultrasonic time reversal based super resolution imaging methods. Time reversal with multiple signal classification (TR-MUSIC) and its phase-coherent form (PC-MUSIC) are two typical imaging methods. In this paper, the effect of Green function on the TR-MUSIC and PC-MUSIC has been investigated. And TFM has also been introduced as a comparison. Point spread function modelling has been implemented to compare the performance of imaging methods under different forms of Green function via simulation. Several experiments have been implemented on metallic samples with defects. The defects can be divided into two categories: point-like targets and extended targets, depending on their relationship with the ultrasonic central wavelength. The perform of imaging methods under different forms of Green function has been compared in terms of lateral resolution, axial resolution, robustness to the noise and imaging for extended targets. A series of parameters have been defined to achieve quantitative comparison. The experimental results show that when compared with TR-MUSIC, PC-MUSIC is sensitive to the Green function. When the proper combination including Green function, the frequency bandwidth as well as the dimension of signal subspace has been setup, the defect can be correctly located and accurately quantitative evaluated based on the PC-MUSIC image.

On T-invariant subvarieties of symplectic Grassmannians and representability of rank 2 symplectic matroids over [formula omitted

For the symplectic Grassmannian SpG(2,2n) of 2-dimensional isotropic subspaces in a 2n-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an n-dimensional torus T on it, we characterize its irreducible T-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank 2 which are representable over C.

Reduced order modeling of structural problems with damage and plasticity

AbstractModel order reduction (MOR) techniques are used across the engineering sciences to reduce the computational complexity of high‐fidelity simulations. MOR methods reduce the computation time by representing the problem using a lower number of degrees of freedom (DOF). The use of reduced order models (ROM) in the analysis of structural problems with damage and plasticity has the potential to significantly reduce computational time and increase efficiency. Of course, the approximation of a problem in a lower dimensional space introduces an approximation error that needs to be kept small enough so that the results of the ROM maintain their validity. One well‐known reduced order modeling approach is the proper orthogonal decomposition (POD). POD is used to extract the dominant modes of the structure, which are then used to solve a problem in the smaller dimensional subspace. To overcome the limitations of the POD regarding nonlinear problems, the discrete empirical interpolation method (DEIM) is employed. An exemplary uncertainty quantification application is used to investigate the methodology. The investigation shows that the POD‐based DEIM can significantly reduce the computational effort of highly nonlinear structural simulations incorporating damage while maintaining a high approximation accuracy.

Statistical performance of quantile tensor regression with convex regularization

In this paper, we consider high-dimensional quantile tensor regression using a general convex decomposable regularizer and analyze the statistical performances of the estimator. The rates are stated in terms of the intrinsic dimension of the estimation problem, which is, roughly speaking, the dimension of the smallest subspace that contains the true coefficient. Previously, convex regularized tensor regression has been studied with a least squares loss, Gaussian tensorial predictors and Gaussian errors, with rates that depend on the Gaussian width of a convex set. Our results extend the previous work to nonsmooth quantile loss. To deal with the non-Gaussian setting, we use the concept of Rademacher complexity with appropriate concentration inequalities instead of the Gaussian width. For the multi-linear nuclear norm penalty, our Orlicz norm bound for the operator norm of a random matrix may be of independent interest. We validate the theoretical guarantees in numerical experiments. We also demonstrate advantage of quantile regression over mean regression, and compare the performance of convex regularization method and nonconvex decomposition method in solving quantile tensor regression problem in simulation studies.

Numerical homogenization of spatial network models

We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.

Model order estimation based on the correntropy of observation eigenvalues

This paper presents a novel approach for estimating the model order in the presence of observation errors. The proposed method is based on the correntropy estimation of eigenvalues in the observation space. By utilizing the bootstrap method to resample the observations, more precise estimations can be obtained. The next step in the algorithm involves partitioning the observation space, generated by the covariance matrix of mixtures, into two subspaces: the signal subspace and the noise subspace. The order of the model, corresponding to the dimension of the signal subspace, is determined using a correntropy estimator based on kernel functions. Theoretical analysis demonstrates the consistency of the proposed algorithm. Comparative evaluations against existing methods in the literature highlight the superior performance of this information-theoretic approach.