Two-dimensional Subspace Research Articles

Face Recognition Method Based on Probabilistic Neural Network Optimizing Two-Dimensional Subspace Analysis

If there is noise in the original image of face recognition, the efficiency of face recognition will be affected. In this paper, a face recognition method based on probabilistic neural network optimizing two-dimensional subspace analysis was proposed. Firstly, discrete wavelet variation was used to preprocess the image, and then two-dimensional linear discriminant analysis was used for feature extraction. Finally, the probabilistic neural network was used to complete the face classification. According to the results of experiments conducted on ORL and Fei general face database and the database collected independently, the recognition rate can also be as high as 98.9% when noise is added, and compared with several new identification methods, this method can achieve better identification performance.

Quantum Teleportation in High Dimensions.

Quantum teleportation allows a "disembodied" transmission of unknown quantum states between distant quantum systems. Yet, all teleportation experiments to date were limited to a two-dimensional subspace of quantized multiple levels of the quantum systems. Here, we propose a scheme for teleportation of arbitrarily high-dimensional photonic quantum states and demonstrate an example of teleporting a qutrit. Measurements over a complete set of 12 qutrit states in mutually unbiased bases yield a teleportation fidelity of 0.75(1), which is well above both the optimal single-copy qutrit state-estimation limit of 1/2 and maximal qubit-qutrit overlap of 2/3, thus confirming a genuine and nonclassical three-dimensional teleportation. Our work will enable advanced quantum technologies in high dimensions, since teleportation plays a central role in quantum repeaters and quantum networks.

Poor man's topological quantum gate based on the Su-Schrieffer-Heeger model

Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. Here, we propose and study a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed $Y$ gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zero-energy modes. With numerical simulations, we demonstrate that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer, nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is non-interacting and non-superconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides.

A Combined Short Pulse-mKDV Equation and its Exact Solutions by Two-Dimensional Invariant Subspaces

In this paper, a multi-component combined short pulse-mKdV equation is constructed as a nonstreching invariant curve flow in the n -dimensional unit sphere S n ( 1 ) = SO ( n + 1 ) / SO( n ) . The invariant subspace method is used to solve the resulting combined short pulse-mKdV equation with n = 1. Through symbolic computation with the aid of mathematical software Maple, new exact solutions are obtained from its two-dimensional invariant subspaces formed in forms of the exponential functions and trigonometric functions.

Direction-of-Arrival Estimation of Circular and Noncircular Wideband Source Signals via Augmented Envelope Alignment

In practical applications of array system, the incident signals may come from either circular or noncircular sources, or both. The problem of direction-of-arrival (DOA) estimation is herein addressed for both circular and noncircular wideband source signals. Based on the concept of effective rank, the aligned augmented effective noise subspace, spanned by the subordinate eigenvectors of the aligned array output augmented covariance matrix, is constructed for each scanning angle. It is shown that the aligned augmented effective noise subspace corresponding to a true DOA of signal is: 1) approximately orthogonal to a frequency-invariant one-dimensional subspace spanned by the aligned augmented transfer vector of the signal in the presence of multiple strictly noncircular sources; and 2) approximately orthogonal to a frequency-invariant two-dimensional subspace spanned by the aligned augmented transfer matrix of the signal in the presence of multiple circular or partially noncircular sources. This enables the noncircularity exploited wideband DOA estimation based on rank-reduction and norm-nulling. The performance of the proposed method has been evaluated and compared with some existing popular wideband DOA estimators by extensive simulations.

Arbitrary d–q current sharing in three‐phase winding sets of multi‐phase machines

This study proposes a technique for arbitrary current sharing between winding sets of any multiple three-phase machine by using a decoupled control in two-dimensional orthogonal vector space decomposition subspaces. With ability to arbitrarily set the d–q currents in all winding sets, both active and reactive power of each winding set can be arbitrarily set as well. Normally, this behaviour requires machine model in multi-stator variables where control implementation may be challenging due to the cross-coupling between winding set variables. On the contrary, in this study, the correlation between multi-stator and the vector space decomposition modelling has been used so that benefits of both modelling approaches are fully utilised: a decoupled control feature thanks to vector space decomposition and an ability to individually control winding set currents, a feature of the multi-stator modelling approach.

A ballistic model-based method for ranging small arms fire using a single acoustic sensor node.

A ballistic model-based method is proposed for ranging small arms fire using a single acoustic sensor node, without a priori knowledge of the two ballistic model parameters: the muzzle speed and ballistic constant of the bullet, except that their actual values lie within a known two-dimensional (2D) parameter subspace. The method assumes that the line of fire misses the sensor node by a small angle, referred to as the miss angle, typically within a few degrees. A closed-form range estimator is derived, which computes the mean value of the feasible range estimates over the 2D parameter subspace for a presumed miss angle, using the measurements of the differential time of arrival and differential angle of arrival of the muzzle blast and ballistic shock wave at the sensor node. The output of the range estimator for a small presumed miss angle provides the range estimate of the point of fire. The performance of the proposed method is evaluated using simulated data for 36 different types of bullets and using real data for two different types of bullets.

Stochastic dynamic response analysis and reliability assessment of non-linear structures under fully non-stationary ground motions

In this paper, stochastic dynamic response analysis and reliability assessment of nonlinear structures under fully non-stationary ground motions is investigated based on the probability density evolution method (PDEM). To handle this problem, a new method is proposed for points selection in high-dimensional random-variate space for PDEM, which is of paramount importance to the tradeoff of accuracy and efficiency. First, the original high-dimensional space is decomposed into several two-dimensional orthogonal subspaces. Then, each subspace is partitioned via the Voronoi cells and the representative points and assigned probabilities in each subspace are specified accordingly. Finally, the representative points and assigned probabilities in each subspace are randomly paired to formulate the representative points and assigned probabilities in high-dimension. A numerical example is studied to validate the proposed method, which indicates that the proposed method is of efficiency and accuracy for high-dimensional stochastic dynamic problem of structures. The problem, which needs to be further investigated, is also pointed out.

A subspace minimization conjugate gradient method based on conic model for unconstrained optimization

In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient.

Методы ортогонального проектирования решения систем линейных уравнений для двумерного крыловского подпространства

Методы ортогонального проектирования решения систем линейных уравнений для двумерного крыловского подпространства

High-dimensional measurement-device-independent quantum key distribution on two-dimensional subspaces

Quantum key distribution (QKD) provides ultimate cryptographic security based on the laws of quantum mechanics. For point-to-point QKD protocols, the security of the generated key is compromised by detector side channel attacks. This problem can be solved with measurement device independent QKD (mdi-QKD). However, mdi-QKD has shown limited performances in terms of the secret key generation rate, due to post-selection in the Bell measurements. We show that high dimensional (Hi-D) encoding (qudits) improves the performance of current mdi-QKD implementations. The scheme is proven to be unconditionally secure even for weak coherent pulses with decoy states, while the secret key rate is derived in the single photon case. Our analysis includes phase errors, imperfect sources and dark counts to mimic real systems. Compared to the standard bidimensional case, we show an improvement in the key generation rate.

A new subspace minimization conjugate gradient method based on tensor model for unconstrained optimization

ABSTRACTA new subspace minimization conjugate gradient method based on tensor model is proposed and analysed. If the objective function is close to a quadratic, we construct a quadratic approximation model in a two-dimensional subspace to generate the search direction; otherwise, we construct a tensor model. It is remarkable that the search direction satisfies the sufficient descent property. We prove the global convergence of the proposed method under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising.

An Efficient Barzilai–Borwein Conjugate Gradient Method for Unconstrained Optimization

The Barzilai–Borwein conjugate gradient methods, which were first proposed by Dai and Kou (Sci China Math 59(8):1511–1524, 2016), are very interesting and very efficient for strictly convex quadratic minimization. In this paper, we present an efficient Barzilai–Borwein conjugate gradient method for unconstrained optimization. Motivated by the Barzilai–Borwein method and the linear conjugate gradient method, we derive a new search direction satisfying the sufficient descent condition based on a quadratic model in a two-dimensional subspace, and design a new strategy for the choice of initial stepsize. A generalized Wolfe line search is also proposed, which is nonmonotone and can avoid a numerical drawback of the original Wolfe line search. Under mild conditions, we establish the global convergence and the R-linear convergence of the proposed method. In particular, we also analyze the convergence for convex functions. Numerical results show that, for the CUTEr library and the test problem collection given by Andrei, the proposed method is superior to two famous conjugate gradient methods, which were proposed by Dai and Kou (SIAM J Optim 23(1):296–320, 2013) and Hager and Zhang (SIAM J Optim 16(1):170–192, 2005), respectively.

The Riesz Basis Property with Brackets for Dirac Systems with Summable Potentials

In the space ℍ = (L2[0, π])2, we study the Dirac operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ generated by the differential expression lP(y) = By′ + Py, where $$ B=\left(\begin{array}{cc}-i& 0\\ {}0& i\end{array}\right),\kern0.5em P(x)=\left(\begin{array}{cc}{p}_1(x)& {p}_2(x)\\ {}{p}_3(x)& {p}_4(x)\end{array}\right),\kern0.5em \mathbf{y}(x)=\left(\begin{array}{c}{y}_1(x)\\ {}{y}_2(x)\end{array}\right), $$ and the regular boundary conditions $$ U\left(\mathbf{y}\right)=\left(\begin{array}{cc}{u}_{11}& {u}_{12}\\ {}{u}_{21}& {u}_{22}\end{array}\right)\left(\begin{array}{c}{y}_1(0)\\ {}{y}_2(0)\end{array}\right)+\left(\begin{array}{cc}{u}_{13}& {u}_{14}\\ {}{u}_{23}& {u}_{24}\end{array}\right)\left(\begin{array}{c}{y}_1\left(\uppi \right)\\ {}{y}_2\left(\uppi \right)\end{array}\right)=0. $$ The elements of the matrix P are assumed to be complex-valued functions summable over [0, π]. We show that the spectrum of the operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ is discrete and consists of eigenvalues {λn}n ∈ ℤ such that $$ {\uplambda}_n={\uplambda}_n^0+o(1) $$ as |n| → ∞, where $$ {\left\{{\uplambda}_n^0\right\}}_{n\in \mathrm{\mathbb{Z}}} $$ is the spectrum of the operator $$ {\mathrm{\mathcal{L}}}_{0,U} $$ with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ forms a Riesz base in the space ℍ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator $$ {\mathrm{\mathcal{L}}}_{0,U} $$ are double, all eigenvalues of the operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator $$ {\mathrm{\mathcal{L}}}_{P,U} $$ is a Riesz base of subspaces (Riesz base with brackets) in the space ℍ.

A Fast Dictionary Learning Algorithm for Image Denoising

The K-SVD is one of the well-known and effective methods to learn a universal and overcomplete dictionary. However, K-SVD is very expensive because many iteration steps are needed. What’s more, when it converts 2D data patches into 1D vectors for training or learning, K-SVD breaks down the inherent geometric structure of the data. To overcome these limitations, employing a subspace partition technique, we propose an efficient and fast algorithm, the fast top-bottom two-dimensional subspace partition algorithm, for learning overcomplete dictionaries. Experimental simulations demonstrate that our dictionary learning approach is effective for image denoising.

Improved optimization methods for image registration problems

In this paper, we propose new multilevel optimization methods for minimizing continuously differentiable functions obtained by discretizing models for image registration problems. These multilevel schemes rely on a novel two-step Gauss-Newton method, in which a second step is computed within each iteration by minimizing a quadratic approximation of the objective function over a certain two-dimensional subspace. Numerical results on image registration problems show that the proposed methods can outperform the standard multilevel Gauss-Newton method.

EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.

Refined asymptotics for the composition of cyclic urns

A cyclic urn is an urn model for balls of types $0,\ldots ,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector. In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $7\le m\le 12$. For $m\ge 13$ we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.

Remarks on the Gaudin model modulo p

We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation. If $k=1$ and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map.

THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

AbstractThis paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.