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Endmember estimation consists of two tasks, that is, determining the number of pure spectral constituents (endmembers) and extracting their spectral signatures. We present a new geometric distance-based method for endmember estimation from hyperspectral images (HSIs), which does not need to know the number of endmembers in advance. Our strategy optimizes the widely used maximum distance analysis (MDA) method from two viewpoints. First, the traditional MDA method performs endmember estimation by computing the maximum distances between any pixel and one specific pixel, line, plane, or affine hull (AH) composed by the endmembers that have been formerly extracted. Instead, our new strategy only requires computing the maximum distance between any pixel and one specific AH. This operation provides a simpler way than MDA to estimate endmembers. Second, our strategy exploits a new distance computation between any pixel and an AH and just needs the normal vector (compared to the traditional MDA method, which uses the normal vector and offset). The new distance computation in our method is much more efficient than that in the traditional MDA method.

Under consideration are some polyhedral properties of the set of schedules for servicing identical requests by parallel devices. The requests satisfy some precedence conditions. Any service interruptions are prohibited. We propose some formalization of the set of schedules as a family of subsets of a finite set, define the polytope of schedules, and find the affine hull and dimension of this polytope. We also obtain the conditions under which the inequalities determining its polyhedral relaxation are the support inequalities.

Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems.

Recently, deep learning techniques have been introduced to address hyperspectral image (HSI) classification problems and have achieved the state-of-the-art performances. In this article, we propose a novel clustering algorithm for HSI based on learning embedding using the set-to-set and sample-to-sample distances (LSSDs). This technique consists of four main components: 1) oversegmentation; 2) generation of set-to-set and sample-to-sample distances; 3) learning embedding by training a siamese network; and 4) density-based spectral clustering. First, the HSI is oversegmented into superpixels by using the entropy rate superpixel (ERS) algorithm. Second, the set-to-set distances are obtained by representing the segmented sets of samples as affine hull (AH) models, whereas the sample-to-sample distances are computed by employing the local covariance matrix representation (LCMR) method. Third, sample pairs with the smallest and largest similarities are extracted according to the two distances. Then, these pairs are fed into the siamese multilayer perceptron (MLP) network and discriminative embeddings are learned by training the network with contrastive loss. Finally, density-based spectral clustering is applied to the deep embedding to obtain clustering results. Experimental results on three real HSIs demonstrate that the proposed method can achieve better performance than the considered baseline methods.

In this paper we propose a novel clustering algorithm that uses local affine/convex hulls for high-dimensional data clustering. In high-dimensional spaces, the sparse and irregular distributions make the nearest-neighbor distances unreliable (hole artifact), and this deteriorates the clustering performance. Therefore, there is a need to fill in these gaps between the nearest samples. To this end, we use local affine/convex hulls of the nearest neighbors of a given sample to fill in the holes, and this greatly improves the Euclidean distance metric and the clustering accuracy. The proposed method can also be seen as the local extension of the well-known iterative subspace clustering algorithms in which the entire cluster is approximated with a single linear/affine subspace. Experimental results show that the proposed method is efficient and it outperforms other subspace clustering algorithms on a wide range of datasets.

Classification based on image set is recently a competitive technique, where each set contains multiple images of a person or an object. As a widely used model, affine hull has shown its power in modeling image set. However, due to the existence of noise and outliers, the over-large affine hull usually matches fails when two hulls overlapped. Aiming at alleviating this handicap, this paper proposes a novel method for image set classification, namely Learning of Reduced Prototypes and Local Metric (LRPLM). Specifically, for each gallery image set, a reduced set of prototypes and an optimal local feature-wise metric are simultaneously learned, which jointly minimize the loss function involved the estimation of classification error probability. In doing so, LRPLM inherits the merits of affine hull with better representation to account for the unseen appearances and makes use of the powerful discriminative ability improved by the local metric. It looks like that LRPLM pulls similar image sets with the same class label “closer” to each other, while pushing dissimilar ones “far away”. Extensive experiments illustrate the considerable effectiveness of LRPLM on three widely used datasets. As we know, classification is a research hotspot in expert and intelligent systems. Different from the previous classification methods, LRPLM focuses on image set-based classification technology, while most of them are single-shot classification technology. Thus, the proposed method can be considered as an expert system technology for medical diagnosis, security monitoring, object categorization, and biometrics recognition applications.

Recently, the authors (Dong et al. in J Global Optim 73(4):801–824, 2019) introduced the multi-step inertial Krasnosel’skiǐ–Mann iteration, where the inertial parameters involve the iterative sequence. Therefore, one has to compute the inertial parameters per iteration. The aim of this article is to present two kinds of inertial parameter arrays which do not depend on the iterative sequence. We first introduce a general Krasnosel’skiǐ–Mann iteration on the affine hull of orbits, based on which one inertial parameter array is presented. Second, we investigate the other inertial parameter array by introducing a modified Krasnosel’skii-Mann iteration. The convergence of the modified Krasnosel’skiǐ–Mann iteration is shown using an exhaustive convergence analysis and the running-average iteration-complexity bound is provided. Finally, we give two numerical examples to illustrate that the multi-step inertial Krasnosel’skiǐ–Mann iteration with inertial parameters proposed in this article behaves better than that with inertial parameters given in [10].

The paper deals with a set of identical requirements processing schedules for parallel machines. The precedence constraints are set, interrupts are prohibited, time is discrete. This set of schedules is the set of feasible solutions to a number of optimization problems which are determined by various objective functions. The paper proposes the set of all schedules as a family of subsets of a finite set and defines the schedule polytope as the convex hull of incidence vectors. The report deals with the affine hull and polyhedral relaxation of the scheduling polytope. Polyhedral relaxation includes nonnegativity constraints, constraints as to the number of machines, and precedence constraints. Using the bH-basis technique, the work shows that constraints for nonnegativity generate facets of the scheduling polytope. Necessary conditions for facet constraints to the number of machines and precedence constraints are found.

Image set classification is recently a competitive technique, of which each gallery set and query set contains multiple images of a person or an object. However, strong noise and outlier could be generated from undesirable environmental conditions in real-world applications, which will result in deterioration of classification accuracy, robustness, and high time complexity of special learning algorithm. To address these problems, this paper presents a novel parameterless method called Candidate Sets Selection and Improved Reverse Training (CSSIRT). It aims at enhancing noise robustness and reducing time complexity while retaining the high classification accuracy via two-stage successive cascaded framework. Specifically, an image set is modeled as a robust reduced affine hull, which provides a better representation to account for the unseen appearances of each image set. The first stage selects the most promising top-N candidate sets and rejects the vast majority of the impossible gallery sets quickly by measuring dissimilarity between two hulls. Since few returned candidate sets from the first stage may intersect the given test one, the second stage utilizes the returned candidate sets to address the issue on sets overlapped by using a novel K-means clustering and normalized frequency histograms fusion based reverse training algorithm. The method we proposed is more robust, has a lower time complexity and higher accuracy. Three kinds of extensive comparisons with the other state-of-the-art methods have corroborated the superiority of our method on a number of challenging datasets.

Existing person re-identification (re-id) models mainly focus on still-image-based module, namely matching person images across non-overlapping camera views. Since video sequence contains much more information than still images and can be easily achieved by tracking algorithms in practical applications, the video re-id has attracted increasing attention in recent years. Distance learning is crucial for a re-id system. However, the computed distances in traditional video-based methods are easily distracted by the randomness of data distribution, especially with small sample size for training. To preferably distinguish different people, a novel regularised hull distance learning video-based person re-id method is proposed. It is advantageous in two aspects: robustness is guaranteed due to expanded video samples by regularised affine hull with limited ones, discriminability is ensured due to penalised hard negative samples more severely. Hence, the discriminability and robustness of the learnt metric are strengthened. Comparisons with the state-of-the-art video-based methods as well as related methods on PRID 2011, iLIDS-VID and MARS datasets demonstrate the superiority of the authors’ method.

For split reductive algebraic groups, we determine the sets of connected components of closed affine Deligne–Lusztig varieties with parahoric level structure.

Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L 2 ( R d ) . We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.

We consider the person re-identification problem, assuming the availability of a sequence of images for each person, commonly referred to as video-based or multi-shot re-identification. We approach this problem from the perspective of learning discriminative distance metric functions. While existing distance metric learning methods typically employ the average feature vector as the data exemplar, this discards the inherent structure of the data. To overcome this issue, we describe the image sequence data using affine hulls. We show that directly computing the distance between the closest points on these affine hulls as in existing recognition algorithms is not sufficiently discriminative in the context of person re-identification. To this end, we incorporate affine hull data modeling into the traditional distance metric learning framework, learning discriminative feature representations directly using affine hulls. We perform extensive experiments on several publicly available data sets to show that the proposed approach improves the performance of existing metric learning algorithms irrespective of the feature space employed to perform metric learning. Furthermore, we advance the state of the art on iLIDS-VID, PRID, and SAIVT, with absolute rank-1 performance improvements of 6.0%, 11.4%, and 6.0% respectively.

Box-totally dual integral (box-TDI) polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the notions of principal box-integer polyhedra and equimodular matrices. A polyhedron is box-integer if its intersection with any integer box $$\{\ell \le x \le u\}$$ is integer. We define principally box-integer polyhedra to be the polyhedra P such that $$ kP $$ is box-integer whenever $$ kP $$ is integer. A rational $$r\times n$$ matrix is equimodular if it has full row rank and its nonzero $$r\times r$$ determinants all have the same absolute value. A face-defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron. Our main result is that the following statements are equivalent. Along our proof, we show that a polyhedral cone is box-TDI if and only if it is box-integer, and that these properties are carried over to its polar. We illustrate these charaterizations by reviewing well known results about box-TDI polyhedra. We also provide several applications. The first one is a new perspective on the equivalence between two results about binary clutters. Secondly, we refute a conjecture of Ding, Zang, and Zhao about box-perfect graphs. Thirdly, we discuss connections with an abstract class of polyhedra having the Integer Caratheodory Property. Finally, we characterize the box-TDIness of the cone of conservative functions of a graph and provide a corresponding box-TDI system.

In image set classification, dual linear regression classification (DLRC) has shown the excellent performance on face image data without the interference of the complex background. However, DLRC could not well identify the data set with the complex background. The complex background means that the background is cluttered and the viewpoint is unusual or the object is partially occluded. This paper proposes a new model, kernelized dual regression (KDR), based on DLRC and the kernel trick which is a useful technique in image classification. Different from DLRC, KDR adopts a block partitioning strategy to extract the local information, which is able to conquer the shortcoming of DLRC. To capture the nonlinear relationship between the training set and test set, KDR tactfully maps these image sets into a high-dimensional feature space by adopting the nonlinear mapping associated with the Gaussian kernel function. In the reproducing kernel Hilbert space (RKHS), KDR can find the joint coefficients by minimizing the distance between training set and test set, and has a closed-form solution. Extensive experiments on four datasets show that KDR could achieve better classification performance than that of DLRC and other existing methods.

The circumcentered Douglas--Rachford method (C--DRM), introduced by Behling, Bello Cruz and Santos, is an acceleration of the well-known Douglas-Rachford method (DRM) for finding the best approximation onto the intersection of finitely many affine subspaces. Inspired by the C--DRM, we introduced the more flexible circumcentered reflection method (CRM) and circumcentered isometry method (CIM). The CIM essentially chooses the closest point to the solution among all of the points in an associated affine hull as its iterate and is a generalization of the CRM. The circumcentered--reflection method introduced by Behling et al. to generalize the C--DRM is a special class of our CRM. We consider the CIM induced by a set of finitely many isometries for finding the best approximation onto the intersection of fixed point sets of the isometries which turns out to be an intersection of finitely many affine subspaces. We extend our previous linear convergence results on CRMs in finite-dimensional spaces from reflections to isometries. In order to better accelerate the symmetric method of alternating projections (MAP), the accelerated symmetric MAP first applies another operator to the initial point. (Similarly, to accelerate the DRM, the C--DRM first applies another operator to the initial point as well.) Motivated by these facts, we show results on the linear convergence of CIMs in Hilbert spaces with first applying another operator to the initial point. In particular, under some restrictions, our results imply that some CRMs attain the known linear convergence rate of the accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs converging to the best approximation in Hilbert spaces with a convergence rate no worse than the sharp convergence rate of MAP. The fact that some CRMs attain the linear convergence rate of MAP or accelerated symmetric MAP is entirely new., 29 pages

Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where $\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over $\mathcal{K}$. This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix $X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal . In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of the interior of the attainable set. As a byproduct we obtain that it is actually smooth on a possibly smaller set, still open and dense.

In affine differential geometry of hypersurface, C. Scharlach et al. found an inequality involving intrinsic and extrinsic curvatures, and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2-dimensional distribution D 2 is integrable. In this paper, we continue to study affine hyperspheres realizing the equality, including parabolic affine hyperspheres. As main results, firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant, or D 2 is integrable. Next, by introducing a well-defined 3-dimensional distribution D 3 when D 2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D 3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality.

If S is a given regular n-simplex, $$n \ge 2$$ , of edge length a, then the distances $$a_1, \ldots , a_{n+1}$$ of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $$\phi _{n+1} (a,a_1,\ldots ,a_{n+1})=0$$ , where $$\begin{aligned} \phi = \phi _t (x, x_1,\ldots ,x_{n+1}) = \left( x^2+x_1^2+\cdots +x_{n+1}^2\right) ^2 - t\left( x^4+x_1^4+\cdots +x_{n+1}^4\right) . \end{aligned}$$ The natural question whether this is essentially the only relation was recently and positively answered by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha. The authors made use of the irreducibility of the polynomial $$\phi $$ in the case when $$n \ge 2$$ , $$t=n+1$$ , $$x= a \ne 0$$ , and $$k = \mathbb {R}$$ , but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of $$\phi $$ without restrictions on t, x, a, and k. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley–Menger determinant that are more general than those established by C. D’Andrea and M. Sombra.

By solving certain (partial) differential equations we give the total classification of Weingarten affine translation surfaces in three dimensional Euclidean space $$\mathbb {E}^3$$ . Explicitly, a Weingarten affine translation surface in Euclidean 3-space is the minimal affine Scherk surface or the surface with flat metric.

We establish a necessary and sufficient condition for a geodesic line on a nondegenerate hypersurface to be a plane curve. We deduce a formula for the affine curvature of a plane geodesic line on the affine hypersurface in terms of the affine fundamental form and the shape operator. The definition of transverse curvature is presented and some of its elementary properties are determined.

In this paper, we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function. We first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method. At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm.

This letter presents a prototype discriminative learning (PDL) method for image set classification. We aim to simultaneously learn prototypes and a linear discriminative projection to drive that in the target subspace each image set can be discriminated with its nearest neighbor prototype. To reveal the unseen appearance variations implicitly in an image set, the prototypes are actually “virtual,” which do not certainly appear in the set but are searched in the corresponding affine hull. Moreover, to enhance the stability and robustness of the learned target subspace, an orthogonality constraint is imposed on the projection. Thus, to optimize the prototypes and the projection jointly, we design a specific gradient descent mechanism by updating the projection on Stiefel manifold and the prototypes in Euclidean space in an alternative optimization manner. Experimental results on four challenging databases demonstrate the superiority of the proposed PDL method.

Cryptographic properties of permutations viz non-linearity, affine equivalence, and mode transform have been studied in the literature, treating them as bijections on ℤn. In this article, the autho...

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L 2(R d ), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.