Abstract We first characterize the region of the n -dimensional Euclidean space for which two optimization problems with the square distance function as common objective function, but different constraints, are equivalent. The affine hull of a certain face of a closed convex set $$C\subseteq {\mathbb {R}}^n$$ C ⊆ R n is the constraint associated to one problem and the whole closed convex set C is the constraint associated to the other problem. Such optimization problems are best approximation problems which can be reformulated in terms of the metric projection. Using the language of the metric projection, we characterize the region of $${\mathbb {R}}^n$$ R n which is metrically projected over a face of C in the same way that it is projected over the affine hull of the face itself. The metric projection over such a face is the one associated to the entire closed convex set, and the metric projection over the affine hull of such a face is the one associated to the affine hull. It turns out that this region is the closure of the inverse image, through the metric projection over the entire closed convex set, of the relative interior of the face. We also characterize analytically the closure of the regions of $${\mathbb {R}}^n$$ R n that are projected over the relative interiors of the faces of a polyhedral set, through the metric projection of the polyhedral set itself. We show that these regions are polyhedral convex sets by explicitly characterizing them through systems of linear inequalities.

In this paper, we analyse a local convergence of augmented Lagrangian method (ALM) for a class of nonlinear circular conic optimization problems. In light of the singular value decomposition, the Debreu theorem and the implicit function theorem, we prove that the sequence generated by ALM converges to a local minimizer in the linear convergence rate under the constraint nondegeneracy condition and the strong second-order sufficient condition, in which the ratio constant is proportional to 1 / τ , where τ is the associated penalty parameter with a given lower threshold. As a byproduct, we also derive explicit expressions of critical cone and its affine hull for the given nonlinear circular conic program.

With the rapid advances in digital imaging and communication technologies, recently image set classification has attracted significant attention and has been widely used in many real-world scenarios. As an effective technology, the class-specific representation theory-based methods have demonstrated their superior performances. However, this type of methods either only uses one gallery set to measure the gallery-to-probe set distance or ignores the inner connection between different metrics, leading to the learned distance metric lacking robustness, and is sensitive to the size of image sets. In this article, we propose a novel joint metric learning-based class-specific representation framework (JMLC), which can jointly learn the related and unrelated metrics. By iteratively modeling probe set and related or unrelated gallery sets as affine hull, we reconstruct this hull sparsely or collaboratively over another image set. With the obtained representation coefficients, the combined metric between the query set and the gallery set can then be calculated. In addition, we also derive the kernel extension of JMLC and propose two new unrelated set constituting strategies. Specifically, kernelized JMLC (KJMLC) embeds the gallery sets and probe sets into the high-dimensional Hilbert space, and in the kernel space, the data become approximately linear separable. Extensive experiments on seven benchmark databases show the superiority of the proposed methods to the state-of-the-art image set classifiers.

Privacy-utility tradeoff remains as one of the fundamental issues of differentially private machine learning. This paper introduces a geometrically inspired kernel-based approach to mitigate the accuracy-loss issue in classification. In this approach, a representation of the affine hull of given data points is learned in Reproducing Kernel Hilbert Spaces (RKHS). This leads to a novel distance measure that hides privacy-sensitive information about individual data points and improves the privacy-utility tradeoff via significantly reducing the risk of membership inference attacks. The effectiveness of the approach is demonstrated through experiments on MNIST dataset, Freiburg groceries dataset, and a real biomedical dataset. It is verified that the approach remains computationally practical. The application of the approach to federated learning is considered and it is observed that the accuracy-loss due to data being distributed is either marginal or not significantly high.

Sparsity preserving projection aided baselined hyperdisk modeling for interpretable machine health monitoring

Cooperative linear regression model for image set classification

We give an exact criterion of a conjecture of L.M.Kelly to hold true which is stated as follows. If there is a finite family $\Sigma$ of mutually skew lines in $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) of every two lines in $\Sigma$, contains at least one more line of $\Sigma$, then we have that $\Sigma$ is entirely contained in a three dimensional space if and only if the arrangement of affine hulls is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, that is, for $(2,5)$-representations of sylvester-gallai designs in $\mathbb{R}^6$, which is answered in the last section.

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.

This article introduces two methods that find compact deep feature models for approximating images in set based face recognition problems. The proposed method treats each image set as a nonlinear face manifold that is composed of linear components. To find linear components of the face manifold, we first split image sets into subsets containing face images which share similar appearances. Then, our first proposed method approximates each subset by using the center of the deep feature representations of images in those subsets. Centers modeling the subsets are learned by using distance metric learning. The second proposed method uses discriminative common vectors to represent image features in the subsets, and entire subset is approximated with an affine hull in this approach. Discriminative common vectors are subset centers that are projected onto a new feature space where the combined within-class variances coming from all subsets are removed. Our proposed methods can also be considered as distance metric learning methods using triplet loss function where the learned subcluster centers are the selected anchors. This procedure yields to applying distance metric learning to quantized data and brings many advantages over using classical distance metric learning methods. We tested proposed methods on various face recognition problems using image sets and some visual object classification problems. Experimental results show that the proposed methods achieve the state-of-the-art accuracies on the most of the tested image datasets.

While convenient in daily life, face recognition technologies also raise privacy concerns for regular users on the social media since they could be used to analyze face images and videos, efficiently and surreptitiously without any security restrictions. In this paper, we investigate the face privacy protection from a technology standpoint based on a new type of customized cloak, which can be applied to all the images of a regular user, to prevent malicious face recognition systems from uncovering their identity. Specifically, we propose a new method, named one person one mask (OPOM), to generate person-specific (class-wise) universal masks by optimizing each training sample in the direction away from the feature subspace of the source identity. To make full use of the limited training images, we investigate several modeling methods, including affine hulls, class centers and convex hulls, to obtain a better description of the feature subspace of source identities. The effectiveness of the proposed method is evaluated on both common and celebrity datasets against black-box face recognition models with different loss functions and network architectures. In addition, we discuss the advantages and potential problems of the proposed method. In particular, we conduct an application study on the privacy protection of a video dataset, Sherlock, to demonstrate the potential practical usage of the proposed method.

ABSTRACT Reliable labelled samples have always played a vital role in the supervised paradigm of hyperspectral imagery (HSI) classification due to the fact that the inclusion of incorrect label information in the training set can seriously degrade the performance of classification methods. Recently, although some inter-class difference-based detection algorithms have been developed to remove mislabelled samples (i.e. noisy labels) in training set, the benefit of contextual information for each sample has not been fully explored yet. In this paper, a training redefinition with entropy-based structure set density (ESSD) method is designed, which consists of following main steps. First, the proposed ESSD method employs an over-segmentation technique to cluster the HSI into many shape-adaptive regions that correspond to sample sets. Then, each sample set is represented with an affine hull (AH) model, which exploits both the similarity and variance of samples within each sample set to adaptively characterize the set. Specifically, considering spectral and spatial weak assumptions among samples in each sample set, the idea of entropy trick-based -nearest neighbour is introduced into each sample set to redefine its structure by removing different class from the sample set. Next, the distance among AH corresponding to each training sample is calculated based on the AH model. Meanwhile, the set-to-set distance is fed to the density peak algorithm to obtain the density of training samples. Finally, a decision-making value is applied to the density of each training sample to cleanse mislabelled samples within noisy training set. Experimental results on real HSI date sets demonstrate the superiority of the proposed method over several well-known training redefinition methods in terms of detection accuracy.

The affine hull of the schedule polytope for servicing identical requests by parallel devices

Endmember estimation plays a key role in hyperspectral image unmixing, often requiring an estimation of the number of endmembers and extracting endmembers. However, most of the existing extraction algorithms require prior knowledge regarding the number of endmembers, being a critical process during unmixing. To bridge this, a new maximum distance analysis (MDA) method is proposed that simultaneously estimates the number and spectral signatures of endmembers without any prior information on the experimental data containing pure pixel spectral signatures and no noise, being based on the assumption that endmembers form a simplex with the greatest volume over all pixel combinations. The simplex includes the farthest pixel point from the coordinate origin in the spectral space, which implies that: (1) the farthest pixel point from any other pixel point must be an endmember, (2) the farthest pixel point from any line must be an endmember, and (3) the farthest pixel point from any plane (or affine hull) must be an endmember. Under this scenario, the farthest pixel point from the coordinate origin is the first endmember, being used to create the aforementioned point, line, plane, and affine hull. The remaining endmembers are extracted by repetitively searching for the pixel points that satisfy the above three assumptions. In addition to behaving as an endmember estimation algorithm by itself, the MDA method can co-operate with existing endmember extraction techniques without the pure pixel assumption via generalizing them into more effective schemes. The conducted experiments validate the effectiveness and efficiency of our method on synthetic and real data.

The Affine Hull of the Schedule Polytope for Servicing Identical Requests by Parallel Devices

A sphere that is tangent to all four face-planes (i.e., the affine hulls of the faces) of a tetrahedron is called a tangent sphere of the tetrahedron. Two tangent spheres are called neighboring if exactly one face-plane separates them. Grace’s theorem states that for a pair of neighboring tangent spheres S and T of a tetrahedron there is a unique sphere such that (1) passes through the three vertices of the tetrahedron lying on the face-plane that separates S and T, and (2) is either externally tangent to both S, T or internally tangent to both S, T. It seems that this theorem is not widely known, and that no elementary proof has been given. The purpose of this article is to present an elementary and direct proof of this theorem in the case of a trirectangular tetrahedron, and to obtain several further results in this direction. Among them is also the confirmation of a slightly generalized form of Grace’s theorem.

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.

A majority of the image set based face recognition methods use a generatively learned model for each person that is learned independently by ignoring the other persons in the gallery set. In contrast to these methods, this paper introduces a novel method that searches for discriminative convex models that best fit to an individual’s face images but at the same time are as far as possible from the images of other persons in the gallery. We learn discriminative convex models for both affine and convex hulls of image sets. During testing, distances from the query set images to these models are computed efficiently by using simple matrix multiplications, and the query set is assigned to the person in the gallery whose image set is closest to the query images. The proposed method significantly outperforms other methods using generatively learned convex models in terms of both accuracy and testing time, and achieves the state-of-the-art results on six of the eight tested datasets. Especially, the accuracy improvement is significant on the challenging PaSC, COX, IJB-C and ESOGU video datasets.

The circumcentered Douglas–Rachford method (C–DRM), introduced by Behling, Bello Cruz and Santos, iterates by taking the circumcenter of associated successive reflections. It 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, Bello Cruz and Santos 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.

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 Carathéodory Property. Finally, we characterize the box-TDIness of the cone of conservative functions of a graph and provide a corresponding box-TDI system.

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.