Subspace Learning by $$\ell ^{0}$$-Induced Sparsity

Subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure. Such methods with sparsity prior, such as sparse subspace clustering (SSC) (Elhamifar and Vidal in IEEE Trans Pattern Anal Mach Intell 35(11):2765–2781, 2013) with the sparsity induced by the \(\ell ^{1}\)-norm, are demonstrated to be effective in subspace clustering. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. However, these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods. In this paper, we propose \(\ell ^{0}\)-induced sparse subspace clustering (\(\ell ^{0}\)-SSC). In contrast to the required assumptions, such as independence or disjointness, on subspaces for most existing sparse subspace clustering methods, we prove that \(\ell ^{0}\)-SSC guarantees the subspace-sparse representation, a key element in subspace clustering, for arbitrary distinct underlying subspaces almost surely under the mild i.i.d. assumption on the data generation. We also present the “no free lunch” theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding \(\ell ^{0}\) sparse representation problem of \(\ell ^{0}\)-SSC. A novel approximate algorithm named Approximate \(\ell ^{0}\)-SSC (A\(\ell ^{0}\)-SSC) is developed which employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of \(\ell ^{0}\)-SSC with theoretical guarantee. The sub-optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results. Extensive experimental results on various data sets demonstrate the superiority of A\(\ell ^{0}\)-SSC compared to other competing clustering methods. Furthermore, we extend \(\ell ^{0}\)-SSC to semi-supervised learning by performing label propagation on the sparse similarity matrix learnt by A\(\ell ^{0}\)-SSC and demonstrate the effectiveness of the resultant semi-supervised learning method termed \(\ell ^{0}\)-sparse subspace label propagation (\(\ell ^{0}\)-SSLP).     

Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with $${\textit{EQ}}_1^{rot}$$ Nonconforming Finite Element

Nonlinear parabolic equation is studied with a linearized Galerkin finite element method. First of all, a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the proof of the temporal error skillfully. On the other hand, the spatial error is derived \(\tau \)-independently with the above achievements. Then, the superclose result of order \(O(h^2+\tau ^2)\) in broken \(H^1\)-norm is deduced without any restriction of \(\tau \). The two typical characters of the \({\textit{EQ}}_1^{rot}\) nonconforming FE (see Lemma 1 below) play an important role in the procedure of proof. At last, numerical results are provided in the last section to confirm the theoretical analysis. Here, h is the subdivision parameter, and \(\tau \), the time step.     

Phase field computations for surface diffusion and void electromigration in {mathbb{R}^3}

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by a degenerate Cahn–Hilliard equation with an external forcing induced by the electric field. We describe the iterative scheme used to solve the resulting nonlinear discrete equations and present some numerical experiments in three space dimensions. The first author was supported by the EPSRC grant EP/C548973/1.     

Anisotropic Nonconforming { EQ}_1^{rot} Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

The main aim of this paper is to study the nonconforming $EQ_1^{rot}$ quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and $L^2$ -norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.     

Integrated damping parameter and control design in structural systems for \bf{\mathcal{H}^2} and {\mathcal{H}^{\infty}} specifications

The paper presents a linear matrix inequality (LMI)-based approach for the simultaneous optimal design of output feedback control gains and damping parameters in structural systems with collocated actuators and sensors. The proposed integrated design is based on simplified $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ norm upper bound calculations for collocated structural systems. Using these upper bound results, the combined design of the damping parameters of the structural system and the output feedback controller to satisfy closed-loop $\mathcal{H}^2$ or $\mathcal{H}^{\infty}$ performance specifications is formulated as an LMI optimization problem with respect to the unknown damping coefficients and feedback gains. Numerical examples motivated from structural and aerospace engineering applications demonstrate the advantages and computational efficiency of the proposed technique for integrated structural and control design. The effectiveness of the proposed integrated design becomes apparent, especially in very large scale structural systems where the use of classical methods for solving Lyapunov and Riccati equations associated with $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ designs are time-consuming or intractable.     

An Improved Minimum-Distance Texture Estimator for Speckled Data Under the $$mathscr {G}^0$$ G 0 Model

Journal of Mathematical Imaging and Vision - The $$mathscr {G}^0$$ distribution is an apt model for speckled data, such as SAR imagery, because it possesses the ability to characterize areas with...     

Unextendible maximally entangled bases in $${\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}$$

The construction of unextendible maximally entangled bases is tightly related to quantum information processing like local state discrimination. We put forward two constructions of UMEBs in \({\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}\)(\(p\le q\)) based on the constructions of UMEBs in \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d}\) and in \({\mathbb {C}}^{p}\otimes {\mathbb {C}}^{q}\), which generalizes the results in Guo (Phys Rev A 94:052302, 2016) by two approaches. Two different 48-member UMEBs in \({\mathbb {C}}^{6}\otimes {\mathbb {C}}^{9}\) have been constructed in detail.     

The categories L-\mathbf{Top}_{0} and L-\mathbf{Sob} as epireflective hulls

We show that the category \(L\) - \(\mathbf{Top}_{0}\) of \(T_{0}\) - \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}\) of \(L\) -topological spaces and the category \(L\) - \(\mathbf{Sob}\) of sober \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}_{0}\) .     

A unified approximate reasoning theory suitable for both propositional calculus system \mathcal{L}^* and predicate calculus system \mathcal{K}^*

The concepts of metric R ₀-algebra and Hilbert cube of type R ₀ are introduced. A unified approximate reasoning theory in propositional caculus system $\mathcal{L}^* $ and predicate calculus system $\mathcal{K}^* $ is established semantically as well as syntactically, and a unified complete theorem is obtained.     

Upper and lower bounds via penalty functions for control problems

A close relationship between penalty functions and duality is shown to exist for a certain class of convex optimal control problems with state and control variable constraints. This relationship can be used to generate upper and lower bounds on the value of the solution to a control problem of the prescribed class.     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

Mutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$

A way of constructing special entangled basis with fixed Schmidt number 2 (SEB2) in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) is proposed, and the conditions mutually unbiased SEB2s (MUSEB2s) satisfy are discussed. In addition, a very easy way of constructing MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\) is presented. We first establish the concrete construction of SEB2 and MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4}\) and \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{8}\), respectively, and then generalize them into \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) and display the condition that MUSEB2s satisfy; we also give general form of two MUSEB2s as examples in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\).     

Improved $${{\ell }^{1}}$$-tracker using robust PCA and random projection

In this paper, we propose an improved \({{\ell }^{1}}\)-tracker in a particle filter framework using robust principal component analysis (robust PCA) and random projection. At first we redesign the template set and its update scheme. Three target templates and several background templates combined with the trivial templates are used to represent the candidate images sparsely. One fixed target template is generated from the image patch in the first frame. The other two are dynamic target templates, called stable target template, and fast changing one used for long time and short time, respectively. Robust PCA is used to generate and update the stable target template, and fast changing target template is initialized by the stable one at certain times. The background templates are used to strengthen the ability of distinguishing background and foreground. Then, the large set of Haar-like features are extracted and compressively sensed with a very sparse measurement matrix for the \({{\ell }^{1}}\)-tracker framework. The compressive sensing theories ensure that the sensed features preserve almost all the information of the original features. Our proposed method is more robust than the original \({{\ell }^{1}}\)-method. Experiments have been done on numerous sequences to demonstrate the better performance of our improved tracker.     

On Projection Matrices $$\mathcal{P}^k \to \mathcal{P}^2 ,k = 3,...,6, $$ and their Applications in Computer Vision

Projection matrices from projective spaces have long been used in multiple-view geometry to model the perspective projection created by the pin-hole camera. In this work we introduce higher-dimensional mappings for the representation of various applications in which the world we view is no longer rigid. We also describe the multi-view constraints from these new projection matrices (where k > 3) and methods for extracting the (non-rigid) structure and motion for each application.     

Transcendental functions and mechanical theorem proving in elementary geometries

We present a new approach to dealing with certain problems about trigonometric and hyperbolic function using Wu's method by transforming the two kinds of transcendental functions into polynomials. This approach has been used for mechanical theorem proving in Euclidean and several Non-Euclidean geometries. Based on our approach, several meta-theorems are established for four Non-Euclidean geometries. These meta-theorems assert that certain kinds of geometry statements that can be confirmed by our approach in one geometry can also be confirmed in three other geometries. We also proved a meta-theorem which permits us to obtain all hyperbolic identities from trigonometric identities through certain kinds of transformation and vice versa.The revision was supported by the NSF grant CCR-8702108.     

$$hbox {TG}^2$$ TG 2 : text-guided transformer GAN for restoring document readability and perceived quality

International Journal on Document Analysis and Recognition (IJDAR) - Most image enhancement methods focused on restoration of digitized textual documents are limited to cases where the text...     

${rm M}^{3}$-Deterministic, Multiscale, Multirobot Platform for Microsystems Packaging: Design and Quasi-Static Precision Evaluation

The dawn of next generation robots and systems era which is quietly emerging, requires miniaturized and integrated sensors, actuators, and entire microrobots. One of the defining characteristics of these microsystems is their multiscale nature, e.g., the span of their size, features, and tolerances across multiple dimensional scales, from the meso to the micro and nanoscales. Another defining characteristic is the need to reliably integrate heterogeneous materials via assembly and packaging, in a cost-effective manner, even in low quantities. Thus, it is argued in this paper that cost-effective manufacturing of complex microsystems requires special precision robotic assembly cells with modular and reconfigurable characteristics. This paper presents recent research aimed at developing theoretical underpinnings for how to construct such a manufacturing platform. M³ is a multirobot system spanning across the macro-meso-microscales and specifically configured to package. The M³ robots are systematically characterized in terms of quasi-static precision measures and assembly plans are generated using kinematic identification, inverse kinematics and visual servoing. The advantage of our approach the fact that high assembly yields for our system are a consequence of a set of so-called precision resolution-repeatability-accuracy (RRA) rules introduced in this paper. Experimental results for packaging of a microelectromechanical systems switch are provided to support our findings.