Spherical Constraint Research Articles

Extrapolated Proximal Subgradient Algorithms for Nonconvex and Nonsmooth Fractional Programs

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, which encompass many important modern optimization problems arising from diverse areas such as the recently proposed scale-invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and that any one of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted fast iterative shrinking-threshold algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka–Łojasiewicz property. Our algorithm exhibits linear convergence for the scale-invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. When the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose another extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples.

On the geometric analysis of a quartic–quadratic optimization problem under a spherical constraint

This paper considers the problem of solving a special quartic–quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of \(\frac{1}{2}\mathbf {z}^{*}A\mathbf {z}+\frac{\beta }{2}\sum _{k=1}^{n}|z_{k}|^{4}\) such that \(\Vert \mathbf {z}\Vert _{2}=1\). This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate the geometric properties of this optimization problem. Fourth-order optimality conditions are derived for characterizing local and global minima. When the matrix in the quadratic term is diagonal, the problem has no spurious local minima and global solutions can be represented explicitly and calculated in \(O(n\log {n})\) operations. When A is a rank one matrix, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of second-order-type algorithms, is established when the coefficient \(\beta \) of the quartic term is either at least \(O(n^{3/2})\) or not larger than O(1). Finally, the Kurdyka–Łojasiewicz exponent of quartic–quadratic problem is estimated and it is shown that the largest exponent is at least 1/4 for a broad class of stationary points.

Omnidirectional stereo depth estimation based on spherical deep network

Omnidirectional depth estimation is an emerging research topic and has received significant attention in recent years. However, the existing methods were developed based on the theory of planar stereo matching; and introduce the nonlinear epipolar constraint and significant distortions of re-projections. In this paper, we propose a novel approach that use spherical CNNs and the epipolar constraint on sphere for omnidirectional depth estimation. We discuss the epipolar constraint for spherical stereo imaging and convert the nonlinear constraint on a planar projection to the linear constraint on a sphere. We then propose a Spherical Convolution Residual Network (SCRN) for omnidirectional depth estimation via the spherical linear epipolar constraint. The input equirectangular projection (ERP) images are sampled to spherical meshes and fed into SCRN to calculate spherical depth maps. For 2D visualization, we design a Planar Refinement Network (PRN) and adopt the cascade learning scheme to improve the accuracy of depth maps. This scheme reduces the errors caused by projection, interpolation, and the limitation of spherical representation. The experiment shows that our full scheme Cascade Spherical Depth Network (CSDNet) results in more accurate and detailed depth maps with lower errors, as compared to recent seminal works. Our approach yields the comparable performance to the other state-of-the-art works on the omnidirectional stereo datasets with less number of parameters. The effectiveness of the spherical network and the cascade learning scheme is validated, and the influence of spherical sampling density is also discussed.

Safe polyhedral tubes for locally convexified MPC

The note considers optimal and model predictive controls for linear systems with strictly convex quadratic stage and terminal cost functions, and under closed polyhedral stage constraints, finitely many open spherical exclusion stage constraints, and closed polyhedral terminal constraints. The computational complexity of exact nonconvex optimal and model predictive controls is alleviated through the local convexification of the nonconvex exclusion constraints, which is obtained via safe polyhedral tubes. The safe polyhedral tubes are constructed via simple algebraic operations and are of paramount importance for the design of locally convexified optimal and model predictive controls via strictly convex quadratic programming.

Network Embedding on Hierarchical Community Structure Network

Network embedding is a method of learning a low-dimensional vector representation of network vertices under the condition of preserving different types of network properties. Previous studies mainly focus on preserving structural information of vertices at a particular scale, like neighbor information or community information, but cannot preserve the hierarchical community structure, which would enable the network to be easily analyzed at various scales. Inspired by the hierarchical structure of galaxies, we propose the Galaxy Network Embedding (GNE) model, which formulates an optimization problem with spherical constraints to describe the hierarchical community structure preserving network embedding. More specifically, we present an approach of embedding communities into a low-dimensional spherical surface, the center of which represents the parent community they belong to. Our experiments reveal that the representations from GNE preserve the hierarchical community structure and show advantages in several applications such as vertex multi-class classification, network visualization, and link prediction. The source code of GNE is available online.

Spherical Formulation of Geometric Motion Segmentation Constraints in Fisheye Cameras

We introduce a visual motion segmentation method employing spherical geometry for fisheye cameras and automated driving. Three commonly used geometric constraints in pin-hole imagery (the positive height, positive depth and epipolar constraints) are reformulated to spherical coordinates, making them invariant to specific camera configurations as long as the camera calibration is known. A fourth constraint, known as the anti-parallel constraint, is added to resolve motion-parallax ambiguity, to support the detection of moving objects undergoing parallel or near-parallel motion with respect to the host vehicle. A final constraint constraint is described, known as the spherical three-view constraint, is described though not employed in our proposed algorithm. Results are presented and analyzed that demonstrate that the proposal is an effective motion segmentation approach for direct employment on fisheye imagery.

(Non-equilibrium) thermodynamics of integrable models: The Generalized Gibbs Ensemble description of the classical Neumann model

We study the motion of a classical particle subject to anisotropic harmonic forces and constrained to move on the sphere. In the integrable-systems literature this problem is known as the Neumann model. We choose the spring constants in a way that makes the connection with the so-called p = 2 spherical disordered system transparent. We tackle the problem in the N → ∞ limit by introducing a soft version in which the spherical constraint is imposed only on average over initial conditions. We show that the Generalized Gibbs Ensemble, constructed with N conserved charges in involution, captures the long-time averages of all relevant observables of the soft model after sudden changes in the parameters (quenches). We reveal the full dynamic phase diagram with four different phases in which the particles' position and momentum are both extended, only the position quasi-condenses or condenses, and both condense. The scaling properties of the fluctuations allow us to establish in which of these cases the strict and soft spherical constraints are equivalent. We thus confirm the validity of the GGE hypothesis for the Neumann model on a large portion of the dynamic phase diagram.

Optimizing the post-processing of online evolution reconstruction in quantum communication

A method of optimizing the post-processing of online evolution reconstruction in quantum communication is proposed and demonstrated for the six-state protocol. The aim of optimization is to promote the accuracy of recovering the expectation value functions of projectors, because it sets the boundary of the reconstruction errors. Before recovery, the measurement records for equal expectation value functions are integrated to increase the amount of available measurement data. After recovery, the raw recovered expectation value functions are normalized twice according to two spherical constraints on them separately to reduce the recovery errors further. The simulated results of reconstructing a time-varying qubit in quantum information show that our method is equivalent to a tenfold increment of available measurement data for online evolution reconstruction.

Spherical Quasi-Physical Model-Based Color-Shift Keying for Visible Light Communication

In this paper, the new family of spherical color-shift keying (SCSK) constellations is proposed for visible light communication (VLC) systems. Specifically, the constant-intensity constraint (CIC) for conventional color-shift keying (CSK) schemes, which originates from the requirement of non-flickering illumination, confines the constellation symbols on a two-dimensional (2D) plane of the three-dimensional (3D) intensity space. To fully utilize the potential degree of freedom offered by the 3D intensity space, we propose to expand the envelope of CSK constellations by replacing the constant intensity with a new spherical intensity constraint, aiming to maximize the minimum Euclidean distance (MED). More explicitly, we design the robust SCSK constellations with and without white color balance (CB), respectively. Particularly, we transform the constellation design problem to the so-called spherical circle packing (SCP) problem, which is resolved based on the spherical quasi-physical (SQP) model. Then, we propose the so-called spherical heuristic quasi-physical algorithm (SHQPA) integrating the binary search algorithm (BSA) and the configuration update algorithms (CUA) to obtain the optimal SCSK constellations. Furthermore, we also study the corresponding bit-to-symbol mapping rules for the generated SCSK constellations. Simulation results show that the new family of SCSK constellations outperforms the conventional CSK counterparts.

Relation between statics and dynamics in the quench of the Ising model to below the critical point.

The standard phase-ordering process is obtained by quenching a system, like the Ising model, to below the critical point. This is usually done with periodic boundary conditions to ensure ergodicity breaking in the low-temperature phase. With this arrangement the infinite system is known to remain permanently out of equilibrium, i.e., there exists a well-defined asymptotic state which is time invariant but different from the ordered ferromagnetic state. In this paper we establish the critical nature of this invariant state by demonstrating numerically that the quench dynamics with periodic and antiperiodic boundary conditions are indistinguishable from each other. However, while the asymptotic state does not coincide with the equilibrium state for the periodic case, it coincides instead with the equilibrium state of the antiperiodic case, which in fact is critical. The specific example of the Ising model is shown to be one instance of a more general phenomenon, since an analogous picture emerges in the spherical model, where boundary conditions are kept fixed to periodic, while the breaking or preserving of ergodicity is managed by imposing the spherical constraint either sharply or smoothly.

Synthesis of 3-PRS Manipulator Based on Revolute and Cone Angle of Spherical Constraints on Range of Rotational Motion of Spherical Joints

This paper presents synthesis of 3-prismatic-revolute-spherical parallel manipulator according to the spherical constraints such as limitation of cone angle on the range of rotational motion of spherical joint because the range of motion of mobile platform of manipulator will be influenced by the spherical constraints such as the range of cone angle of rotational motion of spherical joints. Therefore, while determining the parameters of the 3-PRS manipulator, the limitations on the cone angle of rotational motion of spherical joints are considered. The results of cone angle of rotational motion of each spherical joint in each position $$\left( {{\varvec{n}}_{{{\varvec{ji}}}} } \right)$$ are determined and compared with the spherical constraint condition of spherical joint of the manipulator. The synthesis of the manipulator is carried out by the limitation on range of motion of cone angle of spherical constraints and is solved by using least-square method, and the synthesized multi-positional 3-PRS parallel manipulator is suitable in surgical and space applications.

An elliptic expansion of the potential field source surface model

Context. The potential field source surface model is frequently used as a basis for further scientific investigations where a comprehensive coronal magnetic field is of importance. Its parameters, especially the position and shape of the source surface, are crucial for the interpretation of the state of the interplanetary medium. Improvements have been suggested that introduce one or more additional free parameters to the model, for example, the current sheet source surface model. Aims. Relaxing the spherical constraint of the source surface and allowing it to be elliptical gives modelers the option of deforming it to more accurately match the physical environment of the specific period or location to be analyzed. Methods. A numerical solver is presented that solves Laplace’s equation on a three-dimensional grid using finite differences. The solver is capable of working on structured spherical grids that can be deformed to create elliptical source surfaces. Results. The configurations of the coronal magnetic field are presented using this new solver. Three-dimensional renderings are complemented by Carrington-like synoptic maps of the magnetic configuration at different heights in the solar corona. Differences in the magnetic configuration computed by the spherical and elliptical models are illustrated.

Force balance controls the relaxation time of the gradient descent algorithm in the satisfiable phase.

We numerically study the relaxation dynamics of the single-layer perceptron with the spherical constraint. This is the simplest model of neural networks and serves a prototypical mean-field model of both convex and nonconvex optimization problems. The relaxation time of the gradient descent algorithm rapidly increases near the SAT-UNSAT (satisfiable-unsatisfiable) transition point. We numerically confirm that the first nonzero eigenvalue of the Hessian controls the relaxation time. This first eigenvalue vanishes much faster upon approaching the SAT-UNSAT transition point than the prediction of the Marchenko-Pastur law in random matrix theory derived under the assumption that the set of unsatisfied constraints are uncorrelated. This leads to a nontrivial critical exponent of the relaxation time in the SAT phase. Using a simple scaling analysis, we show that the isolation of this first eigenvalue from the bulk of spectrum is attributed to the force balance at the SAT-UNSAT transition point. Finally, we show that the estimated critical exponent of the relaxation time in the nonconvex region agrees very well with that of frictionless spherical particles, which have been studied in the context of the jamming transition of granular materials.

Gaussian-spherical restricted Boltzmann machines

We consider a special type of restricted Boltzmann machine (RBM), namely a Gaussian-spherical RBM where the visible units have Gaussian priors while the vector of hidden variables is constrained to stay on an sphere. The spherical constraint having the advantage to admit exact asymptotic treatments, various scaling regimes are explicitly identified based solely on the spectral properties of the coupling matrix (also called weight matrix of the RBM). Incidentally these happen to be formally related to similar scaling behaviors obtained in a different context dealing with spatial condensation of zero range processes. More specifically, when the spectrum of the coupling matrix is doubly degenerated an exact treatment can be proposed to deal with finite size effects. Interestingly the known parallel between the ferromagnetic transition of the spherical model and the Bose–Einstein condensation can be made explicit in that case. More importantly this gives us the ability to extract all needed response functions with arbitrary precision for the training algorithm of the RBM. This allows us then to numerically integrate the dynamics of the spectrum of the weight matrix during learning in a precise way. This dynamics reveals in particular a sequential emergence of modes from the Marchenko–Pastur bulk of singular vectors of the coupling matrix.

A trust region algorithm for computing extreme eigenvalues of tensors

<p style='text-indent:20px;'>Eigenvalues and eigenvectors of high order tensors have crucial applications in sciences and engineering. For computing H-eigenvalues and Z-eigenvalues of even order tensors, we transform the tensor eigenvalue problem to a nonlinear optimization with a spherical constraint. Then, a trust region algorithm for the spherically constrained optimization is proposed in this paper. At each iteration, an unconstrained quadratic model function is solved inexactly to produce a trial step. The Cayley transform maps the trial step onto the unit sphere. If the trial step generates a satisfactory actual decrease of the objective function, we accept the trial step as a new iterate. Otherwise, a second order line search process is performed to exploit valuable information contained in the trial step. Global convergence of the proposed trust region algorithm is analyzed. Preliminary numerical experiments illustrate that the novel trust region algorithm is efficient and promising.

On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution.

Blind deconvolution is the problem of recovering a convolutional kernel a0 and an activation signal x0 from their convolution [Formula: see text]. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed.

Minimally Constrained Flight Simulation in Wind Tunnel

Experimental studies on aircraft stability and performance can be enhanced by using a rig in which the aircraft model support approximates free flight within a wind tunnel. Such multi-degree-of-freedom wind tunnel rigs often impose kinematic restrictions on the aircraft model’s translational motion. This study investigates these kinematic effects, with particular attention to a spherical constraint in which the aircraft is held at the end of a fixed-length pivoting arm. Here, the motions of the aircraft and kinematic constraints are derived as differential-algebraic equations and assessed numerically. The impact is found mainly on translational motion with a negligible effect on the aircraft’s rotation. A concept to reduce these kinematic effects on the aircraft’s motion by applying an external force onto the aircraft is proposed. This compensation, which partially accounts for the constraints on the aircraft motion, is shown to reduce the effects of the arm, allowing for improved physical simulation.

Effects of droplet size and surfactants on anchoring in liquid crystal nanodroplets.

Liquid crystal (LC) droplets attract scientific attention for many advanced applications, including, but not limited to optical and sensing devices. To aid experimental advancements, theoretical calculations have been conducted to quantify molecular driving forces responsible for the collective behaviour of LC molecules within micrometer-size spherical droplets. To quantify the LC molecular anchoring within spherical physical constraints, molecular simulations at atomistic resolution would be useful. In an attempt to bridge the gap between computational capabilities and experimental interest, coarse-grained simulations are used here to study nematic LC nanodroplets dispersed in water. A LC phase diagram is generated as a function of droplet size and temperature. The effect of adding surfactants on LC anchoring was quantified, considering surfactants of different molecular features. When few surfactants are present, they self-assemble at the droplet boojums regardless of their molecular features. All surfactants tested shifted LC orientation from bipolar to uniaxial. When the surfactants have a hydrophobic tail of sufficient length, they cause deviations from the spherical symmetry of LC droplets. Increasing the concentration of these surfactants enhances such phenomenon. Simulations were also conducted to assess the ability of the surfactants to prevent the agglomeration between two LC droplets. The results showed that coalescence was inevitable at all conditions and suggested that large enough surfactant concentrations can delay the phenomenon. The results presented could be helpful for designing novel surface-active compounds to develop optical and/or sensing devices at conditions in which mutual solubility between water and LCs is low.

Derivative free methodologies for circuit worst case analysis

In this paper, a new derivative-free method for Worst Case Analysis (WCA) of circuit design is defined. A WCA of a device can be performed by solving a particular minimization problem where the objective function values are obtained by a simulation code and where some variables are subject to a spherical constraint and others to box constraints. In order to efficiently tackle such a problem, the paper defines a new DF algorithm which follows a two blocks Gauss Seidel approach, namely it alternates an approximated minimization with respect to the variables subject to the spherical constraint with an approximated minimization respect to the variables subject to the box constraints. The algorithm is described and its global convergence properties are analyzed. Furthermore it is tested in the WCA of a MOSFET operational amplifier and its computational behaviour is compared with the one of the efficient optimization tool of the WiCkeD suite for circuit analysis. The obtained results seem to indicate that the proposed algorithm is promising in terms of average efficiency, accuracy and robustness.

Type synthesis of non-holonomic spherical constraint underactuated parallel robotics

Space robotics is taken as one of the most promising approaches for on-orbit servicing missions. The underactuated non-holonomic parallel mechanism is a new direction of parallel space robotics, with small volume, lightweight, low cost, low energy consumption, and high flexibility. The existing methods on type synthesis could not generate underactuated non-holonomic parallel mechanisms with the introduction of the non-holonomic spherical pairs in parallel mechanisms. This paper is devoted to type synthesis of underactuated parallel mechanisms with spherical motion pattern accompanied by the generation of the non-holonomic spherical pairs in parallel mechanisms with spherical motion pattern. After the introduction of the traditional roller-sphere contact, this paper proposes a novel roller-sphere contact. After this paper provides the type synthesis of parallel mechanisms with spherical motion pattern, it puts forward the generation method of non-holonomic spherical pairs in parallel underactuated mechanisms with spherical motion pattern. The non-holonomic underactuated joint mechanism of a space robot for on-orbit servicing operations is taken as an example to demonstrate the proposed method.