Closed-form Computation Research Articles

Heterogeneous heatmap distillation framework based on unbiased alignment for lightweight human pose estimation

The growing demand for mobile devices has generated interest in lightweight human pose estimation. Currently, lightweight estimation generally uses heatmap-based methods, which has demonstrated exceptional performance. However, their use of non-differentiable post-processing imposes considerable inference latencies. Conversely, integral-based approaches expedite the inference process by employing a soft-argmax operation but compromise in accuracy. Integrating explicit heatmap knowledge learned using the heatmap-based method into the implicit heatmap generated by the integral-based method, thereby combining the best of both worlds, offers a promising avenue. However, owing to the disparities in supervision and inference processes, the explicit and implicit heatmaps are heterogeneous. Consequently, direct transfer of knowledge presents difficulties in ensuring consistencies in heat value and location. In this paper, we propose a novel Heterogeneous Heatmap Distillation (HHD) framework that effectively tackles these challenges. The framework seamlessly integrates explicit heatmap knowledge that contains high-precision localization information into implicit heatmaps. The framework revolves around an unbiased heatmap alignment scheme encompassing two steps: heterogeneous heatmap normalization and unbiased cropping. Heterogeneous heatmap normalization separately normalizes the output feature maps of both the teacher and student models, alleviating potential heat value bias during the knowledge transfer. Unbiased cropping applies closed-form computation on the normalized teacher and student heatmap to eliminate location bias. Additionally, mirror expansion is implemented to handle potential cases wherein the cropped region extends beyond the image boundary. Extensive experiments demonstrate the efficiency and effectiveness of our methods on the MSCOCO and MPII datasets compared to other integral-based lightweight networks. Our source codes and pre-trained models are available at https://github.com/ducongju/HHD.

3D macro-element for innovative plug-and-play joints

This paper presents the development, implementation, and validation of a macro-element suitable for the linear analysis of innovative 3D plug-and-play joints between tubular columns and lightweight steel truss-girders. The macro-element is based on the component method, accounts for the three-dimensional interaction between the tube faces, and its components have a clear physical meaning. Simplified procedures are developed for the closed-form computation of the stiffness matrix of the macro-element based on the geometric and mechanical properties of the nodal zone. This facilitates practical application in everyday design scenarios. Furthermore, the macro-element’s architecture is implemented in the framework of OpenSees as a standalone beam-to-column joint finite element. Validation of the conceptual design is accomplished through parametric studies, comparing its performance with models generated in higher-order finite element commercial software, Abaqus. This research offers a valuable resource for the linear analysis and design of innovative 3D plug-and-play joint connections in structural engineering, enhancing efficiency and reliability in construction practices.

Uncertainty Visualization of Critical Points of 2D Scalar Fields for Parametric and Nonparametric Probabilistic Models.

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.

Closed-Form Solution of the Unit Normal Loss Integral in 2 Dimensions, with Application in Value-of-Information Analysis.

The unit normal loss integral (UNLI) is widely used in decision analysis and risk modeling, including in the computation of various value-of-information metrics, but its closed-form solution is only applicable to comparisons of 2 strategies.We derive a closed-form solution for 2-dimensional UNLI, extending the applicability of the UNLI to 3-strategy comparisons.Such closed-form computation takes only a fraction of a second and is free from simulation errors that affect the hitherto available methods.In addition to the relevance in 3-strategy model-based and data-driven decision analyses, a particular application is in risk prediction modeling, where the net benefit of a classifier should always be compared with 2 default strategies of treating none and treating all.

General construction and classes of explicit L1-optimal couplings

The main scope of this paper is to give some explicit classes of examples of L1-optimal couplings. Optimal transportation w.r.t. the Kantorovich metric ℓ1 (resp. the Wasserstein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (Dokl. Akad. Nauk SSSR 115 (1957) 1058–1061) and Sudakov (Proc. Steklov Inst. Math. 141 (1979) 1–178) to occur on rays induced by a decomposition of the basic space (and more generally to higher dimensional decompositions in the case of general measures) induced by the corresponding dual potentials. Several papers have given this kind of structural result and established existence and uniqueness of solutions in varying generality. Since the dual problems pose typically too strong challenges to be solved in explicit form, these structural results have so far been applied for the solution of few particular instances. First, we give a self-contained review of some basic optimal coupling results and we propose and investigate in particular some basic principles for the construction of L1-optimal couplings given by a reduction principle and some usable forms of the decomposition method. This reduction principle, together with symmetry properties of the reduced measures, gives a hint to the decomposition of the space into sectors and via the non crossing property of optimal transport leads to the choice of transportation rays. The optimality of the induced transports is then a consequence of the characterization results of optimal couplings. Then, we apply these principles to determine in explicit form L1-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time a general construction of L1-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scalings, and provide an extended class of dual functionals allowing for the closed-form computation of the ℓ1-metric or of accurate lower bounds of it in a variety of examples.

Fiber Uncertainty Visualization for Bivariate Data With Parametric and Nonparametric Noise Models.

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

Moment Dynamics and Observer Design for a Class of Quasilinear Quantum Stochastic Systems

This paper is concerned with a class of open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices pertaining to finite-level systems. The system interacts with external bosonic fields, and its Hamiltonian and coupling operators depend linearly on the system variables. This results in a Hudson--Parthasarathy quantum stochastic differential equation (QSDE) whose drift and dispersion terms are affine and linear functions of the system variables. The quasilinearity of the QSDE leads to tractable dynamics of mean values and higher-order multipoint moments of the system variables driven by vacuum input fields. This allows for the closed-form computation of the quasi-characteristic function of the invariant quantum state of the system and infinite-horizon asymptotic growth rates for a class of cost functionals. The tractability of the moment dynamics is also used for mean square optimal Luenberger observer design in a measurement-based filtering problem for a quasilinear quantum plant, which leads to a Kalman-like quantum filter.

Closed-Form Spectral Computation of Intermodulation Distortion for Shunt-Based Current Measurements

Closed-Form Spectral Computation of Intermodulation Distortion for Shunt-Based Current Measurements

Determination of the available wrench set of a 3-RPRR kinematically-redundant planar parallel manipulator

Parallel robots that incorporate kinematic redundancy in their designs are frequently considered for use in different applications due to their ability to avoid direct kinematic singularities within their workspace. Knowledge of a robot’s wrench generation capabilities constitutes essential information when designing its geometry and planning its optimal use to accomplish various tasks. In the presence of kinematic redundancies, the determination of a robot’s wrench generation capabilities at a given pose of its mobile platform depends not only on the limits of its actuators but also on the configurations of its legs. In this work, an algorithm allowing for the closed-form computation of the available wrench set of the 3-RPRR planar parallel robot is presented. This results in generating the boundary of the available wrench set as a set of faces in the wrench space. Based on this representation, algorithms allowing for the computation of various application-based performance indices are also described. Example numerical results are included to demonstrate how the tools that are presented could be used in the context of robot design.

Development of a redundant anthropomorphic hydraulically actuated manipulator with a roll-pitch-yaw spherical wrist

The demand for redundant hydraulic manipulators that can implement complex heavy-duty tasks in unstructured areas is increasing; however, current manipulator layouts that remarkably differ from human arms make intuitive kinematic operation challenging to achieve. This study proposes a seven-degree-of-freedom (7-DOF) redundant anthropomorphic hydraulically actuated manipulator with a novel roll-pitch-yaw spherical wrist. A hybrid series-parallel mechanism is presented to achieve the spherical wrist design, which consists of two parallel linear hydraulic cylinders to drive the yaw/pitch 2-DOF wrist plate connected serially to the roll structure. Designed as a 1RPRRR-1SPU mechanism (“R”, “P”, “S”, and “U” denote revolute, prismatic, spherical, and universal joints, respectively; the underlined letter indicates the active joint), the 2-DOF parallel structure is partially decoupled to obtain simple forward/inverse kinematic solutions in which a closed-loop subchain “RPRR” is included. The 7-DOF manipulator is then designed, and its third joint axis goes through the spherical center to obtain closed-form inverse kinematic computation. The analytical inverse kinematic solution is drawn by constructing self-motion manifolds. Finally, a physical prototype is developed, and the kinematic analysis is validated via numerical simulation and test results.

Wireless Performance Evaluation of Building Layouts: Closed-Form Computation of Figures of Merit

This paper presents a part of our ground-breaking work on evaluation of buildings in terms of wireless friendliness at the building-design stage. The main goal is to devise building design practices that provide for a good performance of wireless networks deployed in buildings. In this paper, the interference gain (IG) and power gain (PG) are defined as two figures of merit (FoM) of the wireless performance of buildings. The FoMs bridge the gap between building design and wireless communications industries. An approach to derive exact closed-form equations for these FoMs is proposed for the first time. The derived analytic expressions facilitate straightforward and more computationally efficient numerical evaluation of the proposed FoMs as compared to Monte Carlo simulations for well-known indoor propagation models. It is shown that the derived closed-form expression can be readily employed to evaluate the impact of building properties, such as the sizes and the aspect ratios (ARs) of rooms, on the wireless performance. The proposed approach sheds light to architects on evaluation and design of wireless-friendly building layouts.

Full-Wave Computation of the Electric Field in the Partial Element Equivalent Circuit Method Using Taylor Series Expansion of the Retarded Green’s Function

This article presents new analytical formulas for the efficient computation of the full-wave electric field generated by conductive, dielectric, and magnetic media in the framework of the partial element equivalent circuit (PEEC) method. To this aim, the full-wave Green's function is handled by the Taylor series expansion leading to three types of integrals for which new analytical formulas are provided in order to avoid slower numerical integration. An orthogonal (Manhattan type) tessellation of the geometries is assumed, and the electrical quantities, i.e., currents, charges, and magnetization, are expanded in space through rectangular basis functions. The full-wave electric field radiated by charges, currents, and magnetization is computed analytically in the postprocessing step. The proposed closed-form computation of the electric field is tested using two examples, comparing the results obtained by the derived analytical formulas with the results from a finite element method solver.

Online Stochastic Optimization With Time-Varying Distributions

This article studies online stochastic optimization, where the random parameters follow time-varying distributions. In each time slot, after a control variable is determined, a sample drawn from the current distribution is revealed as feedback information. This form of stochastic optimization has broad applications in online learning and signal processing, where the underlying ground-truth is inherently time-varying, e.g., tracking a moving target. Dynamic optimal points are adopted as the performance benchmark to define the regret of algorithms, as opposed to the static optimal point used in stochastic optimization with fixed distributions. Unconstrained stochastic optimization with time-varying distributions is first examined and a projected stochastic gradient descent algorithm is presented. An upper bound on its regret is established with respect to the drift of the dynamic optima, which measures the temporal variations of the underlying distributions. In particular, the algorithm possesses sublinear regret as long as the drift of the optima is sublinear, i.e., the distributions do not vary too drastically. Further, a stochastic saddle point method involving only iterative closed-form computation is proposed for constrained stochastic optimization with time-varying distributions. Upper bounds on its regret and constraint violation are developed with respect to the drift of the optima. Analogously, sublinear regret and sublinear constraint violation can be ensured provided that the drift of the optima is sublinear. Finally, numerical results are presented to corroborate the efficacy of the proposed algorithms and the derived analytical results.

Simulating Nonstationary Spatio-Temporal Poisson Processes using the Inversion Method

We study the problem of simulating a class of nonstationary spatio-temporal Poisson processes. The Poisson intensity function is non-stationary and piecewise linear in both the time dimension and the spatial location dimensions. We propose an exact simulation algorithm based on the inversion method. This simulation algorithm adopts three advantages. First, the entire procedure involves only closed-form computation with no need for numerical integration or numerical inversion of any function. Each step in the algorithm only requires exact arithmetic operations. Second, the proposed algorithm is sample efficient, especially compared to the thinning method when the maximum intensity value is much larger than the minimum intensity value. Third, the algorithm generates arrivals sequentially, one at a time in ascending order, so that they can be conveniently fed into real-time or online decision-making tools.

Frequency-domain analysis of linear systems with periodic jumps: Definition of hybrid transfer function, pole and zero

A frequency-domain analysis is carried out for a class of linear systems in the presence of periodic time-driven jumps. The result is achieved by introducing the notion of transfer function in this context, and by relying on a suitably defined transform for hybrid arcs, whose properties are comprehensively discussed. It is firstly shown that knowledge of the transfer function permits the computation in closed-form of the complete (state and output) response of the hybrid plant, which may be otherwise intractable from the computational point of view. Moreover, the hybrid transfer function resulting from series interconnection of two plants is derived and exploited to introduce and discuss, in the single-input single-output case, the concepts of zero and pole of the hybrid transfer function. Differently from the classical purely continuous or discrete time cases, only a subset of the latter is related to the stability properties of the hybrid system.

Online Convex Optimization With Time-Varying Constraints and Bandit Feedback

In this paper, online convex optimization problem with time-varying constraints is studied from the perspective of an agent taking sequential actions. Both the objective function and the constraint functions are dynamic and unknown a priori to the agent. We first consider the scenario of the gradient feedback, in which, the values and gradients of the objective function and constraint functions at the chosen action are revealed after an action is submitted. We propose a computationally efficient online algorithm, which only involves direct closed-form computations at each time instant. It is shown that the algorithm possesses sublinear regret with respect to the dynamic benchmark sequence and sublinear constraint violations, as long as the drift of the benchmark sequence is sublinear, or in other words, the underlying dynamic optimization problems do not vary too drastically. Furthermore, we investigate the scenario of the bandit feedback, in which, after an action is chosen, only the values of the objective function and the constraint functions at several random points close to the action are announced to the agent. A bandit version of the online algorithm is proposed and we also establish its sublinear expected regret and sublinear expected constraint violations under the assumption that the drift of the benchmark sequence is sublinear. Finally, two numerical examples, namely online quadratic programming and online logistic regression, are presented to corroborate the effectiveness of the proposed algorithms and to confirm the theoretical guarantees.

Collaborative Cloud and Edge Computing for Latency Minimization

By performing data processing at the network edge, mobile edge computing can effectively overcome the deficiencies of network congestion and long latency in cloud computing systems. To improve edge cloud efficiency with limited communication and computation capacities, we investigate the collaboration between cloud computing and edge computing, where the tasks of mobile devices can be partially processed at the edge node and at the cloud server. First, a joint communication and computation resource allocation problem is formulated to minimize the weighted-sum latency of all mobile devices. Then, the closed-form optimal task splitting strategy is derived as a function of the normalized backhaul communication capacity and the normalized cloud computation capacity. Some interesting and useful insights for the optimal task splitting strategy are also highlighted by analyzing four special scenarios. Based on this, we further transform the original joint communication and computation resource allocation problem into an equivalent convex optimization problem and obtain the closed-form computation resource allocation strategy by leveraging the convex optimization theory. Moreover, a necessary condition is also developed to judge whether a task should be processed at the corresponding edge node only, without offloading to the cloud server. Finally, simulation results confirm our theoretical analysis and demonstrate that the proposed collaborative cloud and edge computing scheme can evidently achieve a better delay performance than the conventional schemes.

Millimeter-Wave Beamformed Full-Dimensional MIMO Channel Estimation Based on Atomic Norm Minimization

The millimeter-wave (mmWave) full-dimensional (FD) MIMO system employs planar arrays at both the base station and user equipment and can simultaneously support both azimuth and elevation beamforming. In this paper, we propose atomic-norm-based methods for mmWave FD-MIMO channel estimation under both uniform planar arrays (UPA) and non-uniform planar arrays (NUPA). Unlike existing algorithms such as compressive sensing (CS) or subspace methods, the atomic-norm-based algorithms do not require to discretize the angle spaces of the angle of arrival (AoA) and angle of departure (AoD) into grids, thus provide much better accuracy in estimation. In the UPA case, to reduce the computational complexity, the original large-scale 4D atomic norm minimization problem is approximately reformulated as a semi-definite program (SDP) containing two decoupled two-level Toeplitz matrices. The SDP is then solved via the alternating direction method of multipliers (ADMM) where each iteration involves only closed-form computations. In the NUPA case, the atomic-norm-based formulation for channel estimation becomes nonconvex and a gradient-decent-based algorithm is proposed to solve the problem. Simulation results show that the proposed algorithms achieve better performance than the CS-based and subspace-based algorithms.

Distributed Linearized ADMM for Network Cost Minimization

In this paper, we study a generic network cost minimization problem, in which every node has a local decision vector to determine. Each node incurs a cost depending on its decision vector and each link also incurs a cost depending on the decision vectors of its two end nodes. All nodes cooperate to minimize the overall network cost. The formulated network cost minimization problem has broad applications in distributed signal processing and control over multiagent systems. To obtain a decentralized algorithm for the formulated problem, we resort to the distributed alternating direction method of multipliers (DADMM). However, each iteration of the DADMM involves solving a local optimization problem at each node, leading to intractable computational burden in many circumstances. As such, inspired by recent works on approximated ADMM for consensus optimization problem, we propose a distributed linearized ADMM (DLADMM) algorithm for network cost minimization. In the DLADMM, each iteration only involves closed-form computations and avoids local optimization problems, which greatly reduces the computational complexity compared to the DADMM. We prove that the DLADMM converges to an optimal point when the local cost functions are convex and have Lipschitz continuous gradients. Linear convergence rate of the DLADMM is also established if the local cost functions are further strongly convex. Numerical experiments are conducted to corroborate the effectiveness of the DLADMM and we observe that the DLADMM has similar convergence performance as DADMM does while the former enjoys much lower computational overhead. The impact of network topology, connectivity, and algorithm parameters are also investigated through simulations.

Analytical probabilistic modeling of RBE-weighted dose for ion therapy

Particle therapy is especially prone to uncertainties. This issue is usually addressed with uncertainty quantification and minimization techniques based on scenario sampling. For proton therapy, however, it was recently shown that it is also possible to use closed-form computations based on analytical probabilistic modeling (APM) for this purpose. APM yields unique features compared to sampling-based approaches, motivating further research in this context.This paper demonstrates the application of APM for intensity-modulated carbon ion therapy to quantify the influence of setup and range uncertainties on the RBE-weighted dose. In particular, we derive analytical forms for the nonlinear computations of the expectation value and variance of the RBE-weighted dose by propagating linearly correlated Gaussian input uncertainties through a pencil beam dose calculation algorithm. Both exact and approximation formulas are presented for the expectation value and variance of the RBE-weighted dose and are subsequently studied in-depth for a one-dimensional carbon ion spread-out Bragg peak. With V and B being the number of voxels and pencil beams, respectively, the proposed approximations induce only a marginal loss of accuracy while lowering the computational complexity from order to for the expectation value and from to for the variance of the RBE-weighted dose. Moreover, we evaluated the approximated calculation of the expectation value and standard deviation of the RBE-weighted dose in combination with a probabilistic effect-based optimization on three patient cases considering carbon ions as radiation modality against sampled references. The resulting global γ-pass rates (2 mm,2%) are 99.15% for the expectation value and 94.95% for the standard deviation of the RBE-weighted dose, respectively. We applied the derived analytical model to carbon ion treatment planning, although the concept is in general applicable to other ion species considering a variable RBE.