Bounded Approximation Error Research Articles

Temporal Set Inversion for Animated Implicits

We exploit the temporal coherence of closed-form animated implicit surfaces by locally re-evaluating an octree-like discretization of the implicit field only as and where is necessary to rigorously maintain a global error invariant over time, thereby saving resources in static or slowly-evolving areas far from the motion where per-frame updates are not necessary. We treat implicit surface rendering as a special case of the continuous constraint satisfaction problem of set inversion, which seeks preimages of arbitrary sets under vector-valued functions. From this perspective, we formalize a temporally-coherent set inversion algorithm that localizes changes in the field by range-bounding its time derivatives using interval arithmetic. We implement our algorithm on the GPU using persistent thread scheduling and apply it to the scalar case of implicit surface and swept volume rendering where we achieve significant speedups in complex scenes with localized deformations like those found in games and modelling applications where interactivity is required and bounded-error approximation is acceptable.

Fast ACE (FACE): an error-bounded approximation of Automatic Color Equalization.

We present an efficient algorithm to approximate the Automatic Color Equalization (ACE) of an input color image, with an upper-bound on the introduced approximation error. The computation is based on Summed Area Tables and a carefully optimized partitioning of the plane into rectangular regions, resulting in a pseudo-linear asymptotic complexity with the number of pixels (against a quadratic straightforward computation of ACE). Our experimental evaluation confirms both the speedups and high accuracy, reaching lower approximation errors than existing approaches. We provide a publicly available reference implementation of our algorithm.

Error-bounded Edge-based Remeshing of High-order Tetrahedral Meshes

We present a robust method to generate high-quality high-order tetrahedral meshes with bounded approximation errors and low mesh complexity. The success of our method relies on two key components. The first one is three novel local operations that robustly modify the topology of the high-order tetrahedral mesh while avoiding invalid (flipped or degenerate) elements. In practice, our meshing algorithm follows the edge-based remeshing algorithm that iteratively conducts these local topological operations and a geometric optimization operation to improve mesh quality. The second is a new containment check procedure that robustly judges whether the approximation error between the input mesh and the high-order mesh exceeds the user-specified bound. If one operation causes the error-bounded constraint to be violated, we reject this operation to ensure a bounded approximation error. Besides, the number of tetrahedrons of the high-order mesh is reduced by progressively increasing the target edge length in the edge-based remeshing algorithm. A large number of experimental results have shown the capability and feasibility of our method. Compared to other state-of-the-art methods, our method achieves higher robustness and quality.

A hybrid tree-based algorithm to solve asymmetric distributed constraint optimization problems

Asymmetric distributed constraint optimization problems (ADCOPs) have emerged as an important formalism in multi-agent community due to their ability to capture personal preferences. However, the existing search-based complete algorithms for ADCOPs only exploit local knowledge to calculate lower bounds, which leads to inefficient pruning and prohibits them from solving large scale problems. On the other hand, inference-based complete algorithms (e.g., DPOP) for distributed constraint optimization problems are able to aggregate the global cost promptly but cannot be directly applied into ADCOPs due to a privacy concern. Thus, in this paper, we investigate the possibility of combining inference and search to effectively solve ADCOPs at an acceptable loss of privacy. Specifically, we propose a hybrid complete ADCOP algorithm called PT-ISABB which uses a tailored inference algorithm to provide tight lower bounds and upper bounds, and a tree-based complete search algorithm to guarantee the optimality. Furthermore, we introduce two suboptimal variants of PT-ISABB based on bounded-error approximation mechanisms to enable trade-off between theoretically guaranteed solutions and coordination overheads. We prove the correctness of PT-ISABB and its suboptimal variants. Finally, the experimental results demonstrate that PT-ISABB exhibits great superiorities over other state-of-the-art search-based complete algorithms and its suboptimal variants can quickly find a solution within the user-specified bounded-error.

Limited-complexity controller tuning: A set membership data-driven approach

Data-driven tuning is an alternative to model-based controller design where controllers are directly identified from data, avoiding a plant identification step. In this paper, an approach to tune limited-complexity controllers from data for linear systems is proposed. The controller is parametrized as a linear combination of a large set of basis functions and the proposed algorithm allows to select a sparse subset of bases, guaranteeing a bounded approximation error. A feasibility condition allows to adjust the trade-off between accuracy and sparsity. The controller design is performed by solving a set of linear programming problems, allowing to handle large data-sets. The proposed strategy is evaluated by means of a Monte-Carlo simulation experiment on a flexible transmission benchmark model. Results show that the proposed solution offers similar results than previous approaches for large data-sets, requiring less adjustable parameters. However, for reduced data-sets, the presented algorithm shows better performance than the compared approaches.

Improving Policies via Search in Cooperative Partially Observable Games

Recent superhuman results in games have largely been achieved in a variety of zero-sum settings, such as Go and Poker, in which agents need to compete against others. However, just like humans, real-world AI systems have to coordinate and communicate with other agents in cooperative partially observable environments as well. These settings commonly require participants to both interpret the actions of others and to act in a way that is informative when being interpreted. Those abilities are typically summarized as theory of mind and are seen as crucial for social interactions. In this paper we propose two different search techniques that can be applied to improve an arbitrary agreed-upon policy in a cooperative partially observable game. The first one, single-agent search, effectively converts the problem into a single agent setting by making all but one of the agents play according to the agreed-upon policy. In contrast, in multi-agent search all agents carry out the same common-knowledge search procedure whenever doing so is computationally feasible, and fall back to playing according to the agreed-upon policy otherwise. We prove that these search procedures are theoretically guaranteed to at least maintain the original performance of the agreed-upon policy (up to a bounded approximation error). In the benchmark challenge problem of Hanabi, our search technique greatly improves the performance of every agent we tested and when applied to a policy trained using RL achieves a new state-of-the-art score of 24.61 / 25 in the game, compared to a previous-best of 24.08 / 25.

Activity Organization for Friend-Making Optimization in Online Social Networks

The social presence theory in social psychology suggests that computer-mediated online interactions are inferior to face-to-face, in-person interactions. Thus, it's important to organize social activities for online social network users to meet in person. In this paper, we consider the scenarios of organizing in person friend-making social activities via online social networks (OSNs) and formulate a new research problem, namely, <i>Hop-bounded Maximum Group Friending (HMGF)</i> , that takes into consideration both existing friendships and the likelihood of new friend making in organization of the targeted in person friend-making social activities. To find a set of attendees for such social activities, HMGF is unique and challenging due to the interplay of the group size, the constraint on existing friendships, and the objective of maximizing the likelihood of friend making. We prove that HMGF is NP-Hard, and there exists no approximation algorithm for it unless <inline-formula><tex-math notation="LaTeX">$P=NP$</tex-math></inline-formula> . We also provide an Integer Linear Programming (ILP) formulation for the HMGF problem. The ILP formulation, which can be solved efficiently by a commercial solver to obtain the optimal solution for small HMGF instances, acts as a baseline approach for comparison in the evaluation of the proposed algorithm. We further propose an error-bounded approximation algorithm, <i>MaxGF</i> , to efficiently obtain the solutions very close to the optimal solutions. To boost the performance, we devise two graph-theoretical pruning strategies, namely <i>Neighbor Pruning</i> and <i>Core Pruning</i> , which can effectively avoid redundant graph explorations to improve the performance of HMGF. We also study HMGF on a class of special graphs, <i>threshold graphs</i> , which have properties very similar to many online social networks. We prove that MaxGF can obtain the optimal solution to HMGF on threshold graphs in polynomial time. We conduct a user study to validate our problem formulation and perform extensive experiments on real datasets to demonstrate the efficiency and effectiveness of our proposed algorithm. The experimental results manifest that our proposed algorithms outperform the baselines, including the ILP formulation.

DRIVEN: A framework for efficient Data Retrieval and clustering in Vehicular Networks

The growing interest in data analysis applications for Cyber–Physical Systems stems from the large amounts of data such large distributed systems sense in a continuous fashion. A key research question in this context is how to jointly address the efficiency and effectiveness challenges of such data analysis applications.DRIVEN proposes a way to jointly address these challenges for a data gathering and distance-based clustering tool in the context of vehicular networks. To cope with the limited communication bandwidth (compared to the sensed data volume) of vehicular networks and data transmission’s monetary costs, DRIVEN avoids gathering raw data from vehicles, but rather relies on a streaming-based and error-bounded approximation, through Piecewise Linear Approximation (PLA), to compress the volumes of gathered data. Moreover, a streaming-based approach is also used to cluster the collected data (once the latter is reconstructed from its PLA-approximated form). DRIVEN’s clustering algorithm leverages the inherent ordering of the spatial and temporal data being collected to perform clustering in an online fashion, while data is being retrieved. As we show, based on our prototype implementation using Apache Flink and thorough evaluation with real-world data such as GPS, LiDAR and other vehicular signals, the accuracy loss for the clustering performed on the gathered approximated data can be small (below 10%), even when the raw data is compressed to 5-35% of its original size, and the transferring of historical data itself can be completed in up to one-tenth of the duration observed when gathering raw data.

Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

Graph sampling set selection, where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal, is a fundamental problem in graph signal processing (GSP). Previous works employ an unbiased least-squares (LS) signal reconstruction scheme and select samples via expensive extreme eigenvector computation. Instead, we assume a biased graph Laplacian regularization (GLR) based scheme that solves a system of linear equations for reconstruction. We then choose samples to minimize the condition number of the coefficient matrix---specifically, maximize the smallest eigenvalue $\lambda_{\min}$. Circumventing explicit eigenvalue computation, we maximize instead the lower bound of $\lambda_{\min}$, designated by the smallest left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we first convert the optimization to a dual problem, where we minimize the number of samples needed to align all Gershgorin disc left-ends at a chosen lower-bound target $T$. Algebraically, the dual problem amounts to optimizing two disc operations: i) shifting of disc centers due to sampling, and ii) scaling of disc radii due to a similarity transformation of the matrix. We further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous NP-hard set cover (SC) problem. The reinterpretation enables us to derive a fast approximation scheme from a known SC error-bounded approximation algorithm. We find an appropriate target $T$ efficiently via binary search. Extensive simulation experiments show that our disc-based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen-decomposition-free sampling schemes in reconstruction error.

Practical error-bounded remeshing by adaptive refinement

We propose an efficient and practically robust method to isotropically remesh an input triangular mesh with bounded approximation error. Our method is based on a key observation, that is, when more uniformly distributed vertices are added into the remeshed mesh, the error-bounded constraint is usually satisfied. Then our algorithm adaptively refines the remeshed mesh to satisfy the error-bounded constraint and avoid heavy computational load. To that end, we present an iterative approach that alternates in each iteration a pass to do an edge-based remeshing using the computed edge length field and a pass to adaptively adjust the edge length field. The robustness of our method is demonstrated by performing tests on complex shapes, as well as models containing sharp features or boundaries. Compared to the state-of-the-art error-bounded methods, our technique is much faster and more practically robust.

Fast and Error-Bounded Space-Variant Bilateral Filtering

The traditional space-invariant isotropic kernel utilized by a bilateral filter (BF) frequently leads to blurry edges and gradient reversal artifacts due to the existence of a large amount of outliers in the local averaging window. However, the efficient and accurate estimation of space-variant kernels which adapt to image structures, and the fast realization of the corresponding space-variant bilateral filtering are challenging problems. To address these problems, we present a space-variant BF (SVBF), and its linear time and error-bounded acceleration method. First, we accurately estimate spacevariant anisotropic kernels that vary with image structures in linear time through structure tensor and minimum spanning tree. Second, we perform SVBF in linear time using two error-bounded approximation methods, namely, low-rank tensor approximation via higher-order singular value decomposition and exponential sum approximation. Therefore, the proposed SVBF can efficiently achieve good edge-preserving results. We validate the advantages of the proposed filter in applications including: image denoising, image enhancement, and image focus editing. Experimental results demonstrate that our fast and error-bounded SVBF is superior to state-of-the-art methods.

Explicit Model Predictive Control via Nonlinear Piecewise Approximations

In this paper, a novel identification methodology is proposed to capture general multivariate nonlinear relationships, with focus on the bounded-error approximation of model predictive control for constrained (non)linear systems. The formulation of the identification problem takes, at each iteration, the form of a constrained linear (or quadratic) optimization problem that is mathematically feasible as well as numerically tractable. The efficiency of the proposed method for the derivation of low-complexity explicit model predictive controllers is demonstrated via the constrained control of a thermodynamic power plant.

HOG-Dot: A Parallel Kernel-Based Gradient Extraction for Embedded Image Processing

In this letter, we propose HOG-Dot, a method for the direct computation of the polar image gradients coordinates from the pixels values. The proposed algorithm, to be used as the first step of the Histogram of Oriented Gradient (HOG) pipeline, approximates the exact gradient with its projection onto a versor chosen among the projection plane set. Instead of non linear computations, the HOG-Dot method exploits linear operations while introducing a bounded approximation error with respect to other HOG approaches, thus resulting a more suitable solution for embedded devices. Concerning the state of the art, it also achieves improved accuracy with the mathematical spatial gradient formulation .

Coding Structure and Replication Optimization for Interactive Multiview Video Streaming

Multiview video refers to videos of the same dynamic 3-D scene captured simultaneously by multiple closely spaced cameras from different viewpoints. We study interactive streaming of pre-encoded multiview videos, where, at any time, a client can request any one of many captured views for playback. Moreover, the client can periodically freeze the video in time and switch to neighboring views for a compelling look-around visual effect. We consider distributed content servers to support large-scale interactive multiview video service. These servers collaboratively replicate and access video contents. We study two challenges in this setting: what is an efficient coding structure that supports interactive view switching and, given that, what to replicate in each server in order to minimize the cost incurred by interactive temporal and view switches? We first propose a redundant coding structure that facilitates interactive view-switching, trading off storage with transmission rate. Using the coding structure, we next propose a content replication strategy that takes advantage of indirect hit to lower view-switching cost: in the event that the exact requested view is not available locally, the local server can fetch a different but correlated view from the other servers, so that the remote repository only needs to supply the pre-encoded view differential. We formulate the video content replication problem to minimize the switching cost as an integer linear programming (ILP) problem and show that it is NP-hard. We first propose an LP relaxation and rounding algorithm (termed Minimum Eviction) with bounded approximation error. We then study a more scalable solution based on dynamic programming and Lagrangian optimization (DPLO) with little sacrifice in performance. Simulation results show that our replication algorithms achieve substantially lower switching cost compared to other content replication schemes.

Model-order reduction of ion channel dynamics using approximate bisimulation

We show that in the context of the Iyer et al. (IMW) 67-variable cardiac myocycte model, it is possible to replace the detailed 13-state probabilistic subsystem of the sodium channel dynamics with a much simpler Hodgkin-Huxley (HH)-like two-state abstraction, while only incurring a bounded approximation error. We then extend our technique to the 10-state subsystem of the fast-recovering calcium-independent potassium channel. The basis of our results is the construction of an approximate bisimulation between the HH-type abstraction and the corresponding detailed ion channel model, both of which are input-controlled (voltage in this case) CTMCs.The construction of the appropriate approximate bisimulation, as well as the overall result regarding the behavior of this modified IMW model, involves: (1) Identification of the voltage-dependent parameters of the m and h gates in the HH-type models via a two-step fitting process, carried out over more than 20,000 representative observational traces of the detailed sodium and potassium ion channel models. (2) Proving that the distance between observations of the detailed models and their respective abstraction is bounded. (3) Exploring the sensitivity of the overall IMW model to the HH-type approximations. Our extensive simulation results experimentally validate our findings, for varying IMW-type input stimuli.

Real-time Approximation of Clothoids With Bounded Error for Path Planning Applications

We present a method for real-time computation of clothoid coordinates that guarantees bounded approximation error over a wide range of clothoid parameters provided that the clothoid's orientation change and length areReal-time approximation bounded. It is shown that coordinates of clothoid with any parameters can be computed from those of a single clothoid (with fixed parameters), using appropriate geometrical transformations. A comprehensive analysis is given on how to determine a required set of clothoids and, based on this, how to sample a clothoid in a lookup table in order to achieve required approximation precision. The algorithm is computationally very efficient and therefore suitable for real-time path planning, as well as for other applications that benefit from fast clothoid computation.

무작위 데이터 근사화를 위한 유계오차 B-스플라인 근사법

This paper presents an error-bounded B-spline fitting technique to approximate unorganized data within a prescribed error tolerance. The proposed approach includes two main steps: leastsquares minimization and error-bounded approximation. A B-spline hypervolume is first described as a data representation model, which includes its mathematical definition and the data structure for implementation. Then we present the least-squares minimization technique for the generation of an approximate B-spline model from the given data set, which provides a unique solution to the problem: overdetermined, underdetermined, or ill-conditioned problem. We also explain an algorithm for the error-bounded approximation which recursively refines the initial base model obtained from the least-squares minimization until the Euclidean distance between the model and the given data is within the given error tolerance. The proposed approach is demonstrated with some examples to show its usefulness and a good possibility for various applications.

A Fourier Based Method for Approximating the Joint Detection Probability in MIMO Communications

We propose a numerically efficient technique to approximate the joint detection probability of a coherent multiple input multiple output (MIMO) receiver in the presence of inter-symbol interference (ISI) and additive white Gaussian noise (AWGN). This technique approximates the probability of detection by numerically integrating the product of the characteristic function (CF) of the received filtered signal with the Fourier transform of the multi-dimension decision region. Naturally, the accuracy of the approximation is dependent on the number of points in the numerical integration. The proposed method selects the number of points to integrate over by deriving bounds on the approximation error. As such, a reduction of up to 1/100th of the number points can be achieved when comparing with existing single input single output (SISO) techniques. Simulation results demonstrate that the proposed method is highly accurate with the calculated errors being well within the bounded approximation error. In addition, the results are closely matched with those obtained using the Monte Carlo simulations with a typical error less than 10-4 while requiring only a fraction of the computation time.

Efficient and accurate computation of upper bounds of approximation errors

For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions.Replacing f by p induces a relative error ϵ=p/f−1. In order to ensure the validity of the use of p instead of f, the maximum error, i.e. the supremum norm ‖ϵ‖∞I must be safely bounded above over an interval I, whose width is typically of order 1.Numerical algorithms for supremum norms are efficient, but they cannot offer the required safety. Previous validated approaches often require tedious manual intervention. If they are automated, they have several drawbacks, such as the lack of quality guarantees.In this article, a novel, automated supremum norm algorithm on univariate approximation errors ϵ is proposed, achieving an a priori quality on the result. It focuses on the validation step and paves the way for formally certified supremum norms.Key elements are the use of intermediate approximation polynomials with bounded approximation error and a non-negativity test based on a sum-of-squares expression of polynomials.The new algorithm was implemented in the Sollya tool. The article includes experimental results on real-life examples.

Randomized and Distributed Self-Configuration of Wireless Networks: Two-Layer Markov Random Fields and Near-Optimality

This work studies the near-optimality versus the complexity of distributed configuration management for wireless networks. We first develop a global probabilistic graphical model for a network configuration which characterizes jointly the statistical spatial dependence of a physical- and a logical-configuration. The global model is a Gibbs distribution that results from the internal network properties on node positions, wireless channel and interference; and the external management constraints on physical connectivity and signal quality. A local model is a two-layer Markov Random Field (i.e., a random bond model) that approximates the global model with the local spatial dependence of neighbors. The complexity of the local model is defined through the communication range among nodes which corresponds to the number of neighbors in the two-layer Markov Random Field. The local model is near-optimal when the approximation error to the global model is within a given bound. We analyze the tradeoff between approximation error and complexity. We then derive sufficient conditions on the near-optimality of the local model. For a fast decaying wireless channel with power attenuation factor α > 4 , a node only needs to communicate with O(1) neighbors for a local model to be near optimal. For a slowly decaying channel with a power attenuation factor 2 ≤ α ≤ 4, a node may have to communicate with more than O(N(4-α)/4) neighbors to result in a bounded approximation error. If the communication range is kept to be O(1), a bounded approximation error can also be achieved by reducing the density of active links to O(N(α-4)/(α+4)) for α 4 . The two-layer Markov Random Fields enable a class of randomized distributed algorithms such as the stochastic relaxation that allows a node to self-configure based on information from neighbors. We validate the model, the analysis and the randomized distributed algorithms through simulation.