This paper describes the development of finite abstractions of Max-Plus-Linear (MPL) systems using tropical operations. The idea of tropical abstraction is inspired by the fact that an MPL system is a discrete-event model updating its state with operations in the tropical algebra. The abstract model is a finite-state transition system: we show that the abstract states can be generated by operations on the tropical algebra, and that the generation of transitions can be established by tropical multiplications of matrices. The complexity of the algorithms based on tropical algebra is discussed and their performance is tested on a numerical benchmark against an existing alternative abstraction approach.
The computation of image and of inverse image of a DBM is described in @cite_7 . These computations are used to perform forward and backward reachability analysis, respectively. The worst-case complexity of both procedures is @math , where @math is the number of variables in @math excluding @math . A more detailed explanation about image and inverse image computation of a DBM is in Section 3.3.
@cite_7
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Many database management systems support whole-image matching. However, users may only remember certain subregions of the images. In this paper, we develop Padding and Reduction Algorithms to support subimage queries of arbitrary size based on local color information. The idea is to estimate the best- case lower bound to the dissimilarity measure between the query and the image. By making use of multiresolution representation, this lower bound becomes tighter as the scale becomes finer. Because image contents are usually pre- extracted and stored, a key issue is how to determine the number of levels used in the representation. We address this issue analytically by estimating the CPU and I O costs, and experimentally by comparing the performance and accuracy of the outcomes of various filtering schemes. Our findings suggest that a 3-level hierarchy is preferred. We also study three strategies for searching multiple resolutions. Our studies indicate that the hybrid strategy with horizontal filtering on the coarse level and vertical filtering on remaining levels is the best choice when using Padding and Reduction Algorithms in the preferred 3-level multiresolution representation. The best 10 desired images can be retrieved efficiently and effectively from a collection of a thousand images in about 3.5 seconds.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only. A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide efficient solutions to large-scale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forward-backward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided. Mathematical optimization plays a fundamental role in solving many problems in computer vision (e.g., camera calibration, image alignment, structure from motion). It is generally accepted that second order descent methods are the most robust, fast, and reliable approaches for nonlinear optimization of a general smooth function. However, in the context of computer vision, second order descent methods have two main drawbacks: 1) the function might not be analytically differentiable and numerical approximations are impractical, and 2) the Hessian may be large and not positive definite. Recently, Supervised Descent Method (SDM), a method that learns the “weighted averaged gradients” in a supervised manner has been proposed to solve these issues. However, SDM is a local algorithm and it is likely to average conflicting gradient directions. This paper proposes Global SDM (GSDM), an extension of SDM that divides the search space into regions of similar gradient directions. GSDM provides a better and more efficient strategy to minimize non-linear least squares functions in computer vision problems. We illustrate the effectiveness of GSDM in two problems: non-rigid image alignment and extrinsic camera calibration. The mirror descent algorithm (MDA) generalizes gradient descent by using a Bregman divergence to replace squared Euclidean distance. In this paper, we similarly generalize the alternating direction method of multipliers (ADMM) to Bregman ADMM (BADMM), which allows the choice of different Bregman divergences to exploit the structure of problems. BADMM provides a unified framework for ADMM and its variants, including generalized ADMM, inexact ADMM and Bethe ADMM. We establish the global convergence and the O(1 T) iteration complexity for BADMM. In some cases, BADMM can be faster than ADMM by a factor of O(n ln n) where n is the dimensionality. In solving the linear program of mass transportation problem, BADMM leads to massive parallelism and can easily run on GPU. BADMM is several times faster than highly optimized commercial software Gurobi. We consider dynamic evaluation of algebraic functions such as computing determinant, matrix adjoint, matrix inverse and solving linear system of equations. We show that in the dynamic setup the above problems can be solved faster than evaluating everything from scratch. In the case when rows and columns of the matrix can change we show an algorithm that achieves O(n sup 2 ) arithmetic operations per update and O(1) arithmetic operations per query. Next, we describe two algorithms, with different tradeoffs, for updating the inverse and determinant when single entries of the matrix are changed. The fastest update for the first tradeoff is O(n sup 1.575 ) arithmetic operations per update and O(n sup 0.575 ) arithmetic operations per query. The second tradeoff gives O(n sup 1.495 ) arithmetic operations per update and O(n sup 1.495 ) arithmetic operations per query. We also consider the case when some number of columns or rows can change. We use dynamic determinant computations to solve the following problems in the dynamic setup: computing the number of spanning trees in a graph and testing if an edge in a graph is contained in some perfect matching. These are the first dynamic algorithms for these problems. Next, with the use of dynamic matrix inverse, we solve fully dynamic transitive closure in general directed graphs. The bounds on arithmetic operations for dynamic matrix inverse translate directly to time bounds for dynamic transitive closure. Thus we obtain the first known algorithm with O(n sup 2 ) worst-case update time and constant query time and two algorithms for transitive closure in general digraphs with subquadratic update and query times. Our algorithms for transitive closure are randomized with one-sided error. We also consider for the first time the case when the edges incident with a part of vertices of the graph can be changed. Deep networks are increasingly being applied to problems involving image synthesis, e.g., generating images from textual descriptions and reconstructing an input image from a compact representation. Supervised training of image-synthesis networks typically uses a pixel-wise loss (PL) to indicate the mismatch between a generated image and its corresponding target image. We propose instead to use a loss function that is better calibrated to human perceptual judgments of image quality: the multiscale structural-similarity score (MS-SSIM). Because MS-SSIM is differentiable, it is easily incorporated into gradient-descent learning. We compare the consequences of using MS-SSIM versus PL loss on training deterministic and stochastic autoencoders. For three different architectures, we collected human judgments of the quality of image reconstructions. Observers reliably prefer images synthesized by MS-SSIM-optimized models over those synthesized by PL-optimized models, for two distinct PL measures ( @math and @math distances). We also explore the effect of training objective on image encoding and analyze conditions under which perceptually-optimized representations yield better performance on image classification. Finally, we demonstrate the superiority of perceptually-optimized networks for super-resolution imaging. Just as computer vision has advanced through the use of convolutional architectures that mimic the structure of the mammalian visual system, we argue that significant additional advances can be made in modeling images through the use of training objectives that are well aligned to characteristics of human perception. In this paper, we propose a novel image indexing and retrieval algorithm using local tetra patterns (LTrPs) for content-based image retrieval (CBIR). The standard local binary pattern (LBP) and local ternary pattern (LTP) encode the relationship between the referenced pixel and its surrounding neighbors by computing gray-level difference. The proposed method encodes the relationship between the referenced pixel and its neighbors, based on the directions that are calculated using the first-order derivatives in vertical and horizontal directions. In addition, we propose a generic strategy to compute nth-order LTrP using (n - 1)th-order horizontal and vertical derivatives for efficient CBIR and analyze the effectiveness of our proposed algorithm by combining it with the Gabor transform. The performance of the proposed method is compared with the LBP, the local derivative patterns, and the LTP based on the results obtained using benchmark image databases viz., Corel 1000 database (DB1), Brodatz texture database (DB2), and MIT VisTex database (DB3). Performance analysis shows that the proposed method improves the retrieval result from 70.34 44.9 to 75.9 48.7 in terms of average precision average recall on database DB1, and from 79.97 to 85.30 and 82.23 to 90.02 in terms of average retrieval rate on databases DB2 and DB3, respectively, as compared with the standard LBP. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math where @math The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes ( , 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math , where @math . The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math . Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. Solving inverse problems continues to be a challenge in a wide array of applications ranging from deblurring, image inpainting, source separation etc. Most existing techniques solve such inverse problems by either explicitly or implicitly finding the inverse of the model. The former class of techniques require explicit knowledge of the measurement process which can be unrealistic, and rely on strong analytical regularizers to constrain the solution space, which often do not generalize well. The latter approaches have had remarkable success in part due to deep learning, but require a large collection of source-observation pairs, which can be prohibitively expensive. In this paper, we propose an unsupervised technique to solve inverse problems with generative adversarial networks (GANs). Using a pre-trained GAN in the space of source signals, we show that one can reliably recover solutions to under determined problems in a blind' fashion, i.e., without knowledge of the measurement process. We solve this by making successive estimates on the model and the solution in an iterative fashion. We show promising results in three challenging applications -- blind source separation, image deblurring, and recovering an image from its edge map, and perform better than several baselines. We introduce F-IVM, a unified incremental view maintenance (IVM) approach for a variety of tasks, including gradient computation for learning linear regression models over joins, matrix chain multiplication, and factorized evaluation of conjunctive queries. F-IVM is a higher-order IVM algorithm that reduces the maintenance of the given task to the maintenance of a hierarchy of increasingly simpler views. The views are functions mapping keys, which are tuples of input data values, to payloads, which are elements from a task-specific ring. Whereas the computation over the keys is the same for all tasks, the computation over the payloads depends on the task. F-IVM achieves efficiency by factorizing the computation of the keys, payloads, and updates. We implemented F-IVM as an extension of DBToaster. We show in a range of scenarios that it can outperform classical first-order IVM, DBToaster's fully recursive higher-order IVM, and plain recomputation by orders of magnitude while using less memory. One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions A major problem in imaging applications such as magnetic resonance imaging and synthetic aperture radar is the task of trying to reconstruct an image with the smallest possible set of Fourier samples, every single one of which has a potential time and or power cost. The theory of compressive sensing (CS) points to ways of exploiting inherent sparsity in such images in order to achieve accurate recovery using sub-Nyquist sampling schemes. Traditional CS approaches to this problem consist of solving total-variation (TV) minimization programs with Fourier measurement constraints or other variations thereof. This paper takes a different approach. Since the horizontal and vertical differences of a medical image are each more sparse or compressible than the corresponding TV image, CS methods will be more successful in recovering these differences individually. We develop an algorithm called GradientRec that uses a CS algorithm to recover the horizontal and vertical gradients and then estimates the original image from these gradients. We present two methods of solving the latter inverse problem, i.e., one based on least-square optimization and the other based on a generalized Poisson solver. After a thorough derivation of our complete algorithm, we present the results of various experiments that compare the effectiveness of the proposed method against other leading methods. Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results. We address the inverse problem that arises in compressed sensing of a low-rank matrix. Our approach is to pose the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition that provides an analogy between parsimonious representations of a sparse vector and a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for the vector case are extended to the matrix case through this atomic decomposition. In particular, we propose an efficient and guaranteed algorithm named ADMiRA that extends CoSaMP, its analogue for the vector case. The performance guarantee is given in terms of the rank-restricted isometry property and bounds both the number of iterations and the error in the approximate solution for the general case where the solution is approximately low-rank and the measurements are noisy. With a sparse measurement operator such as the one arising in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. The numerical experiments for the matrix completion problem show that, although the measurement operator in this case does not satisfy the rank-restricted isometry property, ADMiRA is a competitive algorithm for matrix completion. In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise non-linearity) when the normal operator ( @math , where @math is the adjoint of the forward imaging operator, @math ) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a @math image on the GPU. In this paper, we address compressed sensing of a low-rank matrix posing the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition providing an analogy between parsimonious representations of a sparse vector and a low-rank matrix and extending efficient greedy algorithms from the vector to the matrix case. In particular, we propose an efficient and guaranteed algorithm named atomic decomposition for minimum rank approximation (ADMiRA) that extends Needell and Tropp's compressive sampling matching pursuit (CoSaMP) algorithm from the sparse vector to the low-rank matrix case. The performance guarantee is given in terms of the rank-restricted isometry property (R-RIP) and bounds both the number of iterations and the error in the approximate solution for the general case of noisy measurements and approximately low-rank solution. With a sparse measurement operator as in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. Numerical experiments for the matrix completion problem show that, although the R-RIP is not satisfied in this case, ADMiRA is a competitive algorithm for matrix completion. Relevant Component Analysis (RCA) has been proposed for learning distance metrics with contextual constraints for image retrieval. However, RCA has two important disadvantages. One is the lack of exploiting negative constraints which can also be informative, and the other is its incapability of capturing complex nonlinear relationships between data instances with the contextual information. In this paper, we propose two algorithms to overcome these two disadvantages, i.e., Discriminative Component Analysis (DCA) and Kernel DCA. Compared with other complicated methods for distance metric learning, our algorithms are rather simple to understand and very easy to solve. We evaluate the performance of our algorithms on image retrieval in which experimental results show that our algorithms are effective and promising in learning good quality distance metrics for image retrieval. In this paper, we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the diagonal quadratic approximation method DQAM of Mulvey and Ruszczynski [A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 1992, pp. 205–215] and the parallel coordinate descent method PCDM of Richtarik and Takac [Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873]. We show that the two methods are equivalent for feasibility problems up to the selection of a step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8L′ Lω−12 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian. Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n,k,d] linear code having all-symbol (r,δ)-locality, denoted as (r,δ)_a, is considered optimal if it has the actual highest minimum distance of any code of the given parameters n,k,r and δ. A minimum distance bound is given in . The existing results on the existence and the construction of optimal (r, δ)_a linear codes are limited to only two small regions within this special case, namely, i) m=0 and ii) m≥ (v+δ-1)>(δ-1) and δ=2, where m=n mod(r+δ-1) and v=k mod r. This paper investigates the properties and existence conditions for optimal (r,δ)_a linear codes with general r and δ. First, a structure theorem is derived for general optimal (r,δ)_a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no (r,δ)_a linear code can achieve the minimum distance bound in . For four other cases the optimal (r,δ)_a codes are shown to exist over a field of size q≥( n k-1 ), deterministic constructions are proposed. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on optimal (r,δ)_a linear codes, leaving only two cases in which the construction of optimal codes are not yet known.