The Kaczmarz method is an iterative method for solving large systems of equations that projects iterates orthogonally onto the solution space of each equation. In contrast to direct methods such as Gaussian elimination or QR-factorization, this algorithm is efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the Kaczmarz method as described in the literature has its limitations: it considers only equality constraints and does not support soft constraints, which makes it inapplicable to the UI layout problem. In this paper we extend the Kaczmarz method for solving specifications containing soft constraints, using the prioritized IIS detection algorithm. Furthermore, the performance and convergence of the proposed algorithms are evaluated empirically using randomly generated UI layout specifications of various sizes. The results show that these methods offer improvements in performance over standard methods like Matlab's LINPROG, a well-known efficient linear programming solver.
Besides methods for MaxFS there are also some methods to solve the IIS problem. These methods are: deletion filtering, IIS detection and grouping constraints. Deletion filtering @cite_23 removes constraints from the set of constraints and checks the feasibility of the reduced set. IIS detection @cite_39 starts with a single constraint and adds constraints successively. The grouping constraints method @cite_56 was introduced to speed up the aforementioned algorithms by adding or removing groups of constraints simultaneously. Even though these methods deal with the problem of finding a feasible subsystem, it is not possible to apply them directly. The main reason is that they do not consider prioritized constraints as we do in our approach.
@cite_56 @cite_23 @cite_39
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Algorithms and computer-based tools for analyzing infeasible linear and nonlinear programs have been developed in recent years, but few such tools exist for infeasible mixed-integer or integer linear programs. One approach that has proven especially useful for infeasible linear programs is the isolation of an Irreducible Infeasible Set of constraints (IIS), a subset of the constraints defining the overall linear program that is itself infeasible, but for which any proper subset is feasible. Isolating an IIS from the larger model speeds the diagnosis and repair of the model by focussing the analytic effort. This paper describes and tests algorithms for finding small infeasible sets in infeasible mixed-integer and integer linear programs; where possible these small sets are IISs. We propose a new approach to adaptive system identification when the system model is sparse. The approach applies l 1 relaxation, common in compressive sensing, to improve the performance of LMS-type adaptive methods. This results in two new algorithms, the zero-attracting LMS (ZA-LMS) and the reweighted zero-attracting LMS (RZA-LMS). The ZA-LMS is derived via combining a l 1 norm penalty on the coefficients into the quadratic LMS cost function, which generates a zero attractor in the LMS iteration. The zero attractor promotes sparsity in taps during the filtering process, and therefore accelerates convergence when identifying sparse systems. We prove that the ZA-LMS can achieve lower mean square error than the standard LMS. To further improve the filtering performance, the RZA-LMS is developed using a reweighted zero attractor. The performance of the RZA-LMS is superior to that of the ZA-LMS numerically. Experiments demonstrate the advantages of the proposed filters in both convergence rate and steady-state behavior under sparsity assumptions on the true coefficient vector. The RZA-LMS is also shown to be robust when the number of non-zero taps increases. We present an unsupervised method for learning a hierarchy of sparse feature detectors that are invariant to small shifts and distortions. The resulting feature extractor consists of multiple convolution filters, followed by a feature-pooling layer that computes the max of each filter output within adjacent windows, and a point-wise sigmoid non-linearity. A second level of larger and more invariant features is obtained by training the same algorithm on patches of features from the first level. Training a supervised classifier on these features yields 0.64 error on MNIST, and 54 average recognition rate on Caltech 101 with 30 training samples per category. While the resulting architecture is similar to convolutional networks, the layer-wise unsupervised training procedure alleviates the over-parameterization problems that plague purely supervised learning procedures, and yields good performance with very few labeled training samples. Separating an image into reflectance and shading layers poses a challenge for learning approaches because no large corpus of precise and realistic ground truth decompositions exists. The Intrinsic Images in the Wild (IIW) dataset provides a sparse set of relative human reflectance judgments, which serves as a standard benchmark for intrinsic images. A number of methods use IIW to learn statistical dependencies between the images and their reflectance layer. Although learning plays an important role for high performance, we show that a standard signal processing technique achieves performance on par with current state-of-the-art. We propose a loss function for CNN learning of dense reflectance predictions. Our results show a simple pixel-wise decision, without any context or prior knowledge, is sufficient to provide a strong baseline on IIW. This sets a competitive baseline which only two other approaches surpass. We then develop a joint bilateral filtering method that implements strong prior knowledge about reflectance constancy. This filtering operation can be applied to any intrinsic image algorithm and we improve several previous results achieving a new state-of-the-art on IIW. Our findings suggest that the effect of learning-based approaches may have been over-estimated so far. Explicit prior knowledge is still at least as important to obtain high performance in intrinsic image decompositions.