The augmented Lagrangian (AL) method that solves convex optimization problems with linear constraints has drawn more attention recently in imaging applications due to its decomposable structure for composite cost functions and empirical fast convergence rate under weak conditions. However, for problems such as X-ray computed tomography (CT) image reconstruction and large-scale sparse regression with "big data", where there is no efficient way to solve the inner least-squares problem, the AL method can be slow due to the inevitable iterative inner updates. In this paper, we focus on solving regularized (weighted) least-squares problems using a linearized variant of the AL method that replaces the quadratic AL penalty term in the scaled augmented Lagrangian with its separable quadratic surrogate (SQS) function, thus leading to a much simpler ordered-subsets (OS) accelerable splitting-based algorithm, OS-LALM, for X-ray CT image reconstruction. To further accelerate the proposed algorithm, we use a second-order recursive system analysis to design a deterministic downward continuation approach that avoids tedious parameter tuning and provides fast convergence. Experimental results show that the proposed algorithm significantly accelerates the "convergence" of X-ray CT image reconstruction with negligible overhead and greatly reduces the OS artifacts in the reconstructed image when using many subsets for OS acceleration.

When combined properly, search techniques can reveal the full potential of sophisticated branching heuristics. We demonstrate this observation on the well-known class of random 3-SAT formulae. First, a new branching heuristic is presented, which generalizes existing work on this class. Much smaller search trees can be constructed by using this heuristic. Second, we introduce a variant of discrepancy search, called ALDS. Theoretical and practical evidence support that ALDS traverses the search tree in a near-optimal order when combined with the new heuristic. Both techniques, search and heuristic, have been implemented in the look-ahead solver march. The SAT 2009 competition results show that march is by far the strongest complete solver on random k-SAT formulae.

We observe a trend regarding restart strategies used in SAT solvers. A few years ago, most state-of-the-art solvers restarted on average after a few thousands of backtracks. Currently, restarting after a dozen backtracks results in much better performance. The main reason for this trend is that heuristics and data structures have become more restart-friendly. We expect further continuation of this trend, so future SAT solvers will restart even more rapidly. Additionally, we present experimental results to support our observations.

Unsupervised ranking faces one critical challenge in evaluation applications, that is, no ground truth is available. When PageRank and its variants show a good solution in related subjects, they are applicable only for ranking from link-structure data. In this work, we focus on unsupervised ranking from multi-attribute data which is also common in evaluation tasks. To overcome the challenge, we propose five essential meta-rules for the design and assessment of unsupervised ranking approaches: scale and translation invariance, strict monotonicity, linear/nonlinear capacities, smoothness, and explicitness of parameter size. These meta-rules are regarded as high level knowledge for unsupervised ranking tasks. Inspired by the works in [8] and [14], we propose a ranking principal curve (RPC) model, which learns a one-dimensional manifold function to perform unsupervised ranking tasks on multi-attribute observations. Furthermore, the RPC is modeled to be a cubic B\'ezier curve with control points restricted in the interior of a hypercube, thereby complying with all the five meta-rules to infer a reasonable ranking list. With control points as the model parameters, one is able to understand the learned manifold and to interpret the ranking list semantically. Numerical experiments of the presented RPC model are conducted on two open datasets of different ranking applications. In comparison with the state-of-the-art approaches, the new model is able to show more reasonable ranking lists.

Collecting and maintaining accurate world knowledge in a dynamic, complex, adversarial, and stochastic environment such as the RoboCup 3D Soccer Simulation is a challenging task. Knowledge should be learned in real-time with time constraints. We use recently introduced Off-Policy Gradient Descent algorithms within Reinforcement Learning that illustrate learnable knowledge representations for dynamic role assignments. The results show that the agents have learned competitive policies against the top teams from the RoboCup 2012 competitions for three vs three, five vs five, and seven vs seven agents. We have explicitly used subsets of agents to identify the dynamics and the semantics for which the agents learn to maximize their performance measures, and to gather knowledge about different objectives, so that all agents participate effectively and efficiently within the group.

Recent work introduced the cube-and-conquer technique to solve hard SAT instances. It partitions the search space into cubes using a lookahead solver. Each cube is tackled by a conflict-driven clause learning (CDCL) solver. Crucial for strong performance is the cutoff heuristic that decides when to switch from lookahead to CDCL. Yet, this offline heuristic is far from ideal. In this paper, we present a novel hybrid solver that applies the cube and conquer steps simultaneously. A lookahead and a CDCL solver work together on each cube, while communication is restricted to synchronization. Our concurrent cube-and-conquer solver can solve many instances faster than pure lookahead, pure CDCL and offline cube-and-conquer, and can abort early in favor of a pure CDCL search if an instance is not suitable for cube-and-conquer techniques.

In this paper we show that very large mixtures of Gaussians are efficiently learnable in high dimension. More precisely, we prove that a mixture with known identical covariance matrices whose number of components is a polynomial of any fixed degree in the dimension n is polynomially learnable as long as a certain non-degeneracy condition on the means is satisfied. It turns out that this condition is generic in the sense of smoothed complexity, as soon as the dimensionality of the space is high enough. Moreover, we prove that no such condition can possibly exist in low dimension and the problem of learning the parameters is generically hard. In contrast, much of the existing work on Gaussian Mixtures relies on low-dimensional projections and thus hits an artificial barrier. Our main result on mixture recovery relies on a new "Poissonization"-based technique, which transforms a mixture of Gaussians to a linear map of a product distribution. The problem of learning this map can be efficiently solved using some recent results on tensor decompositions and Independent Component Analysis (ICA), thus giving an algorithm for recovering the mixture. In addition, we combine our low-dimensional hardness results for Gaussian mixtures with Poissonization to show how to embed difficult instances of low-dimensional Gaussian mixtures into the ICA setting, thus establishing exponential information-theoretic lower bounds for underdetermined ICA in low dimension. To the best of our knowledge, this is the first such result in the literature. In addition to contributing to the problem of Gaussian mixture learning, we believe that this work is among the first steps toward better understanding the rare phenomenon of the "blessing of dimensionality" in the computational aspects of statistical inference.

We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we recover and, in several cases, improve results for sets of sparse and structured sparse vectors, low-rank matrices and tensors, and smooth manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for data sets taking the form of an infinite union of subspaces of a Hilbert space.

Partition functions arise in a variety of settings, including conditional random fields, logistic regression, and latent gaussian models. In this paper, we consider semistochastic quadratic bound (SQB) methods for maximum likelihood inference based on partition function optimization. Batch methods based on the quadratic bound were recently proposed for this class of problems, and performed favorably in comparison to state-of-the-art techniques. Semistochastic methods fall in between batch algorithms, which use all the data, and stochastic gradient type methods, which use small random selections at each iteration. We build semistochastic quadratic bound-based methods, and prove both global convergence (to a stationary point) under very weak assumptions, and linear convergence rate under stronger assumptions on the objective. To make the proposed methods faster and more stable, we consider inexact subproblem minimization and batch-size selection schemes. The efficacy of SQB methods is demonstrated via comparison with several state-of-the-art techniques on commonly used datasets.

In this paper we consider a problem of searching a space of predictive models for a given training data set. We propose an iterative procedure for deriving a sequence of improving models and a corresponding sequence of sets of non-linear features on the original input space. After a finite number of iterations N, the non-linear features become 2^N -degree polynomials on the original space. We show that in a limit of an infinite number of iterations derived non-linear features must form an associative algebra: a product of two features is equal to a linear combination of features from the same feature space for any given input point. Because each iteration consists of solving a series of convex problems that contain all previous solutions, the likelihood of the models in the sequence is increasing with each iteration while the dimension of the model parameter space is set to a limited controlled value.