A sufficiently large tree is induced by applying the split selection algorithm recursively.

Neural Information Processing Systems http://nips.cc/

We propose a new algorithm called PLUTO for building logistic regression trees to binary response data. PLUTO can capture the nonlinear and interaction patterns in messy data by recursively partitioning the sample space. It fits a simple or a multiple linear logistic regression model in each partition. PLUTO employs the cyclical coordinate descent method for estimation of multiple linear logistic regression models with elastic net penalties, which allows it to deal with high-dimensional data efficiently. The tree structure comprises a graphical description of the data. Together with the logistic regression models, it provides an accurate classifier as well as a piecewise smooth estimate of the probability of "success". PLUTO controls selection bias by: (1) separating split variable selection from split point selection; (2) applying an adjusted chi-squared test to find the split variable instead of exhaustive search. A bootstrap calibration technique is employed to further correct selection bias. Comparison on real datasets shows that on average, the multiple linear PLUTO models predict more accurately than other algorithms.

The split Bregman (SB) method [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-43] is a fast splitting-based algorithm that solves image reconstruction problems with general l1, e.g., total-variation (TV) and compressed sensing (CS), regularizations by introducing a single variable split to decouple the data-fitting term and the regularization term, yielding simple subproblems that are separable (or partially separable) and easy to minimize. Several convergence proofs have been proposed, and these proofs either impose a "full column rank" assumption to the split or assume exact updates in all subproblems. However, these assumptions are impractical in many applications such as the X-ray computed tomography (CT) image reconstructions, where the inner least-squares problem usually cannot be solved efficiently due to the highly shift-variant Hessian. In this paper, we show that when the data-fitting term is quadratic, the SB method is a convergent alternating direction method of multipliers (ADMM), and a straightforward convergence proof with inexact updates is given using [J. Eckstein and D. P. Bertsekas, Mathematical Programming, 55 (1992), pp. 293-318, Theorem 8]. Furthermore, since the SB method is just a special case of an ADMM algorithm, it seems likely that the ADMM algorithm will be faster than the SB method if the augmented Largangian (AL) penalty parameters are selected appropriately. To have a concrete example, we conduct a convergence rate analysis of the ADMM algorithm using two splits for image restoration problems with quadratic data-fitting term and regularization term. According to our analysis, we can show that the two-split ADMM algorithm can be faster than the SB method if the AL penalty parameter of the SB method is suboptimal. Numerical experiments were conducted to verify our analysis.

We propose a new approach based on model relaxation to compute minimum-cardinality diagnoses of a (faulty) system: We obtain a relaxed model of the system by splitting nodes in the system and compile the abstraction of the relaxed model into DNNF. Abstraction is obtained by treating self-contained sub-systems called cones as single components. We then use a novel branch-and-bound search algorithm and compute the abstract minimum-cardinality diagnoses of the system, which are later refined hierarchically, in a careful manner, to get all minimum-cardinality diagnoses of the system. Experiments on ISCAS-85 benchmark circuits show that the new approach is faster than the previous state-of-the-art hierarchical approach, and scales to all circuits in the suite for the first time.

In Bayesian networks, a most probable explanation (MPE) is a most likely instantiation of all network variables given a piece of evidence. Solving (the decision version of) an MPE query is NP-hard. Recent work proposed a branch-and-bound search algorithm that finds exact solutions to MPE queries, where bounds are computed on a relaxed network obtained by a technique known as node splitting. In this work we study the impact of variable and value ordering on such a search algorithm. We study several heuristics based on the entropies of variables and on the notion of  nogoods, and propose a new meta-heuristic that combines their strengths. Experiments indicate that search efficiency is significantly improved, allowing many hard problems to be solved for the first time.