Submodular Optimization with Contention Resolution Extensions

This paper considers optimizing a submodular function subject to a set of downward closed constraints. Previous literature on this problem has often constructed solutions by (1) discovering a fractional solution to the multi-linear extension and (2) rounding this solution to an integral solution via a contention resolution scheme. This line of research has improved results by either optimizing (1) or (2). Diverging from previous work, this paper introduces a principled method called contention resolution extensions of submodular functions. A contention resolution extension combines the contention resolution scheme into a continuous extension of a discrete submodular function. The contention resolution extension can be defined from effectively any contention resolution scheme. In the case where there is a loss in both (1) and (2), by optimizing them together, the losses can be combined resulting in an overall improvement. This paper showcases the concept by demonstrating that for the problem of optimizing a non-monotone submodular subject to the elements forming an independent set in an interval graph, the algorithm gives a .188-approximation. This improves upon the best known 1/(2e)~eq .1839 approximation

Out-of-sample generalizations for supervised manifold learning for classification

Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structure-preserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function (RBF) interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with a progressive procedure. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets

Efficient AUC Optimization for Information Ranking Applications

Adequate evaluation of an information retrieval system to estimate future performance is a crucial task. Area under the ROC curve (AUC) is widely used to evaluate the generalization of a retrieval system. However, the objective function optimized in many retrieval systems is the error rate and not the AUC value. This paper provides an efficient and effective non-linear approach to optimize AUC using additive regression trees, with a special emphasis on the use of multi-class AUC (MAUC) because multiple relevance levels are widely used in many ranking applications. Compared to a conventional linear approach, the performance of the non-linear approach is comparable on binary-relevance benchmark datasets and is better on multi-relevance benchmark datasets.Comment: 12 page

Effects of electrostatic screening on the conformation of single DNA molecules confined in a nanochannel

Single T4-DNA molecules were confined in rectangular-shaped channels with a depth of 300 nm and a width in the range 150-300 nm casted in a poly(dimethylsiloxane) nanofluidic chip. The extensions of the DNA molecules were measured with fluorescence microscopy as a function of the ionic strength and composition of the buffer as well as the DNA intercalation level by the YOYO-1 dye. The data were interpreted with scaling theory for a wormlike polymer in good solvent, including the effects of confinement, charge, and self-avoidance. It was found that the elongation of the DNA molecules with decreasing ionic strength can be interpreted in terms of an increase of the persistence length. Self-avoidance effects on the extension are moderate, due to the small correlation length imposed by the channel cross-sectional diameter. Intercalation of the dye results in an increase of the DNA contour length and a partial neutralization of the DNA charge, but besides effects of electrostatic origin it has no significant effect on the bare bending rigidity. In the presence of divalent cations, the DNA molecules were observed to contract, but they do not collapse into a condensed structure. It is proposed that this contraction results from a divalent counterion mediated attractive force between the segments of the DNA molecule.Comment: 38 pages, 10 figures, accepted for publication in The Journal of Chemical Physic