Proof of Lemma 2. Identifying the second and first order terms in X we get: The last equality allows to conclude the proof. Additionally, assume that either Assumption 2 or Assumption 3 holds. This concludes the proof according to Lemma 1. Here we establish an auxiliary result, controlling the convergence of Neumann iterates to the matrix inverse. Note that Proposition A.1 can easily be extended to the general case by working with M (61) i.e., a M nonlinearity is applied to the activations.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors present a novel approach to learning to rank. In contrast to traditional approaches, the idea is to focus on the number of positive instances that are ranked before the first negative one. Following a large-margin approach leads to primal and dual representations. Compared to similar approaches, the complexity is only linear in the number of instances.

We propose a form of gradient surgery that projects a task's gradient onto the normal plane of the gradient of

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

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

Graph-based algorithms are among the most successful paradigms for solving semi-supervised learning tasks.

For this problem, we propose a novel generalization of random walk, diffusion, or smooth function methods in the literature to a convex p-norm cut function.

Deep structured output learning shows great promise in tasks like semantic image segmentation. We proffer a new, efficient deep structured model learning scheme, in which we show how deep Convolutional Neural Networks (CNNs) can be used to directly estimate the messages in message passing inference for structured prediction with Conditional Random Fields (CRFs). With such CNN message estimators, we obviate the need to learn or evaluate potential functions for message calculation. This confers significant efficiency for learning, since otherwise when performing structured learning for a CRF with CNN potentials it is necessary to undertake expensive inference for every stochastic gradient iteration. The network output dimension of message estimators is the same as the number of classes, rather than exponentially growing in the order of the potentials. Hence it is more scalable for cases that involve a large number of classes. We apply our method to semantic image segmentation and achieve impressive performance, which demonstrates the effectiveness and usefulness of our CNN message learning method.

A key bottleneck in structured output prediction is the need for inference during training and testing, usually requiring some form of dynamic programming. Rather than using approximate inference or tailoring a specialized inference method for a particular structure--standard responses to the scaling challenge-- we propose to embed prediction constraints directly into the learned representation. By eliminating the need for explicit inference a more scalable approach to structured output prediction can be achieved, particularly at test time. We demonstrate the idea for multi-label prediction under subsumption and mutual exclusion constraints, where a relationship to maximum margin structured output prediction can be established. Experiments demonstrate that the benefits of structured output training can still be realized even after inference has been eliminated.