A.1 Proof for Theorem 1 A.1.1 Proof for (I) and (II) First, observe that the constraint in Equation ( 3) can be equivalently replaced by an inequality constraint f Therefore, the Lagrangian multiplier can be restricted to be λ 0. We have L II) follows a straightforward calculation. Proof for (III), the strong duality We first introduce the following lemma, which is a straight forward generalization of the strong Lagrange duality to functional optimization case. The proof of Lemma 1 is standard. However, for completeness, we include it here. Notice that both sets A and B are convex.

To this end, we prove a formula for the gradient of the exponential of matrices, which can be of practical interest on its own.

We consider the problem of finding M best diverse solutions of energy minimization problems for graphical models. Contrary to the sequential method of Batra et al., which greedily finds one solution after another, we infer all M solutions jointly. It was shown recently that such jointly inferred labelings not only have smaller total energy but also qualitatively outperform the sequentially obtained ones. The only obstacle for using this new technique is the complexity of the corresponding inference problem, since it is considerably slower algorithm than the method of Batra et al. In this work we show that the joint inference of M best diverse solutions can be formulated as a submodular energy minimization if the original MAP-inference problem is submodular, hence fast inference techniques can be used. In addition to the theoretical results we provide practical algorithms that outperform the current state-of-the-art and can be used in both submodular and non-submodular case.

We study the estimation of low rank matrices via nonconvex optimization.

In this work, we consider IHT as a solution to the problem of learning sparse discrete distributions. We study the hardness of using IHT on the space of measures.

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

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