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...ructing efficient learning and inference procedures. Notable examples for structured label sets are graphs (in particular trees) and sequences (Collins, 2000; Altun et al., 2003; Taskar et al., 2003; =-=Tsochantaridis et al., 2004-=-). We now overview how the cost-sensitive learning algorithms described in the previous section can be adapted to structured output settings. For concreteness, we focus on an adaptation for sequence p...

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...predicts the label of a single example, we formulate the learning problem as a multivariate prediction of all examples in the dataset. Based on the sparse approximation algorithm for structural SVMs (=-=Tsochantaridis et al., 2004-=-), we propose a method with which the training problem can be solved in polynomial time. We show that thesA Support Vector Method for Multivariate Performance Measures method applies to any performanc...

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... the max-margin formulation for Markov networks (Taskar et al., 2003; Taskar et al., 2004a) and context free grammars (Taskar et al., 2004b), and is similar to other formulations (Altun et al., 2003; =-=Tsochantaridis et al., 2004-=-). As in the univariate prediction, we measure the error of prediction using a loss function ℓ(y (i) ,y). In structured problems, where we are jointly predicting multiple variables, the loss is often ...

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...ze the non-convex objective in training. The use of latent variables is less well-explored in the case of large-margin structured output learning such as Max-Margin Markov Networks or Structural SVMs =-=[7, 8]-=-. These models are non-probabilistic and offer excellent performance in many structured prediction tasks in the fully observed case. Currently, they do not support the use of latent variables, which e...

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...gs are equally bad. The loss function l(Yn, Hn) measures how incorrect Hn is and penalizes the slack variable ξn in proportion. This loss function formulation from (6) is often called marginrescaling =-=[23]-=-. We consider notions of loss that decompose across the M windows: l(Y, H) = ∑M i=1 l(yi, hi). One simple window-specific loss is 0-1: l01(yi, hi) = I(yi ̸= hi) Hence, the constraint from (6) requires...

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... sort on a set, and as such is related to the task of learning structured outputs. Our approach is very different, however, from recent work on structured outputs, such as the large margin methods of =-=[12, 13]-=-. There, structures are also mapped to the reals (through choice of a suitable inner product), but the best output is found by estimating the argmax over all possible outputs. The ranking problem also...

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... out that our method is a natural generalization of the multiclass SVM formulation proposed in [3, 20] and a particularly interesting special case of a more general learning architecture presented in =-=[15]-=-. Comparing our approach with previous work, we would like to point out what we believe to be the main benefits and strengths of our method. (i) Our architecture avoids a greedy decision process, sinc...

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...bounding box within the image: Y≡{(ω, t, l, b, r) | ω ∈{+1, −1}, (t, l, b, r) ∈ R4 }.Forω= −1 the coordinate vector (t, l, b, r) is ignored. We learn this mapping in the structured learning framework =-=[14,15]-=- as g(x) = argmaxy f(x, y) (1) where f(x, y) is a discriminant function that should give a large value to pairs (x, y) that are well matched. The task is therefore to learn the function f, given that ...

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... (especially if applied to the power set of the label space [4]). When structure can be imposed on the label space (e.g. class hierarchy), efficient learning and prediction methods are often possible =-=[5, 6, 7, 8, 9]-=-. Here, we focus on a different type of structure, namely output sparsity, which is not addressed in previous work. Moreover, our method is general enough to take advantage of structured notions of sp...