Vladimir Kolmogorov

Pritish Mohapatra, Michal Rolinek, C. V. Jawahar et al.

The accuracy of information retrieval systems is often measured using complex loss functions such as the average precision (AP) or the normalized discounted cumulative gain (NDCG). Given a set of positive and negative samples, the parameters of a retrieval system can be estimated by minimizing these loss functions. However, the non-differentiability and non-decomposability of these loss functions does not allow for simple gradient based optimization algorithms. This issue is generally circumvented by either optimizing a structured hinge-loss upper bound to the loss function or by using asymptotic methods like the direct-loss minimization framework. Yet, the high computational complexity of loss-augmented inference, which is necessary for both the frameworks, prohibits its use in large training data sets. To alleviate this deficiency, we present a novel quicksort flavored algorithm for a large class of non-decomposable loss functions. We provide a complete characterization of the loss functions that are amenable to our algorithm, and show that it includes both AP and NDCG based loss functions. Furthermore, we prove that no comparison based algorithm can improve upon the computational complexity of our approach asymptotically. We demonstrate the effectiveness of our approach in the context of optimizing the structured hinge loss upper bound of AP and NDCG loss for learning models for a variety of vision tasks. We show that our approach provides significantly better results than simpler decomposable loss functions, while requiring a comparable training time.

Rustem Takhanov, Vladimir Kolmogorov

We consider two models for the sequence labeling (tagging) problem. The first one is a {\em Pattern-Based Conditional Random Field }(\PB), in which the energy of a string (chain labeling) $x=x_1\ldots x_n\in D^n$ is a sum of terms over intervals $[i,j]$ where each term is non-zero only if the substring $x_i\ldots x_j$ equals a prespecified word $w\in Λ$. The second model is a {\em Weighted Context-Free Grammar }(\WCFG) frequently used for natural language processing. \PB and \WCFG encode local and non-local interactions respectively, and thus can be viewed as complementary. We propose a {\em Grammatical Pattern-Based CRF model }(\GPB) that combines the two in a natural way. We argue that it has certain advantages over existing approaches such as the {\em Hybrid model} of Bened{í} and Sanchez that combines {\em $\mbox{$N$-grams}$} and \WCFGs. The focus of this paper is to analyze the complexity of inference tasks in a \GPB such as computing MAP. We present a polynomial-time algorithm for general \GPBs and a faster version for a special case that we call {\em Interaction Grammars}.

Vladimir Kolmogorov, Christoph Lampert, Emilie Morvant et al.

The 38th Annual Workshop of the Austrian Association for Pattern Recognition (ÖAGM) will be held at IST Austria, on May 22-23, 2014. The workshop provides a platform for researchers and industry to discuss traditional and new areas of computer vision. This year the main topic is: Pattern Recognition: interdisciplinary challenges and opportunities.

Rustem Takhanov, Vladimir Kolmogorov

We consider Conditional Random Fields (CRFs) with pattern-based potentials defined on a chain. In this model the energy of a string (labeling) $x_1...x_n$ is the sum of terms over intervals $[i,j]$ where each term is non-zero only if the substring $x_i...x_j$ equals a prespecified pattern $α$. Such CRFs can be naturally applied to many sequence tagging problems. We present efficient algorithms for the three standard inference tasks in a CRF, namely computing (i) the partition function, (ii) marginals, and (iii) computing the MAP. Their complexities are respectively $O(n L)$, $O(n L \ell_{max})$ and $O(n L \min\{|D|,\log (\ell_{max}+1)\})$ where $L$ is the combined length of input patterns, $\ell_{max}$ is the maximum length of a pattern, and $D$ is the input alphabet. This improves on the previous algorithms of (Ye et al., 2009) whose complexities are respectively $O(n L |D|)$, $O(n |Γ| L^2 \ell_{max}^2)$ and $O(n L |D|)$, where $|Γ|$ is the number of input patterns. In addition, we give an efficient algorithm for sampling. Finally, we consider the case of non-positive weights. (Komodakis & Paragios, 2009) gave an $O(n L)$ algorithm for computing the MAP. We present a modification that has the same worst-case complexity but can beat it in the best case.

Vladimir Kolmogorov, Martin Wainwright

Tree-reweighted max-product (TRW) message passing is a modified form of the ordinary max-product algorithm for attempting to find minimal energy configurations in Markov random field with cycles. For a TRW fixed point satisfying the strong tree agreement condition, the algorithm outputs a configuration that is provably optimal. In this paper, we focus on the case of binary variables with pairwise couplings, and establish stronger properties of TRW fixed points that satisfy only the milder condition of weak tree agreement (WTA). First, we demonstrate how it is possible to identify part of the optimal solution|i.e., a provably optimal solution for a subset of nodes| without knowing a complete solution. Second, we show that for submodular functions, a WTA fixed point always yields a globally optimal solution. We establish that for binary variables, any WTA fixed point always achieves the global maximum of the linear programming relaxation underlying the TRW method.

Vladimir Kolmogorov, Thomas Schoenemann

This paper addresses the problem of approximate MAP-MRF inference in general graphical models. Following [36], we consider a family of linear programming relaxations of the problem where each relaxation is specified by a set of nested pairs of factors for which the marginalization constraint needs to be enforced. We develop a generalization of the TRW-S algorithm [9] for this problem, where we use a decomposition into junction chains, monotonic w.r.t. some ordering on the nodes. This generalizes the monotonic chains in [9] in a natural way. We also show how to deal with nested factors in an efficient way. Experiments show an improvement over min-sum diffusion, MPLP and subgradient ascent algorithms on a number of computer vision and natural language processing problems.