Problems of segmentation, denoising, registration and 3D reconstruction are often addressed with the graph cut algorithm. However, solving an unconstrained graph cut problem is NP-hard. For tractable optimization, pairwise potentials have to fulfill the submodularity inequality. In our learning paradigm, pairwise potentials are created as the dot product of a learned vector w with positive feature vectors. In order to constrain such a model to remain tractable, previous approaches have enforced the weight vector to be positive for pairwise potentials in which the labels differ, and set pairwise potentials to zero in the case that the label remains the same. Such constraints are sufficient to guarantee that the resulting pairwise potentials satisfy the submodularity inequality. However, we show that such an approach unnecessarily restricts the capacity of the learned models. Guaranteeing submodularity for all possible inputs, no matter how improbable, reduces inference error to effectively zero, but increases model error. In contrast, we relax the requirement of guaranteed submodularity to solutions that are probably approximately submodular. We show that the conceptually simple strategy of enforcing submodularity on the training examples guarantees with low sample complexity that test images will also yield submodular pairwise potentials. Results are presented in the binary and muticlass settings, showing substantial improvement from the resulting increased model capacity.

中文翻译：

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具有可能子模约束的条件随机场的判别训练

图切割算法通常可以解决分割、去噪、配准和 3D 重建问题。然而，解决一个无约束的图割问题是 NP 难的。对于易处理的优化，成对势必须满足子模不等式。在我们的学习范式中，成对电位被创建为学习向量 w 与正特征向量的点积。为了约束这样的模型以保持易于处理，以前的方法强制权重向量对于标签不同的成对电位为正，并在标签保持不变的情况下将成对电位设置为零。这样的约束足以保证得到的成对势满足子模不等式。然而，我们表明这种方法不必要地限制了学习模型的能力。保证所有可能输入的子模块性，无论多么不可能，将推理误差减少到有效为零，但会增加模型误差。相比之下，我们将保证子模块化的要求放宽到可能近似子模块化的解决方案。我们表明，在训练示例上强制执行子模块性的概念上简单的策略保证了低样本复杂性，测试图像也将产生子模块成对潜力。结果显示在二元和多类设置中，表明模型容量的增加带来了实质性的改进。但会增加模型误差。相比之下，我们将保证子模块化的要求放宽到可能近似子模块化的解决方案。我们表明，在训练示例上强制执行子模块性的概念上简单的策略保证了低样本复杂性，测试图像也将产生子模块成对潜力。结果显示在二元和多类设置中，表明模型容量的增加带来了实质性的改进。但会增加模型误差。相比之下，我们将保证子模块化的要求放宽到可能近似子模块化的解决方案。我们表明，在训练示例上强制执行子模块性的概念上简单的策略保证了低样本复杂性，测试图像也将产生子模块成对潜力。结果显示在二元和多类设置中，表明模型容量的增加带来了实质性的改进。