一种有效的距离度量学习对偶方法
An Efficient Dual Approach to Distance Metric Learning
摘要：
距离度量学习是机器学习中的一个重要研究方向，因为距离度量的大小会影响到许多学习方法的性能。二次马氏度量学习是解决这个问题的一种流行方法，但通常需要解决半定规划（SDP）问题，这在计算上是昂贵的。标准的边界点SDP求解器通常具有$O(D^{6.5})$的复杂度（其中$D$是输入数据的维度），并且因此实际上只能解决表现出少于几千个变量的问题。由于变量的数量是$D (D+1) / 2 $，这意味着实际上可以解决大约几百个维度的问题的大小的限制。因此，流行的二次马氏度量学习方法的复杂性限制了度量学习可以应用的问题的大小。在这里，我们提出了一个更有效的方法来度量学习问题的基础上的拉格朗日对偶制定的问题。所提出的公式更容易实现，因此允许更大的马氏度量学习问题得到解决。该方法的时间复杂度为$O (D ^ 3) $，这是显着低于SDP方法。在各种数据集上的实验表明，该方法达到了与最先进技术相当的精度，但适用于更大的问题。我们还表明，所提出的方法可以应用于解决更一般的Frobenius范数正则化SDP问题近似。
英文摘要：
Distance metric learning is of fundamental interest in machine learning because the distance metric employed can significantly affect the performance of many learning methods. Quadratic Mahalanobis metric learning is a popular approach to the problem, but typically requires solving a semidefinite programming (SDP) problem, which is computationally expensive. Standard interior-point SDP solvers typically have a complexity of $O(D^{6.5})$ (with $D$ the dimension of input data), and can thus only practically solve problems exhibiting less than a few thousand variables. Since the number of variables is $D (D+1) / 2 $, this implies a limit upon the size of problem that can practically be solved of around a few hundred dimensions. The complexity of the popular quadratic Mahalanobis metric learning approach thus limits the size of problem to which metric learning can be applied. Here we propose a significantly more efficient approach to the metric learning problem based on the Lagrange dual formulation of the problem. The proposed formulation is much simpler to implement, and therefore allows much larger Mahalanobis metric learning problems to be solved. The time complexity of the proposed method is $O (D ^ 3) $, which is significantly lower than that of the SDP approach. Experiments on a variety of datasets demonstrate that the proposed method achieves an accuracy comparable to the state-of-the-art, but is applicable to significantly larger problems. We also show that the proposed method can be applied to solve more general Frobenius-norm regularized SDP problems approximately.
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