目标流形的软边分类
Soft-margin classification of object manifolds
摘要：
对单个对象的多次出现做出响应的神经群体在神经响应空间中定义了流形。分类这种流形的能力是感兴趣的，因为对象识别和其他计算任务需要对流形内的变化不敏感的响应。线性分类的对象流形以前研究了最大边缘分类。软边缘分类器是一类较大的算法，并提供了一个额外的正则化参数，用于通过在更少的训练错误和学习更鲁棒的分类器之间进行平衡来优化训练集之外的性能。在这里，我们开发了一个平均场论描述的软边缘分类器应用于对象流形的行为。随着流形分析的复杂性不断增加，从点到球面再到一般流形，平均场论描述了线性分类器范数的期望值，以及场和松弛变量的分布。通过分析学习的分类对噪声的鲁棒性，我们可以预测分类错误的概率及其对正则化的依赖性，证明了有限的最佳选择。该理论描述了一个以前未知的相变，对应于一个非平凡的解决方案的消失，从而提供了一个软版本的著名的最大利润率分类器的分类能力。
英文摘要：
A neural population responding to multiple appearances of a single object defines a manifold in the neural response space. The ability to classify such manifolds is of interest, as object recognition and other computational tasks require a response that is insensitive to variability within a manifold. Linear classification of object manifolds was previously studied for max-margin classifiers. Soft-margin classifiers are a larger class of algorithms and provide an additional regularization parameter used in applications to optimize performance outside the training set by balancing between making fewer training errors and learning more robust classifiers. Here we develop a mean-field theory describing the behavior of soft-margin classifiers applied to object manifolds. Analyzing manifolds with increasing complexity, from points through spheres to general manifolds, a mean-field theory describes the expected value of the linear classifier's norm, as well as the distribution of fields and slack variables. By analyzing the robustness of the learned classification to noise, we can predict the probability of classification errors and their dependence on regularization, demonstrating a finite optimal choice. The theory describes a previously unknown phase transition, corresponding to the disappearance of a non-trivial solution, thus providing a soft version of the well-known classification capacity of max-margin classifiers.
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