多视点降维的张量典型相关分析
Tensor Canonical Correlation Analysis for Multi-view Dimension Reduction
摘要：
典型相关分析（CCA）因其深厚的理论基础和成功的实际应用而被证明是一种有效的两视图降维工具。然而，在多视图学习方面，它仅限于处理由两个视图特征表示的数据的能力，而在许多现实世界的应用中，视图的数量往往更多。虽然同时探索所有可能的特征对的特设方式可以在数值上处理多视图数据，但它忽略了只能通过同时探索所有特征来发现的高阶顺序统计量（相关信息）。因此，在这项工作中，我们开发了张量CCA（TCCA），它直接而自然地推广CCA处理任意数量的视图的数据，通过分析不同视图的协方差张量。TCCA旨在直接最大化多个（两个以上）视图的典型相关性。至关重要的是，我们证明了多视图的典型相关最大化问题相当于找到最好的秩1近似的数据协方差张量，这可以有效地解决使用著名的交替最小二乘（ALS）算法。因此，包含在不同的视图中的高阶相关信息被探索，从而可以获得一个更可靠的公共子空间共享的所有功能。此外，TCCA的非线性扩展。在大规模生物特征结构预测、网络广告分类和网络图像标注等挑战性任务上的实验结果表明了该方法的有效性。
英文摘要：
Canonical correlation analysis (CCA) has proven an effective tool for two-view dimension reduction due to its profound theoretical foundation and success in practical applications. In respect of multi-view learning, however, it is limited by its capability of only handling data represented by two-view features, while in many real-world applications, the number of views is frequently many more. Although the ad hoc way of simultaneously exploring all possible pairs of features can numerically deal with multi-view data, it ignores the high order statistics (correlation information) which can only be discovered by simultaneously exploring all features. Therefore, in this work, we develop tensor CCA (TCCA) which straightforwardly yet naturally generalizes CCA to handle the data of an arbitrary number of views by analyzing the covariance tensor of the different views. TCCA aims to directly maximize the canonical correlation of multiple (more than two) views. Crucially, we prove that the multi-view canonical correlation maximization problem is equivalent to finding the best rank-1 approximation of the data covariance tensor, which can be solved efficiently using the well-known alternating least squares (ALS) algorithm. As a consequence, the high order correlation information contained in the different views is explored and thus a more reliable common subspace shared by all features can be obtained. In addition, a non-linear extension of TCCA is presented. Experiments on various challenge tasks, including large scale biometric structure prediction, internet advertisement classification and web image annotation, demonstrate the effectiveness of the proposed method.
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