In this paper we consider asymptotically exact support recovery in the context of high dimensional and sparse Canonical Correlation Analysis (CCA). Our main results describe four regimes of interest based on information theoretic and computational considerations. In regimes of "low" sparsity we describe a simple, general, and computationally easy method for support recovery, whereas in a regime of "high" sparsity, it turns out that support recovery is information theoretically impossible. For the sake of information theoretic lower bounds, our results also demonstrate a non-trivial requirement on the "minimal" size of the non-zero elements of the canonical vectors that is required for asymptotically consistent support recovery. Subsequently, the regime of "moderate" sparsity is further divided into two sub-regimes. In the lower of the two sparsity regimes, using a sharp analysis of a coordinate thresholding (Deshpande and Montanari, 2014) type method, we show that polynomial time support recovery is possible. In contrast, in the higher end of the moderate sparsity regime, appealing to the "Low Degree Polynomial" Conjecture (Kunisky et al., 2019), we provide evidence that polynomial time support recovery methods are inconsistent. Finally, we carry out numerical experiments to compare the efficacy of various methods discussed.
翻译:在本文中,我们认为,在高维和稀疏的卡纳尼相关关系分析(CCA)背景下,支持恢复是绝对准确的。我们的主要结果描述了基于信息理论和计算考虑的四种感兴趣的制度。在“低”偏差制度中,我们描述了一种简单、一般和计算上容易的恢复支持方法,而在“高”偏差制度中,我们发现支持恢复在理论上是不可能的。为了信息理论较低的范围,我们的结果还表明,对于“最小”的卡纳尼利矢量的非零成份的“最小”规模,存在着一种非三重性的要求,而对于非零度矢量的向量一致支持恢复是必要的。随后,“中等”偏差制度又被进一步分为两个子项。在“高”的“高度”制度下,在对协调临界值的精确分析(Deshpande和Montaaniri,2014年)方法方面,我们展示了多元时间支持复苏的可能性。相比之下,在中度的卡纳尼基(Syal)系统末端,我们讨论到“Colnial-alviolviolal assalalal assal assal assal assal assolal indal) violviolviolviololtial 方法。