Canonical correlation analysis (CCA) is a classical tool for learning linear associations between two data views, and inference on canonical directions is the key for interpreting their coefficients and detecting interesting relationships. We study the problem of hypothesis testing for canonical directions when paired observations are supplemented by unpaired observations of either view. The key challenge is to incorporate the additional marginal information while accounting for its dependence on the paired sample used to estimate the canonical directions. To address this issue, we propose coupled augmentation functions that jointly use the marginal information from both views and construct a one-step estimator based on the resulting corrections. Theoretically, we show that under the null, the proposed estimator is consistent and asymptotically normal even when dimension grows at the rate $d = o(n)$. This result provides a sign-invariant Wald test statistic for detecting important variables in canonical directions. Through numerical studies, we show that the proposed test statistic with semi-paired data improves upon inference using paired data.

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