Nuclear-norm-based matrix completion was originally developed for imputing missing entries in low rank, or approximately low rank matrices. However, it has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the underlying matrix, as would be imposed by rank constraints. In this manuscript we show that nuclear-norm-based matrix completion attains within a log factor of the minimax rate for estimating the mean structure of matrices that are not necessarily low-rank, but lie in a low-dimensional non-linear manifold, when observations are missing completely at random. In particular, we give upper bounds on the rate of convergence as a function of the number of rows, columns, and observed entries in the matrix, as well as the smoothness and dimension of the non-linear embedding. We additionally give a minimax lower bound: This lower bound agrees with our upper bound (up to a logarithmic factor), which shows that nuclear-norm penalization is (up to log terms) minimax rate optimal for these problems.

The Hilbert-Schmidt Independence Criterion (HSIC) is a powerful kernel-based statistic for assessing the generalized dependence between two multivariate variables. However, independence testing based on the HSIC is not directly possible for cluster-correlated data. Such a correlation pattern among the observations arises in many practical situations, e.g., family-based and longitudinal data, and requires proper accommodation. Therefore, we propose a novel HSIC-based independence test to evaluate the dependence between two multivariate variables based on cluster-correlated data. Using the previously proposed empirical HSIC as our test statistic, we derive its asymptotic distribution under the null hypothesis of independence between the two variables but in the presence of sample correlation. Based on both simulation studies and real data analysis, we show that, with clustered data, our approach effectively controls type I error and has a higher statistical power than competing methods.

Graphical modeling has been broadly useful for exploring the dependence structure among features in a dataset. However, the strength of graphical modeling hinges on our ability to encode and estimate conditional dependencies. In particular, commonly used measures such as partial correlation are only meaningful under strongly parametric (in this case, multivariate Gaussian) assumptions. These assumptions are unverifiable, and there is often little reason to believe they hold in practice. In this paper, we instead consider 3 non-parametric measures of conditional dependence. These measures are meaningful without structural assumptions on the multivariate distribution of the data. In addition, we show that for 2 of these measures there are simple, strong plug-in estimators that require only the estimation of a conditional mean. These plug-in estimators (1) are asymptotically linear and non-parametrically efficient, (2) allow incorporation of flexible machine learning techniques for conditional mean estimation, and (3) enable the construction of valid Wald-type confidence intervals. In addition, by leveraging the influence function of these estimators, one can obtain intervals with simultaneous coverage guarantees for all pairs of features.