Vol 27, No 3 (2018)

We study the problem of inference (estimation and uncertainty quantification problems) on the unknown parameter in the biclustering model by using the penalization method. The underlying biclustering structure is that the high-dimensional parameter consists of a few blocks of equal coordinates. The quality of the inference procedures is measured by the local quantity, the oracle rate, which is the best trade-off between the approximation error by a biclustering structure and the complexity of that approximating biclustering structure. The approach is also robust in that the additive errors are assumed to satisfy only certain mild condition (allowing non-iid errors with unknown joint distribution). By using the penalization method, we construct a confidence set and establish its local (oracle) optimality. Interestingly, as we demonstrate, there is (almost) no deceptiveness issue for the uncertainty quantification problem in the biclustering model. Adaptive minimax results for the biclustering, stochastic block model (with implications for network modeling) and graphon scales follow from our local results.

The paper considers a simple Errors-in-Variables (EiV) model Y_i = a + bX_i + εξ_i; Z_i= X_i + σζ_i, where ξ_i, ζ_i are i.i.d. standard Gaussian random variables, X_i ∈ ℝ are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ℝ based on the observations {Y_i, Z_i, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but \({\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2\) with some \(\epsilon_\circ^2, \sigma_\circ^2>0\). Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed.