Beyond identifiability: Learning causal representations with few environments and finite samples

We provide explicit, finite-sample guarantees for learning causal representations from data with a sublinear number of environments. Causal representation learning seeks to provide a rigourous foundation for the general representation learning problem by bridging causal models with latent factor models in order to learn interpretable representations with causal semantics. Despite a blossoming theory of identifiability in causal representation learning, estimation and finite-sample bounds are less well understood. We show that causal representations can be learned with only a logarithmic number of unknown, multi-node interventions, and that the intervention targets need not be carefully designed in advance. Through a careful perturbation analysis, we provide a new analysis of this problem that guarantees consistent recovery of (a) the latent causal graph, (b) the mixing matrix and representations, and (c) \emph{unknown} intervention targets.

Optimal Variable Clustering for High-Dimensional Matrix Valued Data

Matrix valued data has become increasingly prevalent in many applications. Most of the existing clustering methods for this type of data are tailored to the mean model and do not account for the dependence structure of the features, which can be very informative, especially in high-dimensional settings. To extract the information from the dependence structure for clustering, we propose a new latent variable model for the features arranged in matrix form, with some unknown membership matrices representing the clusters for the rows and columns. Under this model, we further propose a class of hierarchical clustering algorithms using the difference of a weighted covariance matrix as the dissimilarity measure. Theoretically, we show that under mild conditions, our algorithm attains clustering consistency in the high-dimensional setting. While this consistency result holds for our algorithm with a broad class of weighted covariance matrices, the conditions for this result depend on the choice of the weight. To investigate how the weight affects the theoretical performance of our algorithm, we establish the minimax lower bound for clustering under our latent variable model. Given these results, we identify the optimal weight in the sense that using this weight guarantees our algorithm to be minimax rate-optimal in terms of the magnitude of some cluster separation metric. The practical implementation of our algorithm with the optimal weight is also discussed. Finally, we conduct simulation studies to evaluate the finite sample performance of our algorithm and apply the method to a genomic dataset.