Sufficient dimension reduction is a powerful tool to extract core information hidden in the high-dimensional data and has potentially many important applications in machine learning tasks. However, the existing nonlinear sufficient dimension reduction methods often lack the scalability necessary for dealing with large-scale data. We propose a new type of stochastic neural network under a rigorous probabilistic framework and show that it can be used for sufficient dimension reduction for large-scale data. The proposed stochastic neural network is trained using an adaptive stochastic gradient Markov chain Monte Carlo algorithm, whose convergence is rigorously studied in the paper as well. Through extensive experiments on real-world classification and regression problems, we show that the proposed method compares favorably with the existing state-of-the-art sufficient dimension reduction methods and is computationally more efficient for large-scale data.

The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of \k{o}-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation.

Sufficient dimension reduction is a powerful tool to extract core information hidden in the high-dimensional data and has potentially many important applications in machine learning tasks. However, the existing nonlinear sufficient dimension reduction methods often lack the scalability necessary for dealing with large-scale data. We propose a new type of stochastic neural network under a rigorous probabilistic framework and show that it can be used for sufficient dimension reduction for large-scale data. The proposed stochastic neural network is trained using an adaptive stochastic gradient Markov chain Monte Carlo algorithm, whose convergence is rigorously studied in the paper as well. Through extensive experiments on real-world classification and regression problems, we show that the proposed method compares favorably with the existing state-of-the-art sufficient dimension reduction methods and is computationally more efficient for large-scale data.

We introduce a sufficient graphical model by applying the recently developed nonlinear sufficient dimension reduction techniques to the evaluation of conditional independence. The graphical model is nonparametric in nature, as it does not make distributional assumptions such as the Gaussian or copula Gaussian assumptions. However, unlike a fully nonparametric graphical model, which relies on the high-dimensional kernel to characterize conditional independence, our graphical model is based on conditional independence given a set of sufficient predictors with a substantially reduced dimension. In this way we avoid the curse of dimensionality that comes with a high-dimensional kernel. We develop the population-level properties, convergence rate, and variable selection consistency of our estimate. By simulation comparisons and an analysis of the DREAM 4 Challenge data set, we demonstrate that our method outperforms the existing methods when the Gaussian or copula Gaussian assumptions are violated, and its performance remains excellent in the high-dimensional setting.