A general theory for nonlinear sufficient dimension reduction: Formulation and estimation

Abstract

In this paper we introduce a general theory for nonlinear sufficient dimension reduction, and explore its ramifications and scope. This theory subsumes recent work employing reproducing kernel Hilbert spaces, and reveals many parallels between linear and nonlinear sufficient dimension reduction. Using these parallels we analyze the properties of existing methods and develop new ones. We begin by characterizing dimension reduction at the general level of σ-fields and proceed to that of classes of functions, leading to the notions of sufficient, complete and central dimension reduction classes. We show that, when it exists, the complete and sufficient class coincides with the central class, and can be unbiasedly and exhaustively estimated by a generalized sliced inverse regression estimator (GSIR). When completeness does not hold, this estimator captures only part of the central class. However, in these cases we show that a generalized sliced average variance estimator (GSAVE) can capture a larger portion of the class. Both estimators require no numerical optimization because they can be computed by spectral decomposition of linear operators. Finally, we compare our estimators with existing methods by simulation and on actual data sets.

Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

“…Particularly, if an algorithm converges to the SDR subspace at a faster rate than SIR, which our preliminary investigation indicates to be possible in the current numerical setting, the convergence rate of the corresponding IRUQ is automatically doubled according to Theorem 4.1. Moreover, nonlinear SDR methods such as those developed in [33] may be considered if a significant dimension reduction is not attainable via linear transformations. Finally, we point out that the IRUQ-CV algorithm can be viewed as a hybrid of the response surface and the MC methods: while the former captures the low-dimensional major trend of the model, the latter is used to handle the high-dimensional residual.…”

“…More generally, dimension reduction may be defined via a nonlinear mapping η = R(ξ ). See [32,33]. Furthermore, the QoI Y need not be a deterministic function of the input ξ : a dimension reduction η = R(ξ ) is sufficient if Y and ξ are conditionally independent given η [34,21].…”

“…See [32,33]. Furthermore, the QoI Y need not be a deterministic function of the input ξ : a dimension reduction η = R(ξ ) is sufficient if Y and ξ are conditionally independent given η [34,21]. However, in this paper we focus on the narrower sense that the relation between the QoI and the input is deterministic and the reduction function R(ξ ) is linear, which is directly relevant to our application at hand.…”

“…Moreover, any subspace that contains S( A T ) is also an SDR subspace. To achieve maximum dimension reduction, we seek the SDR subspace S 0 with lowest possible dimension, which is called the central subspace [34,35].…”

On an Additive Semigraphoid Model for Statistical Networks With Application to Pathway Analysis

“…The matrix [ T ] is called the matrix representation of T . This notation system is adopted from Horn and Johnson (1985) and was used in Li, Artemiou, and Li (2011), Li, Chun, and Zhao (2012), and Lee, Li, and Chiaromonte (2013).…”