  Sufficient dimension reduction (SDR) in regression, which reduces the
dimension by replacing original predictors with a minimal set of their linear
combinations without loss of information, is very helpful when the number of
predictors is large. The standard SDR methods suffer because the estimated
linear combinations usually consist of all original predictors, making it
difficult to interpret. In this paper, we propose a unified method -
coordinate-independent sparse estimation (CISE) - that can simultaneously
achieve sparse sufficient dimension reduction and screen out irrelevant and
redundant variables efficiently. CISE is subspace oriented in the sense that it
incorporates a coordinate-independent penalty term with a broad series of
model-based and model-free SDR approaches. This results in a Grassmann manifold
optimization problem and a fast algorithm is suggested. Under mild conditions,
based on manifold theories and techniques, it can be shown that CISE would
perform asymptotically as well as if the true irrelevant predictors were known,
which is referred to as the oracle property. Simulation studies and a real-data
example demonstrate the effectiveness and efficiency of the proposed approach.
