This paper investigates new ways of inferring nonlinear dependence from measured data. The existence of unique linear and nonlinear subspaces which are structural invariants of general nonlinear mappings is established and necessary and sufficient conditions determining these sub-spaces are derived. The importance of these invariants in an identification context is that they provide a tractable framework for minimising the dimensionality of the nonlinear modelling task. Specifically, once the linear/nonlinear sub-spaces are known, by definition the explanatory variables may be transformed to form two disjoint sub-sets spanning, respectively, the linear and nonlinear sub-spaces. The nonlinear modelling task is confined to the latter sub-set, which will typically have a smaller number of elements than the original set of explanatory variables. Constructive algorithms are proposed for inferring the linear and nonlinear sub-spaces from noisy data. (c) 2006 Elsevier Ltd. All rights reserved.