Identifiability of single-index models and additive-index models

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Least squares estimation in the monotone single index model (2016, methodology): “Plan and Vershynin (2013); Plan et al. (2016). There are several other alternatives which return an estimate of the index in the single index model with a non necessarily monotone ridge function, and these could also be used here.”
ON THE IDENTIFIABILITY OF ADDITIVE INDEX MODELS (2011, methodology): “To illustrate the difficulty in identifying the additive index model, Lin and Kulasekera (2007) consider the special case where all ridge functions are quadratic and conjectured that it might be necessary to require p > (2M − 1)/2 + √ (2M − 1)2 + 8M2/2 to ensure identifiability.”
Estimation for single-index and partially linear single-index integrated models (2016, methodology): “Following the existing identifiability condition, for example, Lin and Kulasekera (2007), we assume for models studied later that ‖θ0‖ = 1 and the first nonzero component of θ0 is positive.”
The EFM approach for single-index models (2011, methodology): “This is a commonly used assumption on the index parameter [see Carroll et al. (1997), Zhu and Xue (2006), Lin and Kulasekera (2007)].”