Understanding when learning is statistically possible yet computationally hard is a central challenge in high-dimensional statistics. In this work, we investigate this question in the context of single- and multi-index models, classes of functions widely studied as benchmarks to probe the ability of machine learning methods to discover features in high-dimensional data. Our main contribution is to show that a Noise Sensitivity Exponent (NSE) - a simple quantity determined by the activation function - governs the existence and magnitude of statistical-to-computational gaps within a broad regime of these models. We first establish that, in single-index models with large additive noise, the onset of a computational bottleneck is fully characterized by the NSE. We then demonstrate that the same exponent controls a statistical-computational gap in the specialization transition of large separable multi-index models, where individual components become learnable. Finally, in hierarchical multi-index models, we show that the NSE governs the optimal computational rate in which different directions are sequentially learned. Taken together, our results identify the NSE as a unifying property linking noise robustness, computational hardness, and feature specialization in high-dimensional learning.

翻译：理解学习在统计上可能但在计算上困难是高维统计学中的一个核心挑战。在这项工作中，我们在单索引和多索引模型的背景下研究这一问题，这些函数类被广泛用作基准，以探究机器学习方法在高维数据中发现特征的能力。我们的主要贡献是证明噪声敏感指数（NSE）——一个由激活函数决定的简单量——在这些模型的广泛范围内控制着统计-计算差距的存在与大小。我们首先表明，在具有大加性噪声的单索引模型中，计算瓶颈的出现完全由NSE刻画。然后，我们证明同一指数控制着大型可分离多索引模型中专门化转变的统计-计算差距，其中单个分量变得可学习。最后，在层次化多索引模型中，我们表明NSE决定了不同方向被顺序学习的最优计算速率。综上所述，我们的结果将NSE识别为一种统一属性，将高维学习中的噪声鲁棒性、计算难度和特征专门化联系起来。