Single index models provide an effective dimension reduction tool in regression, especially for high dimensional data, by projecting a general multivariate predictor onto a direction vector. We propose a novel single-index model for regression models where metric space-valued random object responses are coupled with multivariate Euclidean predictors. The responses in this regression model include complex, non-Euclidean data that lie in abstract metric spaces, including covariance matrices, graph Laplacians of networks, and univariate probability distribution functions. While Fr\'echet regression has proved useful for modeling the conditional mean of such random objects given multivariate Euclidean vectors, it does not provide for regression parameters such as slopes or intercepts, since the metric space-valued responses are not amenable to linear operations. As a consequence, distributional results for Fr\'echet regression have been elusive. We show here that for the case of multivariate Euclidean predictors, the parameters that define a single index and projection vector can be used to substitute for the inherent absence of parameters in Fr\'echet regression. Specifically, we derive the asymptotic distribution of suitable estimates of these parameters, which then can be utilized to test linear hypotheses for the parameters, subject to an identifiability condition. We demonstrate the finite sample performance of estimation and inference for the proposed single index Fr\'echet regression model through simulation studies. The method is illustrated for resting-state functional Magnetic Resonance Imaging (fMRI) data from the ADNI study.

翻译：单指标模型通过将多变量预测变量投影到方向向量上，在回归分析中提供了有效的降维工具，尤其适用于高维数据。我们针对回归模型提出了一种新型单指标模型，其中度量空间取值的随机对象响应与多变量欧几里得预测变量相耦合。该回归模型中的响应包含位于抽象度量空间中的复杂非欧几里得数据，例如协方差矩阵、网络图拉普拉斯算子及单变量概率分布函数。尽管弗雷歇回归已被证明能有效建模此类随机对象在给定多变量欧几里得向量条件下的条件均值，但由于度量空间取值响应无法进行线性运算，该模型未能提供斜率或截距等回归参数。因此，弗雷歇回归的分布结果一直难以获得。本文表明，对于多变量欧几里得预测变量的情况，定义单指标和投影向量的参数可用于替代弗雷歇回归中固有的参数缺失问题。具体而言，我们推导了这些参数合适估计量的渐近分布，进而可在可识别性约束条件下检验关于参数的线性假设。通过模拟研究，我们展示了所提出的单指标弗雷歇回归模型在估计与推断中的有限样本性能。该方法在ADNI研究的静息态功能磁共振成像数据上进行了验证。