Sufficient dimension reduction (SDR) seeks a low-dimensional linear projection of predictors that preserves the conditional distribution of the response. Existing methods target this conditional distribution indirectly, via inverse moments, local forward regression, or neural ensemble regression. We propose FlowSDR, a likelihood-based framework that jointly learns the projection and the conditional density by maximizing a conditional log-likelihood, with the density parameterized by monotone rational-quadratic spline flows. The estimator is Fisher consistent under the SDR model, and its sample objective admits a population interpretation in terms of mutual information. As a complementary model within the same likelihood framework, we introduce the neural Gaussian SDR, a heteroscedastic conditional Gaussian model whose mean and variance are parameterized by shared neural-network functions of the projected predictors. In simulations spanning Gaussian errors, heavy-tailed distributions, two-component mixtures, and settings with tail behavior not captured by mean-variance structure, FlowSDR recovers the central subspace more accurately than existing SDR methods and the neural Gaussian SDR baseline. We further validate these advantages on a face-age prediction task using the UTKFace dataset.

翻译：充分降维（Sufficient Dimension Reduction, SDR）旨在寻找预测变量的低维线性投影，以保留响应的条件分布。现有方法通过逆矩、局部前向回归或神经集成回归间接逼近该条件分布。我们提出FlowSDR——一种基于似然的框架，通过最大化条件对数似然联合学习投影与条件密度，其中密度由单调有理二次样条流参数化。该估计量在SDR模型下具有Fisher一致性，且其样本目标函数在互信息框架下具有总体解释。作为同一似然框架内的互补模型，我们引入神经高斯SDR——一种异方差条件高斯模型，其均值和方差由共享神经网络函数基于投影预测变量参数化。在模拟实验中（涵盖高斯误差、重尾分布、双分量混合分布及均值-方差结构无法捕捉的尾部行为场景），FlowSDR相比现有SDR方法与神经高斯SDR基线更准确地恢复中心子空间。我们进一步通过UTKFace数据集的面部年龄预测任务验证了上述优势。