Sufficient dimension reduction (SDR) seeks the minimal subspace of the predictors that captures the full conditional distribution of the response, which is known as the central subspace (CS). When the response is multivariate, the problem becomes considerably more challenging, particularly when the sample size is limited. Existing methods face different limitations:inverse regression approaches rely on strong distributional assumptions and matrix inversion, and their multi-response extensions suffer from severe slice sparsity; forward regression methods depend on computationally intensive iterative smoothing whose cost grows with the response dimension; and deep learning-based approaches demand large amounts of labeled data. To circumvent these shortcomings, we propose an SDR framework based on the generalized Stein's lemma. Our method constructs a cross-moment matrix between the multivariate response and the marginal score function of the predictors, and recovers the CS via its singular value decomposition. The proposed method does not rely on the linearity condition, avoids matrix inversion as well as iterative smoothing, and can leverage unlabeled data when available. We establish convergence guarantees for the proposed estimator under standard regularity conditions. Moreover, we propose a practical rank-selection algorithm to estimate the dimension of the CS. Extensive simulation studies and a real data application demonstrate that the proposed methods consistently outperform existing approaches across a variety of settings, particularly in moderate-dimensional, label-scarce scenarios with high noise levels.
Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).