An intuitive method for dimensionality reduction is proposed, which is highly effective for finding interesting projections of multivariate data. Following similar intuitive motivation to a number of existing techniques, the proposed method is based on enhancing the nearest neighbour relationships in the data. The proposed projection arises from the spectral decomposition of a matrix designed to encode the local covariance structure in the data, where the local covariance at a point is captured by pairs of its nearest neighbours. We show that under standard regularity conditions this matrix is a consistent estimator of the so-called ``Density Information Matrix'' (DIM); a non-parametric analogue of the Fisher Information Matrix. Spectral decompositions of DIMs have been shown to be connected with the important problems of Independent Components Analysis and, in the supervised context, Sufficient Dimension Reduction. However, existing estimators of the DIM are computationally expensive to compute and only target the DIM of a surrogate density, which is proportional to the square of the true underlying density. In addition, we go on to explore the practical utility of our method in aiding the downstream tasks of cluster analysis and outlier detection.
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.