In recent years, diffusion models, and more generally score-based deep generative models, have achieved remarkable success in various applications, including image and audio generation. In this paper, we view diffusion models as an implicit approach to nonparametric density estimation and study them within a statistical framework to analyze their surprising performance. A key challenge in high-dimensional statistical inference is leveraging low-dimensional structures inherent in the data to mitigate the curse of dimensionality. We assume that the underlying density exhibits a low-dimensional structure by factorizing into low-dimensional components, a property common in examples such as Bayesian networks and Markov random fields. Under suitable assumptions, we demonstrate that an implicit density estimator constructed from diffusion models adapts to the factorization structure and achieves the minimax optimal rate with respect to the total variation distance. In constructing the estimator, we design a sparse weight-sharing neural network architecture, where sparsity and weight-sharing are key features of practical architectures such as convolutional neural networks and recurrent neural networks.

Obtaining rigorous statistical guarantees for generalization under distribution shift remains an open and active research area. We study a setting we call combinatorial distribution shift, where (a) under the test- and training-distributions, the labels $z$ are determined by pairs of features $(x,y)$, (b) the training distribution has coverage of certain marginal distributions over $x$ and $y$ separately, but (c) the test distribution involves examples from a product distribution over $(x,y)$ that is {not} covered by the training distribution. Focusing on the special case where the labels are given by bilinear embeddings into a Hilbert space $H$: $\mathbb{E}[z \mid x,y ]=\langle f_{\star}(x),g_{\star}(y)\rangle_{{H}}$, we aim to extrapolate to a test distribution domain that is $not$ covered in training, i.e., achieving bilinear combinatorial extrapolation. Our setting generalizes a special case of matrix completion from missing-not-at-random data, for which all existing results require the ground-truth matrices to be either exactly low-rank, or to exhibit very sharp spectral cutoffs. In this work, we develop a series of theoretical results that enable bilinear combinatorial extrapolation under gradual spectral decay as observed in typical high-dimensional data, including novel algorithms, generalization guarantees, and linear-algebraic results. A key tool is a novel perturbation bound for the rank-$k$ singular value decomposition approximations between two matrices that depends on the relative spectral gap rather than the absolute spectral gap, a result that may be of broader independent interest.