Score matching methods -- estimate probability densities without computing the normalization constant -- are particularly useful in deep learning. However, computational and memory costs of score matching methods can be prohibitive for high-dimensional data or complex models, particularly due to the derivatives or Hessians of the log density function appearing in the objective function. Some existing approaches modify the objective function to reduce the quadratic computational complexity for Hessian computation. However, the memory bottleneck of score matching methods remains for deep learning. This study improves the memory efficiency of score matching by leveraging deep equilibrium models. We provide a theoretical analysis of deep equilibrium models for scoring matching and applying implicit differentiation to higher-order derivatives. Empirical evaluations demonstrate that our approach enables the development of deep and expressive models with improved performance and comparable computational and memory costs over shallow architectures.

Deep nonnegative matrix factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse data sets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that ß-divergences offer a more suitable alternative. In this article, we develop new models and algorithms for deep NMF using some ß-divergences, with a focus on the Kullback-Leibler divergence. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.