Motivated by the recent relevance of score-based methods in generative modeling, this thesis investigates whether polynomial score matching (SM) provides a computationally efficient alternative to established methods for log-concave density estimation. We establish theoretically that every strictly positive univariate log-concave density can be approximated arbitrarily well on sufficiently large bounded intervals by polynomial-based log-concave densities with respect to a truncated Kullback–Leibler-divergence. Motivated by this result, two polynomial SM frameworks are introduced. A univariate polynomial SM estimator, which constructs globally log-concave estimates based on Chernova (2022), and a multivariate polynomial SM estimator with log-concavity regularization on a finite grid. The performance of these estimators is systematically compared to kernel density estimation (KDE) and log-concave maximum likelihood estimation (MLE) with respect to the score mean squared error (Score MSE), the empirical Kullback–Leibler divergence (EKL), and their computational runtime, using data distributed according to Gaussian, Logistic, Gumbel, and Laplace distributions. The empirical investigations reveal that polynomial SM can be a computationally efficient alternative to log-concave MLE, particularly in lower-dimensional settings. We demonstrate that the proposed SM estimators can achieve a competitive approximation quality in terms of EKL across multiple scenarios. We thereby observe regular and irregular performance shifts with varying polynomial degree. For univariate Gaussian data, the approximation quality in the tails can be highly sensitive to samples outside the training interval. For the multivariate SM estimator, we empirically show that compared to the non-regularized estimator, a log-concavity grid regularization reduces the number of log-concavity violations on a finite test sample. Additionally, for low-dimensional data with medium-to-large sample sizes, the SM estimator achieves substantially shorter runtime than the log-concave MLE. Although the estimator still exhibits a substantial increase in runtime with growing dimension, the training size does not substantially affect the overall runtime.      «

Motivated by the recent relevance of score-based methods in generative modeling, this thesis investigates whether polynomial score matching (SM) provides a computationally efficient alternative to established methods for log-concave density estimation. We establish theoretically that every strictly positive univariate log-concave density can be approximated arbitrarily well on sufficiently large bounded intervals by polynomial-based log-concave densities with respect to a truncated Kullback–Leib...     »