We study nonparametric maximum likelihood estimation of probability densities under a total variation (TV) type penalty, sectional variation norm (also named as Hardy-Krause variation). TV regularization has a long history in regression and density estimation, including results on $L^2$ and KL divergence convergence rates. Here, we revisit this task using the Highly Adaptive Lasso (HAL) framework. We formulate a HAL-based maximum likelihood estimator (HAL-MLE) using the log-spline link function from \citet{kooperberg1992logspline}, and show that in the univariate setting the bounded sectional variation norm assumption underlying HAL coincides with the classical bounded TV assumption. This equivalence directly connects HAL-MLE to existing TV-penalized approaches such as local adaptive splines \citep{mammen1997locally}. We establish three new theoretical results: (i) the univariate HAL-MLE is asymptotically linear, (ii) it admits pointwise asymptotic normality, and (iii) it achieves uniform convergence at rate $n^{-(k+1)/(2k+3)}$ up to logarithmic factors for the smoothness order $k \geq 1$. These results extend existing results from \citet{van2017uniform}, which previously guaranteed only uniform consistency without rates when $k=0$. We will include the uniform convergence for general dimension $d$ in the follow-up work of this paper. The intention of this paper is to provide a unified framework for the TV-penalized density estimation methods, and to connect the HAL-MLE to the existing TV-penalized methods in the univariate case, despite that the general HAL-MLE is defined for multivariate cases.

翻译：我们研究了在总变差（TV）型惩罚项（即截面变差范数，亦称 Hardy-Krause 变差）下概率密度的非参数极大似然估计。TV 正则化在回归和密度估计中有着悠久的历史，包括关于 $L^2$ 和 KL 散度收敛速率的结果。本文中，我们利用高度自适应 Lasso（HAL）框架重新审视这一任务。我们使用 \citet{kooperberg1992logspline} 中的对数样条连接函数构建了一个基于 HAL 的极大似然估计量（HAL-MLE），并证明在单变量设定下，HAL 所基于的有界截面变差范数假设与经典的有界 TV 假设相一致。这一等价性直接将 HAL-MLE 与现有的 TV 惩罚方法（如局部自适应样条 \citep{mammen1997locally}）联系起来。我们建立了三个新的理论结果：（i）单变量 HAL-MLE 是渐近线性的；（ii）它允许逐点渐近正态性；（iii）对于光滑阶 $k \geq 1$，它实现了高达对数因子的 $n^{-(k+1)/(2k+3)}$ 速率的一致收敛。这些结果扩展了 \citet{van2017uniform} 的现有结论，后者先前仅在 $k=0$ 时保证了无速率的一致相合性。我们将在本文的后续工作中包含一般维度 $d$ 下的一致收敛性。本文旨在为 TV 惩罚密度估计方法提供一个统一的框架，并将 HAL-MLE 与单变量情形下现有的 TV 惩罚方法联系起来，尽管一般的 HAL-MLE 是针对多变量情形定义的。