We propose a new approach to non-parametric density estimation, that is based on regularizing a Sobolev norm of the density. This method is provably different from Kernel Density Estimation, and makes the bias of the model clear and interpretable. While there is no closed analytic form for the associated kernel, we show that one can approximate it using sampling. The optimization problem needed to determine the density is non-convex, and standard gradient methods do not perform well. However, we show that with an appropriate initialization and using natural gradients, one can obtain well performing solutions. Finally, while the approach provides unnormalized densities, which prevents the use of log-likelihood for cross validation, we show that one can instead adapt Fisher Divergence based Score Matching methods for this task. We evaluate the resulting method on the comprehensive recent Anomaly Detection benchmark suite, ADBench, and find that it ranks second best, among more than 15 algorithms.

翻译：我们提出一种非参数密度估计的新方法，该方法基于对密度函数的Sobolev范数进行正则化。这一方法在理论上显著区别于核密度估计，使得模型的偏差清晰且可解释。尽管关联核函数没有闭合解析形式，但我们证明可通过采样对其进行近似。确定密度所需的最优化问题是非凸的，标准梯度方法效果不佳。然而，我们表明，通过合适的初始化并利用自然梯度，可以获得表现良好的解。最后，尽管该方法提供的是未归一化密度，导致无法使用对数似然进行交叉验证，但我们证明可改用基于Fisher散度的评分匹配方法完成此任务。我们在最新的综合异常检测基准套件ADBench上评估了该方法，发现在超过15种算法中其排名第二。