We study nonparametric estimation of a probability mass function (PMF) on a large discrete support, where the PMF is multi-modal and heavy-tailed. The core idea is to treat the empirical PMF as a signal on a line graph and apply a data-dependent low-pass filter. Concretely, we form a symmetric tri-diagonal operator, the path graph Laplacian perturbed with a diagonal matrix built from the empirical PMF, then compute the eigenvectors, corresponding to the smallest feq eigenvalues. Projecting the empirical PMF onto this low dimensional subspace produces a smooth, multi-modal estimate that preserves coarse structure while suppressing noise. A light post-processing step of clipping and re-normalizing yields a valid PMF. Because we compute the eigenpairs of a symmetric tridiagonal matrix, the computation is reliable and runs time and memory proportional to the support times the dimension of the desired low-dimensional supspace. We also provide a practical, data-driven rule for selecting the dimension based on an orthogonal-series risk estimate, so the method "just works" with minimal tuning. On synthetic and real heavy-tailed examples, the approach preserves coarse structure while suppressing sampling noise, compares favorably to logspline and Gaussian-KDE baselines in the intended regimes. However, it has known failure modes (e.g., abrupt discontinuities). The method is short to implement, robust across sample sizes, and suitable for automated pipelines and exploratory analysis at scale because of its reliability and speed.

翻译：本研究探讨了在大型离散支撑集上对多峰且重尾的概率质量函数（PMF）进行非参数估计的问题。其核心思想是将经验PMF视为线图上的信号，并应用一个数据依赖的低通滤波器。具体而言，我们构建一个对称的三对角算子——即由经验PMF构建的对角矩阵扰动的路径图拉普拉斯算子，然后计算对应于最小特征值的特征向量。将经验PMF投影到这个低维子空间上，可得到一个平滑、多峰的估计量，该估计量在保留粗粒度结构的同时抑制了噪声。通过一个简单的后处理步骤（截断与重归一化），即可得到一个有效的PMF。由于我们计算的是对称三对角矩阵的特征对，因此计算过程可靠，且运行时间和内存消耗与支撑集大小乘以所需低维子空间维数成正比。我们还基于正交级数风险估计，提出了一种实用的数据驱动规则用于选择子空间维度，从而使该方法在极少调参的情况下即可“直接运行”。在合成与真实的重尾数据示例中，该方法在保留粗粒度结构的同时有效抑制了采样噪声，在预期适用场景下表现优于对数样条和高斯核密度估计基线。然而，该方法也存在已知的失效模式（例如，突变的不连续性）。该方法实现代码简短，对不同样本量具有鲁棒性，且因其可靠性与速度，适用于自动化流程和大规模探索性分析。