The purpose of this paper is twofold. First, we propose a novel algorithm for
estimating parameters in one-dimensional Gaussian mixture models (GMMs). The
algorithm takes advantage of the Hankel structure inherent in the Fourier data
obtained from independent and identically distributed (i.i.d) samples of the
mixture. For GMMs with a unified variance, a singular value ratio functional
using the Fourier data is introduced and used to resolve the variance and
component number simultaneously. The consistency of the estimator is derived.
Compared to classic algorithms such as the method of moments and the maximum
likelihood method, the proposed algorithm does not require prior knowledge of
the number of Gaussian components or good initial guesses. Numerical
experiments demonstrate its superior performance in estimation accuracy and
computational cost. Second, we reveal that there exists a fundamental limit to
the problem of estimating the number of Gaussian components or model order in
the mixture model if the number of i.i.d samples is finite. For the case of a
single variance, we show that the model order can be successfully estimated
only if the minimum separation distance between the component means exceeds a
certain threshold value and can fail if below. We derive a lower bound for this
threshold value, referred to as the computational resolution limit, in terms of
the number of i.i.d samples, the variance, and the number of Gaussian
components. Numerical experiments confirm this phase transition phenomenon in
estimating the model order. Moreover, we demonstrate that our algorithm
achieves better scores in likelihood, AIC, and BIC when compared to the EM
algorithm.

该研究提出了一种新颖的算法用于估计一维高斯混合模型的参数，利用傅里叶数据中的 Hankel 结构，解决了方差和组分数量同时的问题；同时揭示了有限独立同分布样本情况下，估计高斯混合模型的组分数量的基本极限，并验证了算法在似然度、AIC 和 BIC 等方面比 EM 算法表现更好。

一维高斯混合模型参数估计问题的傅里叶方法

A Fourier Approach to the Parameter Estimation Problem for  One-dimensional Gaussian Mixture Models

A regression model with more parameters than data points in the training data
is overparametrized and has the capability to interpolate the training data.
Based on the classical bias-variance tradeoff expressions, it is commonly
assumed that models which interpolate noisy training data are poor to
generalize. In some cases, this is not true. The best models obtained are
overparametrized and the testing error exhibits the double descent behavior as
the model order increases. In this contribution, we provide some analysis to
explain the double descent phenomenon, first reported in the machine learning
literature. We focus on interpolating models derived from the minimum norm
solution to the classical least-squares problem and also briefly discuss model
fitting using ridge regression. We derive a result based on the behavior of the
smallest singular value of the regression matrix that explains the peak
location and the double descent shape of the testing error as a function of
model order.

本篇论文提出了一个回归模型的理论，在训练数据中具有比数据点更多的参数，这种模型被称为过度参数化模型，有能力插值训练数据，最好的模型是过度参数化的，与模型阶数呈双峰形。我们分析了最小二乘问题的最小化的解的内插模型，以及使用岭回归进行模型拟合的情况。同时也提出了一个基于回归矩阵最小奇异值行为的结果，可以解释测试误差随模型阶数的峰值位置和双峰形状。