无需迭代或先验知识，快速准确估计高维混合高斯模型的成分数。
Fast estimation of Gaussian mixture components via centering and singular value thresholding
- 中心化数据后，通过奇异值阈值法统计超过阈值的奇异值个数。
- 在高维、多成分、类别严重不平衡条件下仍能准确恢复真实成分数。
- 计算极快，百万级样本百维数据一分钟内完成，适合大规模数据。
估计成分数量是无监督学习中的基础挑战，尤其在高维数据且成分众多或大小严重失衡时更为突出。本文针对经典的高斯混合模型提出一种简单估算方法：对数据进行中心化，计算中心化矩阵的奇异值，并统计超过阈值的奇异值个数。该方法无需迭代拟合、无需似然计算，也无需预先知道成分数量。在成分中心满足温和分离条件下，可一致地恢复真实成分数。该结果在维度远大于样本量的高维情形下依然成立，且当成分数趋近于维度与样本量的较小值时，即使存在严重失衡也有效。计算上极为高效，例如处理一千万个百维样本仅需一分钟。大量实验验证了其在高维、多成分和严重类别不平衡等挑战性场景下的准确性。
原文摘要 · Abstract (English)
Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.
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