High-dimensional datasets present substantial challenges in statistical
modeling across various disciplines, necessitating effective dimensionality
reduction methods. Deep learning approaches, notable for their capacity to
distill essential features from complex data, facilitate modeling,
visualization, and compression through reduced dimensionality latent feature
spaces, have wide applications from bioinformatics to earth sciences. This
study introduces a novel workflow to evaluate the stability of these latent
spaces, ensuring consistency and reliability in subsequent analyses. Stability,
defined as the invariance of latent spaces to minor data, training
realizations, and parameter perturbations, is crucial yet often overlooked.
Our proposed methodology delineates three stability types, sample,
structural, and inferential, within latent spaces, and introduces a suite of
metrics for comprehensive evaluation. We implement this workflow across 500
autoencoder realizations and three datasets, encompassing both synthetic and
real-world scenarios to explain latent space dynamics. Employing k-means
clustering and the modified Jonker-Volgenant algorithm for class alignment,
alongside anisotropy metrics and convex hull analysis, we introduce adjusted
stress and Jaccard dissimilarity as novel stability indicators.
Our findings highlight inherent instabilities in latent feature spaces and
demonstrate the workflow's efficacy in quantifying and interpreting these
instabilities. This work advances the understanding of latent feature spaces,
promoting improved model interpretability and quality control for more informed
decision-making for diverse analytical workflows that leverage deep learning.

高维数据集在各个学科的统计建模中带来了很大的挑战，需要高效的降维方法。深度学习方法通过降维的潜在特征空间从复杂数据中提取关键特征，有着广泛的应用，从生物信息学到地球科学等领域。本研究通过引入一个新的工作流程来评估这些潜在空间的稳定性，确保后续分析的一致性和可靠性。该工作流程囊括了三种稳定性类型：样本稳定性、结构稳定性和推断稳定性，并引入了一套综合评估指标。研究结果揭示了潜在特征空间固有的不稳定性，并证明了该工作流程在量化和解释这些不稳定性方面的功效。这项工作推进了对潜在特征空间的理解，促进了深度学习在各种分析工作流中的模型可解释性和质量控制，为更加明智的决策提供了基础。

评估深度学习潜在特征空间的稳定性

Evaluating the Stability of Deep Learning Latent Feature Spaces

Dimensionality reduction methods, such as principal component analysis (PCA)
and factor analysis, are central to many problems in data science. There are,
however, serious and well-understood challenges to finding robust low
dimensional approximations for data with significant heteroskedastic noise.
This paper introduces a relaxed version of Minimum Trace Factor Analysis
(MTFA), a convex optimization method with roots dating back to the work of
Ledermann in 1940. This relaxation is particularly effective at not overfitting
to heteroskedastic perturbations and addresses the commonly cited Heywood cases
in factor analysis and the recently identified "curse of ill-conditioning" for
existing spectral methods. We provide theoretical guarantees on the accuracy of
the resulting low rank subspace and the convergence rate of the proposed
algorithm to compute that matrix. We develop a number of interesting
connections to existing methods, including HeteroPCA, Lasso, and Soft-Impute,
to fill an important gap in the already large literature on low rank matrix
estimation. Numerical experiments benchmark our results against several recent
proposals for dealing with heteroskedastic noise.

本文介绍了 Minimum Trace Factor Analysis（MTFA）的放松版本，该方法对于具有显著异方差噪声的数据不会过度拟合，并解决了因子分析中常见的 Heywood 案例以及现有光谱方法中最近发现的 “病态条件的诅咒”。我们提供了有关所得低秩子空间的准确性以及计算该矩阵的拟议算法的收敛速率的理论保证。同时，我们发展了与现有方法（包括 HeteroPCA，Lasso 和 Soft-Impute）之间的一些有趣联系，以填补已经很大的低秩矩阵估计文献中的重要缺口。通过数值实验证明了我们的结果与用于处理异方差噪声的几个最近的提议进行了基准对比。

关于最小迹因子分析的研究 —— 一首换了调的老歌

On Minimum Trace Factor Analysis - An Old Song Sung to a New Tune

We study the fairness of dimensionality reduction methods for
recommendations. We focus on the established method of principal component
analysis (PCA), which identifies latent components and produces a low-rank
approximation via the leading components while discarding the trailing
components. Prior works have defined notions of "fair PCA"; however, these
definitions do not answer the following question: what makes PCA unfair? We
identify two underlying mechanisms of PCA that induce unfairness at the item
level. The first negatively impacts less popular items, due to the fact that
less popular items rely on trailing latent components to recover their values.
The second negatively impacts the highly popular items, since the leading PCA
components specialize in individual popular items instead of capturing
similarities between items. To address these issues, we develop a
polynomial-time algorithm, Item-Weighted PCA, a modification of PCA that uses
item-specific weights in the objective. On a stylized class of matrices, we
prove that Item-Weighted PCA using a specific set of weights minimizes a
popularity-normalized error metric. Our evaluations on real-world datasets show
that Item-Weighted PCA not only improves overall recommendation quality by up
to $0.1$ item-level AUC-ROC but also improves on both popular and less popular
items.

降维方法的公平性研究，重点关注主成分分析（PCA），通过识别潜在组件并通过前导组件生成低秩近似来拟合数据，我们发现 PCA 引起项目级不公平的两个机制，并提出使用具有项目特定权重的目标优化的修改后 PCA 算法，称为 Item-Weighted PCA，并通过实验证明其提高了推荐质量。