Baokun He, Guihong Wan, Haim Schweitzer
Outlier based Robust Principal Component Analysis (RPCA) requires centering of the non-outliers. We show a "bias trick" that automatically centers these non-outliers. Using this bias trick we obtain the first RPCA algorithm that is optimal with respect to centering.
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Data analyses based on linear methods constitute the simplest, most robust, and transparent approaches to the automatic processing of large amounts of data for building supervised or unsupervised machine learning models. Principal covariates regression (PCovR) is an underappreciated method that interpolates between principal component analysis and linear regression, and can be used to conveniently reveal structure-property relations in terms of simple-to-interpret, low-dimensional maps. Here we provide a pedagogic overview of these data analysis schemes, including the use of the kernel trick t…
Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. These algorithms assume that the covariance matrix can be encoded in a density matrix, but a concrete protocol for this encoding has been lacking. Our work aims to address this gap. Assuming amplitude encoding of the data, with the data given by the ensemble $\{p_i,| ψ_i \rangle\}$, then one can easily prepare the ensemble average density matri…
Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately iden…
Implementation of the Principal Component Analysis onto High-Performance Computer Facilities for Hyperspectral Dimensionality Reduction: Results and Comparisons
本文将PCA算法实现于NVIDIA GPU和Kalray众核处理器以加速高光谱降维，并首次与基于FPGA的实现进行全面比较，揭示了各高性能计算平台的优缺点。
Quantum principal component analysis without eigenvector recovery
提出一种基于测量的软PCA框架，用熵正则化费米-狄拉克滤波器替代硬top-k投影，通过量子电路实现无需特征向量恢复的主子空间评分。
Many pattern recognition methods rely on statistical information from centered data, with the eigenanalysis of an empirical central moment, such as the covariance matrix in principal component analysis (PCA), as well as partial least squares regression, canonical-correlation analysis and Fisher discriminant analysis. Recently, many researchers advocate working on non-centered data. This is the case for instance with the singular value decomposition approach, with the (kernel) entropy component analysis, with the information-theoretic learning framework, and even with nonnegative matrix factori…
Functional data typically contains amplitude and phase variation. In many data situations, phase variation is treated as a nuisance effect and is removed during preprocessing, although it may contain valuable information. In this note, we focus on joint principal component analysis (PCA) of amplitude and phase variation. As the space of warping functions has a complex geometric structure, one key element of the analysis is transforming the warping functions to $L^2(\mathcal{T})$. We present different transformation approaches and show how they fit into a general class of transformations. This …
A data table which is arranged according to two factors can often be considered as a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information would consist of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decom…
Filtered Spectral Projection for Quantum Principal Component Analysis
本文提出滤波谱投影算法(FSPA)，通过放大初始重叠以实现主导谱子空间的投影，无需显式估计特征值，且在小间隙和近退化情况下保持鲁棒性，理论证明其最优性。
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent homology) rank functions. For a point pattern $X$ we construct a filtration of spaces by taking the union of balls of radius $a$ centered on points in $X$, $X_a = \cup_{x\in X}B(x,a)$. The rank function $β_k(X):{\{(a,b)\in \mathbb{R}^2: a\leq b\}} \to \mathbb{R}$ is then defined by $β_k(X)(a,b) = rank ( ι_*:H_k(X_a) \to H_k(X_b))$ where $ι_*$ is the induced map…
Principal component analysis in Bayes spaces for sparsely sampled density functions
本文针对每个密度仅有少量样本的稀疏观测情形，提出基于贝叶斯空间和中心化对数比变换的函数型主成分分析方法，并利用MCEM算法估计潜在密度，用于密度数据探索、降维与预处理改进。
Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${\mathbb H}$ with covariance operator $Σ:={\mathbb E}(X\otimes X).$ Let $Σ=\sum_{r\geq 1}μ_r P_r$ be the spectral decomposition of $Σ$ with distinct eigenvalues $μ_1>μ_2> \dots$ and the corresponding spectral projectors $P_1, P_2, \dots.$ Given a sample $X_1,\dots, X_n$ of size $n$ of i.i.d. copies of $X,$ the sample covariance operator is defined as $\hat Σ_n := n^{-1}\sum_{j=1}^n X_j\otimes X_j.$ The main goal of principal component analysis is to estimate spectral projectors $P_1, P_2, \dots$ by their empirical coun…
Spectrum-Aware Debiasing: A Modern Inference Framework with Applications to Principal Components Regression
本文提出了一种考虑频谱的去偏方法，用于高维回归，并应用于主成分回归，提供了首个去偏PCR估计器，同时引入了检验信号与协方差矩阵特征向量对齐的原理性检验。
Calibrated Principal Component Regression
本文提出校准主成分回归(CPCR)，通过在主成分子空间学习低方差先验并采用中心Tikhonov步骤校准模型，以缓解主成分回归在截断偏倚问题上的不足，理论和实验均显示其在过参数化设置中表现更优。
Learnable Faster Kernel-PCA for Nonlinear Fault Detection: Deep Autoencoder-Based Realization
针对 KPCA 核参数依赖试错且在线计算开销大的问题，提出基于深度自编码器的 DAE-FE/DAE-PCA 可学习快速实现，并在 TE 基准上验证其等价性、检测优势和约百倍在线加速。
Correcting Mean Bias in Text Embeddings: A Refined Renormalization with Training-Free Improvements on MMTEB
发现句子嵌入存在一致均值偏差，提出无训练修正方法R2（投影去除均值方向），在MMTEB上38个模型中获得一致分类提升，并分析其与PCA白化的差异。
PCA, SVD, and Centering of Data
本文研究数据中心化步骤对通过 SVD 计算 PCA 的影响，分析中心化与未中心化嵌入的对齐条件及其谱分析意义，强调理解 PCA 计算细节的重要性。
Pre-Warm: Initializing Convolutional Filters from First-Batch Patch Dictionaries
提出Pre-Warm方法，通过聚类训练批次中的局部图像块并利用质心初始化第一层卷积核的一半，在无额外训练成本下显著提升分类精度。
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