Related papers: Principal Subsimplex Analysis
This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions,…
Dimensionality reduction is a fundamental technique in machine learning and data analysis, enabling efficient representation and visualization of high-dimensional data. This paper explores five key methods: Principal Component Analysis…
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are…
In this article, we introduce a procedure for selecting variables in principal components analysis. The procedure was developed to identify a small subset of the original variables that best explain the principal components through…
Compositional observations are an increasingly prevalent data source in spatial statistics. Analysis of such data is typically done on log-ratio transformations or via Dirichlet regression. However, these approaches often make unnecessarily…
Principal component analysis (PCA) has been widely applied to dimensionality reduction and data pre-processing for different applications in engineering, biology and social science. Classical PCA and its variants seek for linear projections…
One of the outstanding analytical problems in X-ray single particle imaging (SPI) is the classification of structural heterogeneity, which is especially difficult given the low signal-to-noise ratios of individual patterns and that even…
Linear dimensionality reduction techniques, notably principal component analysis, are widely used in climate data analysis as a means to aid in the interpretation of datasets of high dimensionality. These linear methods may not be…
Suppose we observe data of the form $Y_i = D_i (S_i + \varepsilon_i) \in \mathbb{R}^p$ or $Y_i = D_i S_i + \varepsilon_i \in \mathbb{R}^p$, $i=1,\ldots,n$, where $D_i \in \mathbb{R}^{p\times p}$ are known diagonal matrices, $\varepsilon_i$…
In large-scale statistical modeling, reducing data size through subsampling is essential for balancing computational efficiency and statistical accuracy. We propose a new method, Principal Component Analysis guided Quantile Sampling…
We propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis. Nonlinearity is achieved by specifying different transformation matrices at different locations of the…
Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply…
This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose…
Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction…
We provide a remedy for two concerns that have dogged the use of principal components in regression: (i) principal components are computed from the predictors alone and do not make apparent use of the response, and (ii) principal components…
The method of the lower deformation energy has been successfully used for the synthesis of mechanisms for quite a while. It has shown to be a versatile, yet powerful method for assisting in the design of mechanisms. Until now, most of the…
Despite many recent advances, reactive synthesis is still not really a practical technique. The grand challenge is to scale from small transition systems, where synthesis performs well, to complex multi-component designs. Compositional…
Sparse PCA provides a linear combination of small number of features that maximizes variance across data. Although Sparse PCA has apparent advantages compared to PCA, such as better interpretability, it is generally thought to be…
Traditional methods for the analysis of compositional data consider the log-ratios between all different pairs of variables with equal weight, typically in the form of aggregated contributions. This is not meaningful in contexts where it is…
Principal component analysis (PCA) is a foundational tool in modern data analysis, and a crucial step in PCA is selecting the number of components to keep. However, classical selection methods (e.g., scree plots, parallel analysis, etc.)…