A Geometric Analysis of PCA
What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $π/4$.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Geometry of Data
Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric con…
Topological Data AnalysisGeometric Structures and Patterns of Meaning: A PHATE Manifold Analysis of Chinese Character Embeddings
We systematically investigate geometric patterns in Chinese character embeddings using PHATE manifold analysis. Through cross-validation across seven embedding models and eight dimensionality reduction methods, we observ…
Dimensionality ReductionComputing the optimal keyboard through a geometric analysis of the English language
In the context of a group project for the course COMSW4995 002 - Geometric Data Analysis, we bring our attention to the design of fast-typing keyboards. Leveraging some geometric tools in an optimization framework allowe…
Understanding the Impact of Geometric Foundation Models on Vision-Language-Action Models
Recent work explores new opportunities at the intersection of vision-language-action models (VLAs) and geometric foundation models (GFMs) for 3D reconstruction, such as VGGT. While the resulting geometric VLAs often show…
3D ReconstructionImproving Semantic Analysis on Point Clouds via Auxiliary Supervision of Local Geometric Priors
Existing deep learning algorithms for point cloud analysis mainly concern discovering semantic patterns from global configuration of local geometries in a supervised learning manner. However, very few explore geometric p…