자료유형  

 학위논문

Control Number  

0017164881

International Standard Book Number  

9798346381570

Dewey Decimal Classification Number  

512.5

Main Entry-Personal Name  

Wang, Yu.

Publication, Distribution, etc. (Imprint  

[S.l.] : Stanford University., 2024

Publication, Distribution, etc. (Imprint  

Ann Arbor : ProQuest Dissertations & Theses, 2024

Physical Description  

123 p.

General Note  

Source: Dissertations Abstracts International, Volume: 86-05, Section: B.

General Note  

Advisor: Donoho, David.

Dissertation Note  

Thesis (Ph.D.)--Stanford University, 2024.

Summary, Etc.  

요약In this introductory chapter, we will set-up the framework for considering spectral-window regression methods in high dimensions.1.1 The regression problem in high-dimensionsRooted deeply in the foundations of statistical theory, linear regression stands as a generative model whose exploration has driven a significant proportion of statistics research, shaping the field throughout its history. In the standard regression problem, we observe n samples (xi , yi) ∈ R p x R. Our goal is to fit a linear model β that achieves low prediction error R(βˆ) on a new point (xnew, ynew) sampled from the same distribution{xi , yi}i≤n:Developed in the early 19th century, the method of Ordinary Least Squares (OLS) forms the cornerstone of linear modeling. This technique estimates the true predictive risk by minimizing the residual sum of squares. Consider the covariate matrix X ∈ R nxp , where each row xi represents an observation, and the response vector y ∈ R n. If Xis nonsingular, minimization of the residual sum-of-squares has a unique solution,The OLS estimator has many attractive properties - not only are there strong theoretical motivations, such as optimality under Gaussian models and being the best linear unbiased estimator, in practice OLS estimators are computationally efficient, simple to interpret, and easy to implement. Under the classical statistics regime, where p is assumed to be fixed, but ntends to ∞, the OLS estimator can consistently recover the true linear relationship, and in practice performs quite well for linear prediction tasks.However, in the current era dominated by big data, machine learning, and data science, the traditional assumptions underlying methods like OLS can often appear unrealistic. Today, we encounter datasets across all scientific disciplines where the number of predictors p is not only comparable to the number of observations n but can significantly exceed it. This is evident in various fields, from finance, where price predictions involve millions of securities, to biology, where datasets might contain millions of measurements but only thousands of samples, and even in image processing, characterized by data with millions of pixels.A large area of research in the last 60 years has focused instead on the proportional regime, where instead of assuming that p stays fixed, we now have that n, p tend to ∞ together such that the aspect ratio p/n converges to γ ∈ (0, ∞). This substantial shift in focus, which embraces the complexities of modern data, has been unified under the field of random matrix theory (RMT),providing a robust framework for understanding and analyzing high-dimensional data structures.From the seminal work of Marchenko and Pastur [1967] and generalized further in works such as Bai and Yin [1993], a key phenomenon that occurs in the proportional regime is eigenvalue spreading (see Figure A.1). Generally speaking, under loose assumptions, if X true population covariance Σ, then the eigenvalues of the empirical covariance Σb ≡ X⊤X/nwill be more spread out than the eigenvalues of the true covariance Σ. In fact, depending on the limiting aspect ratio γ, the smallest eigenvalue can become arbitrarily close to 0.

Subject Added Entry-Topical Term  

Eigenvalues.

Added Entry-Corporate Name  

Stanford University.

Host Item Entry  

Dissertations Abstracts International. 86-05B.

Electronic Location and Access  

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Control Number  

joongbu:655951