Upper bound for smallest eigenvalue of matrix ‖ ϕ ( x ) ‖ 2 ≤...

lilmoore11p8

lilmoore11p8

Answered

2022-07-04

Upper bound for smallest eigenvalue of matrix ϕ ( x ) 2 1
I am reading a paper which claims the following. But I am not sure how to show it rigorously. Any help is appreciated.
For all x X , assume the d-dimensional feature map is bounded such that ϕ ( x ) 2 1. For any data distribution μ consider the matrix
A = E x μ [ ϕ ( x ) ϕ ( x ) ]
Prove that the largest possible minimum eigenvalue σ min min of matrix A satisfies
σ min ( A ) 1 d

Answer & Explanation

eurgylchnj

eurgylchnj

Expert

2022-07-05Added 14 answers

Since A is a d × d positive semidefinite matrix, its eigenvalues are nonnegative and they coincide with the singular values of A. Therefore
σ min ( A ) = λ min ( A ) 1 d i = 1 d λ i ( A ) = 1 d tr ( A ) = 1 d tr ( E ( ϕ ϕ ) ) = 1 d E ( tr ( ϕ ϕ ) ) = 1 d E ( ϕ 2 2 ) 1 d E ( 1 ) = 1 d .

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