Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

T. Tony Cai; Tiefeng Jiang; L. I. Xiaoou

doi:10.1214/21-AAP1668

Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

T. Tony Cai, Tiefeng Jiang, L. I. Xiaoou

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m × m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.

Original language	English (US)
Pages (from-to)	2953-2990
Number of pages	38
Journal	Annals of Applied Probability
Volume	31
Issue number	6
DOIs	https://doi.org/10.1214/21-AAP1668
State	Published - Dec 2021

Bibliographical note

Funding Information:
PROOF OF LEMMA 12. (i) =⇒ (ii): Easily, E[eαZp] ≤ E[eαZp1{Zp≥δ}] + eαδ. Taking lim supp→∞ on both sides and then letting δ ↓ 0, we obtain lim supp→∞ E[eαZp] ≤ 1. On the otherside,liminfp→∞E[eαZp]≥1sinceZp≥0.Hence,limp→∞E[eαZp]=1. (ii) =⇒ (i):Foreachβ>0,weknow1{Zp≥δ}≤eβ(Zp−δ).Thus, [αZ][αZ+β(Z−δ)]−βδ[(α+β)Z] Ee p1{Zp≥δ} ≤E e p p =e E e p . Taking lim supp→∞ on both sides and then letting β → ∞, we obtain limp→∞E[eαZp 1{Zp≥δ}] = 0. (ii) =⇒ (iii): First, E[Zpα] = α0∫∞xα−1P(Zp ≥ x)dx. By the Markov inequality, P(Zp≥x)≤e−βxEeβZpforallx>0andβ>0.ItfollowsthatE[Zα]≤α(EeβZp)× ∫ ∞ α−1 −βx αΓ(α) βZpp 0xedx=βαEeforallβ>0.Theconclusionthenfollowsbyfirstletting p → ∞ and then sending β → ∞. (iii) =⇒ (iv): This is a direct consequence of the Chebyshev inequality and the equality limp→∞E(Zp) = 0. (iii) =⇒ (v): Let α = 2 in (iii), then lim supp Var(Zp) ≤ limp→∞ E(Zp2) = 0. □ Funding. The research of Tony Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056. Tiefeng Jiang’s research was supported by NSF Grants DMS-1406279 and DMS-1916014. Xiaoou Li was partially supported by NSF Grant DMS-1712657.

Publisher Copyright:
© Institute of Mathematical Statistics, 2021

Keywords

Extremal eigenvalues
Maximum of random variables
Minimum of random variables
Random matrix

Publisher link

10.1214/21-AAP1668

Find It

Find It at U of M Libraries

Cite this

@article{b4437ec2dc65485d890f2d93c44880ad,

title = "Asymptotic analysis for extreme eigenvalues of principal minors of random matrices",

abstract = "Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m × m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.",

keywords = "Extremal eigenvalues, Maximum of random variables, Minimum of random variables, Random matrix",

author = "Cai, {T. Tony} and Tiefeng Jiang and Xiaoou, {L. I.}",

note = "Funding Information: PROOF OF LEMMA 12. (i) =⇒ (ii): Easily, E[eαZp] ≤ E[eαZp1{Zp≥δ}] + eαδ. Taking lim supp→∞ on both sides and then letting δ ↓ 0, we obtain lim supp→∞ E[eαZp] ≤ 1. On the otherside,liminfp→∞E[eαZp]≥1sinceZp≥0.Hence,limp→∞E[eαZp]=1. (ii) =⇒ (i):Foreachβ>0,weknow1{Zp≥δ}≤eβ(Zp−δ).Thus, [αZ][αZ+β(Z−δ)]−βδ[(α+β)Z] Ee p1{Zp≥δ} ≤E e p p =e E e p . Taking lim supp→∞ on both sides and then letting β → ∞, we obtain limp→∞E[eαZp 1{Zp≥δ}] = 0. (ii) =⇒ (iii): First, E[Zpα] = α0∫∞xα−1P(Zp ≥ x)dx. By the Markov inequality, P(Zp≥x)≤e−βxEeβZpforallx>0andβ>0.ItfollowsthatE[Zα]≤α(EeβZp)× ∫ ∞ α−1 −βx αΓ(α) βZpp 0xedx=βαEeforallβ>0.Theconclusionthenfollowsbyfirstletting p → ∞ and then sending β → ∞. (iii) =⇒ (iv): This is a direct consequence of the Chebyshev inequality and the equality limp→∞E(Zp) = 0. (iii) =⇒ (v): Let α = 2 in (iii), then lim supp Var(Zp) ≤ limp→∞ E(Zp2) = 0. □ Funding. The research of Tony Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056. Tiefeng Jiang{\textquoteright}s research was supported by NSF Grants DMS-1406279 and DMS-1916014. Xiaoou Li was partially supported by NSF Grant DMS-1712657. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2021",

year = "2021",

month = dec,

doi = "10.1214/21-AAP1668",

language = "English (US)",

volume = "31",

pages = "2953--2990",

journal = "Annals of Applied Probability",

issn = "1050-5164",

publisher = "Institute of Mathematical Statistics",

number = "6",

}

TY - JOUR

T1 - Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

AU - Cai, T. Tony

AU - Jiang, Tiefeng

AU - Xiaoou, L. I.

N1 - Funding Information: PROOF OF LEMMA 12. (i) =⇒ (ii): Easily, E[eαZp] ≤ E[eαZp1{Zp≥δ}] + eαδ. Taking lim supp→∞ on both sides and then letting δ ↓ 0, we obtain lim supp→∞ E[eαZp] ≤ 1. On the otherside,liminfp→∞E[eαZp]≥1sinceZp≥0.Hence,limp→∞E[eαZp]=1. (ii) =⇒ (i):Foreachβ>0,weknow1{Zp≥δ}≤eβ(Zp−δ).Thus, [αZ][αZ+β(Z−δ)]−βδ[(α+β)Z] Ee p1{Zp≥δ} ≤E e p p =e E e p . Taking lim supp→∞ on both sides and then letting β → ∞, we obtain limp→∞E[eαZp 1{Zp≥δ}] = 0. (ii) =⇒ (iii): First, E[Zpα] = α0∫∞xα−1P(Zp ≥ x)dx. By the Markov inequality, P(Zp≥x)≤e−βxEeβZpforallx>0andβ>0.ItfollowsthatE[Zα]≤α(EeβZp)× ∫ ∞ α−1 −βx αΓ(α) βZpp 0xedx=βαEeforallβ>0.Theconclusionthenfollowsbyfirstletting p → ∞ and then sending β → ∞. (iii) =⇒ (iv): This is a direct consequence of the Chebyshev inequality and the equality limp→∞E(Zp) = 0. (iii) =⇒ (v): Let α = 2 in (iii), then lim supp Var(Zp) ≤ limp→∞ E(Zp2) = 0. □ Funding. The research of Tony Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056. Tiefeng Jiang’s research was supported by NSF Grants DMS-1406279 and DMS-1916014. Xiaoou Li was partially supported by NSF Grant DMS-1712657. Publisher Copyright: © Institute of Mathematical Statistics, 2021

PY - 2021/12

Y1 - 2021/12

N2 - Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m × m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.

AB - Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m × m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.

KW - Extremal eigenvalues

KW - Maximum of random variables

KW - Minimum of random variables

KW - Random matrix

UR - http://www.scopus.com/inward/record.url?scp=85121292520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85121292520&partnerID=8YFLogxK

U2 - 10.1214/21-AAP1668

DO - 10.1214/21-AAP1668

M3 - Article

AN - SCOPUS:85121292520

SN - 1050-5164

VL - 31

SP - 2953

EP - 2990

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 6

ER -

Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this