Limit of the smallest eigenvalue of a large dimensional sample covariance matrix

By Z. Bai, Y. Yin. 1993

Subfields: Statistical Learning Theory, Pure Mathematics

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Initialization and Regularization of Factorized Neural Layers (2021, background): “…ensemble with scale 1 { ? n , which roughly aligns with common initialization schemes (He et al., 2015), then its singular values are roughly distributed around 1 and supported on r 0 , 2 s (Bai & Yin, 1993), and so the linear transforms will not signiﬁcantly increase the magnitude of ∇ ˆ W t .”
The limiting distributions of eigenvalues of sample correlation matrices (2004, background): “|4 oo, by Theorem 2.16 from Bai (1999) (also Theorem 3.1 from Yin et al., 1988 and Theorem 2 from Bai et al., 1993) This implies \\Dn -1\\ = maxi”
The asymptotic distributions of the largest entries of sample correlation matrices (2004, unknown): “β,β /∈B α))−P(η α >t)|, and σ(η β,β /∈B α) is the σ-algebra generated by {η β,β /∈B α}. In particular, if η α is independent of {η β,β /∈B α} for each α, then b 3 =0. The following is Lemma 2 from [4]. Lemma A.5. Let {v,v ij, i,j =1,2,...} be a double array of i.i.d. random variables and let α&gt”
A Randomized Sequential Procedure to Determine the Number of Factors (2018, background): “In the context of “classical” Random Matrix Theory, it has been shown that, under the assumptions that Xi,t is iid across i and t and that NT → c ∈ (0,∞), it holds that λ̂(p) − λ(p) =Oa.s.(1)—see Bai and Yin (1993).”
OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings (2012, methodology): “Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: i.e. we show that for any U ∈ Rn×d with orthonormal columns and random sparse Π, all singular values of ΠU lie in [1− ε, 1 + ε] with good probability.”
A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors (2002, methodology): “On the ESD Existing approaches to large-system analysis of randomly spread models generally rely on the evaluation of the ESD of large random matrices [41], [49].”
Universality of Wigner random matrices: a survey of recent results (2010, background): “[9, 69]) is that the Stieltjes transform msc(z) of the semicircle density ̺sc satisfies the following simple quadratic equation: msc(z) + 1 z +msc(z) = 0, (1.”
EFFECT OF HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM (1999, background): “However, when p is proportional to n (say p=n! y 2 (0; 1)), the ratio of the largest and the smallest eigenvalues of Sn tends to (1 + p y)2=(1 py)2 (see, e.g., references Bai, Silverstein and Yin (1988), Bai and Yin (1993), Geman (1980), Silverstein (1985) and Yin, Bai and Krishnaiah (1988)).”
Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “Moreover, the circular law, the Marchenko-Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.”
Smallest singular value of a random rectangular matrix (2009, background): “This result was proved in [23] for Gaussian matrices, and in [2] for matrices with independent and identically distributed entries with finite fourth moment.”
Convergence rates of least squares regression estimators with heavy-tailed errors (2017, background): “ 4. Consider the scaling regime d/n→ λ∈ (0,1). We claim that E|X 11|4+ε∨kξk 4,1 <∞ for some ε>0 guarantees the validity of (4.5). First, (4.3) holds under the ﬁnite fourth moment condition, see [6]. Second, (4.4) holds under the assumed moment conditions. Note that a fourth moment condition on”
Matrix concentration inequalities and free probability (2021, background): “When each |Si | = 1, that is, when X is a rectangular matrix with i.i.d. standard Gaussian entries, such a bound is well known (e.g., [15]) and is in agreement with the classical asymptotics of the smallest singular value in the proportional dimension regime d →∞, m = γ d with γ ∈ (0,1) due to Bai a”
Bounding the smallest singular value of a random matrix without concentration (2013, background): “Various attempts have been made to find conditions on X that ensure that the extremal singular values of Γ satisfy a non-asymptotic version of the classical Bai-Yin Theorem [3], formulated here.”
The LASSO Risk for Gaussian Matrices (2010, background): “A very general result was proved in [6] (see also [4]).”
A Selective Overview of Variable Selection in High Dimensional Feature Space (2009, background): “For studies on the extreme eigenvalues and limiting spectral distributions of large random matrices, see, e.g., Silverstein (1985), Bai and Yin (1993), Bai (1999), Johnstone (2001), and Ledoux (2001, 2005).”
Surprises in High-Dimensional Ridgeless Least Squares Interpolation (2019, unknown): “In the fifth line, we used ‖δi‖2 = ei (XX /n)ei ≤ smax, and in the sixth line, we used smax ≤ 2 almost surely for sufficiently large n, by the Bai-Yin theorem (Bai and Yin, 1993).”