Isotropic local laws for sample covariance and generalized Wigner matrices
By Alex Bloemendal, L. Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin. 2013
202 citations recorded by the source corpus.
We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v , (X^* X - z)^{-1}w\rangle - \langle v , w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v , w \in \mathbb{C}^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $\Im z \geq N^{-1+\varepsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/cd588c3a9858f77ba2b39f71ec51fdd8c193bae9
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Citations in the corpus
- Anisotropic local laws for random matrices (2014, background): “[7] Bloemendal, A; Erdős, L; Knowles, A; Yau, H-T; Yin, J, Isotropic local laws for sample covariance and generalized Wigner matrices, Electron.”
- A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, background): “z}, and if Imz < 1 then L(z) is a 1-dimensional lattice with spacing N−10 plus the point z. Obviously, we have |L(z)| ≤ N10. The following lemma is stated as Definition 5.4 of [16] and Lemma 4.5 of [5]. Lemma A.10. The self-consistent equation D 1 in (A.28) is stable on S(C 1) in the following sen”
- Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “Using a similar approach, optimal local laws for sample covariance matrices with Σ = 1 were obtained in [34]; see also [9, 16].”
- Statistical inference for principal components of spiked covariance matrices (2020, background): “We also remark here that the original isotropic local laws in [19, 48] were stated in much larger domains which also include the bulk and edge regimes of the MP law.”
- Surprises in High-Dimensional Ridgeless Least Squares Interpolation (2019, unknown): “components, will begin to look rotationally invariant in large samples, which is made precise by an isotropic local version of the Marchenko-Pastur theorem from Bloemendal et al. (2014). The proof of the next result is deferred to Appendix A.”
- A Dynamical Approach to Random Matrix Theory (2017, background): “The general case in the form above was given in [21].”
- On the principal components of sample covariance matrices (2014, methodology): “We refer to [10] for the full details of the method, and leave the modifications outlined above to the reader.”
- Singular vector and singular subspace distribution for the matrix denoising model (2018, methodology): “xed. 3.2. Isotropic local laws. The key ingredient in our estimation is a special case of the anisotropic local law derived in [37], which is essentially the isotropic local law previously derived in [12]. Set Π1(z) := m1(z)IM ⊕m2(z)In. (3.6) We will need the isotropic local law outside the spectrum”
- Eigenvector Statistics of Sparse Random Matrices (2016, background): “otropic local law for the Green’s function of Ht, which is a consequence of the small Gaussian component. The isotropic local law was first proved for Wigner matrices and sample covariance matrices in [5,26]. Our result provides a dynamical version of the entry-wise local law, with general initial ma”
- Spiked separable covariance matrices and principal components (2019, methodology): “choose the “real” spikes of Arand Br. We will use the following notion of stochastic domination, which was first introduced in [19] and subsequently used in many works on random matrix theory, such as [7, 8, 9, 20, 21, 30]. It simplifies the presentation of the results and their proofs by systematizin”