• Corpus ID: 235458169

Noise sensitivity for the top eigenvector of a sparse random matrix

@inproceedings{Bordenave2021NoiseSF,
  title={Noise sensitivity for the top eigenvector of a sparse random matrix},
  author={Charles Bordenave and Jaehun Lee},
  year={2021}
}
We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an N × N sparse random symmetric matrix with an average of d non-zero centered entries per row. We resample k randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector v [ k ] . Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove… 

Higher order fluctuations of extremal eigenvalues of sparse random matrices

We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erdős-Rényi graphs G(N, p). Recently, it was shown that the leading order

Resampling Sensitivity of High-Dimensional PCA

It is shown that PCA is sensitive to the input data in the sense that resampling even a negligible portion of the input may completely change the output.

References

SHOWING 1-10 OF 30 REFERENCES

Eigenvector Statistics of Sparse Random Matrices

We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdős-Renyi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged

Noise sensitivity of the top eigenvector of a Wigner matrix

We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an $$N\times N$$ N × N Wigner matrix. Suppose that k randomly

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

It is proved that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble.

Local law and Tracy–Widom limit for sparse random matrices

A local law for the eigenvalue density up to the spectral edges is proved and it is proved that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included.

Largest eigenvalues of sparse inhomogeneous Erdős–Rényi graphs

We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrix

No-gaps delocalization for general random matrices

We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $${\ell_2}$$ℓ2 norm.

Tail bounds for gaps between eigenvalues of sparse random matrices

The first eigenvalue repulsion bound for sparse random matrices is proved, and it is shown that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu.

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

It is shown that for typical instances of Erdős‐Rényi random graphs G(n, p) with constant edge density p∈(0,1) , the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero.

Lectures on the local semicircle law for Wigner matrices

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent