Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs

Chakrabarty, Arijit; Chakraborty, Sukrit; Hazra, Rajat Subhra

doi:10.48550/arxiv.1911.08244

Cited by 2 publications

(2 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3. The inverse curvature 1/K r equals the variance in the central limit theorem derived in [11]. This is in line with the standard folklore of large deviation theory.…”

Section: Putsupporting

confidence: 84%

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

Chakrabarty,

Hazra,

Hollander

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We consider an inhomogeneous Erdős-Rényi random graph GN with vertex set [N ] = {1, . . . , N } for which the pair of vertices i, j ∈ [N ], i = j, is connected by an edge with probability r( i N , j N ), independently of other pairs of vertices. Here, r : [0, 1] 2 → (0, 1) is a symmetric function that plays the role of a reference graphon. Let λN be the maximal eigenvalue of the adjacency matrix of GN . It is known that λN /N satisfies a large deviation principle as N → ∞. The associated rate function ψr is given by a variational formula that involves the rate function Ir of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of ψr, specially when the reference graphon is of rank 1.

show abstract

“…3. The inverse curvature 1/K r equals the variance in the central limit theorem derived in [11]. This is in line with the standard folklore of large deviation theory.…”

Section: Putsupporting

confidence: 84%

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

Chakrabarty,

Hazra,

Hollander

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Gaussian weights on the edges ( [55]); here, the underlying matrix P is thus drawn from GOE, and does not meet the usual assumptions of matrix completion. We finally mention a significant result on inhomogeneous Erdős-Rényi graphs by Chakrabarty, Chakraborty and Hazra [21] complementing [10].…”

Section: Related Workmentioning

confidence: 87%

Detection thresholds in very sparse matrix completion

Bordenave¹,

Coste²,

Nadakuditi³

2020

Preprint

View full text Add to dashboard Cite

Let A be a rectangular matrix of size m ˆn and A1 be the random matrix where each entry of A is multiplied by an independent t0, 1u-Bernoulli random variable with parameter 1{2. This paper is about when, how and why the non-Hermitian eigen-spectra of the matrices A1pA ´A1q ˚and pA ´A1q ˚A1 captures more of the relevant information about the principal component structure of A than the eigen-spectra of AA ˚and A ˚A.We illustrate the application of this striking phenomenon on the matrix completion problem for the setting where the underlying matrix P is low rank, with incoherent singular vectors, and where the matrix A is equal to the matrix P on a (uniformly) random subset of entries of size dn and all other entries of A are equal to zero. We show that the eigenvalues of the asymmetric matrices A1pA´A1q ˚and pA ´A1q ˚A1 with modulus greater than a detection threshold are asymptotically equal to the eigenvalues of P P ˚and P ˚P and that the associated eigenvectors are aligned as well. The central surprise is that by intentionally inducing asymmetry and additional randomness via the A1 matrix, we can extract more information than if we had worked with the singular value decomposition (SVD) of A!The associated detection threshold is asymptotically exact and is non-universal since it explicitly depends on the element-wise distribution of the underlying matrix P . We show that reliable, statistically optimal but not perfect matrix recovery, via a universal data-driven algorithm, is possible above this detection threshold using the information extracted from the asymmetric eigen-decompositions. Averaging the left and right eigenvectors provably improves estimation accuracy but not the detection threshold. Our results encompass the very sparse regime where d is of order 1 where matrix completion via the SVD of A fails or produces unreliable recovery.We define another variant of this asymmetric principal component analysis procedure that bypasses the randomization step and has a detection threshold that is smaller by a constant factor but with a computational cost that is larger by a polynomial factor of the number of observed entries. Both detection thresholds shatter the seeming barrier due to the well-known information theoretical limit dlog n for matrix completion found in the literature.

show abstract

Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs

Cited by 2 publications

References 20 publications

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

Detection thresholds in very sparse matrix completion

Contact Info

Product

Resources

About