English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the following detection problem: given a realization of asymmetric matrix ${\mathbf{X}}$ of dimension $n$, distinguish between thehypothesis that all upper triangular variables are i.i.d. Gaussians variableswith mean 0 and variance $1$ and the hypothesis where ${\mathbf{X}}$ is the sumof such matrix and an independent rank-one perturbation.  This setup applies to the situation where under the alternative, there is aplanted principal submatrix ${\mathbf{B}}$ of size $L$ for which all uppertriangular variables are i.i.d. Gaussians with mean $1$ and variance $1$,whereas all other upper triangular elements of ${\mathbf{X}}$ not in${\mathbf{B}}$ are i.i.d. Gaussians variables with mean 0 and variance $1$. Werefer to this as the `Gaussian hidden clique problem.'  When $L=(1+\epsilon)\sqrt{n}$ ($\epsilon>0$), it is possible to solve thisdetection problem with probability $1-o_n(1)$ by computing the spectrum of${\mathbf{X}}$ and considering the largest eigenvalue of ${\mathbf{X}}$. Weprove that this condition is tight in the following sense: when$L<(1-\epsilon)\sqrt{n}$ no algorithm that examines only the eigenvalues of${\mathbf{X}}$ can detect the existence of a hidden Gaussian clique, with errorprobability vanishing as $n\to\infty$.  We prove this result as an immediate consequence of a more general result onrank-one perturbations of $k$-dimensional Gaussian tensors. In this context weestablish a lower bound on the critical signal-to-noise ratio below which arank-one signal cannot be detected.

On the limitation of spectral methods: From the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors