Sharp threshold for alignment of graph databases with Gaussian weights

Luca Ganassali
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:314-335, 2022.

Abstract

We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where $\pi^*$ is a planted uniform permutation and all pairs of edge weights $(A_{i,j}, B_{\pi^*(i),\pi^*(j)})_{1 \leq i0$, there is an estimator $\hat{\pi}$ – namely the MAP estimator – based on the observation of databases $A,B$ that achieves exact reconstruction with high probability. Conversely, if $n \rho^2 \leq 4 \log n - \log \log n - \omega(1)$, then any estimator $\hat{\pi}$ verifies $\hat{\pi}=\pi$ with probability $o(1)$. This result shows that the information-theoretic threshold for exact recovery is the same as the one obtained for detection in a recent work by \cite{Wu20}: in other words, for Gaussian weighted graph alignment, the problem of reconstruction is not more difficult than that of detection. Though the reconstruction task was already well understood for vector-shaped database alignment (that is taking signal of the form $(u_i, v_{\pi^*(i)})_{1 \leq i\leq n}$ where $(u_i, v_{\pi^*(i)})$ are i.i.d. pairs in $\dR^{d_u} \times \dR^{d_v}$), its formulation for graph (or matrix) databases brings a drastically different problem for which the hard phase is conjectured to be wide. The proofs build upon the analysis of the MAP estimator and the second moment method, together with the study of the correlation structure of energies of permutations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-ganassali22a, title = {Sharp threshold for alignment of graph databases with Gaussian weights}, author = {Ganassali, Luca}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {314--335}, year = {2022}, editor = {Bruna, Joan and Hesthaven, Jan and Zdeborova, Lenka}, volume = {145}, series = {Proceedings of Machine Learning Research}, month = {16--19 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v145/ganassali22a/ganassali22a.pdf}, url = {https://proceedings.mlr.press/v145/ganassali22a.html}, abstract = {We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where $\pi^*$ is a planted uniform permutation and all pairs of edge weights $(A_{i,j}, B_{\pi^*(i),\pi^*(j)})_{1 \leq i0$, there is an estimator $\hat{\pi}$ – namely the MAP estimator – based on the observation of databases $A,B$ that achieves exact reconstruction with high probability. Conversely, if $n \rho^2 \leq 4 \log n - \log \log n - \omega(1)$, then any estimator $\hat{\pi}$ verifies $\hat{\pi}=\pi$ with probability $o(1)$. This result shows that the information-theoretic threshold for exact recovery is the same as the one obtained for detection in a recent work by \cite{Wu20}: in other words, for Gaussian weighted graph alignment, the problem of reconstruction is not more difficult than that of detection. Though the reconstruction task was already well understood for vector-shaped database alignment (that is taking signal of the form $(u_i, v_{\pi^*(i)})_{1 \leq i\leq n}$ where $(u_i, v_{\pi^*(i)})$ are i.i.d. pairs in $\dR^{d_u} \times \dR^{d_v}$), its formulation for graph (or matrix) databases brings a drastically different problem for which the hard phase is conjectured to be wide. The proofs build upon the analysis of the MAP estimator and the second moment method, together with the study of the correlation structure of energies of permutations. } }
Endnote
%0 Conference Paper %T Sharp threshold for alignment of graph databases with Gaussian weights %A Luca Ganassali %B Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2022 %E Joan Bruna %E Jan Hesthaven %E Lenka Zdeborova %F pmlr-v145-ganassali22a %I PMLR %P 314--335 %U https://proceedings.mlr.press/v145/ganassali22a.html %V 145 %X We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where $\pi^*$ is a planted uniform permutation and all pairs of edge weights $(A_{i,j}, B_{\pi^*(i),\pi^*(j)})_{1 \leq i0$, there is an estimator $\hat{\pi}$ – namely the MAP estimator – based on the observation of databases $A,B$ that achieves exact reconstruction with high probability. Conversely, if $n \rho^2 \leq 4 \log n - \log \log n - \omega(1)$, then any estimator $\hat{\pi}$ verifies $\hat{\pi}=\pi$ with probability $o(1)$. This result shows that the information-theoretic threshold for exact recovery is the same as the one obtained for detection in a recent work by \cite{Wu20}: in other words, for Gaussian weighted graph alignment, the problem of reconstruction is not more difficult than that of detection. Though the reconstruction task was already well understood for vector-shaped database alignment (that is taking signal of the form $(u_i, v_{\pi^*(i)})_{1 \leq i\leq n}$ where $(u_i, v_{\pi^*(i)})$ are i.i.d. pairs in $\dR^{d_u} \times \dR^{d_v}$), its formulation for graph (or matrix) databases brings a drastically different problem for which the hard phase is conjectured to be wide. The proofs build upon the analysis of the MAP estimator and the second moment method, together with the study of the correlation structure of energies of permutations.
APA
Ganassali, L.. (2022). Sharp threshold for alignment of graph databases with Gaussian weights. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:314-335 Available from https://proceedings.mlr.press/v145/ganassali22a.html.

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