[Paper Review] Optimal graphon estimation in cut distance

Olga Klopp, Nicolas Verzélen|arXiv (Cornell University)|Mar 15, 2017

Graph theory and applications28 references3 citations

TL;DR

This paper establishes minimax optimal estimation rates for graphons and connection probability matrices under the cut distance metric, showing that the raw adjacency matrix is already optimal for this metric—no further processing can improve convergence rates. Surprisingly, this contrasts with other metrics like $L_2$, where more sophisticated estimators can achieve better rates.

ABSTRACT

Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons. Surprisingly, our results imply that, from the minimax point of view, the raw data, that is, the adjacency matrix of the observed graph, is already optimal and more involved procedures cannot improve the convergence rates for this metric. This phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics.

Motivation & Objective

• To determine the minimax estimation rates for graphons and connection probability matrices under the cut distance metric.
• To investigate whether more complex estimation procedures can outperform the raw adjacency matrix in terms of convergence rates under the cut metric.
• To analyze the estimation of block constant matrices and step-function graphons in both dense and sparse regimes.
• To resolve the identifiability issue in graphon estimation by working within the quotient space of weakly isomorphic graphons.
• To compare the performance of the cut distance with other classical metrics such as $L_1$ and $L_2$ in the context of graphon estimation.

Proposed method

• The authors consider the $W$-random graph model with a graphon $W_0$, where edge probabilities are generated via latent variables $\xi_i \sim \text{Uniform}[0,1]$.
• They define the cut distance $\delta_1$ as a metric on the quotient space $\widetilde{\mathcal{W}}^+$ of weakly isomorphic graphons, enabling estimation of equivalence classes.
• For step-function graphons with $k$ blocks, they construct a reparameterized estimator $\widehat{f}'$ based on empirical group frequencies $\widehat{\lambda}_a$ to align the estimated and true graphons.
• They use measure-preserving maps $\psi$ to reassign latent variables so that the estimated graphon $\widehat{f}'$ is weakly isomorphic to the empirical graphon $\widetilde{f}_{\boldsymbol{\Theta}^*}$.
• The expected $L_1$ distance between $\widehat{f}'$ and $f'$ is bounded using concentration inequalities, leveraging $\mathbb{E}[|\lambda_a - \widehat{\lambda}_a|] \leq \sqrt{\lambda_a(1 - \lambda_a)/n}$.
• They derive minimax rates by bounding $\mathbb{E}_{W^*}[\|\widehat{f}' - f'\|_1] \leq C\rho_n \|W^*\|_2 \sqrt{k/n}$ for $W^* \in \mathcal{W}^+_2[k]$, and similar bounds for $\mathcal{W}^+_1[k,\mu]$.

Experimental results

Research questions

• RQ1 Is the raw adjacency matrix minimax optimal for estimating graphons under the cut distance metric?
• RQ2 How do estimation rates under the cut distance compare to those under $L_1$ or $L_2$ metrics?
• RQ3 Can more sophisticated estimators improve upon the raw adjacency matrix in terms of cut distance risk?
• RQ4 What are the minimax rates for estimating block constant matrices and step-function graphons under the cut distance?
• RQ5 How does the sparsity level $\rho_n$ affect the minimax estimation rates in the cut distance?

Key findings

• The raw adjacency matrix achieves the minimax optimal rate for graphon estimation under the cut distance metric, implying no further processing can improve convergence rates.
• For $W^* \in \mathcal{W}^+_2[k]$, the minimax rate is $\rho_n \|W^*\|_2 \sqrt{k/n}$, and this rate is unimprovable by any estimator.
• For $W^* \in \mathcal{W}^+_1[k,\mu]$, the minimax rate is $\rho_n \|W^*\|_1 \sqrt{k/(\mu n)}$, with the same optimality result.
• The cut distance metric leads to a fundamentally different minimax behavior than $L_2$ or $L_1$ metrics, where more complex estimators can outperform the raw adjacency matrix.
• The identifiability issue due to weak isomorphism is resolved by working in the quotient space $\widetilde{\mathcal{W}}^+$, and the cut distance is invariant under such transformations.
• The analysis shows that the empirical group frequencies $\widehat{\lambda}_a$ are sufficient to construct a minimax optimal estimator when combined with measure-preserving reassignment.