[Paper Review] Computing The Analytic Connectivity of A Uniform Hypergraph

Chufeng Cui, Ziyan Luo+2 more|arXiv (Cornell University)|Nov 4, 2016

Tensor decomposition and applications49 references3 citations

TL;DR

This paper proposes a feasible trust region algorithm (FTR) to compute the analytic connectivity of uniform hypergraphs, a spectral measure of hypergraph connectivity. By exploiting structural properties that localize the analytic connectivity to minimum-degree vertices, FTR reduces computation to solving fewer, structured polynomial optimization problems, achieving global convergence and superior efficiency over existing solvers like SNOPT, especially for large-scale instances.

ABSTRACT

The analytic connectivity, proposed as a substitute of the algebraic connectivity in the setting of hypergraphs, is an important quantity in spectral hypergraph theory. The definition of the analytic connectivity for a uniform hypergraph involves a series of optimization problems (POPs) associated with the Laplacian tensor of the hypergraph with nonnegativity constraints and a sphere constraint, which poses difficulties in computation. To reduce the involved computation, properties on the algebraic connectivity are further exploited, and several important structured uniform hypergraphs are shown to attain their analytic connectivities at vertices of the minimum degrees, hence admit a relatively less computation by solving a small number of POPs. To efficiently solve each involved POP, we propose a feasible trust region algorithm ({ t FTR}) by exploiting their special structures. The global convergence of { t FTR} to the second-order necessary conditions points is established, and numerical results for both small and large size examples with comparison to other existing algorithms for POPs are reported to demonstrate the efficiency of our proposed algorithm.

Motivation & Objective

• Address the lack of efficient algorithms for computing analytic connectivity in uniform hypergraphs, a key spectral measure of hypergraph connectivity.
• Reduce computational cost by identifying that analytic connectivity in certain structured hypergraphs is attained at minimum-degree vertices.
• Develop a specialized optimization algorithm tailored to the nonconvex, constrained polynomial optimization problems arising from the analytic connectivity definition.
• Establish global convergence of the proposed algorithm to second-order necessary conditions points for the underlying optimization problems.
• Demonstrate the practical feasibility of extending Cheeger-type inequalities to hypergraphs through efficient analytic connectivity computation.

Proposed method

• Formulate the analytic connectivity as a constrained polynomial optimization problem involving a Laplacian tensor, nonnegativity constraints, and a sphere constraint.
• Leverage structural symmetry and degree properties of specific hypergraphs (e.g., complete hypergraphs minus an edge) to reduce the number of optimization subproblems to solve.
• Propose a feasible trust region algorithm (FTR) that respects both the nonnegativity and equality constraints, using sequential quadratic subproblems with trust region updates.
• Ensure global convergence of FTR to second-order necessary optimality conditions by exploiting the problem's special structure and using exact Hessian approximations.
• Utilize sparse matrix techniques to efficiently solve the quadratic subproblems, taking advantage of the $O(1/n)$ sparsity in Hessian matrices for large-scale problems.
• Implement and compare FTR against SNOPT and other solvers on both small and large hypergraph instances to validate performance.

Experimental results

Research questions

• RQ1 Which vertices in structured uniform hypergraphs are most likely to achieve the analytic connectivity, enabling reduction in computational load?
• RQ2 Can a specialized optimization algorithm be designed to efficiently solve the nonconvex, constrained polynomial optimization problems arising from analytic connectivity computation?
• RQ3 How does the proposed feasible trust region algorithm (FTR) compare in convergence speed and accuracy to existing solvers like SNOPT for large-scale hypergraph instances?
• RQ4 To what extent does the analytic connectivity serve as a reliable measure of hypergraph connectivity, and can it be used to bound Cheeger numbers in hypergraphs?
• RQ5 What is the scalability and robustness of the FTR algorithm across varying hypergraph sizes and sparsity levels?

Key findings

• The analytic connectivity of several structured uniform hypergraphs, including $K_n^-$, is attained at vertices of minimum degree, reducing the number of required optimization subproblems.
• FTR achieves global convergence to second-order necessary optimality points for the constrained polynomial optimization problems involved in analytic connectivity computation.
• For large-scale problems (up to $n=500$), FTR outperforms SNOPT in computation time, requiring fewer iterations and achieving faster convergence despite high-dimensional instances.
• Numerical results show that FTR and SNOPT produce identical optimal values across all test cases, with FTR achieving a success rate above 70% and often outperforming SNOPT in both iteration count and CPU time.
• The analytic connectivity of $K_n^-$ increases monotonically with $n$, and the computed values closely match the theoretical upper bound $ar{eta} = n - 2 - rac{2}{n-1}$, confirming tightness of the bound.
• The analytic connectivity is a strong indicator of hypergraph connectivity, with high values observed in well-connected hypergraphs like $K_n^-$, validating its utility in spectral hypergraph theory.