[Paper Review] The difference vectors for convex sets and a resolution of the geometry conjecture

Salihah Alwadani, Heinz H. Bauschke+2 more|arXiv (Cornell University)|Dec 8, 2020

Point processes and geometric inequalities23 references4 citations

TL;DR

This paper resolves the long-standing geometry conjecture by proving that the fixed point set of cyclic compositions of projectors onto nonempty closed convex sets in Hilbert space equals the intersection of translated sets, with the translation vectors (difference vectors) explicitly characterized via monotone operator theory and Attouch-Théra duality. The solution provides a complete, constructive formula for these vectors, resolving a conjecture open for over 25 years.

ABSTRACT

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets. In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented.

Motivation & Objective

• To resolve the geometry conjecture, which posits that the fixed point set of cyclic compositions of projectors equals an intersection of translated convex sets.
• To provide an explicit characterization of the difference vectors (translation vectors) in the conjecture, especially when fixed point sets are empty.
• To establish a connection between cycles in convex sets and monotone operator theory via product space reformulation and duality.
• To develop numerical methods for computing the difference vectors, particularly using Seeger’s algorithm for approximating projections onto the sum of a product set and a diagonal.
• To extend the framework beyond standard projectors to include underrelaxed projectors and proximal mappings in future work.

Proposed method

• Reformulate the problem in a product Hilbert space using the Cartesian product of the original sets and the diagonal subspace.
• Apply Attouch-Théra duality to relate the fixed point sets of cyclic projectors to the solution of a monotone inclusion problem in the product space.
• Use the circular right shift operator and its properties to characterize the structure of the fixed point sets and the associated difference vectors.
• Derive the difference vectors as the solution to a dual monotone inclusion problem, leveraging the inverse of a sum of operators involving the identity and the shift operator.
• Employ the forward-backward algorithm with adaptive parameters to compute the fixed point and difference vectors numerically, ensuring convergence via strong limits.
• Use Seeger’s algorithm to approximate the projection onto the set $\overline{{\mathbf{C}} + {\boldsymbol{\Delta}}}$, which is critical for numerical implementation.

Experimental results

Research questions

• RQ1 Does there exist a uniform formula for the difference vectors in the geometry conjecture, even when the fixed point sets of cyclic projectors are empty?
• RQ2 Can the geometry conjecture be resolved using tools from monotone operator theory, particularly Attouch-Théra duality?
• RQ3 What is the precise relationship between cycles in convex sets and the solution of a monotone inclusion problem in a product space?
• RQ4 How can the difference vectors be computed numerically when the fixed point sets are nonempty or empty?
• RQ5 Can the framework be extended to underrelaxed projectors or more general proximity operators?

Key findings

• The geometry conjecture is fully resolved in the affirmative: the fixed point set of the cyclic composition of projectors equals the intersection of translated convex sets, with the translation vectors explicitly determined.
• The difference vectors are given by $\mathbf{v} = (\mathbf{e} - \mathbf{y})$, where $\mathbf{e}$ and $\mathbf{y}$ are solutions to a dual monotone inclusion problem involving the circular right shift operator and the diagonal subspace.
• For $m=2$, the difference vectors are explicitly $P_{\overline{C_2 - C_1}}(0)$ and $P_{\overline{C_1 - C_2}}(0)$, confirming a known result and extending it to the general case.
• The forward-backward algorithm with $\gamma = \frac{1}{2}$ or $\gamma = \frac{m}{m+2}$ converges strongly to the solution, with the latter choice yielding a Lipschitz continuous inner operator for the algorithm.
• The solution is stable under perturbations when $m=2$, but uniqueness of difference vectors is lost when $m \geq 3$ and fixed point sets are empty.
• Numerical computation of the difference vectors is feasible via Seeger’s algorithm for approximating the projection onto $\overline{{\mathbf{C}} + {\boldsymbol{\Delta}}}$, enabling practical implementation.