Optimisation via encodings: a renormalisation group perspective
This work addresses a fundamental bottleneck in optimization for researchers and practitioners, but it is incremental as it builds on existing cover-encoding maps.
The paper tackles the problem of rugged landscapes in NP-complete optimization problems by analyzing cover-encoding maps, which transform landscapes to remove local minima and enrich near-optimal solutions, with results showing strong similarities to renormalisation group schemes.
Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.