M-layer Transformation and Low-Energy State Accessibility at Zero-Temperature

We investigate a connectivity graph transformation in spin glasses that preserves an instance’s energy landscape features while increasing the likelihood that a zero-temperature quench locates lower-energy states. This transformation extends the M-layer construction, whose free energy function provides a Taylor expansion near the Bethe lattice approximation.

In this approach, the same sample of interaction weights is used across all M layers to correlate the M-layer transformed graph with the energy landscape of the original graph, rather than applying independent disorder across the layers. We examine how the connectivity structure between replicas in this type of M-layer transformation, influenced by the permutations used during its construction, can be controlled to affect the probability that a zero-temperature quench reaches configurations corresponding to lower-energy states in the original graph. The analysis is conducted using the 1-step Replica Symmetry Breaking (1-RSB) cavity method.

This transformation is explored in the context of overparameterization in machine learning and broader non-convex combinatorial optimization, where it enables gradient descent and other optimization algorithms to find approximate ground states.