A one-layer convolutional Ising classifier ( kernel
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · High-performance and data-driven computational physics
Scientific Computing for Physicists, Ch Neural networks on physical data: backpropagation, convolution and symmetry
Problem
A one-layer convolutional Ising classifier (3×3 kernel, periodic padding, global average, sigmoid output) is trained on disordered configurations and on ordered configurations that all have m≈+0.99, drawn at T=1.5J/kB (well below Tc), and scores 100% on held-out configurations drawn from the same distribution. Someone argues: "The convolutional layer's weights are shared across every lattice site, so the network is equivariant under translation. Spin inversion si→−si leaves the Ising Hamiltonian unchanged, so it is also a symmetry of the physical system, and therefore the trained network must automatically classify flipped configurations (m≈−0.99) as ordered too. No further test is needed." When the network is actually evaluated on held-out ordered configurations obtained by flipping every spin, it calls every one of them disordered: its mean output falls from about 0.72 on unflipped ordered configurations to about 0.33 on flipped ones, below the threshold of 21.
((a)) Identify the logical error in the argument.
((b)) State precisely what the convolution guarantees by construction, and what it does not guarantee.
((c)) Name two constructions that would fix this misclassification, and how each should be verified.