Restricted Boltzmann Machines as Robust Generative Surrogates for Spectral-Function: An application to the t − t′ − t′′ − J model

Spectral functions encode one- and many-body excitations but are costly to compute at high fidelity. Machine-learning surrogates can offset this cost, yet many are data-hungry and generalize poorly. We explore Re- stricted Boltzmann Machines (RBMs) as generative surrogates that aim to capture latent physical structure rather than only surface-level cor- relations. As a testbed, we use self-consistent Born approximation-based spectra for the t−t′ −t′′ −J model which describes motion of a single-hole in a quantum antiferromagnet. We implement a Gaussian RBM to handle real-valued spectral intensities with 3 input nodes on the visible layer and 301 output nodes encoding the spectral intensity distribution, connected to a tunable hidden layer. Across extensive parallel sweeps, an architec- ture with 42 hidden units provides a strong balance of expressivity and overfitting resistance. Training uses persistent contrastive divergence with weight regularization and momentum, while temperature annealing at in- ference markedly improves reconstruction stability—reducing MSE by an order of magnitude versus fixed-temperature runs. On 1,000 training and 1,000 test spectra, the GRBM attains an average MSE of 5.5×10⁻³ on both splits, with stable error variance and uniform accuracy across (t′, t′′, J), indicating transferable latent representations rather than memo- rization. While absolute accuracy is moderate compared to ensemble re- gressors (e.g., random forests), preliminary evidence shows qualitatively better out-of-distribution reliability—and potentially stronger generaliza- tion than standard neural networks—highlighting RBMs as promising, data-efficient surrogates for spectral prediction.