Janis AIAD

In the lazy regime, training deep networks reduces to kernel regression, and the NTK spectrum controls convergence and stability. Low-rank random-feature (RF-LR) architectures freeze random feature maps and train only narrow readouts of dimension r ≪N per layer—reducing parameters from O(LN^2) to O(LrN) while preserving kernel behavior. We derive an explicit NTK recursion with a visible 1/r factor at each bottleneck layer, sharp depth scaling for the deterministic proxy, condition-number bounds, and in the three-layer case RKHS equivalence with the shallow ReLU kernel. Low rank does not shrink the function class.

We study low-rank neural networks with frozen random features in the mean-field regime. When the mean-field dynamics converges, the limit is shown to be a global minimizer; this holds for gradient-based training under standard i.i.d. initialization, for any depth L \ge 2, without ad-hoc init. The analysis identifies a rank-channel feature learning mechanism: different low-rank channels specialize to distinct spatial locations and progressively capture higher-frequency components.

In this communication, we aim to present the solving of an MBDA’s use case related to optimal assignment, onto IBM online QPUs. The Quantum Approximate Optimization Algorithm (QAOA) (Farhi et al. 2014) is the base of our Variational Quantum Algorithm developed. We compare two methods to account for constraints, first primarily by integrating them into the Cost Hamiltonian with Lagrangian multipliers and second, by adapting the Mixer Hamiltonian according to (Wang et al. 2022) and (Fuchs et al. 2022). For the former, determining the optimal Lagrangian multipliers is generally a challenging task and the integration of constraints into the Cost Hamiltonian can significantly increase the associated circuit depth. The latter method aims to reduce the overall Hilbert space to only feasible solutions, which lets get rid of Lagrangian multipliers but may significantly enlarge the circuit associated to the Mixer Hamiltonian and make the initial state harder. It is then interesting to compare the circuit depth of both methods with respect to how well they are able to statistically put forward optimal solutions against non-optimal and non-feasible ones, still for relatively small sized instances, to fit on current QPUs.