GENERATIVE MODEL VIA QUANTILE ASSIGNMENT

Learning complex high-dimensional distributions is a challenging problem, and modern generative models address it through a variety of architectures and training objectives, often involving auxiliary learned components such as encoders, discriminators, or critics. We introduce neuroSQL, a generative modeling approach whose only learned component is a single neural generator. neuroSQL fixes a lattice of multivariate quantile points in latent space and trains the generator by alternating between a regression step and an assignment step that matches data points to latent quantiles. Under squared Euclidean cost, exact assignment yields the optimal transport coupling between the empirical data measure and the generator's pushforward of the quantile lattice, so the training objective is exactly their squared Wasserstein-2 distance. This explicit objective makes the optimization amenable to analysis: under suitable smoothness and step-size conditions, we prove that the alternating procedure decreases the objective monotonically and reaches a stationary point of it at rate $O(T^{-1/2})$ — a guarantee unavailable to adversarial training, which has no single potential function, or to variational training, which is monotone only on a bound. The resulting algorithm is lightweight, fast, and deterministic given the lattice. Controlled two-dimensional experiments reach held-out sliced Wasserstein distances comparable to an independent real–real reference. A million-observation simulation confirms that block assignment keeps cost-matrix memory bounded and solver cost linear in $n$. When trained on all 60,000 MNIST training images using block assignment, neuroSQL recovers all ten classes with no exact training-image copies, and is competitive with matched-backbone VAE and GLO baselines at lower training cost. Our guarantees are finite-sample optimization guarantees rather than population-consistency results, and within-class coverage remains the method's clearest limitation.