Mapping Networks

Neural networks can often be trained or fine-tuned through random low-dimensional reparameterization, where a small latent vector is mapped into a full parameter update by a frozen random map. This raises a practical question: how large must the latent search space be to reach a low-loss region? We first express the known accessibility transition in an equivalent conic form, centered for compact convex targets at the statistical dimension of the polar cone. Our main theoretical contribution is an orientation-resolved quadratic master formula that predicts the random-slice residual from both the curvature spectrum and the reference-to-solution displacement profile. It yields a self-consistent isotropic-orientation predictor and, in a conservative radius-only specialization, recovers the earlier Gaussian-width quadratic bound. Building on this analysis, we introduce Random Mapping Networks (RaMaN), which instantiate the predicted latent dimension using structured Hadamard or seed-regenerated Gaussian maps. These constructions avoid the O(dP) storage of dense random maps and reduce optimizer-state memory from O(P) to O(d). We also develop matrix-free curvature approximations and sweep-free dimension selection. Across controlled quadratic and neural-curvature experiments, the orientation-resolved predictor closely tracks measured transition locations and outperforms orientation-agnostic approximations when displacement direction matters. End-to-end experiments further show sharp, protocol-dependent training transitions across image and language models.

The cost of storing and transmitting a trained neural network scales with its parameter count, a bottleneck for over-the-air updates, on-device libraries, and other bandwidth-bound deployments. We study an extreme form of model compression in which the deployable artifact is not the weights but a short recipe for regenerating them. Building on Mapping Networks, which express a network's weights as a nonlinear function of a compact trainable latent and a fixed random basis, we observe that only the latent need be stored, because the basis and initialization center are reproducible from an integer seed. A model becomes a seed together with a quantized latent, whose size is set by the latent dimension and bit width rather than the parameter count. We formalize this artifact and introduce a seeded block-wise basis that scales to networks whose projection cannot be held in memory. In our experiments, a mapped model is as accurate as the same network quantized aggressively to a few bits per weight, while taking far fewer bytes to store. Reaching the most aggressive bit widths depends on fine-tuning the latent with quantization in the loop. The results do not depend on the particular random basis, and a structured basis lets the weights be regenerated almost for free even for large networks.

We study the realization map of deep ReLU networks, focusing on when a function determines its parameters up to scaling and permutation. To analyze hidden redundancies beyond these standard symmetries, we introduce a framework based on weighted polyhedral complexes. Our main result shows that for every architecture whose input and hidden layers have width at least two, there exists an open set of identifiable parameters. This implies that the functional dimension of every such architecture is exactly the number of parameters minus the number of hidden neurons. We further show that minimal functional representations can still have non-trivial parameter redundancies. Finally, we establish a generic depth hierarchy, whereby for an open set of parameters the realized function cannot be represented generically by any shallower network.