The paper investigates how to balance approximation accuracy and stability in deep spline superposition networks under a strict layerwise Lipschitz budget. It provides an exact solution to the finite‑depth diagonal balancing problem, shows how to construct spline discretisations that respect the budget, and establishes minimax lower bounds for operators constrained in both first and third derivative norms. The authors also demonstrate that layer errors can accumulate linearly with depth, indicating that the upper bound is not merely a theoretical artifact.
arXiv:2607. 07778v1 Announce Type: new Abstract: Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights.
arXiv:2607. 06781v1 Announce Type: new Abstract: In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations.
arXiv:2607. 21651v1 Announce Type: new Abstract: We prove that the maximum of $n$ real numbers is exactly representable by a ReLU network with two hidden layers for every $n\le 10$.
arXiv:2608.31157v1 Announce Type: new Abstract: Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture...
In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error.
The paper proves that several decision and approximation problems for ReLU neural networks are computationally hard. For any number of layers λ≥2, deciding whether a network’s output is positive (and thus whether it is surjective) is W[ℓ−1]-hard when parameterized by the input dimension d. In particular, for two-layer networks, the related geometric problem of zonotope non‑containment is W[1]-hard in the ambient dimension, and computing or approximating the Lp‑Lipschitz constant is NP‑hard and W[ℓ−1]-hard with respect to d. The results also show that these problems remain hard when parameterized by the number of layers for constant d, implying that naive enumeration algorithms running in n^{(ℓ−1)d}·poly(N) time are essentially optimal under the Exponential Time Hypothesis.
arXiv:2609. 21523v2 Announce Type: replace Abstract: A state may need to be compressed before its exact downstream task is known.
arXiv:2608.24865v1 Announce Type: cross Abstract: Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult e...
arXiv:2609. 25710v1 Announce Type: cross Abstract: The statistical accuracy of neural networks depends on both their approximation power and the complexity of the class fitted from data.
arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.
arXiv:2609.06430v1 Announce Type: new Abstract: We study the identity straight-through estimator (STE) for training a two-layer binary-activation network with hinge loss from the perspective of Stati...
arXiv:2607. 14576v1 Announce Type: new Abstract: We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].