The paper investigates vector‑valued binary affine refinement operators with finite matrix masks and compactly supported continuous piecewise‑linear data. It demonstrates that every finite refinement iterate can be exactly realized by a ReLU network of fixed width and depth linear in the number of iterations, using a universal reflection‑doubling mechanism that replaces two binary transition matrices with a single fixed block matrix and a swap involution. The construction allows exact branch selection via a continuous piecewise‑linear cone switch, propagates full vectorized profiles without decomposing inputs, and handles stage‑dependent forcing while reducing the doubled cascade to a single parity sector through genuine reflection equivariance.
By Boldsaikhan Bolorkhuu, Tsogtgerel Gantumur
arXiv:2608. 02624v1 Announce Type: cross Abstract: Refinement operators generate many functions used in wavelet constructions, subdivision schemes, and geometric modeling.
By Tsogtgerel Gantumur
arXiv:2607. 11897v1 Announce Type: new Abstract: Linear attention replaces softmax attention's growing KV cache with a fixed recurrent state, but this compression limits exact state tracking and long-context memory.
By Tiantian Zhang
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.
By Aleksander Tankman
arXiv:2607. 23390v1 Announce Type: new Abstract: When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map?
By Mojtaba Soltanalian
arXiv:2607. 26988v1 Announce Type: cross Abstract: What types of decision problems can a causally masked, finite-precision transformer solve for inputs of arbitrary length?
By Franz Nowak, Ryan Cotterell, Reda Boumasmoud