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
The paper investigates exact factorization and canonical presentations of languages relative to a fixed finite‑monoid observation. It shows that unique factorization does not guarantee a finite relative presentation property (FRP) by presenting a 36‑element quotient with infinite valid prime‑return rules, and introduces the stronger finite‑state relative presentation property (FSRP). The authors further define prime‑target left‑division determinism (PTLD), prove its implications for factorization and rule bounds, and provide efficient learning algorithms for the canonical PTLD presentation and FSRP controller.
By Takayuki Kuriyama
arXiv:2606. 09047v1 Announce Type: cross Abstract: A classical universal stabilization formula offers the practitioner no design freedom: it is a single, parameter-free object.
By Miroslav Krstic, Luke Bhan
arXiv:2608. 04879v1 Announce Type: new Abstract: Vision Transformers (ViTs) achieve strong image-recognition performance, but their parameter count grows linearly with depth when each block is independently parameterized.
By Grzegorz Gruszczynski, Pawel Olszowiec, Michal Byra, Grzegorz Stefanski, Alberto Presta
arXiv:2608. 07349v1 Announce Type: new Abstract: Learning from heterogeneous representations is usually reduced to feature concatenation, which erases which representation produced an error.
By Yao Wu
arXiv:2607. 01799v1 Announce Type: cross Abstract: Sparse autoencoders (SAEs) decompose internal activations of neural networks into sparse linear combinations of learned features by fitting an overcomplete dictionary $\mathbf{W}\in\mathbb{R}^{m\times n}$ with $m<n$, and inferring a sparse code $\mathbf{x}\in\mathbb{R}^n$ from $\mathbf{h}\approx\mathbf{W}\mathbf{x}$.
By Rodrigo Mendoza-Smith
SCAMP introduces a training‑free, damped Gauss‑Newton method that adjusts only the sparse anchor points in a frozen differentiable decoder, keeping the rest of the state unchanged. By operating solely in the space of the anchors’ Jacobian rows, it solves a system whose size matches the number of anchor constraints rather than the full state, enabling efficient control across diverse text‑to‑motion generators. Applied to seven existing generators, SCAMP achieves anchor errors that match or surpass all released control methods and can close anchors on hosts that originally lacked them.
By Pengcheng Fang, Tengjiao Sun, Xiaoyu Zhan, Yanwen Guo, Hansung Kim, Xiaohao Cai, Dongjie Fu