arXiv:2608. 02624v1 Announce Type: cross Abstract: Refinement operators generate many functions used in wavelet constructions, subdivision schemes, and geometric modeling.
arXiv:2608. 09707v1 Announce Type: cross Abstract: Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research.
arXiv:2607. 07035v1 Announce Type: cross Abstract: The architecture of deep feedforward neural networks is ubiquitous in deep learning, either as a whole system or as a subnetwork of other architectures, and thus its mechanism is a key ingredient of the black box of neural networks.
arXiv:2607. 20586v1 Announce Type: new Abstract: We study vector-valued affine refinement operators of the form [ (W\gamma)(t)=\sum_{j\in\mathbb{Z}} A_j\gamma(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data.
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.
arXiv:2601. 16884v3 Announce Type: replace Abstract: We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks.
arXiv:2606. 30935v1 Announce Type: cross Abstract: While neural network control policies are powerful, their deployment on safety critical systems depends on ensuring that they obey strict constraints.
arXiv:2606. 07728v1 Announce Type: new Abstract: It is well established that ReLU networks define continuous piecewise-linear functions, and that their linear regions are polyhedra in the input space.
The paper introduces the directional linear separability measure (D‑LSM) to quantify how much linear separability is preserved or improved by injective affine maps in neural networks. It characterizes the geometry supporting D‑LSM, proves its invariance under injective affine embeddings, and derives conditions for gated activations (ReLU, GELU, SiLU) to preserve and recover samples. Experiments validate the theoretical bounds, demonstrate affine‑tube constructions that achieve guaranteed recovery, and apply the method to Vision Transformer representations to obtain early post‑activation separability certificates.
arXiv:2607. 05546v1 Announce Type: cross Abstract: We develop a unified function space theory of deep fully connected neural networks.
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:2609.25987v1 Announce Type: new Abstract: Equivariant convolutional neural networks are usually built from a group acting globally on the space of signals. This hypothesis is inappropriate for...
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.