arXiv:2509. 26371v3 Announce Type: replace-cross Abstract: Recently, there has been growing interest in characterizing the function spaces underlying neural networks.
arXiv:2606. 14954v1 Announce Type: cross Abstract: We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers.
arXiv:2608. 15982v1 Announce Type: new Abstract: We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces.
arXiv:2410. 11116v4 Announce Type: replace-cross Abstract: In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces.
arXiv:2503. 24092v2 Announce Type: replace-cross Abstract: Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces.
We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces. In vector-valued Sobolev RKHSs, we derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
arXiv:2607. 05546v1 Announce Type: cross Abstract: We develop a unified function space theory of deep fully connected neural networks.
arXiv:2606. 16028v1 Announce Type: new Abstract: Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models.
arXiv:2607. 11289v1 Announce Type: cross Abstract: Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs.
arXiv:2106. 04770v2 Announce Type: replace Abstract: We study parameter nonuniqueness in continuous-width depth-two fully connected neural networks.
arXiv:2604. 14037v2 Announce Type: replace Abstract: Parameter space is not function space for neural network architectures.
Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector.
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.