Deep Delta Learning
The efficacy of deep residual networks is fundamentally predicated on the identity shortcut connection. While this mechanism effectively mitigates the vanishing gradient problem, it imposes a strictly additive inductive bias on feature transformations, thereby limiting the network's capacity to model complex state transitions. In this paper, we introduce Deep Delta Learning (DDL), a novel architecture that generalizes the standard residual connection by modulating the identity shortcut with a learnable, data-dependent geometric transformation. This transformation, termed the Delta Operator, constitutes a rank-1 perturbation of the identity matrix, parameterized by a reflection direction vector k(X) and a gating scalar β(X). We provide a spectral analysis of this operator, demonstrating that the gate β(X) enables dynamic interpolation between identity mapping, orthogonal projection, and geometric reflection. Furthermore, we restructure the residual update as a synchronous rank-1 injection, where the gate acts as a dynamic step size governing both the erasure of old information and the writing of new features. This unification empowers the network to explicitly control the spectrum of its layer-wise transition operator, enabling the modeling of complex, non-monotonic dynamics while preserving the stable training characteristics of gated residual architectures.
695b2238832867f253525d62
Yifan Zhang
claimed_verified
2026-01-05T13:49:28.701Z
647bf082aba7062fe5c51ca9
https://cdn-avatars.huggingface.co/v1/production/uploads/647bf082aba7062fe5c51ca9/VvKAhQC_LxBcBuy3XROSX.jpeg
Yifan Zhang
yifAI
user
yifAI
695b2238832867f253525d63
Yifeng Liu
claimed_verified
2026-01-05T08:27:25.264Z
653d276681f52ceb4d12bd85
/avatars/56601a25e5f883a8f6dc15f6fd9dcc57.svg
Yifeng Liu
Lewis-Lau
user
Lewis-Lau
695b2238832867f253525d64
Mengdi Wang
695b2238832867f253525d65
Quanquan Gu
Deep Delta Learning
The efficacy of deep residual networks is fundamentally predicated on the identity shortcut connection. While this mechanism effectively mitigates the vanishing gradient problem, it imposes a strictly additive inductive bias on feature transformations, thereby limiting the network's capacity to model complex state transitions. In this paper, we introduce Deep Delta Learning (DDL), a novel architecture that generalizes the standard residual connection by modulating the identity shortcut with a learnable, data-dependent geometric transformation. This transformation, termed the Delta Operator, constitutes a rank-1 perturbation of the identity matrix, parameterized by a reflection direction vector k(X) and a gating scalar β(X). We provide a spectral analysis of this operator, demonstrating that the gate β(X) enables dynamic interpolation between identity mapping, orthogonal projection, and geometric reflection. Furthermore, we restructure the residual update as a synchronous rank-1 injection, where the gate acts as a dynamic step size governing both the erasure of old information and the writing of new features. This unification empowers the network to explicitly control the spectrum of its layer-wise transition operator, enabling the modeling of complex, non-monotonic dynamics while preserving the stable training characteristics of gated residual architectures.
Deep Delta Learning introduces a novel residual connection mechanism that uses a learnable geometric transformation to generalize identity shortcuts, enabling more flexible feature modeling while maintaining stable training properties.
deep residual networks identity shortcut connection vanishing gradient problem residual connection Delta Operator rank-1 perturbation reflection direction vector gating scalar spectral analysis geometric reflection synchronous rank-1 injection layer-wise transition operator non-monotonic dynamics gated residual architectures