Deep Delta Learning

🤗 Upvotes: 22 | cs.LG, cs.AI, cs.CL, cs.CV

Authors:
Yifan Zhang, Yifeng Liu, Mengdi Wang, Quanquan Gu

Title:
Deep Delta Learning

Arxiv:
http://arxiv.org/abs/2601.00417v1

Abstract:
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 $\mathbf{k}(\mathbf{X})$ and a gating scalar $β(\mathbf{X})$. We provide a spectral analysis of this operator, demonstrating that the gate $β(\mathbf{X})$ 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.

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