Can an MLP Absorb Its Own Skip Connection?
This work clarifies the fundamental representational difference between residual and non-residual MLPs for practitioners designing deep architectures.
The paper investigates whether a skip connection around a single-hidden-layer MLP can be absorbed into a residual-free MLP of the same width. It finds that for most activation functions (e.g., ReLU^2, SwiGLU, GELU) absorption is impossible or non-generic, implying that skip-connected and residual-free MLPs represent generically disjoint function classes.
We study when a skip connection around a single-hidden-layer MLP can be absorbed into a residual-free MLP of the same width. We first show that for any architecture whose skip branch is an invertible linear map (including Hyper-Connections and their manifold-constrained variants), the problem reduces to the identity skip case. For homogeneous activations of degree $k \neq 1$, such as ReLU$^2$ and ReGLU, absorption is unconditionally impossible by a degree argument. For gated activations whose gate is differentiable at the origin with $g(0) = 0$, including SwiGLU and GeGLU, a linearization argument gives the same conclusion. These impossibility results extend to arbitrary depth: a composition of $L$ residual blocks using such activations cannot be replicated by any composition of $L$ residual-free blocks of the same width. For ungated ReLU and GELU, the situation is richer. For generic weight matrices, absorption holds at the single-block level if and only if there exists an index set $S$ of size at least $d$ such that $W_{\mathrm{down}}[:,S]\,W_{\mathrm{up}}[S,:] = -I_d$. This condition is non-generic (it fails with probability one under continuous weight distributions), so skip-connected and residual-free MLPs of the same width represent generically disjoint function classes. Whether this disjointness persists for deep compositions of ReLU or GELU blocks remains open.