Abstract: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.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2604.23705 [cs.LG] |
| (or arXiv:2604.23705v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.23705 arXiv-issued DOI via DataCite (pending registration) |
From: Antonij Mijoski [view email]
[v1]
Sun, 26 Apr 2026 13:37:27 UTC (20 KB)
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