TL;DR
This paper introduces Reductive Lie Neurons (ReLNs), a novel neural network architecture that is exactly equivariant to general linear group symmetries, supporting matrix-valued features and Lie algebraic structures for diverse scientific applications.
Contribution
ReLNs provide the first exactly GL(n)-equivariant architecture supporting matrix and Lie algebra features, enabling broad reuse across subgroups with improved efficiency and accuracy.
Findings
ReLNs outperform existing equivariant models in various algebraic tasks.
ReLNs achieve comparable or better accuracy with fewer parameters and less compute.
ReLNs demonstrate versatility across physics, geometry, and uncertainty estimation tasks.
Abstract
Many scientific and geometric problems exhibit general linear symmetries, yet most equivariant neural networks are built for compact groups or simple vector features, limiting their reuse on matrix-valued data such as covariances, inertias, or shape tensors. We introduce Reductive Lie Neurons (ReLNs), an exactly GL(n)-equivariant architecture that natively supports matrix-valued and Lie-algebraic features. ReLNs resolve a central stability issue for reductive Lie algebras by introducing a non-degenerate adjoint (conjugation)-invariant bilinear form, enabling principled nonlinear interactions and invariant feature construction in a single architecture that transfers across subgroups without redesign. We demonstrate ReLNs on algebraic tasks with sl(3) and sp(4) symmetries, Lorentz-equivariant particle physics, uncertainty-aware drone state estimation via joint velocity-covariance…
Peer Reviews
Decision·ICLR 2026 Conference Desk Rejected Submission
1). The paper is generally well-written. I liked the introduction and Figures 1 and 2. They provide nice high-level overviews of the application landscape and the method's position in the field. \ 2). Providing a practical architecture for exact GL(n) equivariance is an original contribution (although direct applications of full GL(n)-equivariance are questionable).
1). Theoretical and conceptual contribution of the work is moderate: \ a). When the Lie algebra is semisimple, the proposed form $\tilde{B}$ almost coincides with the Killing form $B$ considered in Lie Neurons (see Section C.1). But the semisimple Lie algebras are mostly interesting for applications, and even the authors' experiments in Sections 5.1.1 and 5.1.2 consider the tasks with semisimple Lie algebras, and in the experiment in Section 5.2, the authors artificially put the O(1,3)-equivaria
1. The theoretical foundation seems solid. It resolved the Killing form degeneracy issue in a very simple way. 2. ReLN usifies previous approaches as it is a slight generalization of Lie Neurons and reduces to it on semisimple Lie algebras. 3. The drone navigation results show impressive improvement over the baseline.
1. My main issue is contribution. Beyond defining $\tilde{B}$, I don't seem to find any distinction between this work and Lie Neurons (Lin 2024a). Specifically, the Killing form on $\\mathfrak{gl}(n)$ is $B(X,Y) = 2n\\cdot \\mathrm{tr} (XY) - 2\\mathrm{tr}(X) \\mathrm{Y}$. They define $\\tilde{B}(X,Y) = \\mathrm{tr}(X) \\mathrm{Y} + B(X,Y)$. That's basically their main theoretical contribution. All the rest, including ReLN-ReLU nonlinearity and ReLN-Bracket seem to be identical to Lie Neurons,
S1. The paper's use of augmented bilinear form to generalize Lie neurons to reductive Lie algebras is original and technically sound as far as I can confirm. S2. The writing and presentation is overall clear and easy to follow.
W1. Among the four experiments presented, as far as I can understand there are no experiments that involves reductive Lie algebras with degenerate Killing form, which are the key targets of the construction given in the paper (please correct me if I am wrong). From this, I was not able to draw the conclusion that the proposed method actually solves the problem setup given in Section 4.1. W2. While the work extends the applicability of [1] to reductive Lie algebras, the extension might not be as
The proposed idea is elegant and interesting, and can be potentially impactful in some of the applications described by the authors. The manuscript is quite well written, although it provides little background on the advanced mathematical tools leveraged, which might make it less accessible to the general audience (see comments below).
While the paper is well motivated and mentions some exciting applications, I found the experimental validation less interesting and weaker. Indeed, the authors consider 4 tasks, 2 simple synthetic datasets with *semisimple* Lie Algebras in Sec 5.1, an apparently saturated Jet-Tagging benchmark with the reductive gl(n) Lie group in Sec 5.2, and (if I understood correctly) a SO(3) equivariant task in section 5.3. While these experiments seems good proof of concepts, I don't think they provide goo
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