The Spectral Neuron

Researchers Alex Shtoff and colleagues introduced the spectral neuron, a scalar model f(x)=λ_k(A_0 + A_1 x + ... + A_n x_n) with learned real symmetric matrices, offering a middle ground between linear models' coefficient transparency and neural networks' expressive power. The model's nonlinearity is mathematically explicit, enabling shape control such as convexity, concavity, and monotonicity through eigenvalue and semidefinite constraints. The study characterizes expressivity, feature influence, and shape-control properties, with code available online.

Statistics Machine Learning Submitted on 8 Aug 2026 v1 https://arxiv.org/abs/2608.08003v1 , last revised 16 Aug 2026 this version, v2 Title:The Spectral Neuron View PDF /pdf/2608.08003 HTML experimental https://arxiv.org/html/2608.08003v2 Abstract:As machine learned models increase in complexity and expressive power, features of simpler models, such as intrinsic coefficient transparency and control over the shape of the modeled function are lost. On the one edge of the spectrum we have simple linear models that possess coefficient transparency, but have a limited expressive power. On the other edge we have neural networks, that have expressive power that improves with scaling, but are mostly opaque. In this work we develop the \emph{spectral neuron} concept: a scalar model given by $f x =\lambda k A 0 + A 1 x + ... + A n x n $, with learned real symmetric matrices $A 0, ..., A n$. The input enters the model through an affine matrix function, but the prediction is obtained by reading one of its eigenvalues. Thus, the model is nonlinear, but the source of nonlinearity is still mathematically explicit. This gives us a useful middle ground: the model can become more expressive as the matrix dimension grows, while retaining coefficient transparency through the learned matrices. For example, extremal eigenvalues yield convex or concave functions, semidefinite constraints on the coefficient matrices impose monotonicity, and the associated eigenspaces characterize local feature influence. We study coefficient transparency, feature-influence bounds, and shape-control properties of this model family, and then test whether it can be learned and scaled in practice. We develop a systematic study of this model family, bringing together spectral results from several mathematical literatures to characterize its expressivity, coefficient transparency, feature influence, and shape-control properties. Code available at this https URL . Submission history From: Alex Shtoff view email /show-email/791fa4c9/2608.08003 Sat, 8 Aug 2026 08:31:26 UTC 4,699 KB v1 /abs/2608.08003v1 v2 Sun, 16 Aug 2026 04:42:28 UTC 4,701 KB Current browse context: stat.ML References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .