Contravariance Theory Strong Alignment for Minimal Solutions to Hard Tasks

Jul 16, 2026

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A series of NeuroAI results over fifteen years addresses how to compare deep neural networks (DNNs) with the brain and how much convergent evolution to expect across artificial and biological networks. This work proves that for any two minimal DNN solutions to sufficiently hard tasks, weak alignment via affine mappings ensures strong alignment of privileged axes, and hierarchical “alignment zippers” drive privileged axes to emerge from end-to-end task optimization. It formalizes contravariance and shows that for strong tasks, inter-network metric choice matters less and convergence is likely inevitable.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages94

Reading Time3 h 57 min

Introduction

Goal-driven modeling and brain matching

Linear mapping and representation fit

Neuroanatomical consistency and privileged axes

arXiv :2607 .0856 1v 1 [ cs .LG] 9 Jul 2026 Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Dan Yamins*,1 and Aran Nayebi*,2 1 Departments of Computer Science and Psychology, and Wu Tsai Neurosciences Institute, Stanford University, Stanford, CA 94305 USA 2 Machine Learning Department, Neuroscience Institute, and Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213 USA * Equal contribution July 10, 2026 Abstract A series of results from the NeuroAI over the past fifteen years have raised core questions both about how to compare Deep Neural Network (DNN) models to the brain, and about how much convergent evolution to expect between artificial networks and real brain networks. Here, we show that for any two minimal DNN solutions to a sufficiently hard task: (i)“weak” alignment of network representations based on affine mappings guarantees “strong” alignment of privileged axes, and (ii) alignment “zippers” up the network hierarchy, causing the emergence of privileged axes from end-to-end task optimization. These results formalize the notion of contravariance from Cao and Yamins [2024], and illustrate important consequences for the theory of NeuroAI: with sufficiently strong tasks, choice of metric for inter-network comparison is not all that sensitive, and that convergent evolution is probably inevitable. 1 Introduction The field of NeuroAI is concerned with the ability of deep neural networks (DNNs) to model the brain [Zador et al. , 2023] . A clear statement of this goal comes in the NeuroAI Turing Test, which asks for models whose behavior and internal representations are indistinguishable from biological systems up to the variability seen across real individuals [Feather et al. , 2025 , Thobani et al. , 2025] . In one main style of NeuroAI research toward this end, so-called goal-driven modeling [Yaminsand DiCarlo, 2016], these DNN networks are optimized in an end-to-end fashion for some cognitive goal(s)– or a self-supervised proxy for such goals. Then, fixed by the constraints of the optimization process, the networks’ internal activations are compared to brain data to ask to what extent a match has organically emerged from the optimization constraints themselves. This approach has had notable success building principled, quantitatively accurate models of cortical brain areas responsible for vision [Yamins et al. , 2014 , Khaligh-Razavi and Kriegeskorte, 2014], audition [Kell et al. , 2018], somatosensation [Chung et al. , 2026], motor behavior [Sussillo et al. , 2015 , Michaels et al. , 2020], memory and navigation [Nayebi et al. , 2021], human language [Schrimpf et al. , 2021], and agentic decision making [Keller et al. , 2026] . Many of these results have been obtained by the use of linear mapping techniques [Yamins et al. , 2014], in which real units from brain data (electrodes or voxels, as the case may be) are fit, using a Figure 1: Why do these core NeuroAI results arise? What can we infer from them?(A) In a variety of domains, from visual cortex responses to static images and movies, auditory cortex responses to sounds and spoken words, to language-area responses to text, there turns out tobe a correlation across a wide range of DNN models, between a DNN model’s performance on AI tasks and fit of that model to neural data, using linear mapping transforms. What causes these correlations? (Images reproduced, from left to right, from: [Yamins et al. , 2014], [Tang et al. , 2025],[Kell et al. , 2018], and (right panel) by permission from M. Schrimpf.) (B) Within a model, there is also an emergence of neuroanatomical consistency, with model layers – such as, in the visual system, visual areas V1, V4, posterior inferior temporal cortex (PIT) and anterior interior temporal cortex (AIT)– being matched most effectively by different, and anatomically consistently ordered, DNN model layers. Why do the brain system hierarchies emerge in DNN model hierarch

FAQs

What does contravariance theory claim in this paper?

It formalizes how minimal DNN solutions to hard tasks align with brain representations through a contravariant relationship, linking weak representation alignment to stronger alignment of privileged axes and hierarchical internal structure.

How do the results relate affine mappings to privileged axes alignment?

For two minimal DNN solutions to a sufficiently hard task, weak alignment of representations based on affine mappings guarantees strong alignment of privileged axes.

Why does the paper argue that inter-network metric choice is not very sensitive for strong tasks?

With sufficiently strong tasks, the emergence of privileged axes and hierarchical alignment makes the comparison outcome robust, so the choice of metric for inter-network comparison has limited impact.

Jul 16, 2026

Upload by Ophelia