# The Information-Theoretic Benefit of Shared Representations under Orthogonality Constraints

arXiv: https://arxiv.org/abs/2606.16028 · HTML: https://mlsanity.com/paper/2606.16028/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 14 Jun 2026.

## Builds on (most cited first, 25 of 25)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [An Overview of Multi-Task Learning in Deep Neural Networks](https://mlsanity.com/paper/1706.05098/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Phase Transitions in Rate Distortion Theory and Deep Learning](https://mlsanity.com/paper/2008.01011/) (2020)
- [Learning ReLU networks to high uniform accuracy is intractable](https://mlsanity.com/paper/2205.13531/) (2022)
- [Theory-to-Practice Gap for Neural Networks and Neural Operators](https://mlsanity.com/paper/2503.18219/) (2025)
- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [Limitations of Learning Tanh Neural Networks with Finite Precision](https://mlsanity.com/paper/2606.11104/) (2026)

## Abstract (arXiv)

Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exploiting similarity across different problems, instead of solving them individually, can significantly improve overall performance. While the generalization and sample complexity properties of multitask learning have been widely studied, the parametric complexity of joint approximation in comparison to separate approximation remains less well understood. The question is particularly relevant in modern deep learning, where models are increasingly required to satisfy structural constraints such as equivariance, conservation laws, or orthogonality. We prove lower and upper bounds on the description-length for separate and joint approximation classes, respectively, in uniform norm. We build a class of orthogonal functions by composing a shared hard feature, realized by a Rademacher-Haar wavelet series, with Sawtooth-Walsh readouts to enforce orthogonality of output coordinates. The dyadic tree structure of the Rademacher-Haar wavelet concentrates the approximation hardness in the common feature component, while the readouts act as task-specific heads. Using an information-theoretic framework, we obtain a sharp gap between the optimal approximation rates achievable by joint and separate coding. Finally, we realize this separation in a neural network model using Heaviside activations via reduction to triangle-wave approximation. Our results show that even under an orthogonality constraint joint approximation requires strictly fewer bits in compositional architectures, provided the tasks share a latent hard feature. This provides theoretical insight into the description-length-efficiency of compositional multi-output architectures and clarifies how neural networks can retain expressivity under geometric constraints.