mlsanityData to 7 Oct 2026

Learning Functions: When Is Deep Better Than Shallow

arXiv 1603.00988 ↗First posted 3 Mar 2016cs.LGcited by 19

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Abstract (arXiv)

While the universal approximation property holds both for hierarchical and shallow networks, we prove that deep (hierarchical) networks can approximate the class of compositional functions with the same accuracy as shallow networks but with exponentially lower number of training parameters as well as VC-dimension. This theorem settles an old conjecture by Bengio on the role of depth in networks. We then define a general class of scalable, shift-invariant algorithms to show a simple and natural set of requirements that justify deep convolutional networks.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). First posted on arXiv 3 Mar 2016.

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Citing papers per year: 2021: 7; 2022: 2; 2023: 4; 2024: 3; 2025: 2; 2026: 1.

19 papers.

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  1. 2023Loose lips sink ships: Mitigating Length Bias in Reinforcement Learning from Human Feedbackcited by 78
  2. 2021Model Complexity of Deep Learning: A Surveycited by 64
  3. 2021The staircase property: How hierarchical structure can guide deep learningcited by 62
  4. 2021Tensor Methods in Computer Vision and Deep Learningcited by 52
  5. 2024A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Modelscited by 20
  6. 2022Understanding the Evolution of Linear Regions in Deep Reinforcement Learningcited by 7
  7. 2024Deep Learning as Ricci Flowcited by 4
  8. 2021Neural Network Layer Algebra: A Framework to Measure Capacity and Compression in Deep Learningcited by 2
  9. 2023Universal Approximation Property of Hamiltonian Deep Neural Networkscited by 2
  10. 2025Noise Sensitivity and Learning Lower Bounds for Hierarchical Functionscited by 2
  11. 2025Nonlocal techniques for the analysis of deep ReLU neural network approximationscited by 2
  12. 2021Conditional Deep Gaussian Processes: empirical Bayes hyperdata learningcited by 1
7 more
  1. 2022Inference on Time Series Nonparametric Conditional Moment Restrictions Using General Sievescited by 1
  2. 2023Expressivity of Shallow and Deep Neural Networks for Polynomial Approximationcited by 1
  3. 2021Stochastic Deep Model Reference Adaptive Controlcited by 0
  4. 2021Cooperative Deep Q-learning Framework for Environments Providing Image Feedbackcited by 0
  5. 2023Deep neural network approximation of composite functions without the curse of dimensionalitycited by 0
  6. 2024Freely Long-Thinking Transformer (FraiLT)cited by 0
  7. 2026BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equationscited by 0