mlsanityData to 14 Oct 2026

Complexity, Statistical Risk, and Metric Entropy of Deep Nets Using Total Path Variation

arXiv 1902.00800 ↗First posted 2 Feb 2019stat.ML cs.LGcited by 16

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Abstract (arXiv)

For any ReLU network there is a representation in which the sum of the absolute values of the weights into each node is exactly 1, and the input layer variables are multiplied by a value V coinciding with the total variation of the path weights. Implications are given for Gaussian complexity, Rademacher complexity, statistical risk, and metric entropy, all of which are shown to be proportional to V. There is no dependence on the number of nodes per layer, except for the number of inputs d. For estimation with sub-Gaussian noise, the mean square generalization error bounds that can be obtained are of order V √L + log d/√n, where L is the number of layers and n is the sample size.

Key facts. Cited by 16 papers in the map (data through 14 Oct 2026). First posted on arXiv 2 Feb 2019.

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

16 papers.

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  1. 2021What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theorycited by 60
  2. 2022Approximation bounds for norm constrained neural networks with applications to regression and GANscited by 22
  3. 2023A path-norm toolkit for modern networks: consequences, promises and challengescited by 10
  4. 2024Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networkscited by 8
  5. 2024How DNNs break the Curse of Dimensionality: Compositionality and Symmetry Learningcited by 6
  6. 2025Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNetscited by 5
  7. 2021The Rate of Convergence of Variation-Constrained Deep Neural Networkscited by 4
  8. 2021Generalization Performance of Empirical Risk Minimization on Over-parameterized Deep ReLU Netscited by 4
  9. 2023Provable Identifiability of Two-Layer ReLU Neural Networks via LASSO Regularizationcited by 4
  10. 2024A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizationscited by 4
  11. 2021Neural Network Layer Algebra: A Framework to Measure Capacity and Compression in Deep Learningcited by 2
  12. 2024A practical generalization metric for deep networks benchmarkingcited by 1
4 more
  1. 2025Non-Vacuous Generalization Bounds: Can Rescaling Invariances Help?cited by 1
  2. 2026Deep Neural Variation Spaces: A Unifying Perspective on Depth and Complexitycited by 1
  3. 2026Variation Brownian Kernel Ladderscited by 1
  4. 2025Path Regularization: A Near-Complete and Optimal Nonasymptotic Generalization Theory for Multilayer Neural Networks and Double Descent Phenomenoncited by 0