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Deep Neural Variation Spaces: A Unifying Perspective on Depth and Complexity

T0 review · 0 major / 6 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Under a true function-space norm, depth barely enlarges the class of representable ReLU functions and forbids high frequencies along every line.

desk verdict Clean Banach-norm theory for deep nets that actually saturates depth for univariate ReLU and forces frequency control under true function-space cost. read the letter →

arxiv 2607.05546 v1 pith:7HRXTWAE submitted 2026-07-06 stat.ML cs.LGmath.FA

Julia Nakhleh , Robert D. Nowak This is my paper
classification stat.MLcs.LGmath.FA MSC 68T0741A4646E3562G05
keywords deepneuralnetworksvariationspacespathnormdepthsaturationmetricentropyRademachercomplexityReLUfunctionspacenorms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a single recursive variation space for deep fully connected networks that works for both homogeneous and non-homogeneous activations. Functions are measured by how large an absolutely convex combination of activated previous-layer functions they need, so the norm is a genuine Banach norm on the scalar output rather than an additive layer-wise parameter cost. With that control in place, the unit balls remain small at every depth: their Rademacher complexities and metric entropies grow only mildly with depth. In one dimension for ReLU the story is sharper still: every deep unit ball is trapped between the shallow unit ball and twice that ball. Consequently any multivariate function whose variation norm is bounded cannot oscillate rapidly along any straight line. The authors conclude that many famous “depth-separation” examples rely on hidden layer-wise rescaling that a true function norm forbids.

What carries the argument

Deep neural variation spaces VL whose unit balls are the closed absolute convex hulls of normalized activations of the previous layer; the associated path-norm upper bound and the auxiliary functional I that is non-increasing under the positive-part map.

What would settle it

Exhibit a continuous univariate function whose second-derivative total variation exceeds 2 yet still lies in some deep unit ball BL for L greater than or equal to 3 under the paper’s exact choice of domain and dictionary, or prove that no such function exists.

Watch

Extended reading notes

Core claim

Once network complexity is measured by a genuine function-space variation norm rather than width or sum-of-squared weights, depth alone does not create high-frequency or highly oscillatory functions; in the univariate ReLU case it merely multiplies the shallow unit ball by a depth-independent factor of two.

Load-bearing premise

The sharp depth-saturation constant of two is proved only for the compact interval [-1,1] with first-layer weights and biases also restricted to [-1,1]; the same geometry may not hold for other domains or dictionaries.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper introduces deep neural variation spaces VL whose unit balls BL are defined recursively as closed absolutely convex hulls of normalized activations applied to the previous layer, starting from a compact affine dictionary B1. This yields a Banach function-space norm compatible with both homogeneous and many non-homogeneous activations (ReLU, GELU, SiLU, etc.), recovers the path norm / generalized Barron spaces for ReLU, and supplies a representer theorem for norm-penalized data fitting. The authors prove Rademacher and Lp metric-entropy bounds on BL that grow at most mildly with depth, compare the norm to SOSW and several additive layerwise costs, and establish a univariate ReLU depth-saturation result B2 ⊂ BL ⊂ 2B2 (Theorem 4.3) on Ω = W = B = [−1, 1]. As a corollary, multivariate functions in BL cannot exhibit second-derivative total variation larger than 2 along any line through the origin, so high-frequency behavior along individual directions is forbidden under this norm.

Significance. If the results hold, the paper supplies a clean, activation-general function-space language that separates genuine compositional nonlinearity from compounded layerwise rescaling, and shows that several widely cited depth-expressivity phenomena (Telgarsky sawteeth, high-frequency depth separations) rely on the latter once complexity is measured by a true function norm. The univariate saturation theorem and the line-restriction frequency-control corollary are sharp structural statements with complete proofs; the representer theorem and the mild depth dependence of the complexity bounds are useful for regularization and generalization analyses. Complete appendix proofs (Riesz–Markov, Banach–Alaoglu, Ledoux–Talagrand, BV theory) and explicit recovery of prior path-norm / Barron constructions are clear strengths.

minor comments (6)
  1. Abstract and §1: the phrase “some commonly cited expressivity benefits of depth disappear” is accurate for the VL norm, but a short clarifying sentence that the claim is relative to true function norms (not width or SOSW) would reduce possible misreading against the classical depth-separation literature.
  2. Table 1 and Lemma 2.7: the depth-dependent equivalence constants for SELU, absolute value, and bent identity grow exponentially; a one-line remark in the main text that these bases remain modest for typical L would help practitioners.
  3. Theorem 2.8: the width bound Kℓ ≤ N^{L−ℓ} is correctly stated as possibly improvable; citing the tighter N-per-layer bounds of Parhi–Nowak / Shenouda et al. more explicitly in the main text (rather than only in the comparison paragraph) would orient the reader.
  4. Figure 2 / Remark 4.4: the sharpness example fα is clear; adding the numerical V2 values of (fα)+ for the three plotted α would make the approach to the constant 2 more immediate.
  5. §3.2.1 and Appendix A.15: the ℓ1-pyramid example in B3 \ B2 is useful; a brief pointer in the main text that this is currently the main explicit multivariate witness of strict depth increase would strengthen the open-problems discussion.
  6. Notation: the dual use of BL for both the unit ball and (in places) the depth index is mostly clear from context, but a single sentence at the start of §2.2 fixing “BL always denotes the unit ball of VL” would avoid occasional ambiguity.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: definitions are gauge norms of recursive atomic sets; theorems derive consequences without fitted inputs or self-referential predictions.

  1. self citation load bearing [Section 2.4 / Lemma 2.6 and surrounding discussion]
    "Theorem 2.6 shows that, in the ReLU case, our deep spaces VL are equivalent (again, up to possible differences in first-layer normalization) to the generalized Barron space (or neural tree space) of Wojtowytsch et al. (2020)."

    Minor recovery of a prior construction (Wojtowytsch et al.) as a special case of the new spaces. The citation is not used to justify any novel claim; the paper supplies independent proofs of integral representations, representer theorems, and complexity bounds that apply more broadly. Not load-bearing for the depth-saturation or frequency-control results.

full rationale

The paper constructs VL as Banach spaces whose unit balls BL are closed absolutely convex hulls of normalized activations applied to the previous layer (Eqs. 4-6), with base dictionary B1 of affine functions. All subsequent results (integral representations Lemma 2.6, representer Theorem 2.8, Rademacher/entropy bounds Theorems 3.1/3.3, depth saturation Theorem 4.3, frequency control Corollary 4.8) are derived from this definition by standard functional-analytic arguments (Riesz-Markov, Banach-Alaoglu, contraction inequalities, BV characterizations). Self-citations recover known special cases (path norm, generalized Barron spaces) as instances of the new construction rather than load-bearing premises. No parameters are fitted to data and then re-predicted; no uniqueness theorem is imported to force the main claim; the functional I of Lemma 4.2 is explicitly constructed and verified for the stated geometry. The derivation is therefore self-contained against its own definitions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper works entirely inside classical Banach-space and approximation theory. No free parameters are fitted; the only modeling choices are the recursive gauge-norm construction and the concrete compact sets W, B, Ω used for the saturation theorem. Invented entities are the spaces themselves, which are defined explicitly and shown to recover known objects in special cases.

assumptions (3)
  • standard math Standard facts about absolutely convex hulls, gauge norms, Riesz–Markov–Kakutani, Banach–Alaoglu, and BV spaces on intervals
    Used throughout Sections 2–4 and the appendix to define VL and prove integral representations and representer theorems.
  • domain assumption Local Lipschitz continuity and the listed limiting atoms σ0, σ∞ for the activations in Table 2
    Required for the normalized-activation construction and for the depth-dependent equivalence constants of Lemma 2.7.
  • domain assumption Compactness of the domain Ω and of the weight/bias sets W, B
    Guarantees that the unit balls BL are compact in C(Ω) (Proposition 2.1) and that integral representations are well-defined.
invented entities (1)
  • Deep neural variation spaces VL and their unit balls BL independent evidence
    purpose: Provide a single Banach-norm complexity measure for deep networks with general activations
    Defined recursively via normalized activations; shown to recover path norms and generalized Barron spaces for ReLU, and to be equivalent for many practical activations.

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Pith. "Pith review of Deep Neural Variation Spaces: A Unifying Perspective on Depth and Complexity." pith.science (2026). https://pith.science/paper/7HRXTWAE

@misc{pith2026260705546,
  author       = {Pith},
  title        = {Pith review of: Deep Neural Variation Spaces: A Unifying Perspective on Depth and Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HRXTWAE}},
  note         = {Machine review of arXiv:2607.05546}
}
abstract

We develop a unified function space theory of deep fully connected neural networks. Functions in our spaces are defined recursively as $\ell^1$-bounded linear combinations of activated functions from preceding layers, with a dictionary of affine functions at the first layer. Unlike existing theories that are largely specialized to homogeneous activations such as the ReLU, our framework provides a meaningful notion of functional complexity for deep networks with a broad range of homogeneous and non-homogeneous activation functions commonly used in practice. This simple construction unites several seemingly disparate ideas from the literature, including norm-based complexity bounds and variational characterizations of depth, and facilitates novel analyses of what kinds of functions deep norm-constrained networks can represent. To this end, we prove a novel representer theorem for our spaces and establish novel function-space complexity bounds showing that the associated function classes remain qualitatively small at arbitrary depth. In the univariate ReLU case, we prove a "depth saturation" result: depth in this setting yields only a small constant rescaling of the function class, with no added functional diversity. As a consequence, we show that deep norm-controlled ReLU functions in any dimension cannot exhibit high frequencies along any direction. This finding reveals that some commonly cited expressivity benefits of depth disappear once network complexity is controlled by an appropriate function space norm, rather than parameter count or other representational costs that permit compounded rescaling across layers. Overall, our results illustrate how a function space perspective yields new structural insights into the relationship between depth and complexity.

Figures

Figures reproduced from arXiv: 2607.05546 by the authors.

Figure 1
Figure 1. Scaled feature shapes t 7→ σ(st) and normalized features σs(t) := σ(st)/s for the GELU (Fig. 1a) and tanh (Fig. 1b) activation functions. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The functions fα (as defined in (64)) and (fα)+ for α = 0.2, α = 0.1, and α = 0.01, respectively. Each fα is in B2, but as α ↓ 0, the total variation of D2 (fα)+ approaches 2. Therefore, by Theorem 4.1, (fα)+ ∈ B / 2 for all sufficiently small α. The proof, presented in Appendix A.18, proceeds by showing that the base linear class B1 has supf∈B1 I(f) ≤ 2, and then using Theorem 4.2 to argue that this bound is inheri… view at source ↗
Figure 3
Figure 3. Top row: iterated compositions T ◦L of the Telgarsky sawtooth function (68). The L-fold composition T ◦L has 2 L/2 “teeth” and 2 L − 1 total kinks. However, this compositional oscillatory behavior is highly dependent on the scaling of T. Bottom row: if T is rescaled to T /6 (or to aT for any 0 ≤ a < 1/2), iterated composition produces no additional oscillations, but instead progressively shrinks the original functio… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The functions fn(x) = 1 4πn cos(πnx) for n = 1, 2, 10. These functions have ∥D2fn∥TV = − R 1 −1 |f ′′ n (x)| dx = n, so by Theorem 4.1 and Theorem 4.3, they are excluded (for n > 1) from the univariate ReLU classes B uni L at all depths L. Remark 4.7. (73) is a generic…
Figure 5
Figure 5. Figure 5: Example functions in d = 2 whose restriction to some line in the input domain B 2 2 agrees with the highly oscillatory function fn (n = 10) of [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: A class H of disjoint bump functions on [0, 1]. The j th bump is supported on the dyadic interval (343), and has height aj ≍ 1/(log log j). ambient function space norm. Let {σj}∞ j=1 ⊂ C[0, 1] be a sequence of disjoint bump functions, each of height ∥σj∥∞ = 1, such tha…
Figure 7
Figure 7. Figure 7: Modifications of an example function obtained by successively flattening its negative intervals. [PITH_FULL_IMAGE:figures/full_fig_p070_7.png]

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Works this paper leans on

87 extracted references · 5 linked inside Pith

  1. [1]

    SIAM Journal on Mathematics of Data Science , volume=

    What kinds of functions do deep neural networks learn? Insights from variational spline theory , author=. SIAM Journal on Mathematics of Data Science , volume=. 2022 , publisher=

    2022

  2. [2]

    Annals of Statistics , volume=

    Local Rademacher complexities and empirical minimization , author=. Annals of Statistics , volume=

  3. [3]

    CSIAM Transactions on Applied Mathematics , volume=

    On the Banach Spaces Associated with Multi-Layer ReLU Networks: Function Representation, Approximation Theory and Gradient Descent Dynamics , author=. CSIAM Transactions on Applied Mathematics , volume=

  4. [4]

    Calculus of Variations and Partial Differential Equations , volume=

    Representation formulas and pointwise properties for Barron functions , author=. Calculus of Variations and Partial Differential Equations , volume=. 2022 , publisher=

    2022

  5. [5]

    Conference on Learning Theory , pages=

    How do infinite width bounded norm networks look in function space? , author=. Conference on Learning Theory , pages=. 2019 , organization=

    2019

  6. [6]

    The BV space in variational and evolution problems , author=

  7. [7]

    Journal of the ACM (JACM) , volume=

    Scale-sensitive dimensions, uniform convergence, and learnability , author=. Journal of the ACM (JACM) , volume=. 1997 , publisher=

    1997

  8. [8]

    University of California, Irvine , volume=

    High-dimensional probability , author=. University of California, Irvine , volume=

Show all 87 references
  1. [9]

    IEEE transactions on Information Theory , volume=

    Rademacher averages and phase transitions in Glivenko-Cantelli classes , author=. IEEE transactions on Information Theory , volume=. 2002 , publisher=

    2002

  2. [10]

    International Conference on Computational Learning Theory , pages=

    Geometric methods in the analysis of Glivenko-Cantelli classes , author=. International Conference on Computational Learning Theory , pages=. 2001 , organization=

    2001

  3. [11]

    Advances in neural information processing systems , volume=

    Smoothness, low noise and fast rates , author=. Advances in neural information processing systems , volume=

  4. [12]

    The Thirty Seventh Annual Conference on Learning Theory , pages=

    Depth separation in norm-bounded infinite-width neural networks , author=. The Thirty Seventh Annual Conference on Learning Theory , pages=. 2024 , organization=

    2024

  5. [13]

    Journal of Machine Learning Research , volume=

    Banach space representer theorems for neural networks and ridge splines , author=. Journal of Machine Learning Research , volume=

  6. [14]

    Constructive Approximation , volume=

    Characterization of the variation spaces corresponding to shallow neural networks , author=. Constructive Approximation , volume=. 2023 , publisher=

    2023

  7. [15]

    Journal of Machine Learning Research , volume=

    Variation spaces for multi-output neural networks: Insights on multi-task learning and network compression , author=. Journal of Machine Learning Research , volume=

  8. [16]

    Applied and Computational Harmonic Analysis , volume=

    Weighted variation spaces and approximation by shallow ReLU networks , author=. Applied and Computational Harmonic Analysis , volume=. 2025 , publisher=

    2025

  9. [17]

    , author=

    Reproducing kernel Banach spaces for machine learning. , author=. Journal of Machine Learning Research , volume=

  10. [18]

    Applied and Computational Harmonic Analysis , volume=

    Understanding neural networks with reproducing kernel Banach spaces , author=. Applied and Computational Harmonic Analysis , volume=. 2023 , publisher=

    2023

  11. [19]

    arXiv preprint arXiv:2501.03697 , year=

    Deep networks are reproducing kernel chains , author=. arXiv preprint arXiv:2501.03697 , year=

    arXiv

  12. [20]

    Proceedings of the American Mathematical Society , volume=

    Metric entropy of ��-hulls in Banach spaces of type-�� , author=. Proceedings of the American Mathematical Society , volume=. 2017 , publisher=

    2017

  13. [21]

    arXiv preprint arXiv:1806.01528 , year=

    The universal approximation power of finite-width deep ReLU networks , author=. arXiv preprint arXiv:1806.01528 , year=

    arXiv

  14. [22]

    Journal of machine learning research , volume=

    Depth separation beyond radial functions , author=. Journal of machine learning research , volume=

  15. [23]

    International Conference on Learning Representations , year=

    Depth-Width Trade-offs for ReLU Networks via Sharkovsky's Theorem , author=. International Conference on Learning Representations , year=

  16. [24]

    arXiv preprint arXiv:2606.14954 , year=

    Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks , author=. arXiv preprint arXiv:2606.14954 , year=

    arXiv

  17. [25]

    Advances in Neural Information Processing Systems , volume=

    Sharp representation theorems for relu networks with precise dependence on depth , author=. Advances in Neural Information Processing Systems , volume=

  18. [26]

    1991 , publisher=

    Probability in Banach Spaces: isoperimetry and processes , author=. 1991 , publisher=

    1991

  19. [27]

    Bulletin of the American Mathematical Society , volume=

    Extension of range of functions , author=. Bulletin of the American Mathematical Society , volume=

  20. [28]

    2021 , publisher=

    Upper and lower bounds for stochastic processes , author=. 2021 , publisher=

    2021

  21. [29]

    Studia Mathematica , volume=

    The best constants in the Khintchine inequality , author=. Studia Mathematica , volume=. 1981 , publisher=

    1981

  22. [30]

    Boucheron, Stéphane and Lugosi, Gábor and Massart, Pascal , title =

  23. [31]

    Proceedings of the American Mathematical Society , volume=

    Metric entropy of convex hulls in type �� spaces—the critical case , author=. Proceedings of the American Mathematical Society , volume=

  24. [32]

    Conference on Learning Theory , pages=

    Depth separation for neural networks , author=. Conference on Learning Theory , pages=. 2017 , organization=

    2017

  25. [33]

    International Conference on Learning Representations , year=

    Understanding Deep Neural Networks with Rectified Linear Units , author=. International Conference on Learning Representations , year=

  26. [34]

    Conference on learning theory , pages=

    The power of depth for feedforward neural networks , author=. Conference on learning theory , pages=. 2016 , organization=

    2016

  27. [35]

    2009 , publisher=

    Neural network learning: Theoretical foundations , author=. 2009 , publisher=

    2009

  28. [36]

    Advances in neural information processing systems , volume=

    For valid generalization the size of the weights is more important than the size of the network , author=. Advances in neural information processing systems , volume=

  29. [37]

    Proceedings of the 58th Annual ACM Symposium on Theory of Computing , pages=

    Better neural network expressivity: subdividing the simplex , author=. Proceedings of the 58th Annual ACM Symposium on Theory of Computing , pages=

  30. [38]

    Acta numerica , volume=

    Approximation theory of the MLP model in neural networks , author=. Acta numerica , volume=. 1999 , publisher=

    1999

  31. [39]

    Neural networks , volume=

    Multilayer feedforward networks with a nonpolynomial activation function can approximate any function , author=. Neural networks , volume=. 1993 , publisher=

    1993

  32. [40]

    Advances in Neural Information Processing Systems , volume=

    Towards lower bounds on the depth of ReLU neural networks , author=. Advances in Neural Information Processing Systems , volume=

  33. [41]

    Advances in neural information processing systems , volume=

    Self-normalizing neural networks , author=. Advances in neural information processing systems , volume=

  34. [42]

    Applied and Computational Harmonic Analysis , volume=

    Embeddings between Barron spaces with higher-order activation functions , author=. Applied and Computational Harmonic Analysis , volume=. 2024 , publisher=

    2024

  35. [43]

    Conference on learning theory , pages=

    Norm-based capacity control in neural networks , author=. Conference on learning theory , pages=. 2015 , organization=

    2015

  36. [44]

    Advances in neural information processing systems , volume=

    Spectrally-normalized margin bounds for neural networks , author=. Advances in neural information processing systems , volume=

  37. [45]

    International Conference on Computational Learning Theory , pages=

    Some local measures of complexity of convex hulls and generalization bounds , author=. International Conference on Computational Learning Theory , pages=. 2002 , organization=

    2002

  38. [46]

    Journal of machine learning research , volume=

    Rademacher and gaussian complexities: Risk bounds and structural results , author=. Journal of machine learning research , volume=

  39. [47]

    1977 , publisher=

    Vector Measures , author=. 1977 , publisher=

    1977

  40. [48]

    , publisher =

    Dinculeanu, N. , publisher =. Vector

  41. [49]

    IEEE Transactions on Information Theory , volume=

    Bounds on rates of variable-basis and neural-network approximation , author=. IEEE Transactions on Information Theory , volume=. 2002 , publisher=

    2002

  42. [50]

    Computer Intensive Methods in Control and Signal Processing: The Curse of Dimensionality , pages=

    Dimension-independent rates of approximation by neural networks , author=. Computer Intensive Methods in Control and Signal Processing: The Curse of Dimensionality , pages=. 1997 , publisher=

    1997

  43. [51]

    Conference on learning theory , pages=

    Benefits of depth in neural networks , author=. Conference on learning theory , pages=. 2016 , organization=

    2016

  44. [52]

    Neural networks , volume=

    Error bounds for approximations with deep ReLU networks , author=. Neural networks , volume=. 2017 , publisher=

    2017

  45. [53]

    Journal of Machine Learning Research , volume=

    Breaking the curse of dimensionality with convex neural networks , author=. Journal of Machine Learning Research , volume=

  46. [54]

    Acta Mathematica , volume=

    Improved upper bounds for approximation by zonotopes , author=. Acta Mathematica , volume=. 1996 , publisher=

    1996

  47. [55]

    International Conference on Learning Representations (ICLR 2020) , year=

    A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case , author=. International Conference on Learning Representations (ICLR 2020) , year=

    2020

  48. [56]

    IEEE Transactions on Information Theory , volume=

    Near-minimax optimal estimation with shallow ReLU neural networks , author=. IEEE Transactions on Information Theory , volume=. 2022 , publisher=

    2022

  49. [57]

    Journal of Machine Learning Research , volume=

    Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks , author=. Journal of Machine Learning Research , volume=

  50. [58]

    Foundations of Computational Mathematics , volume=

    Sharp bounds on the approximation rates, metric entropy, and n-widths of shallow neural networks , author=. Foundations of Computational Mathematics , volume=. 2024 , publisher=

    2024

  51. [59]

    Dudley, R. M. , year=. Real Analysis and Probability , publisher=

  52. [60]

    arXiv preprint arXiv:1902.00800 , year=

    Complexity, statistical risk, and metric entropy of deep nets using total path variation , author=. arXiv preprint arXiv:1902.00800 , year=

    1902 arXiv

  53. [61]

    arXiv preprint arXiv:2410.06378 , year=

    Covering numbers for deep relu networks with applications to function approximation and nonparametric regression , author=. arXiv preprint arXiv:2410.06378 , year=

  54. [62]

    arXiv preprint arXiv:2403.08750 , year=

    Neural reproducing kernel Banach spaces and representer theorems for deep networks , author=. arXiv preprint arXiv:2403.08750 , year=

    arXiv

  55. [63]

    Journal of Functional Analysis , volume=

    Entropy numbers, s-numbers, and eigenvalue problems , author=. Journal of Functional Analysis , volume=. 1981 , publisher=

    1981

  56. [64]

    Neural Networks , volume=

    Optimal approximation of piecewise smooth functions using deep ReLU neural networks , author=. Neural Networks , volume=. 2018 , publisher=

    2018

  57. [65]

    Constructive Approximation , volume=

    The Barron space and the flow-induced function spaces for neural network models , author=. Constructive Approximation , volume=. 2022 , publisher=

    2022

  58. [66]

    1999 , publisher=

    The volume of convex bodies and Banach space geometry , author=. 1999 , publisher=

    1999

  59. [67]

    Oracle inequalities in empirical risk minimization and sparse recovery problems: Ecole D’Et

    Koltchinskii, Vladimir , volume=. Oracle inequalities in empirical risk minimization and sparse recovery problems: Ecole D’Et. 2011 , publisher=

    2011

  60. [68]

    Conference on learning theory , pages=

    Size-independent sample complexity of neural networks , author=. Conference on learning theory , pages=. 2018 , organization=

    2018

  61. [69]

    Bulletin of the American Mathematical Society , volume=

    Compositional sparsity of learnable functions , author=. Bulletin of the American Mathematical Society , volume=

  62. [70]

    2014 , publisher=

    Understanding machine learning: From theory to algorithms , author=. 2014 , publisher=

    2014

  63. [71]

    Journal of the London Mathematical Society , volume=

    Metric entropy of convex hulls in Banach spaces , author=. Journal of the London Mathematical Society , volume=. 1999 , publisher=

    1999

  64. [72]

    Annals of Statistics , pages=

    Information-theoretic determination of minimax rates of convergence , author=. Annals of Statistics , pages=. 1999 , publisher=

    1999

  65. [73]

    Israel Journal of Mathematics , volume=

    Metric entropy of convex hulls , author=. Israel Journal of Mathematics , volume=. 2001 , publisher=

    2001

  66. [74]

    1981 , publisher=

    I: Functional analysis , author=. 1981 , publisher=

    1981

  67. [75]

    Journal of Approximation Theory , volume=

    Spline solutions to L1 extremal problems in one and several variables , author=. Journal of Approximation Theory , volume=. 1975 , publisher=

    1975

  68. [76]

    1999 , publisher=

    Real analysis: modern techniques and their applications , author=. 1999 , publisher=

    1999

  69. [77]

    , year =

    Rudin, W. , year =. Functional analysis , publisher =

  70. [78]

    2018 , publisher=

    A course in functional analysis and measure theory , author=. 2018 , publisher=

    2018

  71. [79]

    2005 , publisher=

    Convex functional analysis , author=. 2005 , publisher=

    2005

  72. [80]

    2006 , publisher=

    Infinite dimensional analysis: a hitchhiker’s guide , author=. 2006 , publisher=

    2006

  73. [81]

    Advances in Neural Information Processing Systems , volume=

    Network size and size of the weights in memorization with two-layers neural networks , author=. Advances in Neural Information Processing Systems , volume=

  74. [82]

    Journal of Approximation Theory , volume=

    Entropy of convex hulls—some Lorentz norm results , author=. Journal of Approximation Theory , volume=. 2004 , publisher=

    2004

  75. [83]

    2000 , publisher=

    Functions of bounded variation and free discontinuity problems , author=. 2000 , publisher=

    2000

  76. [84]

    2007 , publisher=

    Measure theory , author=. 2007 , publisher=

    2007

  77. [85]

    2017 , publisher=

    A first course in Sobolev spaces , author=. 2017 , publisher=

    2017

  78. [86]

    Advances in neural information processing systems , volume=

    Neural networks with small weights and depth-separation barriers , author=. Advances in neural information processing systems , volume=

  79. [87]

    Journal of Complexity , volume=

    Entropy numbers of convex hulls in Banach spaces and applications , author=. Journal of Complexity , volume=. 2014 , publisher=

    2014

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