Pith. sign in

REVIEW 4 major objections 5 minor 42 references

A New Formulation of Lipschitz Constrained With Functional Gradient Learning for GANs

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims GAN output diversity is governed by a latent-space neighborhood radius, and that a new ε-centered gradient penalty shrinks that radius below standard 0- and 1-centered penalties, yielding more diverse images.

desk verdict The empirical sweep is useful, but the main theorem is unproven due to a dimensionally invalid generator update and an upper-bound argument that does not order the actual radii. read the letter →

arxiv 2501.11236 v1 pith:7DO72O5J submitted 2025-01-20 cs.CV cs.LG

Chang Wan , Ke Fan , Xinwei Sun , Yanwei Fu , Minglu Li , Yunliang Jiang , Zhonglong Zheng This is my paper · ORCID
classification cs.CVcs.LG
keywords generativeadversarialnetworksfunctionalgradientmethodsLipschitzconstraintpenaltyepsilon-centeredlatentneighborhoodsizemodecollapsesynthesisdiversity
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

This paper tries to put a theoretical handle on mode collapse in GANs by tying the diversity of generated images to a quantity it calls the latent N-size: the radius of the neighborhood in latent space whose points are pulled toward the same image mode. The claim is that shrinking this radius, by making the discriminator gradient norm $\|\nabla_x D\|$ large, pushes the generator's outputs to spread over more modes, and so produces more diverse samples. To enlarge the gradient norm in a controllable way, the paper introduces an $\epsilon$-centered gradient penalty that steers $\|\nabla_x D\|$ away from zero by a tunable amount $\epsilon'$ rather than toward 0 or 1. The central result is the ordering of latent N-sizes $r_{R_1} > r_{R_0} > r_{R_\epsilon}$, with the $\epsilon$-centered penalty giving the smallest neighborhood and, the paper argues, the best diversity. A sympathetic reader would care because this offers a parameterized, more stable alternative to composite functional gradient (CFG) GAN training, with reported FID and recall gains on MNIST, CIFAR-10, LSUN, and ImageNet.

What carries the argument

The load-bearing object is the latent N-size, the paper's term for the neighborhood radius $r = \hat{\epsilon}\,\bigl(2\inf_z(\cdots)\bigr)^{-1}$ of Definition 3.1, whose reciprocal measures how much the generator's output moves between consecutive functional-gradient steps; a small $r$ means nearby latent vectors get pushed toward distinct modes. Proposition 3.5 rewrites that radius in terms of the discriminator gradient norm $\|\nabla_x D\|$ plus a penalty term $R$ inserted into the composite functional gradient (CFG) generator update, and the $\epsilon$-centered gradient penalty $R_\epsilon(\theta,\psi) = \frac{\gamma}{2}\,\mathbb{E}\big[\|\nabla_x D_\psi(\hat{x}) - \epsilon\|^2\big]$ with $\|\epsilon\| = \epsilon'$ is the control knob that enlarges the norm. Lemma 3.6 compares the three penalties' contributions to the radius expression, concluding $\|R_1\| < \|R_0\| < \|R_\epsilon\|$, and Theorem 3.7 turns that into the radius ordering $r_{R_1} > r_{R_0} > r_{R_\epsilon}$. The direction of the argument relies on the corollary $\nabla_x D(x) \le 0$, which makes subtracting $\epsilon$ the move that increases $\|\nabla_x D\|$.

What would settle it

Train one architecture under the 1-centered, 0-centered, and $\epsilon$-centered penalties and measure the latent N-size $r$ directly from the trained generator's input-output map, as Definition 3.1 defines it, rather than from the upper-bound expression in Proposition 3.5; if the measured ordering is not $r_{R_1} > r_{R_0} > r_{R_\epsilon}$, the theorem's conclusion is refuted. A cheaper check of the premise is to compare training with $R$ inserted inside the generator update (Eq. C19) against training with $R$ used only as a discriminator regularizer: if the two trajectories differ while the theory only describes the perturbed one, the derivation is ungrounded.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Lipschitz constraint on the discriminator and the diversity of generated samples are linked through the magnitude of the discriminator gradient. Under the CFG update rule, the latent N-size radius $r$ from Definition 3.1 is inversely controlled by $\|\nabla_x D\|$: larger gradient norms shrink the neighborhood that collapses onto a single mode. Because the CFG derivation gives $\nabla_x D(x) \le 0$, subtracting a fixed offset vector $\epsilon$ in the penalty term enlarges the effective norm by $\|\epsilon\|$, and the paper shows through Proposition 3.5, Lemma 3.6, and Theorem 3.7 that the three common penalties order their latent N-sizes as $r_{R_1} > r_{R_0} > r_{R_\epsilon}$. The theorem is offered as the theoretical foundation for why their Li-CFG method, which adds the $\epsilon$-centered gradient penalty on top of CFG, generates more diverse images while training more stably than unpenalized CFG.

Load-bearing premise

The load-bearing premise is that the scalar gradient penalty $R$ can be inserted as a vector term inside the generator's update step, as written in Proposition 3.5 and Appendix C.1, Eq.~(C19), $G_{\theta_{t+1}}(z) = G_{\theta_t}(z) + \sum_m \eta_m \delta(x)\big(\nabla_x D_m(G_{\theta_t}(z)) + R\big)$, even though $R$ is a scalar loss term attached to the discriminator objective; if that insertion is not valid, the latent N-size formula and the ordering $r_{R_1} > r_{R_0} > r_{R_\epsilon}$ do not follow.

Editorial extensions

If this is right

  • If the ordering $r_{R_1} > r_{R_0} > r_{R_\epsilon}$ is correct, the $\epsilon$-centered penalty yields the smallest latent N-size of the three, and the predicted diversity benefit should reproduce across architectures and datasets.
  • The hyper-parameter $\epsilon'$ becomes a diversity dial: the paper's ablation on MNIST and LSUN Bedroom shows $\epsilon' = 0.3$ giving the best FID and $\epsilon' = 5$ failing to converge, which it reads as a too-small neighborhood.
  • Because the penalty is added to the loss rather than to the network, it can be combined with common GAN objectives; the paper reports improved FID and recall for WGAN, LSGAN, HingeGAN, BigGAN, and DDGAN when the $\epsilon$-centered term is added.
  • The theory predicts a diversity-trainability trade-off: driving $\|\nabla_x D\|$ too high makes the latent N-size too small, and the generator collapses to noise instead of learning modes.
  • Stabilizing $\|\nabla_x D\|$ should stabilize FID across the CFG hyper-parameter $\delta(x)$; the paper shows CFG's FID swings widely with $\delta(x)$, while Li-CFG's stays flat.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Lemma 3.6 compares upper-bound expressions rather than the actual radii $r$, a direct measurement of the neighborhood radius from a trained generator's input-output map would be the real test of the ordering; the bound comparison alone cannot rule out that the actual radii order differently.
  • The paper's own limitations section concedes that the relationship between CFG's $\delta(x)$ and the penalty scale $\epsilon'$ is not analyzed theoretically; a natural next step would be a joint condition on $(\delta(x), \epsilon', \gamma)$ that keeps the latent N-size inside the convergent range.
  • The empirical tables show recall improving with the $\epsilon$-centered penalty while precision sometimes drops slightly, suggesting the mechanism trades a little fidelity for broader mode coverage; that trade-off is a testable prediction of the neighborhood picture.
  • The authors report that their reproduced CFG FID scores run noticeably higher than the numbers in the original CFG paper, so the absolute FID comparisons reported here should be read as internally consistent rather than directly comparable across papers.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript proposes Li-CFG, a variant of Composite Functional Gradient (CFG) GAN training that adds a Lipschitz-style gradient penalty to the discriminator. It defines a 'latent N-size' r (Definition 3.1), claims that increasing the discriminator gradient norm reduces r and hence increases sample diversity, introduces an ε-centered gradient penalty (Eq. (9)), and states Theorem 3.7 that the three gradient penalties yield radii satisfying r_R1 > r_R0 > r_Rε. The empirical sections compare Li-CFG with CFG and other GAN baselines on ring/grid, MNIST, CIFAR-10, LSUN, and ImageNet, and also apply the ε-centered penalty to BigGAN and DDGAN.

Significance. The empirical scope is a strength: the authors evaluate across many datasets and architectures, report FID, IS, and precision/recall, and show consistent improvements when adding their ε-centered penalty. If Theorem 3.7 were established, the paper would provide a useful new perspective connecting discriminator gradient penalties to latent-space diversity. However, the central proof is not sound: the derivation rests on an ill-defined update, replaces the quantity in Definition 3.1 by an upper bound, and uses a norm inequality that is false in general. As it stands, the theoretical claims are not established.

major comments (4)
  1. [Appendix C.1, Eq. (C19)] The update in Eq. (C19) writes Gθ_{t+1}(z) = Gθ_t(z) + Σ η_m δ(x)(∇_x D_m(Gθ_t(z)) + R), but R in Eq. (8) is a scalar regularization term in the discriminator objective (an expectation of squared norms). Adding this scalar directly to the gradient vector ∇_x D is undefined, and no derivation is provided to show that the gradient penalty enters the CFG generator update in this additive form. Since Proposition 3.5 and all subsequent norm comparisons use ∥∇_x D + R∥, the formula for r with a gradient penalty is not meaningful without a corrected derivation.
  2. [Proposition 3.5 / Appendix C.1, Eq. (C22)] Definition 3.1 defines r = ε̂(2 inf_z L(z))^{-1} with L(z) the actual ratio of generator discrepancies. In Eq. (C22) the proof shows L(z) ≤ U(z), and Proposition 3.5 then restates r using U(z) in place of L(z). Since r is inversely proportional to the infimum, replacing L by an upper bound gives r_P = ε̂(2 inf U)^{-1} ≤ r, not equality. The later ordering r_R1 > r_R0 > r_Rε compares these upper-bound expressions; the actual radii are not ordered because the tightness of each upper bound can differ independently. Theorem 3.7 therefore does not follow from the proof as written.
  3. [Lemma 3.6, Eq. (C30)] The claimed ordering ∥R1∥ < ∥R0∥ < ∥Rε∥ is false in general. With a = ∥∇_x D∥ and g0 = 1, if a < 1/2 then |a − 1| > a, so ∥R1∥ = ∥ |a − 1| ∥ > ∥R0∥ = ∥a∥, reversing the first inequality. The condition ∇_x D ≤ 0 used elsewhere does not prevent this, because the gradient norm can still be small. Since Lemma 3.6 is the bridge from penalty norms to radii in Theorem 3.7, this is a load-bearing gap.
  4. [Appendix C.2, proof of ∇_x D(x) ≤ 0] The proof claims that g_m(x) ≤ 0 implies ∇_x D(x) ≤ 0. In the CFG formulation, g_m is a vector-valued function on image space, so 'g_m(x) ≤ 0' is undefined, and a scalar inequality cannot be inferred for each component of ∇_x D. The subsequent argument that the ε-centered penalty enlarges ∥∇_x D∥ relies on this unproved sign condition, so the comparison supporting the ε-centered penalty is not established.
minor comments (5)
  1. [Eq. (9) and Appendix D.1] The symbol ε is used both as a vector (∥ε∥_2 = ε′) and as a scalar inside ε′ = sqrt(C N^2 ε^2); please introduce separate symbols for the vector and the scalar norm target.
  2. [Table 3] In the LSUN B block, the 'ours(ε-centered)' row lists only a single FID value (8.78) instead of the IS/FID pair used in all other rows; please fix the formatting.
  3. [References] References [4] and [6] appear to reference the same work by Johnson and Zhang; they should be merged to avoid duplicate citation.
  4. [Definition 3.4] The main text says z2 is distracted 'from a mode', while Appendix C says 'to a mode'; the terminology should be made consistent.
  5. [Appendix C.3, Eqs. (C27)–(C29)] The penalty expressions are inserted as (∥∇D∥−g0)^2, ∥∇D∥^2, and (∥∇D−ε∥)^2 with the expectation and γ/2 omitted, so the quantities compared are not the actual penalties defined in Eqs. (6)–(9); this should be clarified or corrected.

Circularity Check

2 steps flagged · score 7.0 of 10

The central radius ordering r_R1 > r_R0 > r_Rε is built into the definitions of the penalties and an unproven norm inequality, after Proposition 3.5 substitutes an upper bound for the actual latent N-size.

  1. self definitional [Appendix C.1, Proposition 3.5 (with Eq. C22)]
    "Proposition 3.5. Nr (z1) can be defined with discriminator gradient penalty as follows: r = ˆϵ·(2 inf_z { [2∥Gθt (z1) − Gθt (z)∥ + ηmδ(x) NP m=1 (∥∇xDm(Y2)∥ + ∥∇xDm(Y)∥)] / ∥z1−z∥ })^{-1}, where ∥∇xDm(Y2)∥ = ∥∇xDm(Gθt (z2)) +R∥ and ∥∇xDm(Y)∥ = ∥∇xDm(Gθt (z)) + R∥."

    Definition 3.1 defines r using the exact generator-output discrepancy. In the proof, Eq. (C22) only proves that exact expression is ≤ the expression written in Proposition 3.5. Replacing the minimized quantity by an upper bound changes the infimum, so the quantity called r is not the latent N-size of Definition 3.1. Theorem 3.7 then orders these constructed upper-bound expressions, not the actual radii, so the inequality chain r_R1 > r_R0 > r_Rε is an ordering of the paper's own substitution rather than an independently derived prediction.

  2. self definitional [Section 3.3, Lemma 3.6]
    "Lemma 3.6. The norms of the three Gradient Penalties, which dictate the latent N-size, are defined as follows: ∥R1∥ = ∥ (∥∇xDm(Gθt (z))∥ −g0) ∥, ∥R0∥ = ∥ (∥∇xDm(Gθt (z))∥) ∥, ∥Rε∥=∥ (∥∇xDm(Gθt (z))∥ + ∥ε∥) ∥, respectively. The order of magnitude between the norms of three Gradient Penalty is ∥R1∥ < ∥R0∥ < ∥Rε∥. Consequently, the relationship between the latent N-size of three Gradient Penalty is rR1 > rR0 > rRε ."

    With Rε defined as (∥∇xD − ε∥)^2 in Eq. (9), the norm is rewritten in Lemma 3.6 as ∥∇xD∥ + ∥ε∥, so ∥R0∥ < ∥Rε∥ is true by construction and needs no proof. Because Proposition 3.5 makes r inversely proportional to these same norm expressions, the conclusion rR0 > rRε is just the definitional norm ordering restated. The remaining inequality ∥R1∥ < ∥R0∥ is asserted without the required condition ∥∇xD∥ > g0/2 and is false in general, so the only non-definitional ingredient is unsupported.

full rationale

The paper's central theoretical claim, Theorem 3.7, is not independently derived. Proposition 3.5 defines the latent N-size with gradient penalty by replacing the exact Definition 3.1 discrepancy with an upper bound from Eq. (C22), so the radii being compared are constructed quantities. Lemma 3.6 then writes the three penalty norms in a form where the ε-centered norm contains an extra +∥ε∥ by definition and immediately converts that norm ordering into the radius ordering. The ε-vs-0 part of the theorem is therefore equivalent to the way the penalty was defined, while the 0-vs-1 part relies on an unproven inequality that can reverse for small gradient norms. No independent measurement of latent N-size is reported. The empirical FID/recall comparisons and the CFG background from Johnson and Zhang are external evidence and do not themselves make the proof circular, and there is no load-bearing self-citation chain; the circularity is concentrated in the theorem's reduction of the predicted radius ordering to the penalty definitions.

Assumptions & free parameters 3 free parameters · 4 assumptions · 3 invented entities

The central claim rests on a custom definition of latent neighborhood size, an unjustified vector addition of the scalar penalty into the generator update, and the ∇xD ≤ 0 assumption. The ε parameter and the CFG scaling factor δ(x) are tuned manually and directly control the claimed diversity effect.

free parameters (3)
  • ε′ (norm of ε vector) = 0.1, 0.3, 1, 5
    Controls the target norm of the discriminator gradient in the ε-centered penalty; varies per dataset/experiment and directly determines the claimed latent N-size reduction.
  • δ(x) (scaling factor in CFG update) = 0.5, 1, 5, 10, 20
    Scaling factor in the generator update inherited from CFG; the paper treats it as a hyper-parameter and shows sensitivity to it.
  • γ (gradient penalty weight) = 0.1, 1, 10
    Standard loss weight; tuned per dataset in Tables D3-D4.
assumptions (4)
  • domain assumption Generator evolves as Gθt+1(z) = Gθt(z) + Σ ηm δ(x)(∇xDm(Gθt(z)) + R), where R is the gradient penalty term added as a vector.
    From CFG (Johnson and Zhang) but the vector addition of the scalar R is introduced by the paper (Eq. C19).
  • domain assumption The optimal CFG discriminator satisfies ∇xD(x) ≤ 0 during training.
    Derived in Appendix C.2 from an integral inequality; used to argue the ε-centered penalty enlarges the gradient norm.
  • domain assumption Image space is covered by well-separated modes M_i with intra-mode diameter α and inter-mode distances between α and 2α (Definition 3.2).
    Underlies the definitions of attracted/distracted latent vectors; not validated on real data.
  • ad hoc to paper A smaller latent neighborhood radius r implies higher synthesis diversity.
    The diversity claim is inferred from the local attraction behavior; no direct measurement of r or diversity is made.
invented entities (3)
  • latent N-size (latent neighborhood size) r
    purpose: Quantifies how far latent vectors can be while still being attracted to the same image mode; the paper's central quantity linking gradient penalty to diversity.
    Defined in Definition 3.1 and never directly measured; its behavior is inferred from upper bounds.
  • ε-centered gradient penalty independent evidence
    purpose: A new regularizer that centers the discriminator gradient at a vector ε with norm ε′; claimed to enlarge the gradient norm and shrink the latent N-size.
    It is defined as a loss term and can be tested empirically; the paper reports FID/IS improvements, though without error bars.
  • modes attracted / distracted
    purpose: Intuitive descriptions of how latent vectors move toward or away from image modes; used to motivate the definition of r.
    Purely definitional; no independent measurement.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A New Formulation of Lipschitz Constrained With Functional Gradient Learning for GANs." pith.science (2026). https://pith.science/paper/7DO72O5J

@misc{pith2026250111236,
  author       = {Pith},
  title        = {Pith review of: A New Formulation of Lipschitz Constrained With Functional Gradient Learning for GANs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DO72O5J}},
  note         = {Machine review of arXiv:2501.11236}
}
read the original abstract

This paper introduces a promising alternative method for training Generative Adversarial Networks (GANs) on large-scale datasets with clear theoretical guarantees. GANs are typically learned through a minimax game between a generator and a discriminator, which is known to be empirically unstable. Previous learning paradigms have encountered mode collapse issues without a theoretical solution. To address these challenges, we propose a novel Lipschitz-constrained Functional Gradient GANs learning (Li-CFG) method to stabilize the training of GAN and provide a theoretical foundation for effectively increasing the diversity of synthetic samples by reducing the neighborhood size of the latent vector. Specifically, we demonstrate that the neighborhood size of the latent vector can be reduced by increasing the norm of the discriminator gradient, resulting in enhanced diversity of synthetic samples. To efficiently enlarge the norm of the discriminator gradient, we introduce a novel {\epsilon}-centered gradient penalty that amplifies the norm of the discriminator gradient using the hyper-parameter {\epsilon}. In comparison to other constraints, our method enlarging the discriminator norm, thus obtaining the smallest neighborhood size of the latent vector. Extensive experiments on benchmark datasets for image generation demonstrate the efficacy of the Li-CFG method and the {\epsilon}-centered gradient penalty. The results showcase improved stability and increased diversity of synthetic samples.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 25 canonical work pages

  1. [1]

    Conference on Neural Information Processing Systems 27 (2014)

    Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. Conference on Neural Information Processing Systems 27 (2014)

    work page 2014

  2. [2]

    In: International Conference on Machine Learning, pp

    Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: International Conference on Machine Learning, pp. 214–223 (2017). PMLR

    work page 2017

  3. [3]

    arXiv preprint arXiv:1704.00028 (2017)

    Gulrajani, I., Ahmed, F., Arjovsky, M., Dumoulin, V., Courville, A.: Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028 (2017)

    arXiv 2017

  4. [4]

    In: International Conference on Machine Learning, pp

    Johnson, R., Zhang, T.: Composite functional gradient learning of generative 22 adversarial models. In: International Conference on Machine Learning, pp. 2371– 2379 (2018). PMLR

    work page 2018

  5. [5]

    3481– 3490 (2018)

    Mescheder, L., Geiger, A., Nowozin, S.: Which training methods for gans do actually converge? In: International Conference on Machine Learning, pp. 3481– 3490 (2018). PMLR

    work page 2018

  6. [6]

    IEEE Transactions on Pattern Analysis and Machine Intelligence 43(1), 17–32 (2019)

    Johnson, R., Zhang, T.: A framework of composite functional gradient methods for generative adversarial models. IEEE Transactions on Pattern Analysis and Machine Intelligence 43(1), 17–32 (2019)

    work page 2019

  7. [7]

    arXiv preprint arXiv:1705.09367 (2017)

    Roth, K., Lucchi, A., Nowozin, S., Hofmann, T.: Stabilizing training of generative adversarial networks through regularization. arXiv preprint arXiv:1705.09367 (2017)

    arXiv 2017

  8. [8]

    arXiv preprint arXiv:1901.09024 (2019)

    Yang, D., Hong, S., Jang, Y., Zhao, T., Lee, H.: Diversity-sensitive conditional generative adversarial networks. arXiv preprint arXiv:1901.09024 (2019)

    arXiv 2019

Show all 42 references
  1. [9]

    arXiv preprint arXiv:1511.06434 (2015)

    Radford, A., Metz, L., Chintala, S.: Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434 (2015)

    2015 arXiv

  2. [10]

    In: International Conference on Machine Learning, pp

    Zhang, H., Goodfellow, I., Metaxas, D., Odena, A.: Self-attention generative adversarial networks. In: International Conference on Machine Learning, pp. 7354–7363 (2019). PMLR

    2019

  3. [11]

    arXiv preprint arXiv:1710.10196 (2017)

    Karras, T., Aila, T., Laine, S., Lehtinen, J.: Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196 (2017)

    2017 arXiv

  4. [12]

    arXiv preprint arXiv:1809.11096 (2018)

    Brock, A., Donahue, J., Simonyan, K.: Large scale gan training for high fidelity natural image synthesis. arXiv preprint arXiv:1809.11096 (2018)

    2018 arXiv

  5. [13]

    In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp

    Karras, T., Laine, S., Aila, T.: A style-based generator architecture for generative adversarial networks. In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp. 4401–4410 (2019)

    2019

  6. [14]

    arXiv preprint arXiv:1612.02136 (2016)

    Che, T., Li, Y., Jacob, A.P., Bengio, Y., Li, W.: Mode regularized generative adversarial networks. arXiv preprint arXiv:1612.02136 (2016)

    2016 arXiv

  7. [15]

    In: Conference on Neural Information Processing Systems, pp

    Nowozin, S., Cseke, B., Tomioka, R.: f-gan: Training generative neural samplers using variational divergence minimization. In: Conference on Neural Information Processing Systems, pp. 271–279 (2016)

    2016

  8. [16]

    arXiv preprint arXiv:1706.04156 (2017)

    Nagarajan, V., Kolter, J.Z.: Gradient descent gan optimization is locally stable. arXiv preprint arXiv:1706.04156 (2017)

    2017 arXiv

  9. [17]

    arXiv preprint arXiv:1705.10461 (2017) 23

    Mescheder, L., Nowozin, S., Geiger, A.: The numerics of gans. arXiv preprint arXiv:1705.10461 (2017) 23

    2017 arXiv

  10. [18]

    arXiv preprint arXiv:1808.09540 (2018)

    Oberman, A.M., Calder, J.: Lipschitz regularized deep neural networks converge and generalize. arXiv preprint arXiv:1808.09540 (2018)

    2018 arXiv

  11. [19]

    arXiv preprint arXiv:1805.10965 (2018)

    Scaman, K., Virmaux, A.: Lipschitz regularity of deep neural networks: analysis and efficient estimation. arXiv preprint arXiv:1805.10965 (2018)

    2018 arXiv

  12. [20]

    arXiv preprint arXiv:1807.00751 (2018)

    Zhou, Z., Song, Y., Yu, L., Wang, H., Liang, J., Zhang, W., Zhang, Z., Yu, Y.: Understanding the effectiveness of lipschitz-continuity in generative adversarial nets. arXiv preprint arXiv:1807.00751 (2018)

    2018 arXiv

  13. [21]

    In: International Conference on Machine Learning, pp

    Zhou, Z., Liang, J., Song, Y., Yu, L., Wang, H., Zhang, W., Yu, Y., Zhang, Z.: Lipschitz generative adversarial nets. In: International Conference on Machine Learning, pp. 7584–7593 (2019). PMLR

    2019

  14. [22]

    arXiv preprint arXiv:2004.13135 (2020)

    Herrera, C., Krach, F., Teichmann, J.: Estimating full lipschitz constants of deep neural networks. arXiv preprint arXiv:2004.13135 (2020)

    2020 arXiv

  15. [23]

    In: International Conference on Machine Learning, pp

    Kim, H., Papamakarios, G., Mnih, A.: The lipschitz constant of self-attention. In: International Conference on Machine Learning, pp. 5562–5571 (2021). PMLR

    2021

  16. [24]

    arXiv preprint arXiv:1802.05957 (2018)

    Miyato, T., Kataoka, T., Koyama, M., Yoshida, Y.: Spectral normalization for generative adversarial networks. arXiv preprint arXiv:1802.05957 (2018)

    2018 arXiv

  17. [25]

    In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp

    Bhaskara, V.S., Aumentado-Armstrong, T., Jepson, A.D., Levinshtein, A.: Gran- gan: Piecewise gradient normalization for generative adversarial networks. In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 3821–3830 (2022)

    2022

  18. [26]

    In: InternationalConference on Computer Vision, pp

    Wu, Y.-L., Shuai, H.-H., Tam, Z.-R., Chiu, H.-Y.: Gradient normalization for generative adversarial networks. In: InternationalConference on Computer Vision, pp. 6373–6382 (2021)

    2021

  19. [27]

    Conference on Neural Information Processing Systems 35, 8868–8881 (2022)

    Li, Y., Mo, Y., Shi, L., Yan, J.: Improving generative adversarial networks via adversarial learning in latent space. Conference on Neural Information Processing Systems 35, 8868–8881 (2022)

    2022

  20. [28]

    In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp

    Mao, Q., Lee, H.-Y., Tseng, H.-Y., Ma, S., Yang, M.-H.: Mode seeking generative adversarial networks for diverse image synthesis. In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp. 1429–1437 (2019)

    2019

  21. [29]

    arXiv preprint arXiv:1611.02163 (2016)

    Metz, L., Poole, B., Pfau, D., Sohl-Dickstein, J.: Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163 (2016)

    2016 arXiv

  22. [30]

    arXiv preprint arXiv:2106.00563 (2021)

    Li, Y., Shi, L., Yan, J.: Iid-gan: an iid sampling perspective for regularizing mode collapse. arXiv preprint arXiv:2106.00563 (2021)

    2021 arXiv

  23. [31]

    Proceedings of the IEEE 86(11), 2278–2324 (1998) 24

    LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998) 24

    1998

  24. [32]

    Master’s thesis, Department of Computer Science, University of Toronto (2009)

    Krizhevsky, A., Hinton, G., et al.: Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, University of Toronto (2009)

    2009

  25. [33]

    arXiv preprint arXiv:1506.03365 (2015)

    Yu, F., Seff, A., Zhang, Y., Song, S., Funkhouser, T., Xiao, J.: Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365 (2015)

    2015 arXiv

  26. [34]

    In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp

    Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., Fei-Fei, L.: Imagenet: A large- scale hierarchical image database. In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp. 248–255 (2009). Ieee

    2009

  27. [35]

    In: InternationalConference on Computer Vision, pp

    Mao, X., Li, Q., Xie, H., Lau, R.Y., Wang, Z., Paul Smolley, S.: Least squares generative adversarial networks. In: InternationalConference on Computer Vision, pp. 2794–2802 (2017)

    2017

  28. [36]

    arXiv preprint arXiv:1705.02894 (2017)

    Lim, J.H., Ye, J.C.: Geometric gan. arXiv preprint arXiv:1705.02894 (2017)

    2017 arXiv

  29. [37]

    In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp

    He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp. 770–778 (2016)

    2016

  30. [38]

    arXiv preprint arXiv:2112.07804 (2021)

    Xiao, Z., Kreis, K., Vahdat, A.: Tackling the generative learning trilemma with denoising diffusion gans. arXiv preprint arXiv:2112.07804 (2021)

    2021 arXiv

  31. [39]

    Conference on Neural Information Processing Systems 29, 2234–2242 (2016)

    Salimans, T., Goodfellow, I., Zaremba, W., Cheung, V., Radford, A., Chen, X.: Improved techniques for training gans. Conference on Neural Information Processing Systems 29, 2234–2242 (2016)

    2016

  32. [40]

    Conference on Neural Information Processing Systems 30 (2017)

    Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., Hochreiter, S.: Gans trained by a two time-scale update rule converge to a local nash equilibrium. Conference on Neural Information Processing Systems 30 (2017)

    2017

  33. [41]

    Conference on Neural Information Processing Systems 32 (2019)

    Kynk¨ a¨ anniemi, T., Karras, T., Laine, S., Lehtinen, J., Aila, T.: Improved precision and recall metric for assessing generative models. Conference on Neural Information Processing Systems 32 (2019)

    2019

  34. [42]

    ∇θ 1 2 MP m=1 ηmgm (Gm(z)) 2# 0   =   MP m=1 ηmgm (Gm(z)) ∇θ

    Zadorozhnyy, V., Cheng, Q., Ye, Q.: Adaptive weighted discriminator for training generative adversarial networks. In: IEEE/CVF Computer Vision and Pattern Recognition Conference, pp. 4781–4790 (2021) 25 Appendix Table of Contents A Overview 27 B Analysis of the dynamic theory ...

    2021

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.