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Improving Generative Model Self-Training with Geometrically Modified Outputs

Patrick Batsell, Thomas Walker, Richard Baraniuk

arXiv:2609.35512v1cs.LGcs.AI

TL;DR

As high-quality training data becomes scarce, self-training is attractive, but naïve training on generated samples degrades model quality. GMOs reweight generator-Jacobian singular values to strengthen negative guidance, and empirical evaluations report improved performance across self-training frameworks. The approach is evaluated only in the image domain, with selecting α remaining an open consideration.

  • Problem

    High-quality training data is difficult to expand, while naïve self-training on model outputs degrades generative-model quality.

    Source: §1 · p. 2

  • Method

    GMOs modify model outputs by reweighting singular values of the generator’s input-output Jacobian to provide a stronger, targeted negative signal.

    Source: §5 · p. 9

  • Results

    GMOs consistently improve generative performance when integrated into multiple self-training frameworks, including Neon and SIMS.

    Source: §5 · p. 9

  • Takeaways & Limitations

    GMOs provide a geometric output augmentation for strengthening negative signals in self-training algorithms such as Neon and SIMS.

    Source: §5 · p. 9

    GMOs require specifying α, and the experiments cover only the image domain; extending them to other modalities and architectures remains future work.

    Source: §5 · p. 9

Abstract

from arXiv · show

Self-training generative models - the continued improvement of a model using its own outputs - is becoming increasingly important as high-quality training data becomes scarce. However, naively finetuning on model-generated samples leads to degradation through model collapse and the model autophagy disorder. Negative-guidance self-training methods turn this degradation into a useful signal, using a model finetuned on its own outputs to guide the original model toward improved generation. Existing methods, however, take the negative signal in standard model outputs as given. We instead ask whether this signal can be explicitly strengthened. We introduce Geometrically Modified Outputs (GMOs), which reweight the singular values of the generator's input-output Jacobian to increase the influence of its leading singular directions. This geometric modification amplifies the mode-seeking behavior and distortions of standard outputs, providing a stronger and more targeted negative signal for self-training. Across a range of one-step generative models, GMOs consistently improve the performance of negative-guidance methods, including Neon and SIMS, compared with using standard model outputs.

1 INTRODUCTION

Scarcity of high-quality training data has increased interest in self-training, but naïve training on a model’s own outputs degrades quality. This paper strengthens negative-guidance self-training signals with Geometrically Modified Outputs (GMOs).

1 INTRODUCTION

  • High-quality training data is difficult to expand, increasing interest in developing models using only their own outputs.Source: §1 · p. 2
  • Naïve self-training degrades generative-model quality through model autophagy disorder and model collapse, including artifacts and reduced output diversity.Source: §1 · p. 2
  • Negative guidance has produced the strongest self-training results, motivating the question of whether augmenting outputs can amplify its negative signal.Source: §1 · p. 2
  • GMOs improve the effectiveness of both SIMS and Neon self-training algorithms.Source: §1 · p. 2
  • GMOs offer a geometric perspective on self-training, motivated by observations that generator Jacobians’ effective ranks collapse during naïve self-training.Source: §1 · p. 2

2 THE GEOMETRY AND SELF-TRAINING OF GENERATIVE MODELS

The section establishes notation for one-step generative models and their local geometry, then defines self-training and the negative-guidance algorithms considered.

2.1 GENERATIVE MODELS

  • Generative models are trained on high-quality real data, and the paper focuses on one-step inference maps from latent vectors to outputs.Source: §2.1 · p. 2

2.2 THE GEOMETRY OF GENERATIVE MODELS

  • A one-step output is locally decomposed using the input-output Jacobian and an offset, with local geometry defined by the Jacobian’s spectral properties.Source: §2.2 · p. 2

2.3 SELF-TRAINING GENERATIVE MODELS

  • Self-training seeks a model that outperforms its predecessor using only samples from the predecessor’s output distribution, under a budget measured in images.Source: §2.3 · p. 3
  • Naïve continuation on generated samples can degrade models; alternative approaches include self-play, discriminative guidance, negative guidance, and output verification.Source: §2.3 · p. 3
  • The paper asks whether augmenting outputs can strengthen negative guidance and focuses on Neon, while also considering SIMS.Source: §2.3 · p. 3
  • Neon finetunes the base model on generated samples, then extrapolates away from the finetuned parameters; its improvement guarantee relies on mode-seeking samples.Source: §2.3 · p. 3

3 GEOMETRICALLY MODIFIED OUTPUTS

Geometrically Modified Outputs (GMOs) amplify the influence of leading singular directions, motivated by effective-rank collapse and artifacts during naïve self-training. Approximate computation makes them more tractable while retaining the reported spectral-statistic accuracy.

3 GEOMETRICALLY MODIFIED OUTPUTS

  • GMOs amplify the top singular vectors of the generator map, motivated by effective-rank collapse and low-quality artifacts in prominent directions under naïve self-training.The paper reports verifying effective-rank collapse for a MeanFlow model and amplified mode-seeking for GMOs.Source: §3 · p. 3

3.1 GMO GENERATION

  • An α-GMO reweights the generator Jacobian’s singular values, transferring trailing spectral energy to the leading component while preserving total spectral energy.At α=0 the output is unchanged; at α=1 it is entirely determined by the top singular vectors.Source: §3.1 · p. 4

3.2 COMPUTATIONAL COMPLEXITY OF GENERATING GMOS

  • The approximate method avoids materializing the Jacobian, estimates its leading spectral statistics with power iteration, and estimates its squared Frobenius norm with Hutchinson’s estimator.Its total time complexity is O((k + m) · Cf) and memory complexity is O(d + h).Source: §3.2 · p. 5
  • The approximate GMO implementation is around 2 times faster than the exact implementation, with little loss from approximation and power-iteration counts above roughly ten.The authors use 20 iterations and report improved tractability where memory constraints would otherwise prevent computation.Source: Figure 3 · p. 5, §3.2 · p. 6

4 EXPERIMENTS

Across the tested settings, GMOs improve negative-guidance self-training over standard outputs, including under matched-noise comparisons. The experiments report stronger FID performance, preserved or improved robust diversity, transfer across models, and gains with both Neon and SIMS.

4.1 GMOS OUTPERFORM STANDARD OUTPUTS AND GAUSSIAN PERTURBED OUTPUTS

  • At α equal to 0.05, GMOs yield the best-performing model and outperform equal-energy Gaussian perturbations; their perturbations also look more like standard outputs.The Gaussian baseline matches each GMO’s perturbation amplitude but lacks directional guidance from the input-output Jacobian.Source: §4.1 · p. 6

4.2 THE EFFECT OF GMOS ON FID, PRECISION, RECALL, DENSITY AND COVERAGE

  • FID reaches an optimum around α equal to 0.2; as α increases, precision rises and recall falls, while density and coverage are maintained or improved, including IMM coverage +.021 with recall −.007.The authors conclude that GMOs improve image quality and diversity simultaneously rather than trading one for the other.Source: §4.2 · p. 7, Figure 5 · p. 7

4.3 GMOS TRANSFER ACROSS MODEL SIZES AND INFERENCE STRATEGIES

  • GMOs from SiT-B/2 MeanFlow improve a larger SiT-L/2 model with Neon, and one-step IMM GMOs improve two-step IMM on CIFAR10.The SiT-B/2-to-SiT-L/2 transfer is more challenging because FID is more sensitive to merge weight w.Source: §4.3 · p. 7, Figure 6 · p. 8

4.4 GMOS IMPROVE STATE-OF-THE-ART GENERATIVE MODELS

  • Increasing α systematically lowers Neon’s optimal fine-tuning budget B and merge weight w, and GMOs reach the optimal checkpoint sooner.The authors attribute the faster optimum to GMOs inducing mode-seeking degradation faster.Source: §4.4 · p. 8
  • 17–105%: GMOs increase Neon’s FID reduction over the base model across five ImageNet256 architectures, with lower FID than standard-output Neon in all five.The reported average and standard deviation are based on five random seeds and 50,000 output samples.Source: Table 1 · p. 8, §4.4 · p. 8

4.5 GMOS IMPROVE NEGATIVE GUIDANCE BEYOND NEON

  • GMOs are data-space modifications that strengthen both Neon’s weight-space extrapolation and SIMS-style output-space guidance.Neon is the primary algorithm; SIMS keeps the finetuned model in the sampling loop and extrapolates predictions at inference time.Source: §4.5 · p. 9
  • 6.92 ± .05: On IMM, SIMS guidance with GMO-finetuned models reaches this FID, versus 7.59±.06 with standard-output guidance, from a base FID of 8.34±.06.This extends the GMO benefit beyond Neon to output-space guidance.Source: §4.5 · p. 9

5 DISCUSSION

GMOs augment negative-signal construction rather than replace self-training: they modify the synthetic outputs while retaining downstream finetuning and guidance. Their demonstrated scope includes Neon and SIMS, while α selection and extension beyond images remain open considerations.

5 DISCUSSION

  • GMOs modify the synthetic outputs used to train the negative model while retaining downstream finetuning and guidance procedures.They are demonstrated with Neon’s weight-space extrapolation and SIMS-style inference-time guidance.Source: §5 · p. 9
  • The computational trade-off depends on the downstream method: GMOs add offline sample-construction cost while Neon’s merged model requires no additional inference-time evaluations.SIMS and Autoguidance require auxiliary-model predictions during sampling, while DDO uses multiple self-play rounds and reports a training budget of approximately 12% of pretraining compute.Source: §5 · p. 9
  • The authors identify further directions for work on using generator geometry to improve self-training, without specifying them in this passage.Source: §5 · p. 9
  • GMOs require specifying α, and a search-free heuristic for choosing an optimal value for a model and dataset remains an open need.A fixed α = 0.1 without per-model tuning outperforms standard-outputs Neon on four of five ImageNet-trained architectures and is within seed noise on the fifth.Source: §5 · p. 9
  • The reported GMO experiments are limited to the image domain; extension to other modalities and architectures remains future work.Source: §5 · p. 9

A PROOF OF THEOREM 1

The proof expands the generator’s Jacobian-vector product into singular components and relates the GMO construction to a redistribution of normalized spectral energy. The reweighting concentrates energy in leading components while preserving the constructed matrix’s squared Frobenius norm.

A PROOF OF THEOREM 1

  • The proof writes the standard output as s = J_z z + b_z and expands its Jacobian-vector product into singular components before identifying the construction with Algorithm 2.Source: A PROOF OF THEOREM 1 · p. 14

A.1 SPECTRAL ENERGY REDISTRIBUTION

  • The leading component receives exactly the spectral energy removed from trailing components.Source: §A.1 · p. 14
  • The α transformation interpolates normalized squared singular values between the original spectrum and one with all spectral energy in the leading component.The singular directions and affine offset remain unchanged; α = 0 recovers the original output, while α = 1 gives a modified linear term of rank at most one.Source: §A.1 · p. 14
  • The preserved quantity is the constructed matrix’s squared Frobenius norm, not necessarily the generated output’s norm.This controlled local-spectrum concentration is motivated by spectral concentration observed during self-training.Source: §A.1 · p. 15

B GMOS ARE MODE SEEKING

The paper examines whether GMOs amplify mode-seeking behavior and distortions in model outputs. Across increasing α values, their embeddings become more concentrated, while the visual examples suggest that perturbations focus on output distortions.

B GMOS ARE MODE SEEKING

  • For larger α, t-SNE projections of GMO CLIP embeddings become increasingly concentrated, indicating focus on the same regions of the data distribution.Figure 7’s top row shows the embeddings for 2048 GMOs at α values 0.0, 0.1, 0.2, 0.6, and 1.0.Source: B GMOS ARE MODE SEEKING · p. 15, Figure 7 · p. 15
  • The visual examples suggest that GMO perturbations concentrate on output distortions, and outputs without clear distortions can remain relatively unchanged.This supports the paper’s view that GMOs amplify the negative signal in model outputs.Source: B GMOS ARE MODE SEEKING · p. 15

C THE EFFECT OF GMOS ON THE COMPUTE AND WEIGHT-MERGING OF NEON

For Neon, increasing α is associated with lower optimal finetuning budgets and merge weights. Across models, the complete pipeline, including GMO generation, costs between 0.01% and 1.1% of pretraining compute.

C THE EFFECT OF GMOS ON THE COMPUTE AND WEIGHT-MERGING OF NEON

D EXPERIMENTAL DETAILS

The experiments generate and compare standard outputs with GMOs across finetuning compute budgets. The best observed FID occurs at different budgets for standard and GMO datasets.

D EXPERIMENTAL DETAILS

  • For the 2-step IMM model, the best FID occurs at 1.6 × 10^5 compute with standard outputs, 9.6 × 10^4 with its GMOs, and 1.28×10^5 with 1-step-model GMOs.The experiment generates 10,000 GMOs at α = 0.1; the 2-step base model is finetuned with a 2.56 × 10^5 compute budget.Source: D EXPERIMENTAL DETAILS · p. 16
  • For each finetuning dataset, eight checkpoints are evaluated across weight-merging parameters from 0 to 1 in increments of 0.1.The reported curves show the evaluation sweeps that yielded the lowest FID scores.Source: D EXPERIMENTAL DETAILS · p. 16

D.2 TABLE 1

The section describes the hyperparameter sweeps underlying Table 1 and Figure 10, comparing minimum FID and the corresponding weight-merging parameter across compute levels and models.

D.2 TABLE 1

  • Figure 9 provides the complete evaluation curves for the experiment shown in Figure 6’s right panel.The experiment’s reported compute budgets and best-FID comparisons are described separately in the experimental details.Source: Figure 9 · p. 17
  • Each sweep generates 30,000 standard outputs and GMOs, then evaluates finetuned checkpoints using FID with 50,000 samples.The tested α values are 0.05 and 0.1 for MeanFlow SiT-B/2 and AlphaFlow SiT-XL/2; 0.1 and 0.2 for MeanFlow SiT-L/2, AlphaFlow SiT-B/2, and IMM DiT-XL/2.Source: §D.2 · p. 17
  • Finetuning spans 48 and 60 epochs for MeanFlow SiT-B/2 and SiT-L/2, respectively, and 32 and 15 epochs for AlphaFlow SiT-B/2 and SiT-XL/2.IMM DiT-XL/2 is finetuned for 109 epochs; each configuration evaluates multiple checkpoints and sweeps weight-merging values.Source: §D.2 · p. 17
  • Figure 10 shows each model’s minimum FID across weight-merging values at each compute level, alongside the weight-merging value yielding that minimum.The sweep computation times are based on eight NVIDIA RTX A6000 GPUs.Source: §D.2 · pp. 16, 18

D.3 SIMS-STYLE GUIDANCE

The SIMS-style guidance experiments report image-quality and diversity measures for GMO-based self-training and describe the evaluation setup. The accompanying material includes model-license information and additional qualitative examples.

D.3 SIMS-STYLE GUIDANCE

  • The IMM DiT-XL/2 comparison evaluates standard outputs and GMOs using guidance strengths ω ∈ {0.5, 1.0, 1.6}; the α = 0.2 sweep also tests ω = 2.4.The reported evaluation uses 50,000 samples per FID evaluation, one-step sampling, and classifier-free guidance scale 1.5; Table 3 specifies both guided variants use ω = 1.6.Source: §D.3 · p. 18, Table 3 · p. 18
  • Across every architecture, density and coverage are maintained or improved from standard-outputs Neon to GMOs, with the authors reporting simultaneous improvement in image quality and diversity.Table 4 reports precision, recall, density, and coverage across five seeds; Table 5 reports statistical significance analyses for Table 1’s FID results.Source: §D.3 · p. 18, Table 4 · p. 18, Table 5 · p. 19

E MODEL LICENSES

  • The listed model-weight sources and licenses are IMM under CC BY-NC-SA 4.0, MeanFlow under MIT, and AlphaFlow under the Snap Inc. Non-Commercial license.Source: E MODEL LICENSES · p. 19
  • The supplementary qualitative figures show additional GMO examples and comparisons, while Figure 13 juxtaposes exact and approximate top left singular vectors of the input-output Jacobians.The figure caption says Figure 3 quantitatively demonstrates that power iteration can accurately approximate those singular vectors.Source: Figure 11 · p. 21, Figure 12 · pp. 22–23, Figure 13 · p. 24

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