{
  "schemaVersion": 1,
  "id": "atlas-lesson-denoising-generative-objectives",
  "title": "Train a denoiser and sample iteratively",
  "category": "learning",
  "format": "Atlas lesson",
  "description": "Compute a noisy training target and distinguish noise regression from a flow vector field.",
  "canonicalUrl": "https://semiaifoundry.com/learn/atlas/denoising-generative-objectives/",
  "publishedDate": null,
  "revisionDate": "2026-10-07",
  "author": null,
  "topic": "model-lifecycle-multimodal",
  "series": null,
  "license": null,
  "rightsUrl": "https://semiaifoundry.com/ai-atlas/licenses.html",
  "version": "1.3.0",
  "status": "Published educational lesson",
  "evidenceScope": "Mode: worked example. One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed. Numerical examples are authored calculations, except explicitly identified automated core fixtures. Primary sources support the scoped mechanism; no human specialist review or learner outcome is implied.",
  "contentCoverage": "published-lesson",
  "artifactUrl": "https://semiaifoundry.com/ai-atlas/lesson-data/denoising-generative-objectives-d5f093d00f1eca4e.json",
  "interactiveUrl": "https://semiaifoundry.com/ai-atlas/#denoising-generative-objectives",
  "related": [],
  "supersedes": [],
  "supersededBy": [],
  "contentHash": "c03711621d9017474edf5d3b533149e92c1198627ec0cad2ba3f0f650e382755",
  "contentLength": 6713,
  "jsonUrl": "https://semiaifoundry.com/agents/publications/atlas-lesson-denoising-generative-objectives.json",
  "markdownUrl": "https://semiaifoundry.com/agents/publications/atlas-lesson-denoising-generative-objectives.md",
  "sections": [
    {
      "id": "section-1",
      "title": "Train a denoiser and sample iteratively",
      "start": 43,
      "end": 460
    },
    {
      "id": "section-2",
      "title": "The mechanism",
      "start": 460,
      "end": 1440
    },
    {
      "id": "section-3",
      "title": "Science and assumptions",
      "start": 1440,
      "end": 1468
    },
    {
      "id": "section-4",
      "title": "Train a denoiser and sample iteratively · worked relation",
      "start": 1468,
      "end": 2808
    },
    {
      "id": "section-5",
      "title": "Engineering example",
      "start": 2808,
      "end": 3181
    },
    {
      "id": "section-6",
      "title": "Distinctions and tradeoffs",
      "start": 3181,
      "end": 3635
    },
    {
      "id": "section-7",
      "title": "Specifications and interfaces",
      "start": 3635,
      "end": 4089
    },
    {
      "id": "section-8",
      "title": "Interfaces",
      "start": 4089,
      "end": 4325
    },
    {
      "id": "section-9",
      "title": "Check your understanding",
      "start": 4325,
      "end": 4354
    },
    {
      "id": "section-10",
      "title": "Why are the two regression targets different?",
      "start": 4354,
      "end": 4930
    },
    {
      "id": "section-11",
      "title": "Evidence and scope",
      "start": 4930,
      "end": 5579
    },
    {
      "id": "section-12",
      "title": "Sources",
      "start": 5579,
      "end": 5967
    },
    {
      "id": "section-13",
      "title": "Continue exploring",
      "start": 5967,
      "end": 5990
    },
    {
      "id": "section-14",
      "title": "Foundations",
      "start": 5990,
      "end": 6184
    },
    {
      "id": "section-15",
      "title": "Related lessons",
      "start": 6184,
      "end": 6713
    }
  ],
  "citations": [
    {
      "title": "Explore in Atlas",
      "url": "https://semiaifoundry.com/ai-atlas/#denoising-generative-objectives"
    },
    {
      "title": "Denoising Diffusion Probabilistic Models",
      "url": "https://arxiv.org/html/2006.11239v2"
    },
    {
      "title": "Flow Matching for Generative Modeling",
      "url": "https://arxiv.org/html/2210.02747v2"
    },
    {
      "title": "Published lesson data",
      "url": "https://semiaifoundry.com/ai-atlas/lesson-data/denoising-generative-objectives-d5f093d00f1eca4e.json"
    },
    {
      "title": "Atlas releases and licensing",
      "url": "https://semiaifoundry.com/ai-atlas/licenses.html"
    },
    {
      "title": "Loss, gradients & optimizer",
      "url": "https://semiaifoundry.com/learn/atlas/loss-optimizer/"
    },
    {
      "title": "Evaluation & the AGI frontier",
      "url": "https://semiaifoundry.com/learn/atlas/evaluation/"
    },
    {
      "title": "Pair images and captions in a shared representation",
      "url": "https://semiaifoundry.com/learn/atlas/image-text-contrastive-learning/"
    },
    {
      "title": "Patches, projection and visual instruction tuning",
      "url": "https://semiaifoundry.com/learn/atlas/visual-language-bridge/"
    },
    {
      "title": "Audio representation, alignment and transcript loss",
      "url": "https://semiaifoundry.com/learn/atlas/speech-audio-sequence-objectives/"
    },
    {
      "title": "Predict temporal representations without generating pixels",
      "url": "https://semiaifoundry.com/learn/atlas/video-latent-prediction/"
    }
  ],
  "recordHash": "ccde5ebd9e1a4d055bf4a6a35d77103d3e6b7a645869ae176d4237120e18b685",
  "contentFormat": "text/markdown",
  "content": "model-lifecycle-multimodal · Atlas v1.3.0\n\n# Train a denoiser and sample iteratively\n\nCompute a noisy training target and distinguish noise regression from a flow vector field.\n\n[Explore in Atlas](https://semiaifoundry.com/ai-atlas/#denoising-generative-objectives)Mathematical, data and behavior space; logical dimensions are not physical distance\n\nLearning objectiveCompute a noisy training target and distinguish noise regression from a flow vector field.\n\n## The mechanism\n\nGenerative training can operate on continuous values rather than discrete next-token labels. In a DDPM noise-prediction objective, a clean example is mixed with Gaussian noise at a chosen level and the model predicts that noise. Generation later runs a specified reverse sampling process; a single good denoising prediction is not a complete generated sample. Flow matching instead regresses a vector field along a declared probability path. The target, time convention and sampler therefore differ even when both start from noise. Neither mechanism is implemented by the tiny causal decoder.\n\n1.  Specify whether time indexes a discrete diffusion level or a continuous path.\n2.  Construct the noisy/interpolated input and exact regression target from retained random values.\n3.  Compute the declared coordinate-sum or coordinate-mean loss.\n4.  Evaluate samples from a complete stated sampler with fixed seeds and quality metrics separate from regression loss.\n\n## Science and assumptions\n\n### Train a denoiser and sample iteratively · worked relation\n\n$$\nx_t=\\sqrt{\\bar\\alpha_t}x_0+\\sqrt{1-\\bar\\alpha_t}\\epsilon,\\quad L_\\epsilon=\\lVert\\epsilon-\\epsilon_\\theta(x_t,t)\\rVert^2,\\qquad x_t^{flow}=(1-t)x_0+t x_1,\\quad u=x_1-x_0\n$$\n\nCompute a noisy training target and distinguish noise regression from a flow vector field.\n\nSymbols and units\n\nx\\_0\n\nClean diffusion value or explicitly named flow start\n\ndimensionless\n\nx\\_1\n\nData endpoint in the flow example\n\ndimensionless\n\nalpha bar\\_t\n\nCumulative diffusion signal factor\n\ndimensionless\n\nepsilon\n\nSampled standard Gaussian noise\n\ndimensionless\n\nt\n\nDeclared diffusion index or normalized flow time\n\nindex or dimensionless\n\nu\n\nStraight-path target velocity\n\nvalue per normalized time\n\nL\\_epsilon\n\nCoordinate-summed noise error\n\nsquared value\n\nFor DDPM-style training, x0=2, cumulative alpha=0.64 and sampled noise epsilon=−1 give xt=0.8×2+0.6×(−1)=1. A noise prediction −0.5 has squared error 0.25. In a separate straight flow path from noise x0=−1 to data x1=2, time t=0.25 gives xt=−0.25 and target velocity x1−x0=3. The noise target −1 and flow target 3 refer to different constructions.\n\nAssumptions One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed.\n\n## Engineering example\n\nFor DDPM-style training, x0=2, cumulative alpha=0.64 and sampled noise epsilon=−1 give xt=0.8×2+0.6×(−1)=1. A noise prediction −0.5 has squared error 0.25. In a separate straight flow path from noise x0=−1 to data x1=2, time t=0.25 gives xt=−0.25 and target velocity x1−x0=3. The noise target −1 and flow target 3 refer to different constructions.\n\n## Distinctions and tradeoffs\n\nIt is a mistake to conclude: All noise-based objectives must have the same target.\n\nContinuous objectives offer different representation and sampling choices. More sampling steps incur additional work but do not alone guarantee useful or faithful outputs.\n\nTeach modality-specific representation, objective and evaluation choices. Compute a noisy training target and distinguish noise regression from a flow vector field.\n\n## Specifications and interfaces\n\nLearning mode\n\nworked example\n\nThe executable decoder has 423 parameters; worked extensions do not imply an additional simulator.\n\nWorked contract\n\nCompute a noisy training target and distinguish noise regression from a flow vector field.\n\nOne-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed.\n\n### Interfaces\n\nInputs and state Input\n\nSpecify whether time indexes a discrete diffusion level or a continuous path.\n\nResult and check Output\n\nCompute a noisy training target and distinguish noise regression from a flow vector field.\n\n## Check your understanding\n\n### Why are the two regression targets different?\n\n1.  The flow velocity is the model’s next-token probability.\n2.  They supervise different declared paths and predicted quantities.\n3.  All noise-based objectives must have the same target.\n\nAnswer and reasoning\n\nThey supervise different declared paths and predicted quantities.\n\nThey supervise different declared paths and predicted quantities. One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed.\n\n## Evidence and scope\n\nMode: worked example. One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed. Numerical examples are authored calculations, except explicitly identified automated core fixtures. Primary sources support the scoped mechanism; no human specialist review or learner outcome is implied.\n\nModel lifecycle studio: worked example. One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed.\n\n## Sources\n\n-   [Denoising Diffusion Probabilistic Models](https://arxiv.org/html/2006.11239v2)\n-   [Flow Matching for Generative Modeling](https://arxiv.org/html/2210.02747v2)\n\n[Published lesson data](https://semiaifoundry.com/ai-atlas/lesson-data/denoising-generative-objectives-d5f093d00f1eca4e.json) · [Atlas releases and licensing](https://semiaifoundry.com/ai-atlas/licenses.html)\n\n## Continue exploring\n\n### Foundations\n\n-   [Loss, gradients & optimizer](https://semiaifoundry.com/learn/atlas/loss-optimizer/)\n-   [Evaluation & the AGI frontier](https://semiaifoundry.com/learn/atlas/evaluation/)\n\n### Related lessons\n\n-   [Pair images and captions in a shared representation](https://semiaifoundry.com/learn/atlas/image-text-contrastive-learning/)\n-   [Patches, projection and visual instruction tuning](https://semiaifoundry.com/learn/atlas/visual-language-bridge/)\n-   [Audio representation, alignment and transcript loss](https://semiaifoundry.com/learn/atlas/speech-audio-sequence-objectives/)\n-   [Predict temporal representations without generating pixels](https://semiaifoundry.com/learn/atlas/video-latent-prediction/)"
}