{"id":"denoising-generative-objectives","name":"Train a denoiser and sample iteratively","shortName":"Train a denoiser and sample iteratively","role":"Compute a noisy training target and distinguish noise regression from a flow vector field.","description":"Generative 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.","mechanism":["Specify whether time indexes a discrete diffusion level or a continuous path.","Construct the noisy/interpolated input and exact regression target from retained random values.","Compute the declared coordinate-sum or coordinate-mean loss.","Evaluate samples from a complete stated sampler with fixed seeds and quality metrics separate from regression loss."],"learningObjective":"Compute a noisy training target and distinguish noise regression from a flow vector field.","engineeringExample":"For 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.","evidenceNotes":"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.","specs":[{"label":"Learning mode","value":"worked example","context":"The executable decoder has 423 parameters; worked extensions do not imply an additional simulator."},{"label":"Worked contract","value":"Compute a noisy training target and distinguish noise regression from a flow vector field.","context":"One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed."}],"interfaces":[{"name":"Inputs and state","direction":"in","signal":"Specify whether time indexes a discrete diffusion level or a continuous path."},{"name":"Result and check","direction":"out","signal":"Compute a noisy training target and distinguish noise regression from a flow vector field."}],"tradeoff":"Continuous objectives offer different representation and sampling choices. More sampling steps incur additional work but do not alone guarantee useful or faithful outputs.","whyAI":"Teach modality-specific representation, objective and evaluation choices. Compute a noisy training target and distinguish noise regression from a flow vector field.","misconception":"It is a mistake to conclude: All noise-based objectives must have the same target.","check":{"question":"Why are the two regression targets different?","options":["The flow velocity is the model’s next-token probability.","They supervise different declared paths and predicted quantities.","All noise-based objectives must have the same target."],"answer":1,"explanation":"They 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."},"prerequisites":["loss-optimizer","evaluation"],"related":["loss-optimizer","evaluation","image-text-contrastive-learning","visual-language-bridge","speech-audio-sequence-objectives","video-latent-prediction"],"sources":[{"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"}],"level":"model","scale":"Mathematical, data and behavior space; logical dimensions are not physical distance","kind":"model","branch":"model-lifecycle-multimodal","subbranch":"Train a denoiser and sample iteratively","workstream":"W11","visualFamily":"mathematical","labId":"library","labScope":"Model 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.","science":[{"title":"Train a denoiser and sample iteratively · worked relation","equation":"x_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","explanation":"For 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.","assumptions":"One-dimensional authored samples; diffusion and flow use separate time/start conventions explicitly. No complete sampler, likelihood bound or image-quality benchmark is executed."}],"history":[]}