{ "schemaVersion": 1, "slug": "ddpm", "title": "Learn to reverse a noise schedule", "className": "TinyDDPM", "question": "How does a model learn to reverse corruption?", "description": "Inspect signal and noise coefficients, predict the added noise, and replay a complete short reverse chain with the same denoiser.", "category": "Generative models", "tags": [ "DDPM", "Diffusion", "Noise schedule", "Denoising" ], "curator": "TensorViz", "attribution": "Original TensorViz teaching example. PyTorch provides the underlying operators.", "license": "No separate redistribution license has been declared for these project examples.", "limitations": "Untrained denoiser and shortened schedule, not the original image model or 1,000-step beta schedule. The terminal forward marginal is not fully Gaussian. Reverse samples illustrate arithmetic and do not establish image quality. No likelihood evaluation or learned variance; the graph shows one training forward, while the source and panel expose the repeated sampling loop.", "openingStep": "corrupt", "provenance": { "capturedAt": "2026-09-17T06:47:42.907335+00:00", "productRevision": "cf0ab1c9c58c1dc640235b742fa032cade00b43c", "sourceSha256": "41018be9b51142b30ad5eb268182537f953328bcdf478c8859e71ee34b69372f", "python": "3.13.13", "torch": "2.7.1", "execution": { "inputs": { "args": [ { "shape": [ 1, 1, 8, 8 ], "dtype": "float32", "fill": "randn" }, { "shape": [ 1, 1, 8, 8 ], "dtype": "float32", "fill": "randn" }, { "shape": [ 1 ], "dtype": "int64", "fill": "ones" } ] }, "device": "cpu", "mode": "eval", "seed": 0, "deterministic": true }, "outcome": "passed", "outputShape": [ 1, 1, 8, 8 ], "method": "TensorViz multifile projection and isolated execution worker", "scope": "12-step DDPM schedule; tiny time-conditioned U-Net, oracle reconstruction, posterior-mean equivalence, final-step noise and epsilon-MSE gradient checks.", "numericalChecks": [ "Cumulative signal decreases; known noise reconstructs the clean image at early, middle and late steps", "Epsilon-parameterized reverse means match the analytic forward posterior", "The last reverse step adds no stochastic noise and recovers x0 with oracle epsilon", "Time conditioning affects predictions and receives gradients; one fixed-batch SGD step lowers epsilon MSE" ] }, "layers": [ { "id": "input", "label": "input", "op": "Input", "kind": "input", "parent": null, "params": { "shape": "1,1,8,8" } }, { "id": "corrupt", "label": "corrupt", "op": "Module", "kind": "module", "parent": null, "params": {}, "source": { "line": 51, "endLine": 51 } }, { "id": "denoiser", "label": "denoiser", "op": "Module", "kind": "module", "parent": null, "params": {}, "source": { "line": 52, "endLine": 52 } }, { "id": "output", "label": "output", "op": "Output", "kind": "output", "parent": null, "params": {} }, { "id": "python@corrupt#0", "label": 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A training example can jump directly to any timestep; it does not need to run every preceding noising step." }, { "id": "time", "target": "denoiser.time_embedding", "title": "Tell the denoiser which noise level it sees", "note": "The selected timestep becomes eight learned features, added to fine and coarse image features. The same network weights serve every timestep." }, { "id": "predict", "target": "denoiser.bottleneck", "title": "Combine coarse context with a fine-resolution skip", "note": "The tiny U-Net contracts from 8 × 8 to 4 × 4 and expands again. Concatenation preserves fine features while the bottleneck supplies broader context." }, { "id": "reverse", "target": "denoiser.noise", "title": "Predict epsilon, then reuse the network in reverse", "note": "Training minimizes MSE against the known added noise. Sampling starts from Gaussian noise and converts each prediction into the reverse mean, adding posterior noise except at the final step. Choose a reverse-chain checkpoint in the panel." } ], "milestone": { "name": "Denoising diffusion probabilistic models", "year": 2020, "kind": "Training method", "fidelity": "A 12-step teaching schedule with beta from 0.02 to 0.18. Width-eight, two-resolution U-Net with a learned timestep embedding predicts epsilon for an 8 × 8 grayscale input. Reverse sampling uses fixed posterior variance.", "papers": [ { "title": "Denoising Diffusion Probabilistic Models", "url": "https://arxiv.org/abs/2006.11239" } ] }, "experiment": { "kind": "matrices", "title": "Corrupt once; denoise repeatedly.", "description": "Recorded forward marginals and a separate 12-step reverse chain. 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These are coupled marginal examples, not consecutive forward-chain samples. The untrained prediction is compared with the actual added noise; oracle reconstruction uses known noise unavailable during generation.", "matrices": [ { "label": "Clean x₀", "values": [ [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 1, -1, -1, 1, 0, 0 ], [ 0, 0, 1, -1, -1, 1, 0, 0 ], [ 0, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ] }, { "label": "Noisy x12", "values": [ [ -0.34207829619216673, 1.0905410443739676, 0.0773211403338211, 0.31189065217764866, -1.33648874036214, -1.5131858840545633, 0.2884010066199031, 1.6604247272500139 ], [ 1.380109334621287, 1.5095114032203885, 0.6540917344565454, -0.11549066602212958, -0.1721053492948087, 0.34570452378814737, -0.7891953627030713, -1.8247768599541587 ], [ -0.36754512893120983, 0.46628209129802894, 1.4695298107797778, 0.5601800888197005, 0.5365911661954986, 0.0955057884035731, -0.6711805532323472, -0.27944775967722013 ], [ 0.5338612382824123, 0.3018099101548909, 0.5623333509537038, -0.7511708638926539, -1.4241344657594621, 0.7893872248256377, 1.7651124027738543, 0.6639323716932642 ], [ -0.07984478606311077, 1.9767005880892405, -0.45054294740000544, -0.009462929759798788, -0.4855384421529932, 1.7116470078683803, 1.486200904601686, 1.0065744670860377 ], [ -1.2003672244650725, 0.6947239591803909, -1.244891324608374, 0.6702695981288804, 0.044775173145820135, 0.6104395363640877, -0.31777439172482186, 0.031326503726017516 ], [ 0.8365370486789171, 0.41966177023944146, -0.7702635849759012, 0.3358814186426819, -0.8506735181332348, 1.4511302569759175, -1.1352305932876932, 1.4351437207390185 ], [ -0.7658120296084683, -1.2235910417439877, 0.9669336246605403, 0.7667241480403002, 0.5025334740000911, -0.8979873856663928, -1.548817996505195, -1.7224035115599903 ] ] }, { "label": "Untrained estimate of x₀", "values": [ [ -0.6137591532227292, 2.2807920360068286, 0.34102387501308873, 0.8352194219999243, -2.3835154960271616, -2.6215574999508133, 0.7020880906935464, 3.2423167381581446 ], [ 2.7569846541256275, 2.982680594236041, 1.4149197375974893, -0.012659016548372558, -0.22787474452248013, 0.922271898200391, -1.3365674321432688, -3.297634224498934 ], [ -0.6157252747443447, 1.0207464647821762, 3.01094720510529, 1.2810414497380944, 1.2371154299179492, 0.3624015253082683, -1.0859045457873095, -0.28342967602480723 ], [ 1.0159606069688603, 0.7170808227603929, 1.2639154953207394, -1.2222752530599879, -2.4114488062095685, 1.6988161247053406, 3.4575482287885184, 1.4078687738213034 ], [ -0.10283245565055701, 3.988636016109079, -0.6579681938496793, 0.26928947599104114, -0.6612415665838105, 3.5064845153680233, 2.9002480100644386, 2.0736832492213306 ], [ -2.3196844316297502, 1.5554916285807106, -2.3287530776995626, 1.6666191215346606, 0.10573085816179552, 1.49978383585182, -0.4965408199361375, 0.3341682295561344 ], [ 1.7510670851363712, 0.9587885683998851, -1.2998268969434588, 0.8810423591810158, -1.4172531457702777, 3.004933604610901, -2.0353518106987276, 3.0328340556397055 ], [ -1.417911169477785, -2.295075846819784, 2.007402154869709, 1.5180329509729493, 1.0237700496666367, -1.5935734636308065, -2.8956649515918036, -2.9974402095253994 ] ] } ], "vectors": [ { "label": "Signal coefficient √ᾱₜ", "values": [ 0.5264878919644721 ] }, { "label": "Noise coefficient √(1−ᾱₜ)", "values": [ 0.8501826272130044 ] } ], "metrics": [ { "label": "Noise prediction MSE", "value": 1.2756064806444616 }, { "label": "Oracle x₀ maximum error", "value": 2.220446049250313e-16 } ] }, { "id": "reverse-8", "label": "Reverse chain · reached t = 8", "target": "denoiser.noise", "note": "Start from an independent Gaussian sample and reuse the same untrained denoiser for all 12 reverse steps. This records the state after step 9. At t = 1 no new noise is added. The result is an arithmetic demonstration, not a generated image from a trained DDPM.", "matrices": [ { "label": "Starting Gaussian x₁₂", "values": [ [ 1.9249117241824258, -0.5366028899121948, -0.4948297558009618, -0.13026438953864916, -1.0690239065635012, -1.4852549921846954, -0.20972040395576166, -0.3643393493633569 ], [ -0.8590931641963496, -0.8648108757768114, 0.10026898815961566, -0.18629715380226006, -0.04090342607071494, 0.5640495334205501, 0.0932119811642432, 0.35118753942284603 ], [ -1.5138113422662705, -1.088919024636857, 0.8603237984858954, 0.19359456962423252, 0.9502385676055884, -0.4941852932165892, -1.2982792913599286, -1.6032537840232914 ], [ -0.9853345519305775, -0.20140000284536547, -0.2926421379217897, 1.2307429261727634, -1.088178729489697, 1.9944690735204083, 1.0828581822950778, -0.06696558382152167 ], [ 0.6005340298130931, -1.060530793882208, 1.0135638850622695, -1.154920746161395, -0.9905104755689237, 0.3358060081527006, 0.20674134346268438, -0.32822441749453335 ], [ -0.5491030711468649, -1.1775095895486403, -1.994299394993165, 0.13524046978719248, -0.1659011609794157, -0.26304414321759434, 1.2767004972816027, -1.398437840050358 ], [ 0.8845576263062012, -1.2325498547704115, 0.870800224759094, -1.4289663665735974, -1.4446622672233151, 0.17378905361890104, 0.6878057521841164, -0.3673670776851491 ], [ 0.4573816471385191, 0.49997476904525057, -0.9070984646616539, 2.063126396023007, 1.215892289786361, -1.7788980772676535, -1.5280506573622274, 0.7365234344863488 ] ] }, { "label": "Current sample x8", "values": [ [ 4.379289059501534, -0.8504812101012704, -0.45761356318204105, -0.311739690837984, -0.17450494565111793, -1.2851853394204185, -0.3429619745078271, 1.0051539774261902 ], [ -0.9103795886680129, 1.1745770543283585, 0.9358997076810496, 0.6700275731978834, -0.6809898573263689, -1.3280190334666069, -1.0069082758498649, -0.6222881418631324 ], [ -2.3012663832299585, -1.2305226039392265, 2.085736731031756, -0.9490730251671038, 1.3039829892794135, -0.8590365106927715, -2.0650387503732732, -1.9946413845123403 ], [ -0.8871332973004924, -0.02320752801162966, -0.3127603326091232, 2.9819854696322117, -1.4938330510891982, 3.098832516223673, 1.5499197618559948, 0.3234649841138197 ], [ -0.21610669258726886, -1.7355619958837871, 3.8484542954145478, -1.8067381476102413, -0.8307361564753576, -0.24241708131431572, 1.1163242022162478, 0.6571119327322864 ], [ 0.104359956399138, -2.4128998367050993, -2.541901221919718, 0.321206451804348, 0.30222800724953813, -0.7860027856039191, 2.461638210157353, -2.1881183000124382 ], [ 1.138524707814706, -1.8003444762480334, 0.7501869606263278, -2.381490443330813, -0.9254421849995794, 1.7435090767544714, 1.6425476080022894, -1.213913476717083 ], [ 1.8088407406345879, -0.09313925890756079, -0.7725573348868952, 4.2401430655716945, 1.2385413276509467, -1.0129196509290295, -2.6182512081618694, 2.4133967743335516 ] ] }, { "label": "Reverse mean at this step", "values": [ [ 3.796090458514587, -0.593774808482743, -0.6228892251684229, -0.13193379524067236, -0.4024565846354353, -1.3110653951875706, -0.6028845281117335, 0.7559438379594421 ], [ -0.9662254169073178, 0.9013538393565242, 0.3376574375513329, 0.7437732425295802, -0.8449014473933459, -0.4395783416441151, -0.8892340151283585, 0.010519058657869871 ], [ -1.9840198281086803, -1.2685186664846893, 1.450084585876942, -0.4627265011909371, 0.8153331852147776, -0.751183244945424, -2.55299633774815, -1.6661924194204167 ], [ -0.663418162619983, -0.09602267394518822, -0.6868897789527977, 2.8816751404167724, -1.6936090886288502, 3.2708057923549414, 1.6711552916140455, 0.30618808349764653 ], [ -0.27727917749612324, -2.302524787761523, 3.444104164172473, -1.9665940843203864, -1.4439533408382923, -0.8964655782113753, 1.4187235704210266, 0.22627466468494942 ], [ 0.19563855433547528, -2.501967583733189, -2.333037200386036, -0.07790865663965968, 0.47677745400304566, -0.6582216198100198, 2.4141544327190654, -2.329189215318274 ], [ 0.7571604080229485, -2.204156483552705, 1.15506086857324, -2.3115439771170845, -1.1442032698351667, 1.9322040137206944, 2.6067485849766276, -1.1500619470625935 ], [ 1.3413176128397415, -0.08383410191428713, -0.8664173079302631, 3.926404094969817, 1.4554662419290492, -1.071430359446594, -2.0888333061230857, 2.027383572464066 ] ] } ], "vectors": [ { "label": "Completed reverse timesteps", "values": [ 12, 11, 10, 9 ] } ], "metrics": [ { "label": "Posterior variance at this step", "value": 0.11672279509496179 }, { "label": "Remaining denoiser calls", "value": 8 } ] }, { "id": "reverse-4", "label": "Reverse chain · reached t = 4", "target": "denoiser.noise", "note": "Start from an independent Gaussian sample and reuse the same untrained denoiser for all 12 reverse steps. This records the state after step 5. At t = 1 no new noise is added. 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This records the state after step 1. At t = 1 no new noise is added. 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