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Such inefficiency is becoming a critical bottleneck for the adoption of", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 667, 471, 682 ], "spans": [ { "bbox": [ 105, 667, 471, 682 ], "score": 1.0, "content": "DPMs in downstream tasks, leading to an urgent request to design fast samplers for DPMs.", "type": "text" } ], "index": 39 } ], "index": 34, "bbox_fs": [ 105, 558, 507, 682 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 107, 70, 491, 243 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 70, 491, 243 ], "group_id": 0, "lines": [ { "bbox": [ 107, 70, 491, 243 ], "spans": [ { "bbox": [ 107, 70, 491, 243 ], "score": 0.973, "type": "image", "image_path": "4d6dd94407a2a26c38622fe661183fe6ba98b37b4ec1a90af34b958f1acc3ac9.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 70, 491, 127.66666666666666 ], "spans": [], "index": 0 }, { "bbox": [ 107, 127.66666666666666, 491, 185.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 107, 185.33333333333331, 491, 242.99999999999997 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 105, 249, 505, 270 ], "group_id": 0, "lines": [ { "bbox": [ 105, 248, 506, 261 ], "spans": [ { "bbox": [ 105, 248, 506, 261 ], "score": 1.0, "content": "Figure 1: Samples by DDIM [19] with 10, 15, 20, 100 number of function evaluations (NFE), and DPM-Solver", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 258, 488, 271 ], "spans": [ { "bbox": [ 105, 258, 349, 271 ], "score": 1.0, "content": "(ours) with only 10 NFE, using the pre-trained DPMs on ImageNet", "type": "text" }, { "bbox": [ 349, 259, 385, 269 ], "score": 0.86, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 385, 258, 488, 271 ], "score": 1.0, "content": "with classifier guidance [4].", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 107, 281, 505, 401 ], "lines": [ { "bbox": [ 105, 280, 505, 293 ], "spans": [ { "bbox": [ 105, 280, 505, 293 ], "score": 1.0, "content": "Existing fast samplers for DPMs can be divided into two categories. The first category includes", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 291, 505, 304 ], "spans": [ { "bbox": [ 106, 291, 505, 304 ], "score": 1.0, "content": "knowledge distillation [13, 14] and noise level or sample trajectory learning [15–18]. Such methods", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 302, 506, 316 ], "spans": [ { "bbox": [ 105, 302, 506, 316 ], "score": 1.0, "content": "require a possibly expensive training stage before they can be used for efficient sampling. Furthermore,", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 314, 505, 326 ], "spans": [ { "bbox": [ 106, 314, 505, 326 ], "score": 1.0, "content": "their applicability and flexibility might be limited. It might require nontrivial effort to adapt the", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 324, 506, 338 ], "spans": [ { "bbox": [ 105, 324, 506, 338 ], "score": 1.0, "content": "method to different models, datasets, and number of sampling steps. The second category consists", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 335, 506, 348 ], "spans": [ { "bbox": [ 105, 335, 506, 348 ], "score": 1.0, "content": "of training-free [19–21] samplers, which are suitable for all pre-trained DPMs in a simple plug-and-", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 346, 506, 359 ], "spans": [ { "bbox": [ 106, 346, 506, 359 ], "score": 1.0, "content": "play manner. Training-free samplers include adopting implicit [19] or analytical [21] generation", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 357, 506, 370 ], "spans": [ { "bbox": [ 106, 357, 506, 370 ], "score": 1.0, "content": "process, advanced differential equation (DE) solvers [3, 20, 22–24] and dynamic programming [18].", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 367, 506, 381 ], "spans": [ { "bbox": [ 105, 367, 251, 381 ], "score": 1.0, "content": "However, these methods still require", "type": "text" }, { "bbox": [ 252, 369, 273, 378 ], "score": 0.85, "content": "\\sim 5 0", "type": "inline_equation" }, { "bbox": [ 274, 367, 506, 381 ], "score": 1.0, "content": "function evaluations [21] to generate high-quality samples", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 379, 505, 392 ], "spans": [ { "bbox": [ 106, 379, 505, 392 ], "score": 1.0, "content": "(comparable to those generated by plain samplers in about 1000 function evaluations), thereby are", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 389, 194, 403 ], "spans": [ { "bbox": [ 105, 389, 194, 403 ], "score": 1.0, "content": "still time-consuming.", "type": "text" } ], "index": 15 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 406, 505, 537 ], "lines": [ { "bbox": [ 105, 406, 505, 419 ], "spans": [ { "bbox": [ 105, 406, 505, 419 ], "score": 1.0, "content": "In this work, we bring the efficiency of training-free samplers to a new level to produce high-quality", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 417, 505, 430 ], "spans": [ { "bbox": [ 105, 417, 505, 430 ], "score": 1.0, "content": "samples in the “few-step sampling” regime, where the sampling can be done within around 10 steps", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 428, 505, 441 ], "spans": [ { "bbox": [ 106, 428, 505, 441 ], "score": 1.0, "content": "of sequential function evaluations. We tackle the alternative problem of sampling from DPMs as", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 438, 505, 452 ], "spans": [ { "bbox": [ 105, 438, 505, 452 ], "score": 1.0, "content": "solving the corresponding diffusion ordinary differential equations (ODEs) of DPMs, and carefully", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 450, 506, 462 ], "spans": [ { "bbox": [ 105, 450, 506, 462 ], "score": 1.0, "content": "examine the structure of diffusion ODEs. Diffusion ODEs have a semi-linear structure — they consist", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 461, 506, 473 ], "spans": [ { "bbox": [ 105, 461, 506, 473 ], "score": 1.0, "content": "of a linear function of the data variable and a nonlinear function parameterized by neural networks.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 472, 505, 484 ], "spans": [ { "bbox": [ 105, 472, 505, 484 ], "score": 1.0, "content": "Such structure is omitted in previous training-free samplers [3, 20], which directly use black-box", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 482, 505, 495 ], "spans": [ { "bbox": [ 105, 482, 505, 495 ], "score": 1.0, "content": "DE solvers. To utilize the semi-linear structure, we derive an exact formulation of the solutions of", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 492, 505, 507 ], "spans": [ { "bbox": [ 105, 492, 505, 507 ], "score": 1.0, "content": "diffusion ODEs by analytically computing the linear part of the solutions, avoiding the corresponding", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 504, 505, 518 ], "spans": [ { "bbox": [ 105, 504, 505, 518 ], "score": 1.0, "content": "discretization error. Furthermore, by applying change-of-variable, the solutions can be equivalently", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 515, 506, 528 ], "spans": [ { "bbox": [ 105, 515, 506, 528 ], "score": 1.0, "content": "simplified to an exponentially weighted integral of the neural network. Such integral is very special", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 526, 487, 539 ], "spans": [ { "bbox": [ 105, 526, 487, 539 ], "score": 1.0, "content": "and can be efficiently approximated by the numerical methods for exponential integrators [25].", "type": "text" } ], "index": 27 } ], "index": 21.5 }, { "type": "text", "bbox": [ 107, 542, 505, 663 ], "lines": [ { "bbox": [ 106, 543, 505, 555 ], "spans": [ { "bbox": [ 106, 543, 505, 555 ], "score": 1.0, "content": "Based on our formulation of solutions, we propose DPM-Solver, a fast dedicated solver for diffusion", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 553, 506, 567 ], "spans": [ { "bbox": [ 105, 553, 506, 567 ], "score": 1.0, "content": "ODEs by approximating the above integral. Specifically, we propose first-order, second-order and", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 564, 506, 577 ], "spans": [ { "bbox": [ 105, 564, 506, 577 ], "score": 1.0, "content": "third-order versions of DPM-Solver with convergence order guarantees. We further propose an", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 576, 506, 588 ], "spans": [ { "bbox": [ 106, 576, 506, 588 ], "score": 1.0, "content": "adaptive step size schedule for DPM-Solver. In general, DPM-Solver is applicable to both continuous-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 585, 507, 600 ], "spans": [ { "bbox": [ 105, 585, 507, 600 ], "score": 1.0, "content": "time and discrete-time DPMs, and also conditional sampling with classifier guidance [4]. Fig. 1", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 597, 506, 611 ], "spans": [ { "bbox": [ 105, 597, 506, 611 ], "score": 1.0, "content": "demonstrates the speedup performance of a Denoising Diffusion Implicit Models (DDIM) [19]", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 607, 505, 621 ], "spans": [ { "bbox": [ 105, 607, 505, 621 ], "score": 1.0, "content": "baseline and DPM-Solver, which shows that DPM-Solver can generate high-quality samples with as", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 619, 506, 631 ], "spans": [ { "bbox": [ 106, 619, 506, 631 ], "score": 1.0, "content": "few as 10 function evaluations and is much faster than DDIM on the ImageNet 256x256 dataset [26].", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 629, 506, 642 ], "spans": [ { "bbox": [ 105, 629, 506, 642 ], "score": 1.0, "content": "Our additional experimental results show that DPM-Solver can greatly improve the sampling speed of", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 641, 505, 653 ], "spans": [ { "bbox": [ 105, 641, 505, 653 ], "score": 1.0, "content": "both discrete-time and continuous-time DPMs, and it can achieve excellent sample quality in around", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 651, 489, 664 ], "spans": [ { "bbox": [ 105, 651, 489, 664 ], "score": 1.0, "content": "10 function evaluations, which is much faster than all previous training-free samplers of DPMs.", "type": "text" } ], "index": 38 } ], "index": 33 }, { "type": "title", "bbox": [ 108, 682, 281, 696 ], "lines": [ { "bbox": [ 104, 682, 282, 698 ], "spans": [ { "bbox": [ 104, 682, 282, 698 ], "score": 1.0, "content": "2 Diffusion Probabilistic Models", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 105, 711, 500, 722 ], "lines": [ { "bbox": [ 106, 710, 501, 724 ], "spans": [ { "bbox": [ 106, 710, 501, 724 ], "score": 1.0, "content": "We review diffusion probabilistic models and their associated differential equations in this section.", "type": "text" } ], "index": 40 } ], "index": 40 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 742, 309, 750 ], "lines": [ { "bbox": [ 301, 741, 310, 753 ], "spans": [ { "bbox": [ 301, 741, 310, 753 ], "score": 1.0, "content": "2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 107, 70, 491, 243 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 70, 491, 243 ], "group_id": 0, "lines": [ { "bbox": [ 107, 70, 491, 243 ], "spans": [ { "bbox": [ 107, 70, 491, 243 ], "score": 0.973, "type": "image", "image_path": "4d6dd94407a2a26c38622fe661183fe6ba98b37b4ec1a90af34b958f1acc3ac9.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 70, 491, 127.66666666666666 ], "spans": [], "index": 0 }, { "bbox": [ 107, 127.66666666666666, 491, 185.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 107, 185.33333333333331, 491, 242.99999999999997 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 105, 249, 505, 270 ], "group_id": 0, "lines": [ { "bbox": [ 105, 248, 506, 261 ], "spans": [ { "bbox": [ 105, 248, 506, 261 ], "score": 1.0, "content": "Figure 1: Samples by DDIM [19] with 10, 15, 20, 100 number of function evaluations (NFE), and DPM-Solver", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 258, 488, 271 ], "spans": [ { "bbox": [ 105, 258, 349, 271 ], "score": 1.0, "content": "(ours) with only 10 NFE, using the pre-trained DPMs on ImageNet", "type": "text" }, { "bbox": [ 349, 259, 385, 269 ], "score": 0.86, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 385, 258, 488, 271 ], "score": 1.0, "content": "with classifier guidance [4].", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 107, 281, 505, 401 ], "lines": [ { "bbox": [ 105, 280, 505, 293 ], "spans": [ { "bbox": [ 105, 280, 505, 293 ], "score": 1.0, "content": "Existing fast samplers for DPMs can be divided into two categories. The first category includes", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 291, 505, 304 ], "spans": [ { "bbox": [ 106, 291, 505, 304 ], "score": 1.0, "content": "knowledge distillation [13, 14] and noise level or sample trajectory learning [15–18]. Such methods", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 302, 506, 316 ], "spans": [ { "bbox": [ 105, 302, 506, 316 ], "score": 1.0, "content": "require a possibly expensive training stage before they can be used for efficient sampling. Furthermore,", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 314, 505, 326 ], "spans": [ { "bbox": [ 106, 314, 505, 326 ], "score": 1.0, "content": "their applicability and flexibility might be limited. It might require nontrivial effort to adapt the", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 324, 506, 338 ], "spans": [ { "bbox": [ 105, 324, 506, 338 ], "score": 1.0, "content": "method to different models, datasets, and number of sampling steps. The second category consists", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 335, 506, 348 ], "spans": [ { "bbox": [ 105, 335, 506, 348 ], "score": 1.0, "content": "of training-free [19–21] samplers, which are suitable for all pre-trained DPMs in a simple plug-and-", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 346, 506, 359 ], "spans": [ { "bbox": [ 106, 346, 506, 359 ], "score": 1.0, "content": "play manner. Training-free samplers include adopting implicit [19] or analytical [21] generation", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 357, 506, 370 ], "spans": [ { "bbox": [ 106, 357, 506, 370 ], "score": 1.0, "content": "process, advanced differential equation (DE) solvers [3, 20, 22–24] and dynamic programming [18].", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 367, 506, 381 ], "spans": [ { "bbox": [ 105, 367, 251, 381 ], "score": 1.0, "content": "However, these methods still require", "type": "text" }, { "bbox": [ 252, 369, 273, 378 ], "score": 0.85, "content": "\\sim 5 0", "type": "inline_equation" }, { "bbox": [ 274, 367, 506, 381 ], "score": 1.0, "content": "function evaluations [21] to generate high-quality samples", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 379, 505, 392 ], "spans": [ { "bbox": [ 106, 379, 505, 392 ], "score": 1.0, "content": "(comparable to those generated by plain samplers in about 1000 function evaluations), thereby are", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 389, 194, 403 ], "spans": [ { "bbox": [ 105, 389, 194, 403 ], "score": 1.0, "content": "still time-consuming.", "type": "text" } ], "index": 15 } ], "index": 10, "bbox_fs": [ 105, 280, 506, 403 ] }, { "type": "text", "bbox": [ 107, 406, 505, 537 ], "lines": [ { "bbox": [ 105, 406, 505, 419 ], "spans": [ { "bbox": [ 105, 406, 505, 419 ], "score": 1.0, "content": "In this work, we bring the efficiency of training-free samplers to a new level to produce high-quality", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 417, 505, 430 ], "spans": [ { "bbox": [ 105, 417, 505, 430 ], "score": 1.0, "content": "samples in the “few-step sampling” regime, where the sampling can be done within around 10 steps", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 428, 505, 441 ], "spans": [ { "bbox": [ 106, 428, 505, 441 ], "score": 1.0, "content": "of sequential function evaluations. We tackle the alternative problem of sampling from DPMs as", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 438, 505, 452 ], "spans": [ { "bbox": [ 105, 438, 505, 452 ], "score": 1.0, "content": "solving the corresponding diffusion ordinary differential equations (ODEs) of DPMs, and carefully", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 450, 506, 462 ], "spans": [ { "bbox": [ 105, 450, 506, 462 ], "score": 1.0, "content": "examine the structure of diffusion ODEs. Diffusion ODEs have a semi-linear structure — they consist", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 461, 506, 473 ], "spans": [ { "bbox": [ 105, 461, 506, 473 ], "score": 1.0, "content": "of a linear function of the data variable and a nonlinear function parameterized by neural networks.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 472, 505, 484 ], "spans": [ { "bbox": [ 105, 472, 505, 484 ], "score": 1.0, "content": "Such structure is omitted in previous training-free samplers [3, 20], which directly use black-box", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 482, 505, 495 ], "spans": [ { "bbox": [ 105, 482, 505, 495 ], "score": 1.0, "content": "DE solvers. To utilize the semi-linear structure, we derive an exact formulation of the solutions of", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 492, 505, 507 ], "spans": [ { "bbox": [ 105, 492, 505, 507 ], "score": 1.0, "content": "diffusion ODEs by analytically computing the linear part of the solutions, avoiding the corresponding", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 504, 505, 518 ], "spans": [ { "bbox": [ 105, 504, 505, 518 ], "score": 1.0, "content": "discretization error. Furthermore, by applying change-of-variable, the solutions can be equivalently", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 515, 506, 528 ], "spans": [ { "bbox": [ 105, 515, 506, 528 ], "score": 1.0, "content": "simplified to an exponentially weighted integral of the neural network. Such integral is very special", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 526, 487, 539 ], "spans": [ { "bbox": [ 105, 526, 487, 539 ], "score": 1.0, "content": "and can be efficiently approximated by the numerical methods for exponential integrators [25].", "type": "text" } ], "index": 27 } ], "index": 21.5, "bbox_fs": [ 105, 406, 506, 539 ] }, { "type": "text", "bbox": [ 107, 542, 505, 663 ], "lines": [ { "bbox": [ 106, 543, 505, 555 ], "spans": [ { "bbox": [ 106, 543, 505, 555 ], "score": 1.0, "content": "Based on our formulation of solutions, we propose DPM-Solver, a fast dedicated solver for diffusion", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 553, 506, 567 ], "spans": [ { "bbox": [ 105, 553, 506, 567 ], "score": 1.0, "content": "ODEs by approximating the above integral. Specifically, we propose first-order, second-order and", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 564, 506, 577 ], "spans": [ { "bbox": [ 105, 564, 506, 577 ], "score": 1.0, "content": "third-order versions of DPM-Solver with convergence order guarantees. We further propose an", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 576, 506, 588 ], "spans": [ { "bbox": [ 106, 576, 506, 588 ], "score": 1.0, "content": "adaptive step size schedule for DPM-Solver. In general, DPM-Solver is applicable to both continuous-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 585, 507, 600 ], "spans": [ { "bbox": [ 105, 585, 507, 600 ], "score": 1.0, "content": "time and discrete-time DPMs, and also conditional sampling with classifier guidance [4]. Fig. 1", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 597, 506, 611 ], "spans": [ { "bbox": [ 105, 597, 506, 611 ], "score": 1.0, "content": "demonstrates the speedup performance of a Denoising Diffusion Implicit Models (DDIM) [19]", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 607, 505, 621 ], "spans": [ { "bbox": [ 105, 607, 505, 621 ], "score": 1.0, "content": "baseline and DPM-Solver, which shows that DPM-Solver can generate high-quality samples with as", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 619, 506, 631 ], "spans": [ { "bbox": [ 106, 619, 506, 631 ], "score": 1.0, "content": "few as 10 function evaluations and is much faster than DDIM on the ImageNet 256x256 dataset [26].", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 629, 506, 642 ], "spans": [ { "bbox": [ 105, 629, 506, 642 ], "score": 1.0, "content": "Our additional experimental results show that DPM-Solver can greatly improve the sampling speed of", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 641, 505, 653 ], "spans": [ { "bbox": [ 105, 641, 505, 653 ], "score": 1.0, "content": "both discrete-time and continuous-time DPMs, and it can achieve excellent sample quality in around", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 651, 489, 664 ], "spans": [ { "bbox": [ 105, 651, 489, 664 ], "score": 1.0, "content": "10 function evaluations, which is much faster than all previous training-free samplers of DPMs.", "type": "text" } ], "index": 38 } ], "index": 33, "bbox_fs": [ 105, 543, 507, 664 ] }, { "type": "title", "bbox": [ 108, 682, 281, 696 ], "lines": [ { "bbox": [ 104, 682, 282, 698 ], "spans": [ { "bbox": [ 104, 682, 282, 698 ], "score": 1.0, "content": "2 Diffusion Probabilistic Models", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 105, 711, 500, 722 ], "lines": [ { "bbox": [ 106, 710, 501, 724 ], "spans": [ { "bbox": [ 106, 710, 501, 724 ], "score": 1.0, "content": "We review diffusion probabilistic models and their associated differential equations in this section.", "type": "text" } ], "index": 40 } ], "index": 40, "bbox_fs": [ 106, 710, 501, 724 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 72, 289, 84 ], "lines": [ { "bbox": [ 106, 72, 289, 86 ], "spans": [ { "bbox": [ 106, 72, 289, 86 ], "score": 1.0, "content": "2.1 Forward Process and Diffusion SDEs", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 92, 505, 128 ], "lines": [ { "bbox": [ 105, 92, 505, 106 ], "spans": [ { "bbox": [ 105, 92, 204, 106 ], "score": 1.0, "content": "Assume that we have a", "type": "text" }, { "bbox": [ 205, 94, 214, 103 ], "score": 0.82, "content": "D", "type": "inline_equation" }, { "bbox": [ 214, 92, 339, 106 ], "score": 1.0, "content": "-dimensional random variable", "type": "text" }, { "bbox": [ 339, 92, 379, 104 ], "score": 0.92, "content": "\\pmb { x } _ { 0 } \\in \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 380, 92, 505, 106 ], "score": 1.0, "content": "with an unknown distribution", "type": "text" } ], "index": 1 }, { "bbox": [ 107, 103, 505, 118 ], "spans": [ { "bbox": [ 107, 104, 135, 116 ], "score": 0.9, "content": "q _ { 0 } ( { \\pmb x } _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 135, 103, 437, 118 ], "score": 1.0, "content": ". Diffusion Probabilistic Models (DPMs) [1–3, 10] define a forward process", "type": "text" }, { "bbox": [ 438, 104, 483, 117 ], "score": 0.93, "content": "\\{ \\pmb { x } _ { t } \\} _ { t \\in [ 0 , T ] }", "type": "inline_equation" }, { "bbox": [ 484, 103, 505, 118 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 130 ], "spans": [ { "bbox": [ 107, 117, 133, 127 ], "score": 0.88, "content": "T > 0", "type": "inline_equation" }, { "bbox": [ 134, 114, 186, 130 ], "score": 1.0, "content": "starting with", "type": "text" }, { "bbox": [ 186, 118, 198, 127 ], "score": 0.85, "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 198, 114, 270, 130 ], "score": 1.0, "content": ", such that for any", "type": "text" }, { "bbox": [ 270, 116, 308, 128 ], "score": 0.93, "content": "t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 309, 114, 385, 130 ], "score": 1.0, "content": ", the distribution of", "type": "text" }, { "bbox": [ 385, 117, 396, 127 ], "score": 0.86, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 396, 114, 458, 130 ], "score": 1.0, "content": "conditioned on", "type": "text" }, { "bbox": [ 458, 118, 470, 127 ], "score": 0.86, "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 470, 114, 506, 130 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 3 } ], "index": 2 }, { "type": "interline_equation", "bbox": [ 228, 133, 381, 149 ], "lines": [ { "bbox": [ 228, 133, 381, 149 ], "spans": [ { "bbox": [ 228, 133, 381, 149 ], "score": 0.92, "content": "\\begin{array} { r } { q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) = \\mathcal { N } ( \\pmb { x } _ { t } | \\alpha ( t ) \\pmb { x } _ { 0 } , \\sigma ^ { 2 } ( t ) \\pmb { I } ) , } \\end{array}", "type": "interline_equation", "image_path": "b98f04c1bbffa9531e257210ae6112c502489822e557b5a3afdb274392b83d54.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 228, 133, 381, 149 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 153, 505, 221 ], "lines": [ { "bbox": [ 106, 153, 505, 166 ], "spans": [ { "bbox": [ 106, 153, 134, 166 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 154, 202, 166 ], "score": 0.92, "content": "\\alpha ( t ) , \\sigma ( t ) \\in \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 202, 153, 328, 166 ], "score": 1.0, "content": "are differentiable functions of", "type": "text" }, { "bbox": [ 329, 155, 334, 164 ], "score": 0.81, "content": "t", "type": "inline_equation" }, { "bbox": [ 334, 153, 505, 166 ], "score": 1.0, "content": "with bounded derivatives, and we denote", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 141, 178 ], "score": 1.0, "content": "them as", "type": "text" }, { "bbox": [ 142, 168, 166, 177 ], "score": 0.88, "content": "\\alpha _ { t } , \\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 167, 165, 295, 178 ], "score": 1.0, "content": "for simplicity. The choice for", "type": "text" }, { "bbox": [ 295, 167, 307, 176 ], "score": 0.86, "content": "\\alpha _ { t }", "type": "inline_equation" }, { "bbox": [ 307, 165, 326, 178 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 326, 167, 337, 176 ], "score": 0.86, "content": "\\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 337, 165, 506, 178 ], "score": 1.0, "content": "is referred to as the noise schedule of a", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 175, 505, 188 ], "spans": [ { "bbox": [ 105, 175, 150, 188 ], "score": 1.0, "content": "DPM. Let", "type": "text" }, { "bbox": [ 151, 177, 177, 187 ], "score": 0.9, "content": "q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 177, 175, 323, 188 ], "score": 1.0, "content": "denote the marginal distribution of", "type": "text" }, { "bbox": [ 324, 178, 334, 187 ], "score": 0.87, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 335, 175, 505, 188 ], "score": 1.0, "content": ", DPMs choose noise schedules to ensure", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 186, 506, 200 ], "spans": [ { "bbox": [ 105, 186, 124, 200 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 124, 187, 229, 199 ], "score": 0.89, "content": "q _ { T } ( \\pmb { x } _ { T } ) \\overset { \\cdot } { \\approx } \\dot { \\mathcal { N } } ( \\pmb { x } _ { T } | \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )", "type": "inline_equation" }, { "bbox": [ 229, 186, 267, 200 ], "score": 1.0, "content": "for some", "type": "text" }, { "bbox": [ 267, 187, 293, 198 ], "score": 0.9, "content": "\\tilde { \\sigma } > 0", "type": "inline_equation" }, { "bbox": [ 293, 186, 438, 200 ], "score": 1.0, "content": ", and the signal-to-noise-ratio (SNR)", "type": "text" }, { "bbox": [ 438, 186, 465, 199 ], "score": 0.93, "content": "\\alpha _ { t } ^ { 2 } / \\sigma _ { t } ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 466, 186, 506, 200 ], "score": 1.0, "content": "is strictly", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 198, 506, 211 ], "spans": [ { "bbox": [ 106, 198, 173, 211 ], "score": 1.0, "content": "decreasing w.r.t.", "type": "text" }, { "bbox": [ 174, 200, 179, 208 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 179, 198, 506, 211 ], "score": 1.0, "content": "[10]. Moreover, Kingma et al. [10] prove that the following stochastic differential", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 209, 492, 221 ], "spans": [ { "bbox": [ 105, 209, 313, 221 ], "score": 1.0, "content": "equation (SDE) has the same transition distribution", "type": "text" }, { "bbox": [ 313, 209, 357, 221 ], "score": 0.93, "content": "q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 358, 209, 448, 221 ], "score": 1.0, "content": "as in Eq. (2.1) for any", "type": "text" }, { "bbox": [ 449, 209, 487, 221 ], "score": 0.93, "content": "t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 488, 209, 492, 221 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 10 } ], "index": 7.5 }, { "type": "interline_equation", "bbox": [ 213, 226, 397, 240 ], "lines": [ { "bbox": [ 213, 226, 397, 240 ], "spans": [ { "bbox": [ 213, 226, 397, 240 ], "score": 0.91, "content": "\\mathrm { d } \\pmb { x } _ { t } = f ( t ) \\pmb { x } _ { t } \\mathrm { d } t + g ( t ) \\mathrm { d } \\pmb { w } _ { t } , \\quad \\pmb { x } _ { 0 } \\sim q _ { 0 } ( \\pmb { x } _ { 0 } ) ,", "type": "interline_equation", "image_path": "8433318473b1628737ca5404d0a929d6a4570d9919cf8824f87c274bc4a0bbfd.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 213, 226, 397, 240 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 245, 317, 257 ], "lines": [ { "bbox": [ 105, 243, 318, 260 ], "spans": [ { "bbox": [ 105, 243, 133, 260 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 245, 172, 257 ], "score": 0.94, "content": "{ \\pmb w } _ { t } \\in \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 172, 243, 318, 260 ], "score": 1.0, "content": "is the standard Wiener process, and", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "interline_equation", "bbox": [ 206, 263, 405, 289 ], "lines": [ { "bbox": [ 206, 263, 405, 289 ], "spans": [ { "bbox": [ 206, 263, 405, 289 ], "score": 0.93, "content": "f ( t ) = \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } , \\quad g ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } .", "type": "interline_equation", "image_path": "e5d12cd513ddf8ae3fa219a1feddb4594e4cf22e33260b27389e15d81ab87dc7.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 206, 263, 405, 289 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 107, 294, 506, 317 ], "lines": [ { "bbox": [ 106, 294, 505, 307 ], "spans": [ { "bbox": [ 106, 294, 505, 307 ], "score": 1.0, "content": "Under some regularity conditions, Song et al. [3] show that the forward process in Eq. (2.2) has an", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 304, 476, 318 ], "spans": [ { "bbox": [ 105, 304, 255, 318 ], "score": 1.0, "content": "equivalent reverse process from time", "type": "text" }, { "bbox": [ 256, 306, 264, 315 ], "score": 0.84, "content": "T", "type": "inline_equation" }, { "bbox": [ 264, 304, 276, 318 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 276, 306, 282, 315 ], "score": 0.29, "content": "0", "type": "inline_equation" }, { "bbox": [ 282, 304, 439, 318 ], "score": 1.0, "content": ", starting with the marginal distribution", "type": "text" }, { "bbox": [ 439, 306, 471, 317 ], "score": 0.93, "content": "q _ { T } ( { \\pmb x } _ { T } )", "type": "inline_equation" }, { "bbox": [ 471, 304, 476, 318 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 15 } ], "index": 14.5 }, { "type": "interline_equation", "bbox": [ 163, 322, 446, 338 ], "lines": [ { "bbox": [ 163, 322, 446, 338 ], "spans": [ { "bbox": [ 163, 322, 446, 338 ], "score": 0.88, "content": "\\mathrm { d } \\pmb { x } _ { t } = [ f ( t ) \\pmb { x } _ { t } - g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) ] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { \\pmb { w } } _ { t } , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,", "type": "interline_equation", "image_path": "098b1c09edd91e1e8b539459bb92a3c5f191abb42aa1157755522d3dce62bcae.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 163, 322, 446, 338 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 108, 343, 505, 388 ], "lines": [ { "bbox": [ 106, 343, 505, 355 ], "spans": [ { "bbox": [ 106, 343, 133, 355 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 344, 146, 354 ], "score": 0.88, "content": "\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { w } } _ { t }", "type": "inline_equation" }, { "bbox": [ 147, 343, 505, 355 ], "score": 1.0, "content": "is a standard Wiener process in the reverse time. The only unknown term in Eq. (2.4) is", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 354, 505, 367 ], "spans": [ { "bbox": [ 105, 354, 182, 367 ], "score": 1.0, "content": "the score function", "type": "text" }, { "bbox": [ 183, 354, 239, 366 ], "score": 0.92, "content": "\\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 239, 354, 293, 367 ], "score": 1.0, "content": "at each time", "type": "text" }, { "bbox": [ 293, 356, 298, 364 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 299, 354, 469, 367 ], "score": 1.0, "content": ". In practice, DPMs use a neural network", "type": "text" }, { "bbox": [ 469, 355, 505, 366 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 105, 365, 506, 377 ], "spans": [ { "bbox": [ 105, 365, 180, 377 ], "score": 1.0, "content": "parameterized by", "type": "text" }, { "bbox": [ 181, 366, 186, 375 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 187, 365, 345, 377 ], "score": 1.0, "content": "to estimate the scaled score function:", "type": "text" }, { "bbox": [ 345, 365, 419, 377 ], "score": 0.92, "content": "- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 420, 365, 488, 377 ], "score": 1.0, "content": ". The parameter", "type": "text" }, { "bbox": [ 488, 366, 494, 375 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 495, 365, 506, 377 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 376, 331, 389 ], "spans": [ { "bbox": [ 105, 376, 331, 389 ], "score": 1.0, "content": "optimized by minimizing the following objective [2, 3]:", "type": "text" } ], "index": 20 } ], "index": 18.5 }, { "type": "interline_equation", "bbox": [ 165, 393, 445, 453 ], "lines": [ { "bbox": [ 165, 393, 445, 453 ], "spans": [ { "bbox": [ 165, 393, 445, 453 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\theta ; \\omega ( t ) ) : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { t } ( \\mathbf { \\Delta x } _ { t } ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) + \\sigma _ { t } \\nabla _ { \\mathbf { x } } \\log q _ { t } ( \\mathbf { \\Delta x } _ { t } ) \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { 0 } ( \\mathbf { x } _ { 0 } ) } \\mathbb { E } _ { q ( \\epsilon ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) - \\epsilon \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t + C , } \\end{array}", "type": "interline_equation", "image_path": "18da74f3ada9dcef2cf0de53303bacedefeb327430ef720bfe85381fb0de2f45.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 165, 393, 445, 413.0 ], "spans": [], "index": 21 }, { "bbox": [ 165, 413.0, 445, 433.0 ], "spans": [], "index": 22 }, { "bbox": [ 165, 433.0, 445, 453.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 106, 456, 506, 513 ], "lines": [ { "bbox": [ 106, 456, 506, 470 ], "spans": [ { "bbox": [ 106, 456, 133, 470 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 457, 153, 469 ], "score": 0.92, "content": "\\omega ( t )", "type": "inline_equation" }, { "bbox": [ 153, 456, 253, 470 ], "score": 1.0, "content": "is a weighting function,", "type": "text" }, { "bbox": [ 253, 456, 345, 469 ], "score": 0.88, "content": "\\epsilon \\sim q ( \\epsilon ) = \\mathcal { N } ( \\epsilon | \\mathbf { 0 } , I )", "type": "inline_equation" }, { "bbox": [ 345, 456, 348, 470 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 349, 458, 421, 468 ], "score": 0.88, "content": "{ \\pmb x } _ { t } = \\alpha _ { t } { \\pmb x } _ { 0 } + \\sigma _ { t } { \\pmb \\epsilon }", "type": "inline_equation" }, { "bbox": [ 421, 456, 442, 470 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 442, 457, 451, 467 ], "score": 0.86, "content": "C", "type": "inline_equation" }, { "bbox": [ 451, 456, 506, 470 ], "score": 1.0, "content": "is a constant", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 468, 506, 480 ], "spans": [ { "bbox": [ 106, 468, 169, 480 ], "score": 1.0, "content": "independent of", "type": "text" }, { "bbox": [ 169, 469, 174, 478 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 175, 468, 192, 480 ], "score": 1.0, "content": ". As", "type": "text" }, { "bbox": [ 193, 468, 229, 480 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 229, 468, 482, 480 ], "score": 1.0, "content": "can also be regarded as predicting the Gaussian noise added to", "type": "text" }, { "bbox": [ 482, 469, 493, 479 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 493, 468, 506, 480 ], "score": 1.0, "content": ", it", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 477, 507, 493 ], "spans": [ { "bbox": [ 104, 477, 381, 493 ], "score": 1.0, "content": "is usually called the noise prediction model. Since the ground truth of", "type": "text" }, { "bbox": [ 382, 479, 418, 491 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 418, 477, 429, 493 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 429, 479, 503, 491 ], "score": 0.89, "content": "- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 503, 477, 507, 493 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 488, 506, 503 ], "spans": [ { "bbox": [ 105, 488, 303, 503 ], "score": 1.0, "content": "DPMs replace the score function in Eq. (2.4) by", "type": "text" }, { "bbox": [ 303, 489, 361, 501 ], "score": 0.92, "content": "- \\mathbf { \\epsilon } \\mathbf { \\epsilon } \\bar { \\mathbf { \\alpha } } ( \\mathbf { x } _ { t } , t ) / \\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 361, 488, 506, 503 ], "score": 1.0, "content": "and define a parameterized reverse", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 500, 403, 513 ], "spans": [ { "bbox": [ 105, 500, 179, 513 ], "score": 1.0, "content": "process (diffusion", "type": "text" }, { "bbox": [ 180, 501, 199, 511 ], "score": 0.45, "content": "S D E", "type": "inline_equation" }, { "bbox": [ 200, 500, 246, 513 ], "score": 1.0, "content": ") from time", "type": "text" }, { "bbox": [ 246, 502, 254, 510 ], "score": 0.85, "content": "T", "type": "inline_equation" }, { "bbox": [ 254, 500, 266, 513 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 266, 502, 272, 510 ], "score": 0.38, "content": "0", "type": "inline_equation" }, { "bbox": [ 272, 500, 328, 513 ], "score": 1.0, "content": ", starting with", "type": "text" }, { "bbox": [ 329, 500, 399, 513 ], "score": 0.92, "content": "\\pmb { x } _ { T } \\overset { \\cdot } { \\sim } \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )", "type": "inline_equation" }, { "bbox": [ 399, 500, 403, 513 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 28 } ], "index": 26 }, { "type": "interline_equation", "bbox": [ 163, 518, 448, 546 ], "lines": [ { "bbox": [ 163, 518, 448, 546 ], "spans": [ { "bbox": [ 163, 518, 448, 546 ], "score": 0.94, "content": "\\mathrm { d } x _ { t } = \\left[ f ( t ) x _ { t } + \\frac { g ^ { 2 } ( t ) } { \\sigma _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) \\right] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { w } _ { t } , \\quad x _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } I ) .", "type": "interline_equation", "image_path": "65fdde0d1d949853db59a598710255c20a35d6c68ca6401f8b46cd9f514525f2.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 163, 518, 448, 546 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 107, 551, 506, 607 ], "lines": [ { "bbox": [ 106, 551, 505, 564 ], "spans": [ { "bbox": [ 106, 551, 505, 564 ], "score": 1.0, "content": "Samples can be generated from DPMs by solving the diffusion SDE in Eq. (2.5) with numerical", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 563, 505, 574 ], "spans": [ { "bbox": [ 106, 563, 268, 574 ], "score": 1.0, "content": "solvers, which discretize the SDE from", "type": "text" }, { "bbox": [ 268, 563, 276, 572 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 277, 563, 505, 574 ], "score": 1.0, "content": "to 0. Song et al. [3] proved that the traditional ancestral", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 573, 506, 586 ], "spans": [ { "bbox": [ 105, 573, 506, 586 ], "score": 1.0, "content": "sampling method for DPMs [2] can be viewed as a first-order SDE solver for Eq. (2.5). However, these", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 583, 506, 597 ], "spans": [ { "bbox": [ 105, 583, 506, 597 ], "score": 1.0, "content": "first-order methods usually need hundreds of or thousands of function evaluations to converge [3],", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 595, 279, 608 ], "spans": [ { "bbox": [ 106, 595, 279, 608 ], "score": 1.0, "content": "leading to extremely slow sampling speed.", "type": "text" } ], "index": 34 } ], "index": 32 }, { "type": "title", "bbox": [ 108, 619, 278, 632 ], "lines": [ { "bbox": [ 105, 618, 280, 635 ], "spans": [ { "bbox": [ 105, 618, 280, 635 ], "score": 1.0, "content": "2.2 Diffusion (Probability Flow) ODEs", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 106, 640, 506, 696 ], "lines": [ { "bbox": [ 105, 640, 506, 654 ], "spans": [ { "bbox": [ 105, 640, 506, 654 ], "score": 1.0, "content": "When discretizing SDEs, the step size is limited by the randomness of the Wiener process [27, Chap.", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 651, 505, 665 ], "spans": [ { "bbox": [ 105, 651, 505, 665 ], "score": 1.0, "content": "11]. A large step size (small number of steps) often causes non-convergence, especially in high", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 660, 507, 676 ], "spans": [ { "bbox": [ 105, 660, 507, 676 ], "score": 1.0, "content": "dimensional spaces. For faster sampling, one can consider the associated probability flow ODE [3],", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 672, 507, 687 ], "spans": [ { "bbox": [ 105, 672, 325, 687 ], "score": 1.0, "content": "which has the same marginal distribution at each time", "type": "text" }, { "bbox": [ 325, 675, 330, 683 ], "score": 0.8, "content": "t", "type": "inline_equation" }, { "bbox": [ 330, 672, 507, 687 ], "score": 1.0, "content": "as that of the SDE. Specifically, for DPMs,", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 684, 375, 698 ], "spans": [ { "bbox": [ 106, 684, 375, 698 ], "score": 1.0, "content": "Song et al. [3] proved that the probability flow ODE of Eq. (2.4) is", "type": "text" } ], "index": 40 } ], "index": 38 }, { "type": "interline_equation", "bbox": [ 189, 702, 421, 726 ], "lines": [ { "bbox": [ 189, 702, 421, 726 ], "spans": [ { "bbox": [ 189, 702, 421, 726 ], "score": 0.92, "content": "\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = f ( t ) \\pmb { x } _ { t } - \\frac { 1 } { 2 } g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,", "type": "interline_equation", "image_path": "db0329942038b0f1cd9de0b5b9716a202705a11feeab57d38ae6c43085d15745.jpg" } ] } ], "index": 41, "virtual_lines": [ { "bbox": [ 189, 702, 421, 726 ], "spans": [], "index": 41 } ] } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 301, 740, 310, 752 ], "spans": [ { "bbox": [ 301, 740, 310, 752 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 72, 289, 84 ], "lines": [ { "bbox": [ 106, 72, 289, 86 ], "spans": [ { "bbox": [ 106, 72, 289, 86 ], "score": 1.0, "content": "2.1 Forward Process and Diffusion SDEs", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 92, 505, 128 ], "lines": [ { "bbox": [ 105, 92, 505, 106 ], "spans": [ { "bbox": [ 105, 92, 204, 106 ], "score": 1.0, "content": "Assume that we have a", "type": "text" }, { "bbox": [ 205, 94, 214, 103 ], "score": 0.82, "content": "D", "type": "inline_equation" }, { "bbox": [ 214, 92, 339, 106 ], "score": 1.0, "content": "-dimensional random variable", "type": "text" }, { "bbox": [ 339, 92, 379, 104 ], "score": 0.92, "content": "\\pmb { x } _ { 0 } \\in \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 380, 92, 505, 106 ], "score": 1.0, "content": "with an unknown distribution", "type": "text" } ], "index": 1 }, { "bbox": [ 107, 103, 505, 118 ], "spans": [ { "bbox": [ 107, 104, 135, 116 ], "score": 0.9, "content": "q _ { 0 } ( { \\pmb x } _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 135, 103, 437, 118 ], "score": 1.0, "content": ". Diffusion Probabilistic Models (DPMs) [1–3, 10] define a forward process", "type": "text" }, { "bbox": [ 438, 104, 483, 117 ], "score": 0.93, "content": "\\{ \\pmb { x } _ { t } \\} _ { t \\in [ 0 , T ] }", "type": "inline_equation" }, { "bbox": [ 484, 103, 505, 118 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 130 ], "spans": [ { "bbox": [ 107, 117, 133, 127 ], "score": 0.88, "content": "T > 0", "type": "inline_equation" }, { "bbox": [ 134, 114, 186, 130 ], "score": 1.0, "content": "starting with", "type": "text" }, { "bbox": [ 186, 118, 198, 127 ], "score": 0.85, "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 198, 114, 270, 130 ], "score": 1.0, "content": ", such that for any", "type": "text" }, { "bbox": [ 270, 116, 308, 128 ], "score": 0.93, "content": "t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 309, 114, 385, 130 ], "score": 1.0, "content": ", the distribution of", "type": "text" }, { "bbox": [ 385, 117, 396, 127 ], "score": 0.86, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 396, 114, 458, 130 ], "score": 1.0, "content": "conditioned on", "type": "text" }, { "bbox": [ 458, 118, 470, 127 ], "score": 0.86, "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 470, 114, 506, 130 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 3 } ], "index": 2, "bbox_fs": [ 105, 92, 506, 130 ] }, { "type": "interline_equation", "bbox": [ 228, 133, 381, 149 ], "lines": [ { "bbox": [ 228, 133, 381, 149 ], "spans": [ { "bbox": [ 228, 133, 381, 149 ], "score": 0.92, "content": "\\begin{array} { r } { q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) = \\mathcal { N } ( \\pmb { x } _ { t } | \\alpha ( t ) \\pmb { x } _ { 0 } , \\sigma ^ { 2 } ( t ) \\pmb { I } ) , } \\end{array}", "type": "interline_equation", "image_path": "b98f04c1bbffa9531e257210ae6112c502489822e557b5a3afdb274392b83d54.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 228, 133, 381, 149 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 153, 505, 221 ], "lines": [ { "bbox": [ 106, 153, 505, 166 ], "spans": [ { "bbox": [ 106, 153, 134, 166 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 154, 202, 166 ], "score": 0.92, "content": "\\alpha ( t ) , \\sigma ( t ) \\in \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 202, 153, 328, 166 ], "score": 1.0, "content": "are differentiable functions of", "type": "text" }, { "bbox": [ 329, 155, 334, 164 ], "score": 0.81, "content": "t", "type": "inline_equation" }, { "bbox": [ 334, 153, 505, 166 ], "score": 1.0, "content": "with bounded derivatives, and we denote", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 141, 178 ], "score": 1.0, "content": "them as", "type": "text" }, { "bbox": [ 142, 168, 166, 177 ], "score": 0.88, "content": "\\alpha _ { t } , \\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 167, 165, 295, 178 ], "score": 1.0, "content": "for simplicity. The choice for", "type": "text" }, { "bbox": [ 295, 167, 307, 176 ], "score": 0.86, "content": "\\alpha _ { t }", "type": "inline_equation" }, { "bbox": [ 307, 165, 326, 178 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 326, 167, 337, 176 ], "score": 0.86, "content": "\\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 337, 165, 506, 178 ], "score": 1.0, "content": "is referred to as the noise schedule of a", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 175, 505, 188 ], "spans": [ { "bbox": [ 105, 175, 150, 188 ], "score": 1.0, "content": "DPM. Let", "type": "text" }, { "bbox": [ 151, 177, 177, 187 ], "score": 0.9, "content": "q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 177, 175, 323, 188 ], "score": 1.0, "content": "denote the marginal distribution of", "type": "text" }, { "bbox": [ 324, 178, 334, 187 ], "score": 0.87, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 335, 175, 505, 188 ], "score": 1.0, "content": ", DPMs choose noise schedules to ensure", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 186, 506, 200 ], "spans": [ { "bbox": [ 105, 186, 124, 200 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 124, 187, 229, 199 ], "score": 0.89, "content": "q _ { T } ( \\pmb { x } _ { T } ) \\overset { \\cdot } { \\approx } \\dot { \\mathcal { N } } ( \\pmb { x } _ { T } | \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )", "type": "inline_equation" }, { "bbox": [ 229, 186, 267, 200 ], "score": 1.0, "content": "for some", "type": "text" }, { "bbox": [ 267, 187, 293, 198 ], "score": 0.9, "content": "\\tilde { \\sigma } > 0", "type": "inline_equation" }, { "bbox": [ 293, 186, 438, 200 ], "score": 1.0, "content": ", and the signal-to-noise-ratio (SNR)", "type": "text" }, { "bbox": [ 438, 186, 465, 199 ], "score": 0.93, "content": "\\alpha _ { t } ^ { 2 } / \\sigma _ { t } ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 466, 186, 506, 200 ], "score": 1.0, "content": "is strictly", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 198, 506, 211 ], "spans": [ { "bbox": [ 106, 198, 173, 211 ], "score": 1.0, "content": "decreasing w.r.t.", "type": "text" }, { "bbox": [ 174, 200, 179, 208 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 179, 198, 506, 211 ], "score": 1.0, "content": "[10]. Moreover, Kingma et al. [10] prove that the following stochastic differential", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 209, 492, 221 ], "spans": [ { "bbox": [ 105, 209, 313, 221 ], "score": 1.0, "content": "equation (SDE) has the same transition distribution", "type": "text" }, { "bbox": [ 313, 209, 357, 221 ], "score": 0.93, "content": "q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 358, 209, 448, 221 ], "score": 1.0, "content": "as in Eq. (2.1) for any", "type": "text" }, { "bbox": [ 449, 209, 487, 221 ], "score": 0.93, "content": "t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 488, 209, 492, 221 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 10 } ], "index": 7.5, "bbox_fs": [ 105, 153, 506, 221 ] }, { "type": "interline_equation", "bbox": [ 213, 226, 397, 240 ], "lines": [ { "bbox": [ 213, 226, 397, 240 ], "spans": [ { "bbox": [ 213, 226, 397, 240 ], "score": 0.91, "content": "\\mathrm { d } \\pmb { x } _ { t } = f ( t ) \\pmb { x } _ { t } \\mathrm { d } t + g ( t ) \\mathrm { d } \\pmb { w } _ { t } , \\quad \\pmb { x } _ { 0 } \\sim q _ { 0 } ( \\pmb { x } _ { 0 } ) ,", "type": "interline_equation", "image_path": "8433318473b1628737ca5404d0a929d6a4570d9919cf8824f87c274bc4a0bbfd.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 213, 226, 397, 240 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 245, 317, 257 ], "lines": [ { "bbox": [ 105, 243, 318, 260 ], "spans": [ { "bbox": [ 105, 243, 133, 260 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 245, 172, 257 ], "score": 0.94, "content": "{ \\pmb w } _ { t } \\in \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 172, 243, 318, 260 ], "score": 1.0, "content": "is the standard Wiener process, and", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 105, 243, 318, 260 ] }, { "type": "interline_equation", "bbox": [ 206, 263, 405, 289 ], "lines": [ { "bbox": [ 206, 263, 405, 289 ], "spans": [ { "bbox": [ 206, 263, 405, 289 ], "score": 0.93, "content": "f ( t ) = \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } , \\quad g ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } .", "type": "interline_equation", "image_path": "e5d12cd513ddf8ae3fa219a1feddb4594e4cf22e33260b27389e15d81ab87dc7.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 206, 263, 405, 289 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 107, 294, 506, 317 ], "lines": [ { "bbox": [ 106, 294, 505, 307 ], "spans": [ { "bbox": [ 106, 294, 505, 307 ], "score": 1.0, "content": "Under some regularity conditions, Song et al. [3] show that the forward process in Eq. (2.2) has an", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 304, 476, 318 ], "spans": [ { "bbox": [ 105, 304, 255, 318 ], "score": 1.0, "content": "equivalent reverse process from time", "type": "text" }, { "bbox": [ 256, 306, 264, 315 ], "score": 0.84, "content": "T", "type": "inline_equation" }, { "bbox": [ 264, 304, 276, 318 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 276, 306, 282, 315 ], "score": 0.29, "content": "0", "type": "inline_equation" }, { "bbox": [ 282, 304, 439, 318 ], "score": 1.0, "content": ", starting with the marginal distribution", "type": "text" }, { "bbox": [ 439, 306, 471, 317 ], "score": 0.93, "content": "q _ { T } ( { \\pmb x } _ { T } )", "type": "inline_equation" }, { "bbox": [ 471, 304, 476, 318 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 15 } ], "index": 14.5, "bbox_fs": [ 105, 294, 505, 318 ] }, { "type": "interline_equation", "bbox": [ 163, 322, 446, 338 ], "lines": [ { "bbox": [ 163, 322, 446, 338 ], "spans": [ { "bbox": [ 163, 322, 446, 338 ], "score": 0.88, "content": "\\mathrm { d } \\pmb { x } _ { t } = [ f ( t ) \\pmb { x } _ { t } - g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) ] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { \\pmb { w } } _ { t } , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,", "type": "interline_equation", "image_path": "098b1c09edd91e1e8b539459bb92a3c5f191abb42aa1157755522d3dce62bcae.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 163, 322, 446, 338 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 108, 343, 505, 388 ], "lines": [ { "bbox": [ 106, 343, 505, 355 ], "spans": [ { "bbox": [ 106, 343, 133, 355 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 344, 146, 354 ], "score": 0.88, "content": "\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { w } } _ { t }", "type": "inline_equation" }, { "bbox": [ 147, 343, 505, 355 ], "score": 1.0, "content": "is a standard Wiener process in the reverse time. The only unknown term in Eq. (2.4) is", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 354, 505, 367 ], "spans": [ { "bbox": [ 105, 354, 182, 367 ], "score": 1.0, "content": "the score function", "type": "text" }, { "bbox": [ 183, 354, 239, 366 ], "score": 0.92, "content": "\\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 239, 354, 293, 367 ], "score": 1.0, "content": "at each time", "type": "text" }, { "bbox": [ 293, 356, 298, 364 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 299, 354, 469, 367 ], "score": 1.0, "content": ". In practice, DPMs use a neural network", "type": "text" }, { "bbox": [ 469, 355, 505, 366 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 105, 365, 506, 377 ], "spans": [ { "bbox": [ 105, 365, 180, 377 ], "score": 1.0, "content": "parameterized by", "type": "text" }, { "bbox": [ 181, 366, 186, 375 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 187, 365, 345, 377 ], "score": 1.0, "content": "to estimate the scaled score function:", "type": "text" }, { "bbox": [ 345, 365, 419, 377 ], "score": 0.92, "content": "- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 420, 365, 488, 377 ], "score": 1.0, "content": ". The parameter", "type": "text" }, { "bbox": [ 488, 366, 494, 375 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 495, 365, 506, 377 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 376, 331, 389 ], "spans": [ { "bbox": [ 105, 376, 331, 389 ], "score": 1.0, "content": "optimized by minimizing the following objective [2, 3]:", "type": "text" } ], "index": 20 } ], "index": 18.5, "bbox_fs": [ 105, 343, 506, 389 ] }, { "type": "interline_equation", "bbox": [ 165, 393, 445, 453 ], "lines": [ { "bbox": [ 165, 393, 445, 453 ], "spans": [ { "bbox": [ 165, 393, 445, 453 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\theta ; \\omega ( t ) ) : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { t } ( \\mathbf { \\Delta x } _ { t } ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) + \\sigma _ { t } \\nabla _ { \\mathbf { x } } \\log q _ { t } ( \\mathbf { \\Delta x } _ { t } ) \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { 0 } ( \\mathbf { x } _ { 0 } ) } \\mathbb { E } _ { q ( \\epsilon ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) - \\epsilon \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t + C , } \\end{array}", "type": "interline_equation", "image_path": "18da74f3ada9dcef2cf0de53303bacedefeb327430ef720bfe85381fb0de2f45.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 165, 393, 445, 413.0 ], "spans": [], "index": 21 }, { "bbox": [ 165, 413.0, 445, 433.0 ], "spans": [], "index": 22 }, { "bbox": [ 165, 433.0, 445, 453.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 106, 456, 506, 513 ], "lines": [ { "bbox": [ 106, 456, 506, 470 ], "spans": [ { "bbox": [ 106, 456, 133, 470 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 457, 153, 469 ], "score": 0.92, "content": "\\omega ( t )", "type": "inline_equation" }, { "bbox": [ 153, 456, 253, 470 ], "score": 1.0, "content": "is a weighting function,", "type": "text" }, { "bbox": [ 253, 456, 345, 469 ], "score": 0.88, "content": "\\epsilon \\sim q ( \\epsilon ) = \\mathcal { N } ( \\epsilon | \\mathbf { 0 } , I )", "type": "inline_equation" }, { "bbox": [ 345, 456, 348, 470 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 349, 458, 421, 468 ], "score": 0.88, "content": "{ \\pmb x } _ { t } = \\alpha _ { t } { \\pmb x } _ { 0 } + \\sigma _ { t } { \\pmb \\epsilon }", "type": "inline_equation" }, { "bbox": [ 421, 456, 442, 470 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 442, 457, 451, 467 ], "score": 0.86, "content": "C", "type": "inline_equation" }, { "bbox": [ 451, 456, 506, 470 ], "score": 1.0, "content": "is a constant", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 468, 506, 480 ], "spans": [ { "bbox": [ 106, 468, 169, 480 ], "score": 1.0, "content": "independent of", "type": "text" }, { "bbox": [ 169, 469, 174, 478 ], "score": 0.82, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 175, 468, 192, 480 ], "score": 1.0, "content": ". As", "type": "text" }, { "bbox": [ 193, 468, 229, 480 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 229, 468, 482, 480 ], "score": 1.0, "content": "can also be regarded as predicting the Gaussian noise added to", "type": "text" }, { "bbox": [ 482, 469, 493, 479 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 493, 468, 506, 480 ], "score": 1.0, "content": ", it", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 477, 507, 493 ], "spans": [ { "bbox": [ 104, 477, 381, 493 ], "score": 1.0, "content": "is usually called the noise prediction model. Since the ground truth of", "type": "text" }, { "bbox": [ 382, 479, 418, 491 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 418, 477, 429, 493 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 429, 479, 503, 491 ], "score": 0.89, "content": "- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 503, 477, 507, 493 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 488, 506, 503 ], "spans": [ { "bbox": [ 105, 488, 303, 503 ], "score": 1.0, "content": "DPMs replace the score function in Eq. (2.4) by", "type": "text" }, { "bbox": [ 303, 489, 361, 501 ], "score": 0.92, "content": "- \\mathbf { \\epsilon } \\mathbf { \\epsilon } \\bar { \\mathbf { \\alpha } } ( \\mathbf { x } _ { t } , t ) / \\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 361, 488, 506, 503 ], "score": 1.0, "content": "and define a parameterized reverse", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 500, 403, 513 ], "spans": [ { "bbox": [ 105, 500, 179, 513 ], "score": 1.0, "content": "process (diffusion", "type": "text" }, { "bbox": [ 180, 501, 199, 511 ], "score": 0.45, "content": "S D E", "type": "inline_equation" }, { "bbox": [ 200, 500, 246, 513 ], "score": 1.0, "content": ") from time", "type": "text" }, { "bbox": [ 246, 502, 254, 510 ], "score": 0.85, "content": "T", "type": "inline_equation" }, { "bbox": [ 254, 500, 266, 513 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 266, 502, 272, 510 ], "score": 0.38, "content": "0", "type": "inline_equation" }, { "bbox": [ 272, 500, 328, 513 ], "score": 1.0, "content": ", starting with", "type": "text" }, { "bbox": [ 329, 500, 399, 513 ], "score": 0.92, "content": "\\pmb { x } _ { T } \\overset { \\cdot } { \\sim } \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )", "type": "inline_equation" }, { "bbox": [ 399, 500, 403, 513 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 28 } ], "index": 26, "bbox_fs": [ 104, 456, 507, 513 ] }, { "type": "interline_equation", "bbox": [ 163, 518, 448, 546 ], "lines": [ { "bbox": [ 163, 518, 448, 546 ], "spans": [ { "bbox": [ 163, 518, 448, 546 ], "score": 0.94, "content": "\\mathrm { d } x _ { t } = \\left[ f ( t ) x _ { t } + \\frac { g ^ { 2 } ( t ) } { \\sigma _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) \\right] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { w } _ { t } , \\quad x _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } I ) .", "type": "interline_equation", "image_path": "65fdde0d1d949853db59a598710255c20a35d6c68ca6401f8b46cd9f514525f2.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 163, 518, 448, 546 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 107, 551, 506, 607 ], "lines": [ { "bbox": [ 106, 551, 505, 564 ], "spans": [ { "bbox": [ 106, 551, 505, 564 ], "score": 1.0, "content": "Samples can be generated from DPMs by solving the diffusion SDE in Eq. (2.5) with numerical", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 563, 505, 574 ], "spans": [ { "bbox": [ 106, 563, 268, 574 ], "score": 1.0, "content": "solvers, which discretize the SDE from", "type": "text" }, { "bbox": [ 268, 563, 276, 572 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 277, 563, 505, 574 ], "score": 1.0, "content": "to 0. Song et al. [3] proved that the traditional ancestral", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 573, 506, 586 ], "spans": [ { "bbox": [ 105, 573, 506, 586 ], "score": 1.0, "content": "sampling method for DPMs [2] can be viewed as a first-order SDE solver for Eq. (2.5). However, these", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 583, 506, 597 ], "spans": [ { "bbox": [ 105, 583, 506, 597 ], "score": 1.0, "content": "first-order methods usually need hundreds of or thousands of function evaluations to converge [3],", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 595, 279, 608 ], "spans": [ { "bbox": [ 106, 595, 279, 608 ], "score": 1.0, "content": "leading to extremely slow sampling speed.", "type": "text" } ], "index": 34 } ], "index": 32, "bbox_fs": [ 105, 551, 506, 608 ] }, { "type": "title", "bbox": [ 108, 619, 278, 632 ], "lines": [ { "bbox": [ 105, 618, 280, 635 ], "spans": [ { "bbox": [ 105, 618, 280, 635 ], "score": 1.0, "content": "2.2 Diffusion (Probability Flow) ODEs", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 106, 640, 506, 696 ], "lines": [ { "bbox": [ 105, 640, 506, 654 ], "spans": [ { "bbox": [ 105, 640, 506, 654 ], "score": 1.0, "content": "When discretizing SDEs, the step size is limited by the randomness of the Wiener process [27, Chap.", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 651, 505, 665 ], "spans": [ { "bbox": [ 105, 651, 505, 665 ], "score": 1.0, "content": "11]. A large step size (small number of steps) often causes non-convergence, especially in high", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 660, 507, 676 ], "spans": [ { "bbox": [ 105, 660, 507, 676 ], "score": 1.0, "content": "dimensional spaces. For faster sampling, one can consider the associated probability flow ODE [3],", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 672, 507, 687 ], "spans": [ { "bbox": [ 105, 672, 325, 687 ], "score": 1.0, "content": "which has the same marginal distribution at each time", "type": "text" }, { "bbox": [ 325, 675, 330, 683 ], "score": 0.8, "content": "t", "type": "inline_equation" }, { "bbox": [ 330, 672, 507, 687 ], "score": 1.0, "content": "as that of the SDE. Specifically, for DPMs,", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 684, 375, 698 ], "spans": [ { "bbox": [ 106, 684, 375, 698 ], "score": 1.0, "content": "Song et al. [3] proved that the probability flow ODE of Eq. (2.4) is", "type": "text" } ], "index": 40 } ], "index": 38, "bbox_fs": [ 105, 640, 507, 698 ] }, { "type": "interline_equation", "bbox": [ 189, 702, 421, 726 ], "lines": [ { "bbox": [ 189, 702, 421, 726 ], "spans": [ { "bbox": [ 189, 702, 421, 726 ], "score": 0.92, "content": "\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = f ( t ) \\pmb { x } _ { t } - \\frac { 1 } { 2 } g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,", "type": "interline_equation", "image_path": "db0329942038b0f1cd9de0b5b9716a202705a11feeab57d38ae6c43085d15745.jpg" } ] } ], "index": 41, "virtual_lines": [ { "bbox": [ 189, 702, 421, 726 ], "spans": [], "index": 41 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 72, 504, 95 ], "lines": [ { "bbox": [ 106, 73, 505, 86 ], "spans": [ { "bbox": [ 106, 73, 245, 86 ], "score": 1.0, "content": "where the marginal distribution of", "type": "text" }, { "bbox": [ 246, 74, 257, 84 ], "score": 0.86, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 257, 73, 286, 86 ], "score": 1.0, "content": "is also", "type": "text" }, { "bbox": [ 286, 73, 313, 85 ], "score": 0.92, "content": "q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 313, 73, 505, 86 ], "score": 1.0, "content": ". By replacing the score function with the noise", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 478, 97 ], "spans": [ { "bbox": [ 105, 83, 448, 97 ], "score": 1.0, "content": "prediction model, Song et al. [3] defined the following parameterized ODE (diffusion", "type": "text" }, { "bbox": [ 448, 84, 470, 94 ], "score": 0.41, "content": "O D E", "type": "inline_equation" }, { "bbox": [ 470, 83, 478, 97 ], "score": 1.0, "content": "):", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 170, 96, 441, 123 ], "lines": [ { "bbox": [ 170, 96, 441, 123 ], "spans": [ { "bbox": [ 170, 96, 441, 123 ], "score": 0.94, "content": "\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = \\pmb { h } _ { \\theta } ( \\pmb { x } _ { t } , t ) : = f ( t ) \\pmb { x } _ { t } + \\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t ) , \\quad \\pmb { x } _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\mathbf { I } ) .", "type": "interline_equation", "image_path": "772c489895a83788baac27c0b2cc0da51d29a5ff2c72c378ea8301efa3cfe52d.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 170, 96, 441, 105.0 ], "spans": [], "index": 2 }, { "bbox": [ 170, 105.0, 441, 114.0 ], "spans": [], "index": 3 }, { "bbox": [ 170, 114.0, 441, 123.0 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 123, 506, 212 ], "lines": [ { "bbox": [ 106, 124, 506, 137 ], "spans": [ { "bbox": [ 106, 124, 299, 137 ], "score": 1.0, "content": "Samples can be drawn by solving the ODE from", "type": "text" }, { "bbox": [ 299, 124, 308, 134 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 308, 124, 506, 137 ], "score": 1.0, "content": "to 0. Comparing with SDEs, ODEs can be solved", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 135, 506, 147 ], "spans": [ { "bbox": [ 105, 135, 506, 147 ], "score": 1.0, "content": "with larger step sizes as they have no randomness. Furthermore, we can take advantage of efficient", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 146, 505, 158 ], "spans": [ { "bbox": [ 105, 146, 505, 158 ], "score": 1.0, "content": "numerical ODE solvers to accelerate the sampling. Song et al. [3] used the RK45 ODE solver [28]", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 156, 506, 169 ], "spans": [ { "bbox": [ 105, 156, 315, 169 ], "score": 1.0, "content": "for the diffusion ODEs, which generates samples in", "type": "text" }, { "bbox": [ 315, 157, 337, 167 ], "score": 0.86, "content": "\\sim 6 0", "type": "inline_equation" }, { "bbox": [ 338, 156, 506, 169 ], "score": 1.0, "content": "function evaluations to reach comparable", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 167, 506, 182 ], "spans": [ { "bbox": [ 105, 167, 506, 182 ], "score": 1.0, "content": "quality with a 1000-step SDE solver for Eq. (2.5) on the CIFAR-10 dataset [29]. However, existing", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 178, 506, 192 ], "spans": [ { "bbox": [ 105, 178, 457, 192 ], "score": 1.0, "content": "general-purpose ODE solvers still cannot generate satisfactory samples in the few-step", "type": "text" }, { "bbox": [ 457, 179, 479, 189 ], "score": 0.8, "content": "\\sim 1 0", "type": "inline_equation" }, { "bbox": [ 479, 178, 506, 192 ], "score": 1.0, "content": "steps)", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 190, 506, 201 ], "spans": [ { "bbox": [ 106, 190, 506, 201 ], "score": 1.0, "content": "sampling regime. To the best of our knowledge, there is still a lack of training-free samplers for", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 200, 491, 213 ], "spans": [ { "bbox": [ 106, 200, 491, 213 ], "score": 1.0, "content": "DPMs in the few-step sampling regime, and the sampling speed of DPMs is still a critical issue.", "type": "text" } ], "index": 12 } ], "index": 8.5 }, { "type": "title", "bbox": [ 106, 226, 353, 240 ], "lines": [ { "bbox": [ 104, 225, 354, 242 ], "spans": [ { "bbox": [ 104, 225, 354, 242 ], "score": 1.0, "content": "3 Customized Fast Solvers for Diffusion ODEs", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 106, 249, 505, 316 ], "lines": [ { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "As highlighted in Sec. 2.2, discretizing SDEs is generally difficult in high dimensions [27, Chap. 11]", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 262, 505, 273 ], "spans": [ { "bbox": [ 106, 262, 505, 273 ], "score": 1.0, "content": "and it is hard to converge within few steps. In contrast, ODEs are easier to solve, yielding a potential", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 272, 505, 284 ], "spans": [ { "bbox": [ 106, 272, 505, 284 ], "score": 1.0, "content": "for fast samplers. However, as mentioned in Sec. 2.2, the general black-box ODE solver used in", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 283, 506, 295 ], "spans": [ { "bbox": [ 105, 283, 506, 295 ], "score": 1.0, "content": "previous work [3] empirically fails to converge in few steps. This motivates us to design a dedicated", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 293, 505, 306 ], "score": 1.0, "content": "solver for diffusion ODEs to enable fast and high-quality few-step sampling. We start with a detailed", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 305, 334, 316 ], "spans": [ { "bbox": [ 106, 305, 334, 316 ], "score": 1.0, "content": "investigation of the specific structure of diffusion ODEs.", "type": "text" } ], "index": 19 } ], "index": 16.5 }, { "type": "title", "bbox": [ 107, 327, 390, 340 ], "lines": [ { "bbox": [ 104, 327, 392, 342 ], "spans": [ { "bbox": [ 104, 327, 392, 342 ], "score": 1.0, "content": "3.1 Simplified Formulation of Exact Solutions of Diffusion ODEs", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 348, 504, 382 ], "lines": [ { "bbox": [ 106, 348, 504, 361 ], "spans": [ { "bbox": [ 106, 348, 338, 361 ], "score": 1.0, "content": "The key insight of this work is that given an initial value", "type": "text" }, { "bbox": [ 338, 351, 351, 360 ], "score": 0.86, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 351, 348, 382, 361 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 383, 349, 407, 359 ], "score": 0.89, "content": "s > 0", "type": "inline_equation" }, { "bbox": [ 407, 348, 461, 361 ], "score": 1.0, "content": ", the solution", "type": "text" }, { "bbox": [ 461, 351, 473, 360 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 473, 348, 504, 361 ], "score": 1.0, "content": "at each", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 359, 505, 371 ], "spans": [ { "bbox": [ 106, 359, 127, 371 ], "score": 1.0, "content": "time", "type": "text" }, { "bbox": [ 127, 360, 151, 370 ], "score": 0.89, "content": "t < s", "type": "inline_equation" }, { "bbox": [ 152, 359, 505, 371 ], "score": 1.0, "content": "of diffusion ODEs in Eq. (2.7) can be simplified into a very special exact formulation", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 370, 264, 383 ], "spans": [ { "bbox": [ 106, 370, 264, 383 ], "score": 1.0, "content": "which can be efficiently approximated.", "type": "text" } ], "index": 23 } ], "index": 22 }, { "type": "text", "bbox": [ 106, 386, 505, 480 ], "lines": [ { "bbox": [ 105, 386, 505, 399 ], "spans": [ { "bbox": [ 105, 386, 317, 399 ], "score": 1.0, "content": "Our first key observation is that a part of the solution", "type": "text" }, { "bbox": [ 317, 389, 328, 398 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 329, 386, 505, 399 ], "score": 1.0, "content": "can be exactly computed by considering the", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 397, 507, 411 ], "spans": [ { "bbox": [ 104, 397, 507, 411 ], "score": 1.0, "content": "particular structure of diffusion ODEs. The r.h.s. of diffusion ODEs in Eq. (2.7) consists of two parts:", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 408, 506, 426 ], "spans": [ { "bbox": [ 105, 409, 140, 426 ], "score": 1.0, "content": "the part", "type": "text" }, { "bbox": [ 140, 411, 169, 424 ], "score": 0.93, "content": "f ( t ) x _ { t }", "type": "inline_equation" }, { "bbox": [ 169, 409, 260, 426 ], "score": 1.0, "content": "is a linear function of", "type": "text" }, { "bbox": [ 260, 414, 271, 423 ], "score": 0.85, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 272, 409, 350, 426 ], "score": 1.0, "content": ", and the other part", "type": "text" }, { "bbox": [ 351, 408, 406, 426 ], "score": 0.93, "content": "\\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 406, 409, 506, 426 ], "score": 1.0, "content": "is generally a nonlinear", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 425, 505, 438 ], "spans": [ { "bbox": [ 106, 425, 151, 438 ], "score": 1.0, "content": "function of", "type": "text" }, { "bbox": [ 152, 426, 163, 436 ], "score": 0.87, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 163, 425, 283, 438 ], "score": 1.0, "content": "because of the neural network", "type": "text" }, { "bbox": [ 283, 425, 319, 437 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 320, 425, 505, 438 ], "score": 1.0, "content": ". This type of ODE is referred to as semi-linear", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 435, 506, 448 ], "spans": [ { "bbox": [ 105, 435, 506, 448 ], "score": 1.0, "content": "ODE. The black-box ODE solvers adopted by previous work [3] are ignorant of this semi-linear", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 446, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 232, 459 ], "score": 1.0, "content": "structure as they take the whole", "type": "text" }, { "bbox": [ 233, 446, 271, 459 ], "score": 0.93, "content": "h _ { \\theta } ( x _ { t } , \\bar t ) ", "type": "inline_equation" }, { "bbox": [ 271, 447, 505, 459 ], "score": 1.0, "content": "in Eq. (2.7) as the input, which causes discretization errors", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 458, 505, 469 ], "spans": [ { "bbox": [ 106, 458, 482, 469 ], "score": 1.0, "content": "of both the linear and nonlinear term. We note that for semi-linear ODEs, the solution at time", "type": "text" }, { "bbox": [ 483, 459, 488, 467 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 488, 458, 505, 469 ], "score": 1.0, "content": "can", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 469, 379, 480 ], "spans": [ { "bbox": [ 106, 469, 379, 480 ], "score": 1.0, "content": "be exactly formulated by the “variation of constants” formula [30]:", "type": "text" } ], "index": 31 } ], "index": 27.5 }, { "type": "interline_equation", "bbox": [ 188, 481, 423, 510 ], "lines": [ { "bbox": [ 188, 481, 423, 510 ], "spans": [ { "bbox": [ 188, 481, 423, 510 ], "score": 0.93, "content": "\\pmb { x } _ { t } = e ^ { \\int _ { s } ^ { t } f ( \\tau ) \\mathrm { d } \\tau } \\pmb { x } _ { s } + \\int _ { s } ^ { t } \\left( e ^ { \\int _ { \\tau } ^ { t } f ( r ) \\mathrm { d } r } \\frac { g ^ { 2 } \\big ( \\tau \\big ) } { 2 \\sigma _ { \\tau } } \\epsilon _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\right) \\mathrm { d } \\tau .", "type": "interline_equation", "image_path": "6f1153f8f759866e204632c40d87e1d736b7589259f6710fccd3282a51e399e5.jpg" } ] } ], "index": 32.5, "virtual_lines": [ { "bbox": [ 188, 481, 423, 495.5 ], "spans": [], "index": 32 }, { "bbox": [ 188, 495.5, 423, 510.0 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 510, 505, 565 ], "lines": [ { "bbox": [ 106, 510, 505, 523 ], "spans": [ { "bbox": [ 106, 510, 505, 523 ], "score": 1.0, "content": "This formulation decouples the linear part and the nonlinear part. In contrast to black-box ODE", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 522, 504, 532 ], "spans": [ { "bbox": [ 106, 522, 504, 532 ], "score": 1.0, "content": "solvers, the linear part is now exactly computed, which eliminates the approximation error of the", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 532, 504, 544 ], "spans": [ { "bbox": [ 106, 532, 504, 544 ], "score": 1.0, "content": "linear term. However, the integral of the nonlinear part is still complicated because it couples the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 542, 506, 556 ], "spans": [ { "bbox": [ 105, 542, 273, 556 ], "score": 1.0, "content": "coefficients about the noise schedule (i.e.,", "type": "text" }, { "bbox": [ 273, 543, 334, 555 ], "score": 0.95, "content": "f ( \\tau ) , g ( \\tau ) \\bar { , } \\sigma _ { \\tau } )", "type": "inline_equation" }, { "bbox": [ 335, 542, 464, 556 ], "score": 1.0, "content": "and the complex neural network", "type": "text" }, { "bbox": [ 464, 544, 474, 554 ], "score": 0.85, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 475, 542, 506, 556 ], "score": 1.0, "content": ", which", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 554, 217, 567 ], "spans": [ { "bbox": [ 105, 554, 217, 567 ], "score": 1.0, "content": "is still hard to approximate.", "type": "text" } ], "index": 38 } ], "index": 36 }, { "type": "text", "bbox": [ 106, 570, 505, 614 ], "lines": [ { "bbox": [ 105, 569, 505, 583 ], "spans": [ { "bbox": [ 105, 569, 505, 583 ], "score": 1.0, "content": "Our second key observation is that the integral of the nonlinear part can be greatly simplified by", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 581, 505, 594 ], "spans": [ { "bbox": [ 105, 581, 246, 594 ], "score": 1.0, "content": "introducing a special variable. Let", "type": "text" }, { "bbox": [ 247, 581, 317, 593 ], "score": 0.92, "content": "\\lambda _ { t } : = \\log \\bar { ( \\alpha _ { t } / \\sigma _ { t } ) }", "type": "inline_equation" }, { "bbox": [ 317, 581, 447, 594 ], "score": 1.0, "content": "(one half of the log-SNR), then", "type": "text" }, { "bbox": [ 447, 582, 457, 592 ], "score": 0.89, "content": "\\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 458, 581, 505, 594 ], "score": 1.0, "content": "is a strictly", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 591, 505, 605 ], "spans": [ { "bbox": [ 105, 591, 196, 605 ], "score": 1.0, "content": "decreasing function of", "type": "text" }, { "bbox": [ 197, 593, 201, 602 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 202, 591, 487, 605 ], "score": 1.0, "content": "(due to the definition of DPMs as discussed in Sec. 2.1). We can rewrite", "type": "text" }, { "bbox": [ 488, 592, 505, 604 ], "score": 0.91, "content": "g ( t )", "type": "inline_equation" } ], "index": 41 }, { "bbox": [ 105, 603, 166, 616 ], "spans": [ { "bbox": [ 105, 603, 166, 616 ], "score": 1.0, "content": "in Eq. (2.3) as", "type": "text" } ], "index": 42 } ], "index": 40.5 }, { "type": "interline_equation", "bbox": [ 156, 615, 455, 644 ], "lines": [ { "bbox": [ 156, 615, 455, 644 ], "spans": [ { "bbox": [ 156, 615, 455, 644 ], "score": 0.92, "content": "g ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } = 2 \\sigma _ { t } ^ { 2 } \\left( \\frac { \\mathrm { d } \\log { \\sigma _ { t } } } { \\mathrm { d } t } - \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\right) = - 2 \\sigma _ { t } ^ { 2 } \\frac { \\mathrm { d } \\lambda _ { t } } { \\mathrm { d } t } .", "type": "interline_equation", "image_path": "28428e28211e5384d76d1e8330c965590495d098f46b2f50a448a40982e1a8f0.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 156, 615, 455, 624.6666666666666 ], "spans": [], "index": 43 }, { "bbox": [ 156, 624.6666666666666, 455, 634.3333333333333 ], "spans": [], "index": 44 }, { "bbox": [ 156, 634.3333333333333, 455, 643.9999999999999 ], "spans": [], "index": 45 } ] }, { "type": "text", "bbox": [ 105, 644, 414, 657 ], "lines": [ { "bbox": [ 106, 643, 413, 659 ], "spans": [ { "bbox": [ 106, 643, 174, 659 ], "score": 1.0, "content": "Combining with", "type": "text" }, { "bbox": [ 174, 645, 252, 657 ], "score": 0.93, "content": "f ( t ) = \\mathrm { d } \\log \\alpha _ { t } / \\mathrm { d } t", "type": "inline_equation" }, { "bbox": [ 252, 643, 413, 659 ], "score": 1.0, "content": "in Eq. (2.3), we can rewrite Eq. (3.1) as", "type": "text" } ], "index": 46 } ], "index": 46 }, { "type": "interline_equation", "bbox": [ 211, 658, 399, 686 ], "lines": [ { "bbox": [ 211, 658, 399, 686 ], "spans": [ { "bbox": [ 211, 658, 399, 686 ], "score": 0.94, "content": "\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { s } ^ { t } \\left( \\frac { \\mathrm { d } \\lambda _ { \\tau } } { \\mathrm { d } \\tau } \\right) \\frac { \\sigma _ { \\tau } } { \\alpha _ { \\tau } } \\pmb { \\epsilon } _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\mathrm { d } \\tau .", "type": "interline_equation", "image_path": "d804fb43f91c9fdbe50275bbfdd61568c9d23d13e7c4cea3f8e0751dee5a2bb6.jpg" } ] } ], "index": 47.5, "virtual_lines": [ { "bbox": [ 211, 658, 399, 672.0 ], "spans": [], "index": 47 }, { "bbox": [ 211, 672.0, 399, 686.0 ], "spans": [], "index": 48 } ] }, { "type": "text", "bbox": [ 106, 687, 505, 723 ], "lines": [ { "bbox": [ 105, 686, 505, 701 ], "spans": [ { "bbox": [ 105, 686, 121, 701 ], "score": 1.0, "content": "As", "type": "text" }, { "bbox": [ 121, 687, 167, 699 ], "score": 0.91, "content": "\\lambda ( t ) = \\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 168, 686, 318, 701 ], "score": 1.0, "content": "is a strictly decreasing function of", "type": "text" }, { "bbox": [ 318, 689, 324, 698 ], "score": 0.78, "content": "t", "type": "inline_equation" }, { "bbox": [ 324, 686, 439, 701 ], "score": 1.0, "content": ", it has an inverse function", "type": "text" }, { "bbox": [ 439, 687, 460, 699 ], "score": 0.91, "content": "t _ { \\lambda } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 461, 686, 505, 701 ], "score": 1.0, "content": "satisfying", "type": "text" } ], "index": 49 }, { "bbox": [ 107, 695, 502, 715 ], "spans": [ { "bbox": [ 107, 699, 159, 711 ], "score": 0.91, "content": "t = t _ { \\lambda } ( \\lambda ( t ) )", "type": "inline_equation" }, { "bbox": [ 159, 695, 313, 715 ], "score": 1.0, "content": ". We further change the subscripts of", "type": "text" }, { "bbox": [ 313, 701, 321, 709 ], "score": 0.81, "content": "_ { \\textbf { \\em x } }", "type": "inline_equation" }, { "bbox": [ 321, 695, 339, 715 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 340, 700, 350, 710 ], "score": 0.85, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 351, 695, 374, 715 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 374, 700, 379, 709 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 380, 695, 391, 715 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 392, 699, 398, 709 ], "score": 0.83, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 399, 695, 447, 715 ], "score": 1.0, "content": "and denote", "type": "text" }, { "bbox": [ 447, 699, 502, 712 ], "score": 0.91, "content": "\\hat { \\pmb x } _ { \\lambda } : = \\pmb x _ { t _ { \\lambda } ( \\lambda ) }", "type": "inline_equation" } ], "index": 50 }, { "bbox": [ 106, 710, 492, 724 ], "spans": [ { "bbox": [ 106, 711, 232, 724 ], "score": 0.9, "content": "\\hat { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\hat { x } _ { \\lambda } , \\lambda ) : = \\epsilon _ { \\boldsymbol { \\theta } } ( x _ { t _ { \\lambda } ( \\lambda ) } , t _ { \\lambda } ( \\lambda ) )", "type": "inline_equation" }, { "bbox": [ 232, 710, 424, 724 ], "score": 1.0, "content": ". Rewrite Eq. (3.3) by “change-of-variable” for", "type": "text" }, { "bbox": [ 424, 711, 431, 721 ], "score": 0.79, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 431, 710, 492, 724 ], "score": 1.0, "content": ", then we have:", "type": "text" } ], "index": 51 } ], "index": 50 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 301, 741, 310, 752 ], "spans": [ { "bbox": [ 301, 741, 310, 752 ], "score": 1.0, "content": "4", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 72, 504, 95 ], "lines": [ { "bbox": [ 106, 73, 505, 86 ], "spans": [ { "bbox": [ 106, 73, 245, 86 ], "score": 1.0, "content": "where the marginal distribution of", "type": "text" }, { "bbox": [ 246, 74, 257, 84 ], "score": 0.86, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 257, 73, 286, 86 ], "score": 1.0, "content": "is also", "type": "text" }, { "bbox": [ 286, 73, 313, 85 ], "score": 0.92, "content": "q _ { t } ( \\pmb { x } _ { t } )", "type": "inline_equation" }, { "bbox": [ 313, 73, 505, 86 ], "score": 1.0, "content": ". By replacing the score function with the noise", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 478, 97 ], "spans": [ { "bbox": [ 105, 83, 448, 97 ], "score": 1.0, "content": "prediction model, Song et al. [3] defined the following parameterized ODE (diffusion", "type": "text" }, { "bbox": [ 448, 84, 470, 94 ], "score": 0.41, "content": "O D E", "type": "inline_equation" }, { "bbox": [ 470, 83, 478, 97 ], "score": 1.0, "content": "):", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 73, 505, 97 ] }, { "type": "interline_equation", "bbox": [ 170, 96, 441, 123 ], "lines": [ { "bbox": [ 170, 96, 441, 123 ], "spans": [ { "bbox": [ 170, 96, 441, 123 ], "score": 0.94, "content": "\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = \\pmb { h } _ { \\theta } ( \\pmb { x } _ { t } , t ) : = f ( t ) \\pmb { x } _ { t } + \\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t ) , \\quad \\pmb { x } _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\mathbf { I } ) .", "type": "interline_equation", "image_path": "772c489895a83788baac27c0b2cc0da51d29a5ff2c72c378ea8301efa3cfe52d.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 170, 96, 441, 105.0 ], "spans": [], "index": 2 }, { "bbox": [ 170, 105.0, 441, 114.0 ], "spans": [], "index": 3 }, { "bbox": [ 170, 114.0, 441, 123.0 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 123, 506, 212 ], "lines": [ { "bbox": [ 106, 124, 506, 137 ], "spans": [ { "bbox": [ 106, 124, 299, 137 ], "score": 1.0, "content": "Samples can be drawn by solving the ODE from", "type": "text" }, { "bbox": [ 299, 124, 308, 134 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 308, 124, 506, 137 ], "score": 1.0, "content": "to 0. Comparing with SDEs, ODEs can be solved", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 135, 506, 147 ], "spans": [ { "bbox": [ 105, 135, 506, 147 ], "score": 1.0, "content": "with larger step sizes as they have no randomness. Furthermore, we can take advantage of efficient", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 146, 505, 158 ], "spans": [ { "bbox": [ 105, 146, 505, 158 ], "score": 1.0, "content": "numerical ODE solvers to accelerate the sampling. Song et al. [3] used the RK45 ODE solver [28]", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 156, 506, 169 ], "spans": [ { "bbox": [ 105, 156, 315, 169 ], "score": 1.0, "content": "for the diffusion ODEs, which generates samples in", "type": "text" }, { "bbox": [ 315, 157, 337, 167 ], "score": 0.86, "content": "\\sim 6 0", "type": "inline_equation" }, { "bbox": [ 338, 156, 506, 169 ], "score": 1.0, "content": "function evaluations to reach comparable", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 167, 506, 182 ], "spans": [ { "bbox": [ 105, 167, 506, 182 ], "score": 1.0, "content": "quality with a 1000-step SDE solver for Eq. (2.5) on the CIFAR-10 dataset [29]. However, existing", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 178, 506, 192 ], "spans": [ { "bbox": [ 105, 178, 457, 192 ], "score": 1.0, "content": "general-purpose ODE solvers still cannot generate satisfactory samples in the few-step", "type": "text" }, { "bbox": [ 457, 179, 479, 189 ], "score": 0.8, "content": "\\sim 1 0", "type": "inline_equation" }, { "bbox": [ 479, 178, 506, 192 ], "score": 1.0, "content": "steps)", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 190, 506, 201 ], "spans": [ { "bbox": [ 106, 190, 506, 201 ], "score": 1.0, "content": "sampling regime. To the best of our knowledge, there is still a lack of training-free samplers for", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 200, 491, 213 ], "spans": [ { "bbox": [ 106, 200, 491, 213 ], "score": 1.0, "content": "DPMs in the few-step sampling regime, and the sampling speed of DPMs is still a critical issue.", "type": "text" } ], "index": 12 } ], "index": 8.5, "bbox_fs": [ 105, 124, 506, 213 ] }, { "type": "title", "bbox": [ 106, 226, 353, 240 ], "lines": [ { "bbox": [ 104, 225, 354, 242 ], "spans": [ { "bbox": [ 104, 225, 354, 242 ], "score": 1.0, "content": "3 Customized Fast Solvers for Diffusion ODEs", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 106, 249, 505, 316 ], "lines": [ { "bbox": [ 105, 250, 505, 263 ], "spans": [ { "bbox": [ 105, 250, 505, 263 ], "score": 1.0, "content": "As highlighted in Sec. 2.2, discretizing SDEs is generally difficult in high dimensions [27, Chap. 11]", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 262, 505, 273 ], "spans": [ { "bbox": [ 106, 262, 505, 273 ], "score": 1.0, "content": "and it is hard to converge within few steps. In contrast, ODEs are easier to solve, yielding a potential", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 272, 505, 284 ], "spans": [ { "bbox": [ 106, 272, 505, 284 ], "score": 1.0, "content": "for fast samplers. However, as mentioned in Sec. 2.2, the general black-box ODE solver used in", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 283, 506, 295 ], "spans": [ { "bbox": [ 105, 283, 506, 295 ], "score": 1.0, "content": "previous work [3] empirically fails to converge in few steps. This motivates us to design a dedicated", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 293, 505, 306 ], "score": 1.0, "content": "solver for diffusion ODEs to enable fast and high-quality few-step sampling. We start with a detailed", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 305, 334, 316 ], "spans": [ { "bbox": [ 106, 305, 334, 316 ], "score": 1.0, "content": "investigation of the specific structure of diffusion ODEs.", "type": "text" } ], "index": 19 } ], "index": 16.5, "bbox_fs": [ 105, 250, 506, 316 ] }, { "type": "title", "bbox": [ 107, 327, 390, 340 ], "lines": [ { "bbox": [ 104, 327, 392, 342 ], "spans": [ { "bbox": [ 104, 327, 392, 342 ], "score": 1.0, "content": "3.1 Simplified Formulation of Exact Solutions of Diffusion ODEs", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 348, 504, 382 ], "lines": [ { "bbox": [ 106, 348, 504, 361 ], "spans": [ { "bbox": [ 106, 348, 338, 361 ], "score": 1.0, "content": "The key insight of this work is that given an initial value", "type": "text" }, { "bbox": [ 338, 351, 351, 360 ], "score": 0.86, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 351, 348, 382, 361 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 383, 349, 407, 359 ], "score": 0.89, "content": "s > 0", "type": "inline_equation" }, { "bbox": [ 407, 348, 461, 361 ], "score": 1.0, "content": ", the solution", "type": "text" }, { "bbox": [ 461, 351, 473, 360 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 473, 348, 504, 361 ], "score": 1.0, "content": "at each", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 359, 505, 371 ], "spans": [ { "bbox": [ 106, 359, 127, 371 ], "score": 1.0, "content": "time", "type": "text" }, { "bbox": [ 127, 360, 151, 370 ], "score": 0.89, "content": "t < s", "type": "inline_equation" }, { "bbox": [ 152, 359, 505, 371 ], "score": 1.0, "content": "of diffusion ODEs in Eq. (2.7) can be simplified into a very special exact formulation", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 370, 264, 383 ], "spans": [ { "bbox": [ 106, 370, 264, 383 ], "score": 1.0, "content": "which can be efficiently approximated.", "type": "text" } ], "index": 23 } ], "index": 22, "bbox_fs": [ 106, 348, 505, 383 ] }, { "type": "text", "bbox": [ 106, 386, 505, 480 ], "lines": [ { "bbox": [ 105, 386, 505, 399 ], "spans": [ { "bbox": [ 105, 386, 317, 399 ], "score": 1.0, "content": "Our first key observation is that a part of the solution", "type": "text" }, { "bbox": [ 317, 389, 328, 398 ], "score": 0.84, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 329, 386, 505, 399 ], "score": 1.0, "content": "can be exactly computed by considering the", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 397, 507, 411 ], "spans": [ { "bbox": [ 104, 397, 507, 411 ], "score": 1.0, "content": "particular structure of diffusion ODEs. The r.h.s. of diffusion ODEs in Eq. (2.7) consists of two parts:", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 408, 506, 426 ], "spans": [ { "bbox": [ 105, 409, 140, 426 ], "score": 1.0, "content": "the part", "type": "text" }, { "bbox": [ 140, 411, 169, 424 ], "score": 0.93, "content": "f ( t ) x _ { t }", "type": "inline_equation" }, { "bbox": [ 169, 409, 260, 426 ], "score": 1.0, "content": "is a linear function of", "type": "text" }, { "bbox": [ 260, 414, 271, 423 ], "score": 0.85, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 272, 409, 350, 426 ], "score": 1.0, "content": ", and the other part", "type": "text" }, { "bbox": [ 351, 408, 406, 426 ], "score": 0.93, "content": "\\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 406, 409, 506, 426 ], "score": 1.0, "content": "is generally a nonlinear", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 425, 505, 438 ], "spans": [ { "bbox": [ 106, 425, 151, 438 ], "score": 1.0, "content": "function of", "type": "text" }, { "bbox": [ 152, 426, 163, 436 ], "score": 0.87, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 163, 425, 283, 438 ], "score": 1.0, "content": "because of the neural network", "type": "text" }, { "bbox": [ 283, 425, 319, 437 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 320, 425, 505, 438 ], "score": 1.0, "content": ". This type of ODE is referred to as semi-linear", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 435, 506, 448 ], "spans": [ { "bbox": [ 105, 435, 506, 448 ], "score": 1.0, "content": "ODE. The black-box ODE solvers adopted by previous work [3] are ignorant of this semi-linear", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 446, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 232, 459 ], "score": 1.0, "content": "structure as they take the whole", "type": "text" }, { "bbox": [ 233, 446, 271, 459 ], "score": 0.93, "content": "h _ { \\theta } ( x _ { t } , \\bar t ) ", "type": "inline_equation" }, { "bbox": [ 271, 447, 505, 459 ], "score": 1.0, "content": "in Eq. (2.7) as the input, which causes discretization errors", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 458, 505, 469 ], "spans": [ { "bbox": [ 106, 458, 482, 469 ], "score": 1.0, "content": "of both the linear and nonlinear term. We note that for semi-linear ODEs, the solution at time", "type": "text" }, { "bbox": [ 483, 459, 488, 467 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 488, 458, 505, 469 ], "score": 1.0, "content": "can", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 469, 379, 480 ], "spans": [ { "bbox": [ 106, 469, 379, 480 ], "score": 1.0, "content": "be exactly formulated by the “variation of constants” formula [30]:", "type": "text" } ], "index": 31 } ], "index": 27.5, "bbox_fs": [ 104, 386, 507, 480 ] }, { "type": "interline_equation", "bbox": [ 188, 481, 423, 510 ], "lines": [ { "bbox": [ 188, 481, 423, 510 ], "spans": [ { "bbox": [ 188, 481, 423, 510 ], "score": 0.93, "content": "\\pmb { x } _ { t } = e ^ { \\int _ { s } ^ { t } f ( \\tau ) \\mathrm { d } \\tau } \\pmb { x } _ { s } + \\int _ { s } ^ { t } \\left( e ^ { \\int _ { \\tau } ^ { t } f ( r ) \\mathrm { d } r } \\frac { g ^ { 2 } \\big ( \\tau \\big ) } { 2 \\sigma _ { \\tau } } \\epsilon _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\right) \\mathrm { d } \\tau .", "type": "interline_equation", "image_path": "6f1153f8f759866e204632c40d87e1d736b7589259f6710fccd3282a51e399e5.jpg" } ] } ], "index": 32.5, "virtual_lines": [ { "bbox": [ 188, 481, 423, 495.5 ], "spans": [], "index": 32 }, { "bbox": [ 188, 495.5, 423, 510.0 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 510, 505, 565 ], "lines": [ { "bbox": [ 106, 510, 505, 523 ], "spans": [ { "bbox": [ 106, 510, 505, 523 ], "score": 1.0, "content": "This formulation decouples the linear part and the nonlinear part. In contrast to black-box ODE", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 522, 504, 532 ], "spans": [ { "bbox": [ 106, 522, 504, 532 ], "score": 1.0, "content": "solvers, the linear part is now exactly computed, which eliminates the approximation error of the", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 532, 504, 544 ], "spans": [ { "bbox": [ 106, 532, 504, 544 ], "score": 1.0, "content": "linear term. However, the integral of the nonlinear part is still complicated because it couples the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 542, 506, 556 ], "spans": [ { "bbox": [ 105, 542, 273, 556 ], "score": 1.0, "content": "coefficients about the noise schedule (i.e.,", "type": "text" }, { "bbox": [ 273, 543, 334, 555 ], "score": 0.95, "content": "f ( \\tau ) , g ( \\tau ) \\bar { , } \\sigma _ { \\tau } )", "type": "inline_equation" }, { "bbox": [ 335, 542, 464, 556 ], "score": 1.0, "content": "and the complex neural network", "type": "text" }, { "bbox": [ 464, 544, 474, 554 ], "score": 0.85, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 475, 542, 506, 556 ], "score": 1.0, "content": ", which", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 554, 217, 567 ], "spans": [ { "bbox": [ 105, 554, 217, 567 ], "score": 1.0, "content": "is still hard to approximate.", "type": "text" } ], "index": 38 } ], "index": 36, "bbox_fs": [ 105, 510, 506, 567 ] }, { "type": "text", "bbox": [ 106, 570, 505, 614 ], "lines": [ { "bbox": [ 105, 569, 505, 583 ], "spans": [ { "bbox": [ 105, 569, 505, 583 ], "score": 1.0, "content": "Our second key observation is that the integral of the nonlinear part can be greatly simplified by", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 581, 505, 594 ], "spans": [ { "bbox": [ 105, 581, 246, 594 ], "score": 1.0, "content": "introducing a special variable. Let", "type": "text" }, { "bbox": [ 247, 581, 317, 593 ], "score": 0.92, "content": "\\lambda _ { t } : = \\log \\bar { ( \\alpha _ { t } / \\sigma _ { t } ) }", "type": "inline_equation" }, { "bbox": [ 317, 581, 447, 594 ], "score": 1.0, "content": "(one half of the log-SNR), then", "type": "text" }, { "bbox": [ 447, 582, 457, 592 ], "score": 0.89, "content": "\\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 458, 581, 505, 594 ], "score": 1.0, "content": "is a strictly", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 591, 505, 605 ], "spans": [ { "bbox": [ 105, 591, 196, 605 ], "score": 1.0, "content": "decreasing function of", "type": "text" }, { "bbox": [ 197, 593, 201, 602 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 202, 591, 487, 605 ], "score": 1.0, "content": "(due to the definition of DPMs as discussed in Sec. 2.1). We can rewrite", "type": "text" }, { "bbox": [ 488, 592, 505, 604 ], "score": 0.91, "content": "g ( t )", "type": "inline_equation" } ], "index": 41 }, { "bbox": [ 105, 603, 166, 616 ], "spans": [ { "bbox": [ 105, 603, 166, 616 ], "score": 1.0, "content": "in Eq. (2.3) as", "type": "text" } ], "index": 42 } ], "index": 40.5, "bbox_fs": [ 105, 569, 505, 616 ] }, { "type": "interline_equation", "bbox": [ 156, 615, 455, 644 ], "lines": [ { "bbox": [ 156, 615, 455, 644 ], "spans": [ { "bbox": [ 156, 615, 455, 644 ], "score": 0.92, "content": "g ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } = 2 \\sigma _ { t } ^ { 2 } \\left( \\frac { \\mathrm { d } \\log { \\sigma _ { t } } } { \\mathrm { d } t } - \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\right) = - 2 \\sigma _ { t } ^ { 2 } \\frac { \\mathrm { d } \\lambda _ { t } } { \\mathrm { d } t } .", "type": "interline_equation", "image_path": "28428e28211e5384d76d1e8330c965590495d098f46b2f50a448a40982e1a8f0.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 156, 615, 455, 624.6666666666666 ], "spans": [], "index": 43 }, { "bbox": [ 156, 624.6666666666666, 455, 634.3333333333333 ], "spans": [], "index": 44 }, { "bbox": [ 156, 634.3333333333333, 455, 643.9999999999999 ], "spans": [], "index": 45 } ] }, { "type": "text", "bbox": [ 105, 644, 414, 657 ], "lines": [ { "bbox": [ 106, 643, 413, 659 ], "spans": [ { "bbox": [ 106, 643, 174, 659 ], "score": 1.0, "content": "Combining with", "type": "text" }, { "bbox": [ 174, 645, 252, 657 ], "score": 0.93, "content": "f ( t ) = \\mathrm { d } \\log \\alpha _ { t } / \\mathrm { d } t", "type": "inline_equation" }, { "bbox": [ 252, 643, 413, 659 ], "score": 1.0, "content": "in Eq. (2.3), we can rewrite Eq. (3.1) as", "type": "text" } ], "index": 46 } ], "index": 46, "bbox_fs": [ 106, 643, 413, 659 ] }, { "type": "interline_equation", "bbox": [ 211, 658, 399, 686 ], "lines": [ { "bbox": [ 211, 658, 399, 686 ], "spans": [ { "bbox": [ 211, 658, 399, 686 ], "score": 0.94, "content": "\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { s } ^ { t } \\left( \\frac { \\mathrm { d } \\lambda _ { \\tau } } { \\mathrm { d } \\tau } \\right) \\frac { \\sigma _ { \\tau } } { \\alpha _ { \\tau } } \\pmb { \\epsilon } _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\mathrm { d } \\tau .", "type": "interline_equation", "image_path": "d804fb43f91c9fdbe50275bbfdd61568c9d23d13e7c4cea3f8e0751dee5a2bb6.jpg" } ] } ], "index": 47.5, "virtual_lines": [ { "bbox": [ 211, 658, 399, 672.0 ], "spans": [], "index": 47 }, { "bbox": [ 211, 672.0, 399, 686.0 ], "spans": [], "index": 48 } ] }, { "type": "text", "bbox": [ 106, 687, 505, 723 ], "lines": [ { "bbox": [ 105, 686, 505, 701 ], "spans": [ { "bbox": [ 105, 686, 121, 701 ], "score": 1.0, "content": "As", "type": "text" }, { "bbox": [ 121, 687, 167, 699 ], "score": 0.91, "content": "\\lambda ( t ) = \\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 168, 686, 318, 701 ], "score": 1.0, "content": "is a strictly decreasing function of", "type": "text" }, { "bbox": [ 318, 689, 324, 698 ], "score": 0.78, "content": "t", "type": "inline_equation" }, { "bbox": [ 324, 686, 439, 701 ], "score": 1.0, "content": ", it has an inverse function", "type": "text" }, { "bbox": [ 439, 687, 460, 699 ], "score": 0.91, "content": "t _ { \\lambda } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 461, 686, 505, 701 ], "score": 1.0, "content": "satisfying", "type": "text" } ], "index": 49 }, { "bbox": [ 107, 695, 502, 715 ], "spans": [ { "bbox": [ 107, 699, 159, 711 ], "score": 0.91, "content": "t = t _ { \\lambda } ( \\lambda ( t ) )", "type": "inline_equation" }, { "bbox": [ 159, 695, 313, 715 ], "score": 1.0, "content": ". We further change the subscripts of", "type": "text" }, { "bbox": [ 313, 701, 321, 709 ], "score": 0.81, "content": "_ { \\textbf { \\em x } }", "type": "inline_equation" }, { "bbox": [ 321, 695, 339, 715 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 340, 700, 350, 710 ], "score": 0.85, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 351, 695, 374, 715 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 374, 700, 379, 709 ], "score": 0.74, "content": "t", "type": "inline_equation" }, { "bbox": [ 380, 695, 391, 715 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 392, 699, 398, 709 ], "score": 0.83, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 399, 695, 447, 715 ], "score": 1.0, "content": "and denote", "type": "text" }, { "bbox": [ 447, 699, 502, 712 ], "score": 0.91, "content": "\\hat { \\pmb x } _ { \\lambda } : = \\pmb x _ { t _ { \\lambda } ( \\lambda ) }", "type": "inline_equation" } ], "index": 50 }, { "bbox": [ 106, 710, 492, 724 ], "spans": [ { "bbox": [ 106, 711, 232, 724 ], "score": 0.9, "content": "\\hat { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\hat { x } _ { \\lambda } , \\lambda ) : = \\epsilon _ { \\boldsymbol { \\theta } } ( x _ { t _ { \\lambda } ( \\lambda ) } , t _ { \\lambda } ( \\lambda ) )", "type": "inline_equation" }, { "bbox": [ 232, 710, 424, 724 ], "score": 1.0, "content": ". Rewrite Eq. (3.3) by “change-of-variable” for", "type": "text" }, { "bbox": [ 424, 711, 431, 721 ], "score": 0.79, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 431, 710, 492, 724 ], "score": 1.0, "content": ", then we have:", "type": "text" } ], "index": 51 } ], "index": 50, "bbox_fs": [ 105, 686, 505, 724 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 72, 504, 96 ], "lines": [ { "bbox": [ 106, 72, 505, 85 ], "spans": [ { "bbox": [ 106, 72, 416, 85 ], "score": 1.0, "content": "Proposition 3.1 (Exact solution of diffusion ODEs). Given an initial value", "type": "text" }, { "bbox": [ 417, 75, 429, 84 ], "score": 0.85, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 429, 72, 461, 85 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 461, 73, 486, 83 ], "score": 0.88, "content": "s > 0", "type": "inline_equation" }, { "bbox": [ 486, 72, 505, 85 ], "score": 1.0, "content": ", the", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 83, 356, 96 ], "spans": [ { "bbox": [ 106, 83, 141, 96 ], "score": 1.0, "content": "solution", "type": "text" }, { "bbox": [ 141, 86, 152, 95 ], "score": 0.82, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 153, 83, 183, 96 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 184, 83, 220, 96 ], "score": 0.93, "content": "t \\in [ 0 , s ]", "type": "inline_equation" }, { "bbox": [ 220, 83, 356, 96 ], "score": 1.0, "content": "of diffusion ODEs in Eq. (2.7) is:", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 224, 99, 387, 129 ], "lines": [ { "bbox": [ 224, 99, 387, 129 ], "spans": [ { "bbox": [ 224, 99, 387, 129 ], "score": 0.95, "content": "\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { \\lambda _ { s } } ^ { \\lambda _ { t } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .", "type": "interline_equation", "image_path": "9abcfd7effacf68fb53c21394f34e41f5f607171cf80a13282a4bf3971c34831.jpg" } ] } ], "index": 2.5, "virtual_lines": [ { "bbox": [ 224, 99, 387, 114.0 ], "spans": [], "index": 2 }, { "bbox": [ 224, 114.0, 387, 129.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 106, 138, 505, 172 ], "lines": [ { "bbox": [ 104, 137, 506, 153 ], "spans": [ { "bbox": [ 104, 137, 191, 153 ], "score": 1.0, "content": "We call the integral", "type": "text" }, { "bbox": [ 192, 138, 267, 151 ], "score": 0.93, "content": "\\begin{array} { r } { \\int e ^ { - \\lambda } \\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda } \\end{array}", "type": "inline_equation" }, { "bbox": [ 267, 137, 430, 153 ], "score": 1.0, "content": "the exponentially weighted integral of", "type": "text" }, { "bbox": [ 431, 139, 441, 150 ], "score": 0.87, "content": "\\scriptstyle { \\hat { \\epsilon } } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 441, 137, 506, 153 ], "score": 1.0, "content": ", which is very", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 149, 506, 162 ], "spans": [ { "bbox": [ 105, 149, 506, 162 ], "score": 1.0, "content": "special and highly related to the exponential integrators in the literature of ODE solvers [25]. To the", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 160, 496, 173 ], "spans": [ { "bbox": [ 106, 160, 496, 173 ], "score": 1.0, "content": "best of our knowledge, such formulation has not been revealed in prior work of diffusion models.", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 177, 505, 232 ], "lines": [ { "bbox": [ 106, 176, 506, 190 ], "spans": [ { "bbox": [ 106, 176, 506, 190 ], "score": 1.0, "content": "Eq. (3.4) provides a new perspective for approximating the solutions of diffusion ODEs. Specifically,", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 188, 506, 201 ], "spans": [ { "bbox": [ 106, 188, 132, 201 ], "score": 1.0, "content": "given", "type": "text" }, { "bbox": [ 132, 190, 144, 199 ], "score": 0.86, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 144, 188, 177, 201 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 178, 190, 183, 198 ], "score": 0.68, "content": "s", "type": "inline_equation" }, { "bbox": [ 184, 188, 432, 201 ], "score": 1.0, "content": ", According to Eq. (3.4), approximating the solution at time", "type": "text" }, { "bbox": [ 433, 189, 438, 198 ], "score": 0.76, "content": "t", "type": "inline_equation" }, { "bbox": [ 438, 188, 506, 201 ], "score": 1.0, "content": "is equivalent to", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 198, 505, 212 ], "spans": [ { "bbox": [ 106, 198, 360, 212 ], "score": 1.0, "content": "directly approximating the exponentially weighted integral of", "type": "text" }, { "bbox": [ 361, 199, 371, 210 ], "score": 0.88, "content": "\\hat { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 371, 198, 395, 212 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 395, 199, 407, 210 ], "score": 0.88, "content": "\\lambda _ { s }", "type": "inline_equation" }, { "bbox": [ 407, 198, 419, 212 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 419, 199, 429, 210 ], "score": 0.87, "content": "\\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 430, 198, 505, 212 ], "score": 1.0, "content": ", which avoids the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 209, 506, 223 ], "spans": [ { "bbox": [ 105, 209, 506, 223 ], "score": 1.0, "content": "error of the linear terms and is well-studied in the literature of exponential integrators [25, 31]. Based", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 220, 492, 234 ], "spans": [ { "bbox": [ 106, 220, 492, 234 ], "score": 1.0, "content": "on this insight, we propose fast solvers for diffusion ODEs, as detailed in the following sections.", "type": "text" } ], "index": 11 } ], "index": 9 }, { "type": "title", "bbox": [ 106, 244, 300, 257 ], "lines": [ { "bbox": [ 105, 244, 301, 259 ], "spans": [ { "bbox": [ 105, 244, 301, 259 ], "score": 1.0, "content": "3.2 High-Order Solvers for Diffusion ODEs", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 264, 505, 298 ], "lines": [ { "bbox": [ 105, 264, 505, 278 ], "spans": [ { "bbox": [ 105, 264, 505, 278 ], "score": 1.0, "content": "In this section, we propose high-order solvers for diffusion ODEs with convergence order guarantee", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 276, 505, 288 ], "spans": [ { "bbox": [ 106, 276, 505, 288 ], "score": 1.0, "content": "by leveraging our proposed solution formulation Eq. (3.4). The proposed solvers and analysis are", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 287, 467, 299 ], "spans": [ { "bbox": [ 105, 287, 467, 299 ], "score": 1.0, "content": "highly motivated by the methods of exponential integrators [25, 31] in the ODE literature.", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 302, 505, 348 ], "lines": [ { "bbox": [ 103, 300, 507, 318 ], "spans": [ { "bbox": [ 103, 300, 249, 318 ], "score": 1.0, "content": "Specifically, given an initial value", "type": "text" }, { "bbox": [ 250, 305, 264, 314 ], "score": 0.85, "content": "\\mathbf { \\nabla } _ { \\mathbf { x } _ { T } }", "type": "inline_equation" }, { "bbox": [ 264, 300, 297, 318 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 298, 304, 306, 313 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 307, 300, 326, 318 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 326, 303, 356, 314 ], "score": 0.89, "content": "M + 1", "type": "inline_equation" }, { "bbox": [ 356, 300, 402, 318 ], "score": 1.0, "content": "time steps", "type": "text" }, { "bbox": [ 403, 302, 434, 315 ], "score": 0.93, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 434, 300, 507, 318 ], "score": 1.0, "content": "decreasing from", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 313, 505, 327 ], "spans": [ { "bbox": [ 106, 314, 136, 325 ], "score": 0.9, "content": "t _ { 0 } = T", "type": "inline_equation" }, { "bbox": [ 137, 313, 147, 327 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 147, 315, 179, 325 ], "score": 0.9, "content": "t _ { M } = 0", "type": "inline_equation" }, { "bbox": [ 180, 313, 199, 327 ], "score": 1.0, "content": ". Let", "type": "text" }, { "bbox": [ 199, 315, 240, 325 ], "score": 0.89, "content": "\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }", "type": "inline_equation" }, { "bbox": [ 240, 313, 419, 327 ], "score": 1.0, "content": "be the initial value. The proposed solvers use", "type": "text" }, { "bbox": [ 419, 315, 431, 324 ], "score": 0.81, "content": "M", "type": "inline_equation" }, { "bbox": [ 431, 313, 505, 327 ], "score": 1.0, "content": "steps to iteratively", "type": "text" } ], "index": 17 }, { "bbox": [ 102, 320, 510, 342 ], "spans": [ { "bbox": [ 102, 320, 190, 342 ], "score": 1.0, "content": "compute a sequence", "type": "text" }, { "bbox": [ 190, 325, 227, 337 ], "score": 0.92, "content": "\\{ \\tilde { { \\pmb { x } } } _ { t _ { i } } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 228, 320, 416, 342 ], "score": 1.0, "content": "to approximate the true solutions at time steps", "type": "text" }, { "bbox": [ 417, 324, 448, 337 ], "score": 0.92, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 448, 320, 510, 342 ], "score": 1.0, "content": ". In particular,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 335, 348, 349 ], "spans": [ { "bbox": [ 105, 335, 165, 349 ], "score": 1.0, "content": "the last iterate", "type": "text" }, { "bbox": [ 165, 336, 183, 348 ], "score": 0.91, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { M } }", "type": "inline_equation" }, { "bbox": [ 183, 335, 348, 349 ], "score": 1.0, "content": "approximates the true solution at time 0.", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "text", "bbox": [ 107, 352, 505, 386 ], "lines": [ { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 314, 365 ], "score": 1.0, "content": "In order to reduce the approximation error between", "type": "text" }, { "bbox": [ 315, 353, 333, 364 ], "score": 0.91, "content": "\\tilde { \\pmb { x } } _ { t _ { M } }", "type": "inline_equation" }, { "bbox": [ 333, 352, 505, 365 ], "score": 1.0, "content": "and the true solution at time 0, we need to", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 360, 503, 379 ], "spans": [ { "bbox": [ 104, 360, 264, 379 ], "score": 1.0, "content": "reduce the approximation error for each", "type": "text" }, { "bbox": [ 264, 364, 279, 375 ], "score": 0.9, "content": "\\tilde { \\mathbf { x } } _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 279, 360, 480, 379 ], "score": 1.0, "content": "at every step [30]. Starting with the previous value", "type": "text" }, { "bbox": [ 480, 363, 503, 376 ], "score": 0.91, "content": "\\tilde { \\pmb { x } } _ { t _ { i - 1 } }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 105, 374, 447, 388 ], "spans": [ { "bbox": [ 105, 374, 136, 388 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 136, 375, 154, 386 ], "score": 0.89, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 155, 374, 323, 388 ], "score": 1.0, "content": ", according to Eq. (3.4), the exact solution", "type": "text" }, { "bbox": [ 324, 376, 360, 387 ], "score": 0.92, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 360, 374, 392, 388 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 392, 375, 400, 385 ], "score": 0.85, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 400, 374, 447, 388 ], "score": 1.0, "content": "is given by", "type": "text" } ], "index": 22 } ], "index": 21 }, { "type": "interline_equation", "bbox": [ 194, 390, 416, 422 ], "lines": [ { "bbox": [ 194, 390, 416, 422 ], "spans": [ { "bbox": [ 194, 390, 416, 422 ], "score": 0.93, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .", "type": "interline_equation", "image_path": "e941fd95cbf3e1c496583b82dd623ff65158438a8e8529e43b6a17ad2855df95.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 194, 390, 416, 406.0 ], "spans": [], "index": 23 }, { "bbox": [ 194, 406.0, 416, 422.0 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 106, 425, 505, 478 ], "lines": [ { "bbox": [ 104, 423, 506, 441 ], "spans": [ { "bbox": [ 104, 423, 245, 441 ], "score": 1.0, "content": "Therefore, to compute the value", "type": "text" }, { "bbox": [ 246, 426, 260, 438 ], "score": 0.9, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 260, 423, 342, 441 ], "score": 1.0, "content": "for approximating", "type": "text" }, { "bbox": [ 342, 428, 379, 438 ], "score": 0.92, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 379, 423, 506, 441 ], "score": 1.0, "content": ", we need to approximate the", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 436, 505, 455 ], "spans": [ { "bbox": [ 104, 436, 241, 455 ], "score": 1.0, "content": "exponentially weighted integral of", "type": "text" }, { "bbox": [ 241, 441, 252, 452 ], "score": 0.87, "content": "\\hat { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 252, 436, 273, 455 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 274, 441, 295, 453 ], "score": 0.92, "content": "\\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 295, 436, 306, 455 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 307, 441, 320, 452 ], "score": 0.89, "content": "\\lambda _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 320, 436, 356, 455 ], "score": 1.0, "content": ". Denote", "type": "text" }, { "bbox": [ 357, 440, 424, 453 ], "score": 0.92, "content": "h _ { i } : = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 425, 436, 444, 455 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 444, 437, 505, 453 ], "score": 0.91, "content": "\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { { \\mathbf x } } _ { \\lambda } , \\lambda ) \\mathrel { \\mathop : } =", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 104, 450, 507, 469 ], "spans": [ { "bbox": [ 104, 450, 507, 469 ], "score": 1.0, "content": "dnϵˆθ(xˆλ,λ)dλn as the n-th order total derivative of ϵˆθ(xˆλ, λ) w.r.t. λ. For k ≥ 1, the (k − 1)-th order", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 464, 304, 479 ], "spans": [ { "bbox": [ 105, 464, 188, 479 ], "score": 1.0, "content": "Taylor expansion of", "type": "text" }, { "bbox": [ 188, 465, 228, 478 ], "score": 0.93, "content": "\\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda )", "type": "inline_equation" }, { "bbox": [ 228, 464, 252, 479 ], "score": 1.0, "content": "w.r.t.", "type": "text" }, { "bbox": [ 252, 466, 259, 475 ], "score": 0.79, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 260, 464, 270, 479 ], "score": 1.0, "content": "at", "type": "text" }, { "bbox": [ 271, 466, 292, 478 ], "score": 0.92, "content": "\\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 292, 464, 304, 479 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 28 } ], "index": 26.5 }, { "type": "interline_equation", "bbox": [ 162, 482, 448, 516 ], "lines": [ { "bbox": [ 162, 482, 448, 516 ], "spans": [ { "bbox": [ 162, 482, 448, 516 ], "score": 0.94, "content": "\\hat { \\epsilon } _ { \\theta } ( \\hat { x } _ { \\lambda } , \\lambda ) = \\sum _ { n = 0 } ^ { k - 1 } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { x } _ { \\lambda _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) + \\mathcal { O } ( ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { k } ) ,", "type": "interline_equation", "image_path": "976243e2861066edc99111ffc150a8141dda0494f026576eb3e22e90ae405d03.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 162, 482, 448, 493.3333333333333 ], "spans": [], "index": 29 }, { "bbox": [ 162, 493.3333333333333, 448, 504.66666666666663 ], "spans": [], "index": 30 }, { "bbox": [ 162, 504.66666666666663, 448, 516.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 519, 351, 531 ], "lines": [ { "bbox": [ 106, 518, 351, 532 ], "spans": [ { "bbox": [ 106, 518, 351, 532 ], "score": 1.0, "content": "Substituting the above Taylor expansion into Eq. (3.5) yields", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 111, 534, 482, 569 ], "lines": [ { "bbox": [ 111, 534, 482, 569 ], "spans": [ { "bbox": [ 111, 534, 482, 569 ], "score": 0.94, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\sum _ { n = 0 } ^ { k - 1 } \\hat { \\pmb { \\epsilon } } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { { \\pmb { \\lambda } } _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { k + 1 } ) ,", "type": "interline_equation", "image_path": "aa02a3d9e9670da70428ff51d8dccbfd33121e4f699e839306505e69a37a124c.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 111, 534, 482, 545.6666666666666 ], "spans": [], "index": 33 }, { "bbox": [ 111, 545.6666666666666, 482, 557.3333333333333 ], "spans": [], "index": 34 }, { "bbox": [ 111, 557.3333333333333, 482, 568.9999999999999 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 106, 572, 506, 660 ], "lines": [ { "bbox": [ 102, 569, 509, 593 ], "spans": [ { "bbox": [ 102, 569, 179, 593 ], "score": 1.0, "content": "where the integral", "type": "text" }, { "bbox": [ 180, 572, 261, 590 ], "score": 0.93, "content": "\\begin{array} { r } { \\int e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda } \\end{array}", "type": "inline_equation" }, { "bbox": [ 261, 569, 473, 593 ], "score": 1.0, "content": "can be analytically computed by repeatedly applying", "type": "text" }, { "bbox": [ 473, 579, 480, 587 ], "score": 0.75, "content": "n", "type": "inline_equation" }, { "bbox": [ 480, 569, 509, 593 ], "score": 1.0, "content": "times", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 585, 507, 603 ], "spans": [ { "bbox": [ 105, 585, 396, 603 ], "score": 1.0, "content": "of integration-by-parts (see Appendix B.2). Therefore, to approximate", "type": "text" }, { "bbox": [ 396, 590, 433, 601 ], "score": 0.91, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 433, 585, 507, 603 ], "score": 1.0, "content": ", we only need to", "type": "text" } ], "index": 37 }, { "bbox": [ 103, 599, 507, 617 ], "spans": [ { "bbox": [ 103, 599, 176, 617 ], "score": 1.0, "content": "approximate the", "type": "text" }, { "bbox": [ 176, 604, 183, 613 ], "score": 0.77, "content": "n", "type": "inline_equation" }, { "bbox": [ 184, 599, 290, 617 ], "score": 1.0, "content": "-th order total derivatives", "type": "text" }, { "bbox": [ 290, 600, 337, 615 ], "score": 0.94, "content": "\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda )", "type": "inline_equation" }, { "bbox": [ 337, 599, 354, 617 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 354, 603, 401, 614 ], "score": 0.9, "content": "n \\leq k - 1", "type": "inline_equation" }, { "bbox": [ 402, 599, 507, 617 ], "score": 1.0, "content": ", which is a well-studied", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 612, 507, 630 ], "spans": [ { "bbox": [ 104, 612, 332, 630 ], "score": 1.0, "content": "problem in the ODE literature [31, 32]. By dropping the", "type": "text" }, { "bbox": [ 332, 614, 369, 628 ], "score": 0.93, "content": "\\mathcal { O } ( h _ { i } ^ { k + 1 } )", "type": "inline_equation" }, { "bbox": [ 370, 612, 507, 630 ], "score": 1.0, "content": "error term and approximating the", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 506, 640 ], "spans": [ { "bbox": [ 105, 626, 125, 640 ], "score": 1.0, "content": "first", "type": "text" }, { "bbox": [ 126, 628, 156, 639 ], "score": 0.92, "content": "( k - 1 )", "type": "inline_equation" }, { "bbox": [ 156, 626, 461, 640 ], "score": 1.0, "content": "-th total derivatives with the “stiff order conditions” [31, 32], we can derive", "type": "text" }, { "bbox": [ 461, 627, 468, 637 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 468, 626, 506, 640 ], "score": 1.0, "content": "-th-order", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 638, 504, 649 ], "spans": [ { "bbox": [ 106, 638, 497, 649 ], "score": 1.0, "content": "ODE solvers for diffusion ODEs. We name such solvers as DPM-Solver overall, and DPM-Solver-", "type": "text" }, { "bbox": [ 497, 638, 504, 648 ], "score": 0.77, "content": "k", "type": "inline_equation" } ], "index": 41 }, { "bbox": [ 105, 647, 478, 662 ], "spans": [ { "bbox": [ 105, 647, 184, 662 ], "score": 1.0, "content": "for a specific order", "type": "text" }, { "bbox": [ 184, 649, 191, 658 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 191, 647, 250, 662 ], "score": 1.0, "content": ". Here we take", "type": "text" }, { "bbox": [ 250, 649, 275, 658 ], "score": 0.89, "content": "k = 1", "type": "inline_equation" }, { "bbox": [ 275, 647, 478, 662 ], "score": 1.0, "content": "for demonstration. In this case, Eq. (3.6) becomes", "type": "text" } ], "index": 42 } ], "index": 39 }, { "type": "interline_equation", "bbox": [ 165, 663, 446, 722 ], "lines": [ { "bbox": [ 165, 663, 446, 722 ], "spans": [ { "bbox": [ 165, 663, 446, 722 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle { \\boldsymbol { x } } _ { t _ { i - 1 } \\to t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { 2 } ) } \\\\ { \\displaystyle = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) + \\mathcal { O } ( h _ { i } ^ { 2 } ) . } \\end{array}", "type": "interline_equation", "image_path": "e1a7a77f08fcf2344ebba70e31c391e6092d3781368a0af49fae27dbeb5b255e.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 165, 663, 446, 682.6666666666666 ], "spans": [], "index": 43 }, { "bbox": [ 165, 682.6666666666666, 446, 702.3333333333333 ], "spans": [], "index": 44 }, { "bbox": [ 165, 702.3333333333333, 446, 721.9999999999999 ], "spans": [], "index": 45 } ] } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 741, 309, 752 ], "spans": [ { "bbox": [ 302, 741, 309, 752 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 72, 504, 96 ], "lines": [ { "bbox": [ 106, 72, 505, 85 ], "spans": [ { "bbox": [ 106, 72, 416, 85 ], "score": 1.0, "content": "Proposition 3.1 (Exact solution of diffusion ODEs). Given an initial value", "type": "text" }, { "bbox": [ 417, 75, 429, 84 ], "score": 0.85, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 429, 72, 461, 85 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 461, 73, 486, 83 ], "score": 0.88, "content": "s > 0", "type": "inline_equation" }, { "bbox": [ 486, 72, 505, 85 ], "score": 1.0, "content": ", the", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 83, 356, 96 ], "spans": [ { "bbox": [ 106, 83, 141, 96 ], "score": 1.0, "content": "solution", "type": "text" }, { "bbox": [ 141, 86, 152, 95 ], "score": 0.82, "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", "type": "inline_equation" }, { "bbox": [ 153, 83, 183, 96 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 184, 83, 220, 96 ], "score": 0.93, "content": "t \\in [ 0 , s ]", "type": "inline_equation" }, { "bbox": [ 220, 83, 356, 96 ], "score": 1.0, "content": "of diffusion ODEs in Eq. (2.7) is:", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 106, 72, 505, 96 ] }, { "type": "interline_equation", "bbox": [ 224, 99, 387, 129 ], "lines": [ { "bbox": [ 224, 99, 387, 129 ], "spans": [ { "bbox": [ 224, 99, 387, 129 ], "score": 0.95, "content": "\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { \\lambda _ { s } } ^ { \\lambda _ { t } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .", "type": "interline_equation", "image_path": "9abcfd7effacf68fb53c21394f34e41f5f607171cf80a13282a4bf3971c34831.jpg" } ] } ], "index": 2.5, "virtual_lines": [ { "bbox": [ 224, 99, 387, 114.0 ], "spans": [], "index": 2 }, { "bbox": [ 224, 114.0, 387, 129.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 106, 138, 505, 172 ], "lines": [ { "bbox": [ 104, 137, 506, 153 ], "spans": [ { "bbox": [ 104, 137, 191, 153 ], "score": 1.0, "content": "We call the integral", "type": "text" }, { "bbox": [ 192, 138, 267, 151 ], "score": 0.93, "content": "\\begin{array} { r } { \\int e ^ { - \\lambda } \\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda } \\end{array}", "type": "inline_equation" }, { "bbox": [ 267, 137, 430, 153 ], "score": 1.0, "content": "the exponentially weighted integral of", "type": "text" }, { "bbox": [ 431, 139, 441, 150 ], "score": 0.87, "content": "\\scriptstyle { \\hat { \\epsilon } } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 441, 137, 506, 153 ], "score": 1.0, "content": ", which is very", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 149, 506, 162 ], "spans": [ { "bbox": [ 105, 149, 506, 162 ], "score": 1.0, "content": "special and highly related to the exponential integrators in the literature of ODE solvers [25]. To the", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 160, 496, 173 ], "spans": [ { "bbox": [ 106, 160, 496, 173 ], "score": 1.0, "content": "best of our knowledge, such formulation has not been revealed in prior work of diffusion models.", "type": "text" } ], "index": 6 } ], "index": 5, "bbox_fs": [ 104, 137, 506, 173 ] }, { "type": "text", "bbox": [ 106, 177, 505, 232 ], "lines": [ { "bbox": [ 106, 176, 506, 190 ], "spans": [ { "bbox": [ 106, 176, 506, 190 ], "score": 1.0, "content": "Eq. (3.4) provides a new perspective for approximating the solutions of diffusion ODEs. Specifically,", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 188, 506, 201 ], "spans": [ { "bbox": [ 106, 188, 132, 201 ], "score": 1.0, "content": "given", "type": "text" }, { "bbox": [ 132, 190, 144, 199 ], "score": 0.86, "content": "\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }", "type": "inline_equation" }, { "bbox": [ 144, 188, 177, 201 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 178, 190, 183, 198 ], "score": 0.68, "content": "s", "type": "inline_equation" }, { "bbox": [ 184, 188, 432, 201 ], "score": 1.0, "content": ", According to Eq. (3.4), approximating the solution at time", "type": "text" }, { "bbox": [ 433, 189, 438, 198 ], "score": 0.76, "content": "t", "type": "inline_equation" }, { "bbox": [ 438, 188, 506, 201 ], "score": 1.0, "content": "is equivalent to", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 198, 505, 212 ], "spans": [ { "bbox": [ 106, 198, 360, 212 ], "score": 1.0, "content": "directly approximating the exponentially weighted integral of", "type": "text" }, { "bbox": [ 361, 199, 371, 210 ], "score": 0.88, "content": "\\hat { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 371, 198, 395, 212 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 395, 199, 407, 210 ], "score": 0.88, "content": "\\lambda _ { s }", "type": "inline_equation" }, { "bbox": [ 407, 198, 419, 212 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 419, 199, 429, 210 ], "score": 0.87, "content": "\\lambda _ { t }", "type": "inline_equation" }, { "bbox": [ 430, 198, 505, 212 ], "score": 1.0, "content": ", which avoids the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 209, 506, 223 ], "spans": [ { "bbox": [ 105, 209, 506, 223 ], "score": 1.0, "content": "error of the linear terms and is well-studied in the literature of exponential integrators [25, 31]. Based", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 220, 492, 234 ], "spans": [ { "bbox": [ 106, 220, 492, 234 ], "score": 1.0, "content": "on this insight, we propose fast solvers for diffusion ODEs, as detailed in the following sections.", "type": "text" } ], "index": 11 } ], "index": 9, "bbox_fs": [ 105, 176, 506, 234 ] }, { "type": "title", "bbox": [ 106, 244, 300, 257 ], "lines": [ { "bbox": [ 105, 244, 301, 259 ], "spans": [ { "bbox": [ 105, 244, 301, 259 ], "score": 1.0, "content": "3.2 High-Order Solvers for Diffusion ODEs", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 264, 505, 298 ], "lines": [ { "bbox": [ 105, 264, 505, 278 ], "spans": [ { "bbox": [ 105, 264, 505, 278 ], "score": 1.0, "content": "In this section, we propose high-order solvers for diffusion ODEs with convergence order guarantee", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 276, 505, 288 ], "spans": [ { "bbox": [ 106, 276, 505, 288 ], "score": 1.0, "content": "by leveraging our proposed solution formulation Eq. (3.4). The proposed solvers and analysis are", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 287, 467, 299 ], "spans": [ { "bbox": [ 105, 287, 467, 299 ], "score": 1.0, "content": "highly motivated by the methods of exponential integrators [25, 31] in the ODE literature.", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 105, 264, 505, 299 ] }, { "type": "text", "bbox": [ 106, 302, 505, 348 ], "lines": [ { "bbox": [ 103, 300, 507, 318 ], "spans": [ { "bbox": [ 103, 300, 249, 318 ], "score": 1.0, "content": "Specifically, given an initial value", "type": "text" }, { "bbox": [ 250, 305, 264, 314 ], "score": 0.85, "content": "\\mathbf { \\nabla } _ { \\mathbf { x } _ { T } }", "type": "inline_equation" }, { "bbox": [ 264, 300, 297, 318 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 298, 304, 306, 313 ], "score": 0.81, "content": "T", "type": "inline_equation" }, { "bbox": [ 307, 300, 326, 318 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 326, 303, 356, 314 ], "score": 0.89, "content": "M + 1", "type": "inline_equation" }, { "bbox": [ 356, 300, 402, 318 ], "score": 1.0, "content": "time steps", "type": "text" }, { "bbox": [ 403, 302, 434, 315 ], "score": 0.93, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 434, 300, 507, 318 ], "score": 1.0, "content": "decreasing from", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 313, 505, 327 ], "spans": [ { "bbox": [ 106, 314, 136, 325 ], "score": 0.9, "content": "t _ { 0 } = T", "type": "inline_equation" }, { "bbox": [ 137, 313, 147, 327 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 147, 315, 179, 325 ], "score": 0.9, "content": "t _ { M } = 0", "type": "inline_equation" }, { "bbox": [ 180, 313, 199, 327 ], "score": 1.0, "content": ". Let", "type": "text" }, { "bbox": [ 199, 315, 240, 325 ], "score": 0.89, "content": "\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }", "type": "inline_equation" }, { "bbox": [ 240, 313, 419, 327 ], "score": 1.0, "content": "be the initial value. The proposed solvers use", "type": "text" }, { "bbox": [ 419, 315, 431, 324 ], "score": 0.81, "content": "M", "type": "inline_equation" }, { "bbox": [ 431, 313, 505, 327 ], "score": 1.0, "content": "steps to iteratively", "type": "text" } ], "index": 17 }, { "bbox": [ 102, 320, 510, 342 ], "spans": [ { "bbox": [ 102, 320, 190, 342 ], "score": 1.0, "content": "compute a sequence", "type": "text" }, { "bbox": [ 190, 325, 227, 337 ], "score": 0.92, "content": "\\{ \\tilde { { \\pmb { x } } } _ { t _ { i } } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 228, 320, 416, 342 ], "score": 1.0, "content": "to approximate the true solutions at time steps", "type": "text" }, { "bbox": [ 417, 324, 448, 337 ], "score": 0.92, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 448, 320, 510, 342 ], "score": 1.0, "content": ". In particular,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 335, 348, 349 ], "spans": [ { "bbox": [ 105, 335, 165, 349 ], "score": 1.0, "content": "the last iterate", "type": "text" }, { "bbox": [ 165, 336, 183, 348 ], "score": 0.91, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { M } }", "type": "inline_equation" }, { "bbox": [ 183, 335, 348, 349 ], "score": 1.0, "content": "approximates the true solution at time 0.", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 102, 300, 510, 349 ] }, { "type": "text", "bbox": [ 107, 352, 505, 386 ], "lines": [ { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 314, 365 ], "score": 1.0, "content": "In order to reduce the approximation error between", "type": "text" }, { "bbox": [ 315, 353, 333, 364 ], "score": 0.91, "content": "\\tilde { \\pmb { x } } _ { t _ { M } }", "type": "inline_equation" }, { "bbox": [ 333, 352, 505, 365 ], "score": 1.0, "content": "and the true solution at time 0, we need to", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 360, 503, 379 ], "spans": [ { "bbox": [ 104, 360, 264, 379 ], "score": 1.0, "content": "reduce the approximation error for each", "type": "text" }, { "bbox": [ 264, 364, 279, 375 ], "score": 0.9, "content": "\\tilde { \\mathbf { x } } _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 279, 360, 480, 379 ], "score": 1.0, "content": "at every step [30]. Starting with the previous value", "type": "text" }, { "bbox": [ 480, 363, 503, 376 ], "score": 0.91, "content": "\\tilde { \\pmb { x } } _ { t _ { i - 1 } }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 105, 374, 447, 388 ], "spans": [ { "bbox": [ 105, 374, 136, 388 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 136, 375, 154, 386 ], "score": 0.89, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 155, 374, 323, 388 ], "score": 1.0, "content": ", according to Eq. (3.4), the exact solution", "type": "text" }, { "bbox": [ 324, 376, 360, 387 ], "score": 0.92, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 360, 374, 392, 388 ], "score": 1.0, "content": "at time", "type": "text" }, { "bbox": [ 392, 375, 400, 385 ], "score": 0.85, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 400, 374, 447, 388 ], "score": 1.0, "content": "is given by", "type": "text" } ], "index": 22 } ], "index": 21, "bbox_fs": [ 104, 352, 505, 388 ] }, { "type": "interline_equation", "bbox": [ 194, 390, 416, 422 ], "lines": [ { "bbox": [ 194, 390, 416, 422 ], "spans": [ { "bbox": [ 194, 390, 416, 422 ], "score": 0.93, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .", "type": "interline_equation", "image_path": "e941fd95cbf3e1c496583b82dd623ff65158438a8e8529e43b6a17ad2855df95.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 194, 390, 416, 406.0 ], "spans": [], "index": 23 }, { "bbox": [ 194, 406.0, 416, 422.0 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 106, 425, 505, 478 ], "lines": [ { "bbox": [ 104, 423, 506, 441 ], "spans": [ { "bbox": [ 104, 423, 245, 441 ], "score": 1.0, "content": "Therefore, to compute the value", "type": "text" }, { "bbox": [ 246, 426, 260, 438 ], "score": 0.9, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 260, 423, 342, 441 ], "score": 1.0, "content": "for approximating", "type": "text" }, { "bbox": [ 342, 428, 379, 438 ], "score": 0.92, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 379, 423, 506, 441 ], "score": 1.0, "content": ", we need to approximate the", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 436, 505, 455 ], "spans": [ { "bbox": [ 104, 436, 241, 455 ], "score": 1.0, "content": "exponentially weighted integral of", "type": "text" }, { "bbox": [ 241, 441, 252, 452 ], "score": 0.87, "content": "\\hat { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 252, 436, 273, 455 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 274, 441, 295, 453 ], "score": 0.92, "content": "\\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 295, 436, 306, 455 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 307, 441, 320, 452 ], "score": 0.89, "content": "\\lambda _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 320, 436, 356, 455 ], "score": 1.0, "content": ". Denote", "type": "text" }, { "bbox": [ 357, 440, 424, 453 ], "score": 0.92, "content": "h _ { i } : = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 425, 436, 444, 455 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 444, 437, 505, 453 ], "score": 0.91, "content": "\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { { \\mathbf x } } _ { \\lambda } , \\lambda ) \\mathrel { \\mathop : } =", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 104, 450, 507, 469 ], "spans": [ { "bbox": [ 104, 450, 507, 469 ], "score": 1.0, "content": "dnϵˆθ(xˆλ,λ)dλn as the n-th order total derivative of ϵˆθ(xˆλ, λ) w.r.t. λ. For k ≥ 1, the (k − 1)-th order", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 464, 304, 479 ], "spans": [ { "bbox": [ 105, 464, 188, 479 ], "score": 1.0, "content": "Taylor expansion of", "type": "text" }, { "bbox": [ 188, 465, 228, 478 ], "score": 0.93, "content": "\\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda )", "type": "inline_equation" }, { "bbox": [ 228, 464, 252, 479 ], "score": 1.0, "content": "w.r.t.", "type": "text" }, { "bbox": [ 252, 466, 259, 475 ], "score": 0.79, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 260, 464, 270, 479 ], "score": 1.0, "content": "at", "type": "text" }, { "bbox": [ 271, 466, 292, 478 ], "score": 0.92, "content": "\\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 292, 464, 304, 479 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 28 } ], "index": 26.5, "bbox_fs": [ 104, 423, 507, 479 ] }, { "type": "interline_equation", "bbox": [ 162, 482, 448, 516 ], "lines": [ { "bbox": [ 162, 482, 448, 516 ], "spans": [ { "bbox": [ 162, 482, 448, 516 ], "score": 0.94, "content": "\\hat { \\epsilon } _ { \\theta } ( \\hat { x } _ { \\lambda } , \\lambda ) = \\sum _ { n = 0 } ^ { k - 1 } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { x } _ { \\lambda _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) + \\mathcal { O } ( ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { k } ) ,", "type": "interline_equation", "image_path": "976243e2861066edc99111ffc150a8141dda0494f026576eb3e22e90ae405d03.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 162, 482, 448, 493.3333333333333 ], "spans": [], "index": 29 }, { "bbox": [ 162, 493.3333333333333, 448, 504.66666666666663 ], "spans": [], "index": 30 }, { "bbox": [ 162, 504.66666666666663, 448, 516.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 519, 351, 531 ], "lines": [ { "bbox": [ 106, 518, 351, 532 ], "spans": [ { "bbox": [ 106, 518, 351, 532 ], "score": 1.0, "content": "Substituting the above Taylor expansion into Eq. (3.5) yields", "type": "text" } ], "index": 32 } ], "index": 32, "bbox_fs": [ 106, 518, 351, 532 ] }, { "type": "interline_equation", "bbox": [ 111, 534, 482, 569 ], "lines": [ { "bbox": [ 111, 534, 482, 569 ], "spans": [ { "bbox": [ 111, 534, 482, 569 ], "score": 0.94, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\sum _ { n = 0 } ^ { k - 1 } \\hat { \\pmb { \\epsilon } } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { { \\pmb { \\lambda } } _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { k + 1 } ) ,", "type": "interline_equation", "image_path": "aa02a3d9e9670da70428ff51d8dccbfd33121e4f699e839306505e69a37a124c.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 111, 534, 482, 545.6666666666666 ], "spans": [], "index": 33 }, { "bbox": [ 111, 545.6666666666666, 482, 557.3333333333333 ], "spans": [], "index": 34 }, { "bbox": [ 111, 557.3333333333333, 482, 568.9999999999999 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 106, 572, 506, 660 ], "lines": [ { "bbox": [ 102, 569, 509, 593 ], "spans": [ { "bbox": [ 102, 569, 179, 593 ], "score": 1.0, "content": "where the integral", "type": "text" }, { "bbox": [ 180, 572, 261, 590 ], "score": 0.93, "content": "\\begin{array} { r } { \\int e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda } \\end{array}", "type": "inline_equation" }, { "bbox": [ 261, 569, 473, 593 ], "score": 1.0, "content": "can be analytically computed by repeatedly applying", "type": "text" }, { "bbox": [ 473, 579, 480, 587 ], "score": 0.75, "content": "n", "type": "inline_equation" }, { "bbox": [ 480, 569, 509, 593 ], "score": 1.0, "content": "times", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 585, 507, 603 ], "spans": [ { "bbox": [ 105, 585, 396, 603 ], "score": 1.0, "content": "of integration-by-parts (see Appendix B.2). Therefore, to approximate", "type": "text" }, { "bbox": [ 396, 590, 433, 601 ], "score": 0.91, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 433, 585, 507, 603 ], "score": 1.0, "content": ", we only need to", "type": "text" } ], "index": 37 }, { "bbox": [ 103, 599, 507, 617 ], "spans": [ { "bbox": [ 103, 599, 176, 617 ], "score": 1.0, "content": "approximate the", "type": "text" }, { "bbox": [ 176, 604, 183, 613 ], "score": 0.77, "content": "n", "type": "inline_equation" }, { "bbox": [ 184, 599, 290, 617 ], "score": 1.0, "content": "-th order total derivatives", "type": "text" }, { "bbox": [ 290, 600, 337, 615 ], "score": 0.94, "content": "\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda )", "type": "inline_equation" }, { "bbox": [ 337, 599, 354, 617 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 354, 603, 401, 614 ], "score": 0.9, "content": "n \\leq k - 1", "type": "inline_equation" }, { "bbox": [ 402, 599, 507, 617 ], "score": 1.0, "content": ", which is a well-studied", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 612, 507, 630 ], "spans": [ { "bbox": [ 104, 612, 332, 630 ], "score": 1.0, "content": "problem in the ODE literature [31, 32]. By dropping the", "type": "text" }, { "bbox": [ 332, 614, 369, 628 ], "score": 0.93, "content": "\\mathcal { O } ( h _ { i } ^ { k + 1 } )", "type": "inline_equation" }, { "bbox": [ 370, 612, 507, 630 ], "score": 1.0, "content": "error term and approximating the", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 506, 640 ], "spans": [ { "bbox": [ 105, 626, 125, 640 ], "score": 1.0, "content": "first", "type": "text" }, { "bbox": [ 126, 628, 156, 639 ], "score": 0.92, "content": "( k - 1 )", "type": "inline_equation" }, { "bbox": [ 156, 626, 461, 640 ], "score": 1.0, "content": "-th total derivatives with the “stiff order conditions” [31, 32], we can derive", "type": "text" }, { "bbox": [ 461, 627, 468, 637 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 468, 626, 506, 640 ], "score": 1.0, "content": "-th-order", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 638, 504, 649 ], "spans": [ { "bbox": [ 106, 638, 497, 649 ], "score": 1.0, "content": "ODE solvers for diffusion ODEs. We name such solvers as DPM-Solver overall, and DPM-Solver-", "type": "text" }, { "bbox": [ 497, 638, 504, 648 ], "score": 0.77, "content": "k", "type": "inline_equation" } ], "index": 41 }, { "bbox": [ 105, 647, 478, 662 ], "spans": [ { "bbox": [ 105, 647, 184, 662 ], "score": 1.0, "content": "for a specific order", "type": "text" }, { "bbox": [ 184, 649, 191, 658 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 191, 647, 250, 662 ], "score": 1.0, "content": ". Here we take", "type": "text" }, { "bbox": [ 250, 649, 275, 658 ], "score": 0.89, "content": "k = 1", "type": "inline_equation" }, { "bbox": [ 275, 647, 478, 662 ], "score": 1.0, "content": "for demonstration. In this case, Eq. (3.6) becomes", "type": "text" } ], "index": 42 } ], "index": 39, "bbox_fs": [ 102, 569, 509, 662 ] }, { "type": "interline_equation", "bbox": [ 165, 663, 446, 722 ], "lines": [ { "bbox": [ 165, 663, 446, 722 ], "spans": [ { "bbox": [ 165, 663, 446, 722 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle { \\boldsymbol { x } } _ { t _ { i - 1 } \\to t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { 2 } ) } \\\\ { \\displaystyle = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) + \\mathcal { O } ( h _ { i } ^ { 2 } ) . } \\end{array}", "type": "interline_equation", "image_path": "e1a7a77f08fcf2344ebba70e31c391e6092d3781368a0af49fae27dbeb5b255e.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 165, 663, 446, 682.6666666666666 ], "spans": [], "index": 43 }, { "bbox": [ 165, 682.6666666666666, 446, 702.3333333333333 ], "spans": [], "index": 44 }, { "bbox": [ 165, 702.3333333333333, 446, 721.9999999999999 ], "spans": [], "index": 45 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 72, 504, 95 ], "lines": [ { "bbox": [ 105, 71, 505, 87 ], "spans": [ { "bbox": [ 105, 71, 266, 87 ], "score": 1.0, "content": "By dropping the high-order error term", "type": "text" }, { "bbox": [ 267, 72, 293, 85 ], "score": 0.92, "content": "\\mathcal { O } ( h _ { i } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 294, 71, 448, 87 ], "score": 1.0, "content": ", we can obtain an approximation for", "type": "text" }, { "bbox": [ 448, 75, 485, 85 ], "score": 0.89, "content": "\\pmb { x } _ { t _ { i - 1 } t _ { i } }", "type": "inline_equation" }, { "bbox": [ 485, 71, 505, 87 ], "score": 1.0, "content": ". As", "type": "text" } ], "index": 0 }, { "bbox": [ 107, 80, 464, 99 ], "spans": [ { "bbox": [ 107, 84, 132, 94 ], "score": 0.88, "content": "k = 1", "type": "inline_equation" }, { "bbox": [ 132, 80, 283, 99 ], "score": 1.0, "content": "here, we call this solver DPM-Solver-", "type": "text" }, { "bbox": [ 284, 85, 289, 93 ], "score": 0.69, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 289, 80, 464, 99 ], "score": 1.0, "content": ", and the detailed algorithm is as following.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 104, 99, 505, 124 ], "lines": [ { "bbox": [ 104, 95, 504, 117 ], "spans": [ { "bbox": [ 104, 95, 268, 117 ], "score": 1.0, "content": "DPM-Solver-1. Given an initial value", "type": "text" }, { "bbox": [ 268, 102, 282, 111 ], "score": 0.87, "content": "\\mathbf { \\nabla } _ { \\mathbf { x } _ { T } }", "type": "inline_equation" }, { "bbox": [ 283, 95, 300, 117 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 301, 100, 330, 111 ], "score": 0.89, "content": "M + 1", "type": "inline_equation" }, { "bbox": [ 330, 95, 374, 117 ], "score": 1.0, "content": "time steps", "type": "text" }, { "bbox": [ 374, 99, 405, 113 ], "score": 0.93, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 406, 95, 474, 117 ], "score": 1.0, "content": "decreasing from", "type": "text" }, { "bbox": [ 474, 100, 504, 111 ], "score": 0.88, "content": "t _ { 0 } = T", "type": "inline_equation" } ], "index": 2 }, { "bbox": [ 104, 108, 486, 128 ], "spans": [ { "bbox": [ 104, 108, 117, 128 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 117, 113, 149, 123 ], "score": 0.9, "content": "t _ { M } = 0", "type": "inline_equation" }, { "bbox": [ 149, 108, 207, 128 ], "score": 1.0, "content": ". Starting with", "type": "text" }, { "bbox": [ 208, 112, 248, 124 ], "score": 0.92, "content": "\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }", "type": "inline_equation" }, { "bbox": [ 249, 108, 306, 128 ], "score": 1.0, "content": ", the sequence", "type": "text" }, { "bbox": [ 306, 111, 344, 124 ], "score": 0.93, "content": "\\{ \\tilde { { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 344, 108, 486, 128 ], "score": 1.0, "content": "is computed iteratively as follows:", "type": "text" } ], "index": 3 } ], "index": 2.5 }, { "type": "interline_equation", "bbox": [ 149, 129, 461, 154 ], "lines": [ { "bbox": [ 149, 129, 461, 154 ], "spans": [ { "bbox": [ 149, 129, 461, 154 ], "score": 0.92, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\boldsymbol { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) , \\quad \\mathrm { w h e r e ~ } h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } .", "type": "interline_equation", "image_path": "8deaa6e7603de31da2fd31418b2d224dd8ec375a261d169ca354b6f5b2802791.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 149, 129, 461, 154 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 164, 505, 198 ], "lines": [ { "bbox": [ 106, 164, 505, 177 ], "spans": [ { "bbox": [ 106, 164, 123, 177 ], "score": 1.0, "content": "For", "type": "text" }, { "bbox": [ 123, 165, 149, 176 ], "score": 0.9, "content": "k \\geq 2", "type": "inline_equation" }, { "bbox": [ 150, 164, 249, 177 ], "score": 1.0, "content": ", approximating the first", "type": "text" }, { "bbox": [ 249, 165, 256, 174 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 257, 164, 505, 177 ], "score": 1.0, "content": "terms of the Taylor expansion needs additional intermediate", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 176, 505, 188 ], "spans": [ { "bbox": [ 106, 176, 167, 188 ], "score": 1.0, "content": "points between", "type": "text" }, { "bbox": [ 167, 177, 173, 186 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 173, 176, 190, 188 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 190, 177, 196, 186 ], "score": 0.77, "content": "s", "type": "inline_equation" }, { "bbox": [ 196, 176, 505, 188 ], "score": 1.0, "content": "[31]. The derivation is more technical so we defer it to Appendix B. Below we", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 186, 503, 200 ], "spans": [ { "bbox": [ 105, 186, 200, 200 ], "score": 1.0, "content": "propose algorithms for", "type": "text" }, { "bbox": [ 200, 187, 234, 198 ], "score": 0.91, "content": "k = 2 , 3", "type": "inline_equation" }, { "bbox": [ 235, 186, 503, 200 ], "score": 1.0, "content": "and name them as DPM-Solver-2 and DPM-Solver-3, respectively.", "type": "text" } ], "index": 7 } ], "index": 6 }, { "type": "title", "bbox": [ 107, 212, 222, 223 ], "lines": [ { "bbox": [ 106, 211, 224, 225 ], "spans": [ { "bbox": [ 106, 211, 224, 225 ], "score": 1.0, "content": "Algorithm 1 DPM-Solver-2.", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 227, 333, 240 ], "lines": [ { "bbox": [ 105, 226, 332, 244 ], "spans": [ { "bbox": [ 105, 226, 198, 244 ], "score": 1.0, "content": "Require: initial value", "type": "text" }, { "bbox": [ 199, 230, 212, 239 ], "score": 0.86, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }", "type": "inline_equation" }, { "bbox": [ 213, 226, 259, 244 ], "score": 1.0, "content": ", time steps", "type": "text" }, { "bbox": [ 259, 227, 290, 241 ], "score": 0.9, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 291, 226, 321, 244 ], "score": 1.0, "content": ", model", "type": "text" }, { "bbox": [ 321, 230, 332, 240 ], "score": 0.69, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" } ], "index": 9 } ], "index": 9 }, { "type": "title", "bbox": [ 107, 366, 223, 377 ], "lines": [ { "bbox": [ 106, 365, 224, 379 ], "spans": [ { "bbox": [ 106, 365, 224, 379 ], "score": 1.0, "content": "Algorithm 2 DPM-Solver-3.", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 382, 333, 395 ], "lines": [ { "bbox": [ 104, 378, 332, 399 ], "spans": [ { "bbox": [ 104, 378, 198, 399 ], "score": 1.0, "content": "Require: initial value", "type": "text" }, { "bbox": [ 199, 385, 212, 394 ], "score": 0.84, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }", "type": "inline_equation" }, { "bbox": [ 213, 378, 259, 399 ], "score": 1.0, "content": ", time steps", "type": "text" }, { "bbox": [ 259, 381, 290, 395 ], "score": 0.88, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 290, 378, 321, 399 ], "score": 1.0, "content": ", model", "type": "text" }, { "bbox": [ 322, 385, 332, 394 ], "score": 0.67, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 136, 414, 483, 511 ], "lines": [ { "bbox": [ 136, 414, 483, 511 ], "spans": [ { "bbox": [ 136, 414, 483, 511 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\quad _ { s 2 i - 1 } _ { t _ { \\lambda } } ( \\overline { { \\lambda } } _ { t _ { i - 1 } } + r _ { 1 } h _ { i } ) , \\quad s _ { 2 i } t _ { \\lambda } ( \\lambda _ { t _ { i - 1 } } + r _ { 2 } h _ { i } ) } \\\\ & { u _ { 2 i - 1 } \\frac { \\alpha _ { s _ { 2 i - 1 } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i - 1 } } ( e ^ { r _ { 1 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { D _ { 2 i - 1 } \\epsilon _ { \\theta } ( u _ { 2 i - 1 } , s _ { 2 i - 1 } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { u _ { 2 i } \\frac { \\alpha _ { s _ { 2 i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i } } ( e ^ { r _ { 2 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { s _ { 2 i } } r _ { 2 } } { r _ { 1 } } ( \\frac { e ^ { r _ { 2 } h _ { i } } - 1 } { r _ { 2 } h _ { i } } - 1 ) D _ { 2 i - 1 } } \\\\ & { D _ { 2 i } \\epsilon _ { \\theta } ( u _ { 2 i } , s _ { 2 i } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { \\tilde { x } _ { t _ { i } } \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { t _ { i } } } { r _ { 2 } } ( \\frac { e ^ { h _ { i } } - 1 } { h } - 1 ) D _ { 2 i } } \\end{array}", "type": "interline_equation", "image_path": "ad4609832a6419c85fb2132d9d91d2c95beec92e10bd4907c45b4ed7d6c517e6.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 136, 414, 483, 446.3333333333333 ], "spans": [], "index": 12 }, { "bbox": [ 136, 446.3333333333333, 483, 478.66666666666663 ], "spans": [], "index": 13 }, { "bbox": [ 136, 478.66666666666663, 483, 510.99999999999994 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 108, 519, 171, 531 ], "lines": [ { "bbox": [ 106, 515, 171, 534 ], "spans": [ { "bbox": [ 106, 515, 153, 534 ], "score": 1.0, "content": "10: return", "type": "text" }, { "bbox": [ 153, 519, 171, 531 ], "score": 0.81, "content": "\\tilde { \\pmb { x } } _ { t _ { M } }", "type": "inline_equation" } ], "index": 15 } ], "index": 15 }, { "type": "text", "bbox": [ 106, 553, 506, 631 ], "lines": [ { "bbox": [ 106, 554, 506, 566 ], "spans": [ { "bbox": [ 106, 554, 130, 566 ], "score": 1.0, "content": "Here,", "type": "text" }, { "bbox": [ 130, 554, 151, 566 ], "score": 0.9, "content": "t _ { \\lambda } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 151, 554, 252, 566 ], "score": 1.0, "content": "is the inverse function of", "type": "text" }, { "bbox": [ 252, 554, 270, 566 ], "score": 0.91, "content": "\\lambda ( t )", "type": "inline_equation" }, { "bbox": [ 270, 554, 506, 566 ], "score": 1.0, "content": ", which has an analytical formulation for the practical noise", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 565, 506, 577 ], "spans": [ { "bbox": [ 105, 565, 458, 577 ], "score": 1.0, "content": "schedule used in [2, 16], as shown in Appendix D. The chosen intermediate points are", "type": "text" }, { "bbox": [ 459, 565, 489, 576 ], "score": 0.89, "content": "( s _ { i } , \\pmb { u } _ { i } )", "type": "inline_equation" }, { "bbox": [ 490, 565, 506, 577 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 574, 506, 590 ], "spans": [ { "bbox": [ 104, 574, 188, 590 ], "score": 1.0, "content": "DPM-Solver-2 and", "type": "text" }, { "bbox": [ 188, 576, 247, 588 ], "score": 0.92, "content": "\\left( s _ { 2 i - 1 } , { \\pmb u } _ { 2 i - 1 } \\right)", "type": "inline_equation" }, { "bbox": [ 248, 574, 267, 590 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 268, 576, 306, 588 ], "score": 0.92, "content": "( s _ { 2 i } , { \\pmb u } _ { 2 i } )", "type": "inline_equation" }, { "bbox": [ 307, 574, 506, 590 ], "score": 1.0, "content": "for DPM-Solver-3. As shown in the algorithm,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 586, 505, 599 ], "spans": [ { "bbox": [ 105, 586, 161, 599 ], "score": 1.0, "content": "DPM-Solver-", "type": "text" }, { "bbox": [ 161, 587, 168, 597 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 168, 586, 203, 599 ], "score": 1.0, "content": "requires", "type": "text" }, { "bbox": [ 204, 588, 210, 597 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 211, 586, 343, 599 ], "score": 1.0, "content": "function evaluations per step for", "type": "text" }, { "bbox": [ 344, 587, 387, 598 ], "score": 0.93, "content": "k = 1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 388, 586, 505, 599 ], "score": 1.0, "content": ". Despite the more expensive", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 596, 506, 611 ], "spans": [ { "bbox": [ 105, 596, 215, 611 ], "score": 1.0, "content": "steps, higher-order solvers", "type": "text" }, { "bbox": [ 216, 598, 253, 609 ], "score": 0.83, "content": "( k = 2 , 3", "type": "inline_equation" }, { "bbox": [ 253, 596, 506, 611 ], "score": 1.0, "content": ") are usually more efficient since they require much fewer steps", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 609, 506, 621 ], "spans": [ { "bbox": [ 105, 609, 418, 621 ], "score": 1.0, "content": "to converge, due to their higher convergence order. We show that DPM-Solver-", "type": "text" }, { "bbox": [ 418, 609, 425, 618 ], "score": 0.83, "content": "k", "type": "inline_equation" }, { "bbox": [ 425, 609, 435, 621 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 435, 609, 442, 618 ], "score": 0.84, "content": "k", "type": "inline_equation" }, { "bbox": [ 442, 609, 506, 621 ], "score": 1.0, "content": "-th-order solver,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 619, 362, 632 ], "spans": [ { "bbox": [ 105, 619, 362, 632 ], "score": 1.0, "content": "as stated in the following theorem. The proof is in Appendix B.", "type": "text" } ], "index": 22 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 634, 505, 682 ], "lines": [ { "bbox": [ 105, 633, 506, 647 ], "spans": [ { "bbox": [ 105, 633, 218, 647 ], "score": 1.0, "content": "Theorem 3.2 (DPM-Solver-", "type": "text" }, { "bbox": [ 219, 635, 225, 644 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 226, 633, 244, 647 ], "score": 1.0, "content": "as a", "type": "text" }, { "bbox": [ 244, 635, 250, 644 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 251, 633, 354, 647 ], "score": 1.0, "content": "-th-order solver). Assume", "type": "text" }, { "bbox": [ 354, 634, 390, 646 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 391, 633, 506, 647 ], "score": 1.0, "content": "follows the regularity condi-", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 646, 505, 658 ], "spans": [ { "bbox": [ 106, 646, 262, 658 ], "score": 1.0, "content": "tions detailed in Appendix B.1, then for", "type": "text" }, { "bbox": [ 262, 646, 306, 657 ], "score": 0.78, "content": "k = 1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 306, 646, 362, 658 ], "score": 1.0, "content": ", DPM-Solver-", "type": "text" }, { "bbox": [ 362, 647, 369, 655 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 369, 646, 386, 658 ], "score": 1.0, "content": "is a", "type": "text" }, { "bbox": [ 386, 646, 393, 655 ], "score": 0.72, "content": "k", "type": "inline_equation" }, { "bbox": [ 393, 646, 505, 658 ], "score": 1.0, "content": "-th order solver for diffusion", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 655, 507, 671 ], "spans": [ { "bbox": [ 105, 655, 218, 671 ], "score": 1.0, "content": "ODEs, i.e., for the sequence", "type": "text" }, { "bbox": [ 218, 655, 256, 668 ], "score": 0.92, "content": "\\{ \\tilde { \\pmb { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 256, 655, 362, 671 ], "score": 1.0, "content": "computed by DPM-Solver-", "type": "text" }, { "bbox": [ 362, 657, 368, 667 ], "score": 0.71, "content": "k", "type": "inline_equation" }, { "bbox": [ 369, 655, 507, 671 ], "score": 1.0, "content": ", the approximation error at time 0", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 667, 400, 683 ], "spans": [ { "bbox": [ 104, 667, 140, 683 ], "score": 1.0, "content": "satisfies", "type": "text" }, { "bbox": [ 141, 668, 232, 680 ], "score": 0.93, "content": "\\tilde { \\pmb { x } } _ { t _ { M } } - \\pmb { x } _ { 0 } = \\mathcal { O } ( h _ { \\operatorname* { m a x } } ^ { k } )", "type": "inline_equation" }, { "bbox": [ 232, 667, 263, 683 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 263, 668, 395, 681 ], "score": 0.92, "content": "h _ { m a x } = \\mathrm { m a x } _ { 1 \\leq i \\leq M } ( \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } )", "type": "inline_equation" }, { "bbox": [ 395, 667, 400, 683 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 24.5 }, { "type": "text", "bbox": [ 107, 688, 505, 722 ], "lines": [ { "bbox": [ 106, 688, 505, 701 ], "spans": [ { "bbox": [ 106, 688, 189, 701 ], "score": 1.0, "content": "Finally, solvers with", "type": "text" }, { "bbox": [ 189, 689, 214, 700 ], "score": 0.91, "content": "k \\geq 4", "type": "inline_equation" }, { "bbox": [ 214, 688, 505, 701 ], "score": 1.0, "content": "need much more intermediate points as shown by previous work [31, 32]", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 700, 505, 712 ], "spans": [ { "bbox": [ 105, 700, 329, 712 ], "score": 1.0, "content": "for exponential integrators. 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Starting with", "type": "text" }, { "bbox": [ 208, 112, 248, 124 ], "score": 0.92, "content": "\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }", "type": "inline_equation" }, { "bbox": [ 249, 108, 306, 128 ], "score": 1.0, "content": ", the sequence", "type": "text" }, { "bbox": [ 306, 111, 344, 124 ], "score": 0.93, "content": "\\{ \\tilde { { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 344, 108, 486, 128 ], "score": 1.0, "content": "is computed iteratively as follows:", "type": "text" } ], "index": 3 } ], "index": 2.5, "bbox_fs": [ 104, 95, 504, 128 ] }, { "type": "interline_equation", "bbox": [ 149, 129, 461, 154 ], "lines": [ { "bbox": [ 149, 129, 461, 154 ], "spans": [ { "bbox": [ 149, 129, 461, 154 ], "score": 0.92, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\boldsymbol { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) , \\quad \\mathrm { w h e r e ~ } h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } .", "type": "interline_equation", "image_path": "8deaa6e7603de31da2fd31418b2d224dd8ec375a261d169ca354b6f5b2802791.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 149, 129, 461, 154 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 164, 505, 198 ], "lines": [ { "bbox": [ 106, 164, 505, 177 ], "spans": [ { "bbox": [ 106, 164, 123, 177 ], "score": 1.0, "content": "For", "type": "text" }, { "bbox": [ 123, 165, 149, 176 ], "score": 0.9, "content": "k \\geq 2", "type": "inline_equation" }, { "bbox": [ 150, 164, 249, 177 ], "score": 1.0, "content": ", approximating the first", "type": "text" }, { "bbox": [ 249, 165, 256, 174 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 257, 164, 505, 177 ], "score": 1.0, "content": "terms of the Taylor expansion needs additional intermediate", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 176, 505, 188 ], "spans": [ { "bbox": [ 106, 176, 167, 188 ], "score": 1.0, "content": "points between", "type": "text" }, { "bbox": [ 167, 177, 173, 186 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 173, 176, 190, 188 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 190, 177, 196, 186 ], "score": 0.77, "content": "s", "type": "inline_equation" }, { "bbox": [ 196, 176, 505, 188 ], "score": 1.0, "content": "[31]. The derivation is more technical so we defer it to Appendix B. Below we", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 186, 503, 200 ], "spans": [ { "bbox": [ 105, 186, 200, 200 ], "score": 1.0, "content": "propose algorithms for", "type": "text" }, { "bbox": [ 200, 187, 234, 198 ], "score": 0.91, "content": "k = 2 , 3", "type": "inline_equation" }, { "bbox": [ 235, 186, 503, 200 ], "score": 1.0, "content": "and name them as DPM-Solver-2 and DPM-Solver-3, respectively.", "type": "text" } ], "index": 7 } ], "index": 6, "bbox_fs": [ 105, 164, 505, 200 ] }, { "type": "title", "bbox": [ 107, 212, 222, 223 ], "lines": [ { "bbox": [ 106, 211, 224, 225 ], "spans": [ { "bbox": [ 106, 211, 224, 225 ], "score": 1.0, "content": "Algorithm 1 DPM-Solver-2.", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 227, 333, 240 ], "lines": [ { "bbox": [ 105, 226, 332, 244 ], "spans": [ { "bbox": [ 105, 226, 198, 244 ], "score": 1.0, "content": "Require: initial value", "type": "text" }, { "bbox": [ 199, 230, 212, 239 ], "score": 0.86, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }", "type": "inline_equation" }, { "bbox": [ 213, 226, 259, 244 ], "score": 1.0, "content": ", time steps", "type": "text" }, { "bbox": [ 259, 227, 290, 241 ], "score": 0.9, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 291, 226, 321, 244 ], "score": 1.0, "content": ", model", "type": "text" }, { "bbox": [ 321, 230, 332, 240 ], "score": 0.69, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 105, 226, 332, 244 ] }, { "type": "title", "bbox": [ 107, 366, 223, 377 ], "lines": [ { "bbox": [ 106, 365, 224, 379 ], "spans": [ { "bbox": [ 106, 365, 224, 379 ], "score": 1.0, "content": "Algorithm 2 DPM-Solver-3.", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 382, 333, 395 ], "lines": [ { "bbox": [ 104, 378, 332, 399 ], "spans": [ { "bbox": [ 104, 378, 198, 399 ], "score": 1.0, "content": "Require: initial value", "type": "text" }, { "bbox": [ 199, 385, 212, 394 ], "score": 0.84, "content": "\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }", "type": "inline_equation" }, { "bbox": [ 213, 378, 259, 399 ], "score": 1.0, "content": ", time steps", "type": "text" }, { "bbox": [ 259, 381, 290, 395 ], "score": 0.88, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 290, 378, 321, 399 ], "score": 1.0, "content": ", model", "type": "text" }, { "bbox": [ 322, 385, 332, 394 ], "score": 0.67, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" } ], "index": 11 } ], "index": 11, "bbox_fs": [ 104, 378, 332, 399 ] }, { "type": "interline_equation", "bbox": [ 136, 414, 483, 511 ], "lines": [ { "bbox": [ 136, 414, 483, 511 ], "spans": [ { "bbox": [ 136, 414, 483, 511 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\quad _ { s 2 i - 1 } _ { t _ { \\lambda } } ( \\overline { { \\lambda } } _ { t _ { i - 1 } } + r _ { 1 } h _ { i } ) , \\quad s _ { 2 i } t _ { \\lambda } ( \\lambda _ { t _ { i - 1 } } + r _ { 2 } h _ { i } ) } \\\\ & { u _ { 2 i - 1 } \\frac { \\alpha _ { s _ { 2 i - 1 } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i - 1 } } ( e ^ { r _ { 1 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { D _ { 2 i - 1 } \\epsilon _ { \\theta } ( u _ { 2 i - 1 } , s _ { 2 i - 1 } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { u _ { 2 i } \\frac { \\alpha _ { s _ { 2 i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i } } ( e ^ { r _ { 2 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { s _ { 2 i } } r _ { 2 } } { r _ { 1 } } ( \\frac { e ^ { r _ { 2 } h _ { i } } - 1 } { r _ { 2 } h _ { i } } - 1 ) D _ { 2 i - 1 } } \\\\ & { D _ { 2 i } \\epsilon _ { \\theta } ( u _ { 2 i } , s _ { 2 i } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { \\tilde { x } _ { t _ { i } } \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { t _ { i } } } { r _ { 2 } } ( \\frac { e ^ { h _ { i } } - 1 } { h } - 1 ) D _ { 2 i } } \\end{array}", "type": "interline_equation", "image_path": "ad4609832a6419c85fb2132d9d91d2c95beec92e10bd4907c45b4ed7d6c517e6.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 136, 414, 483, 446.3333333333333 ], "spans": [], "index": 12 }, { "bbox": [ 136, 446.3333333333333, 483, 478.66666666666663 ], "spans": [], "index": 13 }, { "bbox": [ 136, 478.66666666666663, 483, 510.99999999999994 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 108, 519, 171, 531 ], "lines": [ { "bbox": [ 106, 515, 171, 534 ], "spans": [ { "bbox": [ 106, 515, 153, 534 ], "score": 1.0, "content": "10: return", "type": "text" }, { "bbox": [ 153, 519, 171, 531 ], "score": 0.81, "content": "\\tilde { \\pmb { x } } _ { t _ { M } }", "type": "inline_equation" } ], "index": 15 } ], "index": 15, "bbox_fs": [ 106, 515, 171, 534 ] }, { "type": "text", "bbox": [ 106, 553, 506, 631 ], "lines": [ { "bbox": [ 106, 554, 506, 566 ], "spans": [ { "bbox": [ 106, 554, 130, 566 ], "score": 1.0, "content": "Here,", "type": "text" }, { "bbox": [ 130, 554, 151, 566 ], "score": 0.9, "content": "t _ { \\lambda } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 151, 554, 252, 566 ], "score": 1.0, "content": "is the inverse function of", "type": "text" }, { "bbox": [ 252, 554, 270, 566 ], "score": 0.91, "content": "\\lambda ( t )", "type": "inline_equation" }, { "bbox": [ 270, 554, 506, 566 ], "score": 1.0, "content": ", which has an analytical formulation for the practical noise", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 565, 506, 577 ], "spans": [ { "bbox": [ 105, 565, 458, 577 ], "score": 1.0, "content": "schedule used in [2, 16], as shown in Appendix D. The chosen intermediate points are", "type": "text" }, { "bbox": [ 459, 565, 489, 576 ], "score": 0.89, "content": "( s _ { i } , \\pmb { u } _ { i } )", "type": "inline_equation" }, { "bbox": [ 490, 565, 506, 577 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 574, 506, 590 ], "spans": [ { "bbox": [ 104, 574, 188, 590 ], "score": 1.0, "content": "DPM-Solver-2 and", "type": "text" }, { "bbox": [ 188, 576, 247, 588 ], "score": 0.92, "content": "\\left( s _ { 2 i - 1 } , { \\pmb u } _ { 2 i - 1 } \\right)", "type": "inline_equation" }, { "bbox": [ 248, 574, 267, 590 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 268, 576, 306, 588 ], "score": 0.92, "content": "( s _ { 2 i } , { \\pmb u } _ { 2 i } )", "type": "inline_equation" }, { "bbox": [ 307, 574, 506, 590 ], "score": 1.0, "content": "for DPM-Solver-3. As shown in the algorithm,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 586, 505, 599 ], "spans": [ { "bbox": [ 105, 586, 161, 599 ], "score": 1.0, "content": "DPM-Solver-", "type": "text" }, { "bbox": [ 161, 587, 168, 597 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 168, 586, 203, 599 ], "score": 1.0, "content": "requires", "type": "text" }, { "bbox": [ 204, 588, 210, 597 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 211, 586, 343, 599 ], "score": 1.0, "content": "function evaluations per step for", "type": "text" }, { "bbox": [ 344, 587, 387, 598 ], "score": 0.93, "content": "k = 1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 388, 586, 505, 599 ], "score": 1.0, "content": ". Despite the more expensive", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 596, 506, 611 ], "spans": [ { "bbox": [ 105, 596, 215, 611 ], "score": 1.0, "content": "steps, higher-order solvers", "type": "text" }, { "bbox": [ 216, 598, 253, 609 ], "score": 0.83, "content": "( k = 2 , 3", "type": "inline_equation" }, { "bbox": [ 253, 596, 506, 611 ], "score": 1.0, "content": ") are usually more efficient since they require much fewer steps", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 609, 506, 621 ], "spans": [ { "bbox": [ 105, 609, 418, 621 ], "score": 1.0, "content": "to converge, due to their higher convergence order. We show that DPM-Solver-", "type": "text" }, { "bbox": [ 418, 609, 425, 618 ], "score": 0.83, "content": "k", "type": "inline_equation" }, { "bbox": [ 425, 609, 435, 621 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 435, 609, 442, 618 ], "score": 0.84, "content": "k", "type": "inline_equation" }, { "bbox": [ 442, 609, 506, 621 ], "score": 1.0, "content": "-th-order solver,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 619, 362, 632 ], "spans": [ { "bbox": [ 105, 619, 362, 632 ], "score": 1.0, "content": "as stated in the following theorem. The proof is in Appendix B.", "type": "text" } ], "index": 22 } ], "index": 19, "bbox_fs": [ 104, 554, 506, 632 ] }, { "type": "text", "bbox": [ 106, 634, 505, 682 ], "lines": [ { "bbox": [ 105, 633, 506, 647 ], "spans": [ { "bbox": [ 105, 633, 218, 647 ], "score": 1.0, "content": "Theorem 3.2 (DPM-Solver-", "type": "text" }, { "bbox": [ 219, 635, 225, 644 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 226, 633, 244, 647 ], "score": 1.0, "content": "as a", "type": "text" }, { "bbox": [ 244, 635, 250, 644 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 251, 633, 354, 647 ], "score": 1.0, "content": "-th-order solver). Assume", "type": "text" }, { "bbox": [ 354, 634, 390, 646 ], "score": 0.92, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 391, 633, 506, 647 ], "score": 1.0, "content": "follows the regularity condi-", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 646, 505, 658 ], "spans": [ { "bbox": [ 106, 646, 262, 658 ], "score": 1.0, "content": "tions detailed in Appendix B.1, then for", "type": "text" }, { "bbox": [ 262, 646, 306, 657 ], "score": 0.78, "content": "k = 1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 306, 646, 362, 658 ], "score": 1.0, "content": ", DPM-Solver-", "type": "text" }, { "bbox": [ 362, 647, 369, 655 ], "score": 0.78, "content": "k", "type": "inline_equation" }, { "bbox": [ 369, 646, 386, 658 ], "score": 1.0, "content": "is a", "type": "text" }, { "bbox": [ 386, 646, 393, 655 ], "score": 0.72, "content": "k", "type": "inline_equation" }, { "bbox": [ 393, 646, 505, 658 ], "score": 1.0, "content": "-th order solver for diffusion", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 655, 507, 671 ], "spans": [ { "bbox": [ 105, 655, 218, 671 ], "score": 1.0, "content": "ODEs, i.e., for the sequence", "type": "text" }, { "bbox": [ 218, 655, 256, 668 ], "score": 0.92, "content": "\\{ \\tilde { \\pmb { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 256, 655, 362, 671 ], "score": 1.0, "content": "computed by DPM-Solver-", "type": "text" }, { "bbox": [ 362, 657, 368, 667 ], "score": 0.71, "content": "k", "type": "inline_equation" }, { "bbox": [ 369, 655, 507, 671 ], "score": 1.0, "content": ", the approximation error at time 0", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 667, 400, 683 ], "spans": [ { "bbox": [ 104, 667, 140, 683 ], "score": 1.0, "content": "satisfies", "type": "text" }, { "bbox": [ 141, 668, 232, 680 ], "score": 0.93, "content": "\\tilde { \\pmb { x } } _ { t _ { M } } - \\pmb { x } _ { 0 } = \\mathcal { O } ( h _ { \\operatorname* { m a x } } ^ { k } )", "type": "inline_equation" }, { "bbox": [ 232, 667, 263, 683 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 263, 668, 395, 681 ], "score": 0.92, "content": "h _ { m a x } = \\mathrm { m a x } _ { 1 \\leq i \\leq M } ( \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } )", "type": "inline_equation" }, { "bbox": [ 395, 667, 400, 683 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 24.5, "bbox_fs": [ 104, 633, 507, 683 ] }, { "type": "text", "bbox": [ 107, 688, 505, 722 ], "lines": [ { "bbox": [ 106, 688, 505, 701 ], "spans": [ { "bbox": [ 106, 688, 189, 701 ], "score": 1.0, "content": "Finally, solvers with", "type": "text" }, { "bbox": [ 189, 689, 214, 700 ], "score": 0.91, "content": "k \\geq 4", "type": "inline_equation" }, { "bbox": [ 214, 688, 505, 701 ], "score": 1.0, "content": "need much more intermediate points as shown by previous work [31, 32]", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 700, 505, 712 ], "spans": [ { "bbox": [ 105, 700, 329, 712 ], "score": 1.0, "content": "for exponential integrators. Therefore, we only consider", "type": "text" }, { "bbox": [ 329, 700, 335, 710 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 336, 700, 505, 712 ], "score": 1.0, "content": "from 1 to 3 in this work, while leaving the", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 710, 259, 723 ], "spans": [ { "bbox": [ 106, 710, 185, 723 ], "score": 1.0, "content": "solvers with higher", "type": "text" }, { "bbox": [ 185, 711, 192, 721 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 192, 710, 259, 723 ], "score": 1.0, "content": "for future study.", "type": "text" } ], "index": 29 } ], "index": 28, "bbox_fs": [ 105, 688, 505, 723 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 72, 210, 84 ], "lines": [ { "bbox": [ 105, 71, 211, 86 ], "spans": [ { "bbox": [ 105, 71, 211, 86 ], "score": 1.0, "content": "3.3 Step Size Schedule", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 92, 505, 181 ], "lines": [ { "bbox": [ 102, 90, 508, 110 ], "spans": [ { "bbox": [ 102, 90, 357, 110 ], "score": 1.0, "content": "The proposed solvers in Sec. 3.2 need to specify the time steps", "type": "text" }, { "bbox": [ 358, 92, 389, 105 ], "score": 0.93, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 389, 90, 508, 110 ], "score": 1.0, "content": "in advance. We propose two", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 116 ], "spans": [ { "bbox": [ 106, 104, 505, 116 ], "score": 1.0, "content": "choices of the time step schedule. One choice is handcrafted, which is to uniformly split the interval", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 128 ], "spans": [ { "bbox": [ 107, 114, 142, 126 ], "score": 0.88, "content": "[ \\lambda _ { T } , \\lambda _ { 0 } ]", "type": "inline_equation" }, { "bbox": [ 142, 114, 163, 128 ], "score": 1.0, "content": ", i.e.", "type": "text" }, { "bbox": [ 163, 114, 271, 128 ], "score": 0.85, "content": "\\begin{array} { r } { \\lambda _ { t _ { i } } = \\dot { \\lambda _ { T } } + \\frac { i } { M } ( \\lambda _ { 0 } - \\lambda _ { T } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 271, 114, 275, 128 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 276, 115, 333, 126 ], "score": 0.83, "content": "i = 0 , \\ldots , M", "type": "inline_equation" }, { "bbox": [ 333, 114, 506, 128 ], "score": 1.0, "content": ". Note that this is different from previous", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 506, 139 ], "spans": [ { "bbox": [ 105, 124, 287, 139 ], "score": 1.0, "content": "work [2, 3] which chooses uniform steps for", "type": "text" }, { "bbox": [ 288, 127, 295, 137 ], "score": 0.85, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 296, 124, 506, 139 ], "score": 1.0, "content": ". Empirically, DPM-Solver with uniform time steps", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 136, 506, 150 ], "spans": [ { "bbox": [ 106, 137, 120, 148 ], "score": 0.89, "content": "\\lambda _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 120, 136, 506, 150 ], "score": 1.0, "content": "can already generate quite good samples in few steps, where results are listed in Appendix E. As", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 146, 506, 161 ], "spans": [ { "bbox": [ 105, 146, 506, 161 ], "score": 1.0, "content": "the other choice, we propose an adaptive step size algorithm, which dynamically adjusts the step size", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 159, 505, 171 ], "spans": [ { "bbox": [ 106, 159, 505, 171 ], "score": 1.0, "content": "by combining different orders of DPM-Solver. The adaptive algorithm is inspired by [20] and we", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 169, 298, 182 ], "spans": [ { "bbox": [ 106, 169, 298, 182 ], "score": 1.0, "content": "defer its implementation details to Appendix C.", "type": "text" } ], "index": 8 } ], "index": 4.5 }, { "type": "text", "bbox": [ 107, 185, 505, 241 ], "lines": [ { "bbox": [ 106, 186, 505, 198 ], "spans": [ { "bbox": [ 106, 186, 505, 198 ], "score": 1.0, "content": "For few-step sampling, we need to use up all the number of function evaluations (NFE). When the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 196, 505, 210 ], "spans": [ { "bbox": [ 105, 196, 505, 210 ], "score": 1.0, "content": "NFE is not divisible by 3, we firstly apply DPM-Solver-3 as much as possible, and then add a single", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 207, 505, 220 ], "spans": [ { "bbox": [ 105, 207, 391, 220 ], "score": 1.0, "content": "step of DPM-Solver-1 or DPM-Solver-2 (dependent on the reminder of", "type": "text" }, { "bbox": [ 391, 208, 402, 217 ], "score": 0.83, "content": "K", "type": "inline_equation" }, { "bbox": [ 402, 207, 505, 220 ], "score": 1.0, "content": "divided by 3), as detailed", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 218, 505, 231 ], "spans": [ { "bbox": [ 105, 218, 505, 231 ], "score": 1.0, "content": "in Appendix D. In the subsequent experiments, we use such combination of solvers with the uniform", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 229, 426, 243 ], "spans": [ { "bbox": [ 105, 229, 194, 243 ], "score": 1.0, "content": "step size schedule for", "type": "text" }, { "bbox": [ 194, 230, 237, 240 ], "score": 0.77, "content": "\\mathrm { N F E } \\leq 2 0", "type": "inline_equation" }, { "bbox": [ 237, 229, 426, 243 ], "score": 1.0, "content": ", and otherwise the adaptive step size schedule.", "type": "text" } ], "index": 13 } ], "index": 11 }, { "type": "title", "bbox": [ 108, 254, 286, 266 ], "lines": [ { "bbox": [ 105, 253, 287, 268 ], "spans": [ { "bbox": [ 105, 253, 287, 268 ], "score": 1.0, "content": "3.4 Sampling from Discrete-Time DPMs", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 273, 506, 365 ], "lines": [ { "bbox": [ 102, 268, 510, 293 ], "spans": [ { "bbox": [ 102, 268, 353, 293 ], "score": 1.0, "content": "Discrete-time DPMs [2] train the noise prediction model at", "type": "text" }, { "bbox": [ 353, 275, 363, 284 ], "score": 0.83, "content": "N", "type": "inline_equation" }, { "bbox": [ 363, 268, 432, 293 ], "score": 1.0, "content": "fixed time steps", "type": "text" }, { "bbox": [ 433, 273, 468, 286 ], "score": 0.93, "content": "\\{ t _ { n } \\} _ { n = 1 } ^ { N }", "type": "inline_equation" }, { "bbox": [ 468, 268, 510, 293 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 289, 298 ], "score": 1.0, "content": "noise prediction model is parameterized by", "type": "text" }, { "bbox": [ 289, 286, 330, 298 ], "score": 0.92, "content": "\\tilde { \\epsilon } _ { \\theta } ( { \\boldsymbol x } _ { n } , n )", "type": "inline_equation" }, { "bbox": [ 330, 285, 347, 298 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 347, 286, 425, 297 ], "score": 0.92, "content": "n = 0 , \\ldots , N - 1", "type": "inline_equation" }, { "bbox": [ 426, 285, 480, 298 ], "score": 1.0, "content": ", where each", "type": "text" }, { "bbox": [ 480, 287, 493, 297 ], "score": 0.86, "content": "{ \\pmb x } _ { n }", "type": "inline_equation" }, { "bbox": [ 494, 285, 506, 298 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 297, 505, 308 ], "spans": [ { "bbox": [ 105, 297, 240, 308 ], "score": 1.0, "content": "corresponding to the value at time", "type": "text" }, { "bbox": [ 241, 297, 261, 308 ], "score": 0.9, "content": "t _ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 261, 297, 505, 308 ], "score": 1.0, "content": ". We can transform the discrete-time noise prediction model to", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 305, 506, 325 ], "spans": [ { "bbox": [ 105, 305, 238, 325 ], "score": 1.0, "content": "the continuous version by letting", "type": "text" }, { "bbox": [ 238, 308, 343, 323 ], "score": 0.91, "content": "\\begin{array} { r } { \\epsilon _ { \\theta } ( x , t ) : = \\tilde { \\epsilon } _ { \\theta } ( x , \\frac { ( N - 1 ) t } { T } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 343, 305, 373, 325 ], "score": 1.0, "content": ", for all", "type": "text" }, { "bbox": [ 374, 308, 447, 322 ], "score": 0.93, "content": "\\pmb { x } \\in \\mathbb { R } ^ { d } , t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 448, 305, 506, 325 ], "score": 1.0, "content": ". Note that the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 320, 506, 333 ], "spans": [ { "bbox": [ 105, 320, 160, 333 ], "score": 1.0, "content": "input time of", "type": "text" }, { "bbox": [ 160, 321, 171, 332 ], "score": 0.87, "content": "\\tilde { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 171, 320, 506, 333 ], "score": 1.0, "content": "may not be integers, but we find that the noise prediction model can still work well,", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 331, 506, 344 ], "spans": [ { "bbox": [ 105, 331, 506, 344 ], "score": 1.0, "content": "and we hypothesize that it is because of the smooth time embeddings (e.g., position embeddings [2]).", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 343, 506, 355 ], "spans": [ { "bbox": [ 106, 343, 506, 355 ], "score": 1.0, "content": "By such reparameterization, the noise prediction model can adopt the continuous-time steps as input,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 352, 333, 366 ], "spans": [ { "bbox": [ 105, 352, 333, 366 ], "score": 1.0, "content": "and thus we can also use DPM-Solver for fast sampling.", "type": "text" } ], "index": 22 } ], "index": 18.5 }, { "type": "title", "bbox": [ 107, 380, 384, 395 ], "lines": [ { "bbox": [ 104, 379, 385, 398 ], "spans": [ { "bbox": [ 104, 379, 385, 398 ], "score": 1.0, "content": "4 Comparison with Existing Fast Sampling Methods", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 106, 405, 505, 439 ], "lines": [ { "bbox": [ 105, 404, 505, 419 ], "spans": [ { "bbox": [ 105, 404, 505, 419 ], "score": 1.0, "content": "Here, we discuss the relationship and highlight the difference between DPM-Solver and existing", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 416, 505, 429 ], "spans": [ { "bbox": [ 106, 416, 505, 429 ], "score": 1.0, "content": "ODE-based fast sampling methods for DPMs. We further briefly discuss the advantage of training-free", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 428, 271, 441 ], "spans": [ { "bbox": [ 106, 428, 271, 441 ], "score": 1.0, "content": "samplers over those training-based ones.", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "title", "bbox": [ 107, 452, 232, 464 ], "lines": [ { "bbox": [ 106, 452, 233, 465 ], "spans": [ { "bbox": [ 106, 452, 233, 465 ], "score": 1.0, "content": "4.1 DDIM as DPM-Solver-1", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 472, 505, 506 ], "lines": [ { "bbox": [ 105, 471, 505, 486 ], "spans": [ { "bbox": [ 105, 471, 505, 486 ], "score": 1.0, "content": "Denoising Diffusion Implicit Models (DDIM) [19] design a deterministic method for fast sampling", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 483, 505, 497 ], "spans": [ { "bbox": [ 105, 483, 273, 497 ], "score": 1.0, "content": "from DPMs. For two adjacent time steps", "type": "text" }, { "bbox": [ 274, 485, 292, 496 ], "score": 0.89, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 292, 483, 310, 497 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 310, 485, 318, 495 ], "score": 0.86, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 319, 483, 450, 497 ], "score": 1.0, "content": ", assume that we have a solution", "type": "text" }, { "bbox": [ 451, 484, 473, 497 ], "score": 0.92, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 473, 483, 505, 497 ], "score": 1.0, "content": "at time", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 495, 351, 507 ], "spans": [ { "bbox": [ 107, 496, 124, 506 ], "score": 0.86, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 125, 495, 282, 507 ], "score": 1.0, "content": ", then a single step of DDIM from time", "type": "text" }, { "bbox": [ 282, 496, 300, 506 ], "score": 0.9, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 300, 495, 331, 507 ], "score": 1.0, "content": "to time", "type": "text" }, { "bbox": [ 332, 496, 340, 506 ], "score": 0.86, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 340, 495, 351, 507 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 30 } ], "index": 29 }, { "type": "interline_equation", "bbox": [ 188, 512, 423, 540 ], "lines": [ { "bbox": [ 188, 512, 423, 540 ], "spans": [ { "bbox": [ 188, 512, 423, 540 ], "score": 0.93, "content": "\\tilde { \\pmb { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\left( \\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } - \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } \\right) \\epsilon _ { \\theta } ( \\tilde { \\pmb { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) .", "type": "interline_equation", "image_path": "3ad93903d54c31c8256e52d597943286bbf9fba385901a6d278538f4eb730eed.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 188, 512, 423, 526.0 ], "spans": [], "index": 31 }, { "bbox": [ 188, 526.0, 423, 540.0 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 107, 545, 505, 606 ], "lines": [ { "bbox": [ 105, 545, 505, 557 ], "spans": [ { "bbox": [ 105, 545, 505, 557 ], "score": 1.0, "content": "Although motivated by entirely different perspectives, we show that the updates of DPM-Solver-1", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 556, 505, 568 ], "spans": [ { "bbox": [ 105, 556, 460, 568 ], "score": 1.0, "content": "and Denoising Diffusion Implicit Models (DDIM) [19] are identical. By the definition of", "type": "text" }, { "bbox": [ 460, 557, 466, 566 ], "score": 0.78, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 467, 556, 505, 568 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 34 }, { "bbox": [ 108, 566, 506, 585 ], "spans": [ { "bbox": [ 108, 567, 174, 585 ], "score": 0.92, "content": "\\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } = e ^ { - \\lambda _ { t _ { i - 1 } } ^ { - } }", "type": "inline_equation" }, { "bbox": [ 174, 566, 193, 581 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 193, 568, 243, 585 ], "score": 0.91, "content": "\\begin{array} { r } { \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } = e ^ { - \\lambda _ { t _ { i } } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 243, 567, 324, 583 ], "score": 1.0, "content": ". Plugging these and", "type": "text" }, { "bbox": [ 324, 569, 390, 582 ], "score": 0.93, "content": "h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 390, 567, 506, 583 ], "score": 1.0, "content": "to Eq. (4.1) results in exactly", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 584, 506, 596 ], "spans": [ { "bbox": [ 105, 584, 506, 596 ], "score": 1.0, "content": "a step of DPM-Solver-1 in Eq. (3.7). However, the semi-linear ODE formulation of DPM-Solver", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 595, 473, 607 ], "spans": [ { "bbox": [ 105, 595, 473, 607 ], "score": 1.0, "content": "allows for principled generalization to higher-order solvers and convergence order analysis.", "type": "text" } ], "index": 37 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 610, 506, 666 ], "lines": [ { "bbox": [ 106, 611, 506, 623 ], "spans": [ { "bbox": [ 106, 611, 506, 623 ], "score": 1.0, "content": "Recent work [13] also show that DDIM is a first-order discretization of diffusion ODEs by differenti-", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 621, 505, 633 ], "spans": [ { "bbox": [ 106, 621, 505, 633 ], "score": 1.0, "content": "ating both sides of Eq. (4.1). However, they cannot explain the difference between DDIM and the", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 632, 506, 645 ], "spans": [ { "bbox": [ 106, 632, 506, 645 ], "score": 1.0, "content": "first-order Euler discretization of diffusion ODEs. In contrast, by showing that DDIM is a special", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 644, 506, 657 ], "spans": [ { "bbox": [ 106, 644, 506, 657 ], "score": 1.0, "content": "case of DPM-Solver, we reveal that DDIM makes full use of the semi-linearity of diffusion ODEs,", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 654, 352, 667 ], "spans": [ { "bbox": [ 106, 654, 352, 667 ], "score": 1.0, "content": "which explains its superiority over traditional Euler methods.", "type": "text" } ], "index": 42 } ], "index": 40 }, { "type": "title", "bbox": [ 106, 679, 353, 691 ], "lines": [ { "bbox": [ 104, 677, 354, 695 ], "spans": [ { "bbox": [ 104, 677, 354, 695 ], "score": 1.0, "content": "4.2 Comparison with Traditional Runge-Kutta Methods", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 106, 699, 503, 722 ], "lines": [ { "bbox": [ 106, 699, 505, 712 ], "spans": [ { "bbox": [ 106, 699, 505, 712 ], "score": 1.0, "content": "One can obtain a high-order solver by directly applying traditional explicit Runge-Kutta (RK) methods", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 709, 505, 724 ], "spans": [ { "bbox": [ 105, 709, 505, 724 ], "score": 1.0, "content": "to the diffusion ODE in Eq. (2.7). Specifically, RK methods write the solution of Eq. (2.7) in the", "type": "text" } ], "index": 45 } ], "index": 44.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 741, 309, 752 ], "spans": [ { "bbox": [ 302, 741, 309, 752 ], "score": 1.0, "content": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 72, 210, 84 ], "lines": [ { "bbox": [ 105, 71, 211, 86 ], "spans": [ { "bbox": [ 105, 71, 211, 86 ], "score": 1.0, "content": "3.3 Step Size Schedule", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 92, 505, 181 ], "lines": [ { "bbox": [ 102, 90, 508, 110 ], "spans": [ { "bbox": [ 102, 90, 357, 110 ], "score": 1.0, "content": "The proposed solvers in Sec. 3.2 need to specify the time steps", "type": "text" }, { "bbox": [ 358, 92, 389, 105 ], "score": 0.93, "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { M }", "type": "inline_equation" }, { "bbox": [ 389, 90, 508, 110 ], "score": 1.0, "content": "in advance. We propose two", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 116 ], "spans": [ { "bbox": [ 106, 104, 505, 116 ], "score": 1.0, "content": "choices of the time step schedule. One choice is handcrafted, which is to uniformly split the interval", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 128 ], "spans": [ { "bbox": [ 107, 114, 142, 126 ], "score": 0.88, "content": "[ \\lambda _ { T } , \\lambda _ { 0 } ]", "type": "inline_equation" }, { "bbox": [ 142, 114, 163, 128 ], "score": 1.0, "content": ", i.e.", "type": "text" }, { "bbox": [ 163, 114, 271, 128 ], "score": 0.85, "content": "\\begin{array} { r } { \\lambda _ { t _ { i } } = \\dot { \\lambda _ { T } } + \\frac { i } { M } ( \\lambda _ { 0 } - \\lambda _ { T } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 271, 114, 275, 128 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 276, 115, 333, 126 ], "score": 0.83, "content": "i = 0 , \\ldots , M", "type": "inline_equation" }, { "bbox": [ 333, 114, 506, 128 ], "score": 1.0, "content": ". Note that this is different from previous", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 124, 506, 139 ], "spans": [ { "bbox": [ 105, 124, 287, 139 ], "score": 1.0, "content": "work [2, 3] which chooses uniform steps for", "type": "text" }, { "bbox": [ 288, 127, 295, 137 ], "score": 0.85, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 296, 124, 506, 139 ], "score": 1.0, "content": ". Empirically, DPM-Solver with uniform time steps", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 136, 506, 150 ], "spans": [ { "bbox": [ 106, 137, 120, 148 ], "score": 0.89, "content": "\\lambda _ { t _ { i } }", "type": "inline_equation" }, { "bbox": [ 120, 136, 506, 150 ], "score": 1.0, "content": "can already generate quite good samples in few steps, where results are listed in Appendix E. As", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 146, 506, 161 ], "spans": [ { "bbox": [ 105, 146, 506, 161 ], "score": 1.0, "content": "the other choice, we propose an adaptive step size algorithm, which dynamically adjusts the step size", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 159, 505, 171 ], "spans": [ { "bbox": [ 106, 159, 505, 171 ], "score": 1.0, "content": "by combining different orders of DPM-Solver. The adaptive algorithm is inspired by [20] and we", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 169, 298, 182 ], "spans": [ { "bbox": [ 106, 169, 298, 182 ], "score": 1.0, "content": "defer its implementation details to Appendix C.", "type": "text" } ], "index": 8 } ], "index": 4.5, "bbox_fs": [ 102, 90, 508, 182 ] }, { "type": "text", "bbox": [ 107, 185, 505, 241 ], "lines": [ { "bbox": [ 106, 186, 505, 198 ], "spans": [ { "bbox": [ 106, 186, 505, 198 ], "score": 1.0, "content": "For few-step sampling, we need to use up all the number of function evaluations (NFE). When the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 196, 505, 210 ], "spans": [ { "bbox": [ 105, 196, 505, 210 ], "score": 1.0, "content": "NFE is not divisible by 3, we firstly apply DPM-Solver-3 as much as possible, and then add a single", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 207, 505, 220 ], "spans": [ { "bbox": [ 105, 207, 391, 220 ], "score": 1.0, "content": "step of DPM-Solver-1 or DPM-Solver-2 (dependent on the reminder of", "type": "text" }, { "bbox": [ 391, 208, 402, 217 ], "score": 0.83, "content": "K", "type": "inline_equation" }, { "bbox": [ 402, 207, 505, 220 ], "score": 1.0, "content": "divided by 3), as detailed", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 218, 505, 231 ], "spans": [ { "bbox": [ 105, 218, 505, 231 ], "score": 1.0, "content": "in Appendix D. In the subsequent experiments, we use such combination of solvers with the uniform", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 229, 426, 243 ], "spans": [ { "bbox": [ 105, 229, 194, 243 ], "score": 1.0, "content": "step size schedule for", "type": "text" }, { "bbox": [ 194, 230, 237, 240 ], "score": 0.77, "content": "\\mathrm { N F E } \\leq 2 0", "type": "inline_equation" }, { "bbox": [ 237, 229, 426, 243 ], "score": 1.0, "content": ", and otherwise the adaptive step size schedule.", "type": "text" } ], "index": 13 } ], "index": 11, "bbox_fs": [ 105, 186, 505, 243 ] }, { "type": "title", "bbox": [ 108, 254, 286, 266 ], "lines": [ { "bbox": [ 105, 253, 287, 268 ], "spans": [ { "bbox": [ 105, 253, 287, 268 ], "score": 1.0, "content": "3.4 Sampling from Discrete-Time DPMs", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 273, 506, 365 ], "lines": [ { "bbox": [ 102, 268, 510, 293 ], "spans": [ { "bbox": [ 102, 268, 353, 293 ], "score": 1.0, "content": "Discrete-time DPMs [2] train the noise prediction model at", "type": "text" }, { "bbox": [ 353, 275, 363, 284 ], "score": 0.83, "content": "N", "type": "inline_equation" }, { "bbox": [ 363, 268, 432, 293 ], "score": 1.0, "content": "fixed time steps", "type": "text" }, { "bbox": [ 433, 273, 468, 286 ], "score": 0.93, "content": "\\{ t _ { n } \\} _ { n = 1 } ^ { N }", "type": "inline_equation" }, { "bbox": [ 468, 268, 510, 293 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 289, 298 ], "score": 1.0, "content": "noise prediction model is parameterized by", "type": "text" }, { "bbox": [ 289, 286, 330, 298 ], "score": 0.92, "content": "\\tilde { \\epsilon } _ { \\theta } ( { \\boldsymbol x } _ { n } , n )", "type": "inline_equation" }, { "bbox": [ 330, 285, 347, 298 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 347, 286, 425, 297 ], "score": 0.92, "content": "n = 0 , \\ldots , N - 1", "type": "inline_equation" }, { "bbox": [ 426, 285, 480, 298 ], "score": 1.0, "content": ", where each", "type": "text" }, { "bbox": [ 480, 287, 493, 297 ], "score": 0.86, "content": "{ \\pmb x } _ { n }", "type": "inline_equation" }, { "bbox": [ 494, 285, 506, 298 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 297, 505, 308 ], "spans": [ { "bbox": [ 105, 297, 240, 308 ], "score": 1.0, "content": "corresponding to the value at time", "type": "text" }, { "bbox": [ 241, 297, 261, 308 ], "score": 0.9, "content": "t _ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 261, 297, 505, 308 ], "score": 1.0, "content": ". We can transform the discrete-time noise prediction model to", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 305, 506, 325 ], "spans": [ { "bbox": [ 105, 305, 238, 325 ], "score": 1.0, "content": "the continuous version by letting", "type": "text" }, { "bbox": [ 238, 308, 343, 323 ], "score": 0.91, "content": "\\begin{array} { r } { \\epsilon _ { \\theta } ( x , t ) : = \\tilde { \\epsilon } _ { \\theta } ( x , \\frac { ( N - 1 ) t } { T } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 343, 305, 373, 325 ], "score": 1.0, "content": ", for all", "type": "text" }, { "bbox": [ 374, 308, 447, 322 ], "score": 0.93, "content": "\\pmb { x } \\in \\mathbb { R } ^ { d } , t \\in [ 0 , T ]", "type": "inline_equation" }, { "bbox": [ 448, 305, 506, 325 ], "score": 1.0, "content": ". Note that the", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 320, 506, 333 ], "spans": [ { "bbox": [ 105, 320, 160, 333 ], "score": 1.0, "content": "input time of", "type": "text" }, { "bbox": [ 160, 321, 171, 332 ], "score": 0.87, "content": "\\tilde { \\epsilon } _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 171, 320, 506, 333 ], "score": 1.0, "content": "may not be integers, but we find that the noise prediction model can still work well,", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 331, 506, 344 ], "spans": [ { "bbox": [ 105, 331, 506, 344 ], "score": 1.0, "content": "and we hypothesize that it is because of the smooth time embeddings (e.g., position embeddings [2]).", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 343, 506, 355 ], "spans": [ { "bbox": [ 106, 343, 506, 355 ], "score": 1.0, "content": "By such reparameterization, the noise prediction model can adopt the continuous-time steps as input,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 352, 333, 366 ], "spans": [ { "bbox": [ 105, 352, 333, 366 ], "score": 1.0, "content": "and thus we can also use DPM-Solver for fast sampling.", "type": "text" } ], "index": 22 } ], "index": 18.5, "bbox_fs": [ 102, 268, 510, 366 ] }, { "type": "title", "bbox": [ 107, 380, 384, 395 ], "lines": [ { "bbox": [ 104, 379, 385, 398 ], "spans": [ { "bbox": [ 104, 379, 385, 398 ], "score": 1.0, "content": "4 Comparison with Existing Fast Sampling Methods", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 106, 405, 505, 439 ], "lines": [ { "bbox": [ 105, 404, 505, 419 ], "spans": [ { "bbox": [ 105, 404, 505, 419 ], "score": 1.0, "content": "Here, we discuss the relationship and highlight the difference between DPM-Solver and existing", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 416, 505, 429 ], "spans": [ { "bbox": [ 106, 416, 505, 429 ], "score": 1.0, "content": "ODE-based fast sampling methods for DPMs. We further briefly discuss the advantage of training-free", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 428, 271, 441 ], "spans": [ { "bbox": [ 106, 428, 271, 441 ], "score": 1.0, "content": "samplers over those training-based ones.", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 105, 404, 505, 441 ] }, { "type": "title", "bbox": [ 107, 452, 232, 464 ], "lines": [ { "bbox": [ 106, 452, 233, 465 ], "spans": [ { "bbox": [ 106, 452, 233, 465 ], "score": 1.0, "content": "4.1 DDIM as DPM-Solver-1", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 472, 505, 506 ], "lines": [ { "bbox": [ 105, 471, 505, 486 ], "spans": [ { "bbox": [ 105, 471, 505, 486 ], "score": 1.0, "content": "Denoising Diffusion Implicit Models (DDIM) [19] design a deterministic method for fast sampling", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 483, 505, 497 ], "spans": [ { "bbox": [ 105, 483, 273, 497 ], "score": 1.0, "content": "from DPMs. For two adjacent time steps", "type": "text" }, { "bbox": [ 274, 485, 292, 496 ], "score": 0.89, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 292, 483, 310, 497 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 310, 485, 318, 495 ], "score": 0.86, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 319, 483, 450, 497 ], "score": 1.0, "content": ", assume that we have a solution", "type": "text" }, { "bbox": [ 451, 484, 473, 497 ], "score": 0.92, "content": "\\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 473, 483, 505, 497 ], "score": 1.0, "content": "at time", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 495, 351, 507 ], "spans": [ { "bbox": [ 107, 496, 124, 506 ], "score": 0.86, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 125, 495, 282, 507 ], "score": 1.0, "content": ", then a single step of DDIM from time", "type": "text" }, { "bbox": [ 282, 496, 300, 506 ], "score": 0.9, "content": "t _ { i - 1 }", "type": "inline_equation" }, { "bbox": [ 300, 495, 331, 507 ], "score": 1.0, "content": "to time", "type": "text" }, { "bbox": [ 332, 496, 340, 506 ], "score": 0.86, "content": "t _ { i }", "type": "inline_equation" }, { "bbox": [ 340, 495, 351, 507 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 30 } ], "index": 29, "bbox_fs": [ 105, 471, 505, 507 ] }, { "type": "interline_equation", "bbox": [ 188, 512, 423, 540 ], "lines": [ { "bbox": [ 188, 512, 423, 540 ], "spans": [ { "bbox": [ 188, 512, 423, 540 ], "score": 0.93, "content": "\\tilde { \\pmb { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\left( \\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } - \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } \\right) \\epsilon _ { \\theta } ( \\tilde { \\pmb { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) .", "type": "interline_equation", "image_path": "3ad93903d54c31c8256e52d597943286bbf9fba385901a6d278538f4eb730eed.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 188, 512, 423, 526.0 ], "spans": [], "index": 31 }, { "bbox": [ 188, 526.0, 423, 540.0 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 107, 545, 505, 606 ], "lines": [ { "bbox": [ 105, 545, 505, 557 ], "spans": [ { "bbox": [ 105, 545, 505, 557 ], "score": 1.0, "content": "Although motivated by entirely different perspectives, we show that the updates of DPM-Solver-1", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 556, 505, 568 ], "spans": [ { "bbox": [ 105, 556, 460, 568 ], "score": 1.0, "content": "and Denoising Diffusion Implicit Models (DDIM) [19] are identical. By the definition of", "type": "text" }, { "bbox": [ 460, 557, 466, 566 ], "score": 0.78, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 467, 556, 505, 568 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 34 }, { "bbox": [ 108, 566, 506, 585 ], "spans": [ { "bbox": [ 108, 567, 174, 585 ], "score": 0.92, "content": "\\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } = e ^ { - \\lambda _ { t _ { i - 1 } } ^ { - } }", "type": "inline_equation" }, { "bbox": [ 174, 566, 193, 581 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 193, 568, 243, 585 ], "score": 0.91, "content": "\\begin{array} { r } { \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } = e ^ { - \\lambda _ { t _ { i } } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 243, 567, 324, 583 ], "score": 1.0, "content": ". Plugging these and", "type": "text" }, { "bbox": [ 324, 569, 390, 582 ], "score": 0.93, "content": "h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }", "type": "inline_equation" }, { "bbox": [ 390, 567, 506, 583 ], "score": 1.0, "content": "to Eq. (4.1) results in exactly", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 584, 506, 596 ], "spans": [ { "bbox": [ 105, 584, 506, 596 ], "score": 1.0, "content": "a step of DPM-Solver-1 in Eq. (3.7). However, the semi-linear ODE formulation of DPM-Solver", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 595, 473, 607 ], "spans": [ { "bbox": [ 105, 595, 473, 607 ], "score": 1.0, "content": "allows for principled generalization to higher-order solvers and convergence order analysis.", "type": "text" } ], "index": 37 } ], "index": 35, "bbox_fs": [ 105, 545, 506, 607 ] }, { "type": "text", "bbox": [ 107, 610, 506, 666 ], "lines": [ { "bbox": [ 106, 611, 506, 623 ], "spans": [ { "bbox": [ 106, 611, 506, 623 ], "score": 1.0, "content": "Recent work [13] also show that DDIM is a first-order discretization of diffusion ODEs by differenti-", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 621, 505, 633 ], "spans": [ { "bbox": [ 106, 621, 505, 633 ], "score": 1.0, "content": "ating both sides of Eq. (4.1). However, they cannot explain the difference between DDIM and the", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 632, 506, 645 ], "spans": [ { "bbox": [ 106, 632, 506, 645 ], "score": 1.0, "content": "first-order Euler discretization of diffusion ODEs. In contrast, by showing that DDIM is a special", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 644, 506, 657 ], "spans": [ { "bbox": [ 106, 644, 506, 657 ], "score": 1.0, "content": "case of DPM-Solver, we reveal that DDIM makes full use of the semi-linearity of diffusion ODEs,", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 654, 352, 667 ], "spans": [ { "bbox": [ 106, 654, 352, 667 ], "score": 1.0, "content": "which explains its superiority over traditional Euler methods.", "type": "text" } ], "index": 42 } ], "index": 40, "bbox_fs": [ 106, 611, 506, 667 ] }, { "type": "title", "bbox": [ 106, 679, 353, 691 ], "lines": [ { "bbox": [ 104, 677, 354, 695 ], "spans": [ { "bbox": [ 104, 677, 354, 695 ], "score": 1.0, "content": "4.2 Comparison with Traditional Runge-Kutta Methods", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 106, 699, 503, 722 ], "lines": [ { "bbox": [ 106, 699, 505, 712 ], "spans": [ { "bbox": [ 106, 699, 505, 712 ], "score": 1.0, "content": "One can obtain a high-order solver by directly applying traditional explicit Runge-Kutta (RK) methods", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 709, 505, 724 ], "spans": [ { "bbox": [ 105, 709, 505, 724 ], "score": 1.0, "content": "to the diffusion ODE in Eq. (2.7). Specifically, RK methods write the solution of Eq. (2.7) in the", "type": "text" } ], "index": 45 } ], "index": 44.5, "bbox_fs": [ 105, 699, 505, 724 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 146, 109, 462, 196 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 78, 505, 108 ], "group_id": 0, "lines": [ { "bbox": [ 105, 77, 505, 90 ], "spans": [ { "bbox": [ 105, 77, 505, 90 ], "score": 1.0, "content": "Table 1: FID ↓ on CIFAR-10 for different orders of Runge-Kutta (RK) methods and DPM-Solvers, varying the", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 88, 505, 99 ], "spans": [ { "bbox": [ 105, 88, 445, 99 ], "score": 1.0, "content": "number of function evaluations (NFE). For RK methods, we evaluate diffusion ODEs w.r.t. both", "type": "text" }, { "bbox": [ 445, 89, 450, 97 ], "score": 0.51, "content": "t", "type": "inline_equation" }, { "bbox": [ 450, 88, 505, 99 ], "score": 1.0, "content": "(Eq. (2.7)) and", "type": "text" } ], "index": 1 }, { "bbox": [ 107, 98, 505, 110 ], "spans": [ { "bbox": [ 107, 99, 113, 107 ], "score": 0.5, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 114, 98, 256, 110 ], "score": 1.0, "content": "(Eq. (E.1)). We use uniform step size in", "type": "text" }, { "bbox": [ 257, 99, 262, 107 ], "score": 0.52, "content": "t", "type": "inline_equation" }, { "bbox": [ 262, 98, 389, 110 ], "score": 1.0, "content": "for RK (t), and uniform step size in", "type": "text" }, { "bbox": [ 389, 99, 396, 107 ], "score": 0.58, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 397, 98, 424, 110 ], "score": 1.0, "content": "for RK", "type": "text" }, { "bbox": [ 424, 99, 436, 108 ], "score": 0.43, "content": "( \\lambda )", "type": "inline_equation" }, { "bbox": [ 437, 98, 505, 110 ], "score": 1.0, "content": "and DPM-Solvers.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 146, 109, 462, 196 ], "group_id": 0, "lines": [ { "bbox": [ 146, 109, 462, 196 ], "spans": [ { "bbox": [ 146, 109, 462, 196 ], "score": 0.98, "html": "
Sampling method\\NFE12182430364248
RK2 (t)16.407.253.903.633.583.593.54
RK2(入)107.8142.0417.717.654.623.583.17
DPM-Solver-25.283.433.022.852.782.722.69
RK3 (t)48.7521.8610.906.965.224.564.12
RK3 (入)34.294.903.503.032.852.742.69
DPM-Solver-36.032.902.752.702.672.652.65
", "type": "table", "image_path": "114b9edc4cc71bed7f9d053944dde80589018ddf790160907d7334f50b5c35fb.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 146, 109, 462, 138.0 ], "spans": [], "index": 3 }, { "bbox": [ 146, 138.0, 462, 167.0 ], "spans": [], "index": 4 }, { "bbox": [ 146, 167.0, 462, 196.0 ], "spans": [], "index": 5 } ] } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 218, 204, 230 ], "lines": [ { "bbox": [ 106, 217, 205, 232 ], "spans": [ { "bbox": [ 106, 217, 205, 232 ], "score": 1.0, "content": "following integral form:", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 154, 236, 456, 265 ], "lines": [ { "bbox": [ 154, 236, 456, 265 ], "spans": [ { "bbox": [ 154, 236, 456, 265 ], "score": 0.94, "content": "{ \\bf { x } } _ { t } = { \\bf { x } } _ { s } + \\int _ { s } ^ { t } h _ { \\theta } ( { \\bf { x } } _ { \\tau } , \\tau ) \\mathrm { { d } } \\tau = { \\bf { x } } _ { s } + \\int _ { s } ^ { t } \\left( f ( \\tau ) { \\bf { x } } _ { \\tau } + \\frac { g ^ { 2 } ( \\tau ) } { 2 \\sigma _ { \\tau } } \\epsilon _ { \\theta } ( { \\bf { x } } _ { \\tau } , \\tau ) \\right) \\mathrm { { d } } \\tau ,", "type": "interline_equation", "image_path": "dbc1e1d3bec4cc742d44654340bd9da788aefebc8fc4b4e2a9e38d739f018ceb.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 154, 236, 456, 245.66666666666666 ], "spans": [], "index": 7 }, { "bbox": [ 154, 245.66666666666666, 456, 255.33333333333331 ], "spans": [], "index": 8 }, { "bbox": [ 154, 255.33333333333331, 456, 265.0 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 106, 271, 505, 370 ], "lines": [ { "bbox": [ 106, 271, 505, 284 ], "spans": [ { "bbox": [ 106, 271, 292, 284 ], "score": 1.0, "content": "and use some intermediate time steps between", "type": "text" }, { "bbox": [ 293, 271, 312, 284 ], "score": 0.85, "content": "[ t , s ]", "type": "inline_equation" }, { "bbox": [ 312, 271, 438, 284 ], "score": 1.0, "content": "and combine the evaluations of", "type": "text" }, { "bbox": [ 439, 272, 451, 282 ], "score": 0.88, "content": "h _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 451, 271, 505, 284 ], "score": 1.0, "content": "at these time", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 282, 506, 295 ], "spans": [ { "bbox": [ 105, 282, 506, 295 ], "score": 1.0, "content": "steps to approximate the whole integral. The approximation error of explicit RK methods depends on", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 294, 118, 304 ], "score": 0.85, "content": "h _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 119, 294, 378, 306 ], "score": 1.0, "content": ", which consists of the error corresponding to both the linear term", "type": "text" }, { "bbox": [ 378, 293, 410, 306 ], "score": 0.93, "content": "f ( \\tau ) x _ { \\tau }", "type": "inline_equation" }, { "bbox": [ 410, 294, 505, 306 ], "score": 1.0, "content": "and the nonlinear noise", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 304, 505, 317 ], "spans": [ { "bbox": [ 105, 304, 177, 317 ], "score": 1.0, "content": "prediction model", "type": "text" }, { "bbox": [ 178, 305, 188, 315 ], "score": 0.82, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 189, 304, 505, 317 ], "score": 1.0, "content": ". However, the error of the linear term may increase exponentially because the", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 314, 506, 329 ], "spans": [ { "bbox": [ 104, 314, 506, 329 ], "score": 1.0, "content": "exact solution of the linear term has an exponential coefficient (as shown in Eq. (3.1)). There are many", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 325, 506, 339 ], "spans": [ { "bbox": [ 105, 325, 506, 339 ], "score": 1.0, "content": "empirical evidence [25, 31] showing that directly using explicit RK methods for semi-linear ODEs", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 337, 505, 349 ], "spans": [ { "bbox": [ 106, 337, 505, 349 ], "score": 1.0, "content": "may suffer from unstable numerical issues for large step size. We also demonstrate the empirical", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 347, 506, 360 ], "spans": [ { "bbox": [ 105, 347, 506, 360 ], "score": 1.0, "content": "difference of the proposed DPM-Solver and the traditional explicit RK methods in Sec. 5.1, which", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 358, 506, 371 ], "spans": [ { "bbox": [ 106, 358, 506, 371 ], "score": 1.0, "content": "shows that DPM-Solver have smaller discretization errors than the RK methods with the same order.", "type": "text" } ], "index": 18 } ], "index": 14 }, { "type": "title", "bbox": [ 106, 384, 343, 397 ], "lines": [ { "bbox": [ 105, 384, 344, 399 ], "spans": [ { "bbox": [ 105, 384, 344, 399 ], "score": 1.0, "content": "4.3 Training-based Fast Sampling Methods for DPMs", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 405, 506, 493 ], "lines": [ { "bbox": [ 105, 404, 506, 419 ], "spans": [ { "bbox": [ 105, 404, 506, 419 ], "score": 1.0, "content": "Samplers that need extra training or optimization include knowledge distillation [13, 14], learning", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 416, 506, 429 ], "spans": [ { "bbox": [ 105, 416, 506, 429 ], "score": 1.0, "content": "the noise level or variance [15, 16, 33], and learning the noise schedule or sample trajectory [17, 18].", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 427, 506, 440 ], "spans": [ { "bbox": [ 105, 427, 506, 440 ], "score": 1.0, "content": "Although the progressive distillation method [13] can obtain a fast sampler within 4 steps, it needs", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 438, 506, 450 ], "spans": [ { "bbox": [ 105, 438, 506, 450 ], "score": 1.0, "content": "further training costs and loses part of the information in the original DPM (e.g., after distillation, the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 449, 506, 462 ], "spans": [ { "bbox": [ 105, 449, 476, 462 ], "score": 1.0, "content": "noise prediction model cannot predict the noise (score function) at every time step between", "type": "text" }, { "bbox": [ 477, 450, 502, 461 ], "score": 0.87, "content": "[ 0 , T ] )", "type": "inline_equation" }, { "bbox": [ 503, 449, 506, 462 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 460, 506, 473 ], "spans": [ { "bbox": [ 105, 460, 506, 473 ], "score": 1.0, "content": "In contrast, training-free samplers can keep all the information of the original model, and thereby can", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 471, 506, 483 ], "spans": [ { "bbox": [ 106, 471, 506, 483 ], "score": 1.0, "content": "be directly extended to the conditional sampling by combining the original model and an external", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 482, 463, 494 ], "spans": [ { "bbox": [ 106, 482, 463, 494 ], "score": 1.0, "content": "classifier [4] (e.g. see Appendix D for the conditional sampling with classifier guidance).", "type": "text" } ], "index": 27 } ], "index": 23.5 }, { "type": "text", "bbox": [ 107, 498, 505, 553 ], "lines": [ { "bbox": [ 106, 498, 505, 510 ], "spans": [ { "bbox": [ 106, 498, 505, 510 ], "score": 1.0, "content": "Beyond directly designing fast samplers for DPMs, several works also propose novel types of", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 509, 505, 521 ], "spans": [ { "bbox": [ 106, 509, 505, 521 ], "score": 1.0, "content": "DPMs which supports faster sampling. For instance, defining a low-dimensional latent variable for", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 518, 506, 534 ], "spans": [ { "bbox": [ 104, 518, 506, 534 ], "score": 1.0, "content": "DPMs [34]; designing special diffusion processes with bounded score functions [35]; combining", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 531, 506, 544 ], "spans": [ { "bbox": [ 105, 531, 506, 544 ], "score": 1.0, "content": "GANs with the reverse process of DPMs [36]. The proposed DPM-Solver may also be suitable for", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 542, 417, 554 ], "spans": [ { "bbox": [ 106, 542, 417, 554 ], "score": 1.0, "content": "accelerating the sampling of these DPMs, and we leave them for future work.", "type": "text" } ], "index": 32 } ], "index": 30 }, { "type": "title", "bbox": [ 107, 570, 191, 584 ], "lines": [ { "bbox": [ 104, 568, 193, 587 ], "spans": [ { "bbox": [ 104, 568, 193, 587 ], "score": 1.0, "content": "5 Experiments", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 107, 596, 505, 673 ], "lines": [ { "bbox": [ 105, 595, 505, 610 ], "spans": [ { "bbox": [ 105, 595, 505, 610 ], "score": 1.0, "content": "In this section, we show that as a training-free sampler, DPM-Solver can greatly speedup the sampling", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 607, 505, 619 ], "spans": [ { "bbox": [ 105, 607, 505, 619 ], "score": 1.0, "content": "of existing pre-trained DPMs, including both continuous-time and discrete-time ones, with both", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 618, 505, 630 ], "spans": [ { "bbox": [ 105, 618, 505, 630 ], "score": 1.0, "content": "linear noise schedule [2, 19] and cosine noise schedule [16]. We vary different number of function", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 629, 505, 642 ], "spans": [ { "bbox": [ 105, 629, 412, 642 ], "score": 1.0, "content": "evaluations (NFE) which is the number of calls to the noise prediction model", "type": "text" }, { "bbox": [ 412, 629, 448, 641 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 448, 629, 505, 642 ], "score": 1.0, "content": ", and compare", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 640, 505, 652 ], "spans": [ { "bbox": [ 105, 640, 505, 652 ], "score": 1.0, "content": "the sample quality between DPM-Solver and other methods. For each experiment, We draw 50K", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 650, 506, 664 ], "spans": [ { "bbox": [ 105, 650, 506, 664 ], "score": 1.0, "content": "samples and use the widely adopted FID score [37] to evaluate the sample quality, where lower FID", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 662, 258, 676 ], "spans": [ { "bbox": [ 105, 662, 258, 676 ], "score": 1.0, "content": "usually implies better sample quality.", "type": "text" } ], "index": 40 } ], "index": 37 }, { "type": "text", "bbox": [ 108, 678, 505, 722 ], "lines": [ { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "Unless explicitly mentioned, we always use the solver combination with the uniform step size", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 689, 505, 701 ], "spans": [ { "bbox": [ 106, 689, 505, 701 ], "score": 1.0, "content": "schedule in Sec. 3.3 if the NFE budget is less than 20, and otherwise the DPM-Solver-3 with the", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 700, 505, 712 ], "spans": [ { "bbox": [ 106, 700, 505, 712 ], "score": 1.0, "content": "adaptive step size schedule in Sec. 3.3. We refer to Appendix D for other implementation details of", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 710, 310, 724 ], "spans": [ { "bbox": [ 105, 710, 310, 724 ], "score": 1.0, "content": "DPM-Solver and Appendix E for detailed settings.", "type": "text" } ], "index": 44 } ], "index": 42.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 742, 308, 750 ], "lines": [ { "bbox": [ 301, 740, 310, 752 ], "spans": [ { "bbox": [ 301, 740, 310, 752 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 9 } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 146, 109, 462, 196 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 78, 505, 108 ], "group_id": 0, "lines": [ { "bbox": [ 105, 77, 505, 90 ], "spans": [ { "bbox": [ 105, 77, 505, 90 ], "score": 1.0, "content": "Table 1: FID ↓ on CIFAR-10 for different orders of Runge-Kutta (RK) methods and DPM-Solvers, varying the", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 88, 505, 99 ], "spans": [ { "bbox": [ 105, 88, 445, 99 ], "score": 1.0, "content": "number of function evaluations (NFE). For RK methods, we evaluate diffusion ODEs w.r.t. both", "type": "text" }, { "bbox": [ 445, 89, 450, 97 ], "score": 0.51, "content": "t", "type": "inline_equation" }, { "bbox": [ 450, 88, 505, 99 ], "score": 1.0, "content": "(Eq. (2.7)) and", "type": "text" } ], "index": 1 }, { "bbox": [ 107, 98, 505, 110 ], "spans": [ { "bbox": [ 107, 99, 113, 107 ], "score": 0.5, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 114, 98, 256, 110 ], "score": 1.0, "content": "(Eq. (E.1)). We use uniform step size in", "type": "text" }, { "bbox": [ 257, 99, 262, 107 ], "score": 0.52, "content": "t", "type": "inline_equation" }, { "bbox": [ 262, 98, 389, 110 ], "score": 1.0, "content": "for RK (t), and uniform step size in", "type": "text" }, { "bbox": [ 389, 99, 396, 107 ], "score": 0.58, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 397, 98, 424, 110 ], "score": 1.0, "content": "for RK", "type": "text" }, { "bbox": [ 424, 99, 436, 108 ], "score": 0.43, "content": "( \\lambda )", "type": "inline_equation" }, { "bbox": [ 437, 98, 505, 110 ], "score": 1.0, "content": "and DPM-Solvers.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 146, 109, 462, 196 ], "group_id": 0, "lines": [ { "bbox": [ 146, 109, 462, 196 ], "spans": [ { "bbox": [ 146, 109, 462, 196 ], "score": 0.98, "html": "
Sampling method\\NFE12182430364248
RK2 (t)16.407.253.903.633.583.593.54
RK2(入)107.8142.0417.717.654.623.583.17
DPM-Solver-25.283.433.022.852.782.722.69
RK3 (t)48.7521.8610.906.965.224.564.12
RK3 (入)34.294.903.503.032.852.742.69
DPM-Solver-36.032.902.752.702.672.652.65
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The approximation error of explicit RK methods depends on", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 293, 505, 306 ], "spans": [ { "bbox": [ 106, 294, 118, 304 ], "score": 0.85, "content": "h _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 119, 294, 378, 306 ], "score": 1.0, "content": ", which consists of the error corresponding to both the linear term", "type": "text" }, { "bbox": [ 378, 293, 410, 306 ], "score": 0.93, "content": "f ( \\tau ) x _ { \\tau }", "type": "inline_equation" }, { "bbox": [ 410, 294, 505, 306 ], "score": 1.0, "content": "and the nonlinear noise", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 304, 505, 317 ], "spans": [ { "bbox": [ 105, 304, 177, 317 ], "score": 1.0, "content": "prediction model", "type": "text" }, { "bbox": [ 178, 305, 188, 315 ], "score": 0.82, "content": "\\epsilon _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 189, 304, 505, 317 ], "score": 1.0, "content": ". However, the error of the linear term may increase exponentially because the", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 314, 506, 329 ], "spans": [ { "bbox": [ 104, 314, 506, 329 ], "score": 1.0, "content": "exact solution of the linear term has an exponential coefficient (as shown in Eq. (3.1)). There are many", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 325, 506, 339 ], "spans": [ { "bbox": [ 105, 325, 506, 339 ], "score": 1.0, "content": "empirical evidence [25, 31] showing that directly using explicit RK methods for semi-linear ODEs", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 337, 505, 349 ], "spans": [ { "bbox": [ 106, 337, 505, 349 ], "score": 1.0, "content": "may suffer from unstable numerical issues for large step size. We also demonstrate the empirical", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 347, 506, 360 ], "spans": [ { "bbox": [ 105, 347, 506, 360 ], "score": 1.0, "content": "difference of the proposed DPM-Solver and the traditional explicit RK methods in Sec. 5.1, which", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 358, 506, 371 ], "spans": [ { "bbox": [ 106, 358, 506, 371 ], "score": 1.0, "content": "shows that DPM-Solver have smaller discretization errors than the RK methods with the same order.", "type": "text" } ], "index": 18 } ], "index": 14, "bbox_fs": [ 104, 271, 506, 371 ] }, { "type": "title", "bbox": [ 106, 384, 343, 397 ], "lines": [ { "bbox": [ 105, 384, 344, 399 ], "spans": [ { "bbox": [ 105, 384, 344, 399 ], "score": 1.0, "content": "4.3 Training-based Fast Sampling Methods for DPMs", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 405, 506, 493 ], "lines": [ { "bbox": [ 105, 404, 506, 419 ], "spans": [ { "bbox": [ 105, 404, 506, 419 ], "score": 1.0, "content": "Samplers that need extra training or optimization include knowledge distillation [13, 14], learning", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 416, 506, 429 ], "spans": [ { "bbox": [ 105, 416, 506, 429 ], "score": 1.0, "content": "the noise level or variance [15, 16, 33], and learning the noise schedule or sample trajectory [17, 18].", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 427, 506, 440 ], "spans": [ { "bbox": [ 105, 427, 506, 440 ], "score": 1.0, "content": "Although the progressive distillation method [13] can obtain a fast sampler within 4 steps, it needs", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 438, 506, 450 ], "spans": [ { "bbox": [ 105, 438, 506, 450 ], "score": 1.0, "content": "further training costs and loses part of the information in the original DPM (e.g., after distillation, the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 449, 506, 462 ], "spans": [ { "bbox": [ 105, 449, 476, 462 ], "score": 1.0, "content": "noise prediction model cannot predict the noise (score function) at every time step between", "type": "text" }, { "bbox": [ 477, 450, 502, 461 ], "score": 0.87, "content": "[ 0 , T ] )", "type": "inline_equation" }, { "bbox": [ 503, 449, 506, 462 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 460, 506, 473 ], "spans": [ { "bbox": [ 105, 460, 506, 473 ], "score": 1.0, "content": "In contrast, training-free samplers can keep all the information of the original model, and thereby can", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 471, 506, 483 ], "spans": [ { "bbox": [ 106, 471, 506, 483 ], "score": 1.0, "content": "be directly extended to the conditional sampling by combining the original model and an external", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 482, 463, 494 ], "spans": [ { "bbox": [ 106, 482, 463, 494 ], "score": 1.0, "content": "classifier [4] (e.g. see Appendix D for the conditional sampling with classifier guidance).", "type": "text" } ], "index": 27 } ], "index": 23.5, "bbox_fs": [ 105, 404, 506, 494 ] }, { "type": "text", "bbox": [ 107, 498, 505, 553 ], "lines": [ { "bbox": [ 106, 498, 505, 510 ], "spans": [ { "bbox": [ 106, 498, 505, 510 ], "score": 1.0, "content": "Beyond directly designing fast samplers for DPMs, several works also propose novel types of", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 509, 505, 521 ], "spans": [ { "bbox": [ 106, 509, 505, 521 ], "score": 1.0, "content": "DPMs which supports faster sampling. For instance, defining a low-dimensional latent variable for", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 518, 506, 534 ], "spans": [ { "bbox": [ 104, 518, 506, 534 ], "score": 1.0, "content": "DPMs [34]; designing special diffusion processes with bounded score functions [35]; combining", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 531, 506, 544 ], "spans": [ { "bbox": [ 105, 531, 506, 544 ], "score": 1.0, "content": "GANs with the reverse process of DPMs [36]. The proposed DPM-Solver may also be suitable for", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 542, 417, 554 ], "spans": [ { "bbox": [ 106, 542, 417, 554 ], "score": 1.0, "content": "accelerating the sampling of these DPMs, and we leave them for future work.", "type": "text" } ], "index": 32 } ], "index": 30, "bbox_fs": [ 104, 498, 506, 554 ] }, { "type": "title", "bbox": [ 107, 570, 191, 584 ], "lines": [ { "bbox": [ 104, 568, 193, 587 ], "spans": [ { "bbox": [ 104, 568, 193, 587 ], "score": 1.0, "content": "5 Experiments", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 107, 596, 505, 673 ], "lines": [ { "bbox": [ 105, 595, 505, 610 ], "spans": [ { "bbox": [ 105, 595, 505, 610 ], "score": 1.0, "content": "In this section, we show that as a training-free sampler, DPM-Solver can greatly speedup the sampling", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 607, 505, 619 ], "spans": [ { "bbox": [ 105, 607, 505, 619 ], "score": 1.0, "content": "of existing pre-trained DPMs, including both continuous-time and discrete-time ones, with both", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 618, 505, 630 ], "spans": [ { "bbox": [ 105, 618, 505, 630 ], "score": 1.0, "content": "linear noise schedule [2, 19] and cosine noise schedule [16]. We vary different number of function", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 629, 505, 642 ], "spans": [ { "bbox": [ 105, 629, 412, 642 ], "score": 1.0, "content": "evaluations (NFE) which is the number of calls to the noise prediction model", "type": "text" }, { "bbox": [ 412, 629, 448, 641 ], "score": 0.93, "content": "\\epsilon _ { \\theta } ( x _ { t } , t )", "type": "inline_equation" }, { "bbox": [ 448, 629, 505, 642 ], "score": 1.0, "content": ", and compare", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 640, 505, 652 ], "spans": [ { "bbox": [ 105, 640, 505, 652 ], "score": 1.0, "content": "the sample quality between DPM-Solver and other methods. For each experiment, We draw 50K", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 650, 506, 664 ], "spans": [ { "bbox": [ 105, 650, 506, 664 ], "score": 1.0, "content": "samples and use the widely adopted FID score [37] to evaluate the sample quality, where lower FID", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 662, 258, 676 ], "spans": [ { "bbox": [ 105, 662, 258, 676 ], "score": 1.0, "content": "usually implies better sample quality.", "type": "text" } ], "index": 40 } ], "index": 37, "bbox_fs": [ 105, 595, 506, 676 ] }, { "type": "text", "bbox": [ 108, 678, 505, 722 ], "lines": [ { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "Unless explicitly mentioned, we always use the solver combination with the uniform step size", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 689, 505, 701 ], "spans": [ { "bbox": [ 106, 689, 505, 701 ], "score": 1.0, "content": "schedule in Sec. 3.3 if the NFE budget is less than 20, and otherwise the DPM-Solver-3 with the", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 700, 505, 712 ], "spans": [ { "bbox": [ 106, 700, 505, 712 ], "score": 1.0, "content": "adaptive step size schedule in Sec. 3.3. We refer to Appendix D for other implementation details of", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 710, 310, 724 ], "spans": [ { "bbox": [ 105, 710, 310, 724 ], "score": 1.0, "content": "DPM-Solver and Appendix E for detailed settings.", "type": "text" } ], "index": 44 } ], "index": 42.5, "bbox_fs": [ 105, 677, 506, 724 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 110, 69, 500, 289 ], "blocks": [ { "type": "image_body", "bbox": [ 110, 69, 500, 289 ], "group_id": 0, "lines": [ { "bbox": [ 110, 69, 500, 289 ], "spans": [ { "bbox": [ 110, 69, 500, 289 ], "score": 0.976, "type": "image", "image_path": "4e1bcabdec913d95ec9ff692dca4450fca481c16e793a3c86ffe1cf469e22305.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 110, 69, 500, 142.33333333333331 ], "spans": [], "index": 0 }, { "bbox": [ 110, 142.33333333333331, 500, 215.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 110, 215.66666666666663, 500, 288.99999999999994 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 295, 505, 356 ], "group_id": 0, "lines": [ { "bbox": [ 106, 296, 505, 307 ], "spans": [ { "bbox": [ 106, 296, 264, 307 ], "score": 1.0, "content": "Figure 2: Sample quality measured by FID", "type": "text" }, { "bbox": [ 264, 297, 271, 306 ], "score": 0.31, "content": "\\downarrow", "type": "inline_equation" }, { "bbox": [ 272, 296, 505, 307 ], "score": 1.0, "content": "of different sampling methods for DPMs on CIFAR-10 with both", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 306, 505, 317 ], "spans": [ { "bbox": [ 106, 306, 428, 317 ], "score": 1.0, "content": "continuous-time and discrete-time models, CelebA 64x64, ImageNet 64x64, ImageNet", "type": "text" }, { "bbox": [ 429, 306, 462, 315 ], "score": 0.42, "content": "1 2 8 \\mathrm { x } 1 2 8", "type": "inline_equation" }, { "bbox": [ 462, 306, 505, 317 ], "score": 1.0, "content": "and LSUN", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 316, 505, 327 ], "spans": [ { "bbox": [ 106, 316, 141, 327 ], "score": 1.0, "content": "bedroom", "type": "text" }, { "bbox": [ 141, 316, 174, 325 ], "score": 0.63, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 174, 316, 505, 327 ], "score": 1.0, "content": "with discrete-time models, varying the number of function evaluations (NFE). The method", "type": "text" } ], "index": 5 }, { "bbox": [ 107, 324, 506, 339 ], "spans": [ { "bbox": [ 107, 325, 140, 336 ], "score": 0.26, "content": "^ { \\dag } { \\bf G } { \\bf G } { \\bf D } { \\bf M }", "type": "inline_equation" }, { "bbox": [ 140, 324, 506, 339 ], "score": 1.0, "content": "[18] needs extra training to optimize the sample trajectory, while other methods are training-free. To", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 335, 507, 347 ], "spans": [ { "bbox": [ 104, 335, 507, 347 ], "score": 1.0, "content": "get the strongest baseline, we use the quadratic step size for DDIM on CelebA, which has a better FID than that", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 346, 288, 357 ], "spans": [ { "bbox": [ 106, 346, 288, 357 ], "score": 1.0, "content": "of the uniform step size in the original paper [19].", "type": "text" } ], "index": 8 } ], "index": 5.5 } ], "index": 3.25 }, { "type": "title", "bbox": [ 107, 366, 364, 378 ], "lines": [ { "bbox": [ 105, 365, 365, 381 ], "spans": [ { "bbox": [ 105, 365, 365, 381 ], "score": 1.0, "content": "5.1 Comparison with Continuous-Time Sampling Methods", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 388, 505, 443 ], "lines": [ { "bbox": [ 106, 388, 505, 401 ], "spans": [ { "bbox": [ 106, 388, 505, 401 ], "score": 1.0, "content": "We firstly compare DPM-Solver with other continuous-time sampling methods for DPMs. The", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 399, 505, 412 ], "spans": [ { "bbox": [ 105, 399, 505, 412 ], "score": 1.0, "content": "compared methods include the Euler-Maruyama discretization for diffusion SDEs [3], the adaptive", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 409, 506, 424 ], "spans": [ { "bbox": [ 105, 409, 506, 424 ], "score": 1.0, "content": "step size solver for diffusion SDEs [20] and the RK methods for diffusion ODEs [3, 28] in Eq. (2.7).", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 419, 506, 435 ], "spans": [ { "bbox": [ 105, 419, 506, 435 ], "score": 1.0, "content": "We compare these methods for sampling from a pre-trained continuous-time “VP deep” model [3] on", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 432, 338, 444 ], "spans": [ { "bbox": [ 106, 432, 338, 444 ], "score": 1.0, "content": "the CIFAR-10 dataset [29] with the linear noise schedule.", "type": "text" } ], "index": 14 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 448, 505, 525 ], "lines": [ { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 505, 461 ], "score": 1.0, "content": "Fig. 2a shows the efficiency of compared solvers. We use uniform time steps with 50, 200, 1000 NFE", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 459, 505, 471 ], "spans": [ { "bbox": [ 105, 459, 505, 471 ], "score": 1.0, "content": "for the diffusion SDE with Euler discretization, and vary the tolerance hyperparameter [3, 20] for the", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 470, 505, 483 ], "spans": [ { "bbox": [ 106, 470, 505, 483 ], "score": 1.0, "content": "adaptive step size SDE solver [20] and RK45 ODE solver [28] to control the NFE. DPM-Solver can", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 481, 506, 494 ], "spans": [ { "bbox": [ 105, 481, 506, 494 ], "score": 1.0, "content": "generate good sample quality within around 10 NFE, while other solvers have large discretization", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 491, 505, 504 ], "spans": [ { "bbox": [ 105, 491, 372, 504 ], "score": 1.0, "content": "error even in 50 NFE, which shows that DPM-Solver can achieve", "type": "text" }, { "bbox": [ 372, 492, 387, 502 ], "score": 0.85, "content": "{ \\sim } 5 ", "type": "inline_equation" }, { "bbox": [ 387, 491, 505, 504 ], "score": 1.0, "content": "speedup of the previous best", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 503, 505, 515 ], "spans": [ { "bbox": [ 105, 503, 505, 515 ], "score": 1.0, "content": "solver. In particular, we achieve 4.70 FID with 10 NFE, 3.75 FID with 12 NFE, 3.24 FID with 15", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 514, 413, 525 ], "spans": [ { "bbox": [ 106, 514, 413, 525 ], "score": 1.0, "content": "NFE, and 2.87 FID with 20 NFE, which is the fastest sampler on CIFAR-10.", "type": "text" } ], "index": 21 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 530, 505, 628 ], "lines": [ { "bbox": [ 106, 530, 506, 542 ], "spans": [ { "bbox": [ 106, 530, 506, 542 ], "score": 1.0, "content": "As an ablation study, we also compare the second-order and third-order DPM-Solver and RK methods,", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 540, 506, 554 ], "spans": [ { "bbox": [ 105, 540, 434, 554 ], "score": 1.0, "content": "as shown in Table 1. We compare RK methods for diffusion ODEs w.r.t. both time", "type": "text" }, { "bbox": [ 434, 542, 440, 551 ], "score": 0.6, "content": "t", "type": "inline_equation" }, { "bbox": [ 440, 540, 506, 554 ], "score": 1.0, "content": "in Eq. (2.7) and", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 551, 506, 565 ], "spans": [ { "bbox": [ 105, 551, 164, 565 ], "score": 1.0, "content": "half-log-SNR", "type": "text" }, { "bbox": [ 164, 552, 172, 562 ], "score": 0.7, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 172, 551, 506, 565 ], "score": 1.0, "content": "by applying change-of-variable (see detailed formulations in Appendix E.1). The", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 562, 505, 576 ], "spans": [ { "bbox": [ 105, 562, 505, 576 ], "score": 1.0, "content": "results show that given the same NFE, the sample quality of DPM-Solver is consistently better than", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 573, 506, 587 ], "spans": [ { "bbox": [ 104, 573, 506, 587 ], "score": 1.0, "content": "RK methods with the same order. The superior efficiency of DPM-Solver is particularly evident in", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 584, 505, 597 ], "spans": [ { "bbox": [ 106, 584, 505, 597 ], "score": 1.0, "content": "the few-step regime under 15 NFE, where RK methods have rather large discretization errors. This", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 594, 506, 609 ], "spans": [ { "bbox": [ 105, 594, 506, 609 ], "score": 1.0, "content": "is mainly because DPM-Solver analytically computes the linear term, avoiding the corresponding", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 606, 506, 619 ], "spans": [ { "bbox": [ 106, 606, 506, 619 ], "score": 1.0, "content": "discretization error. Besides, the higher order DPM-Solver-3 converges faster than DPM-Solver-2,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 618, 307, 630 ], "spans": [ { "bbox": [ 106, 618, 307, 630 ], "score": 1.0, "content": "which matches the order analysis in Theorem 3.2.", "type": "text" } ], "index": 30 } ], "index": 26 }, { "type": "title", "bbox": [ 106, 645, 349, 657 ], "lines": [ { "bbox": [ 104, 643, 351, 661 ], "spans": [ { "bbox": [ 104, 643, 351, 661 ], "score": 1.0, "content": "5.2 Comparison with Discrete-Time Sampling Methods", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 667, 505, 722 ], "lines": [ { "bbox": [ 105, 666, 506, 680 ], "spans": [ { "bbox": [ 105, 666, 506, 680 ], "score": 1.0, "content": "We use the method in Sec. 3.4 for using DPM-Solver in discrete-time DPMs, and then compare", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "DPM-Solver with other discrete-time training-free samplers, including DDPM [2], DDIM [19],", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 688, 506, 702 ], "spans": [ { "bbox": [ 105, 688, 506, 702 ], "score": 1.0, "content": "Analytic-DDPM [21], Analytic-DDIM [21], PNDM [22], FastDPM [38] and Itô-Taylor [24]. We", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 700, 506, 713 ], "spans": [ { "bbox": [ 105, 700, 506, 713 ], "score": 1.0, "content": "also compare with GGDM [18], which uses the same pre-trained model but needs further training for", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 711, 466, 723 ], "spans": [ { "bbox": [ 105, 711, 466, 723 ], "score": 1.0, "content": "the sampling trajectory. We compare the sample quality by varying NFE from 10 to 1000.", "type": "text" } ], "index": 36 } ], "index": 34 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 308, 750 ], "lines": [ { "bbox": [ 302, 741, 309, 752 ], "spans": [ { "bbox": [ 302, 741, 309, 752 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 110, 69, 500, 289 ], "blocks": [ { "type": "image_body", "bbox": [ 110, 69, 500, 289 ], "group_id": 0, "lines": [ { "bbox": [ 110, 69, 500, 289 ], "spans": [ { "bbox": [ 110, 69, 500, 289 ], "score": 0.976, "type": "image", "image_path": "4e1bcabdec913d95ec9ff692dca4450fca481c16e793a3c86ffe1cf469e22305.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 110, 69, 500, 142.33333333333331 ], "spans": [], "index": 0 }, { "bbox": [ 110, 142.33333333333331, 500, 215.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 110, 215.66666666666663, 500, 288.99999999999994 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 295, 505, 356 ], "group_id": 0, "lines": [ { "bbox": [ 106, 296, 505, 307 ], "spans": [ { "bbox": [ 106, 296, 264, 307 ], "score": 1.0, "content": "Figure 2: Sample quality measured by FID", "type": "text" }, { "bbox": [ 264, 297, 271, 306 ], "score": 0.31, "content": "\\downarrow", "type": "inline_equation" }, { "bbox": [ 272, 296, 505, 307 ], "score": 1.0, "content": "of different sampling methods for DPMs on CIFAR-10 with both", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 306, 505, 317 ], "spans": [ { "bbox": [ 106, 306, 428, 317 ], "score": 1.0, "content": "continuous-time and discrete-time models, CelebA 64x64, ImageNet 64x64, ImageNet", "type": "text" }, { "bbox": [ 429, 306, 462, 315 ], "score": 0.42, "content": "1 2 8 \\mathrm { x } 1 2 8", "type": "inline_equation" }, { "bbox": [ 462, 306, 505, 317 ], "score": 1.0, "content": "and LSUN", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 316, 505, 327 ], "spans": [ { "bbox": [ 106, 316, 141, 327 ], "score": 1.0, "content": "bedroom", "type": "text" }, { "bbox": [ 141, 316, 174, 325 ], "score": 0.63, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 174, 316, 505, 327 ], "score": 1.0, "content": "with discrete-time models, varying the number of function evaluations (NFE). The method", "type": "text" } ], "index": 5 }, { "bbox": [ 107, 324, 506, 339 ], "spans": [ { "bbox": [ 107, 325, 140, 336 ], "score": 0.26, "content": "^ { \\dag } { \\bf G } { \\bf G } { \\bf D } { \\bf M }", "type": "inline_equation" }, { "bbox": [ 140, 324, 506, 339 ], "score": 1.0, "content": "[18] needs extra training to optimize the sample trajectory, while other methods are training-free. To", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 335, 507, 347 ], "spans": [ { "bbox": [ 104, 335, 507, 347 ], "score": 1.0, "content": "get the strongest baseline, we use the quadratic step size for DDIM on CelebA, which has a better FID than that", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 346, 288, 357 ], "spans": [ { "bbox": [ 106, 346, 288, 357 ], "score": 1.0, "content": "of the uniform step size in the original paper [19].", "type": "text" } ], "index": 8 } ], "index": 5.5 } ], "index": 3.25 }, { "type": "title", "bbox": [ 107, 366, 364, 378 ], "lines": [ { "bbox": [ 105, 365, 365, 381 ], "spans": [ { "bbox": [ 105, 365, 365, 381 ], "score": 1.0, "content": "5.1 Comparison with Continuous-Time Sampling Methods", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 388, 505, 443 ], "lines": [ { "bbox": [ 106, 388, 505, 401 ], "spans": [ { "bbox": [ 106, 388, 505, 401 ], "score": 1.0, "content": "We firstly compare DPM-Solver with other continuous-time sampling methods for DPMs. The", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 399, 505, 412 ], "spans": [ { "bbox": [ 105, 399, 505, 412 ], "score": 1.0, "content": "compared methods include the Euler-Maruyama discretization for diffusion SDEs [3], the adaptive", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 409, 506, 424 ], "spans": [ { "bbox": [ 105, 409, 506, 424 ], "score": 1.0, "content": "step size solver for diffusion SDEs [20] and the RK methods for diffusion ODEs [3, 28] in Eq. (2.7).", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 419, 506, 435 ], "spans": [ { "bbox": [ 105, 419, 506, 435 ], "score": 1.0, "content": "We compare these methods for sampling from a pre-trained continuous-time “VP deep” model [3] on", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 432, 338, 444 ], "spans": [ { "bbox": [ 106, 432, 338, 444 ], "score": 1.0, "content": "the CIFAR-10 dataset [29] with the linear noise schedule.", "type": "text" } ], "index": 14 } ], "index": 12, "bbox_fs": [ 105, 388, 506, 444 ] }, { "type": "text", "bbox": [ 107, 448, 505, 525 ], "lines": [ { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 505, 461 ], "score": 1.0, "content": "Fig. 2a shows the efficiency of compared solvers. We use uniform time steps with 50, 200, 1000 NFE", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 459, 505, 471 ], "spans": [ { "bbox": [ 105, 459, 505, 471 ], "score": 1.0, "content": "for the diffusion SDE with Euler discretization, and vary the tolerance hyperparameter [3, 20] for the", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 470, 505, 483 ], "spans": [ { "bbox": [ 106, 470, 505, 483 ], "score": 1.0, "content": "adaptive step size SDE solver [20] and RK45 ODE solver [28] to control the NFE. DPM-Solver can", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 481, 506, 494 ], "spans": [ { "bbox": [ 105, 481, 506, 494 ], "score": 1.0, "content": "generate good sample quality within around 10 NFE, while other solvers have large discretization", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 491, 505, 504 ], "spans": [ { "bbox": [ 105, 491, 372, 504 ], "score": 1.0, "content": "error even in 50 NFE, which shows that DPM-Solver can achieve", "type": "text" }, { "bbox": [ 372, 492, 387, 502 ], "score": 0.85, "content": "{ \\sim } 5 ", "type": "inline_equation" }, { "bbox": [ 387, 491, 505, 504 ], "score": 1.0, "content": "speedup of the previous best", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 503, 505, 515 ], "spans": [ { "bbox": [ 105, 503, 505, 515 ], "score": 1.0, "content": "solver. In particular, we achieve 4.70 FID with 10 NFE, 3.75 FID with 12 NFE, 3.24 FID with 15", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 514, 413, 525 ], "spans": [ { "bbox": [ 106, 514, 413, 525 ], "score": 1.0, "content": "NFE, and 2.87 FID with 20 NFE, which is the fastest sampler on CIFAR-10.", "type": "text" } ], "index": 21 } ], "index": 18, "bbox_fs": [ 105, 448, 506, 525 ] }, { "type": "text", "bbox": [ 107, 530, 505, 628 ], "lines": [ { "bbox": [ 106, 530, 506, 542 ], "spans": [ { "bbox": [ 106, 530, 506, 542 ], "score": 1.0, "content": "As an ablation study, we also compare the second-order and third-order DPM-Solver and RK methods,", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 540, 506, 554 ], "spans": [ { "bbox": [ 105, 540, 434, 554 ], "score": 1.0, "content": "as shown in Table 1. We compare RK methods for diffusion ODEs w.r.t. both time", "type": "text" }, { "bbox": [ 434, 542, 440, 551 ], "score": 0.6, "content": "t", "type": "inline_equation" }, { "bbox": [ 440, 540, 506, 554 ], "score": 1.0, "content": "in Eq. (2.7) and", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 551, 506, 565 ], "spans": [ { "bbox": [ 105, 551, 164, 565 ], "score": 1.0, "content": "half-log-SNR", "type": "text" }, { "bbox": [ 164, 552, 172, 562 ], "score": 0.7, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 172, 551, 506, 565 ], "score": 1.0, "content": "by applying change-of-variable (see detailed formulations in Appendix E.1). The", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 562, 505, 576 ], "spans": [ { "bbox": [ 105, 562, 505, 576 ], "score": 1.0, "content": "results show that given the same NFE, the sample quality of DPM-Solver is consistently better than", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 573, 506, 587 ], "spans": [ { "bbox": [ 104, 573, 506, 587 ], "score": 1.0, "content": "RK methods with the same order. The superior efficiency of DPM-Solver is particularly evident in", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 584, 505, 597 ], "spans": [ { "bbox": [ 106, 584, 505, 597 ], "score": 1.0, "content": "the few-step regime under 15 NFE, where RK methods have rather large discretization errors. This", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 594, 506, 609 ], "spans": [ { "bbox": [ 105, 594, 506, 609 ], "score": 1.0, "content": "is mainly because DPM-Solver analytically computes the linear term, avoiding the corresponding", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 606, 506, 619 ], "spans": [ { "bbox": [ 106, 606, 506, 619 ], "score": 1.0, "content": "discretization error. Besides, the higher order DPM-Solver-3 converges faster than DPM-Solver-2,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 618, 307, 630 ], "spans": [ { "bbox": [ 106, 618, 307, 630 ], "score": 1.0, "content": "which matches the order analysis in Theorem 3.2.", "type": "text" } ], "index": 30 } ], "index": 26, "bbox_fs": [ 104, 530, 506, 630 ] }, { "type": "title", "bbox": [ 106, 645, 349, 657 ], "lines": [ { "bbox": [ 104, 643, 351, 661 ], "spans": [ { "bbox": [ 104, 643, 351, 661 ], "score": 1.0, "content": "5.2 Comparison with Discrete-Time Sampling Methods", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 667, 505, 722 ], "lines": [ { "bbox": [ 105, 666, 506, 680 ], "spans": [ { "bbox": [ 105, 666, 506, 680 ], "score": 1.0, "content": "We use the method in Sec. 3.4 for using DPM-Solver in discrete-time DPMs, and then compare", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "DPM-Solver with other discrete-time training-free samplers, including DDPM [2], DDIM [19],", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 688, 506, 702 ], "spans": [ { "bbox": [ 105, 688, 506, 702 ], "score": 1.0, "content": "Analytic-DDPM [21], Analytic-DDIM [21], PNDM [22], FastDPM [38] and Itô-Taylor [24]. We", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 700, 506, 713 ], "spans": [ { "bbox": [ 105, 700, 506, 713 ], "score": 1.0, "content": "also compare with GGDM [18], which uses the same pre-trained model but needs further training for", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 711, 466, 723 ], "spans": [ { "bbox": [ 105, 711, 466, 723 ], "score": 1.0, "content": "the sampling trajectory. We compare the sample quality by varying NFE from 10 to 1000.", "type": "text" } ], "index": 36 } ], "index": 34, "bbox_fs": [ 105, 666, 506, 723 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 72, 505, 182 ], "lines": [ { "bbox": [ 105, 72, 505, 86 ], "spans": [ { "bbox": [ 105, 72, 329, 86 ], "score": 1.0, "content": "Specifically, we use the discrete-time model trained by", "type": "text" }, { "bbox": [ 329, 73, 356, 84 ], "score": 0.9, "content": "L _ { \\mathrm { s i m p l e } }", "type": "inline_equation" }, { "bbox": [ 357, 72, 505, 86 ], "score": 1.0, "content": "in [2] on the CIFAR-10 dataset with", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 506, 96 ], "spans": [ { "bbox": [ 105, 83, 506, 96 ], "score": 1.0, "content": "linear noise schedule; the discrete-time model in [19] on CelebA 64x64 [39] with linear noise", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 93, 506, 109 ], "spans": [ { "bbox": [ 105, 93, 292, 109 ], "score": 1.0, "content": "schedule; the discrete-time model trained by", "type": "text" }, { "bbox": [ 292, 95, 319, 106 ], "score": 0.89, "content": "L _ { \\mathrm { h y b r i d } }", "type": "inline_equation" }, { "bbox": [ 320, 93, 506, 109 ], "score": 1.0, "content": "in [16] on ImageNet 64x64 [26] with cosine", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 105, 505, 118 ], "spans": [ { "bbox": [ 106, 105, 505, 118 ], "score": 1.0, "content": "noise schedule; the discrete-time model with classifier guidance in [4] on ImageNet 128x128 [26]", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 117, 505, 128 ], "spans": [ { "bbox": [ 106, 117, 426, 128 ], "score": 1.0, "content": "with linear noise schedule; the discrete-time model in [4] on LSUN bedroom", "type": "text" }, { "bbox": [ 426, 117, 464, 127 ], "score": 0.29, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 464, 117, 505, 128 ], "score": 1.0, "content": "[40] with", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 127, 506, 139 ], "spans": [ { "bbox": [ 105, 127, 506, 139 ], "score": 1.0, "content": "linear noise schedule. For the models trained on ImageNet, we only use their “mean” model and", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 138, 506, 150 ], "spans": [ { "bbox": [ 106, 138, 506, 150 ], "score": 1.0, "content": "omit the “variance” model. As shown in Fig. 2, on all datasets, DPM-Solver can obtain reasonable", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 149, 505, 161 ], "spans": [ { "bbox": [ 105, 149, 505, 161 ], "score": 1.0, "content": "samples within 12 steps (FID 4.65 on CIFAR-10, FID 3.71 on CelebA 64x64 and FID 19.97 on", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 159, 506, 172 ], "spans": [ { "bbox": [ 106, 159, 268, 172 ], "score": 1.0, "content": "ImageNet 64x64, FID 4.08 on ImageNet", "type": "text" }, { "bbox": [ 268, 160, 304, 171 ], "score": 0.33, "content": "1 2 8 \\mathbf { x } 1 2 8 _ { \\rho }", "type": "inline_equation" }, { "bbox": [ 304, 159, 345, 172 ], "score": 1.0, "content": "), which is", "type": "text" }, { "bbox": [ 345, 160, 383, 171 ], "score": 0.92, "content": "4 \\sim 1 6 \\times", "type": "inline_equation" }, { "bbox": [ 383, 159, 506, 172 ], "score": 1.0, "content": "faster than the previous fastest", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 170, 496, 185 ], "spans": [ { "bbox": [ 105, 170, 496, 185 ], "score": 1.0, "content": "training-free sampler. DPM-Solver even outperforms GGDM, which requires additional training.", "type": "text" } ], "index": 9 } ], "index": 4.5 }, { "type": "title", "bbox": [ 107, 199, 187, 212 ], "lines": [ { "bbox": [ 104, 197, 190, 214 ], "spans": [ { "bbox": [ 104, 197, 190, 214 ], "score": 1.0, "content": "6 Conclusions", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 224, 505, 345 ], "lines": [ { "bbox": [ 105, 223, 506, 237 ], "spans": [ { "bbox": [ 105, 223, 506, 237 ], "score": 1.0, "content": "We tackle the problem of fast and training-free sampling from DPMs. We propose DPM-Solver,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 235, 506, 248 ], "spans": [ { "bbox": [ 105, 235, 506, 248 ], "score": 1.0, "content": "a fast dedicated training-free solver of diffusion ODEs for fast sampling of DPMs in around 10", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 246, 505, 258 ], "spans": [ { "bbox": [ 105, 246, 505, 258 ], "score": 1.0, "content": "steps of function evaluations. DPM-Solver leverages the semi-linearity of diffusion ODEs and", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 505, 269 ], "score": 1.0, "content": "it directly approximates a simplified formulation of exact solutions of diffusion ODEs, which", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 268, 506, 280 ], "spans": [ { "bbox": [ 106, 268, 506, 280 ], "score": 1.0, "content": "consists of an exponentially weighted integral of the noise prediction model. Inspired by numerical", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 279, 506, 291 ], "spans": [ { "bbox": [ 105, 279, 506, 291 ], "score": 1.0, "content": "methods for exponential integrators, we propose first-order, second-order and third-order DPM-", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 290, 505, 302 ], "spans": [ { "bbox": [ 105, 290, 505, 302 ], "score": 1.0, "content": "Solver to approximate the exponentially weighted integral of noise prediction models with theoretical", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 301, 505, 313 ], "spans": [ { "bbox": [ 105, 301, 505, 313 ], "score": 1.0, "content": "convergence guarantee. We propose both handcrafted and adaptive step size schedule, and apply", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 311, 506, 324 ], "spans": [ { "bbox": [ 105, 311, 506, 324 ], "score": 1.0, "content": "DPM-Solver for both continuous-time and discrete-time DPMs. Our experimental results show that", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 322, 506, 335 ], "spans": [ { "bbox": [ 105, 322, 506, 335 ], "score": 1.0, "content": "DPM-Solver can generate high-quality samples in around 10 function evaluations on various datasets,", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 333, 506, 346 ], "spans": [ { "bbox": [ 105, 333, 180, 346 ], "score": 1.0, "content": "and it can achieve", "type": "text" }, { "bbox": [ 180, 334, 217, 344 ], "score": 0.9, "content": "4 \\sim 1 6 \\times", "type": "inline_equation" }, { "bbox": [ 218, 333, 506, 346 ], "score": 1.0, "content": "speedup compared with previous state-of-the-art training-free samplers.", "type": "text" } ], "index": 21 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 349, 505, 416 ], "lines": [ { "bbox": [ 105, 349, 506, 362 ], "spans": [ { "bbox": [ 105, 349, 506, 362 ], "score": 1.0, "content": "Limitations and broader impact Despite the promising speedup performance, DPM-Solver is", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 361, 505, 372 ], "spans": [ { "bbox": [ 106, 361, 505, 372 ], "score": 1.0, "content": "designed for fast sampling, which may be not suitable for accelerating the likelihood evaluations of", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 371, 505, 384 ], "spans": [ { "bbox": [ 105, 371, 505, 384 ], "score": 1.0, "content": "DPMs. Besides, compared to the commonly-used GANs, diffusion models with DPM-Solver are still", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 381, 506, 396 ], "spans": [ { "bbox": [ 105, 381, 506, 396 ], "score": 1.0, "content": "not fast enough for real-time applications. In addition, like other deep generative models, DPMs may", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 393, 505, 406 ], "spans": [ { "bbox": [ 105, 393, 505, 406 ], "score": 1.0, "content": "be used to generate adverse fake contents, and the proposed solver may further amplify the potential", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 404, 410, 417 ], "spans": [ { "bbox": [ 105, 404, 410, 417 ], "score": 1.0, "content": "undesirable influence of deep generative models for malicious applications.", "type": "text" } ], "index": 27 } ], "index": 24.5 }, { "type": "title", "bbox": [ 108, 432, 207, 445 ], "lines": [ { "bbox": [ 106, 431, 208, 448 ], "spans": [ { "bbox": [ 106, 431, 208, 448 ], "score": 1.0, "content": "Acknowledgements", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 107, 457, 506, 545 ], "lines": [ { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "This work was supported by National Key Research and Development Project of China (No.", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 469, 506, 481 ], "spans": [ { "bbox": [ 105, 469, 506, 481 ], "score": 1.0, "content": "2021ZD0110502); NSF of China Projects (Nos. 62061136001, 61620106010, 62076145, U19B2034,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 479, 506, 493 ], "spans": [ { "bbox": [ 105, 479, 506, 493 ], "score": 1.0, "content": "U1811461, U19A2081, 6197222, 62106120); Beijing NSF Project (No. JQ19016); Beijing Outstand-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 491, 505, 503 ], "spans": [ { "bbox": [ 106, 491, 505, 503 ], "score": 1.0, "content": "ing Young Scientist Program NO. BJJWZYJH012019100020098; a grant from Tsinghua Institute", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 501, 505, 513 ], "spans": [ { "bbox": [ 106, 501, 505, 513 ], "score": 1.0, "content": "for Guo Qiang; the NVIDIA NVAIL Program with GPU/DGX Acceleration; the High Performance", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 512, 506, 525 ], "spans": [ { "bbox": [ 105, 512, 506, 525 ], "score": 1.0, "content": "Computing Center, Tsinghua University; the Fundamental Research Funds for the Central Universi-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 522, 505, 537 ], "spans": [ { "bbox": [ 105, 522, 505, 537 ], "score": 1.0, "content": "ties, and the Research Funds of Renmin University of China (22XNKJ13). J.Z is also supported by", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 534, 181, 546 ], "spans": [ { "bbox": [ 106, 534, 181, 546 ], "score": 1.0, "content": "the XPlorer Prize.", "type": "text" } ], "index": 36 } ], "index": 32.5 }, { "type": "title", "bbox": [ 107, 563, 163, 575 ], "lines": [ { "bbox": [ 106, 561, 165, 577 ], "spans": [ { "bbox": [ 106, 561, 165, 577 ], "score": 1.0, "content": "References", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 110, 582, 506, 722 ], "lines": [ { "bbox": [ 110, 581, 505, 598 ], "spans": [ { "bbox": [ 110, 581, 505, 598 ], "score": 1.0, "content": "[1] J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning", "type": "text" } ], "index": 38 }, { "bbox": [ 126, 592, 506, 609 ], "spans": [ { "bbox": [ 126, 592, 506, 609 ], "score": 1.0, "content": "using nonequilibrium thermodynamics,” in International Conference on Machine Learning.", "type": "text" } ], "index": 39 }, { "bbox": [ 126, 604, 250, 619 ], "spans": [ { "bbox": [ 126, 604, 250, 619 ], "score": 1.0, "content": "PMLR, 2015, pp. 2256–2265.", "type": "text" } ], "index": 40 }, { "bbox": [ 111, 625, 506, 639 ], "spans": [ { "bbox": [ 111, 625, 506, 639 ], "score": 1.0, "content": "[2] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” in Advances in Neural", "type": "text" } ], "index": 41 }, { "bbox": [ 127, 636, 383, 650 ], "spans": [ { "bbox": [ 127, 636, 383, 650 ], "score": 1.0, "content": "Information Processing Systems, vol. 33, 2020, pp. 6840–6851.", "type": "text" } ], "index": 42 }, { "bbox": [ 110, 657, 505, 671 ], "spans": [ { "bbox": [ 110, 657, 505, 671 ], "score": 1.0, "content": "[3] Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, “Score-based", "type": "text" } ], "index": 43 }, { "bbox": [ 127, 668, 505, 682 ], "spans": [ { "bbox": [ 127, 668, 505, 682 ], "score": 1.0, "content": "generative modeling through stochastic differential equations,” in International Conference on", "type": "text" } ], "index": 44 }, { "bbox": [ 127, 679, 260, 693 ], "spans": [ { "bbox": [ 127, 679, 260, 693 ], "score": 1.0, "content": "Learning Representations, 2021.", "type": "text" } ], "index": 45 }, { "bbox": [ 111, 699, 505, 714 ], "spans": [ { "bbox": [ 111, 699, 505, 714 ], "score": 1.0, "content": "[4] P. Dhariwal and A. Q. Nichol, “Diffusion models beat GANs on image synthesis,” in Advances", "type": "text" } ], "index": 46 }, { "bbox": [ 126, 711, 424, 724 ], "spans": [ { "bbox": [ 126, 711, 424, 724 ], "score": 1.0, "content": "in Neural Information Processing Systems, vol. 34, 2021, pp. 8780–8794.", "type": "text" } ], "index": 47 } ], "index": 42.5 } ], "page_idx": 9, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 301, 741, 311, 750 ], "lines": [ { "bbox": [ 299, 740, 313, 755 ], "spans": [ { "bbox": [ 299, 740, 313, 755 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 14 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 72, 505, 182 ], "lines": [ { "bbox": [ 105, 72, 505, 86 ], "spans": [ { "bbox": [ 105, 72, 329, 86 ], "score": 1.0, "content": "Specifically, we use the discrete-time model trained by", "type": "text" }, { "bbox": [ 329, 73, 356, 84 ], "score": 0.9, "content": "L _ { \\mathrm { s i m p l e } }", "type": "inline_equation" }, { "bbox": [ 357, 72, 505, 86 ], "score": 1.0, "content": "in [2] on the CIFAR-10 dataset with", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 506, 96 ], "spans": [ { "bbox": [ 105, 83, 506, 96 ], "score": 1.0, "content": "linear noise schedule; the discrete-time model in [19] on CelebA 64x64 [39] with linear noise", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 93, 506, 109 ], "spans": [ { "bbox": [ 105, 93, 292, 109 ], "score": 1.0, "content": "schedule; the discrete-time model trained by", "type": "text" }, { "bbox": [ 292, 95, 319, 106 ], "score": 0.89, "content": "L _ { \\mathrm { h y b r i d } }", "type": "inline_equation" }, { "bbox": [ 320, 93, 506, 109 ], "score": 1.0, "content": "in [16] on ImageNet 64x64 [26] with cosine", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 105, 505, 118 ], "spans": [ { "bbox": [ 106, 105, 505, 118 ], "score": 1.0, "content": "noise schedule; the discrete-time model with classifier guidance in [4] on ImageNet 128x128 [26]", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 117, 505, 128 ], "spans": [ { "bbox": [ 106, 117, 426, 128 ], "score": 1.0, "content": "with linear noise schedule; the discrete-time model in [4] on LSUN bedroom", "type": "text" }, { "bbox": [ 426, 117, 464, 127 ], "score": 0.29, "content": "2 5 6 \\times 2 5 6", "type": "inline_equation" }, { "bbox": [ 464, 117, 505, 128 ], "score": 1.0, "content": "[40] with", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 127, 506, 139 ], "spans": [ { "bbox": [ 105, 127, 506, 139 ], "score": 1.0, "content": "linear noise schedule. For the models trained on ImageNet, we only use their “mean” model and", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 138, 506, 150 ], "spans": [ { "bbox": [ 106, 138, 506, 150 ], "score": 1.0, "content": "omit the “variance” model. As shown in Fig. 2, on all datasets, DPM-Solver can obtain reasonable", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 149, 505, 161 ], "spans": [ { "bbox": [ 105, 149, 505, 161 ], "score": 1.0, "content": "samples within 12 steps (FID 4.65 on CIFAR-10, FID 3.71 on CelebA 64x64 and FID 19.97 on", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 159, 506, 172 ], "spans": [ { "bbox": [ 106, 159, 268, 172 ], "score": 1.0, "content": "ImageNet 64x64, FID 4.08 on ImageNet", "type": "text" }, { "bbox": [ 268, 160, 304, 171 ], "score": 0.33, "content": "1 2 8 \\mathbf { x } 1 2 8 _ { \\rho }", "type": "inline_equation" }, { "bbox": [ 304, 159, 345, 172 ], "score": 1.0, "content": "), which is", "type": "text" }, { "bbox": [ 345, 160, 383, 171 ], "score": 0.92, "content": "4 \\sim 1 6 \\times", "type": "inline_equation" }, { "bbox": [ 383, 159, 506, 172 ], "score": 1.0, "content": "faster than the previous fastest", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 170, 496, 185 ], "spans": [ { "bbox": [ 105, 170, 496, 185 ], "score": 1.0, "content": "training-free sampler. DPM-Solver even outperforms GGDM, which requires additional training.", "type": "text" } ], "index": 9 } ], "index": 4.5, "bbox_fs": [ 105, 72, 506, 185 ] }, { "type": "title", "bbox": [ 107, 199, 187, 212 ], "lines": [ { "bbox": [ 104, 197, 190, 214 ], "spans": [ { "bbox": [ 104, 197, 190, 214 ], "score": 1.0, "content": "6 Conclusions", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 224, 505, 345 ], "lines": [ { "bbox": [ 105, 223, 506, 237 ], "spans": [ { "bbox": [ 105, 223, 506, 237 ], "score": 1.0, "content": "We tackle the problem of fast and training-free sampling from DPMs. We propose DPM-Solver,", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 235, 506, 248 ], "spans": [ { "bbox": [ 105, 235, 506, 248 ], "score": 1.0, "content": "a fast dedicated training-free solver of diffusion ODEs for fast sampling of DPMs in around 10", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 246, 505, 258 ], "spans": [ { "bbox": [ 105, 246, 505, 258 ], "score": 1.0, "content": "steps of function evaluations. DPM-Solver leverages the semi-linearity of diffusion ODEs and", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 505, 269 ], "score": 1.0, "content": "it directly approximates a simplified formulation of exact solutions of diffusion ODEs, which", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 268, 506, 280 ], "spans": [ { "bbox": [ 106, 268, 506, 280 ], "score": 1.0, "content": "consists of an exponentially weighted integral of the noise prediction model. Inspired by numerical", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 279, 506, 291 ], "spans": [ { "bbox": [ 105, 279, 506, 291 ], "score": 1.0, "content": "methods for exponential integrators, we propose first-order, second-order and third-order DPM-", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 290, 505, 302 ], "spans": [ { "bbox": [ 105, 290, 505, 302 ], "score": 1.0, "content": "Solver to approximate the exponentially weighted integral of noise prediction models with theoretical", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 301, 505, 313 ], "spans": [ { "bbox": [ 105, 301, 505, 313 ], "score": 1.0, "content": "convergence guarantee. We propose both handcrafted and adaptive step size schedule, and apply", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 311, 506, 324 ], "spans": [ { "bbox": [ 105, 311, 506, 324 ], "score": 1.0, "content": "DPM-Solver for both continuous-time and discrete-time DPMs. Our experimental results show that", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 322, 506, 335 ], "spans": [ { "bbox": [ 105, 322, 506, 335 ], "score": 1.0, "content": "DPM-Solver can generate high-quality samples in around 10 function evaluations on various datasets,", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 333, 506, 346 ], "spans": [ { "bbox": [ 105, 333, 180, 346 ], "score": 1.0, "content": "and it can achieve", "type": "text" }, { "bbox": [ 180, 334, 217, 344 ], "score": 0.9, "content": "4 \\sim 1 6 \\times", "type": "inline_equation" }, { "bbox": [ 218, 333, 506, 346 ], "score": 1.0, "content": "speedup compared with previous state-of-the-art training-free samplers.", "type": "text" } ], "index": 21 } ], "index": 16, "bbox_fs": [ 105, 223, 506, 346 ] }, { "type": "text", "bbox": [ 107, 349, 505, 416 ], "lines": [ { "bbox": [ 105, 349, 506, 362 ], "spans": [ { "bbox": [ 105, 349, 506, 362 ], "score": 1.0, "content": "Limitations and broader impact Despite the promising speedup performance, DPM-Solver is", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 361, 505, 372 ], "spans": [ { "bbox": [ 106, 361, 505, 372 ], "score": 1.0, "content": "designed for fast sampling, which may be not suitable for accelerating the likelihood evaluations of", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 371, 505, 384 ], "spans": [ { "bbox": [ 105, 371, 505, 384 ], "score": 1.0, "content": "DPMs. Besides, compared to the commonly-used GANs, diffusion models with DPM-Solver are still", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 381, 506, 396 ], "spans": [ { "bbox": [ 105, 381, 506, 396 ], "score": 1.0, "content": "not fast enough for real-time applications. In addition, like other deep generative models, DPMs may", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 393, 505, 406 ], "spans": [ { "bbox": [ 105, 393, 505, 406 ], "score": 1.0, "content": "be used to generate adverse fake contents, and the proposed solver may further amplify the potential", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 404, 410, 417 ], "spans": [ { "bbox": [ 105, 404, 410, 417 ], "score": 1.0, "content": "undesirable influence of deep generative models for malicious applications.", "type": "text" } ], "index": 27 } ], "index": 24.5, "bbox_fs": [ 105, 349, 506, 417 ] }, { "type": "title", "bbox": [ 108, 432, 207, 445 ], "lines": [ { "bbox": [ 106, 431, 208, 448 ], "spans": [ { "bbox": [ 106, 431, 208, 448 ], "score": 1.0, "content": "Acknowledgements", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 107, 457, 506, 545 ], "lines": [ { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "This work was supported by National Key Research and Development Project of China (No.", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 469, 506, 481 ], "spans": [ { "bbox": [ 105, 469, 506, 481 ], "score": 1.0, "content": "2021ZD0110502); NSF of China Projects (Nos. 62061136001, 61620106010, 62076145, U19B2034,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 479, 506, 493 ], "spans": [ { "bbox": [ 105, 479, 506, 493 ], "score": 1.0, "content": "U1811461, U19A2081, 6197222, 62106120); Beijing NSF Project (No. JQ19016); Beijing Outstand-", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 491, 505, 503 ], "spans": [ { "bbox": [ 106, 491, 505, 503 ], "score": 1.0, "content": "ing Young Scientist Program NO. BJJWZYJH012019100020098; a grant from Tsinghua Institute", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 501, 505, 513 ], "spans": [ { "bbox": [ 106, 501, 505, 513 ], "score": 1.0, "content": "for Guo Qiang; the NVIDIA NVAIL Program with GPU/DGX Acceleration; the High Performance", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 512, 506, 525 ], "spans": [ { "bbox": [ 105, 512, 506, 525 ], "score": 1.0, "content": "Computing Center, Tsinghua University; the Fundamental Research Funds for the Central Universi-", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 522, 505, 537 ], "spans": [ { "bbox": [ 105, 522, 505, 537 ], "score": 1.0, "content": "ties, and the Research Funds of Renmin University of China (22XNKJ13). J.Z is also supported by", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 534, 181, 546 ], "spans": [ { "bbox": [ 106, 534, 181, 546 ], "score": 1.0, "content": "the XPlorer Prize.", "type": "text" } ], "index": 36 } ], "index": 32.5, "bbox_fs": [ 105, 457, 506, 546 ] }, { "type": "title", "bbox": [ 107, 563, 163, 575 ], "lines": [ { "bbox": [ 106, 561, 165, 577 ], "spans": [ { "bbox": [ 106, 561, 165, 577 ], "score": 1.0, "content": "References", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "list", "bbox": [ 110, 582, 506, 722 ], "lines": [ { "bbox": [ 110, 581, 505, 598 ], "spans": [ { "bbox": [ 110, 581, 505, 598 ], "score": 1.0, "content": "[1] J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning", "type": "text" } ], "index": 38, "is_list_start_line": true }, { "bbox": [ 126, 592, 506, 609 ], "spans": [ { "bbox": [ 126, 592, 506, 609 ], "score": 1.0, "content": "using nonequilibrium thermodynamics,” in International Conference on Machine Learning.", "type": "text" } ], "index": 39 }, { "bbox": [ 126, 604, 250, 619 ], "spans": [ { "bbox": [ 126, 604, 250, 619 ], "score": 1.0, "content": "PMLR, 2015, pp. 2256–2265.", "type": "text" } ], "index": 40, "is_list_end_line": true }, { "bbox": [ 111, 625, 506, 639 ], "spans": [ { "bbox": [ 111, 625, 506, 639 ], "score": 1.0, "content": "[2] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” in Advances in Neural", "type": "text" } ], "index": 41, "is_list_start_line": true }, { "bbox": [ 127, 636, 383, 650 ], "spans": [ { "bbox": [ 127, 636, 383, 650 ], "score": 1.0, "content": "Information Processing Systems, vol. 33, 2020, pp. 6840–6851.", "type": "text" } ], "index": 42, "is_list_end_line": true }, { "bbox": [ 110, 657, 505, 671 ], "spans": [ { "bbox": [ 110, 657, 505, 671 ], "score": 1.0, "content": "[3] Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, “Score-based", "type": "text" } ], "index": 43, "is_list_start_line": true }, { "bbox": [ 127, 668, 505, 682 ], "spans": [ { "bbox": [ 127, 668, 505, 682 ], "score": 1.0, "content": "generative modeling through stochastic differential equations,” in International Conference on", "type": "text" } ], "index": 44 }, { "bbox": [ 127, 679, 260, 693 ], "spans": [ { "bbox": [ 127, 679, 260, 693 ], "score": 1.0, "content": "Learning Representations, 2021.", "type": "text" } ], "index": 45, "is_list_end_line": true }, { "bbox": [ 111, 699, 505, 714 ], "spans": [ { "bbox": [ 111, 699, 505, 714 ], "score": 1.0, "content": "[4] P. Dhariwal and A. Q. Nichol, “Diffusion models beat GANs on image synthesis,” in Advances", "type": "text" } ], "index": 46, "is_list_start_line": true }, { "bbox": [ 126, 711, 424, 724 ], "spans": [ { "bbox": [ 126, 711, 424, 724 ], "score": 1.0, "content": "in Neural Information Processing Systems, vol. 34, 2021, pp. 8780–8794.", "type": "text" } ], "index": 47, "is_list_end_line": true }, { "bbox": [ 109, 71, 507, 88 ], "spans": [ { "bbox": [ 109, 71, 507, 88 ], "score": 1.0, "content": "[5] C. Meng, Y. Song, J. Song, J. Wu, J.-Y. Zhu, and S. Ermon, “SDEdit: Image synthesis and editing", "type": "text", "cross_page": true } ], "index": 0, "is_list_start_line": true }, { "bbox": [ 124, 82, 507, 99 ], "spans": [ { "bbox": [ 124, 82, 507, 99 ], "score": 1.0, "content": "with stochastic differential equations,” in International Conference on Learning Representations,", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 127, 93, 154, 108 ], "spans": [ { "bbox": [ 127, 93, 154, 108 ], "score": 1.0, "content": "2022.", "type": "text", "cross_page": true } ], "index": 2, "is_list_end_line": true }, { "bbox": [ 110, 114, 505, 128 ], "spans": [ { "bbox": [ 110, 114, 505, 128 ], "score": 1.0, "content": "[6] J. Ho, T. Salimans, A. Gritsenko, W. Chan, M. Norouzi, and D. J. Fleet, “Video diffusion", "type": "text", "cross_page": true } ], "index": 3, "is_list_start_line": true }, { "bbox": [ 127, 124, 327, 138 ], "spans": [ { "bbox": [ 127, 124, 327, 138 ], "score": 1.0, "content": "models,” arXiv preprint arXiv:2204.03458, 2022.", "type": "text", "cross_page": true } ], "index": 4, "is_list_end_line": true }, { "bbox": [ 109, 143, 505, 160 ], "spans": [ { "bbox": [ 109, 143, 505, 160 ], "score": 1.0, "content": "[7] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen, “Hierarchical text-conditional image", "type": "text", "cross_page": true } ], "index": 5, "is_list_start_line": true }, { "bbox": [ 125, 155, 414, 171 ], "spans": [ { "bbox": [ 125, 155, 414, 171 ], "score": 1.0, "content": "generation with CLIP latents,” arXiv preprint arXiv:2204.06125, 2022.", "type": "text", "cross_page": true } ], "index": 6, "is_list_end_line": true }, { "bbox": [ 109, 173, 506, 190 ], "spans": [ { "bbox": [ 109, 173, 506, 190 ], "score": 1.0, "content": "[8] N. Chen, Y. Zhang, H. Zen, R. J. Weiss, M. Norouzi, and W. Chan, “Wavegrad: Estimating", "type": "text", "cross_page": true } ], "index": 7, "is_list_start_line": true }, { "bbox": [ 125, 185, 507, 201 ], "spans": [ { "bbox": [ 125, 185, 507, 201 ], "score": 1.0, "content": "gradients for waveform generation,” in International Conference on Learning Representations,", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 127, 196, 155, 211 ], "spans": [ { "bbox": [ 127, 196, 155, 211 ], "score": 1.0, "content": "2021.", "type": "text", "cross_page": true } ], "index": 9, "is_list_end_line": true }, { "bbox": [ 110, 217, 505, 231 ], "spans": [ { "bbox": [ 110, 217, 505, 231 ], "score": 1.0, "content": "[9] N. Chen, Y. Zhang, H. Zen, R. J. Weiss, M. Norouzi, N. Dehak, and W. Chan, “Wavegrad", "type": "text", "cross_page": true } ], "index": 10, "is_list_start_line": true }, { "bbox": [ 127, 228, 506, 242 ], "spans": [ { "bbox": [ 127, 228, 506, 242 ], "score": 1.0, "content": "2: Iterative refinement for text-to-speech synthesis,” in International Speech Communication", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 126, 239, 270, 253 ], "spans": [ { "bbox": [ 126, 239, 270, 253 ], "score": 1.0, "content": "Association, 2021, pp. 3765–3769.", "type": "text", "cross_page": true } ], "index": 12, "is_list_end_line": true }, { "bbox": [ 105, 258, 506, 272 ], "spans": [ { "bbox": [ 105, 258, 506, 272 ], "score": 1.0, "content": "[10] D. P. Kingma, T. Salimans, B. Poole, and J. Ho, “Variational diffusion models,” in Advances in", "type": "text", "cross_page": true } ], "index": 13, "is_list_start_line": true }, { "bbox": [ 127, 270, 316, 282 ], "spans": [ { "bbox": [ 127, 270, 316, 282 ], "score": 1.0, "content": "Neural Information Processing Systems, 2021.", "type": "text", "cross_page": true } ], "index": 14, "is_list_end_line": true }, { "bbox": [ 104, 288, 507, 303 ], "spans": [ { "bbox": [ 104, 288, 507, 303 ], "score": 1.0, "content": "[11] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. C. Courville,", "type": "text", "cross_page": true } ], "index": 15, "is_list_start_line": true }, { "bbox": [ 126, 299, 505, 315 ], "spans": [ { "bbox": [ 126, 299, 505, 315 ], "score": 1.0, "content": "and Y. Bengio, “Generative adversarial nets,” in Advances in Neural Information Processing", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 127, 311, 288, 325 ], "spans": [ { "bbox": [ 127, 311, 288, 325 ], "score": 1.0, "content": "Systems, vol. 27, 2014, pp. 2672–2680.", "type": "text", "cross_page": true } ], "index": 17, "is_list_end_line": true }, { "bbox": [ 106, 331, 505, 343 ], "spans": [ { "bbox": [ 106, 331, 505, 343 ], "score": 1.0, "content": "[12] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” in International Conference", "type": "text", "cross_page": true } ], "index": 18, "is_list_start_line": true }, { "bbox": [ 126, 341, 274, 354 ], "spans": [ { "bbox": [ 126, 341, 274, 354 ], "score": 1.0, "content": "on Learning Representations, 2014.", "type": "text", "cross_page": true } ], "index": 19, "is_list_end_line": true }, { "bbox": [ 105, 361, 506, 375 ], "spans": [ { "bbox": [ 105, 361, 506, 375 ], "score": 1.0, "content": "[13] T. Salimans and J. Ho, “Progressive distillation for fast sampling of diffusion models,” in", "type": "text", "cross_page": true } ], "index": 20, "is_list_start_line": true }, { "bbox": [ 126, 372, 375, 386 ], "spans": [ { "bbox": [ 126, 372, 375, 386 ], "score": 1.0, "content": "International Conference on Learning Representations, 2022.", "type": "text", "cross_page": true } ], "index": 21, "is_list_end_line": true }, { "bbox": [ 105, 391, 506, 405 ], "spans": [ { "bbox": [ 105, 391, 506, 405 ], "score": 1.0, "content": "[14] E. Luhman and T. Luhman, “Knowledge distillation in iterative generative models for improved", "type": "text", "cross_page": true } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 127, 402, 362, 416 ], "spans": [ { "bbox": [ 127, 402, 362, 416 ], "score": 1.0, "content": "sampling speed,” arXiv preprint arXiv:2101.02388, 2021.", "type": "text", "cross_page": true } ], "index": 23, "is_list_end_line": true }, { "bbox": [ 104, 420, 507, 437 ], "spans": [ { "bbox": [ 104, 420, 507, 437 ], "score": 1.0, "content": "[15] R. San-Roman, E. Nachmani, and L. Wolf, “Noise estimation for generative diffusion models,”", "type": "text", "cross_page": true } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 127, 434, 290, 446 ], "spans": [ { "bbox": [ 127, 434, 290, 446 ], "score": 1.0, "content": "arXiv preprint arXiv:2104.02600, 2021.", "type": "text", "cross_page": true } ], "index": 25, "is_list_end_line": true }, { "bbox": [ 106, 452, 507, 466 ], "spans": [ { "bbox": [ 106, 452, 507, 466 ], "score": 1.0, "content": "[16] A. Q. Nichol and P. Dhariwal, “Improved denoising diffusion probabilistic models,” in Interna-", "type": "text", "cross_page": true } ], "index": 26, "is_list_start_line": true }, { "bbox": [ 126, 464, 424, 478 ], "spans": [ { "bbox": [ 126, 464, 424, 478 ], "score": 1.0, "content": "tional Conference on Machine Learning. PMLR, 2021, pp. 8162–8171.", "type": "text", "cross_page": true } ], "index": 27, "is_list_end_line": true }, { "bbox": [ 106, 484, 505, 496 ], "spans": [ { "bbox": [ 106, 484, 505, 496 ], "score": 1.0, "content": "[17] M. W. Lam, J. Wang, R. Huang, D. Su, and D. Yu, “Bilateral denoising diffusion models,” arXiv", "type": "text", "cross_page": true } ], "index": 28, "is_list_start_line": true }, { "bbox": [ 124, 494, 268, 509 ], "spans": [ { "bbox": [ 124, 494, 268, 509 ], "score": 1.0, "content": "preprint arXiv:2108.11514, 2021.", "type": "text", "cross_page": true } ], "index": 29, "is_list_end_line": true }, { "bbox": [ 106, 514, 506, 528 ], "spans": [ { "bbox": [ 106, 514, 506, 528 ], "score": 1.0, "content": "[18] D. Watson, W. Chan, J. Ho, and M. Norouzi, “Learning fast samplers for diffusion models by dif-", "type": "text", "cross_page": true } ], "index": 30, "is_list_start_line": true }, { "bbox": [ 127, 525, 507, 539 ], "spans": [ { "bbox": [ 127, 525, 507, 539 ], "score": 1.0, "content": "ferentiating through sample quality,” in International Conference on Learning Representations,", "type": "text", "cross_page": true } ], "index": 31 }, { "bbox": [ 126, 534, 155, 549 ], "spans": [ { "bbox": [ 126, 534, 155, 549 ], "score": 1.0, "content": "2022.", "type": "text", "cross_page": true } ], "index": 32, "is_list_end_line": true }, { "bbox": [ 106, 555, 505, 569 ], "spans": [ { "bbox": [ 106, 555, 505, 569 ], "score": 1.0, "content": "[19] J. Song, C. Meng, and S. Ermon, “Denoising diffusion implicit models,” in International", "type": "text", "cross_page": true } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 127, 567, 321, 580 ], "spans": [ { "bbox": [ 127, 567, 321, 580 ], "score": 1.0, "content": "Conference on Learning Representations, 2021.", "type": "text", "cross_page": true } ], "index": 34, "is_list_end_line": true }, { "bbox": [ 106, 586, 506, 600 ], "spans": [ { "bbox": [ 106, 586, 506, 600 ], "score": 1.0, "content": "[20] A. Jolicoeur-Martineau, K. Li, R. Piché-Taillefer, T. Kachman, and I. Mitliagkas, “Gotta go fast", "type": "text", "cross_page": true } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 126, 597, 486, 610 ], "spans": [ { "bbox": [ 126, 597, 486, 610 ], "score": 1.0, "content": "when generating data with score-based models,” arXiv preprint arXiv:2105.14080, 2021.", "type": "text", "cross_page": true } ], "index": 36, "is_list_end_line": true }, { "bbox": [ 106, 617, 506, 631 ], "spans": [ { "bbox": [ 106, 617, 506, 631 ], "score": 1.0, "content": "[21] F. Bao, C. Li, J. Zhu, and B. Zhang, “Analytic-DPM: An analytic estimate of the optimal", "type": "text", "cross_page": true } ], "index": 37, "is_list_start_line": true }, { "bbox": [ 124, 626, 506, 643 ], "spans": [ { "bbox": [ 124, 626, 506, 643 ], "score": 1.0, "content": "reverse variance in diffusion probabilistic models,” in International Conference on Learning", "type": "text", "cross_page": true } ], "index": 38 }, { "bbox": [ 126, 638, 223, 652 ], "spans": [ { "bbox": [ 126, 638, 223, 652 ], "score": 1.0, "content": "Representations, 2022.", "type": "text", "cross_page": true } ], "index": 39, "is_list_end_line": true }, { "bbox": [ 106, 657, 506, 671 ], "spans": [ { "bbox": [ 106, 657, 506, 671 ], "score": 1.0, "content": "[22] L. Liu, Y. Ren, Z. Lin, and Z. Zhao, “Pseudo numerical methods for diffusion models on", "type": "text", "cross_page": true } ], "index": 40, "is_list_start_line": true }, { "bbox": [ 127, 669, 435, 683 ], "spans": [ { "bbox": [ 127, 669, 435, 683 ], "score": 1.0, "content": "manifolds,” in International Conference on Learning Representations, 2022.", "type": "text", "cross_page": true } ], "index": 41, "is_list_end_line": true }, { "bbox": [ 105, 687, 507, 704 ], "spans": [ { "bbox": [ 105, 687, 507, 704 ], "score": 1.0, "content": "[23] V. Popov, I. Vovk, V. Gogoryan, T. Sadekova, M. Kudinov, and J. Wei, “Diffusion-based voice", "type": "text", "cross_page": true } ], "index": 42, "is_list_start_line": true }, { "bbox": [ 125, 698, 507, 715 ], "spans": [ { "bbox": [ 125, 698, 507, 715 ], "score": 1.0, "content": "conversion with fast maximum likelihood sampling scheme,” in International Conference on", "type": "text", "cross_page": true } ], "index": 43 }, { "bbox": [ 126, 711, 261, 725 ], "spans": [ { "bbox": [ 126, 711, 261, 725 ], "score": 1.0, "content": "Learning Representations, 2022.", "type": "text", "cross_page": true } ], "index": 44, "is_list_end_line": true }, { "bbox": [ 105, 72, 506, 88 ], "spans": [ { "bbox": [ 105, 72, 506, 88 ], "score": 1.0, "content": "[24] H. Tachibana, M. Go, M. Inahara, Y. Katayama, and Y. Watanabe, “Itô-Taylor sampling", "type": "text", "cross_page": true } ], "index": 0, "is_list_start_line": true }, { "bbox": [ 126, 83, 506, 99 ], "spans": [ { "bbox": [ 126, 83, 506, 99 ], "score": 1.0, "content": "scheme for denoising diffusion probabilistic models using ideal derivatives,” arXiv preprint", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 127, 94, 232, 108 ], "spans": [ { "bbox": [ 127, 94, 232, 108 ], "score": 1.0, "content": "arXiv:2112.13339, 2021.", "type": "text", "cross_page": true } ], "index": 2, "is_list_end_line": true }, { "bbox": [ 104, 112, 508, 132 ], "spans": [ { "bbox": [ 104, 112, 508, 132 ], "score": 1.0, "content": "[25] M. Hochbruck and A. Ostermann, “Exponential integrators,” Acta Numerica, vol. 19, pp.", "type": "text", "cross_page": true } ], "index": 3, "is_list_start_line": true }, { "bbox": [ 127, 126, 193, 140 ], "spans": [ { "bbox": [ 127, 126, 193, 140 ], "score": 1.0, "content": "209–286, 2010.", "type": "text", "cross_page": true } ], "index": 4, "is_list_end_line": true }, { "bbox": [ 106, 146, 506, 160 ], "spans": [ { "bbox": [ 106, 146, 506, 160 ], "score": 1.0, "content": "[26] J. Deng, W. Dong, R. Socher, L. Li, K. Li, and L. Fei-Fei, “ImageNet: A large-scale hierarchical", "type": "text", "cross_page": true } ], "index": 5, "is_list_start_line": true }, { "bbox": [ 127, 157, 506, 171 ], "spans": [ { "bbox": [ 127, 157, 506, 171 ], "score": 1.0, "content": "image database,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEE,", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 127, 168, 209, 182 ], "spans": [ { "bbox": [ 127, 168, 209, 182 ], "score": 1.0, "content": "2009, pp. 248–255.", "type": "text", "cross_page": true } ], "index": 7, "is_list_end_line": true }, { "bbox": [ 105, 188, 507, 204 ], "spans": [ { "bbox": [ 105, 188, 507, 204 ], "score": 1.0, "content": "[27] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer,", "type": "text", "cross_page": true } ], "index": 8, "is_list_start_line": true }, { "bbox": [ 126, 199, 154, 214 ], "spans": [ { "bbox": [ 126, 199, 154, 214 ], "score": 1.0, "content": "1992.", "type": "text", "cross_page": true } ], "index": 9, "is_list_end_line": true }, { "bbox": [ 106, 221, 507, 234 ], "spans": [ { "bbox": [ 106, 221, 507, 234 ], "score": 1.0, "content": "[28] J. R. Dormand and P. J. Prince, “A family of embedded Runge-Kutta formulae,” Journal of", "type": "text", "cross_page": true } ], "index": 10, "is_list_start_line": true }, { "bbox": [ 125, 231, 416, 246 ], "spans": [ { "bbox": [ 125, 231, 416, 246 ], "score": 1.0, "content": "computational and applied mathematics, vol. 6, no. 1, pp. 19–26, 1980.", "type": "text", "cross_page": true } ], "index": 11, "is_list_end_line": true }, { "bbox": [ 106, 252, 489, 266 ], "spans": [ { "bbox": [ 106, 252, 489, 266 ], "score": 1.0, "content": "[29] A. Krizhevsky, “Learning multiple layers of features from tiny images,” Tech. Rep., 2009.", "type": "text", "cross_page": true } ], "index": 12, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 105, 272, 507, 286 ], "spans": [ { "bbox": [ 105, 272, 507, 286 ], "score": 1.0, "content": "[30] K. Atkinson, W. Han, and D. E. Stewart, Numerical solution of ordinary differential equations.", "type": "text", "cross_page": true } ], "index": 13, "is_list_start_line": true }, { "bbox": [ 127, 284, 273, 296 ], "spans": [ { "bbox": [ 127, 284, 273, 296 ], "score": 1.0, "content": "John Wiley & Sons, 2011, vol. 108.", "type": "text", "cross_page": true } ], "index": 14, "is_list_end_line": true }, { "bbox": [ 104, 303, 506, 318 ], "spans": [ { "bbox": [ 104, 303, 506, 318 ], "score": 1.0, "content": "[31] M. Hochbruck and A. Ostermann, “Explicit exponential Runge-Kutta methods for semilinear", "type": "text", "cross_page": true } ], "index": 15, "is_list_start_line": true }, { "bbox": [ 125, 314, 508, 330 ], "spans": [ { "bbox": [ 125, 314, 508, 330 ], "score": 1.0, "content": "parabolic problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 3, pp. 1069–1090, 2005.", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 106, 335, 506, 349 ], "spans": [ { "bbox": [ 106, 335, 506, 349 ], "score": 1.0, "content": "[32] V. T. Luan, “Efficient exponential Runge-Kutta methods of high order: Construction and", "type": "text", "cross_page": true } ], "index": 17, "is_list_start_line": true }, { "bbox": [ 127, 347, 455, 361 ], "spans": [ { "bbox": [ 127, 347, 455, 361 ], "score": 1.0, "content": "implementation,” BIT Numerical Mathematics, vol. 61, no. 2, pp. 535–560, 2021.", "type": "text", "cross_page": true } ], "index": 18, "is_list_end_line": true }, { "bbox": [ 106, 367, 506, 381 ], "spans": [ { "bbox": [ 106, 367, 506, 381 ], "score": 1.0, "content": "[33] F. Bao, C. Li, J. Sun, J. Zhu, and B. Zhang, “Estimating the optimal covariance with imperfect", "type": "text", "cross_page": true } ], "index": 19, "is_list_start_line": true }, { "bbox": [ 126, 378, 452, 392 ], "spans": [ { "bbox": [ 126, 378, 452, 392 ], "score": 1.0, "content": "mean in diffusion probabilistic models,” arXiv preprint arXiv:2206.07309, 2022.", "type": "text", "cross_page": true } ], "index": 20, "is_list_end_line": true }, { "bbox": [ 105, 397, 506, 413 ], "spans": [ { "bbox": [ 105, 397, 506, 413 ], "score": 1.0, "content": "[34] A. Vahdat, K. Kreis, and J. Kautz, “Score-based generative modeling in latent space,” in", "type": "text", "cross_page": true } ], "index": 21, "is_list_start_line": true }, { "bbox": [ 124, 408, 478, 424 ], "spans": [ { "bbox": [ 124, 408, 478, 424 ], "score": 1.0, "content": "Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 11 287–11 302.", "type": "text", "cross_page": true } ], "index": 22, "is_list_end_line": true }, { "bbox": [ 106, 430, 506, 444 ], "spans": [ { "bbox": [ 106, 430, 506, 444 ], "score": 1.0, "content": "[35] T. Dockhorn, A. Vahdat, and K. Kreis, “Score-based generative modeling with critically-damped", "type": "text", "cross_page": true } ], "index": 23, "is_list_start_line": true }, { "bbox": [ 127, 441, 470, 455 ], "spans": [ { "bbox": [ 127, 441, 470, 455 ], "score": 1.0, "content": "Langevin diffusion,” in International Conference on Learning Representations, 2022.", "type": "text", "cross_page": true } ], "index": 24, "is_list_end_line": true }, { "bbox": [ 104, 460, 506, 477 ], "spans": [ { "bbox": [ 104, 460, 506, 477 ], "score": 1.0, "content": "[36] Z. Xiao, K. Kreis, and A. Vahdat, “Tackling the generative learning trilemma with denoising", "type": "text", "cross_page": true } ], "index": 25, "is_list_start_line": true }, { "bbox": [ 127, 474, 458, 485 ], "spans": [ { "bbox": [ 127, 474, 458, 485 ], "score": 1.0, "content": "diffusion GANs,” in International Conference on Learning Representations, 2022.", "type": "text", "cross_page": true } ], "index": 26, "is_list_end_line": true }, { "bbox": [ 104, 492, 506, 507 ], "spans": [ { "bbox": [ 104, 492, 506, 507 ], "score": 1.0, "content": "[37] M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter, “GANs trained by a two", "type": "text", "cross_page": true } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 127, 504, 506, 518 ], "spans": [ { "bbox": [ 127, 504, 506, 518 ], "score": 1.0, "content": "time-scale update rule converge to a local Nash equilibrium,” in Advances in Neural Information", "type": "text", "cross_page": true } ], "index": 28 }, { "bbox": [ 126, 514, 508, 530 ], "spans": [ { "bbox": [ 126, 514, 508, 530 ], "score": 1.0, "content": "Processing Systems, I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N.", "type": "text", "cross_page": true } ], "index": 29 }, { "bbox": [ 127, 525, 397, 540 ], "spans": [ { "bbox": [ 127, 525, 397, 540 ], "score": 1.0, "content": "Vishwanathan, and R. Garnett, Eds., vol. 30, 2017, pp. 6626–6637.", "type": "text", "cross_page": true } ], "index": 30, "is_list_end_line": true }, { "bbox": [ 106, 546, 506, 560 ], "spans": [ { "bbox": [ 106, 546, 506, 560 ], "score": 1.0, "content": "[38] Z. Kong and W. Ping, “On fast sampling of diffusion probabilistic models,” arXiv preprint", "type": "text", "cross_page": true } ], "index": 31, "is_list_start_line": true }, { "bbox": [ 127, 559, 231, 570 ], "spans": [ { "bbox": [ 127, 559, 231, 570 ], "score": 1.0, "content": "arXiv:2106.00132, 2021.", "type": "text", "cross_page": true } ], "index": 32, "is_list_end_line": true }, { "bbox": [ 104, 576, 506, 593 ], "spans": [ { "bbox": [ 104, 576, 506, 593 ], "score": 1.0, "content": "[39] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings", "type": "text", "cross_page": true } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 125, 588, 455, 603 ], "spans": [ { "bbox": [ 125, 588, 455, 603 ], "score": 1.0, "content": "of the IEEE International Conference on Computer Vision, 2015, pp. 3730–3738.", "type": "text", "cross_page": true } ], "index": 34, "is_list_end_line": true }, { "bbox": [ 105, 608, 506, 622 ], "spans": [ { "bbox": [ 105, 608, 506, 622 ], "score": 1.0, "content": "[40] F. Yu, A. Seff, Y. Zhang, S. Song, T. Funkhouser, and J. Xiao, “LSUN: Construction of", "type": "text", "cross_page": true } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 125, 619, 506, 636 ], "spans": [ { "bbox": [ 125, 619, 506, 636 ], "score": 1.0, "content": "a large-scale image dataset using deep learning with humans in the loop,” arXiv preprint", "type": "text", "cross_page": true } ], "index": 36 }, { "bbox": [ 127, 631, 231, 645 ], "spans": [ { "bbox": [ 127, 631, 231, 645 ], "score": 1.0, "content": "arXiv:1506.03365, 2015.", "type": "text", "cross_page": true } ], "index": 37, "is_list_end_line": true }, { "bbox": [ 105, 652, 506, 665 ], "spans": [ { "bbox": [ 105, 652, 506, 665 ], "score": 1.0, "content": "[41] Y. Song, C. Durkan, I. Murray, and S. Ermon, “Maximum likelihood training of score-based", "type": "text", "cross_page": true } ], "index": 38, "is_list_start_line": true }, { "bbox": [ 125, 660, 507, 679 ], "spans": [ { "bbox": [ 125, 660, 507, 679 ], "score": 1.0, "content": "diffusion models,” in Advances in Neural Information Processing Systems, vol. 34, 2021, pp.", "type": "text", "cross_page": true } ], "index": 39 }, { "bbox": [ 127, 675, 178, 686 ], "spans": [ { "bbox": [ 127, 675, 178, 686 ], "score": 1.0, "content": "1415–1428.", "type": "text", "cross_page": true } ], "index": 40, "is_list_end_line": true }, { "bbox": [ 104, 692, 508, 710 ], "spans": [ { "bbox": [ 104, 692, 508, 710 ], "score": 1.0, "content": "[42] K. Yang, J. Yau, L. Fei-Fei, J. Deng, and O. Russakovsky, “A study of face obfuscation in", "type": "text", "cross_page": true } ], "index": 41, "is_list_start_line": true }, { "bbox": [ 126, 704, 340, 721 ], "spans": [ { "bbox": [ 126, 704, 340, 721 ], "score": 1.0, "content": "ImageNet,” arXiv preprint arXiv:2103.06191, 2021.", "type": "text", "cross_page": true } ], "index": 42, "is_list_end_line": true } ], "index": 42.5, "bbox_fs": [ 110, 581, 506, 724 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 52, 507, 726 ], "lines": [ { "bbox": [ 109, 71, 507, 88 ], "spans": [ { "bbox": [ 109, 71, 507, 88 ], "score": 1.0, "content": "[5] C. Meng, Y. Song, J. Song, J. Wu, J.-Y. Zhu, and S. Ermon, “SDEdit: Image synthesis and editing", "type": "text" } ], "index": 0 }, { "bbox": [ 124, 82, 507, 99 ], "spans": [ { "bbox": [ 124, 82, 507, 99 ], "score": 1.0, "content": "with stochastic differential equations,” in International Conference on Learning Representations,", "type": "text" } ], "index": 1 }, { "bbox": [ 127, 93, 154, 108 ], "spans": [ { "bbox": [ 127, 93, 154, 108 ], "score": 1.0, "content": "2022.", "type": "text" } ], "index": 2 }, { "bbox": [ 110, 114, 505, 128 ], "spans": [ { "bbox": [ 110, 114, 505, 128 ], "score": 1.0, "content": "[6] J. Ho, T. Salimans, A. Gritsenko, W. Chan, M. Norouzi, and D. J. Fleet, “Video diffusion", "type": "text" } ], "index": 3 }, { "bbox": [ 127, 124, 327, 138 ], "spans": [ { "bbox": [ 127, 124, 327, 138 ], "score": 1.0, "content": "models,” arXiv preprint arXiv:2204.03458, 2022.", "type": "text" } ], "index": 4 }, { "bbox": [ 109, 143, 505, 160 ], "spans": [ { "bbox": [ 109, 143, 505, 160 ], "score": 1.0, "content": "[7] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen, “Hierarchical text-conditional image", "type": "text" } ], "index": 5 }, { "bbox": [ 125, 155, 414, 171 ], "spans": [ { "bbox": [ 125, 155, 414, 171 ], "score": 1.0, "content": "generation with CLIP latents,” arXiv preprint arXiv:2204.06125, 2022.", "type": "text" } ], "index": 6 }, { "bbox": [ 109, 173, 506, 190 ], "spans": [ { "bbox": [ 109, 173, 506, 190 ], "score": 1.0, "content": "[8] N. Chen, Y. Zhang, H. Zen, R. J. Weiss, M. Norouzi, and W. Chan, “Wavegrad: Estimating", "type": "text" } ], "index": 7 }, { "bbox": [ 125, 185, 507, 201 ], "spans": [ { "bbox": [ 125, 185, 507, 201 ], "score": 1.0, "content": "gradients for waveform generation,” in International Conference on Learning Representations,", "type": "text" } ], "index": 8 }, { "bbox": [ 127, 196, 155, 211 ], "spans": [ { "bbox": [ 127, 196, 155, 211 ], "score": 1.0, "content": "2021.", "type": "text" } ], "index": 9 }, { "bbox": [ 110, 217, 505, 231 ], "spans": [ { "bbox": [ 110, 217, 505, 231 ], "score": 1.0, "content": "[9] N. Chen, Y. Zhang, H. Zen, R. J. Weiss, M. Norouzi, N. Dehak, and W. Chan, “Wavegrad", "type": "text" } ], "index": 10 }, { "bbox": [ 127, 228, 506, 242 ], "spans": [ { "bbox": [ 127, 228, 506, 242 ], "score": 1.0, "content": "2: Iterative refinement for text-to-speech synthesis,” in International Speech Communication", "type": "text" } ], "index": 11 }, { "bbox": [ 126, 239, 270, 253 ], "spans": [ { "bbox": [ 126, 239, 270, 253 ], "score": 1.0, "content": "Association, 2021, pp. 3765–3769.", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 258, 506, 272 ], "spans": [ { "bbox": [ 105, 258, 506, 272 ], "score": 1.0, "content": "[10] D. P. Kingma, T. Salimans, B. Poole, and J. Ho, “Variational diffusion models,” in Advances in", "type": "text" } ], "index": 13 }, { "bbox": [ 127, 270, 316, 282 ], "spans": [ { "bbox": [ 127, 270, 316, 282 ], "score": 1.0, "content": "Neural Information Processing Systems, 2021.", "type": "text" } ], "index": 14 }, { "bbox": [ 104, 288, 507, 303 ], "spans": [ { "bbox": [ 104, 288, 507, 303 ], "score": 1.0, "content": "[11] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. C. Courville,", "type": "text" } ], "index": 15 }, { "bbox": [ 126, 299, 505, 315 ], "spans": [ { "bbox": [ 126, 299, 505, 315 ], "score": 1.0, "content": "and Y. Bengio, “Generative adversarial nets,” in Advances in Neural Information Processing", "type": "text" } ], "index": 16 }, { "bbox": [ 127, 311, 288, 325 ], "spans": [ { "bbox": [ 127, 311, 288, 325 ], "score": 1.0, "content": "Systems, vol. 27, 2014, pp. 2672–2680.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 331, 505, 343 ], "spans": [ { "bbox": [ 106, 331, 505, 343 ], "score": 1.0, "content": "[12] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” in International Conference", "type": "text" } ], "index": 18 }, { "bbox": [ 126, 341, 274, 354 ], "spans": [ { "bbox": [ 126, 341, 274, 354 ], "score": 1.0, "content": "on Learning Representations, 2014.", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 361, 506, 375 ], "spans": [ { "bbox": [ 105, 361, 506, 375 ], "score": 1.0, "content": "[13] T. Salimans and J. Ho, “Progressive distillation for fast sampling of diffusion models,” in", "type": "text" } ], "index": 20 }, { "bbox": [ 126, 372, 375, 386 ], "spans": [ { "bbox": [ 126, 372, 375, 386 ], "score": 1.0, "content": "International Conference on Learning Representations, 2022.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 391, 506, 405 ], "spans": [ { "bbox": [ 105, 391, 506, 405 ], "score": 1.0, "content": "[14] E. Luhman and T. Luhman, “Knowledge distillation in iterative generative models for improved", "type": "text" } ], "index": 22 }, { "bbox": [ 127, 402, 362, 416 ], "spans": [ { "bbox": [ 127, 402, 362, 416 ], "score": 1.0, "content": "sampling speed,” arXiv preprint arXiv:2101.02388, 2021.", "type": "text" } ], "index": 23 }, { "bbox": [ 104, 420, 507, 437 ], "spans": [ { "bbox": [ 104, 420, 507, 437 ], "score": 1.0, "content": "[15] R. San-Roman, E. Nachmani, and L. Wolf, “Noise estimation for generative diffusion models,”", "type": "text" } ], "index": 24 }, { "bbox": [ 127, 434, 290, 446 ], "spans": [ { "bbox": [ 127, 434, 290, 446 ], "score": 1.0, "content": "arXiv preprint arXiv:2104.02600, 2021.", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 452, 507, 466 ], "spans": [ { "bbox": [ 106, 452, 507, 466 ], "score": 1.0, "content": "[16] A. Q. Nichol and P. Dhariwal, “Improved denoising diffusion probabilistic models,” in Interna-", "type": "text" } ], "index": 26 }, { "bbox": [ 126, 464, 424, 478 ], "spans": [ { "bbox": [ 126, 464, 424, 478 ], "score": 1.0, "content": "tional Conference on Machine Learning. PMLR, 2021, pp. 8162–8171.", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 484, 505, 496 ], "spans": [ { "bbox": [ 106, 484, 505, 496 ], "score": 1.0, "content": "[17] M. W. Lam, J. Wang, R. Huang, D. Su, and D. Yu, “Bilateral denoising diffusion models,” arXiv", "type": "text" } ], "index": 28 }, { "bbox": [ 124, 494, 268, 509 ], "spans": [ { "bbox": [ 124, 494, 268, 509 ], "score": 1.0, "content": "preprint arXiv:2108.11514, 2021.", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 514, 506, 528 ], "spans": [ { "bbox": [ 106, 514, 506, 528 ], "score": 1.0, "content": "[18] D. Watson, W. Chan, J. Ho, and M. Norouzi, “Learning fast samplers for diffusion models by dif-", "type": "text" } ], "index": 30 }, { "bbox": [ 127, 525, 507, 539 ], "spans": [ { "bbox": [ 127, 525, 507, 539 ], "score": 1.0, "content": "ferentiating through sample quality,” in International Conference on Learning Representations,", "type": "text" } ], "index": 31 }, { "bbox": [ 126, 534, 155, 549 ], "spans": [ { "bbox": [ 126, 534, 155, 549 ], "score": 1.0, "content": "2022.", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 555, 505, 569 ], "spans": [ { "bbox": [ 106, 555, 505, 569 ], "score": 1.0, "content": "[19] J. Song, C. Meng, and S. Ermon, “Denoising diffusion implicit models,” in International", "type": "text" } ], "index": 33 }, { "bbox": [ 127, 567, 321, 580 ], "spans": [ { "bbox": [ 127, 567, 321, 580 ], "score": 1.0, "content": "Conference on Learning Representations, 2021.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 586, 506, 600 ], "spans": [ { "bbox": [ 106, 586, 506, 600 ], "score": 1.0, "content": "[20] A. Jolicoeur-Martineau, K. Li, R. Piché-Taillefer, T. Kachman, and I. Mitliagkas, “Gotta go fast", "type": "text" } ], "index": 35 }, { "bbox": [ 126, 597, 486, 610 ], "spans": [ { "bbox": [ 126, 597, 486, 610 ], "score": 1.0, "content": "when generating data with score-based models,” arXiv preprint arXiv:2105.14080, 2021.", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 617, 506, 631 ], "spans": [ { "bbox": [ 106, 617, 506, 631 ], "score": 1.0, "content": "[21] F. Bao, C. Li, J. Zhu, and B. Zhang, “Analytic-DPM: An analytic estimate of the optimal", "type": "text" } ], "index": 37 }, { "bbox": [ 124, 626, 506, 643 ], "spans": [ { "bbox": [ 124, 626, 506, 643 ], "score": 1.0, "content": "reverse variance in diffusion probabilistic models,” in International Conference on Learning", "type": "text" } ], "index": 38 }, { "bbox": [ 126, 638, 223, 652 ], "spans": [ { "bbox": [ 126, 638, 223, 652 ], "score": 1.0, "content": "Representations, 2022.", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 657, 506, 671 ], "spans": [ { "bbox": [ 106, 657, 506, 671 ], "score": 1.0, "content": "[22] L. Liu, Y. Ren, Z. Lin, and Z. Zhao, “Pseudo numerical methods for diffusion models on", "type": "text" } ], "index": 40 }, { "bbox": [ 127, 669, 435, 683 ], "spans": [ { "bbox": [ 127, 669, 435, 683 ], "score": 1.0, "content": "manifolds,” in International Conference on Learning Representations, 2022.", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 687, 507, 704 ], "spans": [ { "bbox": [ 105, 687, 507, 704 ], "score": 1.0, "content": "[23] V. Popov, I. Vovk, V. Gogoryan, T. Sadekova, M. Kudinov, and J. Wei, “Diffusion-based voice", "type": "text" } ], "index": 42 }, { "bbox": [ 125, 698, 507, 715 ], "spans": [ { "bbox": [ 125, 698, 507, 715 ], "score": 1.0, "content": "conversion with fast maximum likelihood sampling scheme,” in International Conference on", "type": "text" } ], "index": 43 }, { "bbox": [ 126, 711, 261, 725 ], "spans": [ { "bbox": [ 126, 711, 261, 725 ], "score": 1.0, "content": "Learning Representations, 2022.", "type": "text" } ], "index": 44 } ], "index": 22 } ], "page_idx": 10, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 741, 311, 751 ], "lines": [ { "bbox": [ 299, 740, 312, 754 ], "spans": [ { "bbox": [ 299, 740, 312, 754 ], "score": 1.0, "content": "11", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 105, 52, 507, 726 ], "lines": [], "index": 22, "bbox_fs": [ 104, 71, 507, 725 ], "lines_deleted": true } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 51, 507, 719 ], "lines": [ { "bbox": [ 105, 72, 506, 88 ], "spans": [ { "bbox": [ 105, 72, 506, 88 ], "score": 1.0, "content": "[24] H. Tachibana, M. Go, M. Inahara, Y. Katayama, and Y. Watanabe, “Itô-Taylor sampling", "type": "text" } ], "index": 0 }, { "bbox": [ 126, 83, 506, 99 ], "spans": [ { "bbox": [ 126, 83, 506, 99 ], "score": 1.0, "content": "scheme for denoising diffusion probabilistic models using ideal derivatives,” arXiv preprint", "type": "text" } ], "index": 1 }, { "bbox": [ 127, 94, 232, 108 ], "spans": [ { "bbox": [ 127, 94, 232, 108 ], "score": 1.0, "content": "arXiv:2112.13339, 2021.", "type": "text" } ], "index": 2 }, { "bbox": [ 104, 112, 508, 132 ], "spans": [ { "bbox": [ 104, 112, 508, 132 ], "score": 1.0, "content": "[25] M. Hochbruck and A. Ostermann, “Exponential integrators,” Acta Numerica, vol. 19, pp.", "type": "text" } ], "index": 3 }, { "bbox": [ 127, 126, 193, 140 ], "spans": [ { "bbox": [ 127, 126, 193, 140 ], "score": 1.0, "content": "209–286, 2010.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 146, 506, 160 ], "spans": [ { "bbox": [ 106, 146, 506, 160 ], "score": 1.0, "content": "[26] J. Deng, W. Dong, R. Socher, L. Li, K. Li, and L. Fei-Fei, “ImageNet: A large-scale hierarchical", "type": "text" } ], "index": 5 }, { "bbox": [ 127, 157, 506, 171 ], "spans": [ { "bbox": [ 127, 157, 506, 171 ], "score": 1.0, "content": "image database,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEE,", "type": "text" } ], "index": 6 }, { "bbox": [ 127, 168, 209, 182 ], "spans": [ { "bbox": [ 127, 168, 209, 182 ], "score": 1.0, "content": "2009, pp. 248–255.", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 188, 507, 204 ], "spans": [ { "bbox": [ 105, 188, 507, 204 ], "score": 1.0, "content": "[27] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer,", "type": "text" } ], "index": 8 }, { "bbox": [ 126, 199, 154, 214 ], "spans": [ { "bbox": [ 126, 199, 154, 214 ], "score": 1.0, "content": "1992.", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 221, 507, 234 ], "spans": [ { "bbox": [ 106, 221, 507, 234 ], "score": 1.0, "content": "[28] J. R. Dormand and P. J. Prince, “A family of embedded Runge-Kutta formulae,” Journal of", "type": "text" } ], "index": 10 }, { "bbox": [ 125, 231, 416, 246 ], "spans": [ { "bbox": [ 125, 231, 416, 246 ], "score": 1.0, "content": "computational and applied mathematics, vol. 6, no. 1, pp. 19–26, 1980.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 252, 489, 266 ], "spans": [ { "bbox": [ 106, 252, 489, 266 ], "score": 1.0, "content": "[29] A. Krizhevsky, “Learning multiple layers of features from tiny images,” Tech. Rep., 2009.", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 272, 507, 286 ], "spans": [ { "bbox": [ 105, 272, 507, 286 ], "score": 1.0, "content": "[30] K. Atkinson, W. Han, and D. E. Stewart, Numerical solution of ordinary differential equations.", "type": "text" } ], "index": 13 }, { "bbox": [ 127, 284, 273, 296 ], "spans": [ { "bbox": [ 127, 284, 273, 296 ], "score": 1.0, "content": "John Wiley & Sons, 2011, vol. 108.", "type": "text" } ], "index": 14 }, { "bbox": [ 104, 303, 506, 318 ], "spans": [ { "bbox": [ 104, 303, 506, 318 ], "score": 1.0, "content": "[31] M. Hochbruck and A. Ostermann, “Explicit exponential Runge-Kutta methods for semilinear", "type": "text" } ], "index": 15 }, { "bbox": [ 125, 314, 508, 330 ], "spans": [ { "bbox": [ 125, 314, 508, 330 ], "score": 1.0, "content": "parabolic problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 3, pp. 1069–1090, 2005.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 335, 506, 349 ], "spans": [ { "bbox": [ 106, 335, 506, 349 ], "score": 1.0, "content": "[32] V. T. Luan, “Efficient exponential Runge-Kutta methods of high order: Construction and", "type": "text" } ], "index": 17 }, { "bbox": [ 127, 347, 455, 361 ], "spans": [ { "bbox": [ 127, 347, 455, 361 ], "score": 1.0, "content": "implementation,” BIT Numerical Mathematics, vol. 61, no. 2, pp. 535–560, 2021.", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 367, 506, 381 ], "spans": [ { "bbox": [ 106, 367, 506, 381 ], "score": 1.0, "content": "[33] F. Bao, C. Li, J. Sun, J. Zhu, and B. Zhang, “Estimating the optimal covariance with imperfect", "type": "text" } ], "index": 19 }, { "bbox": [ 126, 378, 452, 392 ], "spans": [ { "bbox": [ 126, 378, 452, 392 ], "score": 1.0, "content": "mean in diffusion probabilistic models,” arXiv preprint arXiv:2206.07309, 2022.", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 397, 506, 413 ], "spans": [ { "bbox": [ 105, 397, 506, 413 ], "score": 1.0, "content": "[34] A. Vahdat, K. Kreis, and J. Kautz, “Score-based generative modeling in latent space,” in", "type": "text" } ], "index": 21 }, { "bbox": [ 124, 408, 478, 424 ], "spans": [ { "bbox": [ 124, 408, 478, 424 ], "score": 1.0, "content": "Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 11 287–11 302.", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 430, 506, 444 ], "spans": [ { "bbox": [ 106, 430, 506, 444 ], "score": 1.0, "content": "[35] T. Dockhorn, A. Vahdat, and K. Kreis, “Score-based generative modeling with critically-damped", "type": "text" } ], "index": 23 }, { "bbox": [ 127, 441, 470, 455 ], "spans": [ { "bbox": [ 127, 441, 470, 455 ], "score": 1.0, "content": "Langevin diffusion,” in International Conference on Learning Representations, 2022.", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 460, 506, 477 ], "spans": [ { "bbox": [ 104, 460, 506, 477 ], "score": 1.0, "content": "[36] Z. Xiao, K. Kreis, and A. Vahdat, “Tackling the generative learning trilemma with denoising", "type": "text" } ], "index": 25 }, { "bbox": [ 127, 474, 458, 485 ], "spans": [ { "bbox": [ 127, 474, 458, 485 ], "score": 1.0, "content": "diffusion GANs,” in International Conference on Learning Representations, 2022.", "type": "text" } ], "index": 26 }, { "bbox": [ 104, 492, 506, 507 ], "spans": [ { "bbox": [ 104, 492, 506, 507 ], "score": 1.0, "content": "[37] M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter, “GANs trained by a two", "type": "text" } ], "index": 27 }, { "bbox": [ 127, 504, 506, 518 ], "spans": [ { "bbox": [ 127, 504, 506, 518 ], "score": 1.0, "content": "time-scale update rule converge to a local Nash equilibrium,” in Advances in Neural Information", "type": "text" } ], "index": 28 }, { "bbox": [ 126, 514, 508, 530 ], "spans": [ { "bbox": [ 126, 514, 508, 530 ], "score": 1.0, "content": "Processing Systems, I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N.", "type": "text" } ], "index": 29 }, { "bbox": [ 127, 525, 397, 540 ], "spans": [ { "bbox": [ 127, 525, 397, 540 ], "score": 1.0, "content": "Vishwanathan, and R. Garnett, Eds., vol. 30, 2017, pp. 6626–6637.", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 546, 506, 560 ], "spans": [ { "bbox": [ 106, 546, 506, 560 ], "score": 1.0, "content": "[38] Z. Kong and W. Ping, “On fast sampling of diffusion probabilistic models,” arXiv preprint", "type": "text" } ], "index": 31 }, { "bbox": [ 127, 559, 231, 570 ], "spans": [ { "bbox": [ 127, 559, 231, 570 ], "score": 1.0, "content": "arXiv:2106.00132, 2021.", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 576, 506, 593 ], "spans": [ { "bbox": [ 104, 576, 506, 593 ], "score": 1.0, "content": "[39] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings", "type": "text" } ], "index": 33 }, { "bbox": [ 125, 588, 455, 603 ], "spans": [ { "bbox": [ 125, 588, 455, 603 ], "score": 1.0, "content": "of the IEEE International Conference on Computer Vision, 2015, pp. 3730–3738.", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 608, 506, 622 ], "spans": [ { "bbox": [ 105, 608, 506, 622 ], "score": 1.0, "content": "[40] F. Yu, A. Seff, Y. Zhang, S. Song, T. Funkhouser, and J. Xiao, “LSUN: Construction of", "type": "text" } ], "index": 35 }, { "bbox": [ 125, 619, 506, 636 ], "spans": [ { "bbox": [ 125, 619, 506, 636 ], "score": 1.0, "content": "a large-scale image dataset using deep learning with humans in the loop,” arXiv preprint", "type": "text" } ], "index": 36 }, { "bbox": [ 127, 631, 231, 645 ], "spans": [ { "bbox": [ 127, 631, 231, 645 ], "score": 1.0, "content": "arXiv:1506.03365, 2015.", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 652, 506, 665 ], "spans": [ { "bbox": [ 105, 652, 506, 665 ], "score": 1.0, "content": "[41] Y. Song, C. Durkan, I. Murray, and S. Ermon, “Maximum likelihood training of score-based", "type": "text" } ], "index": 38 }, { "bbox": [ 125, 660, 507, 679 ], "spans": [ { "bbox": [ 125, 660, 507, 679 ], "score": 1.0, "content": "diffusion models,” in Advances in Neural Information Processing Systems, vol. 34, 2021, pp.", "type": "text" } ], "index": 39 }, { "bbox": [ 127, 675, 178, 686 ], "spans": [ { "bbox": [ 127, 675, 178, 686 ], "score": 1.0, "content": "1415–1428.", "type": "text" } ], "index": 40 }, { "bbox": [ 104, 692, 508, 710 ], "spans": [ { "bbox": [ 104, 692, 508, 710 ], "score": 1.0, "content": "[42] K. Yang, J. Yau, L. Fei-Fei, J. Deng, and O. Russakovsky, “A study of face obfuscation in", "type": "text" } ], "index": 41 }, { "bbox": [ 126, 704, 340, 721 ], "spans": [ { "bbox": [ 126, 704, 340, 721 ], "score": 1.0, "content": "ImageNet,” arXiv preprint arXiv:2103.06191, 2021.", "type": "text" } ], "index": 42 } ], "index": 21 } ], "page_idx": 11, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 741, 311, 751 ], "lines": [ { "bbox": [ 299, 740, 313, 754 ], "spans": [ { "bbox": [ 299, 740, 313, 754 ], "score": 1.0, "content": "12", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 105, 51, 507, 719 ], "lines": [], "index": 21, "bbox_fs": [ 104, 72, 508, 721 ], "lines_deleted": true } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 71, 156, 84 ], "lines": [ { "bbox": [ 106, 70, 158, 85 ], "spans": [ { "bbox": [ 106, 70, 158, 85 ], "score": 1.0, "content": "Checklist", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 131, 92, 208, 103 ], "lines": [ { "bbox": [ 129, 91, 210, 105 ], "spans": [ { "bbox": [ 129, 91, 210, 105 ], "score": 1.0, "content": "1. For all authors...", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 146, 107, 505, 190 ], "lines": [ { "bbox": [ 145, 106, 505, 119 ], "spans": [ { "bbox": [ 145, 106, 505, 119 ], "score": 1.0, "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", "type": "text" } ], "index": 2 }, { "bbox": [ 162, 117, 288, 130 ], "spans": [ { "bbox": [ 162, 117, 288, 130 ], "score": 1.0, "content": "contributions and scope? [Yes]", "type": "text" } ], "index": 3 }, { "bbox": [ 144, 130, 434, 144 ], "spans": [ { "bbox": [ 144, 130, 434, 144 ], "score": 1.0, "content": "(b) Did you describe the limitations of your work? [Yes] See section 6.", "type": "text" } ], "index": 4 }, { "bbox": [ 144, 142, 506, 158 ], "spans": [ { "bbox": [ 144, 142, 506, 158 ], "score": 1.0, "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] See", "type": "text" } ], "index": 5 }, { "bbox": [ 160, 155, 203, 166 ], "spans": [ { "bbox": [ 160, 155, 203, 166 ], "score": 1.0, "content": "section 6.", "type": "text" } ], "index": 6 }, { "bbox": [ 144, 167, 506, 181 ], "spans": [ { "bbox": [ 144, 167, 506, 181 ], "score": 1.0, "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", "type": "text" } ], "index": 7 }, { "bbox": [ 161, 178, 214, 191 ], "spans": [ { "bbox": [ 161, 178, 214, 191 ], "score": 1.0, "content": "them? [Yes]", "type": "text" } ], "index": 8 } ], "index": 5 }, { "type": "text", "bbox": [ 131, 193, 302, 205 ], "lines": [ { "bbox": [ 129, 192, 304, 207 ], "spans": [ { "bbox": [ 129, 192, 304, 207 ], "score": 1.0, "content": "2. If you are including theoretical results...", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 146, 208, 505, 244 ], "lines": [ { "bbox": [ 145, 207, 506, 223 ], "spans": [ { "bbox": [ 145, 207, 506, 223 ], "score": 1.0, "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Ap-", "type": "text" } ], "index": 10 }, { "bbox": [ 160, 219, 204, 231 ], "spans": [ { "bbox": [ 160, 219, 204, 231 ], "score": 1.0, "content": "pendix B.", "type": "text" } ], "index": 11 }, { "bbox": [ 144, 231, 491, 246 ], "spans": [ { "bbox": [ 144, 231, 491, 246 ], "score": 1.0, "content": "(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B.", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "text", "bbox": [ 131, 247, 241, 258 ], "lines": [ { "bbox": [ 128, 245, 243, 261 ], "spans": [ { "bbox": [ 128, 245, 243, 261 ], "score": 1.0, "content": "3. If you ran experiments...", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 146, 263, 505, 421 ], "lines": [ { "bbox": [ 146, 262, 506, 275 ], "spans": [ { "bbox": [ 146, 262, 506, 275 ], "score": 1.0, "content": "(a) Did you include the code, data, and instructions needed to reproduce the main experi-", "type": "text" } ], "index": 14 }, { "bbox": [ 162, 273, 505, 284 ], "spans": [ { "bbox": [ 162, 273, 505, 284 ], "score": 1.0, "content": "mental results (either in the supplemental material or as a URL)? [Yes] Code is attached", "type": "text" } ], "index": 15 }, { "bbox": [ 161, 284, 360, 297 ], "spans": [ { "bbox": [ 161, 284, 360, 297 ], "score": 1.0, "content": "in the supplemental materials, with the appendix.", "type": "text" } ], "index": 16 }, { "bbox": [ 145, 296, 505, 310 ], "spans": [ { "bbox": [ 145, 296, 505, 310 ], "score": 1.0, "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were", "type": "text" } ], "index": 17 }, { "bbox": [ 161, 307, 505, 320 ], "spans": [ { "bbox": [ 161, 307, 505, 320 ], "score": 1.0, "content": "chosen)? [Yes] Our method is training-free. But we also report the hyperparameters", "type": "text" } ], "index": 18 }, { "bbox": [ 161, 318, 338, 331 ], "spans": [ { "bbox": [ 161, 318, 338, 331 ], "score": 1.0, "content": "for evaluations used in our proposed solver.", "type": "text" } ], "index": 19 }, { "bbox": [ 145, 331, 507, 345 ], "spans": [ { "bbox": [ 145, 331, 507, 345 ], "score": 1.0, "content": "(c) Did you report error bars (e.g., with respect to the random seed after running ex-", "type": "text" } ], "index": 20 }, { "bbox": [ 161, 343, 505, 354 ], "spans": [ { "bbox": [ 161, 343, 505, 354 ], "score": 1.0, "content": "periments multiple times)? [No] We observe that the standard deviation of the FID", "type": "text" } ], "index": 21 }, { "bbox": [ 162, 354, 505, 365 ], "spans": [ { "bbox": [ 162, 354, 505, 365 ], "score": 1.0, "content": "evaluations of DPM-Solver are rather small (mainly less than 0.01) because the FID is", "type": "text" } ], "index": 22 }, { "bbox": [ 162, 364, 505, 376 ], "spans": [ { "bbox": [ 162, 364, 505, 376 ], "score": 1.0, "content": "already averaged over 50K samples, following existing work [18, 20, 21]. The small", "type": "text" } ], "index": 23 }, { "bbox": [ 161, 375, 367, 387 ], "spans": [ { "bbox": [ 161, 375, 367, 387 ], "score": 1.0, "content": "standard deviation does not change the conclusion.", "type": "text" } ], "index": 24 }, { "bbox": [ 146, 388, 505, 401 ], "spans": [ { "bbox": [ 146, 388, 505, 401 ], "score": 1.0, "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", "type": "text" } ], "index": 25 }, { "bbox": [ 162, 399, 506, 412 ], "spans": [ { "bbox": [ 162, 399, 506, 412 ], "score": 1.0, "content": "of GPUs, internal cluster, or cloud provider)? [Yes] The GPU type and amount is", "type": "text" } ], "index": 26 }, { "bbox": [ 162, 410, 259, 422 ], "spans": [ { "bbox": [ 162, 410, 259, 422 ], "score": 1.0, "content": "detailed in Appendix E.", "type": "text" } ], "index": 27 } ], "index": 20.5 }, { "type": "text", "bbox": [ 127, 425, 504, 437 ], "lines": [ { "bbox": [ 129, 423, 506, 439 ], "spans": [ { "bbox": [ 129, 423, 506, 439 ], "score": 1.0, "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 146, 440, 505, 546 ], "lines": [ { "bbox": [ 146, 440, 424, 452 ], "spans": [ { "bbox": [ 146, 440, 424, 452 ], "score": 1.0, "content": "(a) If your work uses existing assets, did you cite the creators? [Yes]", "type": "text" } ], "index": 29 }, { "bbox": [ 145, 453, 424, 465 ], "spans": [ { "bbox": [ 145, 453, 424, 465 ], "score": 1.0, "content": "(b) Did you mention the license of the assets? [Yes] See Appendix E", "type": "text" } ], "index": 30 }, { "bbox": [ 145, 465, 505, 479 ], "spans": [ { "bbox": [ 145, 465, 505, 479 ], "score": 1.0, "content": "(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]", "type": "text" } ], "index": 31 }, { "bbox": [ 161, 475, 368, 489 ], "spans": [ { "bbox": [ 161, 475, 368, 489 ], "score": 1.0, "content": "We include our code in the supplemental materials.", "type": "text" } ], "index": 32 }, { "bbox": [ 145, 489, 505, 502 ], "spans": [ { "bbox": [ 145, 489, 505, 502 ], "score": 1.0, "content": "(d) Did you discuss whether and how consent was obtained from people whose data you’re", "type": "text" } ], "index": 33 }, { "bbox": [ 162, 501, 506, 512 ], "spans": [ { "bbox": [ 162, 501, 506, 512 ], "score": 1.0, "content": "using/curating? [No] All of the datasets used in the experiments are publicly available.", "type": "text" } ], "index": 34 }, { "bbox": [ 145, 513, 506, 526 ], "spans": [ { "bbox": [ 145, 513, 506, 526 ], "score": 1.0, "content": "(e) Did you discuss whether the data you are using/curating contains personally identifiable", "type": "text" } ], "index": 35 }, { "bbox": [ 162, 523, 505, 536 ], "spans": [ { "bbox": [ 162, 523, 505, 536 ], "score": 1.0, "content": "information or offensive content? [Yes] We mentioned the human privacy issues of the", "type": "text" } ], "index": 36 }, { "bbox": [ 162, 534, 297, 547 ], "spans": [ { "bbox": [ 162, 534, 297, 547 ], "score": 1.0, "content": "ImageNet dataset in Appendix E.", "type": "text" } ], "index": 37 } ], "index": 33 }, { "type": "text", "bbox": [ 131, 550, 432, 561 ], "lines": [ { "bbox": [ 128, 548, 433, 564 ], "spans": [ { "bbox": [ 128, 548, 433, 564 ], "score": 1.0, "content": "5. If you used crowdsourcing or conducted research with human subjects...", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 146, 564, 505, 635 ], "lines": [ { "bbox": [ 145, 564, 506, 577 ], "spans": [ { "bbox": [ 145, 564, 506, 577 ], "score": 1.0, "content": "(a) Did you include the full text of instructions given to participants and screenshots, if", "type": "text" } ], "index": 39 }, { "bbox": [ 162, 576, 237, 588 ], "spans": [ { "bbox": [ 162, 576, 237, 588 ], "score": 1.0, "content": "applicable? [N/A]", "type": "text" } ], "index": 40 }, { "bbox": [ 145, 588, 505, 601 ], "spans": [ { "bbox": [ 145, 588, 505, 601 ], "score": 1.0, "content": "(b) Did you describe any potential participant risks, with links to Institutional Review", "type": "text" } ], "index": 41 }, { "bbox": [ 161, 599, 342, 612 ], "spans": [ { "bbox": [ 161, 599, 342, 612 ], "score": 1.0, "content": "Board (IRB) approvals, if applicable? [N/A]", "type": "text" } ], "index": 42 }, { "bbox": [ 147, 612, 505, 625 ], "spans": [ { "bbox": [ 147, 612, 505, 625 ], "score": 1.0, "content": "(c) Did you include the estimated hourly wage paid to participants and the total amount", "type": "text" } ], "index": 43 }, { "bbox": [ 162, 624, 333, 636 ], "spans": [ { "bbox": [ 162, 624, 333, 636 ], "score": 1.0, "content": "spent on participant compensation? 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For all authors...", "type": "text" } ], "index": 1 } ], "index": 1, "bbox_fs": [ 129, 91, 210, 105 ] }, { "type": "list", "bbox": [ 146, 107, 505, 190 ], "lines": [ { "bbox": [ 145, 106, 505, 119 ], "spans": [ { "bbox": [ 145, 106, 505, 119 ], "score": 1.0, "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", "type": "text" } ], "index": 2, "is_list_start_line": true }, { "bbox": [ 162, 117, 288, 130 ], "spans": [ { "bbox": [ 162, 117, 288, 130 ], "score": 1.0, "content": "contributions and scope? [Yes]", "type": "text" } ], "index": 3, "is_list_end_line": true }, { "bbox": [ 144, 130, 434, 144 ], "spans": [ { "bbox": [ 144, 130, 434, 144 ], "score": 1.0, "content": "(b) Did you describe the limitations of your work? [Yes] See section 6.", "type": "text" } ], "index": 4, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 144, 142, 506, 158 ], "spans": [ { "bbox": [ 144, 142, 506, 158 ], "score": 1.0, "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] See", "type": "text" } ], "index": 5, "is_list_start_line": true }, { "bbox": [ 160, 155, 203, 166 ], "spans": [ { "bbox": [ 160, 155, 203, 166 ], "score": 1.0, "content": "section 6.", "type": "text" } ], "index": 6, "is_list_end_line": true }, { "bbox": [ 144, 167, 506, 181 ], "spans": [ { "bbox": [ 144, 167, 506, 181 ], "score": 1.0, "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", "type": "text" } ], "index": 7, "is_list_start_line": true }, { "bbox": [ 161, 178, 214, 191 ], "spans": [ { "bbox": [ 161, 178, 214, 191 ], "score": 1.0, "content": "them? [Yes]", "type": "text" } ], "index": 8, "is_list_end_line": true } ], "index": 5, "bbox_fs": [ 144, 106, 506, 191 ] }, { "type": "text", "bbox": [ 131, 193, 302, 205 ], "lines": [ { "bbox": [ 129, 192, 304, 207 ], "spans": [ { "bbox": [ 129, 192, 304, 207 ], "score": 1.0, "content": "2. If you are including theoretical results...", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 129, 192, 304, 207 ] }, { "type": "text", "bbox": [ 146, 208, 505, 244 ], "lines": [ { "bbox": [ 145, 207, 506, 223 ], "spans": [ { "bbox": [ 145, 207, 506, 223 ], "score": 1.0, "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Ap-", "type": "text" } ], "index": 10 }, { "bbox": [ 160, 219, 204, 231 ], "spans": [ { "bbox": [ 160, 219, 204, 231 ], "score": 1.0, "content": "pendix B.", "type": "text" } ], "index": 11 }, { "bbox": [ 144, 231, 491, 246 ], "spans": [ { "bbox": [ 144, 231, 491, 246 ], "score": 1.0, "content": "(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B.", "type": "text" } ], "index": 12 } ], "index": 11, "bbox_fs": [ 144, 207, 506, 246 ] }, { "type": "text", "bbox": [ 131, 247, 241, 258 ], "lines": [ { "bbox": [ 128, 245, 243, 261 ], "spans": [ { "bbox": [ 128, 245, 243, 261 ], "score": 1.0, "content": "3. If you ran experiments...", "type": "text" } ], "index": 13 } ], "index": 13, "bbox_fs": [ 128, 245, 243, 261 ] }, { "type": "list", "bbox": [ 146, 263, 505, 421 ], "lines": [ { "bbox": [ 146, 262, 506, 275 ], "spans": [ { "bbox": [ 146, 262, 506, 275 ], "score": 1.0, "content": "(a) Did you include the code, data, and instructions needed to reproduce the main experi-", "type": "text" } ], "index": 14, "is_list_start_line": true }, { "bbox": [ 162, 273, 505, 284 ], "spans": [ { "bbox": [ 162, 273, 505, 284 ], "score": 1.0, "content": "mental results (either in the supplemental material or as a URL)? [Yes] Code is attached", "type": "text" } ], "index": 15 }, { "bbox": [ 161, 284, 360, 297 ], "spans": [ { "bbox": [ 161, 284, 360, 297 ], "score": 1.0, "content": "in the supplemental materials, with the appendix.", "type": "text" } ], "index": 16, "is_list_end_line": true }, { "bbox": [ 145, 296, 505, 310 ], "spans": [ { "bbox": [ 145, 296, 505, 310 ], "score": 1.0, "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were", "type": "text" } ], "index": 17, "is_list_start_line": true }, { "bbox": [ 161, 307, 505, 320 ], "spans": [ { "bbox": [ 161, 307, 505, 320 ], "score": 1.0, "content": "chosen)? [Yes] Our method is training-free. But we also report the hyperparameters", "type": "text" } ], "index": 18 }, { "bbox": [ 161, 318, 338, 331 ], "spans": [ { "bbox": [ 161, 318, 338, 331 ], "score": 1.0, "content": "for evaluations used in our proposed solver.", "type": "text" } ], "index": 19, "is_list_end_line": true }, { "bbox": [ 145, 331, 507, 345 ], "spans": [ { "bbox": [ 145, 331, 507, 345 ], "score": 1.0, "content": "(c) Did you report error bars (e.g., with respect to the random seed after running ex-", "type": "text" } ], "index": 20, "is_list_start_line": true }, { "bbox": [ 161, 343, 505, 354 ], "spans": [ { "bbox": [ 161, 343, 505, 354 ], "score": 1.0, "content": "periments multiple times)? [No] We observe that the standard deviation of the FID", "type": "text" } ], "index": 21 }, { "bbox": [ 162, 354, 505, 365 ], "spans": [ { "bbox": [ 162, 354, 505, 365 ], "score": 1.0, "content": "evaluations of DPM-Solver are rather small (mainly less than 0.01) because the FID is", "type": "text" } ], "index": 22 }, { "bbox": [ 162, 364, 505, 376 ], "spans": [ { "bbox": [ 162, 364, 505, 376 ], "score": 1.0, "content": "already averaged over 50K samples, following existing work [18, 20, 21]. 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[Yes]", "type": "text" } ], "index": 29, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 145, 453, 424, 465 ], "spans": [ { "bbox": [ 145, 453, 424, 465 ], "score": 1.0, "content": "(b) Did you mention the license of the assets? [Yes] See Appendix E", "type": "text" } ], "index": 30, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 145, 465, 505, 479 ], "spans": [ { "bbox": [ 145, 465, 505, 479 ], "score": 1.0, "content": "(c) Did you include any new assets either in the supplemental material or as a URL? 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[No] All of the datasets used in the experiments are publicly available.", "type": "text" } ], "index": 34 }, { "bbox": [ 145, 513, 506, 526 ], "spans": [ { "bbox": [ 145, 513, 506, 526 ], "score": 1.0, "content": "(e) Did you discuss whether the data you are using/curating contains personally identifiable", "type": "text" } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 162, 523, 505, 536 ], "spans": [ { "bbox": [ 162, 523, 505, 536 ], "score": 1.0, "content": "information or offensive content? [Yes] We mentioned the human privacy issues of the", "type": "text" } ], "index": 36 }, { "bbox": [ 162, 534, 297, 547 ], "spans": [ { "bbox": [ 162, 534, 297, 547 ], "score": 1.0, "content": "ImageNet dataset in Appendix E.", "type": "text" } ], "index": 37, "is_list_end_line": true } ], "index": 33, "bbox_fs": [ 145, 440, 506, 547 ] }, { "type": "text", "bbox": [ 131, 550, 432, 561 ], "lines": [ { "bbox": [ 128, 548, 433, 564 ], "spans": [ { "bbox": [ 128, 548, 433, 564 ], "score": 1.0, "content": "5. 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