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The Python code is", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 667, 437, 682 ], "spans": [ { "bbox": [ 105, 667, 437, 682 ], "score": 1.0, "content": "available at https://github.com/atong01/conditional-flow-matching.", "type": "text" } ], "index": 43 } ], "index": 36.5, "bbox_fs": [ 104, 513, 506, 682 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 72, 80, 159, 94 ], "lines": [ { "bbox": [ 68, 78, 160, 97 ], "spans": [ { "bbox": [ 68, 78, 160, 97 ], "score": 1.0, "content": "1 Introduction", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 110, 541, 206 ], "lines": [ { "bbox": [ 70, 111, 542, 124 ], "spans": [ { "bbox": [ 70, 111, 542, 124 ], "score": 1.0, "content": "Generative modeling considers the problem of approximating and sampling from a probability distribution.", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 123, 541, 136 ], "spans": [ { "bbox": [ 70, 123, 541, 136 ], "score": 1.0, "content": "Normalizing flows, which have emerged as a competitive generative modeling method, construct an invertible", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 134, 542, 148 ], "spans": [ { "bbox": [ 69, 134, 542, 148 ], "score": 1.0, "content": "and efficiently differentiable mapping between a fixed (e.g., standard normal) distribution and the data", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 145, 542, 160 ], "spans": [ { "bbox": [ 69, 145, 542, 160 ], "score": 1.0, "content": "distribution (Rezende & Mohamed, 2015). While original normalizing flow work specified this mapping as", "type": "text" } ], "index": 4 }, { "bbox": [ 68, 158, 542, 172 ], "spans": [ { "bbox": [ 68, 158, 542, 172 ], "score": 1.0, "content": "a static composition of invertible modules, continuous normalizing flows (CNFs) express the mapping by a", "type": "text" } ], "index": 5 }, { "bbox": [ 68, 169, 542, 184 ], "spans": [ { "bbox": [ 68, 169, 542, 184 ], "score": 1.0, "content": "neural ordinary differential equation (ODE) (Chen et al., 2018). Unfortunately, CNFs have been held back by", "type": "text" } ], "index": 6 }, { "bbox": [ 68, 181, 543, 196 ], "spans": [ { "bbox": [ 68, 181, 543, 196 ], "score": 1.0, "content": "difficulties in training and scaling to large datasets (Chen et al., 2018; Grathwohl et al., 2019; Onken et al.,", "type": "text" } ], "index": 7 }, { "bbox": [ 68, 192, 101, 208 ], "spans": [ { "bbox": [ 68, 192, 101, 208 ], "score": 1.0, "content": "2021).", "type": "text" } ], "index": 8 } ], "index": 4.5 }, { "type": "text", "bbox": [ 70, 212, 541, 367 ], "lines": [ { "bbox": [ 70, 212, 541, 226 ], "spans": [ { "bbox": [ 70, 212, 541, 226 ], "score": 1.0, "content": "Meanwhile, diffusion models, which are the current state of the art on many generative modeling tasks", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 224, 542, 238 ], "spans": [ { "bbox": [ 69, 224, 542, 238 ], "score": 1.0, "content": "(Dhariwal & Nichol, 2021; Austin et al., 2021; Corso et al., 2023; Watson et al., 2022b), approximate a", "type": "text" } ], "index": 10 }, { "bbox": [ 68, 236, 542, 250 ], "spans": [ { "bbox": [ 68, 236, 542, 250 ], "score": 1.0, "content": "stochastic differential equation (SDE) that transforms a simple density to the data distribution. Diffusion", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 246, 542, 263 ], "spans": [ { "bbox": [ 69, 246, 542, 263 ], "score": 1.0, "content": "models owe their success in part to their simple regression training objective, which does not require simulating", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 260, 541, 273 ], "spans": [ { "bbox": [ 70, 260, 541, 273 ], "score": 1.0, "content": "the SDE during training. Recently, (Lipman et al., 2023; Albergo & Vanden-Eijnden, 2023; Liu, 2022) showed", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 272, 542, 285 ], "spans": [ { "bbox": [ 70, 272, 542, 285 ], "score": 1.0, "content": "that CNFs could also be trained using a regression of the ODE’s drift similar to training of diffusion models,", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 283, 541, 297 ], "spans": [ { "bbox": [ 69, 283, 541, 297 ], "score": 1.0, "content": "an objective called flow matching (FM). FM was shown to produce high-quality samples and stabilize CNF", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 297, 540, 308 ], "spans": [ { "bbox": [ 70, 297, 540, 308 ], "score": 1.0, "content": "training, but made the assumption of a Gaussian source distribution, which was later relaxed in generalizations", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 308, 541, 321 ], "spans": [ { "bbox": [ 69, 308, 541, 321 ], "score": 1.0, "content": "of FM to more general manifolds (Chen & Lipman, 2024), arbitrary sources (Pooladian et al., 2023), and", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 319, 541, 334 ], "spans": [ { "bbox": [ 69, 319, 541, 334 ], "score": 1.0, "content": "couplings between source and target samples that are either part of the input data or are inferred using", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 332, 542, 345 ], "spans": [ { "bbox": [ 69, 332, 542, 345 ], "score": 1.0, "content": "optimal transport. The first main contribution of the present paper is to propose a unifying conditional", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 343, 542, 358 ], "spans": [ { "bbox": [ 69, 343, 542, 358 ], "score": 1.0, "content": "flow matching (CFM) framework for FM models with arbitrary transport maps, generalizing existing FM", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 356, 269, 370 ], "spans": [ { "bbox": [ 69, 356, 269, 370 ], "score": 1.0, "content": "and diffusion modeling approaches (Table 1).", "type": "text" } ], "index": 21 } ], "index": 15 }, { "type": "text", "bbox": [ 70, 373, 540, 577 ], "lines": [ { "bbox": [ 69, 374, 540, 386 ], "spans": [ { "bbox": [ 69, 374, 540, 386 ], "score": 1.0, "content": "A major drawback of both CNF (ODE) and diffusion (SDE) models compared to other generative models", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 385, 541, 399 ], "spans": [ { "bbox": [ 69, 385, 541, 399 ], "score": 1.0, "content": "(e.g., variational autoencoders (Kingma & Welling, 2014), (discrete-time) normalizing flows, and generative", "type": "text" } ], "index": 23 }, { "bbox": [ 68, 395, 542, 413 ], "spans": [ { "bbox": [ 68, 395, 542, 413 ], "score": 1.0, "content": "adversarial networks (Goodfellow et al., 2014)), is that integration of the ODE or SDE requires many passes", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 409, 542, 424 ], "spans": [ { "bbox": [ 69, 409, 542, 424 ], "score": 1.0, "content": "through the network to generate a high-quality sample, resulting in a long inference time. This drawback", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 421, 542, 435 ], "spans": [ { "bbox": [ 69, 421, 542, 435 ], "score": 1.0, "content": "has motivated work on enforcing an optimal transport (OT) property in neural ODEs (Tong et al., 2020;", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 433, 542, 446 ], "spans": [ { "bbox": [ 69, 433, 542, 446 ], "score": 1.0, "content": "Finlay et al., 2020; Onken et al., 2021; Liu, 2022; Liu et al., 2023b), yielding straighter flows that can be", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 445, 541, 459 ], "spans": [ { "bbox": [ 69, 445, 541, 459 ], "score": 1.0, "content": "integrated accurately in fewer neural network evaluations. Such regularizations have not yet been studied for", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 456, 541, 471 ], "spans": [ { "bbox": [ 69, 456, 541, 471 ], "score": 1.0, "content": "the full generality of models trained with FM-like objectives, and their properties with regard to solving the", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 469, 542, 483 ], "spans": [ { "bbox": [ 69, 469, 542, 483 ], "score": 1.0, "content": "dynamic optimal transport problem were not empirically evaluated. Our second main contribution is a", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 481, 542, 495 ], "spans": [ { "bbox": [ 69, 481, 542, 495 ], "score": 1.0, "content": "variant of CFM called optimal transport conditional flow matching (OT-CFM) that approximates dynamic", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 493, 541, 506 ], "spans": [ { "bbox": [ 69, 493, 541, 506 ], "score": 1.0, "content": "OT via CNFs. We show that OT-CFM not only improves the efficiency of training and inference, but also", "type": "text" } ], "index": 32 }, { "bbox": [ 68, 503, 541, 520 ], "spans": [ { "bbox": [ 68, 503, 541, 520 ], "score": 1.0, "content": "leads to more accurate OT flows than existing neural OT models based on ODEs (Tong et al., 2020; Finlay", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 517, 541, 531 ], "spans": [ { "bbox": [ 69, 517, 541, 531 ], "score": 1.0, "content": "et al., 2020), SDEs (De Bortoli et al., 2021; Vargas et al., 2021), or input-convex neural networks (Makkuva", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 528, 540, 542 ], "spans": [ { "bbox": [ 69, 528, 540, 542 ], "score": 1.0, "content": "et al., 2020). Furthermore, an entropic variant of OT-CFM can be used to efficiently train a CNF to match", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 542, 541, 554 ], "spans": [ { "bbox": [ 70, 542, 541, 554 ], "score": 1.0, "content": "the probability flow of a Schrödinger bridge. We show that in the case where the true transport", "type": "text" } ], "index": 36 }, { "bbox": [ 70, 553, 541, 566 ], "spans": [ { "bbox": [ 70, 553, 541, 566 ], "score": 1.0, "content": "plan is sampleable, our methods approximate the dynamic OT maps and Schrödinger bridge", "type": "text" } ], "index": 37 }, { "bbox": [ 68, 564, 533, 581 ], "spans": [ { "bbox": [ 68, 564, 533, 581 ], "score": 1.0, "content": "probability flows for arbitrary source and target distributions with simulation-free training.", "type": "text" } ], "index": 38 } ], "index": 30 }, { "type": "text", "bbox": [ 72, 583, 224, 594 ], "lines": [ { "bbox": [ 69, 582, 226, 595 ], "spans": [ { "bbox": [ 69, 582, 226, 595 ], "score": 1.0, "content": "In summary, our contributions are:", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 70, 600, 541, 732 ], "lines": [ { "bbox": [ 69, 600, 542, 614 ], "spans": [ { "bbox": [ 69, 600, 542, 614 ], "score": 1.0, "content": "(1) We introduce a generalized formulation of the recent conditional flow matching framework (§3.1), and", "type": "text" } ], "index": 40 }, { "bbox": [ 87, 613, 541, 627 ], "spans": [ { "bbox": [ 87, 613, 541, 627 ], "score": 1.0, "content": "prove its correctness encompassing many existing flow matching methods (Lipman et al., 2023; Albergo", "type": "text" } ], "index": 41 }, { "bbox": [ 87, 625, 342, 638 ], "spans": [ { "bbox": [ 87, 625, 342, 638 ], "score": 1.0, "content": "& Vanden-Eijnden, 2023; Liu et al., 2022a) (See Table 1).", "type": "text" } ], "index": 42 }, { "bbox": [ 72, 635, 542, 651 ], "spans": [ { "bbox": [ 72, 635, 542, 651 ], "score": 1.0, "content": "(2) We consider a special case of CFM that draws source and target samples according to an optimal transport", "type": "text" } ], "index": 43 }, { "bbox": [ 86, 648, 541, 663 ], "spans": [ { "bbox": [ 86, 648, 541, 663 ], "score": 1.0, "content": "plan, allowing us to solve the dynamic OT and Schrödinger bridge problems in a simulation-free way, using", "type": "text" } ], "index": 44 }, { "bbox": [ 87, 660, 541, 673 ], "spans": [ { "bbox": [ 87, 660, 541, 673 ], "score": 1.0, "content": "only static OT maps between marginal distributions. We show that efficient minibatch approximations", "type": "text" } ], "index": 45 }, { "bbox": [ 86, 672, 542, 687 ], "spans": [ { "bbox": [ 86, 672, 542, 687 ], "score": 1.0, "content": "to the OT map still yield correct solutions to the generative modeling problem while incurring a low", "type": "text" } ], "index": 46 }, { "bbox": [ 86, 684, 290, 699 ], "spans": [ { "bbox": [ 86, 684, 290, 699 ], "score": 1.0, "content": "detriment to the dynamic OT solution (§3.2).", "type": "text" } ], "index": 47 }, { "bbox": [ 72, 696, 541, 710 ], "spans": [ { "bbox": [ 72, 696, 541, 710 ], "score": 1.0, "content": "(3) We evaluate CFM and OT-CFM in experiments on single-cell dynamics, image generation, unsupervised", "type": "text" } ], "index": 48 }, { "bbox": [ 87, 709, 541, 721 ], "spans": [ { "bbox": [ 87, 709, 541, 721 ], "score": 1.0, "content": "image translation, and energy-based models. We show that the OT-CFM objective leads to more efficient", "type": "text" } ], "index": 49 }, { "bbox": [ 87, 720, 541, 733 ], "spans": [ { "bbox": [ 87, 720, 541, 733 ], "score": 1.0, "content": "training and decreases inference time while finding better approximate solutions to the dynamic OT", "type": "text" } ], "index": 50 } ], "index": 45 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 367, 37 ], "lines": [ { "bbox": [ 70, 24, 368, 39 ], "spans": [ { "bbox": [ 70, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 72, 80, 159, 94 ], "lines": [ { "bbox": [ 68, 78, 160, 97 ], "spans": [ { "bbox": [ 68, 78, 160, 97 ], "score": 1.0, "content": "1 Introduction", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 110, 541, 206 ], "lines": [ { "bbox": [ 70, 111, 542, 124 ], "spans": [ { "bbox": [ 70, 111, 542, 124 ], "score": 1.0, "content": "Generative modeling considers the problem of approximating and sampling from a probability distribution.", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 123, 541, 136 ], "spans": [ { "bbox": [ 70, 123, 541, 136 ], "score": 1.0, "content": "Normalizing flows, which have emerged as a competitive generative modeling method, construct an invertible", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 134, 542, 148 ], "spans": [ { "bbox": [ 69, 134, 542, 148 ], "score": 1.0, "content": "and efficiently differentiable mapping between a fixed (e.g., standard normal) distribution and the data", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 145, 542, 160 ], "spans": [ { "bbox": [ 69, 145, 542, 160 ], "score": 1.0, "content": "distribution (Rezende & Mohamed, 2015). While original normalizing flow work specified this mapping as", "type": "text" } ], "index": 4 }, { "bbox": [ 68, 158, 542, 172 ], "spans": [ { "bbox": [ 68, 158, 542, 172 ], "score": 1.0, "content": "a static composition of invertible modules, continuous normalizing flows (CNFs) express the mapping by a", "type": "text" } ], "index": 5 }, { "bbox": [ 68, 169, 542, 184 ], "spans": [ { "bbox": [ 68, 169, 542, 184 ], "score": 1.0, "content": "neural ordinary differential equation (ODE) (Chen et al., 2018). Unfortunately, CNFs have been held back by", "type": "text" } ], "index": 6 }, { "bbox": [ 68, 181, 543, 196 ], "spans": [ { "bbox": [ 68, 181, 543, 196 ], "score": 1.0, "content": "difficulties in training and scaling to large datasets (Chen et al., 2018; Grathwohl et al., 2019; Onken et al.,", "type": "text" } ], "index": 7 }, { "bbox": [ 68, 192, 101, 208 ], "spans": [ { "bbox": [ 68, 192, 101, 208 ], "score": 1.0, "content": "2021).", "type": "text" } ], "index": 8 } ], "index": 4.5, "bbox_fs": [ 68, 111, 543, 208 ] }, { "type": "text", "bbox": [ 70, 212, 541, 367 ], "lines": [ { "bbox": [ 70, 212, 541, 226 ], "spans": [ { "bbox": [ 70, 212, 541, 226 ], "score": 1.0, "content": "Meanwhile, diffusion models, which are the current state of the art on many generative modeling tasks", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 224, 542, 238 ], "spans": [ { "bbox": [ 69, 224, 542, 238 ], "score": 1.0, "content": "(Dhariwal & Nichol, 2021; Austin et al., 2021; Corso et al., 2023; Watson et al., 2022b), approximate a", "type": "text" } ], "index": 10 }, { "bbox": [ 68, 236, 542, 250 ], "spans": [ { "bbox": [ 68, 236, 542, 250 ], "score": 1.0, "content": "stochastic differential equation (SDE) that transforms a simple density to the data distribution. Diffusion", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 246, 542, 263 ], "spans": [ { "bbox": [ 69, 246, 542, 263 ], "score": 1.0, "content": "models owe their success in part to their simple regression training objective, which does not require simulating", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 260, 541, 273 ], "spans": [ { "bbox": [ 70, 260, 541, 273 ], "score": 1.0, "content": "the SDE during training. Recently, (Lipman et al., 2023; Albergo & Vanden-Eijnden, 2023; Liu, 2022) showed", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 272, 542, 285 ], "spans": [ { "bbox": [ 70, 272, 542, 285 ], "score": 1.0, "content": "that CNFs could also be trained using a regression of the ODE’s drift similar to training of diffusion models,", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 283, 541, 297 ], "spans": [ { "bbox": [ 69, 283, 541, 297 ], "score": 1.0, "content": "an objective called flow matching (FM). FM was shown to produce high-quality samples and stabilize CNF", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 297, 540, 308 ], "spans": [ { "bbox": [ 70, 297, 540, 308 ], "score": 1.0, "content": "training, but made the assumption of a Gaussian source distribution, which was later relaxed in generalizations", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 308, 541, 321 ], "spans": [ { "bbox": [ 69, 308, 541, 321 ], "score": 1.0, "content": "of FM to more general manifolds (Chen & Lipman, 2024), arbitrary sources (Pooladian et al., 2023), and", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 319, 541, 334 ], "spans": [ { "bbox": [ 69, 319, 541, 334 ], "score": 1.0, "content": "couplings between source and target samples that are either part of the input data or are inferred using", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 332, 542, 345 ], "spans": [ { "bbox": [ 69, 332, 542, 345 ], "score": 1.0, "content": "optimal transport. The first main contribution of the present paper is to propose a unifying conditional", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 343, 542, 358 ], "spans": [ { "bbox": [ 69, 343, 542, 358 ], "score": 1.0, "content": "flow matching (CFM) framework for FM models with arbitrary transport maps, generalizing existing FM", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 356, 269, 370 ], "spans": [ { "bbox": [ 69, 356, 269, 370 ], "score": 1.0, "content": "and diffusion modeling approaches (Table 1).", "type": "text" } ], "index": 21 } ], "index": 15, "bbox_fs": [ 68, 212, 542, 370 ] }, { "type": "text", "bbox": [ 70, 373, 540, 577 ], "lines": [ { "bbox": [ 69, 374, 540, 386 ], "spans": [ { "bbox": [ 69, 374, 540, 386 ], "score": 1.0, "content": "A major drawback of both CNF (ODE) and diffusion (SDE) models compared to other generative models", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 385, 541, 399 ], "spans": [ { "bbox": [ 69, 385, 541, 399 ], "score": 1.0, "content": "(e.g., variational autoencoders (Kingma & Welling, 2014), (discrete-time) normalizing flows, and generative", "type": "text" } ], "index": 23 }, { "bbox": [ 68, 395, 542, 413 ], "spans": [ { "bbox": [ 68, 395, 542, 413 ], "score": 1.0, "content": "adversarial networks (Goodfellow et al., 2014)), is that integration of the ODE or SDE requires many passes", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 409, 542, 424 ], "spans": [ { "bbox": [ 69, 409, 542, 424 ], "score": 1.0, "content": "through the network to generate a high-quality sample, resulting in a long inference time. This drawback", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 421, 542, 435 ], "spans": [ { "bbox": [ 69, 421, 542, 435 ], "score": 1.0, "content": "has motivated work on enforcing an optimal transport (OT) property in neural ODEs (Tong et al., 2020;", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 433, 542, 446 ], "spans": [ { "bbox": [ 69, 433, 542, 446 ], "score": 1.0, "content": "Finlay et al., 2020; Onken et al., 2021; Liu, 2022; Liu et al., 2023b), yielding straighter flows that can be", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 445, 541, 459 ], "spans": [ { "bbox": [ 69, 445, 541, 459 ], "score": 1.0, "content": "integrated accurately in fewer neural network evaluations. Such regularizations have not yet been studied for", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 456, 541, 471 ], "spans": [ { "bbox": [ 69, 456, 541, 471 ], "score": 1.0, "content": "the full generality of models trained with FM-like objectives, and their properties with regard to solving the", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 469, 542, 483 ], "spans": [ { "bbox": [ 69, 469, 542, 483 ], "score": 1.0, "content": "dynamic optimal transport problem were not empirically evaluated. Our second main contribution is a", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 481, 542, 495 ], "spans": [ { "bbox": [ 69, 481, 542, 495 ], "score": 1.0, "content": "variant of CFM called optimal transport conditional flow matching (OT-CFM) that approximates dynamic", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 493, 541, 506 ], "spans": [ { "bbox": [ 69, 493, 541, 506 ], "score": 1.0, "content": "OT via CNFs. We show that OT-CFM not only improves the efficiency of training and inference, but also", "type": "text" } ], "index": 32 }, { "bbox": [ 68, 503, 541, 520 ], "spans": [ { "bbox": [ 68, 503, 541, 520 ], "score": 1.0, "content": "leads to more accurate OT flows than existing neural OT models based on ODEs (Tong et al., 2020; Finlay", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 517, 541, 531 ], "spans": [ { "bbox": [ 69, 517, 541, 531 ], "score": 1.0, "content": "et al., 2020), SDEs (De Bortoli et al., 2021; Vargas et al., 2021), or input-convex neural networks (Makkuva", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 528, 540, 542 ], "spans": [ { "bbox": [ 69, 528, 540, 542 ], "score": 1.0, "content": "et al., 2020). Furthermore, an entropic variant of OT-CFM can be used to efficiently train a CNF to match", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 542, 541, 554 ], "spans": [ { "bbox": [ 70, 542, 541, 554 ], "score": 1.0, "content": "the probability flow of a Schrödinger bridge. We show that in the case where the true transport", "type": "text" } ], "index": 36 }, { "bbox": [ 70, 553, 541, 566 ], "spans": [ { "bbox": [ 70, 553, 541, 566 ], "score": 1.0, "content": "plan is sampleable, our methods approximate the dynamic OT maps and Schrödinger bridge", "type": "text" } ], "index": 37 }, { "bbox": [ 68, 564, 533, 581 ], "spans": [ { "bbox": [ 68, 564, 533, 581 ], "score": 1.0, "content": "probability flows for arbitrary source and target distributions with simulation-free training.", "type": "text" } ], "index": 38 } ], "index": 30, "bbox_fs": [ 68, 374, 542, 581 ] }, { "type": "text", "bbox": [ 72, 583, 224, 594 ], "lines": [ { "bbox": [ 69, 582, 226, 595 ], "spans": [ { "bbox": [ 69, 582, 226, 595 ], "score": 1.0, "content": "In summary, our contributions are:", "type": "text" } ], "index": 39 } ], "index": 39, "bbox_fs": [ 69, 582, 226, 595 ] }, { "type": "list", "bbox": [ 70, 600, 541, 732 ], "lines": [ { "bbox": [ 69, 600, 542, 614 ], "spans": [ { "bbox": [ 69, 600, 542, 614 ], "score": 1.0, "content": "(1) We introduce a generalized formulation of the recent conditional flow matching framework (§3.1), and", "type": "text" } ], "index": 40, "is_list_start_line": true }, { "bbox": [ 87, 613, 541, 627 ], "spans": [ { "bbox": [ 87, 613, 541, 627 ], "score": 1.0, "content": "prove its correctness encompassing many existing flow matching methods (Lipman et al., 2023; Albergo", "type": "text" } ], "index": 41 }, { "bbox": [ 87, 625, 342, 638 ], "spans": [ { "bbox": [ 87, 625, 342, 638 ], "score": 1.0, "content": "& Vanden-Eijnden, 2023; Liu et al., 2022a) (See Table 1).", "type": "text" } ], "index": 42, "is_list_end_line": true }, { "bbox": [ 72, 635, 542, 651 ], "spans": [ { "bbox": [ 72, 635, 542, 651 ], "score": 1.0, "content": "(2) We consider a special case of CFM that draws source and target samples according to an optimal transport", "type": "text" } ], "index": 43, "is_list_start_line": true }, { "bbox": [ 86, 648, 541, 663 ], "spans": [ { "bbox": [ 86, 648, 541, 663 ], "score": 1.0, "content": "plan, allowing us to solve the dynamic OT and Schrödinger bridge problems in a simulation-free way, using", "type": "text" } ], "index": 44 }, { "bbox": [ 87, 660, 541, 673 ], "spans": [ { "bbox": [ 87, 660, 541, 673 ], "score": 1.0, "content": "only static OT maps between marginal distributions. We show that efficient minibatch approximations", "type": "text" } ], "index": 45 }, { "bbox": [ 86, 672, 542, 687 ], "spans": [ { "bbox": [ 86, 672, 542, 687 ], "score": 1.0, "content": "to the OT map still yield correct solutions to the generative modeling problem while incurring a low", "type": "text" } ], "index": 46 }, { "bbox": [ 86, 684, 290, 699 ], "spans": [ { "bbox": [ 86, 684, 290, 699 ], "score": 1.0, "content": "detriment to the dynamic OT solution (§3.2).", "type": "text" } ], "index": 47, "is_list_end_line": true }, { "bbox": [ 72, 696, 541, 710 ], "spans": [ { "bbox": [ 72, 696, 541, 710 ], "score": 1.0, "content": "(3) We evaluate CFM and OT-CFM in experiments on single-cell dynamics, image generation, unsupervised", "type": "text" } ], "index": 48, "is_list_start_line": true }, { "bbox": [ 87, 709, 541, 721 ], "spans": [ { "bbox": [ 87, 709, 541, 721 ], "score": 1.0, "content": "image translation, and energy-based models. We show that the OT-CFM objective leads to more efficient", "type": "text" } ], "index": 49 }, { "bbox": [ 87, 720, 541, 733 ], "spans": [ { "bbox": [ 87, 720, 541, 733 ], "score": 1.0, "content": "training and decreases inference time while finding better approximate solutions to the dynamic OT", "type": "text" } ], "index": 50 } ], "index": 45, "bbox_fs": [ 69, 600, 542, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 78, 86, 537, 178 ], "blocks": [ { "type": "image_body", "bbox": [ 78, 86, 537, 178 ], "group_id": 0, "lines": [ { "bbox": [ 78, 86, 537, 178 ], "spans": [ { "bbox": [ 78, 86, 537, 178 ], "score": 0.93, "type": "image", "image_path": "dc40f4b609d7170e916c04dfda98d040800f9a274dbe55ca15b7497c77114b0f.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 78, 86, 537, 116.66666666666667 ], "spans": [], "index": 0 }, { "bbox": [ 78, 116.66666666666667, 537, 147.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 78, 147.33333333333334, 537, 178.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 71, 188, 541, 225 ], "group_id": 0, "lines": [ { "bbox": [ 69, 186, 542, 203 ], "spans": [ { "bbox": [ 69, 186, 542, 203 ], "score": 1.0, "content": "Figure 1: Left: Conditional flows from FM (Lipman et al., 2023), I-CFM (§3.2.2), and OT-CFM (§3.2.3).", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 199, 541, 214 ], "spans": [ { "bbox": [ 69, 199, 541, 214 ], "score": 1.0, "content": "Right: Learned flows (green) from moons (blue) to 8gaussians (black) using I-CFM (centre-right) and", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 211, 164, 226 ], "spans": [ { "bbox": [ 70, 211, 164, 226 ], "score": 1.0, "content": "OT-CFM (far right).", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 86, 246, 541, 282 ], "lines": [ { "bbox": [ 86, 245, 541, 260 ], "spans": [ { "bbox": [ 86, 245, 541, 260 ], "score": 1.0, "content": "and Schrödinger bridge problems. For high-dimensional image generation, we also propose improved", "type": "text" } ], "index": 6 }, { "bbox": [ 86, 258, 541, 272 ], "spans": [ { "bbox": [ 86, 258, 541, 272 ], "score": 1.0, "content": "and reproducible training practices for flow-based models that significantly improve the performance of", "type": "text" } ], "index": 7 }, { "bbox": [ 87, 271, 230, 283 ], "spans": [ { "bbox": [ 87, 271, 230, 283 ], "score": 1.0, "content": "algorithms from past work (§5).", "type": "text" } ], "index": 8 } ], "index": 7 }, { "type": "text", "bbox": [ 73, 283, 541, 319 ], "lines": [ { "bbox": [ 70, 282, 541, 296 ], "spans": [ { "bbox": [ 70, 282, 541, 296 ], "score": 1.0, "content": "(4) We release a Python package, torchcfm, that unifies new and existing algorithms for training flow-based", "type": "text" } ], "index": 9 }, { "bbox": [ 86, 294, 542, 308 ], "spans": [ { "bbox": [ 86, 294, 542, 308 ], "score": 1.0, "content": "generative models under a shared interface and provides implementations of our main experiments. The", "type": "text" } ], "index": 10 }, { "bbox": [ 87, 305, 488, 321 ], "spans": [ { "bbox": [ 87, 305, 488, 321 ], "score": 1.0, "content": "Python code is available at https://github.com/atong01/conditional-flow-matching.", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "title", "bbox": [ 70, 333, 362, 348 ], "lines": [ { "bbox": [ 68, 332, 364, 351 ], "spans": [ { "bbox": [ 68, 332, 364, 351 ], "score": 1.0, "content": "2 Background: Optimal transport and neural ODEs", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 70, 360, 541, 421 ], "lines": [ { "bbox": [ 69, 360, 542, 374 ], "spans": [ { "bbox": [ 69, 360, 420, 374 ], "score": 1.0, "content": "Throughout the paper, we consider the setting of a pair of data distributions over", "type": "text" }, { "bbox": [ 420, 361, 432, 370 ], "score": 0.91, "content": "\\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 433, 360, 542, 374 ], "score": 1.0, "content": "with (possibly unknown)", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 371, 540, 386 ], "spans": [ { "bbox": [ 69, 371, 110, 386 ], "score": 1.0, "content": "densities", "type": "text" }, { "bbox": [ 111, 374, 133, 385 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 134, 371, 155, 386 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 155, 374, 178, 385 ], "score": 0.93, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 178, 371, 240, 386 ], "score": 1.0, "content": "(also denoted", "type": "text" }, { "bbox": [ 241, 378, 250, 384 ], "score": 0.65, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 250, 371, 255, 386 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 255, 378, 264, 384 ], "score": 0.76, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 265, 371, 533, 386 ], "score": 1.0, "content": "). 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The unique vector field whose integration map satisfies", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 286, 144, 299 ], "spans": [ { "bbox": [ 69, 286, 144, 299 ], "score": 1.0, "content": "(4) has the form", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": "interline_equation", "bbox": [ 252, 297, 360, 322 ], "lines": [ { "bbox": [ 252, 297, 360, 322 ], "spans": [ { "bbox": [ 252, 297, 360, 322 ], "score": 0.95, "content": "u _ { t } ( x ) = \\frac { \\sigma _ { t } ^ { \\prime } } { \\sigma _ { t } } ( x - \\mu _ { t } ) + \\mu _ { t } ^ { \\prime } ,", "type": "interline_equation", "image_path": "c28418758c1bab01f26718221098457f77d5ddc82fc90f87be2abc0151a8a22d.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 252, 297, 360, 322 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 70, 326, 545, 351 ], "lines": [ { "bbox": [ 69, 325, 542, 339 ], "spans": [ { "bbox": [ 69, 325, 99, 339 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 327, 109, 338 ], "score": 0.88, "content": "\\boldsymbol { \\sigma } _ { t } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 110, 325, 133, 339 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 133, 328, 143, 338 ], "score": 0.87, "content": "\\mu _ { t } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 144, 325, 281, 339 ], "score": 1.0, "content": "denote the time derivative of", "type": "text" }, { "bbox": [ 281, 331, 291, 338 ], "score": 0.88, "content": "\\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 291, 325, 315, 339 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 315, 331, 325, 338 ], "score": 0.87, "content": "\\mu _ { t }", "type": "inline_equation" }, { "bbox": [ 325, 325, 542, 339 ], "score": 1.0, "content": ", respectively, and the vector field u with initial", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 337, 435, 352 ], "spans": [ { "bbox": [ 69, 337, 118, 352 ], "score": 1.0, "content": "conditions", "type": "text" }, { "bbox": [ 118, 339, 161, 351 ], "score": 0.93, "content": "\\mathcal { N } ( \\mu _ { 0 } , \\sigma _ { 0 } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 162, 337, 338, 352 ], "score": 1.0, "content": "generates the Gaussian probability path", "type": "text" }, { "bbox": [ 338, 340, 429, 351 ], "score": 0.93, "content": "p _ { t } ( x ) = \\mathcal { N } ( x \\mid \\mu _ { t } , \\sigma _ { t } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 429, 337, 435, 352 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 14.5 }, { "type": "title", "bbox": [ 70, 363, 268, 376 ], "lines": [ { "bbox": [ 69, 361, 269, 378 ], "spans": [ { "bbox": [ 69, 361, 269, 378 ], "score": 1.0, "content": "2.2 Static and dynamic optimal transport", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 70, 384, 541, 430 ], "lines": [ { "bbox": [ 69, 384, 543, 398 ], "spans": [ { "bbox": [ 69, 384, 543, 398 ], "score": 1.0, "content": "The (static) optimal transport problem seeks a mapping from one measure to another that minimizes a", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 395, 540, 411 ], "spans": [ { "bbox": [ 68, 395, 499, 411 ], "score": 1.0, "content": "displacement cost. 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The corresponding optimization problem", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 421, 81, 432 ], "spans": [ { "bbox": [ 69, 421, 81, 432 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 20 } ], "index": 18.5 }, { "type": "interline_equation", "bbox": [ 215, 429, 396, 455 ], "lines": [ { "bbox": [ 215, 429, 396, 455 ], "spans": [ { "bbox": [ 215, 429, 396, 455 ], "score": 0.95, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 } ^ { 2 } = \\operatorname* { i n f } _ { \\pi \\in \\Pi } \\int _ { \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } } c ( x , y ) ^ { 2 } d \\pi ( x , y ) ,", "type": "interline_equation", "image_path": "0157541f06819aab278c6db74ad4ff4914dcdf85074d47df00fbcabe81364576.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 215, 429, 396, 455 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 70, 459, 541, 507 ], "lines": [ { "bbox": [ 69, 458, 542, 473 ], "spans": [ { "bbox": [ 69, 458, 100, 473 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 462, 108, 470 ], "score": 0.87, "content": "\\amalg", "type": "inline_equation" }, { "bbox": [ 109, 458, 342, 473 ], "score": 1.0, "content": "denotes the set of all joint probability measures on", "type": "text" }, { "bbox": [ 342, 461, 378, 470 ], "score": 0.94, "content": "\\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 379, 458, 475, 473 ], "score": 1.0, "content": "whose marginals are", "type": "text" }, { "bbox": [ 475, 465, 484, 472 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 485, 458, 507, 473 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 508, 465, 517, 471 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 517, 458, 542, 473 ], "score": 1.0, "content": ". For", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 471, 541, 485 ], "spans": [ { "bbox": [ 70, 471, 325, 485 ], "score": 1.0, "content": "compactly supported distributions and for the ground cost", "type": "text" }, { "bbox": [ 325, 474, 399, 484 ], "score": 0.94, "content": "c ( x , y ) = \\| x - y \\|", "type": "inline_equation" }, { "bbox": [ 399, 471, 541, 485 ], "score": 1.0, "content": ", the set of solutions of (6) is not", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 483, 542, 496 ], "spans": [ { "bbox": [ 69, 483, 187, 496 ], "score": 1.0, "content": "empty Villani (2009), and", "type": "text" }, { "bbox": [ 187, 486, 201, 495 ], "score": 0.92, "content": "W _ { 2 }", "type": "inline_equation" }, { "bbox": [ 202, 483, 446, 496 ], "score": 1.0, "content": "is a metric on the space of probability distributions on", "type": "text" }, { "bbox": [ 447, 485, 459, 493 ], "score": 0.92, "content": "\\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 459, 483, 542, 496 ], "score": 1.0, "content": "with finite second", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 496, 110, 508 ], "spans": [ { "bbox": [ 69, 496, 110, 508 ], "score": 1.0, "content": "moment.", "type": "text" } ], "index": 25 } ], "index": 23.5 }, { "type": "text", "bbox": [ 71, 513, 537, 537 ], "lines": [ { "bbox": [ 69, 512, 539, 527 ], "spans": [ { "bbox": [ 69, 512, 529, 527 ], "score": 1.0, "content": "The dynamic form of the 2-Wasserstein distance is defined by an optimization problem over vector fields", "type": "text" }, { "bbox": [ 530, 519, 539, 525 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 70, 525, 252, 537 ], "spans": [ { "bbox": [ 70, 525, 252, 537 ], "score": 1.0, "content": "that transform one measure to the other:", "type": "text" } ], "index": 27 } ], "index": 26.5 }, { "type": "interline_equation", "bbox": [ 207, 542, 403, 570 ], "lines": [ { "bbox": [ 207, 542, 403, 570 ], "spans": [ { "bbox": [ 207, 542, 403, 570 ], "score": 0.94, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 } ^ { 2 } = \\operatorname* { i n f } _ { p _ { t } , u _ { t } } \\int _ { \\mathbb { R } ^ { d } } \\int _ { 0 } ^ { 1 } p _ { t } ( x ) \\| u _ { t } ( x ) \\| ^ { 2 } d t d x ,", "type": "interline_equation", "image_path": "946d2cea5f7a0e2746b74ac33e3913742d50805ba16b9841028e2f2822b31c47.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 207, 542, 403, 570 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 70, 576, 540, 637 ], "lines": [ { "bbox": [ 69, 576, 542, 591 ], "spans": [ { "bbox": [ 69, 576, 93, 591 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 93, 580, 121, 588 ], "score": 0.92, "content": "p _ { t } \\geq 0", "type": "inline_equation" }, { "bbox": [ 121, 576, 300, 591 ], "score": 1.0, "content": "and subject to the boundary conditions", "type": "text" }, { "bbox": [ 301, 582, 333, 588 ], "score": 0.91, "content": "p _ { 0 } = q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 333, 576, 338, 591 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 339, 582, 371, 588 ], "score": 0.9, "content": "p _ { 1 } = q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 371, 576, 542, 591 ], "score": 1.0, "content": ", and the continuity equation (2). The", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 588, 539, 601 ], "spans": [ { "bbox": [ 69, 588, 531, 601 ], "score": 1.0, "content": "equivalence between the dynamic and static optimal transport formulations was first proven in Benamou", "type": "text" }, { "bbox": [ 531, 591, 539, 599 ], "score": 0.49, "content": "\\&", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 69, 600, 541, 614 ], "spans": [ { "bbox": [ 69, 600, 263, 614 ], "score": 1.0, "content": "Brenier (2000) under the assumptions that", "type": "text" }, { "bbox": [ 263, 606, 272, 613 ], "score": 0.9, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 272, 600, 295, 614 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 295, 606, 304, 613 ], "score": 0.91, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 304, 600, 541, 614 ], "score": 1.0, "content": "are compactly supported distributions with bounded", "type": "text" } ], "index": 31 }, { "bbox": [ 70, 613, 541, 626 ], "spans": [ { "bbox": [ 70, 613, 541, 626 ], "score": 1.0, "content": "density. We refer to (Ambrosio & Gigli, 2013, Chapter2) for a recent overview on optimal transport the", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 624, 241, 638 ], "spans": [ { "bbox": [ 69, 624, 241, 638 ], "score": 1.0, "content": "relation between the two formulations.", "type": "text" } ], "index": 33 } ], "index": 31 }, { "type": "text", "bbox": [ 70, 641, 540, 691 ], "lines": [ { "bbox": [ 70, 642, 541, 655 ], "spans": [ { "bbox": [ 70, 642, 336, 655 ], "score": 1.0, "content": "Tong et al. (2020); Finlay et al. (2020) showed that CNFs with", "type": "text" }, { "bbox": [ 336, 644, 348, 653 ], "score": 0.91, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 348, 642, 541, 655 ], "score": 1.0, "content": "regularization approximate dynamic optimal", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 653, 541, 669 ], "spans": [ { "bbox": [ 69, 653, 541, 669 ], "score": 1.0, "content": "transport. For general marginals, however, these models required integrating over and backpropagating", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 666, 541, 680 ], "spans": [ { "bbox": [ 70, 666, 541, 680 ], "score": 1.0, "content": "through tens to hundreds of function evaluations, resulting in both numerical and efficiency issues. We aim", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 677, 448, 693 ], "spans": [ { "bbox": [ 69, 677, 448, 693 ], "score": 1.0, "content": "to avoid these issues by directly regressing to the vector field in a simulation-free way.", "type": "text" } ], "index": 37 } ], "index": 35.5 }, { "type": "text", "bbox": [ 71, 695, 540, 732 ], "lines": [ { "bbox": [ 69, 694, 542, 711 ], "spans": [ { "bbox": [ 69, 694, 542, 711 ], "score": 1.0, "content": "Optimal transport is also related to the Schrödinger bridge (SB) problem (Léonard, 2014b). We show in", "type": "text" } ], "index": 38 }, { "bbox": [ 70, 708, 541, 722 ], "spans": [ { "bbox": [ 70, 708, 541, 722 ], "score": 1.0, "content": "§3.2.4 that a variant of the algorithm we propose recovers the probability flow of the solution to a SB problem", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 720, 258, 733 ], "spans": [ { "bbox": [ 69, 720, 258, 733 ], "score": 1.0, "content": "with a Brownian motion reference process.", "type": "text" } ], "index": 40 } ], "index": 39 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 70, 26, 368, 38 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 9 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 71, 81, 235, 94 ], "lines": [ { "bbox": [ 70, 81, 236, 96 ], "spans": [ { "bbox": [ 70, 81, 236, 96 ], "score": 1.0, "content": "the flow matching (FM) objective:", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 70, 81, 236, 96 ] }, { "type": "interline_equation", "bbox": [ 204, 101, 407, 116 ], "lines": [ { "bbox": [ 204, 101, 407, 116 ], "spans": [ { "bbox": [ 204, 101, 407, 116 ], "score": 0.9, "content": "\\mathcal { L } _ { \\mathrm { F M } } ( \\theta ) : = \\mathbb { E } _ { t \\sim \\mathcal { U } ( 0 , 1 ) , x \\sim p _ { t } ( x ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x ) \\| ^ { 2 } .", "type": "interline_equation", "image_path": "98d08a0c400a24c6ccef812c59b22fbd90a8ad9cb5ebecaf5f365aff2d4b8077.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 204, 101, 407, 116 ], "spans": [], "index": 1 } ] }, { "type": "text", "bbox": [ 70, 122, 541, 172 ], "lines": [ { "bbox": [ 69, 123, 542, 136 ], "spans": [ { "bbox": [ 69, 123, 542, 136 ], "score": 1.0, "content": "Lipman et al. (2023) used a version of this objective with a stochastic regression target to fit ODEs that", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 135, 541, 148 ], "spans": [ { "bbox": [ 69, 135, 181, 148 ], "score": 1.0, "content": "map a Gaussian density", "type": "text" }, { "bbox": [ 181, 140, 190, 147 ], "score": 0.89, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 191, 135, 245, 148 ], "score": 1.0, "content": "to a target", "type": "text" }, { "bbox": [ 245, 140, 254, 147 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 254, 135, 541, 148 ], "score": 1.0, "content": ". However, this objective becomes intractable for general source", "type": "text" } ], "index": 3 }, { "bbox": [ 68, 146, 542, 162 ], "spans": [ { "bbox": [ 68, 146, 542, 162 ], "score": 1.0, "content": "and target distributions. In §3, we develop generalizations that allow more flexible and efficient generative", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 158, 115, 173 ], "spans": [ { "bbox": [ 69, 158, 115, 173 ], "score": 1.0, "content": "modeling.", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 68, 123, 542, 173 ] }, { "type": "text", "bbox": [ 70, 183, 541, 219 ], "lines": [ { "bbox": [ 70, 182, 542, 197 ], "spans": [ { "bbox": [ 70, 182, 542, 197 ], "score": 1.0, "content": "The case of Gaussian marginals. Consider the special case of an ODE whose marginal densities are", "type": "text" } ], "index": 6 }, { "bbox": [ 70, 194, 542, 209 ], "spans": [ { "bbox": [ 70, 194, 117, 209 ], "score": 1.0, "content": "Gaussian:", "type": "text" }, { "bbox": [ 118, 196, 209, 208 ], "score": 0.93, "content": "p _ { t } ( x ) = \\mathcal { N } ( x \\mid \\mu _ { t } , \\sigma _ { t } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 210, 194, 542, 209 ], "score": 1.0, "content": ". While the ODE that generates these marginal densities is not unique, one", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 207, 241, 219 ], "spans": [ { "bbox": [ 69, 207, 241, 219 ], "score": 1.0, "content": "of the simplest is the one that satisfies", "type": "text" } ], "index": 8 } ], "index": 7, "bbox_fs": [ 69, 182, 542, 219 ] }, { "type": "interline_equation", "bbox": [ 241, 226, 369, 253 ], "lines": [ { "bbox": [ 241, 226, 369, 253 ], "spans": [ { "bbox": [ 241, 226, 369, 253 ], "score": 0.95, "content": "\\phi _ { t } ( x _ { 0 } ) = \\mu _ { t } + \\sigma _ { t } \\left( { \\frac { x _ { 0 } - \\mu _ { 0 } } { \\sigma _ { 0 } } } \\right) ,", "type": "interline_equation", "image_path": "a4907985f06e59beb6e681876dec949d4c970253929a1af773e03ec57f657536.jpg" } ] } ], "index": 9, "virtual_lines": [ { "bbox": [ 241, 226, 369, 253 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 71, 258, 253, 271 ], "lines": [ { "bbox": [ 69, 258, 254, 273 ], "spans": [ { "bbox": [ 69, 258, 254, 273 ], "score": 1.0, "content": "which is unique by the following theorem.", "type": "text" } ], "index": 10 } ], "index": 10, "bbox_fs": [ 69, 258, 254, 273 ] }, { "type": "text", "bbox": [ 69, 274, 543, 298 ], "lines": [ { "bbox": [ 69, 273, 542, 288 ], "spans": [ { "bbox": [ 69, 273, 542, 288 ], "score": 1.0, "content": "Theorem 2.1 (Theorem 3 of Lipman et al. (2023)). The unique vector field whose integration map satisfies", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 286, 144, 299 ], "spans": [ { "bbox": [ 69, 286, 144, 299 ], "score": 1.0, "content": "(4) has the form", "type": "text" } ], "index": 12 } ], "index": 11.5, "bbox_fs": [ 69, 273, 542, 299 ] }, { "type": "interline_equation", "bbox": [ 252, 297, 360, 322 ], "lines": [ { "bbox": [ 252, 297, 360, 322 ], "spans": [ { "bbox": [ 252, 297, 360, 322 ], "score": 0.95, "content": "u _ { t } ( x ) = \\frac { \\sigma _ { t } ^ { \\prime } } { \\sigma _ { t } } ( x - \\mu _ { t } ) + \\mu _ { t } ^ { \\prime } ,", "type": "interline_equation", "image_path": "c28418758c1bab01f26718221098457f77d5ddc82fc90f87be2abc0151a8a22d.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 252, 297, 360, 322 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 70, 326, 545, 351 ], "lines": [ { "bbox": [ 69, 325, 542, 339 ], "spans": [ { "bbox": [ 69, 325, 99, 339 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 327, 109, 338 ], "score": 0.88, "content": "\\boldsymbol { \\sigma } _ { t } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 110, 325, 133, 339 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 133, 328, 143, 338 ], "score": 0.87, "content": "\\mu _ { t } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 144, 325, 281, 339 ], "score": 1.0, "content": "denote the time derivative of", "type": "text" }, { "bbox": [ 281, 331, 291, 338 ], "score": 0.88, "content": "\\sigma _ { t }", "type": "inline_equation" }, { "bbox": [ 291, 325, 315, 339 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 315, 331, 325, 338 ], "score": 0.87, "content": "\\mu _ { t }", "type": "inline_equation" }, { "bbox": [ 325, 325, 542, 339 ], "score": 1.0, "content": ", respectively, and the vector field u with initial", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 337, 435, 352 ], "spans": [ { "bbox": [ 69, 337, 118, 352 ], "score": 1.0, "content": "conditions", "type": "text" }, { "bbox": [ 118, 339, 161, 351 ], "score": 0.93, "content": "\\mathcal { N } ( \\mu _ { 0 } , \\sigma _ { 0 } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 162, 337, 338, 352 ], "score": 1.0, "content": "generates the Gaussian probability path", "type": "text" }, { "bbox": [ 338, 340, 429, 351 ], "score": 0.93, "content": "p _ { t } ( x ) = \\mathcal { N } ( x \\mid \\mu _ { t } , \\sigma _ { t } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 429, 337, 435, 352 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 14.5, "bbox_fs": [ 69, 325, 542, 352 ] }, { "type": "title", "bbox": [ 70, 363, 268, 376 ], "lines": [ { "bbox": [ 69, 361, 269, 378 ], "spans": [ { "bbox": [ 69, 361, 269, 378 ], "score": 1.0, "content": "2.2 Static and dynamic optimal transport", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 70, 384, 541, 430 ], "lines": [ { "bbox": [ 69, 384, 543, 398 ], "spans": [ { "bbox": [ 69, 384, 543, 398 ], "score": 1.0, "content": "The (static) optimal transport problem seeks a mapping from one measure to another that minimizes a", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 395, 540, 411 ], "spans": [ { "bbox": [ 68, 395, 499, 411 ], "score": 1.0, "content": "displacement cost. The case of greatest interest is the 2-Wasserstein distance between distributions", "type": "text" }, { "bbox": [ 500, 402, 509, 409 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 509, 395, 531, 411 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 531, 402, 540, 409 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 69, 408, 541, 422 ], "spans": [ { "bbox": [ 69, 408, 84, 422 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 84, 410, 96, 419 ], "score": 0.91, "content": "\\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 97, 408, 285, 422 ], "score": 1.0, "content": "with respect to the Euclidean distance cost", "type": "text" }, { "bbox": [ 286, 411, 358, 421 ], "score": 0.94, "content": "c ( x , y ) = \\| x - y \\|", "type": "inline_equation" }, { "bbox": [ 359, 408, 541, 422 ], "score": 1.0, "content": ". The corresponding optimization problem", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 421, 81, 432 ], "spans": [ { "bbox": [ 69, 421, 81, 432 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 20 } ], "index": 18.5, "bbox_fs": [ 68, 384, 543, 432 ] }, { "type": "interline_equation", "bbox": [ 215, 429, 396, 455 ], "lines": [ { "bbox": [ 215, 429, 396, 455 ], "spans": [ { "bbox": [ 215, 429, 396, 455 ], "score": 0.95, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 } ^ { 2 } = \\operatorname* { i n f } _ { \\pi \\in \\Pi } \\int _ { \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } } c ( x , y ) ^ { 2 } d \\pi ( x , y ) ,", "type": "interline_equation", "image_path": "0157541f06819aab278c6db74ad4ff4914dcdf85074d47df00fbcabe81364576.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 215, 429, 396, 455 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 70, 459, 541, 507 ], "lines": [ { "bbox": [ 69, 458, 542, 473 ], "spans": [ { "bbox": [ 69, 458, 100, 473 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 462, 108, 470 ], "score": 0.87, "content": "\\amalg", "type": "inline_equation" }, { "bbox": [ 109, 458, 342, 473 ], "score": 1.0, "content": "denotes the set of all joint probability measures on", "type": "text" }, { "bbox": [ 342, 461, 378, 470 ], "score": 0.94, "content": "\\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 379, 458, 475, 473 ], "score": 1.0, "content": "whose marginals are", "type": "text" }, { "bbox": [ 475, 465, 484, 472 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 485, 458, 507, 473 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 508, 465, 517, 471 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 517, 458, 542, 473 ], "score": 1.0, "content": ". For", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 471, 541, 485 ], "spans": [ { "bbox": [ 70, 471, 325, 485 ], "score": 1.0, "content": "compactly supported distributions and for the ground cost", "type": "text" }, { "bbox": [ 325, 474, 399, 484 ], "score": 0.94, "content": "c ( x , y ) = \\| x - y \\|", "type": "inline_equation" }, { "bbox": [ 399, 471, 541, 485 ], "score": 1.0, "content": ", the set of solutions of (6) is not", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 483, 542, 496 ], "spans": [ { "bbox": [ 69, 483, 187, 496 ], "score": 1.0, "content": "empty Villani (2009), and", "type": "text" }, { "bbox": [ 187, 486, 201, 495 ], "score": 0.92, "content": "W _ { 2 }", "type": "inline_equation" }, { "bbox": [ 202, 483, 446, 496 ], "score": 1.0, "content": "is a metric on the space of probability distributions on", "type": "text" }, { "bbox": [ 447, 485, 459, 493 ], "score": 0.92, "content": "\\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 459, 483, 542, 496 ], "score": 1.0, "content": "with finite second", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 496, 110, 508 ], "spans": [ { "bbox": [ 69, 496, 110, 508 ], "score": 1.0, "content": "moment.", "type": "text" } ], "index": 25 } ], "index": 23.5, "bbox_fs": [ 69, 458, 542, 508 ] }, { "type": "text", "bbox": [ 71, 513, 537, 537 ], "lines": [ { "bbox": [ 69, 512, 539, 527 ], "spans": [ { "bbox": [ 69, 512, 529, 527 ], "score": 1.0, "content": "The dynamic form of the 2-Wasserstein distance is defined by an optimization problem over vector fields", "type": "text" }, { "bbox": [ 530, 519, 539, 525 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 70, 525, 252, 537 ], "spans": [ { "bbox": [ 70, 525, 252, 537 ], "score": 1.0, "content": "that transform one measure to the other:", "type": "text" } ], "index": 27 } ], "index": 26.5, "bbox_fs": [ 69, 512, 539, 537 ] }, { "type": "interline_equation", "bbox": [ 207, 542, 403, 570 ], "lines": [ { "bbox": [ 207, 542, 403, 570 ], "spans": [ { "bbox": [ 207, 542, 403, 570 ], "score": 0.94, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 } ^ { 2 } = \\operatorname* { i n f } _ { p _ { t } , u _ { t } } \\int _ { \\mathbb { R } ^ { d } } \\int _ { 0 } ^ { 1 } p _ { t } ( x ) \\| u _ { t } ( x ) \\| ^ { 2 } d t d x ,", "type": "interline_equation", "image_path": "946d2cea5f7a0e2746b74ac33e3913742d50805ba16b9841028e2f2822b31c47.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 207, 542, 403, 570 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 70, 576, 540, 637 ], "lines": [ { "bbox": [ 69, 576, 542, 591 ], "spans": [ { "bbox": [ 69, 576, 93, 591 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 93, 580, 121, 588 ], "score": 0.92, "content": "p _ { t } \\geq 0", "type": "inline_equation" }, { "bbox": [ 121, 576, 300, 591 ], "score": 1.0, "content": "and subject to the boundary conditions", "type": "text" }, { "bbox": [ 301, 582, 333, 588 ], "score": 0.91, "content": "p _ { 0 } = q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 333, 576, 338, 591 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 339, 582, 371, 588 ], "score": 0.9, "content": "p _ { 1 } = q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 371, 576, 542, 591 ], "score": 1.0, "content": ", and the continuity equation (2). The", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 588, 539, 601 ], "spans": [ { "bbox": [ 69, 588, 531, 601 ], "score": 1.0, "content": "equivalence between the dynamic and static optimal transport formulations was first proven in Benamou", "type": "text" }, { "bbox": [ 531, 591, 539, 599 ], "score": 0.49, "content": "\\&", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 69, 600, 541, 614 ], "spans": [ { "bbox": [ 69, 600, 263, 614 ], "score": 1.0, "content": "Brenier (2000) under the assumptions that", "type": "text" }, { "bbox": [ 263, 606, 272, 613 ], "score": 0.9, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 272, 600, 295, 614 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 295, 606, 304, 613 ], "score": 0.91, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 304, 600, 541, 614 ], "score": 1.0, "content": "are compactly supported distributions with bounded", "type": "text" } ], "index": 31 }, { "bbox": [ 70, 613, 541, 626 ], "spans": [ { "bbox": [ 70, 613, 541, 626 ], "score": 1.0, "content": "density. We refer to (Ambrosio & Gigli, 2013, Chapter2) for a recent overview on optimal transport the", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 624, 241, 638 ], "spans": [ { "bbox": [ 69, 624, 241, 638 ], "score": 1.0, "content": "relation between the two formulations.", "type": "text" } ], "index": 33 } ], "index": 31, "bbox_fs": [ 69, 576, 542, 638 ] }, { "type": "text", "bbox": [ 70, 641, 540, 691 ], "lines": [ { "bbox": [ 70, 642, 541, 655 ], "spans": [ { "bbox": [ 70, 642, 336, 655 ], "score": 1.0, "content": "Tong et al. (2020); Finlay et al. (2020) showed that CNFs with", "type": "text" }, { "bbox": [ 336, 644, 348, 653 ], "score": 0.91, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 348, 642, 541, 655 ], "score": 1.0, "content": "regularization approximate dynamic optimal", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 653, 541, 669 ], "spans": [ { "bbox": [ 69, 653, 541, 669 ], "score": 1.0, "content": "transport. For general marginals, however, these models required integrating over and backpropagating", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 666, 541, 680 ], "spans": [ { "bbox": [ 70, 666, 541, 680 ], "score": 1.0, "content": "through tens to hundreds of function evaluations, resulting in both numerical and efficiency issues. We aim", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 677, 448, 693 ], "spans": [ { "bbox": [ 69, 677, 448, 693 ], "score": 1.0, "content": "to avoid these issues by directly regressing to the vector field in a simulation-free way.", "type": "text" } ], "index": 37 } ], "index": 35.5, "bbox_fs": [ 69, 642, 541, 693 ] }, { "type": "text", "bbox": [ 71, 695, 540, 732 ], "lines": [ { "bbox": [ 69, 694, 542, 711 ], "spans": [ { "bbox": [ 69, 694, 542, 711 ], "score": 1.0, "content": "Optimal transport is also related to the Schrödinger bridge (SB) problem (Léonard, 2014b). We show in", "type": "text" } ], "index": 38 }, { "bbox": [ 70, 708, 541, 722 ], "spans": [ { "bbox": [ 70, 708, 541, 722 ], "score": 1.0, "content": "§3.2.4 that a variant of the algorithm we propose recovers the probability flow of the solution to a SB problem", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 720, 258, 733 ], "spans": [ { "bbox": [ 69, 720, 258, 733 ], "score": 1.0, "content": "with a Brownian motion reference process.", "type": "text" } ], "index": 40 } ], "index": 39, "bbox_fs": [ 69, 694, 542, 733 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 69, 79, 395, 95 ], "lines": [ { "bbox": [ 67, 78, 397, 98 ], "spans": [ { "bbox": [ 67, 78, 397, 98 ], "score": 1.0, "content": "3 Conditional flow matching: ODEs from static couplings", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "title", "bbox": [ 70, 106, 340, 119 ], "lines": [ { "bbox": [ 69, 104, 341, 122 ], "spans": [ { "bbox": [ 69, 104, 341, 122 ], "score": 1.0, "content": "3.1 Vector fields generating mixtures of probability paths", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 70, 127, 542, 152 ], "lines": [ { "bbox": [ 69, 127, 541, 141 ], "spans": [ { "bbox": [ 69, 127, 268, 141 ], "score": 1.0, "content": "Suppose that the marginal probability path", "type": "text" }, { "bbox": [ 268, 130, 291, 140 ], "score": 0.93, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 291, 127, 441, 141 ], "score": 1.0, "content": "is a mixture of probability paths", "type": "text" }, { "bbox": [ 442, 130, 472, 140 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 472, 127, 541, 141 ], "score": 1.0, "content": "that vary with", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 139, 236, 153 ], "spans": [ { "bbox": [ 69, 139, 190, 153 ], "score": 1.0, "content": "some conditioning variable", "type": "text" }, { "bbox": [ 190, 145, 196, 150 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 196, 139, 236, 153 ], "score": 1.0, "content": ", that is,", "type": "text" } ], "index": 3 } ], "index": 2.5 }, { "type": "interline_equation", "bbox": [ 250, 159, 360, 185 ], "lines": [ { "bbox": [ 250, 159, 360, 185 ], "spans": [ { "bbox": [ 250, 159, 360, 185 ], "score": 0.95, "content": "p _ { t } ( x ) = \\int p _ { t } ( x | z ) q ( z ) d z ,", "type": "interline_equation", "image_path": "6f8a9777f7ba8925c61455cd3344e57d50c471583b6af8f6451d2e65d875ccd7.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 250, 159, 360, 185 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 70, 192, 544, 217 ], "lines": [ { "bbox": [ 70, 192, 542, 207 ], "spans": [ { "bbox": [ 70, 192, 99, 207 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 99, 195, 117, 205 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 117, 192, 441, 207 ], "score": 1.0, "content": "is some distribution over the conditioning variable. If the probability path", "type": "text" }, { "bbox": [ 442, 195, 471, 205 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 472, 192, 542, 207 ], "score": 1.0, "content": "is generated by", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 204, 446, 217 ], "spans": [ { "bbox": [ 69, 204, 140, 217 ], "score": 1.0, "content": "the vector field", "type": "text" }, { "bbox": [ 140, 207, 171, 217 ], "score": 0.95, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 171, 204, 273, 217 ], "score": 1.0, "content": "from initial conditions", "type": "text" }, { "bbox": [ 273, 207, 305, 217 ], "score": 0.95, "content": "p _ { 0 } ( x | z )", "type": "inline_equation" }, { "bbox": [ 305, 204, 446, 217 ], "score": 1.0, "content": "(see §2.1), then the vector field", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "interline_equation", "bbox": [ 244, 225, 367, 251 ], "lines": [ { "bbox": [ 244, 225, 367, 251 ], "spans": [ { "bbox": [ 244, 225, 367, 251 ], "score": 0.95, "content": "u _ { t } ( x ) : = \\mathbb { E } _ { q ( z ) } \\frac { u _ { t } ( x | z ) p _ { t } ( x | z ) } { p _ { t } ( x ) }", "type": "interline_equation", "image_path": "1cce44ebc08f67a5b9e6898fba337b1b218f840da71139957cfb89768363abff.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 244, 225, 367, 251 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 71, 259, 359, 272 ], "lines": [ { "bbox": [ 69, 259, 360, 273 ], "spans": [ { "bbox": [ 69, 259, 206, 273 ], "score": 1.0, "content": "generates the probability path", "type": "text" }, { "bbox": [ 206, 262, 228, 272 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 229, 259, 360, 273 ], "score": 1.0, "content": ", under some mild conditions:", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 78, 275, 540, 288 ], "lines": [ { "bbox": [ 76, 273, 541, 290 ], "spans": [ { "bbox": [ 76, 273, 514, 290 ], "score": 1.0, "content": "heorem 3.1. The marginal vector field (9) generates the probability path (8) from initial conditions", "type": "text" }, { "bbox": [ 515, 277, 538, 288 ], "score": 0.94, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 538, 273, 541, 290 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 70, 296, 539, 321 ], "lines": [ { "bbox": [ 68, 295, 542, 311 ], "spans": [ { "bbox": [ 68, 295, 542, 311 ], "score": 1.0, "content": "All proofs appear in Appendix A. This result extends (Lipman et al., 2023, Theorem 1) to general conditioning", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 309, 316, 322 ], "spans": [ { "bbox": [ 69, 309, 293, 322 ], "score": 1.0, "content": "variables and delineates some minor conditions on", "type": "text" }, { "bbox": [ 293, 311, 311, 322 ], "score": 0.94, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 311, 309, 316, 322 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 10.5 }, { "type": "text", "bbox": [ 70, 333, 541, 406 ], "lines": [ { "bbox": [ 69, 333, 542, 347 ], "spans": [ { "bbox": [ 69, 333, 542, 347 ], "score": 1.0, "content": "A regression objective for mixtures. We are interested in the case where conditional probability paths", "type": "text" } ], "index": 12 }, { "bbox": [ 71, 345, 541, 358 ], "spans": [ { "bbox": [ 71, 347, 101, 358 ], "score": 0.93, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 101, 345, 179, 358 ], "score": 1.0, "content": "and vector fields", "type": "text" }, { "bbox": [ 180, 347, 210, 358 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 210, 345, 541, 358 ], "score": 1.0, "content": "are known and have a simple form, and we wish to recover the vector field", "type": "text" } ], "index": 13 }, { "bbox": [ 71, 357, 540, 370 ], "spans": [ { "bbox": [ 71, 359, 94, 370 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 357, 333, 369 ], "score": 1.0, "content": ", defined by (9), that generates the probability path", "type": "text" }, { "bbox": [ 334, 359, 356, 370 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 357, 357, 540, 369 ], "score": 1.0, "content": ". Exact computation via (9) is generally", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 369, 542, 383 ], "spans": [ { "bbox": [ 69, 369, 210, 383 ], "score": 1.0, "content": "intractable, as the denominator", "type": "text" }, { "bbox": [ 210, 371, 232, 382 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 233, 369, 542, 383 ], "score": 1.0, "content": "is defined by an integral (8) that may be difficult to evaluate. Instead,", "type": "text" } ], "index": 15 }, { "bbox": [ 68, 380, 542, 395 ], "spans": [ { "bbox": [ 68, 380, 437, 395 ], "score": 1.0, "content": "we develop an unbiased stochastic objective for regression of a learned vector field to", "type": "text" }, { "bbox": [ 437, 383, 460, 394 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 460, 380, 542, 395 ], "score": 1.0, "content": ", which generalizes", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 392, 275, 407 ], "spans": [ { "bbox": [ 69, 392, 275, 407 ], "score": 1.0, "content": "the unconditional flow matching objective (3).", "type": "text" } ], "index": 17 } ], "index": 14.5 }, { "type": "text", "bbox": [ 70, 411, 540, 435 ], "lines": [ { "bbox": [ 69, 410, 542, 424 ], "spans": [ { "bbox": [ 69, 410, 88, 424 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 88, 412, 195, 424 ], "score": 0.93, "content": "v _ { \\theta } ( \\cdot , \\cdot ) : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 195, 410, 542, 424 ], "score": 1.0, "content": "be a time-dependent vector field parametrized as a neural network with weights", "type": "text" } ], "index": 18 }, { "bbox": [ 71, 422, 347, 436 ], "spans": [ { "bbox": [ 71, 426, 77, 433 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 77, 422, 347, 436 ], "score": 1.0, "content": ". Define the conditional flow matching (CFM) objective:", "type": "text" } ], "index": 19 } ], "index": 18.5 }, { "type": "interline_equation", "bbox": [ 205, 443, 406, 458 ], "lines": [ { "bbox": [ 205, 443, 406, 458 ], "spans": [ { "bbox": [ 205, 443, 406, 458 ], "score": 0.9, "content": "\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) : = \\mathbb { E } _ { t , q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | z ) \\| ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "b66c7024329a5a5db8326a1220fed947b50892843fc450c0109b7f2954d05ca7.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 205, 443, 406, 458 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 466, 542, 502 ], "lines": [ { "bbox": [ 69, 464, 542, 481 ], "spans": [ { "bbox": [ 69, 464, 409, 481 ], "score": 1.0, "content": "The CFM objective describes how to regress against the marginal vector field", "type": "text" }, { "bbox": [ 409, 468, 432, 479 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 433, 464, 542, 481 ], "score": 1.0, "content": "given by (9) with access", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 478, 541, 491 ], "spans": [ { "bbox": [ 70, 478, 311, 491 ], "score": 1.0, "content": "only to samples from the conditional probability path", "type": "text" }, { "bbox": [ 311, 480, 341, 491 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 342, 478, 471, 491 ], "score": 1.0, "content": "and conditional vector fields", "type": "text" }, { "bbox": [ 471, 480, 502, 491 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 503, 478, 541, 491 ], "score": 1.0, "content": ". This is", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 489, 231, 504 ], "spans": [ { "bbox": [ 69, 489, 231, 504 ], "score": 1.0, "content": "formalized in the following theorem.", "type": "text" } ], "index": 23 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 505, 539, 529 ], "lines": [ { "bbox": [ 69, 504, 542, 520 ], "spans": [ { "bbox": [ 69, 504, 151, 520 ], "score": 1.0, "content": "Theorem 3.2. If", "type": "text" }, { "bbox": [ 151, 507, 192, 518 ], "score": 0.91, "content": "p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 192, 504, 225, 520 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 225, 507, 255, 516 ], "score": 0.91, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 255, 504, 277, 520 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 277, 508, 313, 518 ], "score": 0.93, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 313, 504, 483, 520 ], "score": 1.0, "content": ", then, up to a constant independent of", "type": "text" }, { "bbox": [ 483, 509, 488, 516 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 488, 504, 494, 520 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 495, 509, 520, 518 ], "score": 0.91, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 520, 504, 542, 520 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 24 }, { "bbox": [ 71, 518, 186, 531 ], "spans": [ { "bbox": [ 71, 521, 91, 529 ], "score": 0.89, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 91, 518, 186, 531 ], "score": 1.0, "content": "are equal, and hence", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 249, 530, 362, 543 ], "lines": [ { "bbox": [ 249, 530, 362, 543 ], "spans": [ { "bbox": [ 249, 530, 362, 543 ], "score": 0.91, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } ) .", "type": "interline_equation", "image_path": "c569b68588727e9c2c5b133dec47c768f3fc95d7822ecdfe80cdd262b9c553c3.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 249, 530, 362, 543 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 553, 541, 590 ], "lines": [ { "bbox": [ 70, 553, 541, 567 ], "spans": [ { "bbox": [ 70, 553, 337, 567 ], "score": 1.0, "content": "The CFM objective is useful when the marginal vector field", "type": "text" }, { "bbox": [ 338, 556, 360, 566 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 361, 553, 541, 567 ], "score": 1.0, "content": "is intractable but the conditional vector", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 566, 541, 578 ], "spans": [ { "bbox": [ 70, 566, 92, 578 ], "score": 1.0, "content": "field", "type": "text" }, { "bbox": [ 92, 568, 123, 578 ], "score": 0.95, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 123, 566, 360, 578 ], "score": 1.0, "content": "is tractable. As long as we can efficiently sample from", "type": "text" }, { "bbox": [ 360, 568, 378, 578 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 378, 566, 399, 578 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 399, 568, 429, 578 ], "score": 0.94, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 429, 566, 492, 578 ], "score": 1.0, "content": "and calculate", "type": "text" }, { "bbox": [ 492, 568, 523, 578 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 523, 566, 541, 578 ], "score": 1.0, "content": ", we", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 577, 421, 591 ], "spans": [ { "bbox": [ 69, 577, 259, 591 ], "score": 1.0, "content": "can use this stochastic objective to regress", "type": "text" }, { "bbox": [ 259, 583, 268, 589 ], "score": 0.87, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 269, 577, 394, 591 ], "score": 1.0, "content": "to the marginal vector field", "type": "text" }, { "bbox": [ 394, 579, 417, 590 ], "score": 0.95, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 417, 577, 421, 591 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 29 } ], "index": 28 }, { "type": "text", "bbox": [ 74, 595, 541, 619 ], "lines": [ { "bbox": [ 70, 594, 543, 610 ], "spans": [ { "bbox": [ 70, 594, 543, 610 ], "score": 1.0, "content": "We discuss the variance arising from the stochastic regression target, and ways to reduce it, in §C.1,", "type": "text" } ], "index": 30 }, { "bbox": [ 72, 607, 219, 620 ], "spans": [ { "bbox": [ 72, 607, 219, 620 ], "score": 1.0, "content": "Proposition B.2, Proposition B.3.", "type": "text" } ], "index": 31 } ], "index": 30.5 }, { "type": "title", "bbox": [ 70, 632, 277, 645 ], "lines": [ { "bbox": [ 69, 631, 278, 648 ], "spans": [ { "bbox": [ 69, 631, 278, 648 ], "score": 1.0, "content": "3.2 Sources of conditional probability paths", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 71, 654, 541, 679 ], "lines": [ { "bbox": [ 68, 654, 542, 668 ], "spans": [ { "bbox": [ 68, 654, 416, 668 ], "score": 1.0, "content": "In this section, we introduce several forms of CFM depending on the choices of", "type": "text" }, { "bbox": [ 416, 657, 434, 667 ], "score": 0.85, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 434, 654, 439, 668 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 439, 657, 466, 667 ], "score": 0.9, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation" }, { "bbox": [ 467, 654, 491, 668 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 492, 657, 519, 667 ], "score": 0.94, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation" }, { "bbox": [ 520, 654, 542, 668 ], "score": 1.0, "content": ". All", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 666, 456, 679 ], "spans": [ { "bbox": [ 69, 666, 456, 679 ], "score": 1.0, "content": "of the CFM variants and related objectives from prior work are summarized in Table 1.", "type": "text" } ], "index": 34 } ], "index": 33.5 }, { "type": "text", "bbox": [ 72, 684, 541, 732 ], "lines": [ { "bbox": [ 71, 684, 543, 697 ], "spans": [ { "bbox": [ 71, 684, 543, 697 ], "score": 1.0, "content": "• §3.2.1: We interpret the algorithm of Lipman et al. (2023) (FM from a Gaussian) as a special case of CFM.", "type": "text" } ], "index": 35 }, { "bbox": [ 72, 695, 542, 709 ], "spans": [ { "bbox": [ 72, 695, 403, 709 ], "score": 1.0, "content": "• §3.2.2: We relax the Gaussian source requirement by letting the condition", "type": "text" }, { "bbox": [ 404, 702, 409, 706 ], "score": 0.87, "content": "z", "type": "inline_equation" }, { "bbox": [ 410, 695, 453, 709 ], "score": 1.0, "content": "be a pair", "type": "text" }, { "bbox": [ 454, 698, 487, 709 ], "score": 0.93, "content": "( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 487, 695, 542, 709 ], "score": 1.0, "content": "of an initial", "type": "text" } ], "index": 36 }, { "bbox": [ 81, 707, 543, 722 ], "spans": [ { "bbox": [ 81, 707, 435, 722 ], "score": 1.0, "content": "and a terminal point. 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The marginal vector field (9) generates the probability path (8) from initial conditions", "type": "text" }, { "bbox": [ 515, 277, 538, 288 ], "score": 0.94, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 538, 273, 541, 290 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 76, 273, 541, 290 ] }, { "type": "text", "bbox": [ 70, 296, 539, 321 ], "lines": [ { "bbox": [ 68, 295, 542, 311 ], "spans": [ { "bbox": [ 68, 295, 542, 311 ], "score": 1.0, "content": "All proofs appear in Appendix A. This result extends (Lipman et al., 2023, Theorem 1) to general conditioning", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 309, 316, 322 ], "spans": [ { "bbox": [ 69, 309, 293, 322 ], "score": 1.0, "content": "variables and delineates some minor conditions on", "type": "text" }, { "bbox": [ 293, 311, 311, 322 ], "score": 0.94, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 311, 309, 316, 322 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 10.5, "bbox_fs": [ 68, 295, 542, 322 ] }, { "type": "text", "bbox": [ 70, 333, 541, 406 ], "lines": [ { "bbox": [ 69, 333, 542, 347 ], "spans": [ { "bbox": [ 69, 333, 542, 347 ], "score": 1.0, "content": "A regression objective for mixtures. We are interested in the case where conditional probability paths", "type": "text" } ], "index": 12 }, { "bbox": [ 71, 345, 541, 358 ], "spans": [ { "bbox": [ 71, 347, 101, 358 ], "score": 0.93, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 101, 345, 179, 358 ], "score": 1.0, "content": "and vector fields", "type": "text" }, { "bbox": [ 180, 347, 210, 358 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 210, 345, 541, 358 ], "score": 1.0, "content": "are known and have a simple form, and we wish to recover the vector field", "type": "text" } ], "index": 13 }, { "bbox": [ 71, 357, 540, 370 ], "spans": [ { "bbox": [ 71, 359, 94, 370 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 357, 333, 369 ], "score": 1.0, "content": ", defined by (9), that generates the probability path", "type": "text" }, { "bbox": [ 334, 359, 356, 370 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 357, 357, 540, 369 ], "score": 1.0, "content": ". Exact computation via (9) is generally", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 369, 542, 383 ], "spans": [ { "bbox": [ 69, 369, 210, 383 ], "score": 1.0, "content": "intractable, as the denominator", "type": "text" }, { "bbox": [ 210, 371, 232, 382 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 233, 369, 542, 383 ], "score": 1.0, "content": "is defined by an integral (8) that may be difficult to evaluate. Instead,", "type": "text" } ], "index": 15 }, { "bbox": [ 68, 380, 542, 395 ], "spans": [ { "bbox": [ 68, 380, 437, 395 ], "score": 1.0, "content": "we develop an unbiased stochastic objective for regression of a learned vector field to", "type": "text" }, { "bbox": [ 437, 383, 460, 394 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 460, 380, 542, 395 ], "score": 1.0, "content": ", which generalizes", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 392, 275, 407 ], "spans": [ { "bbox": [ 69, 392, 275, 407 ], "score": 1.0, "content": "the unconditional flow matching objective (3).", "type": "text" } ], "index": 17 } ], "index": 14.5, "bbox_fs": [ 68, 333, 542, 407 ] }, { "type": "text", "bbox": [ 70, 411, 540, 435 ], "lines": [ { "bbox": [ 69, 410, 542, 424 ], "spans": [ { "bbox": [ 69, 410, 88, 424 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 88, 412, 195, 424 ], "score": 0.93, "content": "v _ { \\theta } ( \\cdot , \\cdot ) : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 195, 410, 542, 424 ], "score": 1.0, "content": "be a time-dependent vector field parametrized as a neural network with weights", "type": "text" } ], "index": 18 }, { "bbox": [ 71, 422, 347, 436 ], "spans": [ { "bbox": [ 71, 426, 77, 433 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 77, 422, 347, 436 ], "score": 1.0, "content": ". Define the conditional flow matching (CFM) objective:", "type": "text" } ], "index": 19 } ], "index": 18.5, "bbox_fs": [ 69, 410, 542, 436 ] }, { "type": "interline_equation", "bbox": [ 205, 443, 406, 458 ], "lines": [ { "bbox": [ 205, 443, 406, 458 ], "spans": [ { "bbox": [ 205, 443, 406, 458 ], "score": 0.9, "content": "\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) : = \\mathbb { E } _ { t , q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | z ) \\| ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "b66c7024329a5a5db8326a1220fed947b50892843fc450c0109b7f2954d05ca7.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 205, 443, 406, 458 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 466, 542, 502 ], "lines": [ { "bbox": [ 69, 464, 542, 481 ], "spans": [ { "bbox": [ 69, 464, 409, 481 ], "score": 1.0, "content": "The CFM objective describes how to regress against the marginal vector field", "type": "text" }, { "bbox": [ 409, 468, 432, 479 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 433, 464, 542, 481 ], "score": 1.0, "content": "given by (9) with access", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 478, 541, 491 ], "spans": [ { "bbox": [ 70, 478, 311, 491 ], "score": 1.0, "content": "only to samples from the conditional probability path", "type": "text" }, { "bbox": [ 311, 480, 341, 491 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 342, 478, 471, 491 ], "score": 1.0, "content": "and conditional vector fields", "type": "text" }, { "bbox": [ 471, 480, 502, 491 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 503, 478, 541, 491 ], "score": 1.0, "content": ". This is", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 489, 231, 504 ], "spans": [ { "bbox": [ 69, 489, 231, 504 ], "score": 1.0, "content": "formalized in the following theorem.", "type": "text" } ], "index": 23 } ], "index": 22, "bbox_fs": [ 69, 464, 542, 504 ] }, { "type": "text", "bbox": [ 70, 505, 539, 529 ], "lines": [ { "bbox": [ 69, 504, 542, 520 ], "spans": [ { "bbox": [ 69, 504, 151, 520 ], "score": 1.0, "content": "Theorem 3.2. If", "type": "text" }, { "bbox": [ 151, 507, 192, 518 ], "score": 0.91, "content": "p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 192, 504, 225, 520 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 225, 507, 255, 516 ], "score": 0.91, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 255, 504, 277, 520 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 277, 508, 313, 518 ], "score": 0.93, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 313, 504, 483, 520 ], "score": 1.0, "content": ", then, up to a constant independent of", "type": "text" }, { "bbox": [ 483, 509, 488, 516 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 488, 504, 494, 520 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 495, 509, 520, 518 ], "score": 0.91, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 520, 504, 542, 520 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 24 }, { "bbox": [ 71, 518, 186, 531 ], "spans": [ { "bbox": [ 71, 521, 91, 529 ], "score": 0.89, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 91, 518, 186, 531 ], "score": 1.0, "content": "are equal, and hence", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 69, 504, 542, 531 ] }, { "type": "interline_equation", "bbox": [ 249, 530, 362, 543 ], "lines": [ { "bbox": [ 249, 530, 362, 543 ], "spans": [ { "bbox": [ 249, 530, 362, 543 ], "score": 0.91, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } ) .", "type": "interline_equation", "image_path": "c569b68588727e9c2c5b133dec47c768f3fc95d7822ecdfe80cdd262b9c553c3.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 249, 530, 362, 543 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 553, 541, 590 ], "lines": [ { "bbox": [ 70, 553, 541, 567 ], "spans": [ { "bbox": [ 70, 553, 337, 567 ], "score": 1.0, "content": "The CFM objective is useful when the marginal vector field", "type": "text" }, { "bbox": [ 338, 556, 360, 566 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 361, 553, 541, 567 ], "score": 1.0, "content": "is intractable but the conditional vector", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 566, 541, 578 ], "spans": [ { "bbox": [ 70, 566, 92, 578 ], "score": 1.0, "content": "field", "type": "text" }, { "bbox": [ 92, 568, 123, 578 ], "score": 0.95, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 123, 566, 360, 578 ], "score": 1.0, "content": "is tractable. As long as we can efficiently sample from", "type": "text" }, { "bbox": [ 360, 568, 378, 578 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 378, 566, 399, 578 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 399, 568, 429, 578 ], "score": 0.94, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 429, 566, 492, 578 ], "score": 1.0, "content": "and calculate", "type": "text" }, { "bbox": [ 492, 568, 523, 578 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 523, 566, 541, 578 ], "score": 1.0, "content": ", we", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 577, 421, 591 ], "spans": [ { "bbox": [ 69, 577, 259, 591 ], "score": 1.0, "content": "can use this stochastic objective to regress", "type": "text" }, { "bbox": [ 259, 583, 268, 589 ], "score": 0.87, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 269, 577, 394, 591 ], "score": 1.0, "content": "to the marginal vector field", "type": "text" }, { "bbox": [ 394, 579, 417, 590 ], "score": 0.95, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 417, 577, 421, 591 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 29 } ], "index": 28, "bbox_fs": [ 69, 553, 541, 591 ] }, { "type": "text", "bbox": [ 74, 595, 541, 619 ], "lines": [ { "bbox": [ 70, 594, 543, 610 ], "spans": [ { "bbox": [ 70, 594, 543, 610 ], "score": 1.0, "content": "We discuss the variance arising from the stochastic regression target, and ways to reduce it, in §C.1,", "type": "text" } ], "index": 30 }, { "bbox": [ 72, 607, 219, 620 ], "spans": [ { "bbox": [ 72, 607, 219, 620 ], "score": 1.0, "content": "Proposition B.2, Proposition B.3.", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 70, 594, 543, 620 ] }, { "type": "title", "bbox": [ 70, 632, 277, 645 ], "lines": [ { "bbox": [ 69, 631, 278, 648 ], "spans": [ { "bbox": [ 69, 631, 278, 648 ], "score": 1.0, "content": "3.2 Sources of conditional probability paths", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 71, 654, 541, 679 ], "lines": [ { "bbox": [ 68, 654, 542, 668 ], "spans": [ { "bbox": [ 68, 654, 416, 668 ], "score": 1.0, "content": "In this section, we introduce several forms of CFM depending on the choices of", "type": "text" }, { "bbox": [ 416, 657, 434, 667 ], "score": 0.85, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 434, 654, 439, 668 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 439, 657, 466, 667 ], "score": 0.9, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation" }, { "bbox": [ 467, 654, 491, 668 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 492, 657, 519, 667 ], "score": 0.94, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation" }, { "bbox": [ 520, 654, 542, 668 ], "score": 1.0, "content": ". All", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 666, 456, 679 ], "spans": [ { "bbox": [ 69, 666, 456, 679 ], "score": 1.0, "content": "of the CFM variants and related objectives from prior work are summarized in Table 1.", "type": "text" } ], "index": 34 } ], "index": 33.5, "bbox_fs": [ 68, 654, 542, 679 ] }, { "type": "text", "bbox": [ 72, 684, 541, 732 ], "lines": [ { "bbox": [ 71, 684, 543, 697 ], "spans": [ { "bbox": [ 71, 684, 543, 697 ], "score": 1.0, "content": "• §3.2.1: We interpret the algorithm of Lipman et al. (2023) (FM from a Gaussian) as a special case of CFM.", "type": "text" } ], "index": 35 }, { "bbox": [ 72, 695, 542, 709 ], "spans": [ { "bbox": [ 72, 695, 403, 709 ], "score": 1.0, "content": "• §3.2.2: We relax the Gaussian source requirement by letting the condition", "type": "text" }, { "bbox": [ 404, 702, 409, 706 ], "score": 0.87, "content": "z", "type": "inline_equation" }, { "bbox": [ 410, 695, 453, 709 ], "score": 1.0, "content": "be a pair", "type": "text" }, { "bbox": [ 454, 698, 487, 709 ], "score": 0.93, "content": "( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 487, 695, 542, 709 ], "score": 1.0, "content": "of an initial", "type": "text" } ], "index": 36 }, { "bbox": [ 81, 707, 543, 722 ], "spans": [ { "bbox": [ 81, 707, 435, 722 ], "score": 1.0, "content": "and a terminal point. In the basic form of CFM (I-CFM), we take the distribution", "type": "text" }, { "bbox": [ 435, 710, 453, 721 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 453, 707, 492, 722 ], "score": 1.0, "content": "to equal", "type": "text" }, { "bbox": [ 493, 710, 538, 721 ], "score": 0.94, "content": "q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 538, 707, 543, 722 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 37 }, { "bbox": [ 82, 720, 379, 734 ], "spans": [ { "bbox": [ 82, 720, 379, 734 ], "score": 1.0, "content": "allowing generative modeling with an arbitrary source distribution.", "type": "text" } ], "index": 38 } ], "index": 36.5, "bbox_fs": [ 71, 684, 543, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 73, 97, 479, 172 ], "blocks": [ { "type": "table_caption", "bbox": [ 73, 82, 257, 94 ], "group_id": 0, "lines": [ { "bbox": [ 70, 79, 259, 97 ], "spans": [ { "bbox": [ 70, 79, 259, 97 ], "score": 1.0, "content": "Algorithm 1 Conditional Flow Matching", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 73, 97, 479, 172 ], "group_id": 0, "lines": [ { "bbox": [ 73, 97, 479, 172 ], "spans": [ { "bbox": [ 73, 97, 479, 172 ], "score": 0.78, "html": "
Input: Efficiently samplable q(z), Pt(x|z),and computable ut(x|z) and initial network vo.
while Trainingdo
z~q(z)；t~U(0,1)； x~pt(x|z)
LcFM(θ)←|uθ(t,x)-ut(x|z)|l²
0 ←Update(0,VθLcFm(0)) returnvθ
", "type": "table", "image_path": "ce9b439e19b95b6e7afafb5d4794dcd12cb6fec3bc27b76752b8568e93425bc5.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 73, 97, 479, 122.0 ], "spans": [], "index": 1 }, { "bbox": [ 73, 122.0, 479, 147.0 ], "spans": [], "index": 2 }, { "bbox": [ 73, 147.0, 479, 172.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 70, 221, 539, 304 ], "blocks": [ { "type": "table_caption", "bbox": [ 70, 184, 542, 220 ], "group_id": 1, "lines": [ { "bbox": [ 70, 183, 541, 198 ], "spans": [ { "bbox": [ 70, 183, 541, 198 ], "score": 1.0, "content": "Table 1: Probability path definitions for existing methods which fit in the generalized conditional flow", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 196, 541, 210 ], "spans": [ { "bbox": [ 69, 196, 541, 210 ], "score": 1.0, "content": "matching framework (top) and our newly defined paths (bottom). We define two new probability path", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 207, 438, 222 ], "spans": [ { "bbox": [ 69, 207, 438, 222 ], "score": 1.0, "content": "objectives that can handle general source distributions and optimal transport flows.", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "table_body", "bbox": [ 70, 221, 539, 304 ], "group_id": 1, "lines": [ { "bbox": [ 70, 221, 539, 304 ], "spans": [ { "bbox": [ 70, 221, 539, 304 ], "score": 0.979, "html": "
Probability Pathq（2）μ（2)σtCond. OTMarginal OTGeneral source
Var. Exploding (Song& Ermon, 2019)q(𝑥1)T101-t××
Var.Preserving (Ho et al., 2020)q(x1)α1-t11-a²−t××
Flow Matching (Lipman et al., 2023)q(x1)tx1tσ-t+1× √××
Rectified Flow Liu (2022)q(xo)q(x1)tx1+(1-t)xo√×√
Var.Pres.Stochastic Interpolant Albergo & Vanden-Eijnden (2023)q(xo)q(x1）cos(1πt)𝑥0+sin(πt)𝑥1√×√
Independent CFMq(x0)q(x1)tc1+(1-t)𝑥√×√
(Ours) Optimal Transport CFMπ(xo,x1)tx1+(1-t)x√√√
(Ours)Schrodinger Bridge CFMπ2g²(x0,x1)tx1+(1-t)xoσ√t(1-t)√√√
", "type": "table", "image_path": "35966003199535f7f1cc3f6db0ab48780a49a5d82dba01a898ebd419227d86f1.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 70, 221, 539, 248.66666666666666 ], "spans": [], "index": 7 }, { "bbox": [ 70, 248.66666666666666, 539, 276.3333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 70, 276.3333333333333, 539, 304.0 ], "spans": [], "index": 9 } ] } ], "index": 6.5 }, { "type": "text", "bbox": [ 72, 324, 541, 372 ], "lines": [ { "bbox": [ 72, 324, 541, 337 ], "spans": [ { "bbox": [ 72, 324, 83, 337 ], "score": 1.0, "content": "•", "type": "text" }, { "bbox": [ 83, 325, 110, 336 ], "score": 0.31, "content": "\\ S 3 . 2 . 3", "type": "inline_equation" }, { "bbox": [ 110, 324, 258, 337 ], "score": 1.0, "content": ": We consider joint distributions", "type": "text" }, { "bbox": [ 259, 326, 328, 336 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 329, 324, 541, 337 ], "score": 1.0, "content": "that are given by minibatch optimal transport", "type": "text" } ], "index": 10 }, { "bbox": [ 81, 336, 367, 349 ], "spans": [ { "bbox": [ 81, 336, 367, 349 ], "score": 1.0, "content": "maps, causing the learned flow to be an (approximate) OT flow.", "type": "text" } ], "index": 11 }, { "bbox": [ 71, 348, 541, 362 ], "spans": [ { "bbox": [ 71, 348, 83, 362 ], "score": 1.0, "content": "•", "type": "text" }, { "bbox": [ 83, 350, 109, 360 ], "score": 0.38, "content": "\\ S 3 . 2 . 4", "type": "inline_equation" }, { "bbox": [ 110, 348, 169, 362 ], "score": 1.0, "content": ": we consider", "type": "text" }, { "bbox": [ 169, 350, 187, 361 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 187, 348, 541, 362 ], "score": 1.0, "content": "given by an entropy-regularized OT map and show that the CFM objective with", "type": "text" } ], "index": 12 }, { "bbox": [ 82, 359, 295, 373 ], "spans": [ { "bbox": [ 82, 359, 103, 373 ], "score": 1.0, "content": "this", "type": "text" }, { "bbox": [ 103, 362, 121, 372 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 122, 359, 295, 373 ], "score": 1.0, "content": "solves the Schrödinger bridge problem.", "type": "text" } ], "index": 13 } ], "index": 11.5 }, { "type": "title", "bbox": [ 71, 384, 208, 397 ], "lines": [ { "bbox": [ 69, 383, 209, 398 ], "spans": [ { "bbox": [ 69, 383, 209, 398 ], "score": 1.0, "content": "3.2.1 FM from the Gaussian", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 70, 405, 541, 441 ], "lines": [ { "bbox": [ 70, 403, 541, 417 ], "spans": [ { "bbox": [ 70, 403, 541, 417 ], "score": 1.0, "content": "Lipman et al. (2023) considered the problem of unconditional generative modeling given a training dataset.", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 416, 542, 431 ], "spans": [ { "bbox": [ 69, 416, 183, 431 ], "score": 1.0, "content": "Identifying the condition", "type": "text" }, { "bbox": [ 184, 423, 189, 427 ], "score": 0.89, "content": "z", "type": "inline_equation" }, { "bbox": [ 190, 416, 297, 431 ], "score": 1.0, "content": "with a single datapoint", "type": "text" }, { "bbox": [ 298, 422, 329, 429 ], "score": 0.9, "content": "z : = x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 329, 416, 493, 431 ], "score": 1.0, "content": ", and choosing a smoothing constant", "type": "text" }, { "bbox": [ 493, 420, 518, 428 ], "score": 0.92, "content": "\\sigma > 0", "type": "inline_equation" }, { "bbox": [ 518, 416, 542, 431 ], "score": 1.0, "content": ", one", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 430, 90, 442 ], "spans": [ { "bbox": [ 69, 430, 90, 442 ], "score": 1.0, "content": "sets", "type": "text" } ], "index": 17 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 230, 449, 381, 495 ], "lines": [ { "bbox": [ 230, 449, 381, 495 ], "spans": [ { "bbox": [ 230, 449, 381, 495 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } , ( t \\sigma - t + 1 ) ^ { 2 } ) , } \\\\ { u _ { t } ( x | z ) = \\displaystyle \\frac { x _ { 1 } - ( 1 - \\sigma ) x } { 1 - ( 1 - \\sigma ) t } , } \\end{array}", "type": "interline_equation", "image_path": "7c602eeb8e9184a4193736450c41c757e50c0237def786b286d558829431d14d.jpg" } ] } ], "index": 18.5, "virtual_lines": [ { "bbox": [ 230, 449, 381, 472.0 ], "spans": [], "index": 18 }, { "bbox": [ 230, 472.0, 381, 495.0 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 70, 501, 541, 550 ], "lines": [ { "bbox": [ 69, 501, 542, 515 ], "spans": [ { "bbox": [ 69, 501, 375, 515 ], "score": 1.0, "content": "which is a probability path from the standard normal distribution", "type": "text" }, { "bbox": [ 376, 504, 472, 514 ], "score": 0.89, "content": "( p _ { 0 } ( x | z ) = \\mathcal { N } ( x ; 0 , I ) )", "type": "inline_equation" }, { "bbox": [ 472, 501, 542, 515 ], "score": 1.0, "content": "to a Gaussian", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 512, 542, 528 ], "spans": [ { "bbox": [ 68, 512, 179, 528 ], "score": 1.0, "content": "distribution centered at", "type": "text" }, { "bbox": [ 179, 519, 189, 525 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 190, 512, 302, 528 ], "score": 1.0, "content": "with standard deviation", "type": "text" }, { "bbox": [ 303, 519, 309, 523 ], "score": 0.85, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 309, 512, 315, 528 ], "score": 1.0, "content": "(", "type": "text" }, { "bbox": [ 315, 515, 417, 526 ], "score": 0.9, "content": "p _ { 1 } ( x | z ) = \\mathcal { N } ( x ; x _ { 1 } , \\sigma ^ { 2 } ) )", "type": "inline_equation" }, { "bbox": [ 417, 512, 473, 528 ], "score": 1.0, "content": ". If one sets", "type": "text" }, { "bbox": [ 473, 515, 527, 526 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 527, 512, 542, 528 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 525, 542, 540 ], "spans": [ { "bbox": [ 69, 525, 542, 540 ], "score": 1.0, "content": "be the uniform distribution over the training dataset, the objective introduced by Lipman et al. (2023) is", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 538, 398, 551 ], "spans": [ { "bbox": [ 70, 538, 398, 551 ], "score": 1.0, "content": "equivalent to the CFM objective (10) for this conditional probability path.", "type": "text" } ], "index": 23 } ], "index": 21.5 }, { "type": "text", "bbox": [ 70, 555, 540, 592 ], "lines": [ { "bbox": [ 69, 555, 541, 569 ], "spans": [ { "bbox": [ 69, 555, 351, 569 ], "score": 1.0, "content": "We emphasize that although the conditional probability path", "type": "text" }, { "bbox": [ 351, 558, 381, 568 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 382, 555, 541, 569 ], "score": 1.0, "content": "is an optimal transport path from", "type": "text" } ], "index": 24 }, { "bbox": [ 71, 567, 542, 581 ], "spans": [ { "bbox": [ 71, 569, 102, 580 ], "score": 0.94, "content": "p _ { 0 } ( x | z )", "type": "inline_equation" }, { "bbox": [ 102, 567, 117, 581 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 117, 569, 148, 580 ], "score": 0.94, "content": "p _ { 1 } ( x | z )", "type": "inline_equation" }, { "bbox": [ 148, 567, 235, 581 ], "score": 1.0, "content": ", the marginal path", "type": "text" }, { "bbox": [ 235, 569, 257, 580 ], "score": 0.93, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 258, 567, 504, 581 ], "score": 1.0, "content": "is not in general an OT path from the standard normal", "type": "text" }, { "bbox": [ 504, 569, 527, 580 ], "score": 0.93, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 528, 567, 542, 581 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 577, 193, 594 ], "spans": [ { "bbox": [ 69, 577, 165, 594 ], "score": 1.0, "content": "the data distribution", "type": "text" }, { "bbox": [ 165, 581, 189, 592 ], "score": 0.94, "content": "p _ { 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 189, 577, 193, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "title", "bbox": [ 70, 603, 298, 617 ], "lines": [ { "bbox": [ 68, 601, 299, 620 ], "spans": [ { "bbox": [ 68, 601, 299, 620 ], "score": 1.0, "content": "3.2.2 Basic form of CFM: Independent coupling", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 70, 624, 541, 660 ], "lines": [ { "bbox": [ 69, 623, 541, 637 ], "spans": [ { "bbox": [ 69, 623, 286, 637 ], "score": 1.0, "content": "In the basic form of CFM (I-CFM), we identify", "type": "text" }, { "bbox": [ 286, 630, 292, 635 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 293, 623, 509, 637 ], "score": 1.0, "content": "with a pair of random variables, a source point", "type": "text" }, { "bbox": [ 509, 630, 519, 636 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 520, 623, 541, 637 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 28 }, { "bbox": [ 68, 636, 542, 649 ], "spans": [ { "bbox": [ 68, 636, 137, 649 ], "score": 1.0, "content": "a target point", "type": "text" }, { "bbox": [ 137, 642, 147, 648 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 148, 636, 190, 649 ], "score": 1.0, "content": ", and set", "type": "text" }, { "bbox": [ 191, 639, 269, 649 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 269, 636, 542, 649 ], "score": 1.0, "content": "to be the independent coupling. We let the conditionals be", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 648, 390, 662 ], "spans": [ { "bbox": [ 69, 648, 177, 662 ], "score": 1.0, "content": "Gaussian flows between", "type": "text" }, { "bbox": [ 177, 654, 187, 660 ], "score": 0.89, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 188, 648, 210, 662 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 654, 220, 660 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 221, 648, 331, 662 ], "score": 1.0, "content": "with standard deviation", "type": "text" }, { "bbox": [ 331, 654, 338, 658 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 338, 648, 390, 662 ], "score": 1.0, "content": ", defined by", "type": "text" } ], "index": 30 } ], "index": 29 }, { "type": "interline_equation", "bbox": [ 226, 669, 385, 701 ], "lines": [ { "bbox": [ 226, 669, 385, 701 ], "spans": [ { "bbox": [ 226, 669, 385, 701 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) , } \\\\ { u _ { t } ( x | z ) = x _ { 1 } - x _ { 0 } . } \\end{array}", "type": "interline_equation", "image_path": "df15a84a85891e946d0afbb4241bff6c4a5a11d317c78bd73e3a6a8ea2a45044.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 226, 669, 385, 685.0 ], "spans": [], "index": 31 }, { "bbox": [ 226, 685.0, 385, 701.0 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 69, 708, 543, 733 ], "lines": [ { "bbox": [ 70, 708, 542, 721 ], "spans": [ { "bbox": [ 70, 708, 222, 721 ], "score": 1.0, "content": "We note that the formulation of", "type": "text" }, { "bbox": [ 222, 710, 253, 721 ], "score": 0.93, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 253, 708, 542, 721 ], "score": 1.0, "content": "follows from an application of Theorem 2.1 to the conditional", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 720, 541, 734 ], "spans": [ { "bbox": [ 69, 720, 171, 734 ], "score": 1.0, "content": "probability path with", "type": "text" }, { "bbox": [ 172, 722, 264, 733 ], "score": 0.93, "content": "\\mu _ { t } = t x _ { 1 } + ( 1 - t ) x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 264, 720, 288, 734 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 288, 725, 320, 732 ], "score": 0.91, "content": "\\sigma _ { t } = \\sigma", "type": "inline_equation" }, { "bbox": [ 320, 720, 452, 734 ], "score": 1.0, "content": ". Furthermore, we note that", "type": "text" }, { "bbox": [ 452, 722, 483, 732 ], "score": 0.94, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 483, 720, 541, 734 ], "score": 1.0, "content": "is efficiently", "type": "text" } ], "index": 34 } ], "index": 33.5 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 26, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 73, 97, 479, 172 ], "blocks": [ { "type": "table_caption", "bbox": [ 73, 82, 257, 94 ], "group_id": 0, "lines": [ { "bbox": [ 70, 79, 259, 97 ], "spans": [ { "bbox": [ 70, 79, 259, 97 ], "score": 1.0, "content": "Algorithm 1 Conditional Flow Matching", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 73, 97, 479, 172 ], "group_id": 0, "lines": [ { "bbox": [ 73, 97, 479, 172 ], "spans": [ { "bbox": [ 73, 97, 479, 172 ], "score": 0.78, "html": "
Input: Efficiently samplable q(z), Pt(x|z),and computable ut(x|z) and initial network vo.
while Trainingdo
z~q(z)；t~U(0,1)； x~pt(x|z)
LcFM(θ)←|uθ(t,x)-ut(x|z)|l²
0 ←Update(0,VθLcFm(0)) returnvθ
", "type": "table", "image_path": "ce9b439e19b95b6e7afafb5d4794dcd12cb6fec3bc27b76752b8568e93425bc5.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 73, 97, 479, 122.0 ], "spans": [], "index": 1 }, { "bbox": [ 73, 122.0, 479, 147.0 ], "spans": [], "index": 2 }, { "bbox": [ 73, 147.0, 479, 172.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "table", "bbox": [ 70, 221, 539, 304 ], "blocks": [ { "type": "table_caption", "bbox": [ 70, 184, 542, 220 ], "group_id": 1, "lines": [ { "bbox": [ 70, 183, 541, 198 ], "spans": [ { "bbox": [ 70, 183, 541, 198 ], "score": 1.0, "content": "Table 1: Probability path definitions for existing methods which fit in the generalized conditional flow", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 196, 541, 210 ], "spans": [ { "bbox": [ 69, 196, 541, 210 ], "score": 1.0, "content": "matching framework (top) and our newly defined paths (bottom). We define two new probability path", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 207, 438, 222 ], "spans": [ { "bbox": [ 69, 207, 438, 222 ], "score": 1.0, "content": "objectives that can handle general source distributions and optimal transport flows.", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "table_body", "bbox": [ 70, 221, 539, 304 ], "group_id": 1, "lines": [ { "bbox": [ 70, 221, 539, 304 ], "spans": [ { "bbox": [ 70, 221, 539, 304 ], "score": 0.979, "html": "
Probability Pathq（2）μ（2)σtCond. OTMarginal OTGeneral source
Var. Exploding (Song& Ermon, 2019)q(𝑥1)T101-t××
Var.Preserving (Ho et al., 2020)q(x1)α1-t11-a²−t××
Flow Matching (Lipman et al., 2023)q(x1)tx1tσ-t+1× √××
Rectified Flow Liu (2022)q(xo)q(x1)tx1+(1-t)xo√×√
Var.Pres.Stochastic Interpolant Albergo & Vanden-Eijnden (2023)q(xo)q(x1）cos(1πt)𝑥0+sin(πt)𝑥1√×√
Independent CFMq(x0)q(x1)tc1+(1-t)𝑥√×√
(Ours) Optimal Transport CFMπ(xo,x1)tx1+(1-t)x√√√
(Ours)Schrodinger Bridge CFMπ2g²(x0,x1)tx1+(1-t)xoσ√t(1-t)√√√
", "type": "table", "image_path": "35966003199535f7f1cc3f6db0ab48780a49a5d82dba01a898ebd419227d86f1.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 70, 221, 539, 248.66666666666666 ], "spans": [], "index": 7 }, { "bbox": [ 70, 248.66666666666666, 539, 276.3333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 70, 276.3333333333333, 539, 304.0 ], "spans": [], "index": 9 } ] } ], "index": 6.5 }, { "type": "list", "bbox": [ 72, 324, 541, 372 ], "lines": [ { "bbox": [ 72, 324, 541, 337 ], "spans": [ { "bbox": [ 72, 324, 83, 337 ], "score": 1.0, "content": "•", "type": "text" }, { "bbox": [ 83, 325, 110, 336 ], "score": 0.31, "content": "\\ S 3 . 2 . 3", "type": "inline_equation" }, { "bbox": [ 110, 324, 258, 337 ], "score": 1.0, "content": ": We consider joint distributions", "type": "text" }, { "bbox": [ 259, 326, 328, 336 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 329, 324, 541, 337 ], "score": 1.0, "content": "that are given by minibatch optimal transport", "type": "text" } ], "index": 10, "is_list_start_line": true }, { "bbox": [ 81, 336, 367, 349 ], "spans": [ { "bbox": [ 81, 336, 367, 349 ], "score": 1.0, "content": "maps, causing the learned flow to be an (approximate) OT flow.", "type": "text" } ], "index": 11, "is_list_end_line": true }, { "bbox": [ 71, 348, 541, 362 ], "spans": [ { "bbox": [ 71, 348, 83, 362 ], "score": 1.0, "content": "•", "type": "text" }, { "bbox": [ 83, 350, 109, 360 ], "score": 0.38, "content": "\\ S 3 . 2 . 4", "type": "inline_equation" }, { "bbox": [ 110, 348, 169, 362 ], "score": 1.0, "content": ": we consider", "type": "text" }, { "bbox": [ 169, 350, 187, 361 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 187, 348, 541, 362 ], "score": 1.0, "content": "given by an entropy-regularized OT map and show that the CFM objective with", "type": "text" } ], "index": 12, "is_list_start_line": true }, { "bbox": [ 82, 359, 295, 373 ], "spans": [ { "bbox": [ 82, 359, 103, 373 ], "score": 1.0, "content": "this", "type": "text" }, { "bbox": [ 103, 362, 121, 372 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 122, 359, 295, 373 ], "score": 1.0, "content": "solves the Schrödinger bridge problem.", "type": "text" } ], "index": 13, "is_list_end_line": true } ], "index": 11.5, "bbox_fs": [ 71, 324, 541, 373 ] }, { "type": "title", "bbox": [ 71, 384, 208, 397 ], "lines": [ { "bbox": [ 69, 383, 209, 398 ], "spans": [ { "bbox": [ 69, 383, 209, 398 ], "score": 1.0, "content": "3.2.1 FM from the Gaussian", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 70, 405, 541, 441 ], "lines": [ { "bbox": [ 70, 403, 541, 417 ], "spans": [ { "bbox": [ 70, 403, 541, 417 ], "score": 1.0, "content": "Lipman et al. (2023) considered the problem of unconditional generative modeling given a training dataset.", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 416, 542, 431 ], "spans": [ { "bbox": [ 69, 416, 183, 431 ], "score": 1.0, "content": "Identifying the condition", "type": "text" }, { "bbox": [ 184, 423, 189, 427 ], "score": 0.89, "content": "z", "type": "inline_equation" }, { "bbox": [ 190, 416, 297, 431 ], "score": 1.0, "content": "with a single datapoint", "type": "text" }, { "bbox": [ 298, 422, 329, 429 ], "score": 0.9, "content": "z : = x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 329, 416, 493, 431 ], "score": 1.0, "content": ", and choosing a smoothing constant", "type": "text" }, { "bbox": [ 493, 420, 518, 428 ], "score": 0.92, "content": "\\sigma > 0", "type": "inline_equation" }, { "bbox": [ 518, 416, 542, 431 ], "score": 1.0, "content": ", one", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 430, 90, 442 ], "spans": [ { "bbox": [ 69, 430, 90, 442 ], "score": 1.0, "content": "sets", "type": "text" } ], "index": 17 } ], "index": 16, "bbox_fs": [ 69, 403, 542, 442 ] }, { "type": "interline_equation", "bbox": [ 230, 449, 381, 495 ], "lines": [ { "bbox": [ 230, 449, 381, 495 ], "spans": [ { "bbox": [ 230, 449, 381, 495 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } , ( t \\sigma - t + 1 ) ^ { 2 } ) , } \\\\ { u _ { t } ( x | z ) = \\displaystyle \\frac { x _ { 1 } - ( 1 - \\sigma ) x } { 1 - ( 1 - \\sigma ) t } , } \\end{array}", "type": "interline_equation", "image_path": "7c602eeb8e9184a4193736450c41c757e50c0237def786b286d558829431d14d.jpg" } ] } ], "index": 18.5, "virtual_lines": [ { "bbox": [ 230, 449, 381, 472.0 ], "spans": [], "index": 18 }, { "bbox": [ 230, 472.0, 381, 495.0 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 70, 501, 541, 550 ], "lines": [ { "bbox": [ 69, 501, 542, 515 ], "spans": [ { "bbox": [ 69, 501, 375, 515 ], "score": 1.0, "content": "which is a probability path from the standard normal distribution", "type": "text" }, { "bbox": [ 376, 504, 472, 514 ], "score": 0.89, "content": "( p _ { 0 } ( x | z ) = \\mathcal { N } ( x ; 0 , I ) )", "type": "inline_equation" }, { "bbox": [ 472, 501, 542, 515 ], "score": 1.0, "content": "to a Gaussian", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 512, 542, 528 ], "spans": [ { "bbox": [ 68, 512, 179, 528 ], "score": 1.0, "content": "distribution centered at", "type": "text" }, { "bbox": [ 179, 519, 189, 525 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 190, 512, 302, 528 ], "score": 1.0, "content": "with standard deviation", "type": "text" }, { "bbox": [ 303, 519, 309, 523 ], "score": 0.85, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 309, 512, 315, 528 ], "score": 1.0, "content": "(", "type": "text" }, { "bbox": [ 315, 515, 417, 526 ], "score": 0.9, "content": "p _ { 1 } ( x | z ) = \\mathcal { N } ( x ; x _ { 1 } , \\sigma ^ { 2 } ) )", "type": "inline_equation" }, { "bbox": [ 417, 512, 473, 528 ], "score": 1.0, "content": ". If one sets", "type": "text" }, { "bbox": [ 473, 515, 527, 526 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 527, 512, 542, 528 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 525, 542, 540 ], "spans": [ { "bbox": [ 69, 525, 542, 540 ], "score": 1.0, "content": "be the uniform distribution over the training dataset, the objective introduced by Lipman et al. (2023) is", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 538, 398, 551 ], "spans": [ { "bbox": [ 70, 538, 398, 551 ], "score": 1.0, "content": "equivalent to the CFM objective (10) for this conditional probability path.", "type": "text" } ], "index": 23 } ], "index": 21.5, "bbox_fs": [ 68, 501, 542, 551 ] }, { "type": "text", "bbox": [ 70, 555, 540, 592 ], "lines": [ { "bbox": [ 69, 555, 541, 569 ], "spans": [ { "bbox": [ 69, 555, 351, 569 ], "score": 1.0, "content": "We emphasize that although the conditional probability path", "type": "text" }, { "bbox": [ 351, 558, 381, 568 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 382, 555, 541, 569 ], "score": 1.0, "content": "is an optimal transport path from", "type": "text" } ], "index": 24 }, { "bbox": [ 71, 567, 542, 581 ], "spans": [ { "bbox": [ 71, 569, 102, 580 ], "score": 0.94, "content": "p _ { 0 } ( x | z )", "type": "inline_equation" }, { "bbox": [ 102, 567, 117, 581 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 117, 569, 148, 580 ], "score": 0.94, "content": "p _ { 1 } ( x | z )", "type": "inline_equation" }, { "bbox": [ 148, 567, 235, 581 ], "score": 1.0, "content": ", the marginal path", "type": "text" }, { "bbox": [ 235, 569, 257, 580 ], "score": 0.93, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 258, 567, 504, 581 ], "score": 1.0, "content": "is not in general an OT path from the standard normal", "type": "text" }, { "bbox": [ 504, 569, 527, 580 ], "score": 0.93, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 528, 567, 542, 581 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 577, 193, 594 ], "spans": [ { "bbox": [ 69, 577, 165, 594 ], "score": 1.0, "content": "the data distribution", "type": "text" }, { "bbox": [ 165, 581, 189, 592 ], "score": 0.94, "content": "p _ { 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 189, 577, 193, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 69, 555, 542, 594 ] }, { "type": "title", "bbox": [ 70, 603, 298, 617 ], "lines": [ { "bbox": [ 68, 601, 299, 620 ], "spans": [ { "bbox": [ 68, 601, 299, 620 ], "score": 1.0, "content": "3.2.2 Basic form of CFM: Independent coupling", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 70, 624, 541, 660 ], "lines": [ { "bbox": [ 69, 623, 541, 637 ], "spans": [ { "bbox": [ 69, 623, 286, 637 ], "score": 1.0, "content": "In the basic form of CFM (I-CFM), we identify", "type": "text" }, { "bbox": [ 286, 630, 292, 635 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 293, 623, 509, 637 ], "score": 1.0, "content": "with a pair of random variables, a source point", "type": "text" }, { "bbox": [ 509, 630, 519, 636 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 520, 623, 541, 637 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 28 }, { "bbox": [ 68, 636, 542, 649 ], "spans": [ { "bbox": [ 68, 636, 137, 649 ], "score": 1.0, "content": "a target point", "type": "text" }, { "bbox": [ 137, 642, 147, 648 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 148, 636, 190, 649 ], "score": 1.0, "content": ", and set", "type": "text" }, { "bbox": [ 191, 639, 269, 649 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 269, 636, 542, 649 ], "score": 1.0, "content": "to be the independent coupling. We let the conditionals be", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 648, 390, 662 ], "spans": [ { "bbox": [ 69, 648, 177, 662 ], "score": 1.0, "content": "Gaussian flows between", "type": "text" }, { "bbox": [ 177, 654, 187, 660 ], "score": 0.89, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 188, 648, 210, 662 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 654, 220, 660 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 221, 648, 331, 662 ], "score": 1.0, "content": "with standard deviation", "type": "text" }, { "bbox": [ 331, 654, 338, 658 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 338, 648, 390, 662 ], "score": 1.0, "content": ", defined by", "type": "text" } ], "index": 30 } ], "index": 29, "bbox_fs": [ 68, 623, 542, 662 ] }, { "type": "interline_equation", "bbox": [ 226, 669, 385, 701 ], "lines": [ { "bbox": [ 226, 669, 385, 701 ], "spans": [ { "bbox": [ 226, 669, 385, 701 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) , } \\\\ { u _ { t } ( x | z ) = x _ { 1 } - x _ { 0 } . } \\end{array}", "type": "interline_equation", "image_path": "df15a84a85891e946d0afbb4241bff6c4a5a11d317c78bd73e3a6a8ea2a45044.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 226, 669, 385, 685.0 ], "spans": [], "index": 31 }, { "bbox": [ 226, 685.0, 385, 701.0 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 69, 708, 543, 733 ], "lines": [ { "bbox": [ 70, 708, 542, 721 ], "spans": [ { "bbox": [ 70, 708, 222, 721 ], "score": 1.0, "content": "We note that the formulation of", "type": "text" }, { "bbox": [ 222, 710, 253, 721 ], "score": 0.93, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 253, 708, 542, 721 ], "score": 1.0, "content": "follows from an application of Theorem 2.1 to the conditional", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 720, 541, 734 ], "spans": [ { "bbox": [ 69, 720, 171, 734 ], "score": 1.0, "content": "probability path with", "type": "text" }, { "bbox": [ 172, 722, 264, 733 ], "score": 0.93, "content": "\\mu _ { t } = t x _ { 1 } + ( 1 - t ) x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 264, 720, 288, 734 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 288, 725, 320, 732 ], "score": 0.91, "content": "\\sigma _ { t } = \\sigma", "type": "inline_equation" }, { "bbox": [ 320, 720, 452, 734 ], "score": 1.0, "content": ". Furthermore, we note that", "type": "text" }, { "bbox": [ 452, 722, 483, 732 ], "score": 0.94, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 483, 720, 541, 734 ], "score": 1.0, "content": "is efficiently", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 81, 541, 95 ], "spans": [ { "bbox": [ 69, 81, 137, 95 ], "score": 1.0, "content": "samplable and", "type": "text", "cross_page": true }, { "bbox": [ 138, 87, 147, 93 ], "score": 0.89, "content": "u _ { t }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 147, 81, 374, 95 ], "score": 1.0, "content": "is efficiently computable, thus gradient descent on", "type": "text", "cross_page": true }, { "bbox": [ 375, 85, 400, 93 ], "score": 0.92, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 400, 81, 541, 95 ], "score": 1.0, "content": "is also efficient. For this choice", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 70, 93, 541, 108 ], "spans": [ { "bbox": [ 70, 93, 82, 108 ], "score": 1.0, "content": "of", "type": "text", "cross_page": true }, { "bbox": [ 83, 99, 88, 104 ], "score": 0.86, "content": "z", "type": "inline_equation", "cross_page": true }, { "bbox": [ 88, 93, 93, 108 ], "score": 1.0, "content": ",", "type": "text", "cross_page": true }, { "bbox": [ 94, 96, 121, 106 ], "score": 0.93, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 122, 93, 147, 108 ], "score": 1.0, "content": ", and", "type": "text", "cross_page": true }, { "bbox": [ 147, 96, 175, 106 ], "score": 0.94, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 175, 93, 430, 108 ], "score": 1.0, "content": ", we know the marginal boundary probabilities approach", "type": "text", "cross_page": true }, { "bbox": [ 430, 99, 439, 106 ], "score": 0.89, "content": "q _ { 0 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 439, 93, 462, 108 ], "score": 1.0, "content": "and", "type": "text", "cross_page": true }, { "bbox": [ 462, 99, 471, 106 ], "score": 0.9, "content": "q _ { 1 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 471, 93, 541, 108 ], "score": 1.0, "content": "respectively as", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 71, 105, 326, 119 ], "spans": [ { "bbox": [ 71, 109, 98, 116 ], "score": 0.91, "content": "\\sigma 0", "type": "inline_equation", "cross_page": true }, { "bbox": [ 98, 105, 326, 119 ], "score": 1.0, "content": ". This is made explicit in the following Proposition:", "type": "text", "cross_page": true } ], "index": 2 } ], "index": 33.5, "bbox_fs": [ 69, 708, 542, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 82, 541, 118 ], "lines": [ { "bbox": [ 69, 81, 541, 95 ], "spans": [ { "bbox": [ 69, 81, 137, 95 ], "score": 1.0, "content": "samplable and", "type": "text" }, { "bbox": [ 138, 87, 147, 93 ], "score": 0.89, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 147, 81, 374, 95 ], "score": 1.0, "content": "is efficiently computable, thus gradient descent on", "type": "text" }, { "bbox": [ 375, 85, 400, 93 ], "score": 0.92, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 400, 81, 541, 95 ], "score": 1.0, "content": "is also efficient. For this choice", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 541, 108 ], "spans": [ { "bbox": [ 70, 93, 82, 108 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 83, 99, 88, 104 ], "score": 0.86, "content": "z", "type": "inline_equation" }, { "bbox": [ 88, 93, 93, 108 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 94, 96, 121, 106 ], "score": 0.93, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation" }, { "bbox": [ 122, 93, 147, 108 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 147, 96, 175, 106 ], "score": 0.94, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation" }, { "bbox": [ 175, 93, 430, 108 ], "score": 1.0, "content": ", we know the marginal boundary probabilities approach", "type": "text" }, { "bbox": [ 430, 99, 439, 106 ], "score": 0.89, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 439, 93, 462, 108 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 462, 99, 471, 106 ], "score": 0.9, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 471, 93, 541, 108 ], "score": 1.0, "content": "respectively as", "type": "text" } ], "index": 1 }, { "bbox": [ 71, 105, 326, 119 ], "spans": [ { "bbox": [ 71, 109, 98, 116 ], "score": 0.91, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 98, 105, 326, 119 ], "score": 1.0, "content": ". This is made explicit in the following Proposition:", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 71, 122, 540, 158 ], "lines": [ { "bbox": [ 70, 122, 542, 136 ], "spans": [ { "bbox": [ 70, 122, 217, 136 ], "score": 1.0, "content": "Proposition 3.3. The marginal", "type": "text" }, { "bbox": [ 218, 127, 226, 134 ], "score": 0.88, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 227, 122, 304, 136 ], "score": 1.0, "content": "corresponding to", "type": "text" }, { "bbox": [ 305, 124, 381, 135 ], "score": 0.93, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 382, 122, 420, 136 ], "score": 1.0, "content": "and the", "type": "text" }, { "bbox": [ 420, 124, 486, 135 ], "score": 0.93, "content": "p _ { t } ( x | z ) , u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 486, 122, 542, 136 ], "score": 1.0, "content": "in (14) and", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 133, 542, 149 ], "spans": [ { "bbox": [ 69, 133, 196, 149 ], "score": 1.0, "content": "(15) has boundary conditions", "type": "text" }, { "bbox": [ 196, 136, 287, 146 ], "score": 0.93, "content": "p _ { 1 } = q _ { 1 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 133, 308, 149 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 136, 399, 146 ], "score": 0.92, "content": "p _ { 0 } = q _ { 0 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 399, 133, 432, 149 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 432, 139, 437, 144 ], "score": 0.66, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 438, 133, 542, 149 ], "score": 1.0, "content": "denotes the convolution", "type": "text" } ], "index": 4 }, { "bbox": [ 68, 147, 111, 158 ], "spans": [ { "bbox": [ 68, 147, 111, 158 ], "score": 1.0, "content": "operator.", "type": "text" } ], "index": 5 } ], "index": 4 }, { "type": "text", "bbox": [ 71, 167, 541, 228 ], "lines": [ { "bbox": [ 68, 166, 541, 183 ], "spans": [ { "bbox": [ 68, 166, 143, 183 ], "score": 1.0, "content": "In particular, as", "type": "text" }, { "bbox": [ 144, 171, 170, 178 ], "score": 0.91, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 171, 166, 284, 183 ], "score": 1.0, "content": ", the marginal vector field", "type": "text" }, { "bbox": [ 285, 170, 307, 180 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 308, 166, 517, 183 ], "score": 1.0, "content": "approaches one that transports the distribution", "type": "text" }, { "bbox": [ 518, 170, 541, 180 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 70, 179, 542, 193 ], "spans": [ { "bbox": [ 70, 179, 83, 193 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 83, 182, 106, 192 ], "score": 0.94, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 106, 179, 310, 193 ], "score": 1.0, "content": "and can thus be seen as a generative model of", "type": "text" }, { "bbox": [ 310, 185, 321, 191 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 321, 179, 491, 193 ], "score": 1.0, "content": ". Note that there is no requirement for", "type": "text" }, { "bbox": [ 491, 182, 514, 192 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 514, 179, 542, 193 ], "score": 1.0, "content": "to be", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 191, 541, 205 ], "spans": [ { "bbox": [ 69, 191, 193, 205 ], "score": 1.0, "content": "Gaussian. Conditioning on", "type": "text" }, { "bbox": [ 193, 197, 204, 203 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 204, 191, 541, 205 ], "score": 1.0, "content": "allows us to generalize flow matching to arbitrary source distributions with", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 204, 541, 217 ], "spans": [ { "bbox": [ 70, 204, 541, 217 ], "score": 1.0, "content": "intractable densities. In the case of FM from a Gaussian, while each conditional flow is the dynamic optimal", "type": "text" } ], "index": 9 }, { "bbox": [ 70, 215, 535, 229 ], "spans": [ { "bbox": [ 70, 215, 159, 229 ], "score": 1.0, "content": "transport flow from", "type": "text" }, { "bbox": [ 159, 217, 202, 228 ], "score": 0.94, "content": "\\mathcal { N } ( x _ { 0 } , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 203, 215, 217, 229 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 217, 217, 261, 228 ], "score": 0.94, "content": "\\mathcal { N } ( x _ { 1 } , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 261, 215, 376, 229 ], "score": 1.0, "content": ", the marginal vector field", "type": "text" }, { "bbox": [ 376, 218, 399, 228 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 399, 215, 535, 229 ], "score": 1.0, "content": "is not necessarily an OT flow.", "type": "text" } ], "index": 10 } ], "index": 8 }, { "type": "text", "bbox": [ 70, 239, 541, 312 ], "lines": [ { "bbox": [ 70, 239, 541, 254 ], "spans": [ { "bbox": [ 70, 239, 541, 254 ], "score": 1.0, "content": "Connection with related Rectified Flow and Stochastic interpolants methods. We note that", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 252, 541, 266 ], "spans": [ { "bbox": [ 69, 252, 356, 266 ], "score": 1.0, "content": "I-CFM is closely related to the algorithms proposed by Albergo", "type": "text" }, { "bbox": [ 356, 254, 364, 262 ], "score": 0.79, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 365, 252, 541, 266 ], "score": 1.0, "content": "Vanden-Eijnden (2023); Liu (2022). In", "type": "text" } ], "index": 12 }, { "bbox": [ 68, 262, 543, 279 ], "spans": [ { "bbox": [ 68, 262, 287, 279 ], "score": 1.0, "content": "the case where the conditional probability path", "type": "text" }, { "bbox": [ 288, 269, 296, 276 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 297, 262, 347, 279 ], "score": 1.0, "content": "is a Dirac", "type": "text" }, { "bbox": [ 348, 266, 404, 276 ], "score": 0.77, "content": "( i . e . , \\sigma = 0 )", "type": "inline_equation" }, { "bbox": [ 405, 262, 543, 279 ], "score": 1.0, "content": "), I-CFM is equivalent to (Liu,", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 275, 541, 289 ], "spans": [ { "bbox": [ 69, 275, 317, 289 ], "score": 1.0, "content": "2022). Furthermore, if we consider the Gaussian mean", "type": "text" }, { "bbox": [ 317, 277, 447, 289 ], "score": 0.93, "content": "\\begin{array} { r } { \\mu _ { t } = \\cos ( \\frac { 1 } { 2 } \\pi t ) x _ { 0 } + \\sin ( \\frac { 1 } { 2 } \\pi t ) x _ { 1 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 447, 275, 541, 289 ], "score": 1.0, "content": "instead of the linear", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 288, 539, 302 ], "spans": [ { "bbox": [ 69, 288, 531, 302 ], "score": 1.0, "content": "interpolation, I-CFM would be equivalent to the variance preserving stochastic interpolant in Albergo", "type": "text" }, { "bbox": [ 531, 290, 539, 298 ], "score": 0.27, "content": "\\&", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 70, 300, 352, 313 ], "spans": [ { "bbox": [ 70, 300, 352, 313 ], "score": 1.0, "content": "Vanden-Eijnden (2023), which has also been further generalized.", "type": "text" } ], "index": 16 } ], "index": 13.5 }, { "type": "text", "bbox": [ 70, 324, 541, 361 ], "lines": [ { "bbox": [ 70, 324, 541, 338 ], "spans": [ { "bbox": [ 70, 324, 541, 338 ], "score": 1.0, "content": "Connection to FM from the Gaussian. There exists a set of conditional probability paths conditioned", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 336, 541, 350 ], "spans": [ { "bbox": [ 69, 336, 85, 350 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 85, 342, 95, 348 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 96, 336, 118, 350 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 119, 338, 174, 349 ], "score": 0.94, "content": "x _ { 0 } \\sim \\mathcal { N } ( 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 175, 336, 429, 350 ], "score": 1.0, "content": "that have an equivalent probability flow to the marginal", "type": "text" }, { "bbox": [ 430, 342, 438, 348 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 439, 336, 541, 350 ], "score": 1.0, "content": "of flow matching from", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 347, 435, 362 ], "spans": [ { "bbox": [ 69, 347, 298, 362 ], "score": 1.0, "content": "the Gaussian (§3.2.1), which is only conditioned on", "type": "text" }, { "bbox": [ 298, 353, 308, 360 ], "score": 0.91, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 309, 347, 435, 362 ], "score": 1.0, "content": ". These paths are defined by", "type": "text" } ], "index": 19 } ], "index": 18 }, { "type": "interline_equation", "bbox": [ 193, 369, 418, 383 ], "lines": [ { "bbox": [ 193, 369, 418, 383 ], "spans": [ { "bbox": [ 193, 369, 418, 383 ], "score": 0.89, "content": "p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , ( \\sigma t ) ^ { 2 } + 2 \\sigma t ( 1 - t ) ) .", "type": "interline_equation", "image_path": "ed6a002948dd1cb476d6f3e0c3af14d3b7132803b00b70d860930e3b9fa455dd.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 193, 369, 418, 383 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 391, 514, 404 ], "lines": [ { "bbox": [ 70, 391, 516, 405 ], "spans": [ { "bbox": [ 70, 391, 516, 405 ], "score": 1.0, "content": "Proposition B.1 states an equivalence between I-CFM with these paths and the objective from §3.2.1.", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "title", "bbox": [ 72, 416, 214, 429 ], "lines": [ { "bbox": [ 69, 415, 215, 431 ], "spans": [ { "bbox": [ 69, 415, 215, 431 ], "score": 1.0, "content": "3.2.3 Optimal transport CFM", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 436, 540, 484 ], "lines": [ { "bbox": [ 69, 436, 540, 449 ], "spans": [ { "bbox": [ 69, 436, 540, 449 ], "score": 1.0, "content": "In this section, we present our second main contribution. The formulation in the previous section can readily", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 449, 541, 462 ], "spans": [ { "bbox": [ 70, 449, 206, 462 ], "score": 1.0, "content": "be generalized to distributions", "type": "text" }, { "bbox": [ 207, 451, 275, 461 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 275, 449, 318, 462 ], "score": 1.0, "content": "in which", "type": "text" }, { "bbox": [ 318, 454, 328, 460 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 329, 449, 351, 462 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 351, 454, 361, 460 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 361, 449, 504, 462 ], "score": 1.0, "content": "are not independent, as long as", "type": "text" }, { "bbox": [ 504, 451, 522, 461 ], "score": 0.94, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 522, 449, 541, 462 ], "score": 1.0, "content": "has", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 460, 541, 475 ], "spans": [ { "bbox": [ 69, 460, 116, 475 ], "score": 1.0, "content": "marginals", "type": "text" }, { "bbox": [ 116, 462, 138, 473 ], "score": 0.94, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 139, 460, 160, 475 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 161, 462, 184, 473 ], "score": 0.93, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 184, 460, 316, 475 ], "score": 1.0, "content": ". Therefore, we propose to set", "type": "text" }, { "bbox": [ 316, 462, 334, 473 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 334, 460, 541, 475 ], "score": 1.0, "content": "to be the 2-Wasserstein optimal transport map", "type": "text" } ], "index": 25 }, { "bbox": [ 71, 471, 246, 486 ], "spans": [ { "bbox": [ 71, 478, 77, 482 ], "score": 0.89, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 78, 471, 246, 486 ], "score": 1.0, "content": "achieving the infimum in (6), namely,", "type": "text" } ], "index": 26 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 268, 484, 344, 497 ], "lines": [ { "bbox": [ 268, 484, 344, 497 ], "spans": [ { "bbox": [ 268, 484, 344, 497 ], "score": 0.93, "content": "q ( z ) : = \\pi ( x _ { 0 } , x _ { 1 } ) .", "type": "interline_equation", "image_path": "a161670994a9ce0b334b2c59b724bfabd4ec932f87553f8049949c3686cb7661.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 268, 484, 344, 497 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 70, 502, 541, 550 ], "lines": [ { "bbox": [ 69, 502, 541, 516 ], "spans": [ { "bbox": [ 69, 502, 125, 516 ], "score": 1.0, "content": "In this case,", "type": "text" }, { "bbox": [ 126, 508, 131, 513 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 131, 502, 300, 516 ], "score": 1.0, "content": "is still a tuple of points, but instead of", "type": "text" }, { "bbox": [ 300, 508, 325, 515 ], "score": 0.91, "content": "x _ { 0 } , x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 326, 502, 541, 516 ], "score": 1.0, "content": "being sampled independently from their marginal", "type": "text" } ], "index": 28 }, { "bbox": [ 70, 514, 541, 527 ], "spans": [ { "bbox": [ 70, 514, 432, 527 ], "score": 1.0, "content": "distributions, they are sampled jointly according to the optimal transport map", "type": "text" }, { "bbox": [ 432, 520, 439, 524 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 439, 514, 541, 527 ], "score": 1.0, "content": ". We call this method", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 526, 542, 540 ], "spans": [ { "bbox": [ 69, 526, 297, 540 ], "score": 1.0, "content": "optimal transport CFM (OT-CFM). If one uses the", "type": "text" }, { "bbox": [ 297, 528, 327, 539 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 327, 526, 419, 540 ], "score": 1.0, "content": "defined by (14) and", "type": "text" }, { "bbox": [ 419, 528, 450, 539 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 450, 526, 542, 540 ], "score": 1.0, "content": "in (15), OT-CFM is", "type": "text" } ], "index": 30 }, { "bbox": [ 70, 538, 349, 551 ], "spans": [ { "bbox": [ 70, 538, 349, 551 ], "score": 1.0, "content": "equivalent to dynamic optimal transport in the following sense.", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 70, 554, 540, 591 ], "lines": [ { "bbox": [ 70, 554, 541, 568 ], "spans": [ { "bbox": [ 70, 554, 339, 568 ], "score": 1.0, "content": "Proposition 3.4. The results of Proposition 3.3 also hold for", "type": "text" }, { "bbox": [ 339, 556, 357, 567 ], "score": 0.92, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 357, 554, 541, 568 ], "score": 1.0, "content": "in (17). Furthermore, assuming regularity", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 565, 542, 581 ], "spans": [ { "bbox": [ 69, 565, 128, 581 ], "score": 1.0, "content": "properties of", "type": "text" }, { "bbox": [ 128, 572, 137, 578 ], "score": 0.7, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 138, 565, 144, 581 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 144, 572, 153, 578 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 153, 565, 295, 581 ], "score": 1.0, "content": ", and the optimal transport plan", "type": "text" }, { "bbox": [ 295, 572, 302, 576 ], "score": 0.86, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 302, 565, 320, 581 ], "score": 1.0, "content": ", as", "type": "text" }, { "bbox": [ 320, 568, 352, 576 ], "score": 0.92, "content": "\\sigma ^ { 2 } \\to 0", "type": "inline_equation" }, { "bbox": [ 352, 565, 434, 581 ], "score": 1.0, "content": "the marginal path", "type": "text" }, { "bbox": [ 434, 572, 443, 578 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 443, 565, 487, 581 ], "score": 1.0, "content": "and field", "type": "text" }, { "bbox": [ 487, 572, 496, 578 ], "score": 0.88, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 496, 565, 542, 581 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 33 }, { "bbox": [ 68, 577, 407, 594 ], "spans": [ { "bbox": [ 68, 577, 110, 594 ], "score": 1.0, "content": "(7), i.e.,", "type": "text" }, { "bbox": [ 110, 583, 119, 590 ], "score": 0.88, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 120, 577, 360, 594 ], "score": 1.0, "content": "solves the dynamic optimal transport problem between", "type": "text" }, { "bbox": [ 360, 583, 369, 590 ], "score": 0.87, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 369, 577, 392, 594 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 392, 583, 401, 590 ], "score": 0.88, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 401, 577, 407, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 34 } ], "index": 33 }, { "type": "text", "bbox": [ 70, 600, 540, 672 ], "lines": [ { "bbox": [ 69, 599, 541, 614 ], "spans": [ { "bbox": [ 69, 599, 541, 614 ], "score": 1.0, "content": "We consider two cases: (1) when the data set is small enough and we know the static optimal transport plan", "type": "text" } ], "index": 35 }, { "bbox": [ 68, 611, 541, 626 ], "spans": [ { "bbox": [ 68, 611, 541, 626 ], "score": 1.0, "content": "(e.g. single cell data). (2) when the data is too large (or continuous) (e.g. image data) and the static OT", "type": "text" } ], "index": 36 }, { "bbox": [ 70, 624, 540, 636 ], "spans": [ { "bbox": [ 70, 624, 540, 636 ], "score": 1.0, "content": "plan is computationally infeasible to determine exactly. In the first case we are able to extend the transport", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 636, 540, 650 ], "spans": [ { "bbox": [ 69, 636, 540, 650 ], "score": 1.0, "content": "map to unseen data similar to the task presented in Bunne et al. (2023). In the second case we show an", "type": "text" } ], "index": 38 }, { "bbox": [ 68, 647, 542, 662 ], "spans": [ { "bbox": [ 68, 647, 542, 662 ], "score": 1.0, "content": "approximation with minibatch OT improves over a random plan in terms of generative modelling performance", "type": "text" } ], "index": 39 }, { "bbox": [ 70, 660, 153, 673 ], "spans": [ { "bbox": [ 70, 660, 153, 673 ], "score": 1.0, "content": "and training time.", "type": "text" } ], "index": 40 } ], "index": 37.5 }, { "type": "text", "bbox": [ 71, 684, 540, 732 ], "lines": [ { "bbox": [ 70, 684, 541, 698 ], "spans": [ { "bbox": [ 70, 684, 394, 698 ], "score": 1.0, "content": "Minibatch OT approximation. For large datasets, the transport plan", "type": "text" }, { "bbox": [ 395, 690, 401, 694 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 401, 684, 541, 698 ], "score": 1.0, "content": "can be difficult to compute and", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 696, 542, 709 ], "spans": [ { "bbox": [ 69, 696, 542, 709 ], "score": 1.0, "content": "store due to OT’s cubic time and quadratic memory complexity in the number of samples (Cuturi, 2013;", "type": "text" } ], "index": 42 }, { "bbox": [ 70, 708, 542, 721 ], "spans": [ { "bbox": [ 70, 708, 542, 721 ], "score": 1.0, "content": "Tong et al., 2020). Therefore, we rely on a minibatch OT approximation similar to Fatras et al. (2021b).", "type": "text" } ], "index": 43 }, { "bbox": [ 71, 721, 540, 732 ], "spans": [ { "bbox": [ 71, 721, 540, 732 ], "score": 1.0, "content": "Although minibatch OT incurs an error relative to the exact OT solution, it has been successfully used in", "type": "text" } ], "index": 44 } ], "index": 42.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 9 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 82, 541, 118 ], "lines": [], "index": 1, "bbox_fs": [ 69, 81, 541, 119 ], "lines_deleted": true }, { "type": "text", "bbox": [ 71, 122, 540, 158 ], "lines": [ { "bbox": [ 70, 122, 542, 136 ], "spans": [ { "bbox": [ 70, 122, 217, 136 ], "score": 1.0, "content": "Proposition 3.3. The marginal", "type": "text" }, { "bbox": [ 218, 127, 226, 134 ], "score": 0.88, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 227, 122, 304, 136 ], "score": 1.0, "content": "corresponding to", "type": "text" }, { "bbox": [ 305, 124, 381, 135 ], "score": 0.93, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 382, 122, 420, 136 ], "score": 1.0, "content": "and the", "type": "text" }, { "bbox": [ 420, 124, 486, 135 ], "score": 0.93, "content": "p _ { t } ( x | z ) , u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 486, 122, 542, 136 ], "score": 1.0, "content": "in (14) and", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 133, 542, 149 ], "spans": [ { "bbox": [ 69, 133, 196, 149 ], "score": 1.0, "content": "(15) has boundary conditions", "type": "text" }, { "bbox": [ 196, 136, 287, 146 ], "score": 0.93, "content": "p _ { 1 } = q _ { 1 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 133, 308, 149 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 136, 399, 146 ], "score": 0.92, "content": "p _ { 0 } = q _ { 0 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 399, 133, 432, 149 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 432, 139, 437, 144 ], "score": 0.66, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 438, 133, 542, 149 ], "score": 1.0, "content": "denotes the convolution", "type": "text" } ], "index": 4 }, { "bbox": [ 68, 147, 111, 158 ], "spans": [ { "bbox": [ 68, 147, 111, 158 ], "score": 1.0, "content": "operator.", "type": "text" } ], "index": 5 } ], "index": 4, "bbox_fs": [ 68, 122, 542, 158 ] }, { "type": "text", "bbox": [ 71, 167, 541, 228 ], "lines": [ { "bbox": [ 68, 166, 541, 183 ], "spans": [ { "bbox": [ 68, 166, 143, 183 ], "score": 1.0, "content": "In particular, as", "type": "text" }, { "bbox": [ 144, 171, 170, 178 ], "score": 0.91, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 171, 166, 284, 183 ], "score": 1.0, "content": ", the marginal vector field", "type": "text" }, { "bbox": [ 285, 170, 307, 180 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 308, 166, 517, 183 ], "score": 1.0, "content": "approaches one that transports the distribution", "type": "text" }, { "bbox": [ 518, 170, 541, 180 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 70, 179, 542, 193 ], "spans": [ { "bbox": [ 70, 179, 83, 193 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 83, 182, 106, 192 ], "score": 0.94, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 106, 179, 310, 193 ], "score": 1.0, "content": "and can thus be seen as a generative model of", "type": "text" }, { "bbox": [ 310, 185, 321, 191 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 321, 179, 491, 193 ], "score": 1.0, "content": ". Note that there is no requirement for", "type": "text" }, { "bbox": [ 491, 182, 514, 192 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 514, 179, 542, 193 ], "score": 1.0, "content": "to be", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 191, 541, 205 ], "spans": [ { "bbox": [ 69, 191, 193, 205 ], "score": 1.0, "content": "Gaussian. Conditioning on", "type": "text" }, { "bbox": [ 193, 197, 204, 203 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 204, 191, 541, 205 ], "score": 1.0, "content": "allows us to generalize flow matching to arbitrary source distributions with", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 204, 541, 217 ], "spans": [ { "bbox": [ 70, 204, 541, 217 ], "score": 1.0, "content": "intractable densities. In the case of FM from a Gaussian, while each conditional flow is the dynamic optimal", "type": "text" } ], "index": 9 }, { "bbox": [ 70, 215, 535, 229 ], "spans": [ { "bbox": [ 70, 215, 159, 229 ], "score": 1.0, "content": "transport flow from", "type": "text" }, { "bbox": [ 159, 217, 202, 228 ], "score": 0.94, "content": "\\mathcal { N } ( x _ { 0 } , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 203, 215, 217, 229 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 217, 217, 261, 228 ], "score": 0.94, "content": "\\mathcal { N } ( x _ { 1 } , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 261, 215, 376, 229 ], "score": 1.0, "content": ", the marginal vector field", "type": "text" }, { "bbox": [ 376, 218, 399, 228 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 399, 215, 535, 229 ], "score": 1.0, "content": "is not necessarily an OT flow.", "type": "text" } ], "index": 10 } ], "index": 8, "bbox_fs": [ 68, 166, 542, 229 ] }, { "type": "text", "bbox": [ 70, 239, 541, 312 ], "lines": [ { "bbox": [ 70, 239, 541, 254 ], "spans": [ { "bbox": [ 70, 239, 541, 254 ], "score": 1.0, "content": "Connection with related Rectified Flow and Stochastic interpolants methods. We note that", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 252, 541, 266 ], "spans": [ { "bbox": [ 69, 252, 356, 266 ], "score": 1.0, "content": "I-CFM is closely related to the algorithms proposed by Albergo", "type": "text" }, { "bbox": [ 356, 254, 364, 262 ], "score": 0.79, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 365, 252, 541, 266 ], "score": 1.0, "content": "Vanden-Eijnden (2023); Liu (2022). In", "type": "text" } ], "index": 12 }, { "bbox": [ 68, 262, 543, 279 ], "spans": [ { "bbox": [ 68, 262, 287, 279 ], "score": 1.0, "content": "the case where the conditional probability path", "type": "text" }, { "bbox": [ 288, 269, 296, 276 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 297, 262, 347, 279 ], "score": 1.0, "content": "is a Dirac", "type": "text" }, { "bbox": [ 348, 266, 404, 276 ], "score": 0.77, "content": "( i . e . , \\sigma = 0 )", "type": "inline_equation" }, { "bbox": [ 405, 262, 543, 279 ], "score": 1.0, "content": "), I-CFM is equivalent to (Liu,", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 275, 541, 289 ], "spans": [ { "bbox": [ 69, 275, 317, 289 ], "score": 1.0, "content": "2022). Furthermore, if we consider the Gaussian mean", "type": "text" }, { "bbox": [ 317, 277, 447, 289 ], "score": 0.93, "content": "\\begin{array} { r } { \\mu _ { t } = \\cos ( \\frac { 1 } { 2 } \\pi t ) x _ { 0 } + \\sin ( \\frac { 1 } { 2 } \\pi t ) x _ { 1 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 447, 275, 541, 289 ], "score": 1.0, "content": "instead of the linear", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 288, 539, 302 ], "spans": [ { "bbox": [ 69, 288, 531, 302 ], "score": 1.0, "content": "interpolation, I-CFM would be equivalent to the variance preserving stochastic interpolant in Albergo", "type": "text" }, { "bbox": [ 531, 290, 539, 298 ], "score": 0.27, "content": "\\&", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 70, 300, 352, 313 ], "spans": [ { "bbox": [ 70, 300, 352, 313 ], "score": 1.0, "content": "Vanden-Eijnden (2023), which has also been further generalized.", "type": "text" } ], "index": 16 } ], "index": 13.5, "bbox_fs": [ 68, 239, 543, 313 ] }, { "type": "text", "bbox": [ 70, 324, 541, 361 ], "lines": [ { "bbox": [ 70, 324, 541, 338 ], "spans": [ { "bbox": [ 70, 324, 541, 338 ], "score": 1.0, "content": "Connection to FM from the Gaussian. There exists a set of conditional probability paths conditioned", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 336, 541, 350 ], "spans": [ { "bbox": [ 69, 336, 85, 350 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 85, 342, 95, 348 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 96, 336, 118, 350 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 119, 338, 174, 349 ], "score": 0.94, "content": "x _ { 0 } \\sim \\mathcal { N } ( 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 175, 336, 429, 350 ], "score": 1.0, "content": "that have an equivalent probability flow to the marginal", "type": "text" }, { "bbox": [ 430, 342, 438, 348 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 439, 336, 541, 350 ], "score": 1.0, "content": "of flow matching from", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 347, 435, 362 ], "spans": [ { "bbox": [ 69, 347, 298, 362 ], "score": 1.0, "content": "the Gaussian (§3.2.1), which is only conditioned on", "type": "text" }, { "bbox": [ 298, 353, 308, 360 ], "score": 0.91, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 309, 347, 435, 362 ], "score": 1.0, "content": ". These paths are defined by", "type": "text" } ], "index": 19 } ], "index": 18, "bbox_fs": [ 69, 324, 541, 362 ] }, { "type": "interline_equation", "bbox": [ 193, 369, 418, 383 ], "lines": [ { "bbox": [ 193, 369, 418, 383 ], "spans": [ { "bbox": [ 193, 369, 418, 383 ], "score": 0.89, "content": "p _ { t } ( x | z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , ( \\sigma t ) ^ { 2 } + 2 \\sigma t ( 1 - t ) ) .", "type": "interline_equation", "image_path": "ed6a002948dd1cb476d6f3e0c3af14d3b7132803b00b70d860930e3b9fa455dd.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 193, 369, 418, 383 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 391, 514, 404 ], "lines": [ { "bbox": [ 70, 391, 516, 405 ], "spans": [ { "bbox": [ 70, 391, 516, 405 ], "score": 1.0, "content": "Proposition B.1 states an equivalence between I-CFM with these paths and the objective from §3.2.1.", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 70, 391, 516, 405 ] }, { "type": "title", "bbox": [ 72, 416, 214, 429 ], "lines": [ { "bbox": [ 69, 415, 215, 431 ], "spans": [ { "bbox": [ 69, 415, 215, 431 ], "score": 1.0, "content": "3.2.3 Optimal transport CFM", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 436, 540, 484 ], "lines": [ { "bbox": [ 69, 436, 540, 449 ], "spans": [ { "bbox": [ 69, 436, 540, 449 ], "score": 1.0, "content": "In this section, we present our second main contribution. The formulation in the previous section can readily", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 449, 541, 462 ], "spans": [ { "bbox": [ 70, 449, 206, 462 ], "score": 1.0, "content": "be generalized to distributions", "type": "text" }, { "bbox": [ 207, 451, 275, 461 ], "score": 0.94, "content": "q ( z ) = q ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 275, 449, 318, 462 ], "score": 1.0, "content": "in which", "type": "text" }, { "bbox": [ 318, 454, 328, 460 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 329, 449, 351, 462 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 351, 454, 361, 460 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 361, 449, 504, 462 ], "score": 1.0, "content": "are not independent, as long as", "type": "text" }, { "bbox": [ 504, 451, 522, 461 ], "score": 0.94, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 522, 449, 541, 462 ], "score": 1.0, "content": "has", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 460, 541, 475 ], "spans": [ { "bbox": [ 69, 460, 116, 475 ], "score": 1.0, "content": "marginals", "type": "text" }, { "bbox": [ 116, 462, 138, 473 ], "score": 0.94, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 139, 460, 160, 475 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 161, 462, 184, 473 ], "score": 0.93, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 184, 460, 316, 475 ], "score": 1.0, "content": ". Therefore, we propose to set", "type": "text" }, { "bbox": [ 316, 462, 334, 473 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 334, 460, 541, 475 ], "score": 1.0, "content": "to be the 2-Wasserstein optimal transport map", "type": "text" } ], "index": 25 }, { "bbox": [ 71, 471, 246, 486 ], "spans": [ { "bbox": [ 71, 478, 77, 482 ], "score": 0.89, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 78, 471, 246, 486 ], "score": 1.0, "content": "achieving the infimum in (6), namely,", "type": "text" } ], "index": 26 } ], "index": 24.5, "bbox_fs": [ 69, 436, 541, 486 ] }, { "type": "interline_equation", "bbox": [ 268, 484, 344, 497 ], "lines": [ { "bbox": [ 268, 484, 344, 497 ], "spans": [ { "bbox": [ 268, 484, 344, 497 ], "score": 0.93, "content": "q ( z ) : = \\pi ( x _ { 0 } , x _ { 1 } ) .", "type": "interline_equation", "image_path": "a161670994a9ce0b334b2c59b724bfabd4ec932f87553f8049949c3686cb7661.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 268, 484, 344, 497 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 70, 502, 541, 550 ], "lines": [ { "bbox": [ 69, 502, 541, 516 ], "spans": [ { "bbox": [ 69, 502, 125, 516 ], "score": 1.0, "content": "In this case,", "type": "text" }, { "bbox": [ 126, 508, 131, 513 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 131, 502, 300, 516 ], "score": 1.0, "content": "is still a tuple of points, but instead of", "type": "text" }, { "bbox": [ 300, 508, 325, 515 ], "score": 0.91, "content": "x _ { 0 } , x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 326, 502, 541, 516 ], "score": 1.0, "content": "being sampled independently from their marginal", "type": "text" } ], "index": 28 }, { "bbox": [ 70, 514, 541, 527 ], "spans": [ { "bbox": [ 70, 514, 432, 527 ], "score": 1.0, "content": "distributions, they are sampled jointly according to the optimal transport map", "type": "text" }, { "bbox": [ 432, 520, 439, 524 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 439, 514, 541, 527 ], "score": 1.0, "content": ". We call this method", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 526, 542, 540 ], "spans": [ { "bbox": [ 69, 526, 297, 540 ], "score": 1.0, "content": "optimal transport CFM (OT-CFM). If one uses the", "type": "text" }, { "bbox": [ 297, 528, 327, 539 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 327, 526, 419, 540 ], "score": 1.0, "content": "defined by (14) and", "type": "text" }, { "bbox": [ 419, 528, 450, 539 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 450, 526, 542, 540 ], "score": 1.0, "content": "in (15), OT-CFM is", "type": "text" } ], "index": 30 }, { "bbox": [ 70, 538, 349, 551 ], "spans": [ { "bbox": [ 70, 538, 349, 551 ], "score": 1.0, "content": "equivalent to dynamic optimal transport in the following sense.", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 69, 502, 542, 551 ] }, { "type": "text", "bbox": [ 70, 554, 540, 591 ], "lines": [ { "bbox": [ 70, 554, 541, 568 ], "spans": [ { "bbox": [ 70, 554, 339, 568 ], "score": 1.0, "content": "Proposition 3.4. The results of Proposition 3.3 also hold for", "type": "text" }, { "bbox": [ 339, 556, 357, 567 ], "score": 0.92, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 357, 554, 541, 568 ], "score": 1.0, "content": "in (17). Furthermore, assuming regularity", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 565, 542, 581 ], "spans": [ { "bbox": [ 69, 565, 128, 581 ], "score": 1.0, "content": "properties of", "type": "text" }, { "bbox": [ 128, 572, 137, 578 ], "score": 0.7, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 138, 565, 144, 581 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 144, 572, 153, 578 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 153, 565, 295, 581 ], "score": 1.0, "content": ", and the optimal transport plan", "type": "text" }, { "bbox": [ 295, 572, 302, 576 ], "score": 0.86, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 302, 565, 320, 581 ], "score": 1.0, "content": ", as", "type": "text" }, { "bbox": [ 320, 568, 352, 576 ], "score": 0.92, "content": "\\sigma ^ { 2 } \\to 0", "type": "inline_equation" }, { "bbox": [ 352, 565, 434, 581 ], "score": 1.0, "content": "the marginal path", "type": "text" }, { "bbox": [ 434, 572, 443, 578 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 443, 565, 487, 581 ], "score": 1.0, "content": "and field", "type": "text" }, { "bbox": [ 487, 572, 496, 578 ], "score": 0.88, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 496, 565, 542, 581 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 33 }, { "bbox": [ 68, 577, 407, 594 ], "spans": [ { "bbox": [ 68, 577, 110, 594 ], "score": 1.0, "content": "(7), i.e.,", "type": "text" }, { "bbox": [ 110, 583, 119, 590 ], "score": 0.88, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 120, 577, 360, 594 ], "score": 1.0, "content": "solves the dynamic optimal transport problem between", "type": "text" }, { "bbox": [ 360, 583, 369, 590 ], "score": 0.87, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 369, 577, 392, 594 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 392, 583, 401, 590 ], "score": 0.88, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 401, 577, 407, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 34 } ], "index": 33, "bbox_fs": [ 68, 554, 542, 594 ] }, { "type": "text", "bbox": [ 70, 600, 540, 672 ], "lines": [ { "bbox": [ 69, 599, 541, 614 ], "spans": [ { "bbox": [ 69, 599, 541, 614 ], "score": 1.0, "content": "We consider two cases: (1) when the data set is small enough and we know the static optimal transport plan", "type": "text" } ], "index": 35 }, { "bbox": [ 68, 611, 541, 626 ], "spans": [ { "bbox": [ 68, 611, 541, 626 ], "score": 1.0, "content": "(e.g. single cell data). (2) when the data is too large (or continuous) (e.g. image data) and the static OT", "type": "text" } ], "index": 36 }, { "bbox": [ 70, 624, 540, 636 ], "spans": [ { "bbox": [ 70, 624, 540, 636 ], "score": 1.0, "content": "plan is computationally infeasible to determine exactly. In the first case we are able to extend the transport", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 636, 540, 650 ], "spans": [ { "bbox": [ 69, 636, 540, 650 ], "score": 1.0, "content": "map to unseen data similar to the task presented in Bunne et al. (2023). In the second case we show an", "type": "text" } ], "index": 38 }, { "bbox": [ 68, 647, 542, 662 ], "spans": [ { "bbox": [ 68, 647, 542, 662 ], "score": 1.0, "content": "approximation with minibatch OT improves over a random plan in terms of generative modelling performance", "type": "text" } ], "index": 39 }, { "bbox": [ 70, 660, 153, 673 ], "spans": [ { "bbox": [ 70, 660, 153, 673 ], "score": 1.0, "content": "and training time.", "type": "text" } ], "index": 40 } ], "index": 37.5, "bbox_fs": [ 68, 599, 542, 673 ] }, { "type": "text", "bbox": [ 71, 684, 540, 732 ], "lines": [ { "bbox": [ 70, 684, 541, 698 ], "spans": [ { "bbox": [ 70, 684, 394, 698 ], "score": 1.0, "content": "Minibatch OT approximation. For large datasets, the transport plan", "type": "text" }, { "bbox": [ 395, 690, 401, 694 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 401, 684, 541, 698 ], "score": 1.0, "content": "can be difficult to compute and", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 696, 542, 709 ], "spans": [ { "bbox": [ 69, 696, 542, 709 ], "score": 1.0, "content": "store due to OT’s cubic time and quadratic memory complexity in the number of samples (Cuturi, 2013;", "type": "text" } ], "index": 42 }, { "bbox": [ 70, 708, 542, 721 ], "spans": [ { "bbox": [ 70, 708, 542, 721 ], "score": 1.0, "content": "Tong et al., 2020). Therefore, we rely on a minibatch OT approximation similar to Fatras et al. (2021b).", "type": "text" } ], "index": 43 }, { "bbox": [ 71, 721, 540, 732 ], "spans": [ { "bbox": [ 71, 721, 540, 732 ], "score": 1.0, "content": "Although minibatch OT incurs an error relative to the exact OT solution, it has been successfully used in", "type": "text" } ], "index": 44 }, { "bbox": [ 69, 82, 543, 96 ], "spans": [ { "bbox": [ 69, 82, 543, 96 ], "score": 1.0, "content": "many applications like domain adaptation or generative modeling (Damodaran et al., 2018; Genevay et al.,", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 66, 91, 545, 115 ], "spans": [ { "bbox": [ 66, 91, 250, 115 ], "score": 1.0, "content": "2018). Specifically, for each batch of data", "type": "text", "cross_page": true }, { "bbox": [ 251, 95, 339, 109 ], "score": 0.93, "content": "( \\{ x _ { 0 } ^ { ( i ) } \\} _ { i = 1 } ^ { B } , \\{ x _ { 1 } ^ { ( i ) } \\} _ { i = 1 } ^ { B } )", "type": "inline_equation", "cross_page": true }, { "bbox": [ 339, 91, 545, 115 ], "score": 1.0, "content": "seen during training, we sample pairs of points", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 69, 108, 543, 122 ], "spans": [ { "bbox": [ 69, 108, 190, 122 ], "score": 1.0, "content": "from the joint distribution", "type": "text", "cross_page": true }, { "bbox": [ 191, 113, 216, 119 ], "score": 0.85, "content": "\\pi _ { \\mathrm { b a t c h } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 217, 108, 543, 122 ], "score": 1.0, "content": "given by the OT plan between the source and target points in the batch.", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 69, 120, 542, 133 ], "spans": [ { "bbox": [ 69, 120, 542, 133 ], "score": 1.0, "content": "(The OT batch size need not match the optimization batch size, but we keep them equal for simplicity.) Thus,", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 69, 132, 541, 145 ], "spans": [ { "bbox": [ 69, 132, 541, 145 ], "score": 1.0, "content": "we solve a minibatch approximation of dynamic optimal transport. However, when the OT batch size equals", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 69, 143, 542, 158 ], "spans": [ { "bbox": [ 69, 143, 155, 158 ], "score": 1.0, "content": "the support size of", "type": "text", "cross_page": true }, { "bbox": [ 155, 146, 185, 156 ], "score": 0.93, "content": "\\left( q _ { 0 } , q _ { 1 } \\right)", "type": "inline_equation", "cross_page": true }, { "bbox": [ 186, 143, 542, 158 ], "score": 1.0, "content": ", we recover exact OT and therefore, by Proposition 3.4, learn the exact dynamic", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 69, 155, 541, 169 ], "spans": [ { "bbox": [ 69, 155, 541, 169 ], "score": 1.0, "content": "optimal transport. We show empirically that the batch size can be much smaller than the full dataset size", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 68, 167, 542, 182 ], "spans": [ { "bbox": [ 68, 167, 542, 182 ], "score": 1.0, "content": "and still give good performance, which aligns with prior studies (Fatras et al., 2020; 2021a). Concurrently, a", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 69, 180, 411, 193 ], "spans": [ { "bbox": [ 69, 180, 411, 193 ], "score": 1.0, "content": "similar framework and theoretical results appeared in Pooladian et al. (2023).", "type": "text", "cross_page": true } ], "index": 8 } ], "index": 42.5, "bbox_fs": [ 69, 684, 542, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 192 ], "lines": [ { "bbox": [ 69, 82, 543, 96 ], "spans": [ { "bbox": [ 69, 82, 543, 96 ], "score": 1.0, "content": "many applications like domain adaptation or generative modeling (Damodaran et al., 2018; Genevay et al.,", "type": "text" } ], "index": 0 }, { "bbox": [ 66, 91, 545, 115 ], "spans": [ { "bbox": [ 66, 91, 250, 115 ], "score": 1.0, "content": "2018). Specifically, for each batch of data", "type": "text" }, { "bbox": [ 251, 95, 339, 109 ], "score": 0.93, "content": "( \\{ x _ { 0 } ^ { ( i ) } \\} _ { i = 1 } ^ { B } , \\{ x _ { 1 } ^ { ( i ) } \\} _ { i = 1 } ^ { B } )", "type": "inline_equation" }, { "bbox": [ 339, 91, 545, 115 ], "score": 1.0, "content": "seen during training, we sample pairs of points", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 108, 543, 122 ], "spans": [ { "bbox": [ 69, 108, 190, 122 ], "score": 1.0, "content": "from the joint distribution", "type": "text" }, { "bbox": [ 191, 113, 216, 119 ], "score": 0.85, "content": "\\pi _ { \\mathrm { b a t c h } }", "type": "inline_equation" }, { "bbox": [ 217, 108, 543, 122 ], "score": 1.0, "content": "given by the OT plan between the source and target points in the batch.", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 120, 542, 133 ], "spans": [ { "bbox": [ 69, 120, 542, 133 ], "score": 1.0, "content": "(The OT batch size need not match the optimization batch size, but we keep them equal for simplicity.) Thus,", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 132, 541, 145 ], "spans": [ { "bbox": [ 69, 132, 541, 145 ], "score": 1.0, "content": "we solve a minibatch approximation of dynamic optimal transport. However, when the OT batch size equals", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 143, 542, 158 ], "spans": [ { "bbox": [ 69, 143, 155, 158 ], "score": 1.0, "content": "the support size of", "type": "text" }, { "bbox": [ 155, 146, 185, 156 ], "score": 0.93, "content": "\\left( q _ { 0 } , q _ { 1 } \\right)", "type": "inline_equation" }, { "bbox": [ 186, 143, 542, 158 ], "score": 1.0, "content": ", we recover exact OT and therefore, by Proposition 3.4, learn the exact dynamic", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 155, 541, 169 ], "spans": [ { "bbox": [ 69, 155, 541, 169 ], "score": 1.0, "content": "optimal transport. We show empirically that the batch size can be much smaller than the full dataset size", "type": "text" } ], "index": 6 }, { "bbox": [ 68, 167, 542, 182 ], "spans": [ { "bbox": [ 68, 167, 542, 182 ], "score": 1.0, "content": "and still give good performance, which aligns with prior studies (Fatras et al., 2020; 2021a). Concurrently, a", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 180, 411, 193 ], "spans": [ { "bbox": [ 69, 180, 411, 193 ], "score": 1.0, "content": "similar framework and theoretical results appeared in Pooladian et al. (2023).", "type": "text" } ], "index": 8 } ], "index": 4 }, { "type": "title", "bbox": [ 72, 203, 217, 217 ], "lines": [ { "bbox": [ 69, 203, 218, 218 ], "spans": [ { "bbox": [ 69, 203, 218, 218 ], "score": 1.0, "content": "3.2.4 Schrödinger bridge CFM", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 71, 223, 540, 284 ], "lines": [ { "bbox": [ 70, 224, 542, 237 ], "spans": [ { "bbox": [ 70, 224, 542, 237 ], "score": 1.0, "content": "Recently, there has been significant effort in learning diffusion models with general source distributions,", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 235, 541, 250 ], "spans": [ { "bbox": [ 69, 235, 541, 250 ], "score": 1.0, "content": "formulated as a Schrödinger bridge problem (De Bortoli et al., 2021; Vargas et al., 2021; Chen et al., 2022)", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 247, 541, 262 ], "spans": [ { "bbox": [ 69, 247, 541, 262 ], "score": 1.0, "content": "or bridge matching Peluchetti (2023); Liu et al. (2022b); Ye et al. (2022). Here we show that SB-CFM, an", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 259, 541, 274 ], "spans": [ { "bbox": [ 69, 259, 541, 274 ], "score": 1.0, "content": "entropic variant of OT-CFM, can be used to train an ODE to match the probability flow of a Schrödinger", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 271, 289, 285 ], "spans": [ { "bbox": [ 70, 271, 289, 285 ], "score": 1.0, "content": "bridge with a Brownian motion reference process.", "type": "text" } ], "index": 14 } ], "index": 12 }, { "type": "text", "bbox": [ 71, 289, 540, 326 ], "lines": [ { "bbox": [ 70, 290, 541, 303 ], "spans": [ { "bbox": [ 70, 290, 90, 303 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 90, 295, 106, 302 ], "score": 0.88, "content": "\\scriptstyle p _ { \\mathrm { r e f } }", "type": "inline_equation" }, { "bbox": [ 106, 290, 306, 303 ], "score": 1.0, "content": "be the standard Wiener process scaled by", "type": "text" }, { "bbox": [ 306, 295, 312, 300 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 313, 290, 437, 303 ], "score": 1.0, "content": "with initial-time marginal", "type": "text" }, { "bbox": [ 437, 292, 511, 302 ], "score": 0.94, "content": "p _ { \\mathrm { r e f } } ( x _ { 0 } ) = q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 511, 290, 541, 303 ], "score": 1.0, "content": ". The", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 302, 541, 316 ], "spans": [ { "bbox": [ 70, 302, 355, 316 ], "score": 1.0, "content": "Schrödinger bridge problem (Schrödinger, 1932) seeks the process", "type": "text" }, { "bbox": [ 356, 307, 362, 312 ], "score": 0.87, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 362, 302, 438, 316 ], "score": 1.0, "content": "that is closest to", "type": "text" }, { "bbox": [ 439, 307, 454, 314 ], "score": 0.89, "content": "\\scriptstyle p _ { \\mathrm { r e f } }", "type": "inline_equation" }, { "bbox": [ 454, 302, 541, 316 ], "score": 1.0, "content": "while having initial", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 313, 451, 327 ], "spans": [ { "bbox": [ 69, 313, 378, 327 ], "score": 1.0, "content": "and terminal marginal distributions specified by the data distribution", "type": "text" }, { "bbox": [ 378, 316, 401, 326 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 402, 313, 423, 327 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 424, 316, 447, 326 ], "score": 0.93, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 447, 313, 451, 327 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 17 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 218, 335, 393, 356 ], "lines": [ { "bbox": [ 218, 335, 393, 356 ], "spans": [ { "bbox": [ 218, 335, 393, 356 ], "score": 0.93, "content": "\\pi ^ { * } : = \\underset { \\pi ( x _ { 0 } ) = q ( x _ { 0 } ) , \\pi ( x _ { 1 } ) = q ( x _ { 1 } ) } { \\arg \\operatorname* { m i n } } \\ \\mathrm { K L } ( \\pi \\| p _ { \\mathrm { r e f } } ) .", "type": "interline_equation", "image_path": "6cb9758f417cc33791c87a51fe525e85eda7abbbfe014f48b3c7f015bba0d9ff.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 218, 335, 393, 356 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 71, 364, 209, 375 ], "lines": [ { "bbox": [ 70, 364, 209, 377 ], "spans": [ { "bbox": [ 70, 364, 209, 377 ], "score": 1.0, "content": "We define the joint distribution", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 263, 376, 349, 388 ], "lines": [ { "bbox": [ 263, 376, 349, 388 ], "spans": [ { "bbox": [ 263, 376, 349, 388 ], "score": 0.93, "content": "q ( z ) : = \\pi _ { 2 \\sigma ^ { 2 } } ( x _ { 0 } , x _ { 1 } )", "type": "interline_equation", "image_path": "b604b5d908d43fa5899b1564481e8cd670d77e631cffa7cc78cc6266c8b6d918.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 263, 376, 349, 388 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 72, 393, 539, 442 ], "lines": [ { "bbox": [ 70, 393, 541, 406 ], "spans": [ { "bbox": [ 70, 393, 100, 406 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 399, 119, 405 ], "score": 0.89, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 119, 393, 541, 406 ], "score": 1.0, "content": "is the solution of the entropy-regularized optimal transport problem (Cuturi, 2013) with cost", "type": "text" } ], "index": 21 }, { "bbox": [ 72, 405, 542, 419 ], "spans": [ { "bbox": [ 72, 407, 114, 418 ], "score": 0.94, "content": "\\| x _ { 0 } - x _ { 1 } \\|", "type": "inline_equation" }, { "bbox": [ 115, 405, 240, 419 ], "score": 1.0, "content": "and entropy regularization", "type": "text" }, { "bbox": [ 240, 407, 276, 415 ], "score": 0.93, "content": "\\lambda = 2 \\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 277, 405, 542, 419 ], "score": 1.0, "content": "(see (33) for the background on entropic OT). We set the", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 416, 542, 431 ], "spans": [ { "bbox": [ 69, 416, 415, 431 ], "score": 1.0, "content": "conditional path distribution to be a Brownian bridge with diffusion scale", "type": "text" }, { "bbox": [ 415, 423, 421, 427 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 422, 416, 466, 431 ], "score": 1.0, "content": "between", "type": "text" }, { "bbox": [ 466, 423, 477, 429 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 477, 416, 501, 431 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 502, 423, 512, 429 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 512, 416, 542, 431 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 429, 264, 443 ], "spans": [ { "bbox": [ 69, 429, 264, 443 ], "score": 1.0, "content": "probability path and generating vector field", "type": "text" } ], "index": 24 } ], "index": 22.5 }, { "type": "interline_equation", "bbox": [ 184, 448, 427, 493 ], "lines": [ { "bbox": [ 184, 448, 427, 493 ], "spans": [ { "bbox": [ 184, 448, 427, 493 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x \\mid z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , t ( 1 - t ) \\sigma ^ { 2 } ) } \\\\ { u _ { t } ( x \\mid z ) = \\displaystyle \\frac { 1 - 2 t } { 2 t ( 1 - t ) } ( x - ( t x _ { 1 } + ( 1 - t ) x _ { 0 } ) ) + ( x _ { 1 } - x _ { 0 } ) , } \\end{array}", "type": "interline_equation", "image_path": "1b848168d9b7f320f4d27cce7c1b0e9e4bde4ff1eeac6194568c3633501f2705.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 184, 448, 427, 463.0 ], "spans": [], "index": 25 }, { "bbox": [ 184, 463.0, 427, 478.0 ], "spans": [], "index": 26 }, { "bbox": [ 184, 478.0, 427, 493.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 71, 498, 538, 546 ], "lines": [ { "bbox": [ 70, 498, 541, 512 ], "spans": [ { "bbox": [ 70, 498, 98, 512 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 99, 504, 108, 510 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 108, 498, 406, 512 ], "score": 1.0, "content": "is computed by (5) as the vector field generating the probability path", "type": "text" }, { "bbox": [ 407, 501, 437, 511 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 437, 498, 541, 512 ], "score": 1.0, "content": ". The marginal coupling", "type": "text" } ], "index": 28 }, { "bbox": [ 71, 510, 541, 524 ], "spans": [ { "bbox": [ 71, 516, 89, 522 ], "score": 0.75, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 89, 510, 113, 524 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 113, 513, 144, 523 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 144, 510, 175, 524 ], "score": 1.0, "content": "define", "type": "text" }, { "bbox": [ 176, 513, 199, 523 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 199, 510, 541, 524 ], "score": 1.0, "content": ", which is approximated by the regression objective in Alg. 4. The solution of", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 520, 541, 537 ], "spans": [ { "bbox": [ 69, 520, 541, 537 ], "score": 1.0, "content": "the SB is known to be the map which is the solution of the entropically-regularized OT problem, motivating", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 533, 164, 548 ], "spans": [ { "bbox": [ 69, 533, 164, 548 ], "score": 1.0, "content": "the next proposition.", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 70, 549, 538, 575 ], "lines": [ { "bbox": [ 69, 549, 541, 564 ], "spans": [ { "bbox": [ 69, 549, 273, 564 ], "score": 1.0, "content": "Proposition 3.5. The marginal vector field", "type": "text" }, { "bbox": [ 273, 552, 296, 563 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 297, 549, 541, 564 ], "score": 1.0, "content": "defined by (19) and (21) generates the same marginal", "type": "text" } ], "index": 32 }, { "bbox": [ 70, 561, 339, 575 ], "spans": [ { "bbox": [ 70, 561, 207, 575 ], "score": 1.0, "content": "probability path as the solution", "type": "text" }, { "bbox": [ 208, 565, 218, 572 ], "score": 0.85, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 219, 561, 250, 575 ], "score": 1.0, "content": "to the", "type": "text" }, { "bbox": [ 251, 565, 264, 572 ], "score": 0.36, "content": "S B", "type": "inline_equation" }, { "bbox": [ 264, 561, 339, 575 ], "score": 1.0, "content": "problem in (18).", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "text", "bbox": [ 71, 583, 541, 620 ], "lines": [ { "bbox": [ 69, 583, 542, 597 ], "spans": [ { "bbox": [ 69, 583, 387, 597 ], "score": 1.0, "content": "While we define SB-CFM with an entropic regularization coefficient of", "type": "text" }, { "bbox": [ 387, 586, 421, 594 ], "score": 0.93, "content": "\\varepsilon = 2 \\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 422, 583, 542, 597 ], "score": 1.0, "content": ", the flow still matches the", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 595, 542, 609 ], "spans": [ { "bbox": [ 69, 595, 193, 609 ], "score": 1.0, "content": "marginals for any choice of", "type": "text" }, { "bbox": [ 193, 601, 198, 606 ], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation" }, { "bbox": [ 199, 595, 386, 609 ], "score": 1.0, "content": ". Interestingly, we recover OT-CFM when", "type": "text" }, { "bbox": [ 387, 599, 412, 606 ], "score": 0.91, "content": "\\varepsilon 0", "type": "inline_equation" }, { "bbox": [ 412, 595, 494, 609 ], "score": 1.0, "content": "and I-CFM when", "type": "text" }, { "bbox": [ 495, 601, 525, 606 ], "score": 0.89, "content": "\\varepsilon \\to \\infty", "type": "inline_equation" }, { "bbox": [ 525, 595, 542, 609 ], "score": 1.0, "content": ". A", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 607, 381, 621 ], "spans": [ { "bbox": [ 69, 607, 381, 621 ], "score": 1.0, "content": "similar result was proven in a concurrent work Pooladian et al. (2023).", "type": "text" } ], "index": 36 } ], "index": 35 }, { "type": "title", "bbox": [ 71, 634, 163, 648 ], "lines": [ { "bbox": [ 69, 633, 164, 650 ], "spans": [ { "bbox": [ 69, 633, 164, 650 ], "score": 1.0, "content": "4 Related work", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 70, 660, 540, 732 ], "lines": [ { "bbox": [ 70, 659, 541, 673 ], "spans": [ { "bbox": [ 70, 659, 541, 673 ], "score": 1.0, "content": "Simulation-free continuous-time modeling. Simulation-free training is common in stochastic flow", "type": "text" } ], "index": 38 }, { "bbox": [ 68, 672, 541, 686 ], "spans": [ { "bbox": [ 68, 672, 541, 686 ], "score": 1.0, "content": "models where backpropagating through the simulation is numerically challenging and has high variance (Li", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 684, 541, 698 ], "spans": [ { "bbox": [ 69, 684, 541, 698 ], "score": 1.0, "content": "et al., 2020). While these diffusion models have recently achieved exceptional generative performance on many", "type": "text" } ], "index": 40 }, { "bbox": [ 70, 697, 541, 709 ], "spans": [ { "bbox": [ 70, 697, 541, 709 ], "score": 1.0, "content": "tasks (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; 2020; Ho et al., 2020; Song et al., 2021b; Dhariwal &", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 706, 542, 723 ], "spans": [ { "bbox": [ 69, 706, 542, 723 ], "score": 1.0, "content": "Nichol, 2021; Watson et al., 2022b), their simulation requires an inherently costly SDE simulation with many", "type": "text" } ], "index": 42 }, { "bbox": [ 69, 720, 542, 734 ], "spans": [ { "bbox": [ 69, 720, 542, 734 ], "score": 1.0, "content": "follow-up works to improve inference efficiency (Lu et al., 2022; Salimans & Ho, 2022; Watson et al., 2022a;", "type": "text" } ], "index": 43 } ], "index": 40.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 761 ], "spans": [ { "bbox": [ 302, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 192 ], "lines": [], "index": 4, "bbox_fs": [ 66, 82, 545, 193 ], "lines_deleted": true }, { "type": "title", "bbox": [ 72, 203, 217, 217 ], "lines": [ { "bbox": [ 69, 203, 218, 218 ], "spans": [ { "bbox": [ 69, 203, 218, 218 ], "score": 1.0, "content": "3.2.4 Schrödinger bridge CFM", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 71, 223, 540, 284 ], "lines": [ { "bbox": [ 70, 224, 542, 237 ], "spans": [ { "bbox": [ 70, 224, 542, 237 ], "score": 1.0, "content": "Recently, there has been significant effort in learning diffusion models with general source distributions,", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 235, 541, 250 ], "spans": [ { "bbox": [ 69, 235, 541, 250 ], "score": 1.0, "content": "formulated as a Schrödinger bridge problem (De Bortoli et al., 2021; Vargas et al., 2021; Chen et al., 2022)", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 247, 541, 262 ], "spans": [ { "bbox": [ 69, 247, 541, 262 ], "score": 1.0, "content": "or bridge matching Peluchetti (2023); Liu et al. (2022b); Ye et al. (2022). Here we show that SB-CFM, an", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 259, 541, 274 ], "spans": [ { "bbox": [ 69, 259, 541, 274 ], "score": 1.0, "content": "entropic variant of OT-CFM, can be used to train an ODE to match the probability flow of a Schrödinger", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 271, 289, 285 ], "spans": [ { "bbox": [ 70, 271, 289, 285 ], "score": 1.0, "content": "bridge with a Brownian motion reference process.", "type": "text" } ], "index": 14 } ], "index": 12, "bbox_fs": [ 69, 224, 542, 285 ] }, { "type": "text", "bbox": [ 71, 289, 540, 326 ], "lines": [ { "bbox": [ 70, 290, 541, 303 ], "spans": [ { "bbox": [ 70, 290, 90, 303 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 90, 295, 106, 302 ], "score": 0.88, "content": "\\scriptstyle p _ { \\mathrm { r e f } }", "type": "inline_equation" }, { "bbox": [ 106, 290, 306, 303 ], "score": 1.0, "content": "be the standard Wiener process scaled by", "type": "text" }, { "bbox": [ 306, 295, 312, 300 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 313, 290, 437, 303 ], "score": 1.0, "content": "with initial-time marginal", "type": "text" }, { "bbox": [ 437, 292, 511, 302 ], "score": 0.94, "content": "p _ { \\mathrm { r e f } } ( x _ { 0 } ) = q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 511, 290, 541, 303 ], "score": 1.0, "content": ". The", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 302, 541, 316 ], "spans": [ { "bbox": [ 70, 302, 355, 316 ], "score": 1.0, "content": "Schrödinger bridge problem (Schrödinger, 1932) seeks the process", "type": "text" }, { "bbox": [ 356, 307, 362, 312 ], "score": 0.87, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 362, 302, 438, 316 ], "score": 1.0, "content": "that is closest to", "type": "text" }, { "bbox": [ 439, 307, 454, 314 ], "score": 0.89, "content": "\\scriptstyle p _ { \\mathrm { r e f } }", "type": "inline_equation" }, { "bbox": [ 454, 302, 541, 316 ], "score": 1.0, "content": "while having initial", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 313, 451, 327 ], "spans": [ { "bbox": [ 69, 313, 378, 327 ], "score": 1.0, "content": "and terminal marginal distributions specified by the data distribution", "type": "text" }, { "bbox": [ 378, 316, 401, 326 ], "score": 0.93, "content": "q ( x _ { 0 } )", "type": "inline_equation" }, { "bbox": [ 402, 313, 423, 327 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 424, 316, 447, 326 ], "score": 0.93, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 447, 313, 451, 327 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 17 } ], "index": 16, "bbox_fs": [ 69, 290, 541, 327 ] }, { "type": "interline_equation", "bbox": [ 218, 335, 393, 356 ], "lines": [ { "bbox": [ 218, 335, 393, 356 ], "spans": [ { "bbox": [ 218, 335, 393, 356 ], "score": 0.93, "content": "\\pi ^ { * } : = \\underset { \\pi ( x _ { 0 } ) = q ( x _ { 0 } ) , \\pi ( x _ { 1 } ) = q ( x _ { 1 } ) } { \\arg \\operatorname* { m i n } } \\ \\mathrm { K L } ( \\pi \\| p _ { \\mathrm { r e f } } ) .", "type": "interline_equation", "image_path": "6cb9758f417cc33791c87a51fe525e85eda7abbbfe014f48b3c7f015bba0d9ff.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 218, 335, 393, 356 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 71, 364, 209, 375 ], "lines": [ { "bbox": [ 70, 364, 209, 377 ], "spans": [ { "bbox": [ 70, 364, 209, 377 ], "score": 1.0, "content": "We define the joint distribution", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 70, 364, 209, 377 ] }, { "type": "interline_equation", "bbox": [ 263, 376, 349, 388 ], "lines": [ { "bbox": [ 263, 376, 349, 388 ], "spans": [ { "bbox": [ 263, 376, 349, 388 ], "score": 0.93, "content": "q ( z ) : = \\pi _ { 2 \\sigma ^ { 2 } } ( x _ { 0 } , x _ { 1 } )", "type": "interline_equation", "image_path": "b604b5d908d43fa5899b1564481e8cd670d77e631cffa7cc78cc6266c8b6d918.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 263, 376, 349, 388 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 72, 393, 539, 442 ], "lines": [ { "bbox": [ 70, 393, 541, 406 ], "spans": [ { "bbox": [ 70, 393, 100, 406 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 399, 119, 405 ], "score": 0.89, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 119, 393, 541, 406 ], "score": 1.0, "content": "is the solution of the entropy-regularized optimal transport problem (Cuturi, 2013) with cost", "type": "text" } ], "index": 21 }, { "bbox": [ 72, 405, 542, 419 ], "spans": [ { "bbox": [ 72, 407, 114, 418 ], "score": 0.94, "content": "\\| x _ { 0 } - x _ { 1 } \\|", "type": "inline_equation" }, { "bbox": [ 115, 405, 240, 419 ], "score": 1.0, "content": "and entropy regularization", "type": "text" }, { "bbox": [ 240, 407, 276, 415 ], "score": 0.93, "content": "\\lambda = 2 \\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 277, 405, 542, 419 ], "score": 1.0, "content": "(see (33) for the background on entropic OT). We set the", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 416, 542, 431 ], "spans": [ { "bbox": [ 69, 416, 415, 431 ], "score": 1.0, "content": "conditional path distribution to be a Brownian bridge with diffusion scale", "type": "text" }, { "bbox": [ 415, 423, 421, 427 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 422, 416, 466, 431 ], "score": 1.0, "content": "between", "type": "text" }, { "bbox": [ 466, 423, 477, 429 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 477, 416, 501, 431 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 502, 423, 512, 429 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 512, 416, 542, 431 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 429, 264, 443 ], "spans": [ { "bbox": [ 69, 429, 264, 443 ], "score": 1.0, "content": "probability path and generating vector field", "type": "text" } ], "index": 24 } ], "index": 22.5, "bbox_fs": [ 69, 393, 542, 443 ] }, { "type": "interline_equation", "bbox": [ 184, 448, 427, 493 ], "lines": [ { "bbox": [ 184, 448, 427, 493 ], "spans": [ { "bbox": [ 184, 448, 427, 493 ], "score": 0.92, "content": "\\begin{array} { l } { p _ { t } ( x \\mid z ) = \\mathcal { N } ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , t ( 1 - t ) \\sigma ^ { 2 } ) } \\\\ { u _ { t } ( x \\mid z ) = \\displaystyle \\frac { 1 - 2 t } { 2 t ( 1 - t ) } ( x - ( t x _ { 1 } + ( 1 - t ) x _ { 0 } ) ) + ( x _ { 1 } - x _ { 0 } ) , } \\end{array}", "type": "interline_equation", "image_path": "1b848168d9b7f320f4d27cce7c1b0e9e4bde4ff1eeac6194568c3633501f2705.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 184, 448, 427, 463.0 ], "spans": [], "index": 25 }, { "bbox": [ 184, 463.0, 427, 478.0 ], "spans": [], "index": 26 }, { "bbox": [ 184, 478.0, 427, 493.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 71, 498, 538, 546 ], "lines": [ { "bbox": [ 70, 498, 541, 512 ], "spans": [ { "bbox": [ 70, 498, 98, 512 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 99, 504, 108, 510 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 108, 498, 406, 512 ], "score": 1.0, "content": "is computed by (5) as the vector field generating the probability path", "type": "text" }, { "bbox": [ 407, 501, 437, 511 ], "score": 0.95, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 437, 498, 541, 512 ], "score": 1.0, "content": ". The marginal coupling", "type": "text" } ], "index": 28 }, { "bbox": [ 71, 510, 541, 524 ], "spans": [ { "bbox": [ 71, 516, 89, 522 ], "score": 0.75, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 89, 510, 113, 524 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 113, 513, 144, 523 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 144, 510, 175, 524 ], "score": 1.0, "content": "define", "type": "text" }, { "bbox": [ 176, 513, 199, 523 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 199, 510, 541, 524 ], "score": 1.0, "content": ", which is approximated by the regression objective in Alg. 4. The solution of", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 520, 541, 537 ], "spans": [ { "bbox": [ 69, 520, 541, 537 ], "score": 1.0, "content": "the SB is known to be the map which is the solution of the entropically-regularized OT problem, motivating", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 533, 164, 548 ], "spans": [ { "bbox": [ 69, 533, 164, 548 ], "score": 1.0, "content": "the next proposition.", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 69, 498, 541, 548 ] }, { "type": "text", "bbox": [ 70, 549, 538, 575 ], "lines": [ { "bbox": [ 69, 549, 541, 564 ], "spans": [ { "bbox": [ 69, 549, 273, 564 ], "score": 1.0, "content": "Proposition 3.5. The marginal vector field", "type": "text" }, { "bbox": [ 273, 552, 296, 563 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 297, 549, 541, 564 ], "score": 1.0, "content": "defined by (19) and (21) generates the same marginal", "type": "text" } ], "index": 32 }, { "bbox": [ 70, 561, 339, 575 ], "spans": [ { "bbox": [ 70, 561, 207, 575 ], "score": 1.0, "content": "probability path as the solution", "type": "text" }, { "bbox": [ 208, 565, 218, 572 ], "score": 0.85, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 219, 561, 250, 575 ], "score": 1.0, "content": "to the", "type": "text" }, { "bbox": [ 251, 565, 264, 572 ], "score": 0.36, "content": "S B", "type": "inline_equation" }, { "bbox": [ 264, 561, 339, 575 ], "score": 1.0, "content": "problem in (18).", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 69, 549, 541, 575 ] }, { "type": "text", "bbox": [ 71, 583, 541, 620 ], "lines": [ { "bbox": [ 69, 583, 542, 597 ], "spans": [ { "bbox": [ 69, 583, 387, 597 ], "score": 1.0, "content": "While we define SB-CFM with an entropic regularization coefficient of", "type": "text" }, { "bbox": [ 387, 586, 421, 594 ], "score": 0.93, "content": "\\varepsilon = 2 \\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 422, 583, 542, 597 ], "score": 1.0, "content": ", the flow still matches the", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 595, 542, 609 ], "spans": [ { "bbox": [ 69, 595, 193, 609 ], "score": 1.0, "content": "marginals for any choice of", "type": "text" }, { "bbox": [ 193, 601, 198, 606 ], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation" }, { "bbox": [ 199, 595, 386, 609 ], "score": 1.0, "content": ". Interestingly, we recover OT-CFM when", "type": "text" }, { "bbox": [ 387, 599, 412, 606 ], "score": 0.91, "content": "\\varepsilon 0", "type": "inline_equation" }, { "bbox": [ 412, 595, 494, 609 ], "score": 1.0, "content": "and I-CFM when", "type": "text" }, { "bbox": [ 495, 601, 525, 606 ], "score": 0.89, "content": "\\varepsilon \\to \\infty", "type": "inline_equation" }, { "bbox": [ 525, 595, 542, 609 ], "score": 1.0, "content": ". A", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 607, 381, 621 ], "spans": [ { "bbox": [ 69, 607, 381, 621 ], "score": 1.0, "content": "similar result was proven in a concurrent work Pooladian et al. (2023).", "type": "text" } ], "index": 36 } ], "index": 35, "bbox_fs": [ 69, 583, 542, 621 ] }, { "type": "title", "bbox": [ 71, 634, 163, 648 ], "lines": [ { "bbox": [ 69, 633, 164, 650 ], "spans": [ { "bbox": [ 69, 633, 164, 650 ], "score": 1.0, "content": "4 Related work", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 70, 660, 540, 732 ], "lines": [ { "bbox": [ 70, 659, 541, 673 ], "spans": [ { "bbox": [ 70, 659, 541, 673 ], "score": 1.0, "content": "Simulation-free continuous-time modeling. Simulation-free training is common in stochastic flow", "type": "text" } ], "index": 38 }, { "bbox": [ 68, 672, 541, 686 ], "spans": [ { "bbox": [ 68, 672, 541, 686 ], "score": 1.0, "content": "models where backpropagating through the simulation is numerically challenging and has high variance (Li", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 684, 541, 698 ], "spans": [ { "bbox": [ 69, 684, 541, 698 ], "score": 1.0, "content": "et al., 2020). While these diffusion models have recently achieved exceptional generative performance on many", "type": "text" } ], "index": 40 }, { "bbox": [ 70, 697, 541, 709 ], "spans": [ { "bbox": [ 70, 697, 541, 709 ], "score": 1.0, "content": "tasks (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; 2020; Ho et al., 2020; Song et al., 2021b; Dhariwal &", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 706, 542, 723 ], "spans": [ { "bbox": [ 69, 706, 542, 723 ], "score": 1.0, "content": "Nichol, 2021; Watson et al., 2022b), their simulation requires an inherently costly SDE simulation with many", "type": "text" } ], "index": 42 }, { "bbox": [ 69, 720, 542, 734 ], "spans": [ { "bbox": [ 69, 720, 542, 734 ], "score": 1.0, "content": "follow-up works to improve inference efficiency (Lu et al., 2022; Salimans & Ho, 2022; Watson et al., 2022a;", "type": "text" } ], "index": 43 } ], "index": 40.5, "bbox_fs": [ 68, 659, 542, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 70, 128, 539, 224 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 128 ], "group_id": 0, "lines": [ { "bbox": [ 69, 80, 542, 93 ], "spans": [ { "bbox": [ 69, 80, 448, 93 ], "score": 1.0, "content": "Table 2: Comparison of neural optimal transport methods over four distribution pairs (", "type": "text" }, { "bbox": [ 448, 84, 472, 93 ], "score": 0.88, "content": "\\mu \\pm \\sigma", "type": "inline_equation" }, { "bbox": [ 473, 80, 542, 93 ], "score": 1.0, "content": "over five seeds)", "type": "text" } ], "index": 0 }, { "bbox": [ 68, 91, 539, 105 ], "spans": [ { "bbox": [ 68, 91, 539, 105 ], "score": 1.0, "content": "in terms of fit (2-Wasserstein), optimal transport performance (normalized path energy), and runtime. ‘—", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 104, 542, 118 ], "spans": [ { "bbox": [ 69, 104, 542, 118 ], "score": 1.0, "content": "indicates a method that requires a Gaussian source. Best in bold. CFM and RF models are trained on a", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 117, 391, 129 ], "spans": [ { "bbox": [ 69, 117, 391, 129 ], "score": 1.0, "content": "single CPU core, other baselines are trained with a GPU and two CPUs.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "table_body", "bbox": [ 70, 128, 539, 224 ], "group_id": 0, "lines": [ { "bbox": [ 70, 128, 539, 224 ], "spans": [ { "bbox": [ 70, 128, 539, 224 ], "score": 0.979, "html": "
Dataset→N→8gaussiansmoons-→8gaussiansN→moonsN-→scurveAvg.train time
Algorithm ↓Metric→wNPEWNPEwNPEWNPE(x10²s)
OT-CFM1.262±0.3480.018±0.0141.923±0.3910.053±0.0350.239±0.0480.087±0.0610.264±0.0930.027±0.0261.129±0.335
I-CFM1.284±0.3840.222±0.0321.977±0.2662.738±0.1810.338±0.1090.841±0.1480.333±0.0600.867±0.1170.630±0.365
2-RF (Liu, 2022)1.436±0.3440.069±0.0272.211±0.4230.149±0.1010.278±0.0260.076±0.0670.395±0.1110.112±0.0850.862±0.166
3-RF (Liu, 2022)1.337±0.3670.055±0.0432.700±0.5870.123±0.1120.305±0.0260.084±0.0510.395±0.0820.129±0.0750.954±0.116
FM (Lipman et al., 2023)1.062±0.1960.174±0.0300.246±0.0770.778±0.1440.377±0.0990.772±0.0810.708±0.370
Reg.CNF (Finlay et al., 2020)1.144±0.0750.274±0.0600.376±0.0400.620±0.0880.581±0.1950.586±0.5038.021±3.288
CNF (Chen et al., 2018)1.055±0.0590.151±0.0640.387±0.0652.937±1.9730.645±0.34310.548±8.10018.810±12.677
ICNN (Makkuva et al., 2020)1.771±0.3980.747±0.0292.193±0.1360.832±0.0040.532±0.0460.267±0.0100.753±0.0680.344±0.0452.912±0.626
", "type": "table", "image_path": "59ae1b8a396f7d49e4d88fc89f2f1498a6ec874ffb6963f4e2765c1073e3f351.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 70, 128, 539, 160.0 ], "spans": [], "index": 4 }, { "bbox": [ 70, 160.0, 539, 192.0 ], "spans": [], "index": 5 }, { "bbox": [ 70, 192.0, 539, 224.0 ], "spans": [], "index": 6 } ] } ], "index": 3.25 }, { "type": "text", "bbox": [ 71, 245, 540, 293 ], "lines": [ { "bbox": [ 69, 244, 542, 259 ], "spans": [ { "bbox": [ 69, 244, 542, 259 ], "score": 1.0, "content": "Song et al., 2021a; Bao et al., 2022). These methods generally consider a simple Gaussian diffusion process,", "type": "text" } ], "index": 7 }, { "bbox": [ 70, 257, 541, 270 ], "spans": [ { "bbox": [ 70, 257, 541, 270 ], "score": 1.0, "content": "and do not consider generalizing the source distribution. Other works consider general source distributions", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 270, 541, 282 ], "spans": [ { "bbox": [ 70, 270, 541, 282 ], "score": 1.0, "content": "but this makes optimization and inference more challenging, needing multiple iterations or other tricks to", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 281, 408, 295 ], "spans": [ { "bbox": [ 69, 281, 408, 295 ], "score": 1.0, "content": "perform well (Wang et al., 2021; De Bortoli et al., 2021; Vargas et al., 2021).", "type": "text" } ], "index": 10 } ], "index": 8.5 }, { "type": "text", "bbox": [ 70, 299, 540, 383 ], "lines": [ { "bbox": [ 69, 299, 541, 312 ], "spans": [ { "bbox": [ 69, 299, 541, 312 ], "score": 1.0, "content": "Prior work considering simulation-free training of CNFs considers algorithms that are equivalent to CFM with", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 310, 541, 325 ], "spans": [ { "bbox": [ 69, 310, 541, 325 ], "score": 1.0, "content": "Gaussian source distribution (Rozen et al., 2021; Ben-Hamu et al., 2022; Lipman et al., 2023) or independent", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 321, 541, 336 ], "spans": [ { "bbox": [ 69, 321, 131, 336 ], "score": 1.0, "content": "samples from", "type": "text" }, { "bbox": [ 131, 328, 140, 335 ], "score": 0.84, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 141, 321, 146, 336 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 146, 328, 155, 335 ], "score": 0.85, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 155, 321, 541, 336 ], "score": 1.0, "content": "(Albergo & Vanden-Eijnden, 2023; Albergo et al., 2023; Neklyudov et al., 2023). Recent", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 335, 541, 348 ], "spans": [ { "bbox": [ 69, 335, 541, 348 ], "score": 1.0, "content": "work also studies Schrödinger bridges from unpaired samples (Shi et al., 2022) and regularization of flows", "type": "text" } ], "index": 14 }, { "bbox": [ 70, 347, 541, 360 ], "spans": [ { "bbox": [ 70, 347, 541, 360 ], "score": 1.0, "content": "using dynamic OT (Liu et al., 2023b). We also note the work Pooladian et al. (2023), concurrent with the", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 358, 541, 374 ], "spans": [ { "bbox": [ 69, 358, 541, 374 ], "score": 1.0, "content": "preprint version of this paper. Other concurrent works explore various solutions to approximate Schrödinger", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 369, 357, 385 ], "spans": [ { "bbox": [ 69, 369, 357, 385 ], "score": 1.0, "content": "bridges (Somnath et al., 2023; Shi et al., 2023; Liu et al., 2023a).", "type": "text" } ], "index": 17 } ], "index": 14 }, { "type": "text", "bbox": [ 70, 397, 540, 456 ], "lines": [ { "bbox": [ 70, 397, 542, 410 ], "spans": [ { "bbox": [ 70, 397, 542, 410 ], "score": 1.0, "content": "Dynamic optimal transport. There are a variety of methods that consider dynamic OT between", "type": "text" } ], "index": 18 }, { "bbox": [ 70, 409, 541, 421 ], "spans": [ { "bbox": [ 70, 409, 541, 421 ], "score": 1.0, "content": "continuous distributions with neural networks; however, these require constrained architectures (Leygonie", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 420, 542, 434 ], "spans": [ { "bbox": [ 68, 420, 542, 434 ], "score": 1.0, "content": "et al., 2019; Makkuva et al., 2020; Bunne et al., 2022) or use a regularized CNF, which is challenging to", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 432, 543, 446 ], "spans": [ { "bbox": [ 69, 432, 543, 446 ], "score": 1.0, "content": "optimize (Tong et al., 2020; Finlay et al., 2020; Onken et al., 2021; Huguet et al., 2022a). With our work it is", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 445, 415, 457 ], "spans": [ { "bbox": [ 69, 445, 415, 457 ], "score": 1.0, "content": "possible to achieve optimal transport flows without either of these constraints.", "type": "text" } ], "index": 22 } ], "index": 20 }, { "type": "title", "bbox": [ 71, 473, 159, 487 ], "lines": [ { "bbox": [ 69, 470, 160, 491 ], "spans": [ { "bbox": [ 69, 470, 160, 491 ], "score": 1.0, "content": "5 Experiments", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 71, 500, 540, 536 ], "lines": [ { "bbox": [ 68, 498, 542, 514 ], "spans": [ { "bbox": [ 68, 498, 542, 514 ], "score": 1.0, "content": "In this section we empirically evaluate the I-CFM, OT-CFM, and SB-CFM objectives, as well as algorithms", "type": "text" } ], "index": 24 }, { "bbox": [ 70, 513, 541, 525 ], "spans": [ { "bbox": [ 70, 513, 541, 525 ], "score": 1.0, "content": "from prior work, with respect to both optimal transport and generative modeling criteria. All experiment", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 523, 231, 537 ], "spans": [ { "bbox": [ 69, 523, 231, 537 ], "score": 1.0, "content": "details can be found in Appendix E.", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "title", "bbox": [ 70, 551, 394, 563 ], "lines": [ { "bbox": [ 69, 549, 396, 566 ], "spans": [ { "bbox": [ 69, 549, 396, 566 ], "score": 1.0, "content": "5.1 Low-dimensional data: Optimal transport and faster convergence", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 70, 573, 541, 609 ], "lines": [ { "bbox": [ 70, 573, 541, 586 ], "spans": [ { "bbox": [ 70, 573, 541, 586 ], "score": 1.0, "content": "We evaluate how well various models perform dynamic optimal transport and generative modeling in low", "type": "text" } ], "index": 28 }, { "bbox": [ 70, 585, 541, 599 ], "spans": [ { "bbox": [ 70, 585, 541, 599 ], "score": 1.0, "content": "dimensions. We train ODEs mapping between four pairs of two-dimensional datasets: between a standard", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 596, 420, 610 ], "spans": [ { "bbox": [ 69, 596, 420, 610 ], "score": 1.0, "content": "Gaussian and 8gaussians, moons, and scurve and between moons and 8gaussians.", "type": "text" } ], "index": 30 } ], "index": 29 }, { "type": "text", "bbox": [ 70, 622, 541, 732 ], "lines": [ { "bbox": [ 70, 623, 541, 636 ], "spans": [ { "bbox": [ 70, 623, 541, 636 ], "score": 1.0, "content": "OT-CFM approximates dynamic OT. To measure how well a model solves the OT problem we", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 635, 541, 649 ], "spans": [ { "bbox": [ 69, 635, 435, 649 ], "score": 1.0, "content": "use normalized path energy (NPE), defined via the 2-Wasserstein distance as", "type": "text" }, { "bbox": [ 436, 636, 541, 648 ], "score": 0.88, "content": "\\mathrm { N P E } ( v _ { \\theta } ) ~ = ~ | \\mathrm { P E } ( v _ { \\theta } ) ~ -", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 71, 646, 543, 663 ], "spans": [ { "bbox": [ 71, 650, 170, 661 ], "score": 0.93, "content": "W _ { 2 } ^ { 2 } ( q _ { 0 } , q _ { 1 } ) \\vert / W _ { 2 } ^ { 2 } ( q _ { 0 } , q _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 170, 646, 320, 663 ], "score": 1.0, "content": ", where the path energy (PE) is", "type": "text" }, { "bbox": [ 320, 647, 496, 662 ], "score": 0.93, "content": "\\begin{array} { r } { \\mathrm { P E } ( v _ { \\theta } ) = \\mathbb { E } _ { x ( 0 ) \\sim q ( x _ { 0 } ) } \\int _ { 0 } ^ { 1 } \\| v _ { \\theta } ( t , x ( t ) ) \\| ^ { 2 } d t } \\end{array}", "type": "inline_equation" }, { "bbox": [ 496, 646, 543, 663 ], "score": 1.0, "content": ". Table 2", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 660, 541, 673 ], "spans": [ { "bbox": [ 69, 660, 541, 673 ], "score": 1.0, "content": "summarizes our results showing that OT-CFM flows generalize better to the test set and are very close to", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 671, 542, 686 ], "spans": [ { "bbox": [ 69, 671, 474, 686 ], "score": 1.0, "content": "the dynamic OT paths as measured by normalized path energy. We find transforming moons", "type": "text" }, { "bbox": [ 474, 674, 484, 683 ], "score": 0.68, "content": "", "type": "inline_equation" }, { "bbox": [ 484, 671, 542, 686 ], "score": 1.0, "content": "8gaussians to", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 684, 541, 697 ], "spans": [ { "bbox": [ 70, 684, 541, 697 ], "score": 1.0, "content": "be particularly challenging to learn for I-CFM as compared to OT-CFM; the learned paths are depicted in", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 695, 541, 710 ], "spans": [ { "bbox": [ 69, 695, 541, 710 ], "score": 1.0, "content": "Fig. 1 (bottom). Although OT-CFM uses a minibatch OT map, we find that OT-CFM requires surprisingly", "type": "text" } ], "index": 37 }, { "bbox": [ 70, 708, 541, 721 ], "spans": [ { "bbox": [ 70, 708, 541, 721 ], "score": 1.0, "content": "small batches to approximate the OT map well, suggesting some generalization advantages of the network", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 720, 177, 734 ], "spans": [ { "bbox": [ 69, 720, 177, 734 ], "score": 1.0, "content": "optimization (Fig. D.2).", "type": "text" } ], "index": 39 } ], "index": 35 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 367, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 70, 128, 539, 224 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 128 ], "group_id": 0, "lines": [ { "bbox": [ 69, 80, 542, 93 ], "spans": [ { "bbox": [ 69, 80, 448, 93 ], "score": 1.0, "content": "Table 2: Comparison of neural optimal transport methods over four distribution pairs (", "type": "text" }, { "bbox": [ 448, 84, 472, 93 ], "score": 0.88, "content": "\\mu \\pm \\sigma", "type": "inline_equation" }, { "bbox": [ 473, 80, 542, 93 ], "score": 1.0, "content": "over five seeds)", "type": "text" } ], "index": 0 }, { "bbox": [ 68, 91, 539, 105 ], "spans": [ { "bbox": [ 68, 91, 539, 105 ], "score": 1.0, "content": "in terms of fit (2-Wasserstein), optimal transport performance (normalized path energy), and runtime. ‘—", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 104, 542, 118 ], "spans": [ { "bbox": [ 69, 104, 542, 118 ], "score": 1.0, "content": "indicates a method that requires a Gaussian source. Best in bold. CFM and RF models are trained on a", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 117, 391, 129 ], "spans": [ { "bbox": [ 69, 117, 391, 129 ], "score": 1.0, "content": "single CPU core, other baselines are trained with a GPU and two CPUs.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "table_body", "bbox": [ 70, 128, 539, 224 ], "group_id": 0, "lines": [ { "bbox": [ 70, 128, 539, 224 ], "spans": [ { "bbox": [ 70, 128, 539, 224 ], "score": 0.979, "html": "
Dataset→N→8gaussiansmoons-→8gaussiansN→moonsN-→scurveAvg.train time
Algorithm ↓Metric→wNPEWNPEwNPEWNPE(x10²s)
OT-CFM1.262±0.3480.018±0.0141.923±0.3910.053±0.0350.239±0.0480.087±0.0610.264±0.0930.027±0.0261.129±0.335
I-CFM1.284±0.3840.222±0.0321.977±0.2662.738±0.1810.338±0.1090.841±0.1480.333±0.0600.867±0.1170.630±0.365
2-RF (Liu, 2022)1.436±0.3440.069±0.0272.211±0.4230.149±0.1010.278±0.0260.076±0.0670.395±0.1110.112±0.0850.862±0.166
3-RF (Liu, 2022)1.337±0.3670.055±0.0432.700±0.5870.123±0.1120.305±0.0260.084±0.0510.395±0.0820.129±0.0750.954±0.116
FM (Lipman et al., 2023)1.062±0.1960.174±0.0300.246±0.0770.778±0.1440.377±0.0990.772±0.0810.708±0.370
Reg.CNF (Finlay et al., 2020)1.144±0.0750.274±0.0600.376±0.0400.620±0.0880.581±0.1950.586±0.5038.021±3.288
CNF (Chen et al., 2018)1.055±0.0590.151±0.0640.387±0.0652.937±1.9730.645±0.34310.548±8.10018.810±12.677
ICNN (Makkuva et al., 2020)1.771±0.3980.747±0.0292.193±0.1360.832±0.0040.532±0.0460.267±0.0100.753±0.0680.344±0.0452.912±0.626
", "type": "table", "image_path": "59ae1b8a396f7d49e4d88fc89f2f1498a6ec874ffb6963f4e2765c1073e3f351.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 70, 128, 539, 160.0 ], "spans": [], "index": 4 }, { "bbox": [ 70, 160.0, 539, 192.0 ], "spans": [], "index": 5 }, { "bbox": [ 70, 192.0, 539, 224.0 ], "spans": [], "index": 6 } ] } ], "index": 3.25 }, { "type": "text", "bbox": [ 71, 245, 540, 293 ], "lines": [ { "bbox": [ 69, 244, 542, 259 ], "spans": [ { "bbox": [ 69, 244, 542, 259 ], "score": 1.0, "content": "Song et al., 2021a; Bao et al., 2022). These methods generally consider a simple Gaussian diffusion process,", "type": "text" } ], "index": 7 }, { "bbox": [ 70, 257, 541, 270 ], "spans": [ { "bbox": [ 70, 257, 541, 270 ], "score": 1.0, "content": "and do not consider generalizing the source distribution. Other works consider general source distributions", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 270, 541, 282 ], "spans": [ { "bbox": [ 70, 270, 541, 282 ], "score": 1.0, "content": "but this makes optimization and inference more challenging, needing multiple iterations or other tricks to", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 281, 408, 295 ], "spans": [ { "bbox": [ 69, 281, 408, 295 ], "score": 1.0, "content": "perform well (Wang et al., 2021; De Bortoli et al., 2021; Vargas et al., 2021).", "type": "text" } ], "index": 10 } ], "index": 8.5, "bbox_fs": [ 69, 244, 542, 295 ] }, { "type": "text", "bbox": [ 70, 299, 540, 383 ], "lines": [ { "bbox": [ 69, 299, 541, 312 ], "spans": [ { "bbox": [ 69, 299, 541, 312 ], "score": 1.0, "content": "Prior work considering simulation-free training of CNFs considers algorithms that are equivalent to CFM with", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 310, 541, 325 ], "spans": [ { "bbox": [ 69, 310, 541, 325 ], "score": 1.0, "content": "Gaussian source distribution (Rozen et al., 2021; Ben-Hamu et al., 2022; Lipman et al., 2023) or independent", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 321, 541, 336 ], "spans": [ { "bbox": [ 69, 321, 131, 336 ], "score": 1.0, "content": "samples from", "type": "text" }, { "bbox": [ 131, 328, 140, 335 ], "score": 0.84, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 141, 321, 146, 336 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 146, 328, 155, 335 ], "score": 0.85, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 155, 321, 541, 336 ], "score": 1.0, "content": "(Albergo & Vanden-Eijnden, 2023; Albergo et al., 2023; Neklyudov et al., 2023). Recent", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 335, 541, 348 ], "spans": [ { "bbox": [ 69, 335, 541, 348 ], "score": 1.0, "content": "work also studies Schrödinger bridges from unpaired samples (Shi et al., 2022) and regularization of flows", "type": "text" } ], "index": 14 }, { "bbox": [ 70, 347, 541, 360 ], "spans": [ { "bbox": [ 70, 347, 541, 360 ], "score": 1.0, "content": "using dynamic OT (Liu et al., 2023b). We also note the work Pooladian et al. (2023), concurrent with the", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 358, 541, 374 ], "spans": [ { "bbox": [ 69, 358, 541, 374 ], "score": 1.0, "content": "preprint version of this paper. Other concurrent works explore various solutions to approximate Schrödinger", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 369, 357, 385 ], "spans": [ { "bbox": [ 69, 369, 357, 385 ], "score": 1.0, "content": "bridges (Somnath et al., 2023; Shi et al., 2023; Liu et al., 2023a).", "type": "text" } ], "index": 17 } ], "index": 14, "bbox_fs": [ 69, 299, 541, 385 ] }, { "type": "text", "bbox": [ 70, 397, 540, 456 ], "lines": [ { "bbox": [ 70, 397, 542, 410 ], "spans": [ { "bbox": [ 70, 397, 542, 410 ], "score": 1.0, "content": "Dynamic optimal transport. There are a variety of methods that consider dynamic OT between", "type": "text" } ], "index": 18 }, { "bbox": [ 70, 409, 541, 421 ], "spans": [ { "bbox": [ 70, 409, 541, 421 ], "score": 1.0, "content": "continuous distributions with neural networks; however, these require constrained architectures (Leygonie", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 420, 542, 434 ], "spans": [ { "bbox": [ 68, 420, 542, 434 ], "score": 1.0, "content": "et al., 2019; Makkuva et al., 2020; Bunne et al., 2022) or use a regularized CNF, which is challenging to", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 432, 543, 446 ], "spans": [ { "bbox": [ 69, 432, 543, 446 ], "score": 1.0, "content": "optimize (Tong et al., 2020; Finlay et al., 2020; Onken et al., 2021; Huguet et al., 2022a). With our work it is", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 445, 415, 457 ], "spans": [ { "bbox": [ 69, 445, 415, 457 ], "score": 1.0, "content": "possible to achieve optimal transport flows without either of these constraints.", "type": "text" } ], "index": 22 } ], "index": 20, "bbox_fs": [ 68, 397, 543, 457 ] }, { "type": "title", "bbox": [ 71, 473, 159, 487 ], "lines": [ { "bbox": [ 69, 470, 160, 491 ], "spans": [ { "bbox": [ 69, 470, 160, 491 ], "score": 1.0, "content": "5 Experiments", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 71, 500, 540, 536 ], "lines": [ { "bbox": [ 68, 498, 542, 514 ], "spans": [ { "bbox": [ 68, 498, 542, 514 ], "score": 1.0, "content": "In this section we empirically evaluate the I-CFM, OT-CFM, and SB-CFM objectives, as well as algorithms", "type": "text" } ], "index": 24 }, { "bbox": [ 70, 513, 541, 525 ], "spans": [ { "bbox": [ 70, 513, 541, 525 ], "score": 1.0, "content": "from prior work, with respect to both optimal transport and generative modeling criteria. All experiment", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 523, 231, 537 ], "spans": [ { "bbox": [ 69, 523, 231, 537 ], "score": 1.0, "content": "details can be found in Appendix E.", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 68, 498, 542, 537 ] }, { "type": "title", "bbox": [ 70, 551, 394, 563 ], "lines": [ { "bbox": [ 69, 549, 396, 566 ], "spans": [ { "bbox": [ 69, 549, 396, 566 ], "score": 1.0, "content": "5.1 Low-dimensional data: Optimal transport and faster convergence", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 70, 573, 541, 609 ], "lines": [ { "bbox": [ 70, 573, 541, 586 ], "spans": [ { "bbox": [ 70, 573, 541, 586 ], "score": 1.0, "content": "We evaluate how well various models perform dynamic optimal transport and generative modeling in low", "type": "text" } ], "index": 28 }, { "bbox": [ 70, 585, 541, 599 ], "spans": [ { "bbox": [ 70, 585, 541, 599 ], "score": 1.0, "content": "dimensions. We train ODEs mapping between four pairs of two-dimensional datasets: between a standard", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 596, 420, 610 ], "spans": [ { "bbox": [ 69, 596, 420, 610 ], "score": 1.0, "content": "Gaussian and 8gaussians, moons, and scurve and between moons and 8gaussians.", "type": "text" } ], "index": 30 } ], "index": 29, "bbox_fs": [ 69, 573, 541, 610 ] }, { "type": "text", "bbox": [ 70, 622, 541, 732 ], "lines": [ { "bbox": [ 70, 623, 541, 636 ], "spans": [ { "bbox": [ 70, 623, 541, 636 ], "score": 1.0, "content": "OT-CFM approximates dynamic OT. To measure how well a model solves the OT problem we", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 635, 541, 649 ], "spans": [ { "bbox": [ 69, 635, 435, 649 ], "score": 1.0, "content": "use normalized path energy (NPE), defined via the 2-Wasserstein distance as", "type": "text" }, { "bbox": [ 436, 636, 541, 648 ], "score": 0.88, "content": "\\mathrm { N P E } ( v _ { \\theta } ) ~ = ~ | \\mathrm { P E } ( v _ { \\theta } ) ~ -", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 71, 646, 543, 663 ], "spans": [ { "bbox": [ 71, 650, 170, 661 ], "score": 0.93, "content": "W _ { 2 } ^ { 2 } ( q _ { 0 } , q _ { 1 } ) \\vert / W _ { 2 } ^ { 2 } ( q _ { 0 } , q _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 170, 646, 320, 663 ], "score": 1.0, "content": ", where the path energy (PE) is", "type": "text" }, { "bbox": [ 320, 647, 496, 662 ], "score": 0.93, "content": "\\begin{array} { r } { \\mathrm { P E } ( v _ { \\theta } ) = \\mathbb { E } _ { x ( 0 ) \\sim q ( x _ { 0 } ) } \\int _ { 0 } ^ { 1 } \\| v _ { \\theta } ( t , x ( t ) ) \\| ^ { 2 } d t } \\end{array}", "type": "inline_equation" }, { "bbox": [ 496, 646, 543, 663 ], "score": 1.0, "content": ". Table 2", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 660, 541, 673 ], "spans": [ { "bbox": [ 69, 660, 541, 673 ], "score": 1.0, "content": "summarizes our results showing that OT-CFM flows generalize better to the test set and are very close to", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 671, 542, 686 ], "spans": [ { "bbox": [ 69, 671, 474, 686 ], "score": 1.0, "content": "the dynamic OT paths as measured by normalized path energy. We find transforming moons", "type": "text" }, { "bbox": [ 474, 674, 484, 683 ], "score": 0.68, "content": "", "type": "inline_equation" }, { "bbox": [ 484, 671, 542, 686 ], "score": 1.0, "content": "8gaussians to", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 684, 541, 697 ], "spans": [ { "bbox": [ 70, 684, 541, 697 ], "score": 1.0, "content": "be particularly challenging to learn for I-CFM as compared to OT-CFM; the learned paths are depicted in", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 695, 541, 710 ], "spans": [ { "bbox": [ 69, 695, 541, 710 ], "score": 1.0, "content": "Fig. 1 (bottom). Although OT-CFM uses a minibatch OT map, we find that OT-CFM requires surprisingly", "type": "text" } ], "index": 37 }, { "bbox": [ 70, 708, 541, 721 ], "spans": [ { "bbox": [ 70, 708, 541, 721 ], "score": 1.0, "content": "small batches to approximate the OT map well, suggesting some generalization advantages of the network", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 720, 177, 734 ], "spans": [ { "bbox": [ 69, 720, 177, 734 ], "score": 1.0, "content": "optimization (Fig. D.2).", "type": "text" } ], "index": 39 } ], "index": 35, "bbox_fs": [ 69, 623, 543, 734 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 75, 77, 541, 158 ], "blocks": [ { "type": "image_body", "bbox": [ 75, 77, 541, 158 ], "group_id": 0, "lines": [ { "bbox": [ 75, 77, 541, 158 ], "spans": [ { "bbox": [ 75, 77, 541, 158 ], "score": 0.961, "type": "image", "image_path": "817d0bdb712f61197d4553da5ac9f114ed7ad99a7a36875aa40379dabcae2a64.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 75, 77, 541, 104.0 ], "spans": [], "index": 0 }, { "bbox": [ 75, 104.0, 541, 131.0 ], "spans": [], "index": 1 }, { "bbox": [ 75, 131.0, 541, 158.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 71, 165, 542, 200 ], "group_id": 0, "lines": [ { "bbox": [ 68, 164, 543, 179 ], "spans": [ { "bbox": [ 68, 164, 543, 179 ], "score": 1.0, "content": "Figure 2: Left: OT-CFM trains faster, in terms of validation set error, than CFM and FM models. Right:", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 176, 542, 191 ], "spans": [ { "bbox": [ 69, 176, 542, 191 ], "score": 1.0, "content": "With different ODE integrators, OT-CFM reduces the error for a fixed number of function evaluations during", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 188, 114, 203 ], "spans": [ { "bbox": [ 69, 188, 114, 203 ], "score": 1.0, "content": "inference.", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 70, 228, 301, 277 ], "lines": [ { "bbox": [ 70, 227, 303, 242 ], "spans": [ { "bbox": [ 70, 227, 303, 242 ], "score": 1.0, "content": "Table 3: Schrödinger bridge flow comparison, showing", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 241, 302, 253 ], "spans": [ { "bbox": [ 69, 241, 302, 253 ], "score": 1.0, "content": "average error over flow time to ground truth averaged", "type": "text" } ], "index": 7 }, { "bbox": [ 70, 253, 302, 264 ], "spans": [ { "bbox": [ 70, 253, 302, 264 ], "score": 1.0, "content": "over 5 models for SB-CFM and 5 dynamics from DSB", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 265, 179, 277 ], "spans": [ { "bbox": [ 70, 265, 179, 277 ], "score": 1.0, "content": "(De Bortoli et al., 2021).", "type": "text" } ], "index": 9 } ], "index": 7.5 }, { "type": "table", "bbox": [ 312, 219, 538, 288 ], "blocks": [ { "type": "table_body", "bbox": [ 312, 219, 538, 288 ], "group_id": 0, "lines": [ { "bbox": [ 312, 219, 538, 288 ], "spans": [ { "bbox": [ 312, 219, 538, 288 ], "score": 0.975, "html": "
Dataset ↓Alg.→SB-CFMDSB
N-→8gaussians0.454 ± 0.1641.440 ± 0.720
moons-→8gaussians1.377 ± 0.2292.407 ± 1.025
N→moons0.283 ± 0.0480.333 ± 0.129
N-→scurve0.297 ± 0.0640.383 ± 0.134
", "type": "table", "image_path": "5486ade4f62605355541a4fab528927b509ed7800f37eee864e71deea2bbff1f.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 312, 219, 538, 236.25 ], "spans": [], "index": 10 }, { "bbox": [ 312, 236.25, 538, 253.5 ], "spans": [], "index": 11 }, { "bbox": [ 312, 253.5, 538, 270.75 ], "spans": [], "index": 12 }, { "bbox": [ 312, 270.75, 538, 288.0 ], "spans": [], "index": 13 } ] } ], "index": 11.5 }, { "type": "text", "bbox": [ 70, 314, 541, 386 ], "lines": [ { "bbox": [ 70, 313, 542, 327 ], "spans": [ { "bbox": [ 70, 313, 542, 327 ], "score": 1.0, "content": "OT-CFM yields faster training. By conditioning on minibatch optimal transport flows, OT-CFM is", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 326, 541, 339 ], "spans": [ { "bbox": [ 69, 326, 541, 339 ], "score": 1.0, "content": "substantially easier to train, which we posit is due to the variance reduction of the conditional flow. In", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 338, 540, 350 ], "spans": [ { "bbox": [ 70, 338, 540, 350 ], "score": 1.0, "content": "Fig. 2 (left), we evaluate the performance over time of OT-CFM against CFM and FM objectives. For the", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 349, 541, 363 ], "spans": [ { "bbox": [ 69, 349, 541, 363 ], "score": 1.0, "content": "same number of steps OT-CFM has better performance on the validation set. In Table D.1, we compare the", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 361, 541, 375 ], "spans": [ { "bbox": [ 69, 361, 541, 375 ], "score": 1.0, "content": "training times for various Neural OT methods whose performance can be seen in Table 2. Simulation-free", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 374, 417, 388 ], "spans": [ { "bbox": [ 69, 374, 417, 388 ], "score": 1.0, "content": "optimization is significantly faster to train with equal or superior performance.", "type": "text" } ], "index": 19 } ], "index": 16.5 }, { "type": "text", "bbox": [ 70, 405, 540, 501 ], "lines": [ { "bbox": [ 70, 405, 541, 418 ], "spans": [ { "bbox": [ 70, 405, 541, 418 ], "score": 1.0, "content": "OT-CFM yields faster inference. We next evaluate the quality of samples during inference time. In", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 417, 541, 430 ], "spans": [ { "bbox": [ 70, 417, 541, 430 ], "score": 1.0, "content": "Fig. 2 (right), we compare the quality of samples for different number of function evaluations (NFEs) across", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 429, 541, 442 ], "spans": [ { "bbox": [ 69, 429, 541, 442 ], "score": 1.0, "content": "different flow matching objectives. In this experiment we sample from the source distribution test set and", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 440, 541, 453 ], "spans": [ { "bbox": [ 69, 440, 541, 453 ], "score": 1.0, "content": "simulate the ODE over time for different solvers. We find that OT-CFM consistently requires fewer evaluations", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 453, 540, 465 ], "spans": [ { "bbox": [ 69, 453, 540, 465 ], "score": 1.0, "content": "to achieve the same quality and achieves better quality with the same NFEs. This is consistent with previous", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 465, 542, 478 ], "spans": [ { "bbox": [ 69, 465, 542, 478 ], "score": 1.0, "content": "work, which found OT paths lead to faster, higher quality inference in regularized CNFs (Finlay et al., 2020;", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 476, 541, 490 ], "spans": [ { "bbox": [ 69, 476, 541, 490 ], "score": 1.0, "content": "Onken et al., 2021) and flow matching vs. standard variance-preserving and variance-exploding probability", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 488, 195, 503 ], "spans": [ { "bbox": [ 69, 488, 195, 503 ], "score": 1.0, "content": "paths (Lipman et al., 2023).", "type": "text" } ], "index": 27 } ], "index": 23.5 }, { "type": "text", "bbox": [ 71, 520, 540, 580 ], "lines": [ { "bbox": [ 70, 520, 541, 533 ], "spans": [ { "bbox": [ 70, 520, 541, 533 ], "score": 1.0, "content": "SB-CFM reproduces Schrödinger bridge flows. There are a number of methods which theoretically", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 531, 541, 545 ], "spans": [ { "bbox": [ 69, 531, 541, 545 ], "score": 1.0, "content": "converge to a Schrödinger bridge between two datasets. In Table 3 we compare SB-CFM and the diffusion", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 543, 541, 558 ], "spans": [ { "bbox": [ 69, 543, 541, 558 ], "score": 1.0, "content": "Schrödinger bridge (DSB) method introduced in De Bortoli et al. (2021) on the quality of the learnt Schrödinger", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 555, 541, 570 ], "spans": [ { "bbox": [ 69, 555, 541, 570 ], "score": 1.0, "content": "bridges based on the average 2-Wasserstein distance to ground truth Schrödinger bridge samples over 18 time", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 568, 417, 582 ], "spans": [ { "bbox": [ 69, 568, 417, 582 ], "score": 1.0, "content": "steps. Furthermore, SB-CFM is also significantly faster than DSB (Table D.1).", "type": "text" } ], "index": 32 } ], "index": 30 }, { "type": "title", "bbox": [ 71, 600, 271, 613 ], "lines": [ { "bbox": [ 69, 599, 272, 615 ], "spans": [ { "bbox": [ 69, 599, 272, 615 ], "score": 1.0, "content": "5.2 Application to single-cell interpolation", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 70, 624, 541, 732 ], "lines": [ { "bbox": [ 69, 624, 542, 639 ], "spans": [ { "bbox": [ 69, 624, 542, 639 ], "score": 1.0, "content": "As a specific application, we consider the task of single-cell trajectory interpolation. In this task we use", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 635, 542, 650 ], "spans": [ { "bbox": [ 69, 635, 374, 650 ], "score": 1.0, "content": "leave-one-out validation over the timepoints. From times data at times", "type": "text" }, { "bbox": [ 375, 639, 449, 649 ], "score": 0.87, "content": "[ 0 , t - 1 ] , [ t + 1 , T ]", "type": "inline_equation" }, { "bbox": [ 449, 635, 542, 650 ], "score": 1.0, "content": "we try to interpolate", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 648, 543, 662 ], "spans": [ { "bbox": [ 69, 648, 176, 662 ], "score": 1.0, "content": "its distribution at time", "type": "text" }, { "bbox": [ 176, 652, 180, 658 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 180, 648, 543, 662 ], "score": 1.0, "content": "following the setup of Schiebinger et al. (2019); Tong et al. (2020); Huguet et al.", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 660, 541, 673 ], "spans": [ { "bbox": [ 69, 660, 541, 673 ], "score": 1.0, "content": "(2022a). Low error means we model individual cells well, which is useful in a number of downstream tasks", "type": "text" } ], "index": 37 }, { "bbox": [ 68, 672, 543, 686 ], "spans": [ { "bbox": [ 68, 672, 543, 686 ], "score": 1.0, "content": "such as gene regulatory network inference (Aliee et al., 2021; Yeo et al., 2021). Following Huguet et al.", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 684, 542, 698 ], "spans": [ { "bbox": [ 69, 684, 542, 698 ], "score": 1.0, "content": "(2022b), we repurpose the CITE-seq and Multiome datasets from a recent NeurIPS competition for this", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 695, 542, 709 ], "spans": [ { "bbox": [ 69, 695, 542, 709 ], "score": 1.0, "content": "task (Burkhardt et al., 2022). We also include the Embryoid body data from Moon et al. (2019); Tong et al.", "type": "text" } ], "index": 40 }, { "bbox": [ 69, 708, 542, 722 ], "spans": [ { "bbox": [ 69, 708, 542, 722 ], "score": 1.0, "content": "(2020). Table 4 shows the average earth mover’s distance (1-Wasserstein) on left–out timepoints for three", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 718, 487, 735 ], "spans": [ { "bbox": [ 69, 718, 487, 735 ], "score": 1.0, "content": "datasets. On all three datasets OT-CFM outperforms other methods and baselines on average.", "type": "text" } ], "index": 42 } ], "index": 38 } ], "page_idx": 9, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 367, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "10", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 75, 77, 541, 158 ], "blocks": [ { "type": "image_body", "bbox": [ 75, 77, 541, 158 ], "group_id": 0, "lines": [ { "bbox": [ 75, 77, 541, 158 ], "spans": [ { "bbox": [ 75, 77, 541, 158 ], "score": 0.961, "type": "image", "image_path": "817d0bdb712f61197d4553da5ac9f114ed7ad99a7a36875aa40379dabcae2a64.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 75, 77, 541, 104.0 ], "spans": [], "index": 0 }, { "bbox": [ 75, 104.0, 541, 131.0 ], "spans": [], "index": 1 }, { "bbox": [ 75, 131.0, 541, 158.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 71, 165, 542, 200 ], "group_id": 0, "lines": [ { "bbox": [ 68, 164, 543, 179 ], "spans": [ { "bbox": [ 68, 164, 543, 179 ], "score": 1.0, "content": "Figure 2: Left: OT-CFM trains faster, in terms of validation set error, than CFM and FM models. Right:", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 176, 542, 191 ], "spans": [ { "bbox": [ 69, 176, 542, 191 ], "score": 1.0, "content": "With different ODE integrators, OT-CFM reduces the error for a fixed number of function evaluations during", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 188, 114, 203 ], "spans": [ { "bbox": [ 69, 188, 114, 203 ], "score": 1.0, "content": "inference.", "type": "text" } ], "index": 5 } ], "index": 4 } ], "index": 2.5 }, { "type": "text", "bbox": [ 70, 228, 301, 277 ], "lines": [ { "bbox": [ 70, 227, 303, 242 ], "spans": [ { "bbox": [ 70, 227, 303, 242 ], "score": 1.0, "content": "Table 3: Schrödinger bridge flow comparison, showing", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 241, 302, 253 ], "spans": [ { "bbox": [ 69, 241, 302, 253 ], "score": 1.0, "content": "average error over flow time to ground truth averaged", "type": "text" } ], "index": 7 }, { "bbox": [ 70, 253, 302, 264 ], "spans": [ { "bbox": [ 70, 253, 302, 264 ], "score": 1.0, "content": "over 5 models for SB-CFM and 5 dynamics from DSB", "type": "text" } ], "index": 8 }, { "bbox": [ 70, 265, 179, 277 ], "spans": [ { "bbox": [ 70, 265, 179, 277 ], "score": 1.0, "content": "(De Bortoli et al., 2021).", "type": "text" } ], "index": 9 } ], "index": 7.5, "bbox_fs": [ 69, 227, 303, 277 ] }, { "type": "table", "bbox": [ 312, 219, 538, 288 ], "blocks": [ { "type": "table_body", "bbox": [ 312, 219, 538, 288 ], "group_id": 0, "lines": [ { "bbox": [ 312, 219, 538, 288 ], "spans": [ { "bbox": [ 312, 219, 538, 288 ], "score": 0.975, "html": "
Dataset ↓Alg.→SB-CFMDSB
N-→8gaussians0.454 ± 0.1641.440 ± 0.720
moons-→8gaussians1.377 ± 0.2292.407 ± 1.025
N→moons0.283 ± 0.0480.333 ± 0.129
N-→scurve0.297 ± 0.0640.383 ± 0.134
", "type": "table", "image_path": "5486ade4f62605355541a4fab528927b509ed7800f37eee864e71deea2bbff1f.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 312, 219, 538, 236.25 ], "spans": [], "index": 10 }, { "bbox": [ 312, 236.25, 538, 253.5 ], "spans": [], "index": 11 }, { "bbox": [ 312, 253.5, 538, 270.75 ], "spans": [], "index": 12 }, { "bbox": [ 312, 270.75, 538, 288.0 ], "spans": [], "index": 13 } ] } ], "index": 11.5 }, { "type": "text", "bbox": [ 70, 314, 541, 386 ], "lines": [ { "bbox": [ 70, 313, 542, 327 ], "spans": [ { "bbox": [ 70, 313, 542, 327 ], "score": 1.0, "content": "OT-CFM yields faster training. By conditioning on minibatch optimal transport flows, OT-CFM is", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 326, 541, 339 ], "spans": [ { "bbox": [ 69, 326, 541, 339 ], "score": 1.0, "content": "substantially easier to train, which we posit is due to the variance reduction of the conditional flow. In", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 338, 540, 350 ], "spans": [ { "bbox": [ 70, 338, 540, 350 ], "score": 1.0, "content": "Fig. 2 (left), we evaluate the performance over time of OT-CFM against CFM and FM objectives. For the", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 349, 541, 363 ], "spans": [ { "bbox": [ 69, 349, 541, 363 ], "score": 1.0, "content": "same number of steps OT-CFM has better performance on the validation set. In Table D.1, we compare the", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 361, 541, 375 ], "spans": [ { "bbox": [ 69, 361, 541, 375 ], "score": 1.0, "content": "training times for various Neural OT methods whose performance can be seen in Table 2. Simulation-free", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 374, 417, 388 ], "spans": [ { "bbox": [ 69, 374, 417, 388 ], "score": 1.0, "content": "optimization is significantly faster to train with equal or superior performance.", "type": "text" } ], "index": 19 } ], "index": 16.5, "bbox_fs": [ 69, 313, 542, 388 ] }, { "type": "text", "bbox": [ 70, 405, 540, 501 ], "lines": [ { "bbox": [ 70, 405, 541, 418 ], "spans": [ { "bbox": [ 70, 405, 541, 418 ], "score": 1.0, "content": "OT-CFM yields faster inference. We next evaluate the quality of samples during inference time. In", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 417, 541, 430 ], "spans": [ { "bbox": [ 70, 417, 541, 430 ], "score": 1.0, "content": "Fig. 2 (right), we compare the quality of samples for different number of function evaluations (NFEs) across", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 429, 541, 442 ], "spans": [ { "bbox": [ 69, 429, 541, 442 ], "score": 1.0, "content": "different flow matching objectives. In this experiment we sample from the source distribution test set and", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 440, 541, 453 ], "spans": [ { "bbox": [ 69, 440, 541, 453 ], "score": 1.0, "content": "simulate the ODE over time for different solvers. We find that OT-CFM consistently requires fewer evaluations", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 453, 540, 465 ], "spans": [ { "bbox": [ 69, 453, 540, 465 ], "score": 1.0, "content": "to achieve the same quality and achieves better quality with the same NFEs. This is consistent with previous", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 465, 542, 478 ], "spans": [ { "bbox": [ 69, 465, 542, 478 ], "score": 1.0, "content": "work, which found OT paths lead to faster, higher quality inference in regularized CNFs (Finlay et al., 2020;", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 476, 541, 490 ], "spans": [ { "bbox": [ 69, 476, 541, 490 ], "score": 1.0, "content": "Onken et al., 2021) and flow matching vs. standard variance-preserving and variance-exploding probability", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 488, 195, 503 ], "spans": [ { "bbox": [ 69, 488, 195, 503 ], "score": 1.0, "content": "paths (Lipman et al., 2023).", "type": "text" } ], "index": 27 } ], "index": 23.5, "bbox_fs": [ 69, 405, 542, 503 ] }, { "type": "text", "bbox": [ 71, 520, 540, 580 ], "lines": [ { "bbox": [ 70, 520, 541, 533 ], "spans": [ { "bbox": [ 70, 520, 541, 533 ], "score": 1.0, "content": "SB-CFM reproduces Schrödinger bridge flows. There are a number of methods which theoretically", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 531, 541, 545 ], "spans": [ { "bbox": [ 69, 531, 541, 545 ], "score": 1.0, "content": "converge to a Schrödinger bridge between two datasets. In Table 3 we compare SB-CFM and the diffusion", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 543, 541, 558 ], "spans": [ { "bbox": [ 69, 543, 541, 558 ], "score": 1.0, "content": "Schrödinger bridge (DSB) method introduced in De Bortoli et al. (2021) on the quality of the learnt Schrödinger", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 555, 541, 570 ], "spans": [ { "bbox": [ 69, 555, 541, 570 ], "score": 1.0, "content": "bridges based on the average 2-Wasserstein distance to ground truth Schrödinger bridge samples over 18 time", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 568, 417, 582 ], "spans": [ { "bbox": [ 69, 568, 417, 582 ], "score": 1.0, "content": "steps. Furthermore, SB-CFM is also significantly faster than DSB (Table D.1).", "type": "text" } ], "index": 32 } ], "index": 30, "bbox_fs": [ 69, 520, 541, 582 ] }, { "type": "title", "bbox": [ 71, 600, 271, 613 ], "lines": [ { "bbox": [ 69, 599, 272, 615 ], "spans": [ { "bbox": [ 69, 599, 272, 615 ], "score": 1.0, "content": "5.2 Application to single-cell interpolation", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 70, 624, 541, 732 ], "lines": [ { "bbox": [ 69, 624, 542, 639 ], "spans": [ { "bbox": [ 69, 624, 542, 639 ], "score": 1.0, "content": "As a specific application, we consider the task of single-cell trajectory interpolation. In this task we use", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 635, 542, 650 ], "spans": [ { "bbox": [ 69, 635, 374, 650 ], "score": 1.0, "content": "leave-one-out validation over the timepoints. From times data at times", "type": "text" }, { "bbox": [ 375, 639, 449, 649 ], "score": 0.87, "content": "[ 0 , t - 1 ] , [ t + 1 , T ]", "type": "inline_equation" }, { "bbox": [ 449, 635, 542, 650 ], "score": 1.0, "content": "we try to interpolate", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 648, 543, 662 ], "spans": [ { "bbox": [ 69, 648, 176, 662 ], "score": 1.0, "content": "its distribution at time", "type": "text" }, { "bbox": [ 176, 652, 180, 658 ], "score": 0.77, "content": "t", "type": "inline_equation" }, { "bbox": [ 180, 648, 543, 662 ], "score": 1.0, "content": "following the setup of Schiebinger et al. (2019); Tong et al. (2020); Huguet et al.", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 660, 541, 673 ], "spans": [ { "bbox": [ 69, 660, 541, 673 ], "score": 1.0, "content": "(2022a). Low error means we model individual cells well, which is useful in a number of downstream tasks", "type": "text" } ], "index": 37 }, { "bbox": [ 68, 672, 543, 686 ], "spans": [ { "bbox": [ 68, 672, 543, 686 ], "score": 1.0, "content": "such as gene regulatory network inference (Aliee et al., 2021; Yeo et al., 2021). Following Huguet et al.", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 684, 542, 698 ], "spans": [ { "bbox": [ 69, 684, 542, 698 ], "score": 1.0, "content": "(2022b), we repurpose the CITE-seq and Multiome datasets from a recent NeurIPS competition for this", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 695, 542, 709 ], "spans": [ { "bbox": [ 69, 695, 542, 709 ], "score": 1.0, "content": "task (Burkhardt et al., 2022). We also include the Embryoid body data from Moon et al. (2019); Tong et al.", "type": "text" } ], "index": 40 }, { "bbox": [ 69, 708, 542, 722 ], "spans": [ { "bbox": [ 69, 708, 542, 722 ], "score": 1.0, "content": "(2020). Table 4 shows the average earth mover’s distance (1-Wasserstein) on left–out timepoints for three", "type": "text" } ], "index": 41 }, { "bbox": [ 69, 718, 487, 735 ], "spans": [ { "bbox": [ 69, 718, 487, 735 ], "score": 1.0, "content": "datasets. On all three datasets OT-CFM outperforms other methods and baselines on average.", "type": "text" } ], "index": 42 } ], "index": 38, "bbox_fs": [ 68, 624, 543, 735 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 222, 82, 538, 164 ], "blocks": [ { "type": "table_body", "bbox": [ 222, 82, 538, 164 ], "group_id": 0, "lines": [ { "bbox": [ 222, 82, 538, 164 ], "spans": [ { "bbox": [ 222, 82, 538, 164 ], "score": 0.975, "html": "
Algorithm↓Dataset→CiteEBMulti
TrajectoryNet (Tong et al., 2020)*0.848±-
Reg.CNF (Finlay et al., 2020)*0.825 ±-
DSB (De Bortoli et al., 2021)0.953 ± 0.1400.862 ± 0.0231.079 ± 0.117
I-CFM0.965 ± 0.1110.872 ± 0.0871.085 ± 0.099
SB-CFM1.067 ± 0.1071.221 ± 0.3801.129 ± 0.363
OT-CFM0.882 ± 0.0580.790 ± 0.0680.937 ± 0.054
", "type": "table", "image_path": "faf942357d4a5fd2b46b0b76ef33d159cf8484fe0fa8d6a89125c48a079d4449.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 222, 82, 538, 109.33333333333333 ], "spans": [], "index": 7 }, { "bbox": [ 222, 109.33333333333333, 538, 136.66666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 222, 136.66666666666666, 538, 164.0 ], "spans": [], "index": 9 } ] } ], "index": 8 }, { "type": "image", "bbox": [ 85, 188, 525, 342 ], "blocks": [ { "type": "image_caption", "bbox": [ 70, 80, 213, 164 ], "group_id": 0, "lines": [ { "bbox": [ 70, 80, 213, 93 ], "spans": [ { "bbox": [ 70, 80, 213, 93 ], "score": 1.0, "content": "Table 4: Single-cell comparison", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 92, 213, 105 ], "spans": [ { "bbox": [ 70, 92, 213, 105 ], "score": 1.0, "content": "over three datasets averaged over", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 105, 214, 116 ], "spans": [ { "bbox": [ 70, 105, 214, 116 ], "score": 1.0, "content": "leaving out intermediate time-", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 117, 213, 128 ], "spans": [ { "bbox": [ 69, 117, 213, 128 ], "score": 1.0, "content": "points measuring EMD to left out", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 128, 214, 141 ], "spans": [ { "bbox": [ 70, 128, 214, 141 ], "score": 1.0, "content": "distribution following Tong et al.", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 140, 213, 153 ], "spans": [ { "bbox": [ 70, 140, 107, 153 ], "score": 1.0, "content": "(2020).", "type": "text" }, { "bbox": [ 108, 143, 113, 150 ], "score": 0.35, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 113, 140, 213, 153 ], "score": 1.0, "content": "Indicates values taken", "type": "text" } ], "index": 5 }, { "bbox": [ 70, 153, 190, 164 ], "spans": [ { "bbox": [ 70, 153, 190, 164 ], "score": 1.0, "content": "from aforementioned work.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "image_body", "bbox": [ 85, 188, 525, 342 ], "group_id": 0, "lines": [ { "bbox": [ 85, 188, 525, 342 ], "spans": [ { "bbox": [ 85, 188, 525, 342 ], "score": 0.965, "type": "image", "image_path": "1b93a6cf11ee01011406d3be79a25ca143af523b955f5993ef59d2c9177345f2.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 85, 188, 525, 239.33333333333334 ], "spans": [], "index": 10 }, { "bbox": [ 85, 239.33333333333334, 525, 290.6666666666667 ], "spans": [], "index": 11 }, { "bbox": [ 85, 290.6666666666667, 525, 342.0 ], "spans": [], "index": 12 } ] }, { "type": "image_caption", "bbox": [ 70, 359, 540, 407 ], "group_id": 0, "lines": [ { "bbox": [ 69, 359, 541, 373 ], "spans": [ { "bbox": [ 69, 359, 541, 373 ], "score": 1.0, "content": "Figure 3: Left: Fréchet inception distance (FID) scores on CIFAR-10 for different numbers of training steps", "type": "text" } ], "index": 13 }, { "bbox": [ 68, 371, 541, 385 ], "spans": [ { "bbox": [ 68, 371, 541, 385 ], "score": 1.0, "content": "using a dopri5 adaptive solver. Right: FID scores on CIFAR-10 using Euler integration for various numbers", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 384, 540, 397 ], "spans": [ { "bbox": [ 69, 384, 282, 397 ], "score": 1.0, "content": "of function evaluations (NFE) per sample after", "type": "text" }, { "bbox": [ 282, 384, 305, 394 ], "score": 0.43, "content": "4 0 0 \\mathrm { k }", "type": "inline_equation" }, { "bbox": [ 305, 384, 540, 397 ], "score": 1.0, "content": "training steps. In both cases, OT-CFM outperforms", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 395, 411, 408 ], "spans": [ { "bbox": [ 69, 395, 411, 408 ], "score": 1.0, "content": "I-CFM and FM models, showing the benefits of minibatch optimal transport.", "type": "text" } ], "index": 16 } ], "index": 14.5 } ], "index": 11 }, { "type": "title", "bbox": [ 70, 434, 359, 446 ], "lines": [ { "bbox": [ 69, 432, 360, 448 ], "spans": [ { "bbox": [ 69, 432, 360, 448 ], "score": 1.0, "content": "5.3 High-dimensional data: Lower-cost training and inference", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "text", "bbox": [ 70, 457, 540, 517 ], "lines": [ { "bbox": [ 69, 457, 542, 471 ], "spans": [ { "bbox": [ 69, 457, 542, 471 ], "score": 1.0, "content": "We perform an experiment on unconditional CIFAR-10 generation from a Gaussian source to examine how", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 468, 543, 483 ], "spans": [ { "bbox": [ 69, 468, 543, 483 ], "score": 1.0, "content": "OT-CFM performs in the high-dimensional image setting. We use a similar setup to that of Lipman et al.", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 480, 542, 495 ], "spans": [ { "bbox": [ 68, 480, 542, 495 ], "score": 1.0, "content": "(2023), including the time-dependent U-Net architecture from Nichol & Dhariwal (2021) that is commonly", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 493, 540, 506 ], "spans": [ { "bbox": [ 70, 493, 540, 506 ], "score": 1.0, "content": "used in diffusion models. We were not able to reproduce the results reported from Lipman et al. (2023) with", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 504, 500, 519 ], "spans": [ { "bbox": [ 69, 504, 500, 519 ], "score": 1.0, "content": "the parameters specified in the paper.3 Therefore, we selected different training hyperparameters.", "type": "text" } ], "index": 22 } ], "index": 20 }, { "type": "text", "bbox": [ 70, 523, 541, 595 ], "lines": [ { "bbox": [ 69, 522, 543, 537 ], "spans": [ { "bbox": [ 69, 522, 499, 537 ], "score": 1.0, "content": "The main differences with Lipman et al. (2023) are that we use a constant learning rate, set to", "type": "text" }, { "bbox": [ 500, 524, 538, 534 ], "score": 0.93, "content": "2 \\times 1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 538, 522, 543, 537 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 68, 533, 543, 550 ], "spans": [ { "bbox": [ 68, 533, 255, 550 ], "score": 1.0, "content": "instead of a linearly decreasing one (from", "type": "text" }, { "bbox": [ 255, 536, 293, 546 ], "score": 0.92, "content": "5 \\times 1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 294, 533, 309, 550 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 309, 537, 330, 545 ], "score": 0.88, "content": "1 0 ^ { - 8 }", "type": "inline_equation" }, { "bbox": [ 330, 533, 543, 550 ], "score": 1.0, "content": "). To prevent training instabilities and variance,", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 546, 542, 560 ], "spans": [ { "bbox": [ 69, 546, 542, 560 ], "score": 1.0, "content": "we clip the gradient norm to 1 and rely on exponential moving average with a decay of 0.9999. Regarding the", "type": "text" } ], "index": 25 }, { "bbox": [ 68, 558, 541, 572 ], "spans": [ { "bbox": [ 68, 558, 541, 572 ], "score": 1.0, "content": "architecture, we used the same as Lipman et al. (2023), but with a smaller number of channels (128 instead", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 570, 541, 584 ], "spans": [ { "bbox": [ 69, 570, 288, 584 ], "score": 1.0, "content": "of 256), leading to much faster training, as well as", "type": "text" }, { "bbox": [ 288, 573, 306, 581 ], "score": 0.29, "content": "1 0 \\%", "type": "inline_equation" }, { "bbox": [ 307, 570, 541, 584 ], "score": 1.0, "content": "dropout. Furthermore, our batch size was 128 instead", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 582, 275, 596 ], "spans": [ { "bbox": [ 69, 582, 275, 596 ], "score": 1.0, "content": "of 256, which leads to a reduced memory cost.", "type": "text" } ], "index": 28 } ], "index": 25.5 }, { "type": "text", "bbox": [ 70, 600, 540, 660 ], "lines": [ { "bbox": [ 70, 601, 541, 613 ], "spans": [ { "bbox": [ 70, 601, 541, 613 ], "score": 1.0, "content": "We train our OT-CFM, as well as I-CFM and the original FM, with this new training procedure and report", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 612, 541, 626 ], "spans": [ { "bbox": [ 69, 612, 541, 626 ], "score": 1.0, "content": "the Fréchet inception distance (FID) in Table 5. In Fig. 3 (left), we show the FID over training time with the", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 624, 542, 638 ], "spans": [ { "bbox": [ 69, 624, 542, 638 ], "score": 1.0, "content": "Dormand-Prince fifth-order adaptive solver (Hairer et al., 1993, dopri5) using a relative and absolute error", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 635, 542, 650 ], "spans": [ { "bbox": [ 69, 635, 125, 650 ], "score": 1.0, "content": "threshold of", "type": "text" }, { "bbox": [ 125, 638, 146, 646 ], "score": 0.91, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 146, 635, 542, 650 ], "score": 1.0, "content": "similarly to Lipman et al. (2023), and in Fig. 3 (right), we present the FID as a function of", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 648, 366, 662 ], "spans": [ { "bbox": [ 69, 648, 366, 662 ], "score": 1.0, "content": "the numbers of function evaluations (NFE) using Euler integration.", "type": "text" } ], "index": 33 } ], "index": 31 }, { "type": "text", "bbox": [ 71, 667, 129, 678 ], "lines": [ { "bbox": [ 70, 666, 131, 678 ], "spans": [ { "bbox": [ 70, 666, 131, 678 ], "score": 1.0, "content": "We find that:", "type": "text" } ], "index": 34 } ], "index": 34 } ], "page_idx": 10, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 70, 693, 541, 732 ], "lines": [ { "bbox": [ 79, 690, 544, 707 ], "spans": [ { "bbox": [ 79, 690, 520, 707 ], "score": 1.0, "content": "3Specifically, we find that the number generated samples for FID calculation, the value of the smoothing constant", "type": "text" }, { "bbox": [ 520, 698, 538, 703 ], "score": 0.84, "content": "\\sigma _ { \\mathrm { m i n } }", "type": "inline_equation" }, { "bbox": [ 538, 690, 544, 707 ], "score": 1.0, "content": "", "type": "text", "height": 17, "width": 6 } ] }, { "bbox": [ 69, 703, 541, 714 ], "spans": [ { "bbox": [ 69, 703, 341, 714 ], "score": 1.0, "content": "any data augmentation used, the standard deviation of the distribution", "type": "text" }, { "bbox": [ 342, 707, 350, 713 ], "score": 0.87, "content": "p _ { 0 }", "type": "inline_equation" }, { "bbox": [ 351, 703, 541, 714 ], "score": 1.0, "content": ", and the batch size used during evaluation (which", "type": "text" } ] }, { "bbox": [ 69, 713, 541, 723 ], "spans": [ { "bbox": [ 69, 713, 541, 723 ], "score": 1.0, "content": "can affect the function evaluation count with adaptive integrators) are not specified in Lipman et al. (2023)’s manuscript. In", "type": "text" } ] }, { "bbox": [ 69, 722, 375, 733 ], "spans": [ { "bbox": [ 69, 722, 375, 733 ], "score": 1.0, "content": "addition, contradictory information is given about the number of training epochs.", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 222, 82, 538, 164 ], "blocks": [ { "type": "table_body", "bbox": [ 222, 82, 538, 164 ], "group_id": 0, "lines": [ { "bbox": [ 222, 82, 538, 164 ], "spans": [ { "bbox": [ 222, 82, 538, 164 ], "score": 0.975, "html": "
Algorithm↓Dataset→CiteEBMulti
TrajectoryNet (Tong et al., 2020)*0.848±-
Reg.CNF (Finlay et al., 2020)*0.825 ±-
DSB (De Bortoli et al., 2021)0.953 ± 0.1400.862 ± 0.0231.079 ± 0.117
I-CFM0.965 ± 0.1110.872 ± 0.0871.085 ± 0.099
SB-CFM1.067 ± 0.1071.221 ± 0.3801.129 ± 0.363
OT-CFM0.882 ± 0.0580.790 ± 0.0680.937 ± 0.054
", "type": "table", "image_path": "faf942357d4a5fd2b46b0b76ef33d159cf8484fe0fa8d6a89125c48a079d4449.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 222, 82, 538, 109.33333333333333 ], "spans": [], "index": 7 }, { "bbox": [ 222, 109.33333333333333, 538, 136.66666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 222, 136.66666666666666, 538, 164.0 ], "spans": [], "index": 9 } ] } ], "index": 8 }, { "type": "image", "bbox": [ 85, 188, 525, 342 ], "blocks": [ { "type": "image_caption", "bbox": [ 70, 80, 213, 164 ], "group_id": 0, "lines": [ { "bbox": [ 70, 80, 213, 93 ], "spans": [ { "bbox": [ 70, 80, 213, 93 ], "score": 1.0, "content": "Table 4: Single-cell comparison", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 92, 213, 105 ], "spans": [ { "bbox": [ 70, 92, 213, 105 ], "score": 1.0, "content": "over three datasets averaged over", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 105, 214, 116 ], "spans": [ { "bbox": [ 70, 105, 214, 116 ], "score": 1.0, "content": "leaving out intermediate time-", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 117, 213, 128 ], "spans": [ { "bbox": [ 69, 117, 213, 128 ], "score": 1.0, "content": "points measuring EMD to left out", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 128, 214, 141 ], "spans": [ { "bbox": [ 70, 128, 214, 141 ], "score": 1.0, "content": "distribution following Tong et al.", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 140, 213, 153 ], "spans": [ { "bbox": [ 70, 140, 107, 153 ], "score": 1.0, "content": "(2020).", "type": "text" }, { "bbox": [ 108, 143, 113, 150 ], "score": 0.35, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 113, 140, 213, 153 ], "score": 1.0, "content": "Indicates values taken", "type": "text" } ], "index": 5 }, { "bbox": [ 70, 153, 190, 164 ], "spans": [ { "bbox": [ 70, 153, 190, 164 ], "score": 1.0, "content": "from aforementioned work.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "image_body", "bbox": [ 85, 188, 525, 342 ], "group_id": 0, "lines": [ { "bbox": [ 85, 188, 525, 342 ], "spans": [ { "bbox": [ 85, 188, 525, 342 ], "score": 0.965, "type": "image", "image_path": "1b93a6cf11ee01011406d3be79a25ca143af523b955f5993ef59d2c9177345f2.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 85, 188, 525, 239.33333333333334 ], "spans": [], "index": 10 }, { "bbox": [ 85, 239.33333333333334, 525, 290.6666666666667 ], "spans": [], "index": 11 }, { "bbox": [ 85, 290.6666666666667, 525, 342.0 ], "spans": [], "index": 12 } ] }, { "type": "image_caption", "bbox": [ 70, 359, 540, 407 ], "group_id": 0, "lines": [ { "bbox": [ 69, 359, 541, 373 ], "spans": [ { "bbox": [ 69, 359, 541, 373 ], "score": 1.0, "content": "Figure 3: Left: Fréchet inception distance (FID) scores on CIFAR-10 for different numbers of training steps", "type": "text" } ], "index": 13 }, { "bbox": [ 68, 371, 541, 385 ], "spans": [ { "bbox": [ 68, 371, 541, 385 ], "score": 1.0, "content": "using a dopri5 adaptive solver. Right: FID scores on CIFAR-10 using Euler integration for various numbers", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 384, 540, 397 ], "spans": [ { "bbox": [ 69, 384, 282, 397 ], "score": 1.0, "content": "of function evaluations (NFE) per sample after", "type": "text" }, { "bbox": [ 282, 384, 305, 394 ], "score": 0.43, "content": "4 0 0 \\mathrm { k }", "type": "inline_equation" }, { "bbox": [ 305, 384, 540, 397 ], "score": 1.0, "content": "training steps. In both cases, OT-CFM outperforms", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 395, 411, 408 ], "spans": [ { "bbox": [ 69, 395, 411, 408 ], "score": 1.0, "content": "I-CFM and FM models, showing the benefits of minibatch optimal transport.", "type": "text" } ], "index": 16 } ], "index": 14.5 } ], "index": 11 }, { "type": "title", "bbox": [ 70, 434, 359, 446 ], "lines": [ { "bbox": [ 69, 432, 360, 448 ], "spans": [ { "bbox": [ 69, 432, 360, 448 ], "score": 1.0, "content": "5.3 High-dimensional data: Lower-cost training and inference", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "text", "bbox": [ 70, 457, 540, 517 ], "lines": [ { "bbox": [ 69, 457, 542, 471 ], "spans": [ { "bbox": [ 69, 457, 542, 471 ], "score": 1.0, "content": "We perform an experiment on unconditional CIFAR-10 generation from a Gaussian source to examine how", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 468, 543, 483 ], "spans": [ { "bbox": [ 69, 468, 543, 483 ], "score": 1.0, "content": "OT-CFM performs in the high-dimensional image setting. We use a similar setup to that of Lipman et al.", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 480, 542, 495 ], "spans": [ { "bbox": [ 68, 480, 542, 495 ], "score": 1.0, "content": "(2023), including the time-dependent U-Net architecture from Nichol & Dhariwal (2021) that is commonly", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 493, 540, 506 ], "spans": [ { "bbox": [ 70, 493, 540, 506 ], "score": 1.0, "content": "used in diffusion models. We were not able to reproduce the results reported from Lipman et al. (2023) with", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 504, 500, 519 ], "spans": [ { "bbox": [ 69, 504, 500, 519 ], "score": 1.0, "content": "the parameters specified in the paper.3 Therefore, we selected different training hyperparameters.", "type": "text" } ], "index": 22 } ], "index": 20, "bbox_fs": [ 68, 457, 543, 519 ] }, { "type": "text", "bbox": [ 70, 523, 541, 595 ], "lines": [ { "bbox": [ 69, 522, 543, 537 ], "spans": [ { "bbox": [ 69, 522, 499, 537 ], "score": 1.0, "content": "The main differences with Lipman et al. (2023) are that we use a constant learning rate, set to", "type": "text" }, { "bbox": [ 500, 524, 538, 534 ], "score": 0.93, "content": "2 \\times 1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 538, 522, 543, 537 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 68, 533, 543, 550 ], "spans": [ { "bbox": [ 68, 533, 255, 550 ], "score": 1.0, "content": "instead of a linearly decreasing one (from", "type": "text" }, { "bbox": [ 255, 536, 293, 546 ], "score": 0.92, "content": "5 \\times 1 0 ^ { - 4 }", "type": "inline_equation" }, { "bbox": [ 294, 533, 309, 550 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 309, 537, 330, 545 ], "score": 0.88, "content": "1 0 ^ { - 8 }", "type": "inline_equation" }, { "bbox": [ 330, 533, 543, 550 ], "score": 1.0, "content": "). To prevent training instabilities and variance,", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 546, 542, 560 ], "spans": [ { "bbox": [ 69, 546, 542, 560 ], "score": 1.0, "content": "we clip the gradient norm to 1 and rely on exponential moving average with a decay of 0.9999. Regarding the", "type": "text" } ], "index": 25 }, { "bbox": [ 68, 558, 541, 572 ], "spans": [ { "bbox": [ 68, 558, 541, 572 ], "score": 1.0, "content": "architecture, we used the same as Lipman et al. (2023), but with a smaller number of channels (128 instead", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 570, 541, 584 ], "spans": [ { "bbox": [ 69, 570, 288, 584 ], "score": 1.0, "content": "of 256), leading to much faster training, as well as", "type": "text" }, { "bbox": [ 288, 573, 306, 581 ], "score": 0.29, "content": "1 0 \\%", "type": "inline_equation" }, { "bbox": [ 307, 570, 541, 584 ], "score": 1.0, "content": "dropout. Furthermore, our batch size was 128 instead", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 582, 275, 596 ], "spans": [ { "bbox": [ 69, 582, 275, 596 ], "score": 1.0, "content": "of 256, which leads to a reduced memory cost.", "type": "text" } ], "index": 28 } ], "index": 25.5, "bbox_fs": [ 68, 522, 543, 596 ] }, { "type": "text", "bbox": [ 70, 600, 540, 660 ], "lines": [ { "bbox": [ 70, 601, 541, 613 ], "spans": [ { "bbox": [ 70, 601, 541, 613 ], "score": 1.0, "content": "We train our OT-CFM, as well as I-CFM and the original FM, with this new training procedure and report", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 612, 541, 626 ], "spans": [ { "bbox": [ 69, 612, 541, 626 ], "score": 1.0, "content": "the Fréchet inception distance (FID) in Table 5. In Fig. 3 (left), we show the FID over training time with the", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 624, 542, 638 ], "spans": [ { "bbox": [ 69, 624, 542, 638 ], "score": 1.0, "content": "Dormand-Prince fifth-order adaptive solver (Hairer et al., 1993, dopri5) using a relative and absolute error", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 635, 542, 650 ], "spans": [ { "bbox": [ 69, 635, 125, 650 ], "score": 1.0, "content": "threshold of", "type": "text" }, { "bbox": [ 125, 638, 146, 646 ], "score": 0.91, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 146, 635, 542, 650 ], "score": 1.0, "content": "similarly to Lipman et al. (2023), and in Fig. 3 (right), we present the FID as a function of", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 648, 366, 662 ], "spans": [ { "bbox": [ 69, 648, 366, 662 ], "score": 1.0, "content": "the numbers of function evaluations (NFE) using Euler integration.", "type": "text" } ], "index": 33 } ], "index": 31, "bbox_fs": [ 69, 601, 542, 662 ] }, { "type": "text", "bbox": [ 71, 667, 129, 678 ], "lines": [ { "bbox": [ 70, 666, 131, 678 ], "spans": [ { "bbox": [ 70, 666, 131, 678 ], "score": 1.0, "content": "We find that:", "type": "text" } ], "index": 34 } ], "index": 34, "bbox_fs": [ 70, 666, 131, 678 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 80, 279, 248 ], "lines": [ { "bbox": [ 70, 81, 280, 92 ], "spans": [ { "bbox": [ 70, 81, 280, 92 ], "score": 1.0, "content": "Table 5: FID score and number of function eval-", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 280, 104 ], "spans": [ { "bbox": [ 70, 93, 280, 104 ], "score": 1.0, "content": "uations (NFE) for different ODE solvers: fixed-", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 105, 280, 117 ], "spans": [ { "bbox": [ 69, 105, 280, 117 ], "score": 1.0, "content": "step Euler integration with 100 and 1000 steps", "type": "text" } ], "index": 2 }, { "bbox": [ 70, 117, 280, 129 ], "spans": [ { "bbox": [ 70, 117, 280, 129 ], "score": 1.0, "content": "and adaptive integration (dopri5). The adap-", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 129, 280, 141 ], "spans": [ { "bbox": [ 70, 129, 280, 141 ], "score": 1.0, "content": "tive solver is significantly better than the Euler", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 141, 279, 153 ], "spans": [ { "bbox": [ 69, 141, 279, 153 ], "score": 1.0, "content": "solver in fewer steps. First three results are from", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 153, 280, 165 ], "spans": [ { "bbox": [ 69, 153, 280, 165 ], "score": 1.0, "content": "Lipman et al. (2023) and fourth from Albergo &", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 164, 280, 177 ], "spans": [ { "bbox": [ 69, 164, 280, 177 ], "score": 1.0, "content": "Vanden-Eijnden (2023). The fifth line is our re-", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 176, 280, 189 ], "spans": [ { "bbox": [ 69, 176, 280, 189 ], "score": 1.0, "content": "produced results following Lipman et al. (2023)’s", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 189, 280, 200 ], "spans": [ { "bbox": [ 69, 189, 280, 200 ], "score": 1.0, "content": "training procedure. We have run OT-FM, S.I.", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 200, 279, 213 ], "spans": [ { "bbox": [ 69, 200, 279, 213 ], "score": 1.0, "content": "and VP-FM following our training procedure", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 212, 279, 224 ], "spans": [ { "bbox": [ 69, 212, 279, 224 ], "score": 1.0, "content": "and we have denoted them (ours). The two last", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 225, 279, 236 ], "spans": [ { "bbox": [ 69, 225, 279, 236 ], "score": 1.0, "content": "rows report the results of our proposed methos", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 236, 168, 247 ], "spans": [ { "bbox": [ 70, 236, 168, 247 ], "score": 1.0, "content": "I-CFM and OT-CFM.", "type": "text" } ], "index": 13 } ], "index": 6.5 }, { "type": "table", "bbox": [ 285, 83, 550, 256 ], "blocks": [ { "type": "table_body", "bbox": [ 285, 83, 550, 256 ], "group_id": 0, "lines": [ { "bbox": [ 285, 83, 550, 256 ], "spans": [ { "bbox": [ 285, 83, 550, 256 ], "score": 0.979, "html": "
NFE / sample → Algorithm↓1001000Adaptive
FIDFIDFIDNFE
DDPM7.48274
OT-FM (reported)6.35142
VP-FM (reported)8.06183
S.I. (reported)10.27
OT-FM (reproduced)13.74212.49111.527139.83
VP-FM (ours)7.7724.0484.335525.92
OT-FM (ours)4.6403.8223.655143.00
S.I. (ours)4.4884.1324.009146.12
I-CFM (ours)4.4613.6433.659146.42
OT-CFM (ours)4.4433.7413.577133.94
", "type": "table", "image_path": "325620c997f772f93744bfdd2c2616c2815d838ccc7f8bd84c8dc55df6307eda.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 285, 83, 550, 140.66666666666666 ], "spans": [], "index": 14 }, { "bbox": [ 285, 140.66666666666666, 550, 198.33333333333331 ], "spans": [], "index": 15 }, { "bbox": [ 285, 198.33333333333331, 550, 255.99999999999997 ], "spans": [], "index": 16 } ] } ], "index": 15 }, { "type": "text", "bbox": [ 71, 287, 542, 395 ], "lines": [ { "bbox": [ 71, 287, 541, 301 ], "spans": [ { "bbox": [ 71, 287, 541, 301 ], "score": 1.0, "content": "• With improved hyperparameters, we achieve a significantly better FID with the FM training objective", "type": "text" } ], "index": 17 }, { "bbox": [ 81, 299, 358, 313 ], "spans": [ { "bbox": [ 81, 299, 358, 313 ], "score": 1.0, "content": "than the one reported by Lipman et al. (2023) at a lower cost.", "type": "text" } ], "index": 18 }, { "bbox": [ 72, 311, 511, 325 ], "spans": [ { "bbox": [ 72, 311, 511, 325 ], "score": 1.0, "content": "• For a short computation budget, OT-CFM outperforms FM and (non-OT) I-CFM (Table 5, left).", "type": "text" } ], "index": 19 }, { "bbox": [ 76, 322, 542, 336 ], "spans": [ { "bbox": [ 76, 322, 542, 336 ], "score": 1.0, "content": "After a long training time, all methods achieve similar performance at a high number of function evaluations", "type": "text" } ], "index": 20 }, { "bbox": [ 82, 335, 542, 349 ], "spans": [ { "bbox": [ 82, 335, 542, 349 ], "score": 1.0, "content": "using fixed-step ODE integration, but OT-CFM performs significantly better with a small number of", "type": "text" } ], "index": 21 }, { "bbox": [ 82, 347, 541, 361 ], "spans": [ { "bbox": [ 82, 347, 541, 361 ], "score": 1.0, "content": "function evaluations (i.e., allows more efficient inference), indicating straighter, easily integrable flows", "type": "text" } ], "index": 22 }, { "bbox": [ 81, 356, 154, 373 ], "spans": [ { "bbox": [ 81, 356, 154, 373 ], "score": 1.0, "content": "(Table 5, right).", "type": "text" } ], "index": 23 }, { "bbox": [ 79, 370, 541, 384 ], "spans": [ { "bbox": [ 79, 370, 541, 384 ], "score": 1.0, "content": "FM and I-CFM are equivalently computationally efficient per iteration and OT-CFM comes with a low", "type": "text" } ], "index": 24 }, { "bbox": [ 85, 381, 294, 398 ], "spans": [ { "bbox": [ 85, 384, 108, 393 ], "score": 0.39, "content": "( < 1 \\%", "type": "inline_equation" }, { "bbox": [ 109, 381, 294, 398 ], "score": 1.0, "content": ") computational overhead during training.", "type": "text" } ], "index": 25 } ], "index": 21 }, { "type": "text", "bbox": [ 71, 401, 541, 437 ], "lines": [ { "bbox": [ 69, 398, 543, 415 ], "spans": [ { "bbox": [ 69, 398, 543, 415 ], "score": 1.0, "content": "Our new training procedures, available at https://github.com/atong01/conditional-flow-matching,", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 412, 540, 425 ], "spans": [ { "bbox": [ 70, 412, 540, 425 ], "score": 1.0, "content": "allow us to outperform the previous reported results from Lipman et al. (2023), while the results with our", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 425, 427, 437 ], "spans": [ { "bbox": [ 70, 425, 427, 437 ], "score": 1.0, "content": "OT-CFM are state-of-the-art for simulation-free neural ODE training algorithms.", "type": "text" } ], "index": 28 } ], "index": 27 }, { "type": "title", "bbox": [ 71, 457, 268, 470 ], "lines": [ { "bbox": [ 69, 456, 270, 473 ], "spans": [ { "bbox": [ 69, 456, 270, 473 ], "score": 1.0, "content": "5.4 OT-CFM for unsupervised translation", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 71, 482, 540, 531 ], "lines": [ { "bbox": [ 69, 482, 541, 495 ], "spans": [ { "bbox": [ 69, 482, 541, 495 ], "score": 1.0, "content": "We show how CFM can be used to learn a mapping between two unpaired datasets in high-dimensional space", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 494, 542, 508 ], "spans": [ { "bbox": [ 69, 494, 404, 508 ], "score": 1.0, "content": "using the CelebA dataset (Liu et al., 2015; Sun et al., 2014), which consists of", "type": "text" }, { "bbox": [ 405, 497, 435, 505 ], "score": 0.41, "content": "\\sim 2 0 0 \\mathrm { k }", "type": "inline_equation" }, { "bbox": [ 435, 494, 542, 508 ], "score": 1.0, "content": "images of faces together", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 506, 541, 520 ], "spans": [ { "bbox": [ 69, 506, 541, 520 ], "score": 1.0, "content": "with 40 binary attribute annotations. For each attribute, we wish to learn an invertible mapping between", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 519, 377, 532 ], "spans": [ { "bbox": [ 69, 519, 322, 532 ], "score": 1.0, "content": "images with and without the attribute (e.g., ‘not smiling’", "type": "text" }, { "bbox": [ 322, 522, 333, 528 ], "score": 0.69, "content": "", "type": "inline_equation" }, { "bbox": [ 333, 519, 377, 532 ], "score": 1.0, "content": "‘smiling’).", "type": "text" } ], "index": 33 } ], "index": 31.5 }, { "type": "text", "bbox": [ 70, 536, 541, 644 ], "lines": [ { "bbox": [ 69, 536, 543, 549 ], "spans": [ { "bbox": [ 69, 536, 543, 549 ], "score": 1.0, "content": "To reduce dimensionality, we first train a VAE on the images and encode them as 128-dimensional latent vectors.", "type": "text" } ], "index": 34 }, { "bbox": [ 70, 549, 541, 561 ], "spans": [ { "bbox": [ 70, 549, 541, 561 ], "score": 1.0, "content": "For each attribute, we learn a flow to map between the embeddings of images without the attribute and those", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 559, 541, 574 ], "spans": [ { "bbox": [ 69, 559, 541, 574 ], "score": 1.0, "content": "of images with the attribute. After the CNF is learned, we push forward a held-out set of negative vectors by", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 572, 542, 586 ], "spans": [ { "bbox": [ 69, 572, 542, 586 ], "score": 1.0, "content": "the CNF and compare them to the held-out positive vectors and vice versa. As a metric of divergence, we use", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 583, 541, 597 ], "spans": [ { "bbox": [ 69, 583, 369, 597 ], "score": 1.0, "content": "maximum mean discrepancy (MMD) with a broad Gaussian kernel", "type": "text" }, { "bbox": [ 369, 586, 481, 597 ], "score": 0.91, "content": "\\mathrm { ' e x p } ( - \\| x - y \\| ^ { 2 } / ( 2 \\cdot 1 2 8 ) ) \\rangle", "type": "inline_equation" }, { "bbox": [ 482, 583, 541, 597 ], "score": 1.0, "content": ". The results", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 596, 541, 610 ], "spans": [ { "bbox": [ 69, 596, 541, 610 ], "score": 1.0, "content": "aggregated over all attributes are shown in Table 6, showing that OT-CFM discovers a better mapping than", "type": "text" } ], "index": 39 }, { "bbox": [ 70, 608, 542, 621 ], "spans": [ { "bbox": [ 70, 608, 294, 621 ], "score": 1.0, "content": "other methods. Although MMD is lower for larger", "type": "text" }, { "bbox": [ 294, 613, 300, 618 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 300, 608, 523, 621 ], "score": 1.0, "content": ", we found that the alignment is less natural when", "type": "text" }, { "bbox": [ 523, 613, 529, 618 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 530, 608, 542, 621 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 40 }, { "bbox": [ 69, 619, 541, 633 ], "spans": [ { "bbox": [ 69, 619, 283, 633 ], "score": 1.0, "content": "large, and performance begins to degrade when", "type": "text" }, { "bbox": [ 284, 623, 308, 630 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 308, 619, 541, 633 ], "score": 1.0, "content": ". Fig. E.1 shows several visualizations of the learned", "type": "text" } ], "index": 41 }, { "bbox": [ 70, 632, 125, 645 ], "spans": [ { "bbox": [ 70, 632, 125, 645 ], "score": 1.0, "content": "trajectories.", "type": "text" } ], "index": 42 } ], "index": 38 }, { "type": "text", "bbox": [ 70, 650, 536, 674 ], "lines": [ { "bbox": [ 69, 648, 539, 664 ], "spans": [ { "bbox": [ 69, 648, 539, 664 ], "score": 1.0, "content": "Finally, while here we work in a latent space, future work should consider learning flows directly in image", "type": "text" } ], "index": 43 }, { "bbox": [ 69, 662, 411, 676 ], "spans": [ { "bbox": [ 69, 662, 411, 676 ], "score": 1.0, "content": "space, where GAN-based approaches (Zhu et al., 2017) continue to dominate.", "type": "text" } ], "index": 44 } ], "index": 43.5 }, { "type": "title", "bbox": [ 71, 695, 272, 708 ], "lines": [ { "bbox": [ 69, 694, 274, 710 ], "spans": [ { "bbox": [ 69, 694, 274, 710 ], "score": 1.0, "content": "5.5 Additional experiments and extensions", "type": "text" } ], "index": 45 } ], "index": 45 }, { "type": "text", "bbox": [ 70, 720, 517, 732 ], "lines": [ { "bbox": [ 69, 718, 520, 735 ], "spans": [ { "bbox": [ 69, 718, 520, 735 ], "score": 1.0, "content": "We present numerous other extensions, applications, and evaluations of CFM in Appendix D, notably:", "type": "text" } ], "index": 46 } ], "index": 46 } ], "page_idx": 11, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "12", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 80, 279, 248 ], "lines": [ { "bbox": [ 70, 81, 280, 92 ], "spans": [ { "bbox": [ 70, 81, 280, 92 ], "score": 1.0, "content": "Table 5: FID score and number of function eval-", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 280, 104 ], "spans": [ { "bbox": [ 70, 93, 280, 104 ], "score": 1.0, "content": "uations (NFE) for different ODE solvers: fixed-", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 105, 280, 117 ], "spans": [ { "bbox": [ 69, 105, 280, 117 ], "score": 1.0, "content": "step Euler integration with 100 and 1000 steps", "type": "text" } ], "index": 2 }, { "bbox": [ 70, 117, 280, 129 ], "spans": [ { "bbox": [ 70, 117, 280, 129 ], "score": 1.0, "content": "and adaptive integration (dopri5). The adap-", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 129, 280, 141 ], "spans": [ { "bbox": [ 70, 129, 280, 141 ], "score": 1.0, "content": "tive solver is significantly better than the Euler", "type": "text" } ], "index": 4 }, { "bbox": [ 69, 141, 279, 153 ], "spans": [ { "bbox": [ 69, 141, 279, 153 ], "score": 1.0, "content": "solver in fewer steps. First three results are from", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 153, 280, 165 ], "spans": [ { "bbox": [ 69, 153, 280, 165 ], "score": 1.0, "content": "Lipman et al. (2023) and fourth from Albergo &", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 164, 280, 177 ], "spans": [ { "bbox": [ 69, 164, 280, 177 ], "score": 1.0, "content": "Vanden-Eijnden (2023). The fifth line is our re-", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 176, 280, 189 ], "spans": [ { "bbox": [ 69, 176, 280, 189 ], "score": 1.0, "content": "produced results following Lipman et al. (2023)’s", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 189, 280, 200 ], "spans": [ { "bbox": [ 69, 189, 280, 200 ], "score": 1.0, "content": "training procedure. We have run OT-FM, S.I.", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 200, 279, 213 ], "spans": [ { "bbox": [ 69, 200, 279, 213 ], "score": 1.0, "content": "and VP-FM following our training procedure", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 212, 279, 224 ], "spans": [ { "bbox": [ 69, 212, 279, 224 ], "score": 1.0, "content": "and we have denoted them (ours). The two last", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 225, 279, 236 ], "spans": [ { "bbox": [ 69, 225, 279, 236 ], "score": 1.0, "content": "rows report the results of our proposed methos", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 236, 168, 247 ], "spans": [ { "bbox": [ 70, 236, 168, 247 ], "score": 1.0, "content": "I-CFM and OT-CFM.", "type": "text" } ], "index": 13 } ], "index": 6.5, "bbox_fs": [ 69, 81, 280, 247 ] }, { "type": "table", "bbox": [ 285, 83, 550, 256 ], "blocks": [ { "type": "table_body", "bbox": [ 285, 83, 550, 256 ], "group_id": 0, "lines": [ { "bbox": [ 285, 83, 550, 256 ], "spans": [ { "bbox": [ 285, 83, 550, 256 ], "score": 0.979, "html": "
NFE / sample → Algorithm↓1001000Adaptive
FIDFIDFIDNFE
DDPM7.48274
OT-FM (reported)6.35142
VP-FM (reported)8.06183
S.I. (reported)10.27
OT-FM (reproduced)13.74212.49111.527139.83
VP-FM (ours)7.7724.0484.335525.92
OT-FM (ours)4.6403.8223.655143.00
S.I. (ours)4.4884.1324.009146.12
I-CFM (ours)4.4613.6433.659146.42
OT-CFM (ours)4.4433.7413.577133.94
", "type": "table", "image_path": "325620c997f772f93744bfdd2c2616c2815d838ccc7f8bd84c8dc55df6307eda.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 285, 83, 550, 140.66666666666666 ], "spans": [], "index": 14 }, { "bbox": [ 285, 140.66666666666666, 550, 198.33333333333331 ], "spans": [], "index": 15 }, { "bbox": [ 285, 198.33333333333331, 550, 255.99999999999997 ], "spans": [], "index": 16 } ] } ], "index": 15 }, { "type": "list", "bbox": [ 71, 287, 542, 395 ], "lines": [ { "bbox": [ 71, 287, 541, 301 ], "spans": [ { "bbox": [ 71, 287, 541, 301 ], "score": 1.0, "content": "• With improved hyperparameters, we achieve a significantly better FID with the FM training objective", "type": "text" } ], "index": 17, "is_list_start_line": true }, { "bbox": [ 81, 299, 358, 313 ], "spans": [ { "bbox": [ 81, 299, 358, 313 ], "score": 1.0, "content": "than the one reported by Lipman et al. (2023) at a lower cost.", "type": "text" } ], "index": 18, "is_list_end_line": true }, { "bbox": [ 72, 311, 511, 325 ], "spans": [ { "bbox": [ 72, 311, 511, 325 ], "score": 1.0, "content": "• For a short computation budget, OT-CFM outperforms FM and (non-OT) I-CFM (Table 5, left).", "type": "text" } ], "index": 19, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 76, 322, 542, 336 ], "spans": [ { "bbox": [ 76, 322, 542, 336 ], "score": 1.0, "content": "After a long training time, all methods achieve similar performance at a high number of function evaluations", "type": "text" } ], "index": 20, "is_list_start_line": true }, { "bbox": [ 82, 335, 542, 349 ], "spans": [ { "bbox": [ 82, 335, 542, 349 ], "score": 1.0, "content": "using fixed-step ODE integration, but OT-CFM performs significantly better with a small number of", "type": "text" } ], "index": 21 }, { "bbox": [ 82, 347, 541, 361 ], "spans": [ { "bbox": [ 82, 347, 541, 361 ], "score": 1.0, "content": "function evaluations (i.e., allows more efficient inference), indicating straighter, easily integrable flows", "type": "text" } ], "index": 22 }, { "bbox": [ 81, 356, 154, 373 ], "spans": [ { "bbox": [ 81, 356, 154, 373 ], "score": 1.0, "content": "(Table 5, right).", "type": "text" } ], "index": 23, "is_list_end_line": true }, { "bbox": [ 79, 370, 541, 384 ], "spans": [ { "bbox": [ 79, 370, 541, 384 ], "score": 1.0, "content": "FM and I-CFM are equivalently computationally efficient per iteration and OT-CFM comes with a low", "type": "text" } ], "index": 24 }, { "bbox": [ 85, 381, 294, 398 ], "spans": [ { "bbox": [ 85, 384, 108, 393 ], "score": 0.39, "content": "( < 1 \\%", "type": "inline_equation" }, { "bbox": [ 109, 381, 294, 398 ], "score": 1.0, "content": ") computational overhead during training.", "type": "text" } ], "index": 25, "is_list_end_line": true } ], "index": 21, "bbox_fs": [ 71, 287, 542, 398 ] }, { "type": "text", "bbox": [ 71, 401, 541, 437 ], "lines": [ { "bbox": [ 69, 398, 543, 415 ], "spans": [ { "bbox": [ 69, 398, 543, 415 ], "score": 1.0, "content": "Our new training procedures, available at https://github.com/atong01/conditional-flow-matching,", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 412, 540, 425 ], "spans": [ { "bbox": [ 70, 412, 540, 425 ], "score": 1.0, "content": "allow us to outperform the previous reported results from Lipman et al. (2023), while the results with our", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 425, 427, 437 ], "spans": [ { "bbox": [ 70, 425, 427, 437 ], "score": 1.0, "content": "OT-CFM are state-of-the-art for simulation-free neural ODE training algorithms.", "type": "text" } ], "index": 28 } ], "index": 27, "bbox_fs": [ 69, 398, 543, 437 ] }, { "type": "title", "bbox": [ 71, 457, 268, 470 ], "lines": [ { "bbox": [ 69, 456, 270, 473 ], "spans": [ { "bbox": [ 69, 456, 270, 473 ], "score": 1.0, "content": "5.4 OT-CFM for unsupervised translation", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 71, 482, 540, 531 ], "lines": [ { "bbox": [ 69, 482, 541, 495 ], "spans": [ { "bbox": [ 69, 482, 541, 495 ], "score": 1.0, "content": "We show how CFM can be used to learn a mapping between two unpaired datasets in high-dimensional space", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 494, 542, 508 ], "spans": [ { "bbox": [ 69, 494, 404, 508 ], "score": 1.0, "content": "using the CelebA dataset (Liu et al., 2015; Sun et al., 2014), which consists of", "type": "text" }, { "bbox": [ 405, 497, 435, 505 ], "score": 0.41, "content": "\\sim 2 0 0 \\mathrm { k }", "type": "inline_equation" }, { "bbox": [ 435, 494, 542, 508 ], "score": 1.0, "content": "images of faces together", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 506, 541, 520 ], "spans": [ { "bbox": [ 69, 506, 541, 520 ], "score": 1.0, "content": "with 40 binary attribute annotations. For each attribute, we wish to learn an invertible mapping between", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 519, 377, 532 ], "spans": [ { "bbox": [ 69, 519, 322, 532 ], "score": 1.0, "content": "images with and without the attribute (e.g., ‘not smiling’", "type": "text" }, { "bbox": [ 322, 522, 333, 528 ], "score": 0.69, "content": "", "type": "inline_equation" }, { "bbox": [ 333, 519, 377, 532 ], "score": 1.0, "content": "‘smiling’).", "type": "text" } ], "index": 33 } ], "index": 31.5, "bbox_fs": [ 69, 482, 542, 532 ] }, { "type": "text", "bbox": [ 70, 536, 541, 644 ], "lines": [ { "bbox": [ 69, 536, 543, 549 ], "spans": [ { "bbox": [ 69, 536, 543, 549 ], "score": 1.0, "content": "To reduce dimensionality, we first train a VAE on the images and encode them as 128-dimensional latent vectors.", "type": "text" } ], "index": 34 }, { "bbox": [ 70, 549, 541, 561 ], "spans": [ { "bbox": [ 70, 549, 541, 561 ], "score": 1.0, "content": "For each attribute, we learn a flow to map between the embeddings of images without the attribute and those", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 559, 541, 574 ], "spans": [ { "bbox": [ 69, 559, 541, 574 ], "score": 1.0, "content": "of images with the attribute. After the CNF is learned, we push forward a held-out set of negative vectors by", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 572, 542, 586 ], "spans": [ { "bbox": [ 69, 572, 542, 586 ], "score": 1.0, "content": "the CNF and compare them to the held-out positive vectors and vice versa. As a metric of divergence, we use", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 583, 541, 597 ], "spans": [ { "bbox": [ 69, 583, 369, 597 ], "score": 1.0, "content": "maximum mean discrepancy (MMD) with a broad Gaussian kernel", "type": "text" }, { "bbox": [ 369, 586, 481, 597 ], "score": 0.91, "content": "\\mathrm { ' e x p } ( - \\| x - y \\| ^ { 2 } / ( 2 \\cdot 1 2 8 ) ) \\rangle", "type": "inline_equation" }, { "bbox": [ 482, 583, 541, 597 ], "score": 1.0, "content": ". The results", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 596, 541, 610 ], "spans": [ { "bbox": [ 69, 596, 541, 610 ], "score": 1.0, "content": "aggregated over all attributes are shown in Table 6, showing that OT-CFM discovers a better mapping than", "type": "text" } ], "index": 39 }, { "bbox": [ 70, 608, 542, 621 ], "spans": [ { "bbox": [ 70, 608, 294, 621 ], "score": 1.0, "content": "other methods. Although MMD is lower for larger", "type": "text" }, { "bbox": [ 294, 613, 300, 618 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 300, 608, 523, 621 ], "score": 1.0, "content": ", we found that the alignment is less natural when", "type": "text" }, { "bbox": [ 523, 613, 529, 618 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 530, 608, 542, 621 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 40 }, { "bbox": [ 69, 619, 541, 633 ], "spans": [ { "bbox": [ 69, 619, 283, 633 ], "score": 1.0, "content": "large, and performance begins to degrade when", "type": "text" }, { "bbox": [ 284, 623, 308, 630 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 308, 619, 541, 633 ], "score": 1.0, "content": ". Fig. E.1 shows several visualizations of the learned", "type": "text" } ], "index": 41 }, { "bbox": [ 70, 632, 125, 645 ], "spans": [ { "bbox": [ 70, 632, 125, 645 ], "score": 1.0, "content": "trajectories.", "type": "text" } ], "index": 42 } ], "index": 38, "bbox_fs": [ 69, 536, 543, 645 ] }, { "type": "text", "bbox": [ 70, 650, 536, 674 ], "lines": [ { "bbox": [ 69, 648, 539, 664 ], "spans": [ { "bbox": [ 69, 648, 539, 664 ], "score": 1.0, "content": "Finally, while here we work in a latent space, future work should consider learning flows directly in image", "type": "text" } ], "index": 43 }, { "bbox": [ 69, 662, 411, 676 ], "spans": [ { "bbox": [ 69, 662, 411, 676 ], "score": 1.0, "content": "space, where GAN-based approaches (Zhu et al., 2017) continue to dominate.", "type": "text" } ], "index": 44 } ], "index": 43.5, "bbox_fs": [ 69, 648, 539, 676 ] }, { "type": "title", "bbox": [ 71, 695, 272, 708 ], "lines": [ { "bbox": [ 69, 694, 274, 710 ], "spans": [ { "bbox": [ 69, 694, 274, 710 ], "score": 1.0, "content": "5.5 Additional experiments and extensions", "type": "text" } ], "index": 45 } ], "index": 45 }, { "type": "text", "bbox": [ 70, 720, 517, 732 ], "lines": [ { "bbox": [ 69, 718, 520, 735 ], "spans": [ { "bbox": [ 69, 718, 520, 735 ], "score": 1.0, "content": "We present numerous other extensions, applications, and evaluations of CFM in Appendix D, notably:", "type": "text" } ], "index": 46 } ], "index": 46, "bbox_fs": [ 69, 718, 520, 735 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 80, 303, 164 ], "lines": [ { "bbox": [ 70, 80, 302, 93 ], "spans": [ { "bbox": [ 70, 80, 190, 93 ], "score": 1.0, "content": "Table 6: MMD (in units of", "type": "text" }, { "bbox": [ 190, 82, 210, 91 ], "score": 0.91, "content": "1 0 ^ { - 3 }", "type": "inline_equation" }, { "bbox": [ 211, 80, 302, 93 ], "score": 1.0, "content": ") between target and", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 303, 105 ], "spans": [ { "bbox": [ 70, 93, 303, 105 ], "score": 1.0, "content": "transformed source samples of CelebA latent vectors.", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 104, 302, 116 ], "spans": [ { "bbox": [ 70, 104, 302, 116 ], "score": 1.0, "content": "Mean and standard deviation over 40 attributes and", "type": "text" } ], "index": 2 }, { "bbox": [ 70, 116, 303, 128 ], "spans": [ { "bbox": [ 70, 116, 189, 128 ], "score": 1.0, "content": "both translation directions (", "type": "text" }, { "bbox": [ 189, 119, 221, 127 ], "score": 0.87, "content": "- +", "type": "inline_equation" }, { "bbox": [ 221, 116, 303, 128 ], "score": 1.0, "content": ") for each attribute.", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 128, 303, 140 ], "spans": [ { "bbox": [ 69, 128, 303, 140 ], "score": 1.0, "content": "‘Identity’ refers to performing no translation and treat-", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 141, 302, 153 ], "spans": [ { "bbox": [ 70, 141, 302, 153 ], "score": 1.0, "content": "ing source samples as approximate samples from the", "type": "text" } ], "index": 5 }, { "bbox": [ 70, 153, 102, 165 ], "spans": [ { "bbox": [ 70, 153, 102, 165 ], "score": 1.0, "content": "target.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "table", "bbox": [ 312, 92, 537, 154 ], "blocks": [ { "type": "table_body", "bbox": [ 312, 92, 537, 154 ], "group_id": 0, "lines": [ { "bbox": [ 312, 92, 537, 154 ], "spans": [ { "bbox": [ 312, 92, 537, 154 ], "score": 0.979, "html": "
Algorithm↓g=0.1σ=0.3σ=1
Identity9.17 ± 5.689.17± 5.689.17± 5.68
I-CFM4.85 ± 5.093.44± 2.031.59 ± 0.83
OT-CFM2.81± 2.621.91 ± 1.301.04 ± 0.60
", "type": "table", "image_path": "e359d78a45b8c5e465faa584ccaae77b79bfa6ae6031bfd6e49e337246babf65.jpg" } ] } ], "index": 8.5, "virtual_lines": [ { "bbox": [ 312, 92, 537, 107.5 ], "spans": [], "index": 7 }, { "bbox": [ 312, 107.5, 537, 123.0 ], "spans": [], "index": 8 }, { "bbox": [ 312, 123.0, 537, 138.5 ], "spans": [], "index": 9 }, { "bbox": [ 312, 138.5, 537, 154.0 ], "spans": [], "index": 10 } ] } ], "index": 8.5 }, { "type": "text", "bbox": [ 71, 184, 540, 219 ], "lines": [ { "bbox": [ 70, 183, 542, 197 ], "spans": [ { "bbox": [ 70, 183, 542, 197 ], "score": 1.0, "content": "OT-CFM reduces variance in the regression target. To accompany the theoretical results in §C.1,", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 195, 541, 209 ], "spans": [ { "bbox": [ 69, 195, 541, 209 ], "score": 1.0, "content": "in §D.1 we empirically study the variance of the stochastic regression objective in (OT-)CFM. The results", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 208, 436, 221 ], "spans": [ { "bbox": [ 69, 208, 436, 221 ], "score": 1.0, "content": "suggest an explanation for the faster convergence of models trained with OT-CFM.", "type": "text" } ], "index": 13 } ], "index": 12 }, { "type": "text", "bbox": [ 70, 231, 540, 255 ], "lines": [ { "bbox": [ 70, 231, 541, 244 ], "spans": [ { "bbox": [ 70, 231, 235, 244 ], "score": 1.0, "content": "Energy-based CFM. 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Algorithm↓g=0.1σ=0.3σ=1
Identity9.17 ± 5.689.17± 5.689.17± 5.68
I-CFM4.85 ± 5.093.44± 2.031.59 ± 0.83
OT-CFM2.81± 2.621.91 ± 1.301.04 ± 0.60
", "type": "table", "image_path": "e359d78a45b8c5e465faa584ccaae77b79bfa6ae6031bfd6e49e337246babf65.jpg" } ] } ], "index": 8.5, "virtual_lines": [ { "bbox": [ 312, 92, 537, 107.5 ], "spans": [], "index": 7 }, { "bbox": [ 312, 107.5, 537, 123.0 ], "spans": [], "index": 8 }, { "bbox": [ 312, 123.0, 537, 138.5 ], "spans": [], "index": 9 }, { "bbox": [ 312, 138.5, 537, 154.0 ], "spans": [], "index": 10 } ] } ], "index": 8.5 }, { "type": "text", "bbox": [ 71, 184, 540, 219 ], "lines": [ { "bbox": [ 70, 183, 542, 197 ], "spans": [ { "bbox": [ 70, 183, 542, 197 ], "score": 1.0, "content": "OT-CFM reduces variance in the regression target. To accompany the theoretical results in §C.1,", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 195, 541, 209 ], "spans": [ { "bbox": [ 69, 195, 541, 209 ], "score": 1.0, "content": "in §D.1 we empirically study the variance of the stochastic regression objective in (OT-)CFM. 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Our approach to training continuous normalizing flows and conditional flow", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 354, 541, 369 ], "spans": [ { "bbox": [ 68, 354, 541, 369 ], "score": 1.0, "content": "models does not require integration over time during training. We have shown that lifting the static optimal", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 367, 541, 380 ], "spans": [ { "bbox": [ 69, 367, 541, 380 ], "score": 1.0, "content": "transport problem to the dynamic setting leads to simulation-free solutions to the dynamic OT and SB", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 380, 541, 393 ], "spans": [ { "bbox": [ 70, 380, 541, 393 ], "score": 1.0, "content": "problems, while also allowing more efficient training and inference of flow models by lowering variance of the", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 390, 543, 405 ], "spans": [ { "bbox": [ 69, 390, 543, 405 ], "score": 1.0, "content": "objective and straightening flows. One limitation of CFM is that it requires closed-form conditional flows,", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 403, 542, 416 ], "spans": [ { "bbox": [ 69, 403, 489, 416 ], "score": 1.0, "content": "which hinders its application to situations where we want to regularize the marginal vector field", "type": "text" }, { "bbox": [ 489, 405, 512, 415 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 512, 403, 542, 416 ], "score": 1.0, "content": "based", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 415, 541, 428 ], "spans": [ { "bbox": [ 69, 415, 541, 428 ], "score": 1.0, "content": "on prior information (Tong et al., 2020). 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We expect future work to overcome these limitations and", "type": "text" } ], "index": 28 }, { "bbox": [ 70, 452, 496, 463 ], "spans": [ { "bbox": [ 70, 452, 496, 463 ], "score": 1.0, "content": "hope that ideas from conditional flow matching will improve high-dimensional generative models.", "type": "text" } ], "index": 29 } ], "index": 24, "bbox_fs": [ 68, 331, 543, 463 ] }, { "type": "title", "bbox": [ 72, 477, 200, 491 ], "lines": [ { "bbox": [ 70, 476, 201, 493 ], "spans": [ { "bbox": [ 70, 476, 201, 493 ], "score": 1.0, "content": "Contribution statement", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 71, 502, 540, 550 ], "lines": [ { "bbox": [ 69, 502, 541, 515 ], "spans": [ { "bbox": [ 69, 502, 541, 515 ], "score": 1.0, "content": "A.T. initially conceived the idea. Y.Z., G.H., and N.M. led the development of the theory. High-dimensional", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 514, 541, 529 ], "spans": [ { "bbox": [ 69, 514, 541, 529 ], "score": 1.0, "content": "experiments and open-source code were led by A.T. and K.F. Additional experiments were contributed by", "type": "text" } ], "index": 32 }, { "bbox": [ 68, 525, 542, 541 ], "spans": [ { "bbox": [ 68, 525, 542, 541 ], "score": 1.0, "content": "Y.Z., J.R., G.H., N.M., and A.T. All authors contributed to designing the experiments. N.M. and A.T. drove", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 537, 523, 552 ], "spans": [ { "bbox": [ 69, 537, 523, 552 ], "score": 1.0, "content": "the writing of the paper, with contributions from all other authors. G.W. and Y.B. guided the project.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 68, 502, 542, 552 ] }, { "type": "title", "bbox": [ 72, 565, 170, 579 ], "lines": [ { "bbox": [ 69, 563, 171, 582 ], "spans": [ { "bbox": [ 69, 563, 171, 582 ], "score": 1.0, "content": "Acknowledgments", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 71, 590, 540, 674 ], "lines": [ { "bbox": [ 69, 589, 541, 604 ], "spans": [ { "bbox": [ 69, 589, 541, 604 ], "score": 1.0, "content": "We would like to thank Stefano Massaroli for productive conversations as well as thank Xinyu Yuan, Marco", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 602, 542, 615 ], "spans": [ { "bbox": [ 69, 602, 542, 615 ], "score": 1.0, "content": "Jiralerspong, Tara Akhound-Sadegh and Joey Bose for their helpful comments and feedback on the manuscript.", "type": "text" } ], "index": 37 }, { "bbox": [ 70, 614, 541, 628 ], "spans": [ { "bbox": [ 70, 614, 541, 628 ], "score": 1.0, "content": "We are also grateful to the anonymous reviewers for suggesting numerous improvements. This research was", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 627, 541, 640 ], "spans": [ { "bbox": [ 69, 627, 541, 640 ], "score": 1.0, "content": "enabled in part by compute resources provided by Mila (mila.quebec) and NVIDIA Corporation. The authors", "type": "text" } ], "index": 39 }, { "bbox": [ 68, 637, 541, 653 ], "spans": [ { "bbox": [ 68, 637, 541, 653 ], "score": 1.0, "content": "acknowledge funding from CIFAR, Genentech, Samsung, and IBM. 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Building normalizing flows with stochastic interpolants.", "type": "text" } ], "index": 44 }, { "bbox": [ 80, 720, 384, 733 ], "spans": [ { "bbox": [ 80, 720, 384, 733 ], "score": 1.0, "content": "International Conference on Learning Representations (ICLR), 2023.", "type": "text" } ], "index": 45 } ], "index": 44.5, "bbox_fs": [ 69, 707, 540, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 66, 78, 543, 732 ], "lines": [ { "bbox": [ 68, 80, 542, 98 ], "spans": [ { "bbox": [ 68, 80, 542, 98 ], "score": 1.0, "content": "Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. 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Path integral sampler: a stochastic control approach for sampling.", "type": "text" } ], "index": 32 }, { "bbox": [ 79, 557, 384, 570 ], "spans": [ { "bbox": [ 79, 557, 384, 570 ], "score": 1.0, "content": "International Conference on Learning Representations (ICLR), 2022.", "type": "text" } ], "index": 33 }, { "bbox": [ 68, 574, 542, 591 ], "spans": [ { "bbox": [ 68, 574, 542, 591 ], "score": 1.0, "content": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using", "type": "text" } ], "index": 34 }, { "bbox": [ 79, 587, 517, 601 ], "spans": [ { "bbox": [ 79, 587, 517, 601 ], "score": 1.0, "content": "cycle-consistent adversarial networks. International Conference on Computer Vision (ICCV), 2017.", "type": "text" } ], "index": 35 } ], "index": 17.5 }, { "type": "title", "bbox": [ 72, 621, 199, 635 ], "lines": [ { "bbox": [ 69, 619, 200, 638 ], "spans": [ { "bbox": [ 69, 619, 200, 638 ], "score": 1.0, "content": "A Proofs of theorems", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 72, 646, 544, 660 ], "lines": [ { "bbox": [ 69, 644, 542, 661 ], "spans": [ { "bbox": [ 69, 644, 514, 661 ], "score": 1.0, "content": "Theorem 3.1. The marginal vector field (9) generates the probability path (8) from initial conditions", "type": "text" }, { "bbox": [ 515, 649, 538, 659 ], "score": 0.92, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 538, 644, 542, 661 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 69, 671, 501, 684 ], "lines": [ { "bbox": [ 70, 670, 501, 685 ], "spans": [ { "bbox": [ 70, 670, 320, 685 ], "score": 1.0, "content": "Proof of Theorem 3.1. To verify this, we first check that", "type": "text" }, { "bbox": [ 320, 677, 329, 684 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 329, 670, 351, 685 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 352, 677, 361, 683 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 361, 670, 501, 685 ], "score": 1.0, "content": "satisfy the continuity equation.", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 70, 691, 272, 704 ], "lines": [ { "bbox": [ 70, 691, 271, 705 ], "spans": [ { "bbox": [ 70, 691, 271, 705 ], "score": 1.0, "content": "We start with the derivative w.r.t. time of (8)", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "interline_equation", "bbox": [ 213, 707, 342, 730 ], "lines": [ { "bbox": [ 213, 707, 342, 730 ], "spans": [ { "bbox": [ 213, 707, 342, 730 ], "score": 0.94, "content": "\\frac { d } { d t } p _ { t } ( x ) = \\frac { d } { d t } \\int p _ { t } ( x | z ) q ( z ) d z", "type": "interline_equation", "image_path": "9dae90ae30766d1aab982b503c566a1916f588585a4543a1699abf1292592e66.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 213, 707, 342, 730 ], "spans": [], "index": 40 } ] } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 761 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "18", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 65, 74, 543, 609 ], "lines": [], "index": 17.5, "bbox_fs": [ 68, 82, 543, 601 ], "lines_deleted": true }, { "type": "title", "bbox": [ 72, 621, 199, 635 ], "lines": [ { "bbox": [ 69, 619, 200, 638 ], "spans": [ { "bbox": [ 69, 619, 200, 638 ], "score": 1.0, "content": "A Proofs of theorems", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 72, 646, 544, 660 ], "lines": [ { "bbox": [ 69, 644, 542, 661 ], "spans": [ { "bbox": [ 69, 644, 514, 661 ], "score": 1.0, "content": "Theorem 3.1. The marginal vector field (9) generates the probability path (8) from initial conditions", "type": "text" }, { "bbox": [ 515, 649, 538, 659 ], "score": 0.92, "content": "p _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 538, 644, 542, 661 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 69, 644, 542, 661 ] }, { "type": "text", "bbox": [ 69, 671, 501, 684 ], "lines": [ { "bbox": [ 70, 670, 501, 685 ], "spans": [ { "bbox": [ 70, 670, 320, 685 ], "score": 1.0, "content": "Proof of Theorem 3.1. To verify this, we first check that", "type": "text" }, { "bbox": [ 320, 677, 329, 684 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 329, 670, 351, 685 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 352, 677, 361, 683 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 361, 670, 501, 685 ], "score": 1.0, "content": "satisfy the continuity equation.", "type": "text" } ], "index": 38 } ], "index": 38, "bbox_fs": [ 70, 670, 501, 685 ] }, { "type": "text", "bbox": [ 70, 691, 272, 704 ], "lines": [ { "bbox": [ 70, 691, 271, 705 ], "spans": [ { "bbox": [ 70, 691, 271, 705 ], "score": 1.0, "content": "We start with the derivative w.r.t. time of (8)", "type": "text" } ], "index": 39 } ], "index": 39, "bbox_fs": [ 70, 691, 271, 705 ] }, { "type": "interline_equation", "bbox": [ 213, 707, 342, 730 ], "lines": [ { "bbox": [ 213, 707, 342, 730 ], "spans": [ { "bbox": [ 213, 707, 342, 730 ], "score": 0.94, "content": "\\frac { d } { d t } p _ { t } ( x ) = \\frac { d } { d t } \\int p _ { t } ( x | z ) q ( z ) d z", "type": "interline_equation", "image_path": "9dae90ae30766d1aab982b503c566a1916f588585a4543a1699abf1292592e66.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 213, 707, 342, 730 ], "spans": [], "index": 40 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 82, 144, 95 ], "lines": [ { "bbox": [ 70, 81, 144, 96 ], "spans": [ { "bbox": [ 70, 81, 144, 96 ], "score": 1.0, "content": "by Leibniz Rule,", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 70, 122, 201, 135 ], "lines": [ { "bbox": [ 69, 120, 202, 137 ], "spans": [ { "bbox": [ 69, 120, 95, 137 ], "score": 1.0, "content": "since", "type": "text" }, { "bbox": [ 95, 124, 123, 135 ], "score": 0.92, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation" }, { "bbox": [ 124, 120, 169, 137 ], "score": 1.0, "content": "generates", "type": "text" }, { "bbox": [ 170, 123, 198, 135 ], "score": 0.9, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation" }, { "bbox": [ 198, 120, 202, 137 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 71, 162, 240, 174 ], "lines": [ { "bbox": [ 70, 162, 242, 176 ], "spans": [ { "bbox": [ 70, 162, 242, 176 ], "score": 1.0, "content": "exchanging the derivative and integral,", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "text", "bbox": [ 70, 203, 116, 217 ], "lines": [ { "bbox": [ 69, 201, 117, 218 ], "spans": [ { "bbox": [ 69, 201, 117, 218 ], "score": 1.0, "content": "Using (9),", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 70, 241, 354, 255 ], "lines": [ { "bbox": [ 69, 239, 352, 258 ], "spans": [ { "bbox": [ 69, 239, 220, 258 ], "score": 1.0, "content": "satisfying the continuity equation", "type": "text" }, { "bbox": [ 221, 242, 352, 255 ], "score": 0.9, "content": "\\begin{array} { r } { \\frac { d } { d t } p _ { t } ( x ) + \\mathrm { d i v } \\left( u _ { t } ( x ) p _ { t } ( x ) \\right) = 0 . } \\end{array}", "type": "inline_equation" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 70, 261, 543, 285 ], "lines": [ { "bbox": [ 69, 259, 543, 275 ], "spans": [ { "bbox": [ 69, 259, 151, 275 ], "score": 1.0, "content": "Theorem 3.2. If", "type": "text" }, { "bbox": [ 151, 263, 192, 274 ], "score": 0.92, "content": "p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 192, 259, 225, 275 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 225, 262, 255, 272 ], "score": 0.92, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 255, 259, 277, 275 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 277, 263, 313, 274 ], "score": 0.94, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 313, 259, 483, 275 ], "score": 1.0, "content": ", then, up to a constant independent of", "type": "text" }, { "bbox": [ 483, 264, 488, 271 ], "score": 0.86, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 488, 259, 494, 275 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 494, 264, 520, 273 ], "score": 0.89, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 520, 259, 543, 275 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 5 }, { "bbox": [ 71, 273, 186, 286 ], "spans": [ { "bbox": [ 71, 276, 91, 285 ], "score": 0.85, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 91, 273, 186, 286 ], "score": 1.0, "content": "are equal, and hence", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "interline_equation", "bbox": [ 249, 285, 362, 298 ], "lines": [ { "bbox": [ 249, 285, 362, 298 ], "spans": [ { "bbox": [ 249, 285, 362, 298 ], "score": 0.91, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } ) .", "type": "interline_equation", "image_path": "9c0965549b125db4cca422e6d9c9ef97f09beea264dbd1f6edfe8fe6da3e5eb7.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 249, 285, 362, 298 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 310, 541, 347 ], "lines": [ { "bbox": [ 70, 309, 541, 324 ], "spans": [ { "bbox": [ 70, 309, 541, 324 ], "score": 1.0, "content": "Proof of Theorem 3.2. For this proof we need (8), (9) and the existence and exchange of many integrals. As", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 321, 541, 336 ], "spans": [ { "bbox": [ 69, 321, 245, 336 ], "score": 1.0, "content": "in Lipman et al. (2023) we assume that", "type": "text" }, { "bbox": [ 245, 328, 250, 335 ], "score": 0.76, "content": "q", "type": "inline_equation" }, { "bbox": [ 250, 321, 255, 336 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 256, 325, 286, 335 ], "score": 0.92, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 286, 321, 479, 336 ], "score": 1.0, "content": "are decreasing to zero at sufficient speed as", "type": "text" }, { "bbox": [ 479, 325, 520, 335 ], "score": 0.94, "content": "\\| x \\| \\to \\infty", "type": "inline_equation" }, { "bbox": [ 521, 321, 541, 336 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 333, 201, 348 ], "spans": [ { "bbox": [ 69, 333, 92, 348 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 93, 337, 140, 346 ], "score": 0.92, "content": "u _ { t } , v _ { t } , \\nabla _ { \\theta } v _ { t }", "type": "inline_equation" }, { "bbox": [ 141, 333, 201, 348 ], "score": 1.0, "content": "are bounded.", "type": "text" } ], "index": 10 } ], "index": 9 }, { "type": "interline_equation", "bbox": [ 111, 353, 503, 438 ], "lines": [ { "bbox": [ 111, 353, 503, 438 ], "spans": [ { "bbox": [ 111, 353, 503, 438 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\qquad \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x ) \\| ^ { 2 } = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. + \\| u _ { t } ( x ) \\| ^ { 2 } \\right) } \\\\ & { \\qquad = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. \\right) } \\\\ & { \\nabla _ { \\theta } \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | z ) \\| ^ { 2 } = } \\\\ & { \\qquad \\nabla _ { \\theta } \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. + \\| u _ { t } ( x | z ) \\| ^ { 2 } \\right) } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\nabla _ { \\theta } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. \\right) } \\end{array}", "type": "interline_equation", "image_path": "522fac0ef10a5e21590e12da08de63d472502cea0673462b40d546d5d8a45a62.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 353, 503, 381.3333333333333 ], "spans": [], "index": 11 }, { "bbox": [ 111, 381.3333333333333, 503, 409.66666666666663 ], "spans": [], "index": 12 }, { "bbox": [ 111, 409.66666666666663, 503, 437.99999999999994 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 70, 443, 369, 456 ], "lines": [ { "bbox": [ 70, 442, 370, 457 ], "spans": [ { "bbox": [ 70, 442, 241, 457 ], "score": 1.0, "content": "By bilinearity of the 2-norm and since", "type": "text" }, { "bbox": [ 241, 448, 250, 455 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 251, 442, 331, 457 ], "score": 1.0, "content": "is independent of", "type": "text" }, { "bbox": [ 331, 446, 336, 453 ], "score": 0.88, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 337, 442, 370, 457 ], "score": 1.0, "content": ". Next,", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 196, 463, 415, 534 ], "lines": [ { "bbox": [ 196, 463, 415, 534 ], "spans": [ { "bbox": [ 196, 463, 415, 534 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { p _ { t } ( x ) } \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } = \\int \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } p _ { t } ( x ) d x } \\\\ { \\displaystyle \\qquad = \\int \\int \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } p _ { t } ( x | z ) q ( z ) d z d x } \\\\ { \\displaystyle \\qquad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } } \\end{array}", "type": "interline_equation", "image_path": "cd0debe4af74a248c3c8bb823a65a747ce997c70630fa593e8be1f5c4c5e2d8f.jpg" } ] } ], "index": 16.5, "virtual_lines": [ { "bbox": [ 196, 463, 415, 480.75 ], "spans": [], "index": 15 }, { "bbox": [ 196, 480.75, 415, 498.5 ], "spans": [], "index": 16 }, { "bbox": [ 196, 498.5, 415, 516.25 ], "spans": [], "index": 17 }, { "bbox": [ 196, 516.25, 415, 534.0 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 70, 539, 105, 551 ], "lines": [ { "bbox": [ 69, 537, 106, 554 ], "spans": [ { "bbox": [ 69, 537, 106, 554 ], "score": 1.0, "content": "Finally,", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 153, 560, 457, 659 ], "lines": [ { "bbox": [ 153, 560, 457, 659 ], "spans": [ { "bbox": [ 153, 560, 457, 659 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { E } _ { p _ { t } ( x ) } \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. = \\displaystyle \\int \\left. v _ { \\theta } ( t , x ) , \\frac { \\int u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) d z } { p _ { t } ( x ) } \\right. p _ { t } ( x ) d x } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\displaystyle \\int \\left. v _ { \\theta } ( t , x ) , \\int u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) d z \\right. d x } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\iint \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. p _ { t } ( x | z ) q ( z ) d z d x } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. } \\end{array}", "type": "interline_equation", "image_path": "eafe0d6ff95ca0977c8d67eb1a52ffc94c6ec56093eb4fe1e8680b2c9bbb4f05.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 153, 560, 457, 593.0 ], "spans": [], "index": 20 }, { "bbox": [ 153, 593.0, 457, 626.0 ], "spans": [], "index": 21 }, { "bbox": [ 153, 626.0, 457, 659.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 70, 664, 538, 690 ], "lines": [ { "bbox": [ 69, 663, 535, 679 ], "spans": [ { "bbox": [ 69, 663, 535, 679 ], "score": 1.0, "content": "Where we first substitute (9) then change the order of integration for the final equality. Since at all times", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 676, 541, 691 ], "spans": [ { "bbox": [ 70, 676, 142, 691 ], "score": 1.0, "content": "the gradients of", "type": "text" }, { "bbox": [ 142, 680, 163, 689 ], "score": 0.92, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 163, 676, 185, 691 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 680, 210, 689 ], "score": 0.92, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 211, 676, 259, 691 ], "score": 1.0, "content": "are equal,", "type": "text" }, { "bbox": [ 259, 679, 369, 689 ], "score": 0.9, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } )", "type": "inline_equation" }, { "bbox": [ 529, 676, 541, 690 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 24 } ], "index": 23.5 }, { "type": "text", "bbox": [ 70, 695, 541, 732 ], "lines": [ { "bbox": [ 69, 695, 542, 710 ], "spans": [ { "bbox": [ 69, 695, 218, 710 ], "score": 1.0, "content": "Proposition 3.3. The marginal", "type": "text" }, { "bbox": [ 218, 702, 226, 708 ], "score": 0.87, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 227, 695, 304, 710 ], "score": 1.0, "content": "corresponding to", "type": "text" }, { "bbox": [ 305, 698, 381, 709 ], "score": 0.93, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 382, 695, 420, 710 ], "score": 1.0, "content": "and the", "type": "text" }, { "bbox": [ 421, 698, 486, 709 ], "score": 0.93, "content": "p _ { t } ( x | z ) , u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 486, 695, 542, 710 ], "score": 1.0, "content": "in (14) and", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 708, 541, 722 ], "spans": [ { "bbox": [ 69, 708, 196, 722 ], "score": 1.0, "content": "(15) has boundary conditions", "type": "text" }, { "bbox": [ 196, 710, 287, 721 ], "score": 0.92, "content": "p _ { 1 } = q _ { 1 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 708, 307, 722 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 710, 399, 721 ], "score": 0.9, "content": "p _ { 0 } = q _ { 0 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 399, 708, 432, 722 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 432, 713, 437, 718 ], "score": 0.73, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 438, 708, 541, 722 ], "score": 1.0, "content": "denotes the convolution", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 721, 110, 732 ], "spans": [ { "bbox": [ 69, 721, 110, 732 ], "score": 1.0, "content": "operator.", "type": "text" } ], "index": 27 } ], "index": 26 } ], "page_idx": 18, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 70, 26, 369, 38 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 761 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "19", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 531, 242, 541, 253 ], "lines": [ { "bbox": [ 532, 244, 540, 252 ], "spans": [ { "bbox": [ 532, 244, 540, 252 ], "score": 0.999, "content": "□", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 82, 144, 95 ], "lines": [ { "bbox": [ 70, 81, 144, 96 ], "spans": [ { "bbox": [ 70, 81, 144, 96 ], "score": 1.0, "content": "by Leibniz Rule,", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 70, 81, 144, 96 ] }, { "type": "text", "bbox": [ 70, 122, 201, 135 ], "lines": [ { "bbox": [ 69, 120, 202, 137 ], "spans": [ { "bbox": [ 69, 120, 95, 137 ], "score": 1.0, "content": "since", "type": "text" }, { "bbox": [ 95, 124, 123, 135 ], "score": 0.92, "content": "\\boldsymbol { u } _ { t } ( \\cdot | \\boldsymbol { z } )", "type": "inline_equation" }, { "bbox": [ 124, 120, 169, 137 ], "score": 1.0, "content": "generates", "type": "text" }, { "bbox": [ 170, 123, 198, 135 ], "score": 0.9, "content": "p _ { t } ( \\cdot | z )", "type": "inline_equation" }, { "bbox": [ 198, 120, 202, 137 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 1 } ], "index": 1, "bbox_fs": [ 69, 120, 202, 137 ] }, { "type": "text", "bbox": [ 71, 162, 240, 174 ], "lines": [ { "bbox": [ 70, 162, 242, 176 ], "spans": [ { "bbox": [ 70, 162, 242, 176 ], "score": 1.0, "content": "exchanging the derivative and integral,", "type": "text" } ], "index": 2 } ], "index": 2, "bbox_fs": [ 70, 162, 242, 176 ] }, { "type": "text", "bbox": [ 70, 203, 116, 217 ], "lines": [ { "bbox": [ 69, 201, 117, 218 ], "spans": [ { "bbox": [ 69, 201, 117, 218 ], "score": 1.0, "content": "Using (9),", "type": "text" } ], "index": 3 } ], "index": 3, "bbox_fs": [ 69, 201, 117, 218 ] }, { "type": "text", "bbox": [ 70, 241, 354, 255 ], "lines": [ { "bbox": [ 69, 239, 352, 258 ], "spans": [ { "bbox": [ 69, 239, 220, 258 ], "score": 1.0, "content": "satisfying the continuity equation", "type": "text" }, { "bbox": [ 221, 242, 352, 255 ], "score": 0.9, "content": "\\begin{array} { r } { \\frac { d } { d t } p _ { t } ( x ) + \\mathrm { d i v } \\left( u _ { t } ( x ) p _ { t } ( x ) \\right) = 0 . } \\end{array}", "type": "inline_equation" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 69, 239, 352, 258 ] }, { "type": "text", "bbox": [ 70, 261, 543, 285 ], "lines": [ { "bbox": [ 69, 259, 543, 275 ], "spans": [ { "bbox": [ 69, 259, 151, 275 ], "score": 1.0, "content": "Theorem 3.2. If", "type": "text" }, { "bbox": [ 151, 263, 192, 274 ], "score": 0.92, "content": "p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 192, 259, 225, 275 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 225, 262, 255, 272 ], "score": 0.92, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 255, 259, 277, 275 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 277, 263, 313, 274 ], "score": 0.94, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 313, 259, 483, 275 ], "score": 1.0, "content": ", then, up to a constant independent of", "type": "text" }, { "bbox": [ 483, 264, 488, 271 ], "score": 0.86, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 488, 259, 494, 275 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 494, 264, 520, 273 ], "score": 0.89, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 520, 259, 543, 275 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 5 }, { "bbox": [ 71, 273, 186, 286 ], "spans": [ { "bbox": [ 71, 276, 91, 285 ], "score": 0.85, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 91, 273, 186, 286 ], "score": 1.0, "content": "are equal, and hence", "type": "text" } ], "index": 6 } ], "index": 5.5, "bbox_fs": [ 69, 259, 543, 286 ] }, { "type": "interline_equation", "bbox": [ 249, 285, 362, 298 ], "lines": [ { "bbox": [ 249, 285, 362, 298 ], "spans": [ { "bbox": [ 249, 285, 362, 298 ], "score": 0.91, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } ) .", "type": "interline_equation", "image_path": "9c0965549b125db4cca422e6d9c9ef97f09beea264dbd1f6edfe8fe6da3e5eb7.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 249, 285, 362, 298 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 310, 541, 347 ], "lines": [ { "bbox": [ 70, 309, 541, 324 ], "spans": [ { "bbox": [ 70, 309, 541, 324 ], "score": 1.0, "content": "Proof of Theorem 3.2. For this proof we need (8), (9) and the existence and exchange of many integrals. As", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 321, 541, 336 ], "spans": [ { "bbox": [ 69, 321, 245, 336 ], "score": 1.0, "content": "in Lipman et al. (2023) we assume that", "type": "text" }, { "bbox": [ 245, 328, 250, 335 ], "score": 0.76, "content": "q", "type": "inline_equation" }, { "bbox": [ 250, 321, 255, 336 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 256, 325, 286, 335 ], "score": 0.92, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 286, 321, 479, 336 ], "score": 1.0, "content": "are decreasing to zero at sufficient speed as", "type": "text" }, { "bbox": [ 479, 325, 520, 335 ], "score": 0.94, "content": "\\| x \\| \\to \\infty", "type": "inline_equation" }, { "bbox": [ 521, 321, 541, 336 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 333, 201, 348 ], "spans": [ { "bbox": [ 69, 333, 92, 348 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 93, 337, 140, 346 ], "score": 0.92, "content": "u _ { t } , v _ { t } , \\nabla _ { \\theta } v _ { t }", "type": "inline_equation" }, { "bbox": [ 141, 333, 201, 348 ], "score": 1.0, "content": "are bounded.", "type": "text" } ], "index": 10 } ], "index": 9, "bbox_fs": [ 69, 309, 541, 348 ] }, { "type": "interline_equation", "bbox": [ 111, 353, 503, 438 ], "lines": [ { "bbox": [ 111, 353, 503, 438 ], "spans": [ { "bbox": [ 111, 353, 503, 438 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\qquad \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x ) \\| ^ { 2 } = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. + \\| u _ { t } ( x ) \\| ^ { 2 } \\right) } \\\\ & { \\qquad = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { t } ( x ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. \\right) } \\\\ & { \\nabla _ { \\theta } \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | z ) \\| ^ { 2 } = } \\\\ & { \\qquad \\nabla _ { \\theta } \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. + \\| u _ { t } ( x | z ) \\| ^ { 2 } \\right) } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\nabla _ { \\theta } \\left( \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } - 2 \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. \\right) } \\end{array}", "type": "interline_equation", "image_path": "522fac0ef10a5e21590e12da08de63d472502cea0673462b40d546d5d8a45a62.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 353, 503, 381.3333333333333 ], "spans": [], "index": 11 }, { "bbox": [ 111, 381.3333333333333, 503, 409.66666666666663 ], "spans": [], "index": 12 }, { "bbox": [ 111, 409.66666666666663, 503, 437.99999999999994 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 70, 443, 369, 456 ], "lines": [ { "bbox": [ 70, 442, 370, 457 ], "spans": [ { "bbox": [ 70, 442, 241, 457 ], "score": 1.0, "content": "By bilinearity of the 2-norm and since", "type": "text" }, { "bbox": [ 241, 448, 250, 455 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 251, 442, 331, 457 ], "score": 1.0, "content": "is independent of", "type": "text" }, { "bbox": [ 331, 446, 336, 453 ], "score": 0.88, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 337, 442, 370, 457 ], "score": 1.0, "content": ". Next,", "type": "text" } ], "index": 14 } ], "index": 14, "bbox_fs": [ 70, 442, 370, 457 ] }, { "type": "interline_equation", "bbox": [ 196, 463, 415, 534 ], "lines": [ { "bbox": [ 196, 463, 415, 534 ], "spans": [ { "bbox": [ 196, 463, 415, 534 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { p _ { t } ( x ) } \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } = \\int \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } p _ { t } ( x ) d x } \\\\ { \\displaystyle \\qquad = \\int \\int \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } p _ { t } ( x | z ) q ( z ) d z d x } \\\\ { \\displaystyle \\qquad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\| v _ { \\theta } ( t , x ) \\| ^ { 2 } } \\end{array}", "type": "interline_equation", "image_path": "cd0debe4af74a248c3c8bb823a65a747ce997c70630fa593e8be1f5c4c5e2d8f.jpg" } ] } ], "index": 16.5, "virtual_lines": [ { "bbox": [ 196, 463, 415, 480.75 ], "spans": [], "index": 15 }, { "bbox": [ 196, 480.75, 415, 498.5 ], "spans": [], "index": 16 }, { "bbox": [ 196, 498.5, 415, 516.25 ], "spans": [], "index": 17 }, { "bbox": [ 196, 516.25, 415, 534.0 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 70, 539, 105, 551 ], "lines": [ { "bbox": [ 69, 537, 106, 554 ], "spans": [ { "bbox": [ 69, 537, 106, 554 ], "score": 1.0, "content": "Finally,", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 69, 537, 106, 554 ] }, { "type": "interline_equation", "bbox": [ 153, 560, 457, 659 ], "lines": [ { "bbox": [ 153, 560, 457, 659 ], "spans": [ { "bbox": [ 153, 560, 457, 659 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { E } _ { p _ { t } ( x ) } \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x ) \\right. = \\displaystyle \\int \\left. v _ { \\theta } ( t , x ) , \\frac { \\int u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) d z } { p _ { t } ( x ) } \\right. p _ { t } ( x ) d x } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\displaystyle \\int \\left. v _ { \\theta } ( t , x ) , \\int u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) d z \\right. d x } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\iint \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. p _ { t } ( x | z ) q ( z ) d z d x } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { q ( z ) , p _ { t } ( x | z ) } \\left. v _ { \\theta } ( t , x ) , u _ { t } ( x | z ) \\right. } \\end{array}", "type": "interline_equation", "image_path": "eafe0d6ff95ca0977c8d67eb1a52ffc94c6ec56093eb4fe1e8680b2c9bbb4f05.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 153, 560, 457, 593.0 ], "spans": [], "index": 20 }, { "bbox": [ 153, 593.0, 457, 626.0 ], "spans": [], "index": 21 }, { "bbox": [ 153, 626.0, 457, 659.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 70, 664, 538, 690 ], "lines": [ { "bbox": [ 69, 663, 535, 679 ], "spans": [ { "bbox": [ 69, 663, 535, 679 ], "score": 1.0, "content": "Where we first substitute (9) then change the order of integration for the final equality. Since at all times", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 676, 541, 691 ], "spans": [ { "bbox": [ 70, 676, 142, 691 ], "score": 1.0, "content": "the gradients of", "type": "text" }, { "bbox": [ 142, 680, 163, 689 ], "score": 0.92, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" }, { "bbox": [ 163, 676, 185, 691 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 680, 210, 689 ], "score": 0.92, "content": "{ \\mathcal { L } } _ { \\mathrm { C F M } }", "type": "inline_equation" }, { "bbox": [ 211, 676, 259, 691 ], "score": 1.0, "content": "are equal,", "type": "text" }, { "bbox": [ 259, 679, 369, 689 ], "score": 0.9, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { C F M } } ( \\boldsymbol { \\theta } )", "type": "inline_equation" }, { "bbox": [ 529, 676, 541, 690 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 24 } ], "index": 23.5, "bbox_fs": [ 69, 663, 541, 691 ] }, { "type": "text", "bbox": [ 70, 695, 541, 732 ], "lines": [ { "bbox": [ 69, 695, 542, 710 ], "spans": [ { "bbox": [ 69, 695, 218, 710 ], "score": 1.0, "content": "Proposition 3.3. The marginal", "type": "text" }, { "bbox": [ 218, 702, 226, 708 ], "score": 0.87, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 227, 695, 304, 710 ], "score": 1.0, "content": "corresponding to", "type": "text" }, { "bbox": [ 305, 698, 381, 709 ], "score": 0.93, "content": "q ( z ) = q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 382, 695, 420, 710 ], "score": 1.0, "content": "and the", "type": "text" }, { "bbox": [ 421, 698, 486, 709 ], "score": 0.93, "content": "p _ { t } ( x | z ) , u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 486, 695, 542, 710 ], "score": 1.0, "content": "in (14) and", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 708, 541, 722 ], "spans": [ { "bbox": [ 69, 708, 196, 722 ], "score": 1.0, "content": "(15) has boundary conditions", "type": "text" }, { "bbox": [ 196, 710, 287, 721 ], "score": 0.92, "content": "p _ { 1 } = q _ { 1 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 708, 307, 722 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 710, 399, 721 ], "score": 0.9, "content": "p _ { 0 } = q _ { 0 } * \\mathcal { N } ( x \\mid 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 399, 708, 432, 722 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 432, 713, 437, 718 ], "score": 0.73, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 438, 708, 541, 722 ], "score": 1.0, "content": "denotes the convolution", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 721, 110, 732 ], "spans": [ { "bbox": [ 69, 721, 110, 732 ], "score": 1.0, "content": "operator.", "type": "text" } ], "index": 27 } ], "index": 26, "bbox_fs": [ 69, 695, 542, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 107 ], "lines": [ { "bbox": [ 69, 80, 542, 96 ], "spans": [ { "bbox": [ 69, 80, 453, 96 ], "score": 1.0, "content": "Proof of Proposition 3.3. We start with (8) to show the result of the lemma. We note that", "type": "text" }, { "bbox": [ 453, 84, 542, 95 ], "score": 0.92, "content": "q ( z ) = q ( ( x _ { 0 } , x _ { 1 } ) ) =", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 71, 96, 117, 106 ], "spans": [ { "bbox": [ 71, 96, 117, 106 ], "score": 0.92, "content": "q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 186, 116, 425, 195 ], "lines": [ { "bbox": [ 186, 116, 425, 195 ], "spans": [ { "bbox": [ 186, 116, 425, 195 ], "score": 0.95, "content": "\\begin{array} { l } { p _ { t } ( x ) = \\displaystyle \\int p _ { t } ( x | z ) q ( z ) d z } \\\\ { \\displaystyle = \\int N ( x | t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) q ( ( x _ { 0 } , x _ { 1 } ) ) d ( x _ { 0 } , x _ { 1 } ) } \\\\ { \\displaystyle = \\iint \\mathcal { N } ( x | t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) q ( x _ { 1 } ) d x _ { 0 } d x _ { 1 } } \\end{array}", "type": "interline_equation", "image_path": "2921a602cfc356120f8cabeb3b6ed53837df6390d740b4356561a634357072a2.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 186, 116, 425, 131.8 ], "spans": [], "index": 2 }, { "bbox": [ 186, 131.8, 425, 147.60000000000002 ], "spans": [], "index": 3 }, { "bbox": [ 186, 147.60000000000002, 425, 163.40000000000003 ], "spans": [], "index": 4 }, { "bbox": [ 186, 163.40000000000003, 425, 179.20000000000005 ], "spans": [], "index": 5 }, { "bbox": [ 186, 179.20000000000005, 425, 195.00000000000006 ], "spans": [], "index": 6 } ] }, { "type": "text", "bbox": [ 70, 202, 306, 215 ], "lines": [ { "bbox": [ 69, 201, 306, 216 ], "spans": [ { "bbox": [ 69, 201, 127, 216 ], "score": 1.0, "content": "evaluated at", "type": "text" }, { "bbox": [ 128, 205, 159, 214 ], "score": 0.93, "content": "i = 0 , 1", "type": "inline_equation" }, { "bbox": [ 159, 201, 279, 216 ], "score": 1.0, "content": "respectively. Therefore, at", "type": "text" }, { "bbox": [ 280, 205, 302, 212 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 302, 201, 306, 216 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "interline_equation", "bbox": [ 213, 224, 398, 292 ], "lines": [ { "bbox": [ 213, 224, 398, 292 ], "spans": [ { "bbox": [ 213, 224, 398, 292 ], "score": 0.94, "content": "\\begin{array} { l } { p _ { 0 } ( x ) = \\displaystyle \\iint \\mathcal { N } ( x | x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) q ( x _ { 1 } ) d x _ { 0 } d x _ { 1 } } \\\\ { \\displaystyle \\qquad = \\int \\mathcal { N } ( x | x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) d x _ { 0 } } \\\\ { \\displaystyle = q ( x _ { 0 } ) * \\mathcal { N } ( x | 0 , \\sigma ^ { 2 } ) . } \\end{array}", "type": "interline_equation", "image_path": "5ec06312604b630ae75f79fc8d20f93b699e655ab765bdf98880cbb9eb0c3954.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 213, 224, 398, 241.0 ], "spans": [], "index": 8 }, { "bbox": [ 213, 241.0, 398, 258.0 ], "spans": [], "index": 9 }, { "bbox": [ 213, 258.0, 398, 275.0 ], "spans": [], "index": 10 }, { "bbox": [ 213, 275.0, 398, 292.0 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 71, 300, 185, 313 ], "lines": [ { "bbox": [ 70, 300, 186, 313 ], "spans": [ { "bbox": [ 70, 300, 159, 313 ], "score": 1.0, "content": "This is also true for", "type": "text" }, { "bbox": [ 160, 304, 182, 311 ], "score": 0.92, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 182, 300, 186, 313 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 70, 324, 541, 360 ], "lines": [ { "bbox": [ 69, 322, 541, 338 ], "spans": [ { "bbox": [ 69, 322, 339, 338 ], "score": 1.0, "content": "Proposition 3.4. The results of Proposition 3.3 also hold for", "type": "text" }, { "bbox": [ 339, 326, 357, 336 ], "score": 0.92, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 357, 322, 541, 338 ], "score": 1.0, "content": "in (17). Furthermore, assuming regularity", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 334, 542, 351 ], "spans": [ { "bbox": [ 69, 334, 128, 351 ], "score": 1.0, "content": "properties of", "type": "text" }, { "bbox": [ 129, 342, 137, 348 ], "score": 0.8, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 138, 334, 144, 351 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 144, 341, 153, 348 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 153, 334, 295, 351 ], "score": 1.0, "content": ", and the optimal transport plan", "type": "text" }, { "bbox": [ 295, 341, 302, 346 ], "score": 0.85, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 302, 334, 320, 351 ], "score": 1.0, "content": ", as", "type": "text" }, { "bbox": [ 320, 337, 352, 346 ], "score": 0.92, "content": "\\sigma ^ { 2 } \\to 0", "type": "inline_equation" }, { "bbox": [ 352, 334, 434, 351 ], "score": 1.0, "content": "the marginal path", "type": "text" }, { "bbox": [ 434, 341, 443, 348 ], "score": 0.88, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 443, 334, 487, 351 ], "score": 1.0, "content": "and field", "type": "text" }, { "bbox": [ 487, 342, 496, 347 ], "score": 0.87, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 496, 334, 542, 351 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 347, 406, 362 ], "spans": [ { "bbox": [ 69, 347, 110, 362 ], "score": 1.0, "content": "(7), i.e.,", "type": "text" }, { "bbox": [ 110, 353, 119, 359 ], "score": 0.89, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 120, 347, 360, 362 ], "score": 1.0, "content": "solves the dynamic optimal transport problem between", "type": "text" }, { "bbox": [ 360, 353, 369, 360 ], "score": 0.86, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 370, 347, 392, 362 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 392, 353, 401, 360 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 401, 347, 406, 362 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "text", "bbox": [ 70, 376, 540, 437 ], "lines": [ { "bbox": [ 70, 376, 542, 390 ], "spans": [ { "bbox": [ 70, 376, 391, 390 ], "score": 1.0, "content": "Proof of Proposition 3.4. We will assume certain regularity conditions on", "type": "text" }, { "bbox": [ 391, 382, 400, 389 ], "score": 0.86, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 401, 376, 406, 390 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 406, 382, 415, 389 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 416, 376, 440, 390 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 440, 382, 446, 387 ], "score": 0.89, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 447, 376, 542, 390 ], "score": 1.0, "content": "to allow reduction to", "type": "text" } ], "index": 16 }, { "bbox": [ 70, 389, 541, 402 ], "spans": [ { "bbox": [ 70, 389, 541, 402 ], "score": 1.0, "content": "known results. We leave it to future work to determine which of these conditions are necessary and which", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 401, 540, 413 ], "spans": [ { "bbox": [ 69, 401, 493, 413 ], "score": 1.0, "content": "are redundant with other conditions. However, because we are concerned with approximation of", "type": "text" }, { "bbox": [ 494, 403, 516, 413 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 517, 401, 540, 413 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 414, 541, 426 ], "spans": [ { "bbox": [ 69, 414, 541, 426 ], "score": 1.0, "content": "neural networks, which are typically smooth, results that relax the regularity assumptions may be vacuous in", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 426, 110, 437 ], "spans": [ { "bbox": [ 68, 426, 110, 437 ], "score": 1.0, "content": "practice.", "type": "text" } ], "index": 20 } ], "index": 18 }, { "type": "text", "bbox": [ 70, 442, 541, 491 ], "lines": [ { "bbox": [ 69, 441, 542, 457 ], "spans": [ { "bbox": [ 69, 441, 207, 457 ], "score": 1.0, "content": "Preliminaries. We assume that", "type": "text" }, { "bbox": [ 207, 448, 216, 455 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 216, 441, 238, 457 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 239, 448, 247, 455 ], "score": 0.9, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 248, 441, 542, 457 ], "score": 1.0, "content": "are compactly supported and admit bounded densities with respect", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 454, 541, 467 ], "spans": [ { "bbox": [ 70, 454, 541, 467 ], "score": 1.0, "content": "to the Lebesgue measure. Then the conditions for Brenier’s theorem (Brenier, 1991) are satisfied. By Brenier’s", "type": "text" } ], "index": 22 }, { "bbox": [ 68, 465, 543, 480 ], "spans": [ { "bbox": [ 68, 465, 187, 480 ], "score": 1.0, "content": "theorem, the optimal joint", "type": "text" }, { "bbox": [ 187, 472, 193, 477 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 193, 465, 369, 480 ], "score": 1.0, "content": "is unique and is supported on the graph", "type": "text" }, { "bbox": [ 370, 469, 408, 479 ], "score": 0.93, "content": "( x , T ( x ) )", "type": "inline_equation" }, { "bbox": [ 408, 465, 483, 480 ], "score": 1.0, "content": "of a Monge map", "type": "text" }, { "bbox": [ 483, 468, 538, 477 ], "score": 0.92, "content": "T : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 538, 465, 543, 480 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 478, 424, 492 ], "spans": [ { "bbox": [ 69, 478, 90, 492 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 91, 480, 113, 491 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 113, 478, 424, 492 ], "score": 1.0, "content": "is equal to McCann’s interpolation (Peyré & Cuturi, 2019, Chapter 7)", "type": "text" } ], "index": 24 } ], "index": 22.5 }, { "type": "interline_equation", "bbox": [ 250, 502, 361, 515 ], "lines": [ { "bbox": [ 250, 502, 361, 515 ], "spans": [ { "bbox": [ 250, 502, 361, 515 ], "score": 0.92, "content": "p _ { t } = ( ( 1 - t ) \\mathrm { I d } + t T ) _ { \\# } p _ { 0 } .", "type": "interline_equation", "image_path": "2f68f1dda8545ee5860e1a0e0a143e044e713622fc68f4792707ae088291ea2f.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 250, 502, 361, 515 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 70, 524, 542, 585 ], "lines": [ { "bbox": [ 69, 525, 543, 538 ], "spans": [ { "bbox": [ 69, 525, 183, 538 ], "score": 1.0, "content": "In addition, we know that", "type": "text" }, { "bbox": [ 183, 527, 204, 537 ], "score": 0.94, "content": "T ( x )", "type": "inline_equation" }, { "bbox": [ 205, 525, 476, 538 ], "score": 1.0, "content": "can be parameterized as the gradient of a convex function, i.e.,", "type": "text" }, { "bbox": [ 476, 527, 538, 537 ], "score": 0.94, "content": "T ( x ) = \\nabla \\psi ( x )", "type": "inline_equation" }, { "bbox": [ 539, 525, 543, 538 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 536, 543, 551 ], "spans": [ { "bbox": [ 69, 536, 177, 551 ], "score": 1.0, "content": "This characterization of", "type": "text" }, { "bbox": [ 178, 540, 185, 547 ], "score": 0.9, "content": "T", "type": "inline_equation" }, { "bbox": [ 186, 536, 433, 551 ], "score": 1.0, "content": "implies that the conditional probability paths, given by", "type": "text" }, { "bbox": [ 434, 539, 538, 550 ], "score": 0.92, "content": "\\phi _ { t } ( x ) = x + t ( T ( x ) - x )", "type": "inline_equation" }, { "bbox": [ 538, 536, 543, 551 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 27 }, { "bbox": [ 68, 547, 540, 564 ], "spans": [ { "bbox": [ 68, 547, 151, 564 ], "score": 1.0, "content": "do not cross, i.e.,", "type": "text" }, { "bbox": [ 151, 551, 252, 561 ], "score": 0.91, "content": "p _ { t } ( x | x _ { 0 } , T ( x _ { 0 } ) ) = p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 252, 547, 284, 564 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 284, 551, 306, 561 ], "score": 0.93, "content": "( t , x )", "type": "inline_equation" }, { "bbox": [ 307, 547, 485, 564 ], "score": 1.0, "content": ".4 It is known that the probability path", "type": "text" }, { "bbox": [ 486, 551, 540, 562 ], "score": 0.93, "content": "p _ { t } = [ \\phi _ { t } ] _ { \\# } p _ { 0 }", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 69, 560, 542, 575 ], "spans": [ { "bbox": [ 69, 560, 241, 575 ], "score": 1.0, "content": "and its associated vector field, given by", "type": "text" }, { "bbox": [ 242, 563, 333, 573 ], "score": 0.93, "content": "u _ { t } ( \\phi _ { t } ( x ) ) = T ( x ) - x", "type": "inline_equation" }, { "bbox": [ 333, 560, 542, 575 ], "score": 1.0, "content": ", solve the optimal transport problem (Benamou", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 572, 220, 586 ], "spans": [ { "bbox": [ 69, 572, 220, 586 ], "score": 1.0, "content": "& Brenier, 2000, Proposition 1.1).", "type": "text" } ], "index": 30 } ], "index": 28 }, { "type": "text", "bbox": [ 70, 590, 541, 627 ], "lines": [ { "bbox": [ 69, 590, 542, 604 ], "spans": [ { "bbox": [ 69, 590, 240, 604 ], "score": 1.0, "content": "We assume that the induced marginals", "type": "text" }, { "bbox": [ 240, 596, 249, 603 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 249, 590, 542, 604 ], "score": 1.0, "content": "have bounded densities with respect to Lebesgue measure and that", "type": "text" } ], "index": 31 }, { "bbox": [ 71, 601, 542, 616 ], "spans": [ { "bbox": [ 71, 605, 79, 613 ], "score": 0.91, "content": "T", "type": "inline_equation" }, { "bbox": [ 79, 601, 237, 616 ], "score": 1.0, "content": "is almost everywhere continuous in", "type": "text" }, { "bbox": [ 237, 608, 243, 613 ], "score": 0.87, "content": "x", "type": "inline_equation" }, { "bbox": [ 244, 601, 369, 616 ], "score": 1.0, "content": ", which implies the same for", "type": "text" }, { "bbox": [ 369, 605, 379, 615 ], "score": 0.91, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 379, 601, 445, 616 ], "score": 1.0, "content": ". Injectivity of", "type": "text" }, { "bbox": [ 445, 605, 455, 615 ], "score": 0.92, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 455, 601, 542, 616 ], "score": 1.0, "content": "and noncrossing of", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 614, 358, 629 ], "spans": [ { "bbox": [ 69, 614, 132, 629 ], "score": 1.0, "content": "paths implies", "type": "text" }, { "bbox": [ 132, 617, 172, 627 ], "score": 0.95, "content": "\\boldsymbol u _ { t } ( \\phi _ { t } ( \\boldsymbol x ) )", "type": "inline_equation" }, { "bbox": [ 172, 614, 330, 629 ], "score": 1.0, "content": "is almost everywhere continuous in", "type": "text" }, { "bbox": [ 331, 617, 354, 627 ], "score": 0.95, "content": "\\phi _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 354, 614, 358, 629 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 33 } ], "index": 32 }, { "type": "text", "bbox": [ 69, 632, 540, 657 ], "lines": [ { "bbox": [ 69, 632, 542, 646 ], "spans": [ { "bbox": [ 69, 632, 256, 646 ], "score": 1.0, "content": "Formal statement of the result. Denote by", "type": "text" }, { "bbox": [ 256, 635, 280, 645 ], "score": 0.93, "content": "p _ { t } ^ { \\sigma } ( x )", "type": "inline_equation" }, { "bbox": [ 281, 632, 285, 646 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 286, 635, 318, 645 ], "score": 0.94, "content": "p _ { t } ^ { \\sigma } ( x | z )", "type": "inline_equation" }, { "bbox": [ 318, 632, 494, 646 ], "score": 1.0, "content": "the densities in Proposition 3.3, and by", "type": "text" }, { "bbox": [ 495, 635, 519, 645 ], "score": 0.94, "content": "p _ { t } ^ { \\sigma } ( x )", "type": "inline_equation" }, { "bbox": [ 519, 632, 542, 646 ], "score": 1.0, "content": ". We", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 643, 317, 658 ], "spans": [ { "bbox": [ 69, 643, 151, 658 ], "score": 1.0, "content": "will show that for", "type": "text" }, { "bbox": [ 151, 650, 160, 657 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 160, 643, 221, 658 ], "score": 1.0, "content": "-almost every", "type": "text" }, { "bbox": [ 221, 650, 227, 654 ], "score": 0.89, "content": "x", "type": "inline_equation" }, { "bbox": [ 228, 643, 308, 658 ], "score": 1.0, "content": "and almost every", "type": "text" }, { "bbox": [ 308, 648, 312, 655 ], "score": 0.86, "content": "t", "type": "inline_equation" }, { "bbox": [ 313, 643, 317, 658 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "interline_equation", "bbox": [ 201, 666, 411, 693 ], "lines": [ { "bbox": [ 201, 666, 411, 693 ], "spans": [ { "bbox": [ 201, 666, 411, 693 ], "score": 0.93, "content": "u _ { t } ( x ) = \\operatorname* { l i m } _ { \\sigma 0 } u _ { t } ( x ) = \\operatorname* { l i m } _ { \\sigma 0 } \\frac { \\mathbb { E } _ { z \\sim q ( z ) } p _ { t } ^ { \\sigma } ( x | z ) u _ { t } ( x | z ) } { p _ { t } ^ { \\sigma } ( x ) } .", "type": "interline_equation", "image_path": "e9ae1b9351803747d93eb8df5c16f6380352c5ea027ed22fda4f6ff51ed16c4c.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 201, 666, 411, 693 ], "spans": [], "index": 36 } ] } ], "page_idx": 19, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 70, 702, 543, 733 ], "lines": [ { "bbox": [ 80, 700, 543, 716 ], "spans": [ { "bbox": [ 80, 700, 213, 716 ], "score": 1.0, "content": "4Proof: If the paths from distinct", "type": "text" }, { "bbox": [ 213, 707, 222, 712 ], "score": 0.87, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 222, 700, 242, 716 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 242, 705, 251, 714 ], "score": 0.86, "content": "x _ { 0 } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 251, 700, 289, 716 ], "score": 1.0, "content": "cross, so", "type": "text" }, { "bbox": [ 289, 705, 349, 713 ], "score": 0.9, "content": "\\phi _ { t } ( x _ { 0 } ) = \\phi _ { t } ( x _ { 0 } ^ { \\prime } )", "type": "inline_equation" }, { "bbox": [ 349, 700, 374, 716 ], "score": 1.0, "content": ", then", "type": "text" }, { "bbox": [ 374, 704, 538, 713 ], "score": 0.9, "content": "( 1 - t ) x _ { 0 } + t \\nabla \\psi ( x _ { 0 } ) = ( 1 - t ) x _ { 0 } ^ { \\prime } + t \\nabla \\psi ( x _ { 0 } ^ { \\prime } )", "type": "inline_equation" }, { "bbox": [ 539, 700, 543, 716 ], "score": 1.0, "content": ".", "type": "text" } ] }, { "bbox": [ 70, 712, 540, 725 ], "spans": [ { "bbox": [ 70, 712, 165, 725 ], "score": 1.0, "content": "Taking dot product with", "type": "text" }, { "bbox": [ 165, 714, 194, 723 ], "score": 0.85, "content": "x _ { 0 } - x _ { 0 } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 194, 712, 198, 725 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 198, 714, 384, 723 ], "score": 0.86, "content": "( t - 1 ) \\| x _ { 0 } - x _ { 0 } ^ { \\prime } \\| ^ { 2 } = t \\langle \\nabla \\psi ( x _ { 0 } ) - \\nabla \\psi ( x _ { 0 } ^ { \\prime } ) , x _ { 0 } - x _ { 0 } ^ { \\prime } \\rangle", "type": "inline_equation" }, { "bbox": [ 385, 712, 459, 725 ], "score": 1.0, "content": ". However, we have", "type": "text" }, { "bbox": [ 459, 714, 540, 723 ], "score": 0.88, "content": "( t - 1 ) \\| x _ { 0 } - x _ { 0 } ^ { \\prime } \\| ^ { 2 } < 0", "type": "inline_equation" } ] }, { "bbox": [ 69, 721, 338, 734 ], "spans": [ { "bbox": [ 69, 721, 87, 734 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 87, 723, 212, 733 ], "score": 0.9, "content": "t \\langle \\nabla \\psi ( x _ { 0 } ) - \\nabla \\psi ( x _ { 0 } ^ { \\prime } ) , x _ { 0 } - x _ { 0 } ^ { \\prime } \\rangle \\geq 0", "type": "inline_equation" }, { "bbox": [ 212, 721, 273, 734 ], "score": 1.0, "content": "by convexity of", "type": "text" }, { "bbox": [ 274, 724, 280, 732 ], "score": 0.88, "content": "\\psi", "type": "inline_equation" }, { "bbox": [ 280, 721, 338, 734 ], "score": 1.0, "content": ", contradiction.", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 71, 26, 368, 38 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "20", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 531, 301, 541, 312 ], "lines": [ { "bbox": [ 532, 303, 540, 311 ], "spans": [ { "bbox": [ 532, 303, 540, 311 ], "score": 0.994, "content": "□", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 107 ], "lines": [ { "bbox": [ 69, 80, 542, 96 ], "spans": [ { "bbox": [ 69, 80, 453, 96 ], "score": 1.0, "content": "Proof of Proposition 3.3. We start with (8) to show the result of the lemma. We note that", "type": "text" }, { "bbox": [ 453, 84, 542, 95 ], "score": 0.92, "content": "q ( z ) = q ( ( x _ { 0 } , x _ { 1 } ) ) =", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 71, 96, 117, 106 ], "spans": [ { "bbox": [ 71, 96, 117, 106 ], "score": 0.92, "content": "q ( x _ { 0 } ) q ( x _ { 1 } )", "type": "inline_equation" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 69, 80, 542, 106 ] }, { "type": "interline_equation", "bbox": [ 186, 116, 425, 195 ], "lines": [ { "bbox": [ 186, 116, 425, 195 ], "spans": [ { "bbox": [ 186, 116, 425, 195 ], "score": 0.95, "content": "\\begin{array} { l } { p _ { t } ( x ) = \\displaystyle \\int p _ { t } ( x | z ) q ( z ) d z } \\\\ { \\displaystyle = \\int N ( x | t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) q ( ( x _ { 0 } , x _ { 1 } ) ) d ( x _ { 0 } , x _ { 1 } ) } \\\\ { \\displaystyle = \\iint \\mathcal { N } ( x | t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) q ( x _ { 1 } ) d x _ { 0 } d x _ { 1 } } \\end{array}", "type": "interline_equation", "image_path": "2921a602cfc356120f8cabeb3b6ed53837df6390d740b4356561a634357072a2.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 186, 116, 425, 131.8 ], "spans": [], "index": 2 }, { "bbox": [ 186, 131.8, 425, 147.60000000000002 ], "spans": [], "index": 3 }, { "bbox": [ 186, 147.60000000000002, 425, 163.40000000000003 ], "spans": [], "index": 4 }, { "bbox": [ 186, 163.40000000000003, 425, 179.20000000000005 ], "spans": [], "index": 5 }, { "bbox": [ 186, 179.20000000000005, 425, 195.00000000000006 ], "spans": [], "index": 6 } ] }, { "type": "text", "bbox": [ 70, 202, 306, 215 ], "lines": [ { "bbox": [ 69, 201, 306, 216 ], "spans": [ { "bbox": [ 69, 201, 127, 216 ], "score": 1.0, "content": "evaluated at", "type": "text" }, { "bbox": [ 128, 205, 159, 214 ], "score": 0.93, "content": "i = 0 , 1", "type": "inline_equation" }, { "bbox": [ 159, 201, 279, 216 ], "score": 1.0, "content": "respectively. Therefore, at", "type": "text" }, { "bbox": [ 280, 205, 302, 212 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 302, 201, 306, 216 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 7 } ], "index": 7, "bbox_fs": [ 69, 201, 306, 216 ] }, { "type": "interline_equation", "bbox": [ 213, 224, 398, 292 ], "lines": [ { "bbox": [ 213, 224, 398, 292 ], "spans": [ { "bbox": [ 213, 224, 398, 292 ], "score": 0.94, "content": "\\begin{array} { l } { p _ { 0 } ( x ) = \\displaystyle \\iint \\mathcal { N } ( x | x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) q ( x _ { 1 } ) d x _ { 0 } d x _ { 1 } } \\\\ { \\displaystyle \\qquad = \\int \\mathcal { N } ( x | x _ { 0 } , \\sigma ^ { 2 } ) q ( x _ { 0 } ) d x _ { 0 } } \\\\ { \\displaystyle = q ( x _ { 0 } ) * \\mathcal { N } ( x | 0 , \\sigma ^ { 2 } ) . } \\end{array}", "type": "interline_equation", "image_path": "5ec06312604b630ae75f79fc8d20f93b699e655ab765bdf98880cbb9eb0c3954.jpg" } ] } ], "index": 9.5, "virtual_lines": [ { "bbox": [ 213, 224, 398, 241.0 ], "spans": [], "index": 8 }, { "bbox": [ 213, 241.0, 398, 258.0 ], "spans": [], "index": 9 }, { "bbox": [ 213, 258.0, 398, 275.0 ], "spans": [], "index": 10 }, { "bbox": [ 213, 275.0, 398, 292.0 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 71, 300, 185, 313 ], "lines": [ { "bbox": [ 70, 300, 186, 313 ], "spans": [ { "bbox": [ 70, 300, 159, 313 ], "score": 1.0, "content": "This is also true for", "type": "text" }, { "bbox": [ 160, 304, 182, 311 ], "score": 0.92, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 182, 300, 186, 313 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 70, 300, 186, 313 ] }, { "type": "text", "bbox": [ 70, 324, 541, 360 ], "lines": [ { "bbox": [ 69, 322, 541, 338 ], "spans": [ { "bbox": [ 69, 322, 339, 338 ], "score": 1.0, "content": "Proposition 3.4. The results of Proposition 3.3 also hold for", "type": "text" }, { "bbox": [ 339, 326, 357, 336 ], "score": 0.92, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 357, 322, 541, 338 ], "score": 1.0, "content": "in (17). Furthermore, assuming regularity", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 334, 542, 351 ], "spans": [ { "bbox": [ 69, 334, 128, 351 ], "score": 1.0, "content": "properties of", "type": "text" }, { "bbox": [ 129, 342, 137, 348 ], "score": 0.8, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 138, 334, 144, 351 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 144, 341, 153, 348 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 153, 334, 295, 351 ], "score": 1.0, "content": ", and the optimal transport plan", "type": "text" }, { "bbox": [ 295, 341, 302, 346 ], "score": 0.85, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 302, 334, 320, 351 ], "score": 1.0, "content": ", as", "type": "text" }, { "bbox": [ 320, 337, 352, 346 ], "score": 0.92, "content": "\\sigma ^ { 2 } \\to 0", "type": "inline_equation" }, { "bbox": [ 352, 334, 434, 351 ], "score": 1.0, "content": "the marginal path", "type": "text" }, { "bbox": [ 434, 341, 443, 348 ], "score": 0.88, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 443, 334, 487, 351 ], "score": 1.0, "content": "and field", "type": "text" }, { "bbox": [ 487, 342, 496, 347 ], "score": 0.87, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 496, 334, 542, 351 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 347, 406, 362 ], "spans": [ { "bbox": [ 69, 347, 110, 362 ], "score": 1.0, "content": "(7), i.e.,", "type": "text" }, { "bbox": [ 110, 353, 119, 359 ], "score": 0.89, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 120, 347, 360, 362 ], "score": 1.0, "content": "solves the dynamic optimal transport problem between", "type": "text" }, { "bbox": [ 360, 353, 369, 360 ], "score": 0.86, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 370, 347, 392, 362 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 392, 353, 401, 360 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 401, 347, 406, 362 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 69, 322, 542, 362 ] }, { "type": "text", "bbox": [ 70, 376, 540, 437 ], "lines": [ { "bbox": [ 70, 376, 542, 390 ], "spans": [ { "bbox": [ 70, 376, 391, 390 ], "score": 1.0, "content": "Proof of Proposition 3.4. We will assume certain regularity conditions on", "type": "text" }, { "bbox": [ 391, 382, 400, 389 ], "score": 0.86, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 401, 376, 406, 390 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 406, 382, 415, 389 ], "score": 0.87, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 416, 376, 440, 390 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 440, 382, 446, 387 ], "score": 0.89, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 447, 376, 542, 390 ], "score": 1.0, "content": "to allow reduction to", "type": "text" } ], "index": 16 }, { "bbox": [ 70, 389, 541, 402 ], "spans": [ { "bbox": [ 70, 389, 541, 402 ], "score": 1.0, "content": "known results. We leave it to future work to determine which of these conditions are necessary and which", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 401, 540, 413 ], "spans": [ { "bbox": [ 69, 401, 493, 413 ], "score": 1.0, "content": "are redundant with other conditions. However, because we are concerned with approximation of", "type": "text" }, { "bbox": [ 494, 403, 516, 413 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 517, 401, 540, 413 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 414, 541, 426 ], "spans": [ { "bbox": [ 69, 414, 541, 426 ], "score": 1.0, "content": "neural networks, which are typically smooth, results that relax the regularity assumptions may be vacuous in", "type": "text" } ], "index": 19 }, { "bbox": [ 68, 426, 110, 437 ], "spans": [ { "bbox": [ 68, 426, 110, 437 ], "score": 1.0, "content": "practice.", "type": "text" } ], "index": 20 } ], "index": 18, "bbox_fs": [ 68, 376, 542, 437 ] }, { "type": "text", "bbox": [ 70, 442, 541, 491 ], "lines": [ { "bbox": [ 69, 441, 542, 457 ], "spans": [ { "bbox": [ 69, 441, 207, 457 ], "score": 1.0, "content": "Preliminaries. We assume that", "type": "text" }, { "bbox": [ 207, 448, 216, 455 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 216, 441, 238, 457 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 239, 448, 247, 455 ], "score": 0.9, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 248, 441, 542, 457 ], "score": 1.0, "content": "are compactly supported and admit bounded densities with respect", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 454, 541, 467 ], "spans": [ { "bbox": [ 70, 454, 541, 467 ], "score": 1.0, "content": "to the Lebesgue measure. Then the conditions for Brenier’s theorem (Brenier, 1991) are satisfied. By Brenier’s", "type": "text" } ], "index": 22 }, { "bbox": [ 68, 465, 543, 480 ], "spans": [ { "bbox": [ 68, 465, 187, 480 ], "score": 1.0, "content": "theorem, the optimal joint", "type": "text" }, { "bbox": [ 187, 472, 193, 477 ], "score": 0.9, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 193, 465, 369, 480 ], "score": 1.0, "content": "is unique and is supported on the graph", "type": "text" }, { "bbox": [ 370, 469, 408, 479 ], "score": 0.93, "content": "( x , T ( x ) )", "type": "inline_equation" }, { "bbox": [ 408, 465, 483, 480 ], "score": 1.0, "content": "of a Monge map", "type": "text" }, { "bbox": [ 483, 468, 538, 477 ], "score": 0.92, "content": "T : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 538, 465, 543, 480 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 478, 424, 492 ], "spans": [ { "bbox": [ 69, 478, 90, 492 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 91, 480, 113, 491 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 113, 478, 424, 492 ], "score": 1.0, "content": "is equal to McCann’s interpolation (Peyré & Cuturi, 2019, Chapter 7)", "type": "text" } ], "index": 24 } ], "index": 22.5, "bbox_fs": [ 68, 441, 543, 492 ] }, { "type": "interline_equation", "bbox": [ 250, 502, 361, 515 ], "lines": [ { "bbox": [ 250, 502, 361, 515 ], "spans": [ { "bbox": [ 250, 502, 361, 515 ], "score": 0.92, "content": "p _ { t } = ( ( 1 - t ) \\mathrm { I d } + t T ) _ { \\# } p _ { 0 } .", "type": "interline_equation", "image_path": "2f68f1dda8545ee5860e1a0e0a143e044e713622fc68f4792707ae088291ea2f.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 250, 502, 361, 515 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 70, 524, 542, 585 ], "lines": [ { "bbox": [ 69, 525, 543, 538 ], "spans": [ { "bbox": [ 69, 525, 183, 538 ], "score": 1.0, "content": "In addition, we know that", "type": "text" }, { "bbox": [ 183, 527, 204, 537 ], "score": 0.94, "content": "T ( x )", "type": "inline_equation" }, { "bbox": [ 205, 525, 476, 538 ], "score": 1.0, "content": "can be parameterized as the gradient of a convex function, i.e.,", "type": "text" }, { "bbox": [ 476, 527, 538, 537 ], "score": 0.94, "content": "T ( x ) = \\nabla \\psi ( x )", "type": "inline_equation" }, { "bbox": [ 539, 525, 543, 538 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 536, 543, 551 ], "spans": [ { "bbox": [ 69, 536, 177, 551 ], "score": 1.0, "content": "This characterization of", "type": "text" }, { "bbox": [ 178, 540, 185, 547 ], "score": 0.9, "content": "T", "type": "inline_equation" }, { "bbox": [ 186, 536, 433, 551 ], "score": 1.0, "content": "implies that the conditional probability paths, given by", "type": "text" }, { "bbox": [ 434, 539, 538, 550 ], "score": 0.92, "content": "\\phi _ { t } ( x ) = x + t ( T ( x ) - x )", "type": "inline_equation" }, { "bbox": [ 538, 536, 543, 551 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 27 }, { "bbox": [ 68, 547, 540, 564 ], "spans": [ { "bbox": [ 68, 547, 151, 564 ], "score": 1.0, "content": "do not cross, i.e.,", "type": "text" }, { "bbox": [ 151, 551, 252, 561 ], "score": 0.91, "content": "p _ { t } ( x | x _ { 0 } , T ( x _ { 0 } ) ) = p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 252, 547, 284, 564 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 284, 551, 306, 561 ], "score": 0.93, "content": "( t , x )", "type": "inline_equation" }, { "bbox": [ 307, 547, 485, 564 ], "score": 1.0, "content": ".4 It is known that the probability path", "type": "text" }, { "bbox": [ 486, 551, 540, 562 ], "score": 0.93, "content": "p _ { t } = [ \\phi _ { t } ] _ { \\# } p _ { 0 }", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 69, 560, 542, 575 ], "spans": [ { "bbox": [ 69, 560, 241, 575 ], "score": 1.0, "content": "and its associated vector field, given by", "type": "text" }, { "bbox": [ 242, 563, 333, 573 ], "score": 0.93, "content": "u _ { t } ( \\phi _ { t } ( x ) ) = T ( x ) - x", "type": "inline_equation" }, { "bbox": [ 333, 560, 542, 575 ], "score": 1.0, "content": ", solve the optimal transport problem (Benamou", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 572, 220, 586 ], "spans": [ { "bbox": [ 69, 572, 220, 586 ], "score": 1.0, "content": "& Brenier, 2000, Proposition 1.1).", "type": "text" } ], "index": 30 } ], "index": 28, "bbox_fs": [ 68, 525, 543, 586 ] }, { "type": "text", "bbox": [ 70, 590, 541, 627 ], "lines": [ { "bbox": [ 69, 590, 542, 604 ], "spans": [ { "bbox": [ 69, 590, 240, 604 ], "score": 1.0, "content": "We assume that the induced marginals", "type": "text" }, { "bbox": [ 240, 596, 249, 603 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 249, 590, 542, 604 ], "score": 1.0, "content": "have bounded densities with respect to Lebesgue measure and that", "type": "text" } ], "index": 31 }, { "bbox": [ 71, 601, 542, 616 ], "spans": [ { "bbox": [ 71, 605, 79, 613 ], "score": 0.91, "content": "T", "type": "inline_equation" }, { "bbox": [ 79, 601, 237, 616 ], "score": 1.0, "content": "is almost everywhere continuous in", "type": "text" }, { "bbox": [ 237, 608, 243, 613 ], "score": 0.87, "content": "x", "type": "inline_equation" }, { "bbox": [ 244, 601, 369, 616 ], "score": 1.0, "content": ", which implies the same for", "type": "text" }, { "bbox": [ 369, 605, 379, 615 ], "score": 0.91, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 379, 601, 445, 616 ], "score": 1.0, "content": ". Injectivity of", "type": "text" }, { "bbox": [ 445, 605, 455, 615 ], "score": 0.92, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 455, 601, 542, 616 ], "score": 1.0, "content": "and noncrossing of", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 614, 358, 629 ], "spans": [ { "bbox": [ 69, 614, 132, 629 ], "score": 1.0, "content": "paths implies", "type": "text" }, { "bbox": [ 132, 617, 172, 627 ], "score": 0.95, "content": "\\boldsymbol u _ { t } ( \\phi _ { t } ( \\boldsymbol x ) )", "type": "inline_equation" }, { "bbox": [ 172, 614, 330, 629 ], "score": 1.0, "content": "is almost everywhere continuous in", "type": "text" }, { "bbox": [ 331, 617, 354, 627 ], "score": 0.95, "content": "\\phi _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 354, 614, 358, 629 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 33 } ], "index": 32, "bbox_fs": [ 69, 590, 542, 629 ] }, { "type": "text", "bbox": [ 69, 632, 540, 657 ], "lines": [ { "bbox": [ 69, 632, 542, 646 ], "spans": [ { "bbox": [ 69, 632, 256, 646 ], "score": 1.0, "content": "Formal statement of the result. Denote by", "type": "text" }, { "bbox": [ 256, 635, 280, 645 ], "score": 0.93, "content": "p _ { t } ^ { \\sigma } ( x )", "type": "inline_equation" }, { "bbox": [ 281, 632, 285, 646 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 286, 635, 318, 645 ], "score": 0.94, "content": "p _ { t } ^ { \\sigma } ( x | z )", "type": "inline_equation" }, { "bbox": [ 318, 632, 494, 646 ], "score": 1.0, "content": "the densities in Proposition 3.3, and by", "type": "text" }, { "bbox": [ 495, 635, 519, 645 ], "score": 0.94, "content": "p _ { t } ^ { \\sigma } ( x )", "type": "inline_equation" }, { "bbox": [ 519, 632, 542, 646 ], "score": 1.0, "content": ". We", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 643, 317, 658 ], "spans": [ { "bbox": [ 69, 643, 151, 658 ], "score": 1.0, "content": "will show that for", "type": "text" }, { "bbox": [ 151, 650, 160, 657 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 160, 643, 221, 658 ], "score": 1.0, "content": "-almost every", "type": "text" }, { "bbox": [ 221, 650, 227, 654 ], "score": 0.89, "content": "x", "type": "inline_equation" }, { "bbox": [ 228, 643, 308, 658 ], "score": 1.0, "content": "and almost every", "type": "text" }, { "bbox": [ 308, 648, 312, 655 ], "score": 0.86, "content": "t", "type": "inline_equation" }, { "bbox": [ 313, 643, 317, 658 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 69, 632, 542, 658 ] }, { "type": "interline_equation", "bbox": [ 201, 666, 411, 693 ], "lines": [ { "bbox": [ 201, 666, 411, 693 ], "spans": [ { "bbox": [ 201, 666, 411, 693 ], "score": 0.93, "content": "u _ { t } ( x ) = \\operatorname* { l i m } _ { \\sigma 0 } u _ { t } ( x ) = \\operatorname* { l i m } _ { \\sigma 0 } \\frac { \\mathbb { E } _ { z \\sim q ( z ) } p _ { t } ^ { \\sigma } ( x | z ) u _ { t } ( x | z ) } { p _ { t } ^ { \\sigma } ( x ) } .", "type": "interline_equation", "image_path": "e9ae1b9351803747d93eb8df5c16f6380352c5ea027ed22fda4f6ff51ed16c4c.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 201, 666, 411, 693 ], "spans": [], "index": 36 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 118 ], "lines": [ { "bbox": [ 70, 81, 541, 96 ], "spans": [ { "bbox": [ 70, 81, 254, 96 ], "score": 1.0, "content": "Proof. It suffices to show the equality for", "type": "text" }, { "bbox": [ 254, 87, 260, 92 ], "score": 0.89, "content": "x", "type": "inline_equation" }, { "bbox": [ 260, 81, 340, 96 ], "score": 1.0, "content": "in the support of", "type": "text" }, { "bbox": [ 340, 87, 349, 94 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 349, 81, 502, 96 ], "score": 1.0, "content": ", or in the image of the support of", "type": "text" }, { "bbox": [ 502, 87, 511, 94 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 511, 81, 541, 96 ], "score": 1.0, "content": "under", "type": "text" } ], "index": 0 }, { "bbox": [ 71, 93, 542, 108 ], "spans": [ { "bbox": [ 71, 97, 81, 106 ], "score": 0.9, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 81, 93, 141, 108 ], "score": 1.0, "content": ". We identify", "type": "text" }, { "bbox": [ 141, 99, 147, 104 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 147, 93, 284, 108 ], "score": 1.0, "content": "in the expectation with a pair", "type": "text" }, { "bbox": [ 285, 96, 332, 106 ], "score": 0.93, "content": "( x _ { 0 } , T ( x _ { 0 } ) )", "type": "inline_equation" }, { "bbox": [ 333, 93, 366, 108 ], "score": 1.0, "content": "due to", "type": "text" }, { "bbox": [ 367, 96, 456, 106 ], "score": 0.93, "content": "q ( x _ { 0 } , x _ { 1 } ) = \\pi ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 456, 93, 542, 108 ], "score": 1.0, "content": "having support on", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 105, 257, 119 ], "spans": [ { "bbox": [ 70, 105, 257, 119 ], "score": 1.0, "content": "the graph of the Monge map. Noting that", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "interline_equation", "bbox": [ 203, 126, 407, 145 ], "lines": [ { "bbox": [ 203, 126, 407, 145 ], "spans": [ { "bbox": [ 203, 126, 407, 145 ], "score": 0.91, "content": "u _ { t } \\Big ( x | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) = u _ { 0 } ( x _ { 0 } ) = T ( x _ { 0 } ) - x _ { 0 } \\quad \\forall x ,", "type": "interline_equation", "image_path": "0021616da43d2f3fb56af62b581bd4c6079c31ea7c8cb11ee2ef6274e91e9717.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 203, 126, 407, 145 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 71, 151, 212, 163 ], "lines": [ { "bbox": [ 70, 151, 212, 164 ], "spans": [ { "bbox": [ 70, 151, 212, 164 ], "score": 1.0, "content": "we see that (23) is equivalent to", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 196, 171, 415, 204 ], "lines": [ { "bbox": [ 196, 171, 415, 204 ], "spans": [ { "bbox": [ 196, 171, 415, 204 ], "score": 0.93, "content": "u _ { 0 } \\big ( \\phi _ { t } ^ { - 1 } ( x ) \\big ) = \\operatorname* { l i m } _ { \\sigma 0 } \\frac { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } \\Big ( x | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) u _ { 0 } ( x _ { 0 } ) } { p _ { t } ^ { \\sigma } ( x ) } .", "type": "interline_equation", "image_path": "d5006f9b4642faf6320ebdf94b5127eefa2700469f4ac1b9880560c7567afd55.jpg" } ] } ], "index": 5.5, "virtual_lines": [ { "bbox": [ 196, 171, 415, 187.5 ], "spans": [], "index": 5 }, { "bbox": [ 196, 187.5, 415, 204.0 ], "spans": [], "index": 6 } ] }, { "type": "text", "bbox": [ 70, 210, 541, 242 ], "lines": [ { "bbox": [ 70, 209, 541, 225 ], "spans": [ { "bbox": [ 70, 209, 390, 225 ], "score": 1.0, "content": "By the same argument as in the proof of Proposition 3.3, we have that", "type": "text" }, { "bbox": [ 390, 212, 471, 223 ], "score": 0.93, "content": "p _ { t } ^ { \\sigma } = p _ { t } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 471, 209, 541, 225 ], "score": 1.0, "content": ", by integration", "type": "text" } ], "index": 7 }, { "bbox": [ 68, 223, 469, 242 ], "spans": [ { "bbox": [ 68, 223, 92, 241 ], "score": 1.0, "content": "over", "type": "text" }, { "bbox": [ 92, 231, 103, 237 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 103, 223, 223, 241 ], "score": 1.0, "content": "of the conditional equality", "type": "text" }, { "bbox": [ 224, 224, 464, 242 ], "score": 0.89, "content": "p _ { t } \\Big ( \\cdot | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) = \\mathcal { N } ( \\phi _ { t } ( x _ { 0 } ) , \\sigma ^ { 2 } ) = \\delta _ { \\phi _ { t } ( x _ { 0 } ) } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 465, 223, 469, 241 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 8 } ], "index": 7.5 }, { "type": "text", "bbox": [ 71, 246, 244, 258 ], "lines": [ { "bbox": [ 69, 245, 245, 259 ], "spans": [ { "bbox": [ 69, 245, 245, 259 ], "score": 1.0, "content": "Symmetry of the Gaussian implies that", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "interline_equation", "bbox": [ 192, 266, 418, 279 ], "lines": [ { "bbox": [ 192, 266, 418, 279 ], "spans": [ { "bbox": [ 192, 266, 418, 279 ], "score": 0.84, "content": "p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) ) = p _ { t } ^ { \\sigma } ( \\phi _ { t } ( x _ { 0 } ) | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) .", "type": "interline_equation", "image_path": "b9d97a303aa04a476a017b48a560a4aa02c8ccb4a167f48c52e100378656a372.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 192, 266, 418, 279 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 70, 285, 114, 297 ], "lines": [ { "bbox": [ 69, 284, 115, 298 ], "spans": [ { "bbox": [ 69, 284, 115, 298 ], "score": 1.0, "content": "Therefore", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 118, 303, 494, 389 ], "lines": [ { "bbox": [ 118, 303, 494, 389 ], "spans": [ { "bbox": [ 118, 303, 494, 389 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) ) u _ { 0 } ( x _ { 0 } ) = \\mathbb { E } _ { q ( x _ { 0 } ) } \\left[ p _ { t } ^ { \\sigma } ( \\phi _ { t } ( x _ { 0 } ) | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) u _ { 0 } ( x _ { 0 } ) \\right] } \\\\ & { [ \\mathrm { b y ~ c h a n g e ~ o f ~ v a r i a b l e s ~ } x ^ { \\prime } = \\phi _ { t } ( x _ { 0 } ) ] = \\mathbb { E } _ { p _ { t } ( x ^ { \\prime } ) } \\left[ p _ { t } ^ { \\sigma } ( x ^ { \\prime } | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ^ { \\prime } ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { p _ { t } ^ { \\sigma } ( x ^ { \\prime } | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) } \\left[ p _ { t } ( x ^ { \\prime } ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ^ { \\prime } ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\Delta x \\sim N ( 0 , \\sigma ^ { 2 } ) } \\left[ p _ { t } ( x + \\Delta x ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x + \\Delta x ) ) \\right] } \\\\ & { \\qquad = \\left( p _ { t } ( \\cdot ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( \\cdot ) ) * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\right) ( x ) . } \\end{array}", "type": "interline_equation", "image_path": "ee725341be1424536a55d43bbfd30ce2a3a01c52f162fb5ae288cabd00af3bc1.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 118, 303, 494, 331.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 118, 331.6666666666667, 494, 360.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 118, 360.33333333333337, 494, 389.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 71, 394, 539, 433 ], "lines": [ { "bbox": [ 68, 390, 543, 411 ], "spans": [ { "bbox": [ 68, 390, 174, 411 ], "score": 1.0, "content": "The standard fact that", "type": "text" }, { "bbox": [ 175, 395, 250, 408 ], "score": 0.93, "content": "\\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\xrightarrow { \\sigma \\to 0 } \\delta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 251, 390, 385, 411 ], "score": 1.0, "content": "in distribution implies that if", "type": "text" }, { "bbox": [ 386, 398, 392, 408 ], "score": 0.89, "content": "f", "type": "inline_equation" }, { "bbox": [ 392, 390, 543, 411 ], "score": 1.0, "content": "is a bounded, almost everywhere", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 407, 542, 422 ], "spans": [ { "bbox": [ 69, 407, 288, 422 ], "score": 1.0, "content": "continuous, compactly supported function, then", "type": "text" }, { "bbox": [ 288, 410, 371, 420 ], "score": 0.93, "content": "( f * \\mathcal { N } ) ( x ) \\to f ( x )", "type": "inline_equation" }, { "bbox": [ 371, 407, 495, 422 ], "score": 1.0, "content": "pointwise for almost every", "type": "text" }, { "bbox": [ 495, 413, 501, 418 ], "score": 0.88, "content": "x", "type": "inline_equation" }, { "bbox": [ 501, 407, 542, 422 ], "score": 1.0, "content": ". By the", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 419, 498, 435 ], "spans": [ { "bbox": [ 69, 419, 124, 435 ], "score": 1.0, "content": "hypotheses,", "type": "text" }, { "bbox": [ 124, 425, 133, 432 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 133, 419, 155, 435 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 156, 421, 216, 432 ], "score": 0.94, "content": "p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } )", "type": "inline_equation" }, { "bbox": [ 216, 419, 414, 435 ], "score": 1.0, "content": "have this property. It follows that, for every", "type": "text" }, { "bbox": [ 415, 423, 418, 430 ], "score": 0.85, "content": "t", "type": "inline_equation" }, { "bbox": [ 419, 419, 486, 435 ], "score": 1.0, "content": "and almost all", "type": "text" }, { "bbox": [ 487, 425, 493, 430 ], "score": 0.86, "content": "x", "type": "inline_equation" }, { "bbox": [ 493, 419, 498, 435 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 17 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 145, 438, 465, 513 ], "lines": [ { "bbox": [ 145, 438, 465, 513 ], "spans": [ { "bbox": [ 145, 438, 465, 513 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\operatorname* { l i m } _ { \\sigma \\to 0 } \\frac { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) u _ { 0 } ( x _ { 0 } ) } { p _ { t } ^ { \\sigma } ( x ) } = \\operatorname* { l i m } _ { \\sigma \\to 0 } \\frac { \\big ( ( p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } ) ) * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\big ) ( x ) } { ( p _ { t } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) ) ( x ) } } } \\\\ & { } & { \\quad = \\frac { ( p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } ) ) ( x ) } { p _ { t } ( x ) } } \\\\ & { } & { \\quad = u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ) ) , } \\end{array}", "type": "interline_equation", "image_path": "1eaa0b1000279ea37156828e8c1407e86c40e2d4840d9c1642707d525f7050b4.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 145, 438, 465, 463.0 ], "spans": [], "index": 18 }, { "bbox": [ 145, 463.0, 465, 488.0 ], "spans": [], "index": 19 }, { "bbox": [ 145, 488.0, 465, 513.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 518, 151, 530 ], "lines": [ { "bbox": [ 69, 517, 153, 532 ], "spans": [ { "bbox": [ 69, 517, 153, 532 ], "score": 1.0, "content": "which proves (24).", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 72, 536, 541, 561 ], "lines": [ { "bbox": [ 68, 535, 543, 551 ], "spans": [ { "bbox": [ 68, 535, 273, 551 ], "score": 1.0, "content": "Proposition 3.5. The marginal vector field", "type": "text" }, { "bbox": [ 273, 539, 296, 549 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 297, 535, 543, 551 ], "score": 1.0, "content": "defined by (19) and (21) generates the same marginal", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 548, 340, 563 ], "spans": [ { "bbox": [ 69, 548, 207, 563 ], "score": 1.0, "content": "probability path as the solution", "type": "text" }, { "bbox": [ 208, 551, 218, 559 ], "score": 0.86, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 219, 548, 250, 563 ], "score": 1.0, "content": "to the", "type": "text" }, { "bbox": [ 251, 551, 264, 559 ], "score": 0.57, "content": "S B", "type": "inline_equation" }, { "bbox": [ 264, 548, 340, 563 ], "score": 1.0, "content": "problem in (18).", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 70, 573, 541, 598 ], "lines": [ { "bbox": [ 70, 573, 542, 587 ], "spans": [ { "bbox": [ 70, 573, 542, 587 ], "score": 1.0, "content": "Proof of Proposition 3.5. Using Theorem 2.4 of Léonard (2014a), De Bortoli et al. (2021) showed that the", "type": "text" } ], "index": 24 }, { "bbox": [ 70, 586, 414, 598 ], "spans": [ { "bbox": [ 70, 586, 216, 598 ], "score": 1.0, "content": "initial and terminal marginals of", "type": "text" }, { "bbox": [ 216, 588, 227, 595 ], "score": 0.91, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 227, 586, 414, 598 ], "score": 1.0, "content": "are the solution to the static OT problem", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 199, 606, 410, 618 ], "lines": [ { "bbox": [ 199, 606, 410, 618 ], "spans": [ { "bbox": [ 199, 606, 410, 618 ], "score": 0.86, "content": "\\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) = \\arg \\operatorname* { m i n } \\mathrm { K L } ( \\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) \\| p _ { \\mathrm { r e f } } ( x _ { 0 } , x _ { 1 } ) ) ,", "type": "interline_equation", "image_path": "3bebe2eb757084a7c3d5c19fa8c7e828ab24f50ed610f892e1beb63199f88a46.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 199, 606, 410, 618 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 624, 345, 638 ], "lines": [ { "bbox": [ 69, 623, 345, 639 ], "spans": [ { "bbox": [ 69, 623, 246, 639 ], "score": 1.0, "content": "while the conditional path distributions", "type": "text" }, { "bbox": [ 247, 627, 301, 637 ], "score": 0.94, "content": "\\pi ^ { * } ( - | x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 301, 623, 345, 639 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "interline_equation", "bbox": [ 200, 646, 409, 658 ], "lines": [ { "bbox": [ 200, 646, 409, 658 ], "spans": [ { "bbox": [ 200, 646, 409, 658 ], "score": 0.85, "content": "\\begin{array} { r } { \\mathbb { E } _ { x _ { 0 } , x _ { 1 } \\sim \\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) } \\mathrm { K L } ( \\pi ^ { * } ( - | x _ { 0 } , x _ { 1 } ) | | p _ { \\mathrm { r e f } } ( - | x _ { 0 } , x _ { 1 } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "a1454d206d8f30f597234fbd96440ec6087d9c95bb206c16d985fa22c086d952.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 200, 646, 409, 658 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 70, 670, 541, 733 ], "lines": [ { "bbox": [ 70, 670, 540, 683 ], "spans": [ { "bbox": [ 70, 670, 203, 683 ], "score": 1.0, "content": "The optimization problem for", "type": "text" }, { "bbox": [ 203, 673, 246, 683 ], "score": 0.95, "content": "\\pi ^ { * } ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 247, 670, 540, 683 ], "score": 1.0, "content": "is equivalent to the entropy-regularized optimal transport problem", "type": "text" } ], "index": 29 }, { "bbox": [ 68, 682, 544, 714 ], "spans": [ { "bbox": [ 68, 682, 136, 696 ], "score": 1.0, "content": "with optimum", "type": "text" }, { "bbox": [ 72, 696, 142, 710 ], "score": 0.91, "content": "{ \\frac { c ( x _ { 0 } , x _ { 1 } ) ^ { \\alpha } } { 2 \\sigma ^ { 2 } } } + \\mathrm { c o n s t } .", "type": "inline_equation" }, { "bbox": [ 136, 688, 155, 694 ], "score": 0.89, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 142, 690, 177, 714 ], "score": 1.0, "content": "., where", "type": "text" }, { "bbox": [ 155, 682, 466, 696 ], "score": 1.0, "content": ", as observed by De Bortoli et al. (2021). (The key observation is that", "type": "text" }, { "bbox": [ 177, 698, 248, 709 ], "score": 0.95, "content": "c ( x , y ) = \\| x - y \\|", "type": "inline_equation" }, { "bbox": [ 249, 690, 269, 714 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 270, 699, 295, 706 ], "score": 0.9, "content": "\\alpha = 2", "type": "inline_equation" }, { "bbox": [ 295, 690, 544, 714 ], "score": 1.0, "content": ".) The divergences between conditional path distributions", "type": "text" }, { "bbox": [ 466, 685, 541, 695 ], "score": 0.91, "content": "\\log p _ { \\mathrm { r e f } } ( x _ { 0 } , x _ { 1 } ) =", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 68, 707, 543, 722 ], "spans": [ { "bbox": [ 68, 707, 309, 722 ], "score": 1.0, "content": "are optimized by Brownian bridges with diffusion scale", "type": "text" }, { "bbox": [ 310, 713, 316, 718 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 317, 707, 363, 722 ], "score": 1.0, "content": "pinned at", "type": "text" }, { "bbox": [ 363, 713, 374, 720 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 374, 707, 396, 722 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 396, 713, 407, 720 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 407, 707, 543, 722 ], "score": 1.0, "content": ", which are well-known to have", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 720, 541, 734 ], "spans": [ { "bbox": [ 69, 720, 186, 734 ], "score": 1.0, "content": "marginal probability path", "type": "text" }, { "bbox": [ 187, 725, 195, 732 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 196, 720, 437, 734 ], "score": 1.0, "content": "in (20), and, by (5), are generated by the vector fields", "type": "text" }, { "bbox": [ 437, 725, 447, 732 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 447, 720, 484, 734 ], "score": 1.0, "content": "in (21).", "type": "text" }, { "bbox": [ 530, 720, 541, 732 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 32 } ], "index": 30.5 } ], "page_idx": 20, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 70, 26, 368, 38 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 530, 518, 541, 529 ], "lines": [ { "bbox": [ 532, 520, 540, 529 ], "spans": [ { "bbox": [ 532, 520, 540, 529 ], "score": 0.999, "content": "□", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 299, 751, 310, 761 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "21", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 70, 81, 541, 118 ], "lines": [ { "bbox": [ 70, 81, 541, 96 ], "spans": [ { "bbox": [ 70, 81, 254, 96 ], "score": 1.0, "content": "Proof. It suffices to show the equality for", "type": "text" }, { "bbox": [ 254, 87, 260, 92 ], "score": 0.89, "content": "x", "type": "inline_equation" }, { "bbox": [ 260, 81, 340, 96 ], "score": 1.0, "content": "in the support of", "type": "text" }, { "bbox": [ 340, 87, 349, 94 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 349, 81, 502, 96 ], "score": 1.0, "content": ", or in the image of the support of", "type": "text" }, { "bbox": [ 502, 87, 511, 94 ], "score": 0.91, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 511, 81, 541, 96 ], "score": 1.0, "content": "under", "type": "text" } ], "index": 0 }, { "bbox": [ 71, 93, 542, 108 ], "spans": [ { "bbox": [ 71, 97, 81, 106 ], "score": 0.9, "content": "\\phi _ { t }", "type": "inline_equation" }, { "bbox": [ 81, 93, 141, 108 ], "score": 1.0, "content": ". We identify", "type": "text" }, { "bbox": [ 141, 99, 147, 104 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 147, 93, 284, 108 ], "score": 1.0, "content": "in the expectation with a pair", "type": "text" }, { "bbox": [ 285, 96, 332, 106 ], "score": 0.93, "content": "( x _ { 0 } , T ( x _ { 0 } ) )", "type": "inline_equation" }, { "bbox": [ 333, 93, 366, 108 ], "score": 1.0, "content": "due to", "type": "text" }, { "bbox": [ 367, 96, 456, 106 ], "score": 0.93, "content": "q ( x _ { 0 } , x _ { 1 } ) = \\pi ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 456, 93, 542, 108 ], "score": 1.0, "content": "having support on", "type": "text" } ], "index": 1 }, { "bbox": [ 70, 105, 257, 119 ], "spans": [ { "bbox": [ 70, 105, 257, 119 ], "score": 1.0, "content": "the graph of the Monge map. Noting that", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 70, 81, 542, 119 ] }, { "type": "interline_equation", "bbox": [ 203, 126, 407, 145 ], "lines": [ { "bbox": [ 203, 126, 407, 145 ], "spans": [ { "bbox": [ 203, 126, 407, 145 ], "score": 0.91, "content": "u _ { t } \\Big ( x | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) = u _ { 0 } ( x _ { 0 } ) = T ( x _ { 0 } ) - x _ { 0 } \\quad \\forall x ,", "type": "interline_equation", "image_path": "0021616da43d2f3fb56af62b581bd4c6079c31ea7c8cb11ee2ef6274e91e9717.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 203, 126, 407, 145 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 71, 151, 212, 163 ], "lines": [ { "bbox": [ 70, 151, 212, 164 ], "spans": [ { "bbox": [ 70, 151, 212, 164 ], "score": 1.0, "content": "we see that (23) is equivalent to", "type": "text" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 70, 151, 212, 164 ] }, { "type": "interline_equation", "bbox": [ 196, 171, 415, 204 ], "lines": [ { "bbox": [ 196, 171, 415, 204 ], "spans": [ { "bbox": [ 196, 171, 415, 204 ], "score": 0.93, "content": "u _ { 0 } \\big ( \\phi _ { t } ^ { - 1 } ( x ) \\big ) = \\operatorname* { l i m } _ { \\sigma 0 } \\frac { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } \\Big ( x | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) u _ { 0 } ( x _ { 0 } ) } { p _ { t } ^ { \\sigma } ( x ) } .", "type": "interline_equation", "image_path": "d5006f9b4642faf6320ebdf94b5127eefa2700469f4ac1b9880560c7567afd55.jpg" } ] } ], "index": 5.5, "virtual_lines": [ { "bbox": [ 196, 171, 415, 187.5 ], "spans": [], "index": 5 }, { "bbox": [ 196, 187.5, 415, 204.0 ], "spans": [], "index": 6 } ] }, { "type": "text", "bbox": [ 70, 210, 541, 242 ], "lines": [ { "bbox": [ 70, 209, 541, 225 ], "spans": [ { "bbox": [ 70, 209, 390, 225 ], "score": 1.0, "content": "By the same argument as in the proof of Proposition 3.3, we have that", "type": "text" }, { "bbox": [ 390, 212, 471, 223 ], "score": 0.93, "content": "p _ { t } ^ { \\sigma } = p _ { t } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 471, 209, 541, 225 ], "score": 1.0, "content": ", by integration", "type": "text" } ], "index": 7 }, { "bbox": [ 68, 223, 469, 242 ], "spans": [ { "bbox": [ 68, 223, 92, 241 ], "score": 1.0, "content": "over", "type": "text" }, { "bbox": [ 92, 231, 103, 237 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 103, 223, 223, 241 ], "score": 1.0, "content": "of the conditional equality", "type": "text" }, { "bbox": [ 224, 224, 464, 242 ], "score": 0.89, "content": "p _ { t } \\Big ( \\cdot | \\big ( x _ { 0 } , T ( x _ { 0 } ) \\big ) \\Big ) = \\mathcal { N } ( \\phi _ { t } ( x _ { 0 } ) , \\sigma ^ { 2 } ) = \\delta _ { \\phi _ { t } ( x _ { 0 } ) } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 465, 223, 469, 241 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 8 } ], "index": 7.5, "bbox_fs": [ 68, 209, 541, 242 ] }, { "type": "text", "bbox": [ 71, 246, 244, 258 ], "lines": [ { "bbox": [ 69, 245, 245, 259 ], "spans": [ { "bbox": [ 69, 245, 245, 259 ], "score": 1.0, "content": "Symmetry of the Gaussian implies that", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 69, 245, 245, 259 ] }, { "type": "interline_equation", "bbox": [ 192, 266, 418, 279 ], "lines": [ { "bbox": [ 192, 266, 418, 279 ], "spans": [ { "bbox": [ 192, 266, 418, 279 ], "score": 0.84, "content": "p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) ) = p _ { t } ^ { \\sigma } ( \\phi _ { t } ( x _ { 0 } ) | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) .", "type": "interline_equation", "image_path": "b9d97a303aa04a476a017b48a560a4aa02c8ccb4a167f48c52e100378656a372.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 192, 266, 418, 279 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 70, 285, 114, 297 ], "lines": [ { "bbox": [ 69, 284, 115, 298 ], "spans": [ { "bbox": [ 69, 284, 115, 298 ], "score": 1.0, "content": "Therefore", "type": "text" } ], "index": 11 } ], "index": 11, "bbox_fs": [ 69, 284, 115, 298 ] }, { "type": "interline_equation", "bbox": [ 118, 303, 494, 389 ], "lines": [ { "bbox": [ 118, 303, 494, 389 ], "spans": [ { "bbox": [ 118, 303, 494, 389 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) ) u _ { 0 } ( x _ { 0 } ) = \\mathbb { E } _ { q ( x _ { 0 } ) } \\left[ p _ { t } ^ { \\sigma } ( \\phi _ { t } ( x _ { 0 } ) | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) u _ { 0 } ( x _ { 0 } ) \\right] } \\\\ & { [ \\mathrm { b y ~ c h a n g e ~ o f ~ v a r i a b l e s ~ } x ^ { \\prime } = \\phi _ { t } ( x _ { 0 } ) ] = \\mathbb { E } _ { p _ { t } ( x ^ { \\prime } ) } \\left[ p _ { t } ^ { \\sigma } ( x ^ { \\prime } | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ^ { \\prime } ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { p _ { t } ^ { \\sigma } ( x ^ { \\prime } | ( \\phi _ { t } ^ { - 1 } ( x ) , T ( \\phi _ { t } ^ { - 1 } ( x ) ) ) ) } \\left[ p _ { t } ( x ^ { \\prime } ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ^ { \\prime } ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\Delta x \\sim N ( 0 , \\sigma ^ { 2 } ) } \\left[ p _ { t } ( x + \\Delta x ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x + \\Delta x ) ) \\right] } \\\\ & { \\qquad = \\left( p _ { t } ( \\cdot ) u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( \\cdot ) ) * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\right) ( x ) . } \\end{array}", "type": "interline_equation", "image_path": "ee725341be1424536a55d43bbfd30ce2a3a01c52f162fb5ae288cabd00af3bc1.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 118, 303, 494, 331.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 118, 331.6666666666667, 494, 360.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 118, 360.33333333333337, 494, 389.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 71, 394, 539, 433 ], "lines": [ { "bbox": [ 68, 390, 543, 411 ], "spans": [ { "bbox": [ 68, 390, 174, 411 ], "score": 1.0, "content": "The standard fact that", "type": "text" }, { "bbox": [ 175, 395, 250, 408 ], "score": 0.93, "content": "\\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\xrightarrow { \\sigma \\to 0 } \\delta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 251, 390, 385, 411 ], "score": 1.0, "content": "in distribution implies that if", "type": "text" }, { "bbox": [ 386, 398, 392, 408 ], "score": 0.89, "content": "f", "type": "inline_equation" }, { "bbox": [ 392, 390, 543, 411 ], "score": 1.0, "content": "is a bounded, almost everywhere", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 407, 542, 422 ], "spans": [ { "bbox": [ 69, 407, 288, 422 ], "score": 1.0, "content": "continuous, compactly supported function, then", "type": "text" }, { "bbox": [ 288, 410, 371, 420 ], "score": 0.93, "content": "( f * \\mathcal { N } ) ( x ) \\to f ( x )", "type": "inline_equation" }, { "bbox": [ 371, 407, 495, 422 ], "score": 1.0, "content": "pointwise for almost every", "type": "text" }, { "bbox": [ 495, 413, 501, 418 ], "score": 0.88, "content": "x", "type": "inline_equation" }, { "bbox": [ 501, 407, 542, 422 ], "score": 1.0, "content": ". By the", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 419, 498, 435 ], "spans": [ { "bbox": [ 69, 419, 124, 435 ], "score": 1.0, "content": "hypotheses,", "type": "text" }, { "bbox": [ 124, 425, 133, 432 ], "score": 0.89, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 133, 419, 155, 435 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 156, 421, 216, 432 ], "score": 0.94, "content": "p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } )", "type": "inline_equation" }, { "bbox": [ 216, 419, 414, 435 ], "score": 1.0, "content": "have this property. It follows that, for every", "type": "text" }, { "bbox": [ 415, 423, 418, 430 ], "score": 0.85, "content": "t", "type": "inline_equation" }, { "bbox": [ 419, 419, 486, 435 ], "score": 1.0, "content": "and almost all", "type": "text" }, { "bbox": [ 487, 425, 493, 430 ], "score": 0.86, "content": "x", "type": "inline_equation" }, { "bbox": [ 493, 419, 498, 435 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 17 } ], "index": 16, "bbox_fs": [ 68, 390, 543, 435 ] }, { "type": "interline_equation", "bbox": [ 145, 438, 465, 513 ], "lines": [ { "bbox": [ 145, 438, 465, 513 ], "spans": [ { "bbox": [ 145, 438, 465, 513 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\operatorname* { l i m } _ { \\sigma \\to 0 } \\frac { \\mathbb { E } _ { q ( x _ { 0 } ) } p _ { t } ^ { \\sigma } ( x | ( x _ { 0 } , T ( x _ { 0 } ) ) u _ { 0 } ( x _ { 0 } ) } { p _ { t } ^ { \\sigma } ( x ) } = \\operatorname* { l i m } _ { \\sigma \\to 0 } \\frac { \\big ( ( p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } ) ) * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) \\big ) ( x ) } { ( p _ { t } * \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) ) ( x ) } } } \\\\ & { } & { \\quad = \\frac { ( p _ { t } \\cdot ( u _ { 0 } \\circ \\phi _ { t } ^ { - 1 } ) ) ( x ) } { p _ { t } ( x ) } } \\\\ & { } & { \\quad = u _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ) ) , } \\end{array}", "type": "interline_equation", "image_path": "1eaa0b1000279ea37156828e8c1407e86c40e2d4840d9c1642707d525f7050b4.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 145, 438, 465, 463.0 ], "spans": [], "index": 18 }, { "bbox": [ 145, 463.0, 465, 488.0 ], "spans": [], "index": 19 }, { "bbox": [ 145, 488.0, 465, 513.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 70, 518, 151, 530 ], "lines": [ { "bbox": [ 69, 517, 153, 532 ], "spans": [ { "bbox": [ 69, 517, 153, 532 ], "score": 1.0, "content": "which proves (24).", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 69, 517, 153, 532 ] }, { "type": "text", "bbox": [ 72, 536, 541, 561 ], "lines": [ { "bbox": [ 68, 535, 543, 551 ], "spans": [ { "bbox": [ 68, 535, 273, 551 ], "score": 1.0, "content": "Proposition 3.5. The marginal vector field", "type": "text" }, { "bbox": [ 273, 539, 296, 549 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 297, 535, 543, 551 ], "score": 1.0, "content": "defined by (19) and (21) generates the same marginal", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 548, 340, 563 ], "spans": [ { "bbox": [ 69, 548, 207, 563 ], "score": 1.0, "content": "probability path as the solution", "type": "text" }, { "bbox": [ 208, 551, 218, 559 ], "score": 0.86, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 219, 548, 250, 563 ], "score": 1.0, "content": "to the", "type": "text" }, { "bbox": [ 251, 551, 264, 559 ], "score": 0.57, "content": "S B", "type": "inline_equation" }, { "bbox": [ 264, 548, 340, 563 ], "score": 1.0, "content": "problem in (18).", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 68, 535, 543, 563 ] }, { "type": "text", "bbox": [ 70, 573, 541, 598 ], "lines": [ { "bbox": [ 70, 573, 542, 587 ], "spans": [ { "bbox": [ 70, 573, 542, 587 ], "score": 1.0, "content": "Proof of Proposition 3.5. Using Theorem 2.4 of Léonard (2014a), De Bortoli et al. (2021) showed that the", "type": "text" } ], "index": 24 }, { "bbox": [ 70, 586, 414, 598 ], "spans": [ { "bbox": [ 70, 586, 216, 598 ], "score": 1.0, "content": "initial and terminal marginals of", "type": "text" }, { "bbox": [ 216, 588, 227, 595 ], "score": 0.91, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 227, 586, 414, 598 ], "score": 1.0, "content": "are the solution to the static OT problem", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 70, 573, 542, 598 ] }, { "type": "interline_equation", "bbox": [ 199, 606, 410, 618 ], "lines": [ { "bbox": [ 199, 606, 410, 618 ], "spans": [ { "bbox": [ 199, 606, 410, 618 ], "score": 0.86, "content": "\\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) = \\arg \\operatorname* { m i n } \\mathrm { K L } ( \\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) \\| p _ { \\mathrm { r e f } } ( x _ { 0 } , x _ { 1 } ) ) ,", "type": "interline_equation", "image_path": "3bebe2eb757084a7c3d5c19fa8c7e828ab24f50ed610f892e1beb63199f88a46.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 199, 606, 410, 618 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 624, 345, 638 ], "lines": [ { "bbox": [ 69, 623, 345, 639 ], "spans": [ { "bbox": [ 69, 623, 246, 639 ], "score": 1.0, "content": "while the conditional path distributions", "type": "text" }, { "bbox": [ 247, 627, 301, 637 ], "score": 0.94, "content": "\\pi ^ { * } ( - | x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 301, 623, 345, 639 ], "score": 1.0, "content": "minimize", "type": "text" } ], "index": 27 } ], "index": 27, "bbox_fs": [ 69, 623, 345, 639 ] }, { "type": "interline_equation", "bbox": [ 200, 646, 409, 658 ], "lines": [ { "bbox": [ 200, 646, 409, 658 ], "spans": [ { "bbox": [ 200, 646, 409, 658 ], "score": 0.85, "content": "\\begin{array} { r } { \\mathbb { E } _ { x _ { 0 } , x _ { 1 } \\sim \\pi ^ { * } ( x _ { 0 } , x _ { 1 } ) } \\mathrm { K L } ( \\pi ^ { * } ( - | x _ { 0 } , x _ { 1 } ) | | p _ { \\mathrm { r e f } } ( - | x _ { 0 } , x _ { 1 } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "a1454d206d8f30f597234fbd96440ec6087d9c95bb206c16d985fa22c086d952.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 200, 646, 409, 658 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 70, 670, 541, 733 ], "lines": [ { "bbox": [ 70, 670, 540, 683 ], "spans": [ { "bbox": [ 70, 670, 203, 683 ], "score": 1.0, "content": "The optimization problem for", "type": "text" }, { "bbox": [ 203, 673, 246, 683 ], "score": 0.95, "content": "\\pi ^ { * } ( x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 247, 670, 540, 683 ], "score": 1.0, "content": "is equivalent to the entropy-regularized optimal transport problem", "type": "text" } ], "index": 29 }, { "bbox": [ 68, 682, 544, 714 ], "spans": [ { "bbox": [ 68, 682, 136, 696 ], "score": 1.0, "content": "with optimum", "type": "text" }, { "bbox": [ 72, 696, 142, 710 ], "score": 0.91, "content": "{ \\frac { c ( x _ { 0 } , x _ { 1 } ) ^ { \\alpha } } { 2 \\sigma ^ { 2 } } } + \\mathrm { c o n s t } .", "type": "inline_equation" }, { "bbox": [ 136, 688, 155, 694 ], "score": 0.89, "content": "\\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 142, 690, 177, 714 ], "score": 1.0, "content": "., where", "type": "text" }, { "bbox": [ 155, 682, 466, 696 ], "score": 1.0, "content": ", as observed by De Bortoli et al. (2021). (The key observation is that", "type": "text" }, { "bbox": [ 177, 698, 248, 709 ], "score": 0.95, "content": "c ( x , y ) = \\| x - y \\|", "type": "inline_equation" }, { "bbox": [ 249, 690, 269, 714 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 270, 699, 295, 706 ], "score": 0.9, "content": "\\alpha = 2", "type": "inline_equation" }, { "bbox": [ 295, 690, 544, 714 ], "score": 1.0, "content": ".) The divergences between conditional path distributions", "type": "text" }, { "bbox": [ 466, 685, 541, 695 ], "score": 0.91, "content": "\\log p _ { \\mathrm { r e f } } ( x _ { 0 } , x _ { 1 } ) =", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 68, 707, 543, 722 ], "spans": [ { "bbox": [ 68, 707, 309, 722 ], "score": 1.0, "content": "are optimized by Brownian bridges with diffusion scale", "type": "text" }, { "bbox": [ 310, 713, 316, 718 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 317, 707, 363, 722 ], "score": 1.0, "content": "pinned at", "type": "text" }, { "bbox": [ 363, 713, 374, 720 ], "score": 0.9, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 374, 707, 396, 722 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 396, 713, 407, 720 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 407, 707, 543, 722 ], "score": 1.0, "content": ", which are well-known to have", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 720, 541, 734 ], "spans": [ { "bbox": [ 69, 720, 186, 734 ], "score": 1.0, "content": "marginal probability path", "type": "text" }, { "bbox": [ 187, 725, 195, 732 ], "score": 0.9, "content": "p _ { t }", "type": "inline_equation" }, { "bbox": [ 196, 720, 437, 734 ], "score": 1.0, "content": "in (20), and, by (5), are generated by the vector fields", "type": "text" }, { "bbox": [ 437, 725, 447, 732 ], "score": 0.9, "content": "u _ { t }", "type": "inline_equation" }, { "bbox": [ 447, 720, 484, 734 ], "score": 1.0, "content": "in (21).", "type": "text" }, { "bbox": [ 530, 720, 541, 732 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 32 } ], "index": 30.5, "bbox_fs": [ 68, 670, 544, 734 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 71, 80, 252, 95 ], "lines": [ { "bbox": [ 69, 79, 252, 96 ], "spans": [ { "bbox": [ 69, 79, 252, 96 ], "score": 1.0, "content": "B Additional theoretical results", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 105, 542, 131 ], "lines": [ { "bbox": [ 68, 104, 542, 120 ], "spans": [ { "bbox": [ 68, 104, 199, 120 ], "score": 1.0, "content": "Proposition B.1. For any", "type": "text" }, { "bbox": [ 199, 109, 232, 118 ], "score": 0.93, "content": "\\sigma \\in \\mathbb { R } _ { + }", "type": "inline_equation" }, { "bbox": [ 232, 104, 542, 120 ], "score": 1.0, "content": "conditional flow matching with conditional probability paths given by", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 118, 485, 131 ], "spans": [ { "bbox": [ 69, 118, 280, 131 ], "score": 1.0, "content": "(16) has an equivalent marginal probability flow", "type": "text" }, { "bbox": [ 281, 120, 303, 130 ], "score": 0.93, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 303, 118, 485, 131 ], "score": 1.0, "content": "to Lipman et al. (2023)’s flow matching.", "type": "text" } ], "index": 2 } ], "index": 1.5 }, { "type": "text", "bbox": [ 70, 141, 541, 226 ], "lines": [ { "bbox": [ 70, 141, 541, 155 ], "spans": [ { "bbox": [ 70, 141, 541, 155 ], "score": 1.0, "content": "Proof. To prove the proposition, we use the fact that the Gaussian family can be generated by location-scale", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 153, 542, 168 ], "spans": [ { "bbox": [ 69, 153, 231, 168 ], "score": 1.0, "content": "transformations (see, e.g., Lehmann", "type": "text" }, { "bbox": [ 232, 156, 240, 164 ], "score": 0.29, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 240, 153, 459, 168 ], "score": 1.0, "content": "Casella, 2006), i.e., we can express any Gaussian", "type": "text" }, { "bbox": [ 460, 156, 527, 167 ], "score": 0.93, "content": "Z _ { 0 } \\sim \\mathcal { N } ( \\mu _ { 0 } , \\sigma _ { 0 } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 527, 153, 542, 168 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 4 }, { "bbox": [ 71, 165, 542, 180 ], "spans": [ { "bbox": [ 71, 169, 137, 178 ], "score": 0.93, "content": "Z _ { 0 } = \\mu _ { 0 } + \\sigma _ { 0 } Z", "type": "inline_equation" }, { "bbox": [ 137, 165, 169, 180 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 169, 168, 224, 178 ], "score": 0.94, "content": "Z \\sim { \\mathcal { N } } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 224, 165, 336, 180 ], "score": 1.0, "content": ". Recall that the density", "type": "text" }, { "bbox": [ 336, 168, 359, 178 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 359, 165, 421, 180 ], "score": 1.0, "content": "has the form", "type": "text" }, { "bbox": [ 421, 168, 524, 179 ], "score": 0.91, "content": "\\begin{array} { r } { p _ { t } ( x ) = \\int p _ { t } ( x | z ) q ( z ) d z } \\end{array}", "type": "inline_equation" }, { "bbox": [ 524, 165, 542, 180 ], "score": 1.0, "content": ", to", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 177, 541, 191 ], "spans": [ { "bbox": [ 69, 177, 541, 191 ], "score": 1.0, "content": "show the equivalence between the flow from FM and source conditional flow matching, we have to show", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 190, 542, 203 ], "spans": [ { "bbox": [ 69, 190, 92, 203 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 93, 192, 128, 203 ], "score": 0.94, "content": "p _ { t } ( x | x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 129, 190, 451, 203 ], "score": 1.0, "content": "is the same for both methods, that is we show that CFM with variance", "type": "text" }, { "bbox": [ 451, 191, 529, 203 ], "score": 0.94, "content": "( \\sigma t ) ^ { 2 } + 2 \\sigma t ( 1 - t )", "type": "inline_equation" }, { "bbox": [ 529, 190, 542, 203 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 200, 540, 216 ], "spans": [ { "bbox": [ 69, 200, 212, 216 ], "score": 1.0, "content": "equivalent to FM with variance", "type": "text" }, { "bbox": [ 212, 204, 267, 214 ], "score": 0.94, "content": "( t \\sigma - t + 1 ) ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 267, 200, 321, 216 ], "score": 1.0, "content": "(12). Since", "type": "text" }, { "bbox": [ 322, 204, 371, 214 ], "score": 0.94, "content": "p _ { t } ( x | x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 372, 200, 384, 216 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 385, 204, 540, 214 ], "score": 0.91, "content": "\\mathcal { N } ( t x _ { 1 } + ( 1 - t ) x _ { 0 } ) , \\sigma t ( \\sigma t - 2 t + 2 ) )", "type": "inline_equation" } ], "index": 8 }, { "bbox": [ 69, 214, 277, 227 ], "spans": [ { "bbox": [ 69, 214, 220, 227 ], "score": 1.0, "content": "we can write the random variable", "type": "text" }, { "bbox": [ 220, 216, 262, 226 ], "score": 0.94, "content": "X | X _ { 0 } , X _ { 1 }", "type": "inline_equation" }, { "bbox": [ 262, 214, 277, 227 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 9 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 192, 231, 418, 244 ], "lines": [ { "bbox": [ 192, 231, 418, 244 ], "spans": [ { "bbox": [ 192, 231, 418, 244 ], "score": 0.87, "content": "X | X _ { 0 } , X _ { 1 } = t x _ { 1 } + ( 1 - t ) x _ { 0 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) ^ { 1 / 2 } Z ,", "type": "interline_equation", "image_path": "9b71e968a839ed7b0f614bbcbf048808c2bea4afed728f05ba3ac6707e5d72b6.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 192, 231, 418, 244 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 69, 248, 326, 261 ], "lines": [ { "bbox": [ 69, 246, 327, 263 ], "spans": [ { "bbox": [ 69, 246, 99, 263 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 251, 153, 261 ], "score": 0.95, "content": "Z \\sim { \\mathcal { N } } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 154, 246, 271, 263 ], "score": 1.0, "content": ". Without conditioning on", "type": "text" }, { "bbox": [ 271, 252, 284, 261 ], "score": 0.92, "content": "X _ { 0 }", "type": "inline_equation" }, { "bbox": [ 285, 246, 327, 263 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 199, 266, 411, 279 ], "lines": [ { "bbox": [ 199, 266, 411, 279 ], "spans": [ { "bbox": [ 199, 266, 411, 279 ], "score": 0.87, "content": "X | X _ { 1 } = t x _ { 1 } + ( 1 - t ) X _ { 0 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) ^ { 1 / 2 } Z .", "type": "interline_equation", "image_path": "0b1c279ec78db4d0f4e2da1a5d5b41db67fb357ac5106c924198fe4f8d3e6384.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 199, 266, 411, 279 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 70, 283, 541, 320 ], "lines": [ { "bbox": [ 70, 284, 541, 297 ], "spans": [ { "bbox": [ 70, 284, 138, 297 ], "score": 1.0, "content": "By assumption", "type": "text" }, { "bbox": [ 138, 286, 197, 296 ], "score": 0.95, "content": "X _ { 0 } \\sim { \\mathcal { N } } ( 0 , I ) ", "type": "inline_equation" }, { "bbox": [ 197, 284, 224, 297 ], "score": 1.0, "content": ", thus", "type": "text" }, { "bbox": [ 224, 286, 249, 296 ], "score": 0.94, "content": "X | X _ { 1 }", "type": "inline_equation" }, { "bbox": [ 249, 284, 541, 297 ], "score": 1.0, "content": "is Gaussian, since a linear transformation of Gaussian distributions", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 295, 541, 309 ], "spans": [ { "bbox": [ 69, 295, 541, 309 ], "score": 1.0, "content": "is also Gaussian. To define its distribution, we only have to define its expectation and variance. By linearity", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 307, 435, 321 ], "spans": [ { "bbox": [ 69, 307, 173, 321 ], "score": 1.0, "content": "of expectation, we find", "type": "text" }, { "bbox": [ 174, 309, 241, 320 ], "score": 0.94, "content": "{ \\bf E } ( X \\vert X _ { 1 } ) = t x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 241, 307, 352, 321 ], "score": 1.0, "content": ", and by independence of", "type": "text" }, { "bbox": [ 353, 310, 366, 319 ], "score": 0.92, "content": "X _ { 0 }", "type": "inline_equation" }, { "bbox": [ 366, 307, 388, 321 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 388, 310, 396, 317 ], "score": 0.9, "content": "Z", "type": "inline_equation" }, { "bbox": [ 396, 307, 435, 321 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 180, 323, 429, 357 ], "lines": [ { "bbox": [ 180, 323, 429, 357 ], "spans": [ { "bbox": [ 180, 323, 429, 357 ], "score": 0.91, "content": "\\begin{array} { r } { \\mathrm { V a r } ( X | X _ { 1 } ) = ( 1 - t ) ^ { 2 } \\mathrm { V a r } ( X _ { 0 } ) + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) \\mathrm { V a r } ( Z ) } \\\\ { = ( 1 - t ) ^ { 2 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) = ( t \\sigma - t + 1 ) ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "e330016069293841f54927039e275ef02cef9daae5f7a8d758a0607509dfef6e.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 180, 323, 429, 334.3333333333333 ], "spans": [], "index": 16 }, { "bbox": [ 180, 334.3333333333333, 429, 345.66666666666663 ], "spans": [], "index": 17 }, { "bbox": [ 180, 345.66666666666663, 429, 356.99999999999994 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 70, 358, 390, 371 ], "lines": [ { "bbox": [ 69, 358, 391, 372 ], "spans": [ { "bbox": [ 69, 358, 391, 372 ], "score": 1.0, "content": "hence the flow from source conditional flow matching is the same as FM.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 70, 376, 529, 389 ], "lines": [ { "bbox": [ 68, 375, 529, 392 ], "spans": [ { "bbox": [ 68, 375, 168, 392 ], "score": 1.0, "content": "Proposition B.2. If", "type": "text" }, { "bbox": [ 169, 382, 175, 387 ], "score": 0.77, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 176, 375, 475, 392 ], "score": 1.0, "content": "is a Monge map, the objective variance of OT-CFM goes to zero as", "type": "text" }, { "bbox": [ 475, 380, 502, 387 ], "score": 0.9, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 502, 375, 529, 392 ], "score": 1.0, "content": ", i.e.,", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 221, 394, 390, 408 ], "lines": [ { "bbox": [ 221, 394, 390, 408 ], "spans": [ { "bbox": [ 221, 394, 390, 408 ], "score": 0.89, "content": "\\mathbb { E } _ { q ( z ) } \\| u _ { t } ( x | z ) - u _ { t } ( x ) \\| ^ { 2 } \\to 0 \\mathrm { ~ } a s \\sigma \\to 0", "type": "interline_equation", "image_path": "56d15a5f3047959efe1a64f7a4949451bc42e1384bcb034145208eee5169639d.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 221, 394, 390, 408 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 70, 411, 155, 425 ], "lines": [ { "bbox": [ 69, 411, 156, 426 ], "spans": [ { "bbox": [ 69, 411, 87, 426 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 87, 414, 118, 424 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 118, 411, 156, 426 ], "score": 1.0, "content": "in (15).", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 436, 542, 472 ], "lines": [ { "bbox": [ 70, 435, 542, 450 ], "spans": [ { "bbox": [ 70, 435, 380, 450 ], "score": 1.0, "content": "Proof. This follows from a basic fact about the transport plan", "type": "text" }, { "bbox": [ 380, 442, 386, 446 ], "score": 0.86, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 387, 435, 501, 450 ], "score": 1.0, "content": ". Specifically, that as", "type": "text" }, { "bbox": [ 501, 439, 538, 447 ], "score": 0.87, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 538, 435, 542, 450 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 71, 447, 542, 462 ], "spans": [ { "bbox": [ 71, 450, 202, 461 ], "score": 0.92, "content": "D _ { \\mathrm { K L } } ( p _ { t } ( x | z ^ { i } ) \\| p _ { t } ( x | z ^ { j } ) ) \\ \\to \\ \\infty", "type": "inline_equation" }, { "bbox": [ 202, 447, 239, 462 ], "score": 1.0, "content": "for an", "type": "text" }, { "bbox": [ 239, 452, 253, 460 ], "score": 0.91, "content": "t , x", "type": "inline_equation" }, { "bbox": [ 253, 447, 333, 462 ], "score": 1.0, "content": "for two distinct", "type": "text" }, { "bbox": [ 333, 450, 360, 460 ], "score": 0.43, "content": "z ^ { i } , \\ z ^ { j }", "type": "inline_equation" }, { "bbox": [ 360, 447, 451, 462 ], "score": 1.0, "content": ". This means that", "type": "text" }, { "bbox": [ 452, 450, 523, 461 ], "score": 0.93, "content": "p _ { t } ( x | z ) \\ = \\ p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 523, 447, 542, 462 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 460, 158, 474 ], "spans": [ { "bbox": [ 69, 460, 90, 474 ], "score": 1.0, "content": "any", "type": "text" }, { "bbox": [ 90, 464, 113, 472 ], "score": 0.92, "content": "t , x , z", "type": "inline_equation" }, { "bbox": [ 114, 460, 158, 474 ], "score": 1.0, "content": "therefore", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 232, 477, 379, 507 ], "lines": [ { "bbox": [ 232, 477, 379, 507 ], "spans": [ { "bbox": [ 232, 477, 379, 507 ], "score": 0.92, "content": "\\begin{array} { c } { { u _ { t } ( x ) = \\mathbb { E } _ { q ( z ) } u _ { t } ( x | z ) p _ { t } ( x | z ) / p _ { t } ( x ) } } \\\\ { { = u _ { t } ( x | z ) } } \\end{array}", "type": "interline_equation", "image_path": "5141741fab4dc50e83ce082f541d61df41b8ff42f943e7e85fa4110351ced3d2.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 232, 477, 379, 507 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 528, 543, 553 ], "lines": [ { "bbox": [ 69, 527, 543, 543 ], "spans": [ { "bbox": [ 69, 527, 285, 543 ], "score": 1.0, "content": "Proposition B.3. The conditional vector field", "type": "text" }, { "bbox": [ 285, 531, 317, 541 ], "score": 0.94, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 317, 527, 543, 543 ], "score": 1.0, "content": "defined by (26) converges to marginal vector field", "type": "text" } ], "index": 27 }, { "bbox": [ 71, 540, 309, 555 ], "spans": [ { "bbox": [ 71, 542, 94, 553 ], "score": 0.92, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 540, 309, 555 ], "score": 1.0, "content": "defined by (9) as m goes to population size, i.e.,", "type": "text" } ], "index": 28 } ], "index": 27.5 }, { "type": "interline_equation", "bbox": [ 254, 558, 357, 570 ], "lines": [ { "bbox": [ 254, 558, 357, 570 ], "spans": [ { "bbox": [ 254, 558, 357, 570 ], "score": 0.9, "content": "\\| u _ { t } ( x | \\bar { z } ) - u _ { t } ( x ) \\| ^ { 2 } \\to 0", "type": "interline_equation", "image_path": "7d367087f15c5c249661695ee30a8fa75ec64c5e28abf6dd872a565e1202ae99.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 254, 558, 357, 570 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 70, 575, 125, 588 ], "lines": [ { "bbox": [ 69, 574, 127, 589 ], "spans": [ { "bbox": [ 69, 574, 83, 589 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 83, 577, 122, 588 ], "score": 0.93, "content": "m | \\mathcal { X } |", "type": "inline_equation" }, { "bbox": [ 122, 574, 127, 589 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 71, 599, 223, 613 ], "lines": [ { "bbox": [ 70, 598, 223, 614 ], "spans": [ { "bbox": [ 70, 598, 117, 614 ], "score": 1.0, "content": "Proof. As", "type": "text" }, { "bbox": [ 117, 602, 158, 612 ], "score": 0.93, "content": "| z | \\to | \\mathcal { X } |", "type": "inline_equation" }, { "bbox": [ 158, 598, 223, 614 ], "score": 1.0, "content": ", by definition,", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "interline_equation", "bbox": [ 227, 616, 383, 717 ], "lines": [ { "bbox": [ 227, 616, 383, 717 ], "spans": [ { "bbox": [ 227, 616, 383, 717 ], "score": 0.94, "content": "\\begin{array} { l } { { u _ { t } ( x | \\bar { z } ) = \\displaystyle \\frac { \\sum _ { i } ^ { m } u _ { t } ( x | z ^ { i } ) p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { \\sum _ { i } ^ { m } p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } } } \\\\ { { \\ \\ \\qquad = \\displaystyle \\frac { \\sum _ { z \\in \\mathcal { X } } u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) } { \\sum _ { z \\in \\mathcal { X } } p _ { t } ( x | z ) q ( z ) } } } \\\\ { { \\ \\qquad = \\displaystyle \\mathbb { E } _ { q ( z ) } \\frac { u _ { t } ( x | z ) p _ { t } ( x | z ) } { p _ { t } ( x ) } } } \\\\ { { \\ \\qquad = u _ { t } ( x ) } } \\end{array}", "type": "interline_equation", "image_path": "ff2120fa6963a49eb7488fafb6ed711ab0957662477c7ea997490422c72b07d3.jpg" } ] } ], "index": 34.5, "virtual_lines": [ { "bbox": [ 227, 616, 383, 632.8333333333334 ], "spans": [], "index": 32 }, { "bbox": [ 227, 632.8333333333334, 383, 649.6666666666667 ], "spans": [], "index": 33 }, { "bbox": [ 227, 649.6666666666667, 383, 666.5000000000001 ], "spans": [], "index": 34 }, { "bbox": [ 227, 666.5000000000001, 383, 683.3333333333335 ], "spans": [], "index": 35 }, { "bbox": [ 227, 683.3333333333335, 383, 700.1666666666669 ], "spans": [], "index": 36 }, { "bbox": [ 227, 700.1666666666669, 383, 717.0000000000002 ], "spans": [], "index": 37 } ] } ], "page_idx": 21, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 26, 368, 38 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 530, 359, 541, 370 ], "lines": [ { "bbox": [ 531, 360, 541, 371 ], "spans": [ { "bbox": [ 531, 360, 541, 371 ], "score": 0.998, "content": "□", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 530, 510, 541, 521 ], "lines": [ { "bbox": [ 532, 512, 541, 521 ], "spans": [ { "bbox": [ 532, 512, 541, 521 ], "score": 0.999, "content": "□", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 530, 720, 541, 731 ], "lines": [ { "bbox": [ 531, 722, 541, 732 ], "spans": [ { "bbox": [ 531, 722, 541, 732 ], "score": 0.998, "content": "□", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 299, 751, 311, 761 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "22", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 71, 80, 252, 95 ], "lines": [ { "bbox": [ 69, 79, 252, 96 ], "spans": [ { "bbox": [ 69, 79, 252, 96 ], "score": 1.0, "content": "B Additional theoretical results", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 105, 542, 131 ], "lines": [ { "bbox": [ 68, 104, 542, 120 ], "spans": [ { "bbox": [ 68, 104, 199, 120 ], "score": 1.0, "content": "Proposition B.1. For any", "type": "text" }, { "bbox": [ 199, 109, 232, 118 ], "score": 0.93, "content": "\\sigma \\in \\mathbb { R } _ { + }", "type": "inline_equation" }, { "bbox": [ 232, 104, 542, 120 ], "score": 1.0, "content": "conditional flow matching with conditional probability paths given by", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 118, 485, 131 ], "spans": [ { "bbox": [ 69, 118, 280, 131 ], "score": 1.0, "content": "(16) has an equivalent marginal probability flow", "type": "text" }, { "bbox": [ 281, 120, 303, 130 ], "score": 0.93, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 303, 118, 485, 131 ], "score": 1.0, "content": "to Lipman et al. (2023)’s flow matching.", "type": "text" } ], "index": 2 } ], "index": 1.5, "bbox_fs": [ 68, 104, 542, 131 ] }, { "type": "text", "bbox": [ 70, 141, 541, 226 ], "lines": [ { "bbox": [ 70, 141, 541, 155 ], "spans": [ { "bbox": [ 70, 141, 541, 155 ], "score": 1.0, "content": "Proof. To prove the proposition, we use the fact that the Gaussian family can be generated by location-scale", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 153, 542, 168 ], "spans": [ { "bbox": [ 69, 153, 231, 168 ], "score": 1.0, "content": "transformations (see, e.g., Lehmann", "type": "text" }, { "bbox": [ 232, 156, 240, 164 ], "score": 0.29, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 240, 153, 459, 168 ], "score": 1.0, "content": "Casella, 2006), i.e., we can express any Gaussian", "type": "text" }, { "bbox": [ 460, 156, 527, 167 ], "score": 0.93, "content": "Z _ { 0 } \\sim \\mathcal { N } ( \\mu _ { 0 } , \\sigma _ { 0 } ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 527, 153, 542, 168 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 4 }, { "bbox": [ 71, 165, 542, 180 ], "spans": [ { "bbox": [ 71, 169, 137, 178 ], "score": 0.93, "content": "Z _ { 0 } = \\mu _ { 0 } + \\sigma _ { 0 } Z", "type": "inline_equation" }, { "bbox": [ 137, 165, 169, 180 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 169, 168, 224, 178 ], "score": 0.94, "content": "Z \\sim { \\mathcal { N } } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 224, 165, 336, 180 ], "score": 1.0, "content": ". Recall that the density", "type": "text" }, { "bbox": [ 336, 168, 359, 178 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 359, 165, 421, 180 ], "score": 1.0, "content": "has the form", "type": "text" }, { "bbox": [ 421, 168, 524, 179 ], "score": 0.91, "content": "\\begin{array} { r } { p _ { t } ( x ) = \\int p _ { t } ( x | z ) q ( z ) d z } \\end{array}", "type": "inline_equation" }, { "bbox": [ 524, 165, 542, 180 ], "score": 1.0, "content": ", to", "type": "text" } ], "index": 5 }, { "bbox": [ 69, 177, 541, 191 ], "spans": [ { "bbox": [ 69, 177, 541, 191 ], "score": 1.0, "content": "show the equivalence between the flow from FM and source conditional flow matching, we have to show", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 190, 542, 203 ], "spans": [ { "bbox": [ 69, 190, 92, 203 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 93, 192, 128, 203 ], "score": 0.94, "content": "p _ { t } ( x | x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 129, 190, 451, 203 ], "score": 1.0, "content": "is the same for both methods, that is we show that CFM with variance", "type": "text" }, { "bbox": [ 451, 191, 529, 203 ], "score": 0.94, "content": "( \\sigma t ) ^ { 2 } + 2 \\sigma t ( 1 - t )", "type": "inline_equation" }, { "bbox": [ 529, 190, 542, 203 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 7 }, { "bbox": [ 69, 200, 540, 216 ], "spans": [ { "bbox": [ 69, 200, 212, 216 ], "score": 1.0, "content": "equivalent to FM with variance", "type": "text" }, { "bbox": [ 212, 204, 267, 214 ], "score": 0.94, "content": "( t \\sigma - t + 1 ) ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 267, 200, 321, 216 ], "score": 1.0, "content": "(12). Since", "type": "text" }, { "bbox": [ 322, 204, 371, 214 ], "score": 0.94, "content": "p _ { t } ( x | x _ { 0 } , x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 372, 200, 384, 216 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 385, 204, 540, 214 ], "score": 0.91, "content": "\\mathcal { N } ( t x _ { 1 } + ( 1 - t ) x _ { 0 } ) , \\sigma t ( \\sigma t - 2 t + 2 ) )", "type": "inline_equation" } ], "index": 8 }, { "bbox": [ 69, 214, 277, 227 ], "spans": [ { "bbox": [ 69, 214, 220, 227 ], "score": 1.0, "content": "we can write the random variable", "type": "text" }, { "bbox": [ 220, 216, 262, 226 ], "score": 0.94, "content": "X | X _ { 0 } , X _ { 1 }", "type": "inline_equation" }, { "bbox": [ 262, 214, 277, 227 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 9 } ], "index": 6, "bbox_fs": [ 69, 141, 542, 227 ] }, { "type": "interline_equation", "bbox": [ 192, 231, 418, 244 ], "lines": [ { "bbox": [ 192, 231, 418, 244 ], "spans": [ { "bbox": [ 192, 231, 418, 244 ], "score": 0.87, "content": "X | X _ { 0 } , X _ { 1 } = t x _ { 1 } + ( 1 - t ) x _ { 0 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) ^ { 1 / 2 } Z ,", "type": "interline_equation", "image_path": "9b71e968a839ed7b0f614bbcbf048808c2bea4afed728f05ba3ac6707e5d72b6.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 192, 231, 418, 244 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 69, 248, 326, 261 ], "lines": [ { "bbox": [ 69, 246, 327, 263 ], "spans": [ { "bbox": [ 69, 246, 99, 263 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 251, 153, 261 ], "score": 0.95, "content": "Z \\sim { \\mathcal { N } } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 154, 246, 271, 263 ], "score": 1.0, "content": ". Without conditioning on", "type": "text" }, { "bbox": [ 271, 252, 284, 261 ], "score": 0.92, "content": "X _ { 0 }", "type": "inline_equation" }, { "bbox": [ 285, 246, 327, 263 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 11 } ], "index": 11, "bbox_fs": [ 69, 246, 327, 263 ] }, { "type": "interline_equation", "bbox": [ 199, 266, 411, 279 ], "lines": [ { "bbox": [ 199, 266, 411, 279 ], "spans": [ { "bbox": [ 199, 266, 411, 279 ], "score": 0.87, "content": "X | X _ { 1 } = t x _ { 1 } + ( 1 - t ) X _ { 0 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) ^ { 1 / 2 } Z .", "type": "interline_equation", "image_path": "0b1c279ec78db4d0f4e2da1a5d5b41db67fb357ac5106c924198fe4f8d3e6384.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 199, 266, 411, 279 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 70, 283, 541, 320 ], "lines": [ { "bbox": [ 70, 284, 541, 297 ], "spans": [ { "bbox": [ 70, 284, 138, 297 ], "score": 1.0, "content": "By assumption", "type": "text" }, { "bbox": [ 138, 286, 197, 296 ], "score": 0.95, "content": "X _ { 0 } \\sim { \\mathcal { N } } ( 0 , I ) ", "type": "inline_equation" }, { "bbox": [ 197, 284, 224, 297 ], "score": 1.0, "content": ", thus", "type": "text" }, { "bbox": [ 224, 286, 249, 296 ], "score": 0.94, "content": "X | X _ { 1 }", "type": "inline_equation" }, { "bbox": [ 249, 284, 541, 297 ], "score": 1.0, "content": "is Gaussian, since a linear transformation of Gaussian distributions", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 295, 541, 309 ], "spans": [ { "bbox": [ 69, 295, 541, 309 ], "score": 1.0, "content": "is also Gaussian. To define its distribution, we only have to define its expectation and variance. By linearity", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 307, 435, 321 ], "spans": [ { "bbox": [ 69, 307, 173, 321 ], "score": 1.0, "content": "of expectation, we find", "type": "text" }, { "bbox": [ 174, 309, 241, 320 ], "score": 0.94, "content": "{ \\bf E } ( X \\vert X _ { 1 } ) = t x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 241, 307, 352, 321 ], "score": 1.0, "content": ", and by independence of", "type": "text" }, { "bbox": [ 353, 310, 366, 319 ], "score": 0.92, "content": "X _ { 0 }", "type": "inline_equation" }, { "bbox": [ 366, 307, 388, 321 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 388, 310, 396, 317 ], "score": 0.9, "content": "Z", "type": "inline_equation" }, { "bbox": [ 396, 307, 435, 321 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 69, 284, 541, 321 ] }, { "type": "interline_equation", "bbox": [ 180, 323, 429, 357 ], "lines": [ { "bbox": [ 180, 323, 429, 357 ], "spans": [ { "bbox": [ 180, 323, 429, 357 ], "score": 0.91, "content": "\\begin{array} { r } { \\mathrm { V a r } ( X | X _ { 1 } ) = ( 1 - t ) ^ { 2 } \\mathrm { V a r } ( X _ { 0 } ) + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) \\mathrm { V a r } ( Z ) } \\\\ { = ( 1 - t ) ^ { 2 } + ( \\sigma t ( \\sigma t - 2 t + 2 ) ) = ( t \\sigma - t + 1 ) ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "e330016069293841f54927039e275ef02cef9daae5f7a8d758a0607509dfef6e.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 180, 323, 429, 334.3333333333333 ], "spans": [], "index": 16 }, { "bbox": [ 180, 334.3333333333333, 429, 345.66666666666663 ], "spans": [], "index": 17 }, { "bbox": [ 180, 345.66666666666663, 429, 356.99999999999994 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 70, 358, 390, 371 ], "lines": [ { "bbox": [ 69, 358, 391, 372 ], "spans": [ { "bbox": [ 69, 358, 391, 372 ], "score": 1.0, "content": "hence the flow from source conditional flow matching is the same as FM.", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 69, 358, 391, 372 ] }, { "type": "text", "bbox": [ 70, 376, 529, 389 ], "lines": [ { "bbox": [ 68, 375, 529, 392 ], "spans": [ { "bbox": [ 68, 375, 168, 392 ], "score": 1.0, "content": "Proposition B.2. If", "type": "text" }, { "bbox": [ 169, 382, 175, 387 ], "score": 0.77, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 176, 375, 475, 392 ], "score": 1.0, "content": "is a Monge map, the objective variance of OT-CFM goes to zero as", "type": "text" }, { "bbox": [ 475, 380, 502, 387 ], "score": 0.9, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 502, 375, 529, 392 ], "score": 1.0, "content": ", i.e.,", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 68, 375, 529, 392 ] }, { "type": "interline_equation", "bbox": [ 221, 394, 390, 408 ], "lines": [ { "bbox": [ 221, 394, 390, 408 ], "spans": [ { "bbox": [ 221, 394, 390, 408 ], "score": 0.89, "content": "\\mathbb { E } _ { q ( z ) } \\| u _ { t } ( x | z ) - u _ { t } ( x ) \\| ^ { 2 } \\to 0 \\mathrm { ~ } a s \\sigma \\to 0", "type": "interline_equation", "image_path": "56d15a5f3047959efe1a64f7a4949451bc42e1384bcb034145208eee5169639d.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 221, 394, 390, 408 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 70, 411, 155, 425 ], "lines": [ { "bbox": [ 69, 411, 156, 426 ], "spans": [ { "bbox": [ 69, 411, 87, 426 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 87, 414, 118, 424 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 118, 411, 156, 426 ], "score": 1.0, "content": "in (15).", "type": "text" } ], "index": 22 } ], "index": 22, "bbox_fs": [ 69, 411, 156, 426 ] }, { "type": "text", "bbox": [ 70, 436, 542, 472 ], "lines": [ { "bbox": [ 70, 435, 542, 450 ], "spans": [ { "bbox": [ 70, 435, 380, 450 ], "score": 1.0, "content": "Proof. This follows from a basic fact about the transport plan", "type": "text" }, { "bbox": [ 380, 442, 386, 446 ], "score": 0.86, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 387, 435, 501, 450 ], "score": 1.0, "content": ". Specifically, that as", "type": "text" }, { "bbox": [ 501, 439, 538, 447 ], "score": 0.87, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 538, 435, 542, 450 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 71, 447, 542, 462 ], "spans": [ { "bbox": [ 71, 450, 202, 461 ], "score": 0.92, "content": "D _ { \\mathrm { K L } } ( p _ { t } ( x | z ^ { i } ) \\| p _ { t } ( x | z ^ { j } ) ) \\ \\to \\ \\infty", "type": "inline_equation" }, { "bbox": [ 202, 447, 239, 462 ], "score": 1.0, "content": "for an", "type": "text" }, { "bbox": [ 239, 452, 253, 460 ], "score": 0.91, "content": "t , x", "type": "inline_equation" }, { "bbox": [ 253, 447, 333, 462 ], "score": 1.0, "content": "for two distinct", "type": "text" }, { "bbox": [ 333, 450, 360, 460 ], "score": 0.43, "content": "z ^ { i } , \\ z ^ { j }", "type": "inline_equation" }, { "bbox": [ 360, 447, 451, 462 ], "score": 1.0, "content": ". This means that", "type": "text" }, { "bbox": [ 452, 450, 523, 461 ], "score": 0.93, "content": "p _ { t } ( x | z ) \\ = \\ p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 523, 447, 542, 462 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 460, 158, 474 ], "spans": [ { "bbox": [ 69, 460, 90, 474 ], "score": 1.0, "content": "any", "type": "text" }, { "bbox": [ 90, 464, 113, 472 ], "score": 0.92, "content": "t , x , z", "type": "inline_equation" }, { "bbox": [ 114, 460, 158, 474 ], "score": 1.0, "content": "therefore", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 69, 435, 542, 474 ] }, { "type": "interline_equation", "bbox": [ 232, 477, 379, 507 ], "lines": [ { "bbox": [ 232, 477, 379, 507 ], "spans": [ { "bbox": [ 232, 477, 379, 507 ], "score": 0.92, "content": "\\begin{array} { c } { { u _ { t } ( x ) = \\mathbb { E } _ { q ( z ) } u _ { t } ( x | z ) p _ { t } ( x | z ) / p _ { t } ( x ) } } \\\\ { { = u _ { t } ( x | z ) } } \\end{array}", "type": "interline_equation", "image_path": "5141741fab4dc50e83ce082f541d61df41b8ff42f943e7e85fa4110351ced3d2.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 232, 477, 379, 507 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 70, 528, 543, 553 ], "lines": [ { "bbox": [ 69, 527, 543, 543 ], "spans": [ { "bbox": [ 69, 527, 285, 543 ], "score": 1.0, "content": "Proposition B.3. The conditional vector field", "type": "text" }, { "bbox": [ 285, 531, 317, 541 ], "score": 0.94, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 317, 527, 543, 543 ], "score": 1.0, "content": "defined by (26) converges to marginal vector field", "type": "text" } ], "index": 27 }, { "bbox": [ 71, 540, 309, 555 ], "spans": [ { "bbox": [ 71, 542, 94, 553 ], "score": 0.92, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 540, 309, 555 ], "score": 1.0, "content": "defined by (9) as m goes to population size, i.e.,", "type": "text" } ], "index": 28 } ], "index": 27.5, "bbox_fs": [ 69, 527, 543, 555 ] }, { "type": "interline_equation", "bbox": [ 254, 558, 357, 570 ], "lines": [ { "bbox": [ 254, 558, 357, 570 ], "spans": [ { "bbox": [ 254, 558, 357, 570 ], "score": 0.9, "content": "\\| u _ { t } ( x | \\bar { z } ) - u _ { t } ( x ) \\| ^ { 2 } \\to 0", "type": "interline_equation", "image_path": "7d367087f15c5c249661695ee30a8fa75ec64c5e28abf6dd872a565e1202ae99.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 254, 558, 357, 570 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 70, 575, 125, 588 ], "lines": [ { "bbox": [ 69, 574, 127, 589 ], "spans": [ { "bbox": [ 69, 574, 83, 589 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 83, 577, 122, 588 ], "score": 0.93, "content": "m | \\mathcal { X } |", "type": "inline_equation" }, { "bbox": [ 122, 574, 127, 589 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 69, 574, 127, 589 ] }, { "type": "text", "bbox": [ 71, 599, 223, 613 ], "lines": [ { "bbox": [ 70, 598, 223, 614 ], "spans": [ { "bbox": [ 70, 598, 117, 614 ], "score": 1.0, "content": "Proof. As", "type": "text" }, { "bbox": [ 117, 602, 158, 612 ], "score": 0.93, "content": "| z | \\to | \\mathcal { X } |", "type": "inline_equation" }, { "bbox": [ 158, 598, 223, 614 ], "score": 1.0, "content": ", by definition,", "type": "text" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 70, 598, 223, 614 ] }, { "type": "interline_equation", "bbox": [ 227, 616, 383, 717 ], "lines": [ { "bbox": [ 227, 616, 383, 717 ], "spans": [ { "bbox": [ 227, 616, 383, 717 ], "score": 0.94, "content": "\\begin{array} { l } { { u _ { t } ( x | \\bar { z } ) = \\displaystyle \\frac { \\sum _ { i } ^ { m } u _ { t } ( x | z ^ { i } ) p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { \\sum _ { i } ^ { m } p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } } } \\\\ { { \\ \\ \\qquad = \\displaystyle \\frac { \\sum _ { z \\in \\mathcal { X } } u _ { t } ( x | z ) p _ { t } ( x | z ) q ( z ) } { \\sum _ { z \\in \\mathcal { X } } p _ { t } ( x | z ) q ( z ) } } } \\\\ { { \\ \\qquad = \\displaystyle \\mathbb { E } _ { q ( z ) } \\frac { u _ { t } ( x | z ) p _ { t } ( x | z ) } { p _ { t } ( x ) } } } \\\\ { { \\ \\qquad = u _ { t } ( x ) } } \\end{array}", "type": "interline_equation", "image_path": "ff2120fa6963a49eb7488fafb6ed711ab0957662477c7ea997490422c72b07d3.jpg" } ] } ], "index": 34.5, "virtual_lines": [ { "bbox": [ 227, 616, 383, 632.8333333333334 ], "spans": [], "index": 32 }, { "bbox": [ 227, 632.8333333333334, 383, 649.6666666666667 ], "spans": [], "index": 33 }, { "bbox": [ 227, 649.6666666666667, 383, 666.5000000000001 ], "spans": [], "index": 34 }, { "bbox": [ 227, 666.5000000000001, 383, 683.3333333333335 ], "spans": [], "index": 35 }, { "bbox": [ 227, 683.3333333333335, 383, 700.1666666666669 ], "spans": [], "index": 36 }, { "bbox": [ 227, 700.1666666666669, 383, 717.0000000000002 ], "spans": [], "index": 37 } ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 71, 80, 208, 95 ], "lines": [ { "bbox": [ 69, 79, 210, 97 ], "spans": [ { "bbox": [ 69, 79, 210, 97 ], "score": 1.0, "content": "C Algorithm extensions", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 70, 110, 541, 195 ], "lines": [ { "bbox": [ 69, 111, 541, 124 ], "spans": [ { "bbox": [ 69, 111, 432, 124 ], "score": 1.0, "content": "In Alg. 1 we presented the general algorithm for conditional flow matching given", "type": "text" }, { "bbox": [ 432, 113, 450, 123 ], "score": 0.89, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 450, 111, 455, 124 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 456, 113, 486, 123 ], "score": 0.93, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 487, 111, 492, 124 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 492, 113, 523, 123 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 523, 111, 541, 124 ], "score": 1.0, "content": ". In", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 122, 541, 137 ], "spans": [ { "bbox": [ 69, 122, 541, 137 ], "score": 1.0, "content": "Table 1 we presented a number of settings of these leading to interesting probability paths. In practice, we may", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 135, 541, 148 ], "spans": [ { "bbox": [ 69, 135, 144, 148 ], "score": 1.0, "content": "wish to compute", "type": "text" }, { "bbox": [ 145, 137, 163, 147 ], "score": 0.93, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 163, 135, 541, 148 ], "score": 1.0, "content": "on the fly. Therefore in Alg. 2, Alg. 3, and Alg. 4, we give algorithms for the simplified", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 146, 540, 160 ], "spans": [ { "bbox": [ 69, 146, 540, 160 ], "score": 1.0, "content": "conditional flow matching, and minibatch versions of OT conditional flow matching and Schrödinger bridge", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 158, 541, 171 ], "spans": [ { "bbox": [ 70, 158, 541, 171 ], "score": 1.0, "content": "conditional flow matching. 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Informally, as", "type": "text" }, { "bbox": [ 321, 254, 348, 261 ], "score": 0.9, "content": "\\sigma 0", "type": "inline_equation" }, { "bbox": [ 348, 251, 353, 265 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 353, 253, 479, 264 ], "score": 0.92, "content": "\\mathbb { E } _ { x , t , z } \\| u _ { t } ( x | z ) - u _ { t } ( x ) \\| ^ { 2 } \\to 0", "type": "inline_equation" }, { "bbox": [ 479, 251, 542, 265 ], "score": 1.0, "content": "for OT-CFM", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 263, 541, 276 ], "spans": [ { "bbox": [ 69, 263, 541, 276 ], "score": 1.0, "content": "and SB-CFM, which is not true of previous probability paths in Table 1 (See Proposition B.2 for a precise", "type": "text" } ], "index": 11 }, { "bbox": [ 68, 275, 542, 289 ], "spans": [ { "bbox": [ 68, 275, 542, 289 ], "score": 1.0, "content": "statement). As flow models get larger, more powerful, and more costly, reducing objective variance, and", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 287, 541, 300 ], "spans": [ { "bbox": [ 70, 287, 541, 300 ], "score": 1.0, "content": "thereby faster training may lead to significant cost savings (Watson et al., 2022b). To this end we also explore", "type": "text" } ], "index": 13 }, { "bbox": [ 68, 298, 542, 313 ], "spans": [ { "bbox": [ 68, 298, 542, 313 ], "score": 1.0, "content": "reducing the variance of the objective by averaging over a batch. This is not feasible in score matching where", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 311, 541, 325 ], "spans": [ { "bbox": [ 69, 311, 541, 325 ], "score": 1.0, "content": "the flow conditioned on multiple datapoints is complex. Our CFM framework naturally extends from a pair", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 322, 542, 336 ], "spans": [ { "bbox": [ 69, 322, 542, 336 ], "score": 1.0, "content": "of datapoints to a batch of pairs. Instead of conditioning on a single pair of datapoints we can condition on a", "type": "text" } ], "index": 16 }, { "bbox": [ 68, 334, 542, 349 ], "spans": [ { "bbox": [ 68, 334, 542, 349 ], "score": 1.0, "content": "batch of pairs. As the batch increases in size, we trade higher cost in computing the target for lower variance", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 345, 542, 361 ], "spans": [ { "bbox": [ 68, 345, 542, 361 ], "score": 1.0, "content": "in the target as the batch size increases, the variance in the target goes to zero (see Proposition B.3 for a", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 359, 155, 372 ], "spans": [ { "bbox": [ 69, 359, 155, 372 ], "score": 1.0, "content": "precise statement).", "type": "text" } ], "index": 19 } ], "index": 14 }, { "type": "text", "bbox": [ 69, 376, 537, 402 ], "lines": [ { "bbox": [ 69, 375, 540, 389 ], "spans": [ { "bbox": [ 69, 375, 540, 389 ], "score": 1.0, "content": "As formalized in Proposition B.3, we can reduce variance in the target by averaging over multiple datapoints.", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 387, 457, 402 ], "spans": [ { "bbox": [ 69, 387, 206, 402 ], "score": 1.0, "content": "Specifically, in this case we let", "type": "text" }, { "bbox": [ 206, 390, 307, 401 ], "score": 0.92, "content": "\\bar { z } : = \\{ z ^ { i } : = ( x _ { 0 } ^ { i } , x _ { 1 } ^ { i } ) \\} _ { i = 1 } ^ { m }", "type": "inline_equation" }, { "bbox": [ 308, 387, 342, 402 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 342, 390, 351, 398 ], "score": 0.91, "content": "z ^ { i }", "type": "inline_equation" }, { "bbox": [ 351, 387, 416, 402 ], "score": 1.0, "content": "are i.i.d. from", "type": "text" }, { "bbox": [ 417, 390, 434, 401 ], "score": 0.95, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 435, 387, 457, 402 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 21 } ], "index": 20.5 }, { "type": "interline_equation", "bbox": [ 227, 416, 383, 476 ], "lines": [ { "bbox": [ 227, 416, 383, 476 ], "spans": [ { "bbox": [ 227, 416, 383, 476 ], "score": 0.93, "content": "\\begin{array} { l } { p _ { t } ( x | \\bar { z } ) = \\frac { \\sum _ { i } ^ { m } p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { \\sum _ { i } ^ { m } q ( z ^ { i } ) } } \\\\ { u _ { t } ( x | \\bar { z } ) = \\frac { \\sum _ { i } ^ { m } u _ { t } ( x | z ^ { i } ) p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { p _ { t } ( x | \\bar { z } ) } } \\end{array}", "type": "interline_equation", "image_path": "0f1452102ebd7833299bf09adf4b5109a609ce24f87b2ebc7d5dd26654454f89.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 227, 416, 383, 431.0 ], "spans": [], "index": 22 }, { "bbox": [ 227, 431.0, 383, 446.0 ], "spans": [], "index": 23 }, { "bbox": [ 227, 446.0, 383, 461.0 ], "spans": [], "index": 24 }, { "bbox": [ 227, 461.0, 383, 476.0 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 73, 488, 541, 513 ], "lines": [ { "bbox": [ 70, 489, 541, 501 ], "spans": [ { "bbox": [ 70, 489, 142, 501 ], "score": 1.0, "content": "It takes roughly", "type": "text" }, { "bbox": [ 143, 494, 152, 498 ], "score": 0.89, "content": "m", "type": "inline_equation" }, { "bbox": [ 152, 489, 363, 501 ], "score": 1.0, "content": "times as long to compute the conditional target", "type": "text" }, { "bbox": [ 363, 491, 395, 501 ], "score": 0.95, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 396, 489, 541, 501 ], "score": 1.0, "content": "but reduces the variance. As the", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 500, 479, 513 ], "spans": [ { "bbox": [ 69, 500, 251, 513 ], "score": 1.0, "content": "evaluation and backpropagation through", "type": "text" }, { "bbox": [ 251, 506, 261, 512 ], "score": 0.89, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 261, 500, 479, 513 ], "score": 1.0, "content": "gets more difficult this tradeoff can be beneficial.", "type": "text" } ], "index": 27 } ], "index": 26.5 }, { "type": "title", "bbox": [ 72, 533, 219, 546 ], "lines": [ { "bbox": [ 69, 531, 221, 549 ], "spans": [ { "bbox": [ 69, 531, 221, 549 ], "score": 1.0, "content": "C.2 Modeling energy functions", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 71, 557, 540, 594 ], "lines": [ { "bbox": [ 68, 555, 542, 572 ], "spans": [ { "bbox": [ 68, 555, 424, 572 ], "score": 1.0, "content": "If we have access to an energy function two (unnormalized) energy functions", "type": "text" }, { "bbox": [ 425, 558, 507, 571 ], "score": 0.94, "content": "\\mathcal { R } _ { \\{ 0 , 1 \\} } : \\mathbb { R } ^ { d } \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 508, 555, 542, 572 ], "score": 1.0, "content": "at the", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 568, 541, 583 ], "spans": [ { "bbox": [ 70, 568, 222, 583 ], "score": 1.0, "content": "endpoints instead of i.i.d. samples", "type": "text" }, { "bbox": [ 223, 571, 274, 582 ], "score": 0.94, "content": "X _ { t } \\sim q _ { t } ( x _ { t } )", "type": "inline_equation" }, { "bbox": [ 275, 568, 541, 583 ], "score": 1.0, "content": ", then the objective must be slightly modified. We formulate", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 580, 302, 595 ], "spans": [ { "bbox": [ 69, 580, 302, 595 ], "score": 1.0, "content": "the Energy Conditional Flow Matching Objective as", "type": "text" } ], "index": 31 } ], "index": 30 }, { "type": "interline_equation", "bbox": [ 142, 608, 468, 635 ], "lines": [ { "bbox": [ 142, 608, 468, 635 ], "spans": [ { "bbox": [ 142, 608, 468, 635 ], "score": 0.92, "content": "\\mathcal { L } _ { \\mathrm { E C F M } } = \\mathbb { E } _ { t , \\hat { q } _ { 0 } ( x _ { 0 } ) , \\hat { q } _ { 1 } ( x _ { 1 } ) , p _ { t } ( x | x _ { 0 } , x _ { 1 } ) } \\left[ \\frac { R _ { 0 } ( x _ { 0 } ) R _ { 1 } ( x _ { 1 } ) } { \\hat { q } _ { 0 } ( x _ { 0 } ) \\hat { q } _ { 1 } ( x _ { 1 } ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | x _ { 0 } , x _ { 1 } ) \\| _ { 2 } ^ { 2 } \\right]", "type": "interline_equation", "image_path": "25e9dc760dd42b048a116da829278f4aaed6fad4ed02bf1d66da6905a92125e9.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 142, 608, 468, 635 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 70, 648, 540, 673 ], "lines": [ { "bbox": [ 69, 648, 541, 661 ], "spans": [ { "bbox": [ 69, 648, 541, 661 ], "score": 1.0, "content": "We can use this object to train a flow which matches the energies without access to samples. This is formalized", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 660, 182, 674 ], "spans": [ { "bbox": [ 69, 660, 182, 674 ], "score": 1.0, "content": "in the following theorem.", "type": "text" } ], "index": 34 } ], "index": 33.5 }, { "type": "text", "bbox": [ 69, 679, 538, 704 ], "lines": [ { "bbox": [ 68, 677, 539, 695 ], "spans": [ { "bbox": [ 68, 677, 223, 695 ], "score": 1.0, "content": "Proposition C.1. Assuming that", "type": "text" }, { "bbox": [ 223, 682, 305, 693 ], "score": 0.93, "content": "\\hat { q } _ { \\{ 0 , 1 \\} } ( x ) , p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 306, 677, 338, 695 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 338, 682, 365, 690 ], "score": 0.9, "content": "x \\in \\mathcal { X }", "type": "inline_equation" }, { "bbox": [ 366, 677, 388, 695 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 388, 682, 424, 692 ], "score": 0.94, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 424, 677, 519, 695 ], "score": 1.0, "content": "then the gradients of", "type": "text" }, { "bbox": [ 519, 682, 539, 691 ], "score": 0.9, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" } ], "index": 35 }, { "bbox": [ 70, 692, 403, 705 ], "spans": [ { "bbox": [ 70, 692, 89, 705 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 90, 695, 96, 702 ], "score": 0.61, "content": "\\mathcal { L }", "type": "inline_equation" }, { "bbox": [ 97, 692, 190, 705 ], "score": 1.0, "content": "ECFM with respect to", "type": "text" }, { "bbox": [ 190, 695, 195, 702 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 196, 692, 393, 705 ], "score": 1.0, "content": "are equal up to some multiplicative constant", "type": "text" }, { "bbox": [ 393, 697, 398, 702 ], "score": 0.84, "content": "c", "type": "inline_equation" }, { "bbox": [ 399, 692, 403, 705 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 36 } ], "index": 35.5 }, { "type": "interline_equation", "bbox": [ 245, 720, 366, 733 ], "lines": [ { "bbox": [ 245, 720, 366, 733 ], "spans": [ { "bbox": [ 245, 720, 366, 733 ], "score": 0.92, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = c \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { E C F M } } ( \\boldsymbol { \\theta } )", "type": "interline_equation", "image_path": "f6e6f36332c4cb82822b098874005c611e48bafc36b6e4a1f5888c00e1869f15.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 245, 720, 366, 733 ], "spans": [], "index": 37 } ] } ], "page_idx": 22, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 26, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "23", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 71, 80, 208, 95 ], "lines": [ { "bbox": [ 69, 79, 210, 97 ], "spans": [ { "bbox": [ 69, 79, 210, 97 ], "score": 1.0, "content": "C Algorithm extensions", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 70, 110, 541, 195 ], "lines": [ { "bbox": [ 69, 111, 541, 124 ], "spans": [ { "bbox": [ 69, 111, 432, 124 ], "score": 1.0, "content": "In Alg. 1 we presented the general algorithm for conditional flow matching given", "type": "text" }, { "bbox": [ 432, 113, 450, 123 ], "score": 0.89, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 450, 111, 455, 124 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 456, 113, 486, 123 ], "score": 0.93, "content": "p _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 487, 111, 492, 124 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 492, 113, 523, 123 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 523, 111, 541, 124 ], "score": 1.0, "content": ". 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As flow models get larger, more powerful, and more costly, reducing objective variance, and", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 287, 541, 300 ], "spans": [ { "bbox": [ 70, 287, 541, 300 ], "score": 1.0, "content": "thereby faster training may lead to significant cost savings (Watson et al., 2022b). To this end we also explore", "type": "text" } ], "index": 13 }, { "bbox": [ 68, 298, 542, 313 ], "spans": [ { "bbox": [ 68, 298, 542, 313 ], "score": 1.0, "content": "reducing the variance of the objective by averaging over a batch. This is not feasible in score matching where", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 311, 541, 325 ], "spans": [ { "bbox": [ 69, 311, 541, 325 ], "score": 1.0, "content": "the flow conditioned on multiple datapoints is complex. Our CFM framework naturally extends from a pair", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 322, 542, 336 ], "spans": [ { "bbox": [ 69, 322, 542, 336 ], "score": 1.0, "content": "of datapoints to a batch of pairs. Instead of conditioning on a single pair of datapoints we can condition on a", "type": "text" } ], "index": 16 }, { "bbox": [ 68, 334, 542, 349 ], "spans": [ { "bbox": [ 68, 334, 542, 349 ], "score": 1.0, "content": "batch of pairs. As the batch increases in size, we trade higher cost in computing the target for lower variance", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 345, 542, 361 ], "spans": [ { "bbox": [ 68, 345, 542, 361 ], "score": 1.0, "content": "in the target as the batch size increases, the variance in the target goes to zero (see Proposition B.3 for a", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 359, 155, 372 ], "spans": [ { "bbox": [ 69, 359, 155, 372 ], "score": 1.0, "content": "precise statement).", "type": "text" } ], "index": 19 } ], "index": 14, "bbox_fs": [ 68, 240, 542, 372 ] }, { "type": "text", "bbox": [ 69, 376, 537, 402 ], "lines": [ { "bbox": [ 69, 375, 540, 389 ], "spans": [ { "bbox": [ 69, 375, 540, 389 ], "score": 1.0, "content": "As formalized in Proposition B.3, we can reduce variance in the target by averaging over multiple datapoints.", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 387, 457, 402 ], "spans": [ { "bbox": [ 69, 387, 206, 402 ], "score": 1.0, "content": "Specifically, in this case we let", "type": "text" }, { "bbox": [ 206, 390, 307, 401 ], "score": 0.92, "content": "\\bar { z } : = \\{ z ^ { i } : = ( x _ { 0 } ^ { i } , x _ { 1 } ^ { i } ) \\} _ { i = 1 } ^ { m }", "type": "inline_equation" }, { "bbox": [ 308, 387, 342, 402 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 342, 390, 351, 398 ], "score": 0.91, "content": "z ^ { i }", "type": "inline_equation" }, { "bbox": [ 351, 387, 416, 402 ], "score": 1.0, "content": "are i.i.d. from", "type": "text" }, { "bbox": [ 417, 390, 434, 401 ], "score": 0.95, "content": "q ( z )", "type": "inline_equation" }, { "bbox": [ 435, 387, 457, 402 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 21 } ], "index": 20.5, "bbox_fs": [ 69, 375, 540, 402 ] }, { "type": "interline_equation", "bbox": [ 227, 416, 383, 476 ], "lines": [ { "bbox": [ 227, 416, 383, 476 ], "spans": [ { "bbox": [ 227, 416, 383, 476 ], "score": 0.93, "content": "\\begin{array} { l } { p _ { t } ( x | \\bar { z } ) = \\frac { \\sum _ { i } ^ { m } p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { \\sum _ { i } ^ { m } q ( z ^ { i } ) } } \\\\ { u _ { t } ( x | \\bar { z } ) = \\frac { \\sum _ { i } ^ { m } u _ { t } ( x | z ^ { i } ) p _ { t } ( x | z ^ { i } ) q ( z ^ { i } ) } { p _ { t } ( x | \\bar { z } ) } } \\end{array}", "type": "interline_equation", "image_path": "0f1452102ebd7833299bf09adf4b5109a609ce24f87b2ebc7d5dd26654454f89.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 227, 416, 383, 431.0 ], "spans": [], "index": 22 }, { "bbox": [ 227, 431.0, 383, 446.0 ], "spans": [], "index": 23 }, { "bbox": [ 227, 446.0, 383, 461.0 ], "spans": [], "index": 24 }, { "bbox": [ 227, 461.0, 383, 476.0 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 73, 488, 541, 513 ], "lines": [ { "bbox": [ 70, 489, 541, 501 ], "spans": [ { "bbox": [ 70, 489, 142, 501 ], "score": 1.0, "content": "It takes roughly", "type": "text" }, { "bbox": [ 143, 494, 152, 498 ], "score": 0.89, "content": "m", "type": "inline_equation" }, { "bbox": [ 152, 489, 363, 501 ], "score": 1.0, "content": "times as long to compute the conditional target", "type": "text" }, { "bbox": [ 363, 491, 395, 501 ], "score": 0.95, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 396, 489, 541, 501 ], "score": 1.0, "content": "but reduces the variance. As the", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 500, 479, 513 ], "spans": [ { "bbox": [ 69, 500, 251, 513 ], "score": 1.0, "content": "evaluation and backpropagation through", "type": "text" }, { "bbox": [ 251, 506, 261, 512 ], "score": 0.89, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 261, 500, 479, 513 ], "score": 1.0, "content": "gets more difficult this tradeoff can be beneficial.", "type": "text" } ], "index": 27 } ], "index": 26.5, "bbox_fs": [ 69, 489, 541, 513 ] }, { "type": "title", "bbox": [ 72, 533, 219, 546 ], "lines": [ { "bbox": [ 69, 531, 221, 549 ], "spans": [ { "bbox": [ 69, 531, 221, 549 ], "score": 1.0, "content": "C.2 Modeling energy functions", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 71, 557, 540, 594 ], "lines": [ { "bbox": [ 68, 555, 542, 572 ], "spans": [ { "bbox": [ 68, 555, 424, 572 ], "score": 1.0, "content": "If we have access to an energy function two (unnormalized) energy functions", "type": "text" }, { "bbox": [ 425, 558, 507, 571 ], "score": 0.94, "content": "\\mathcal { R } _ { \\{ 0 , 1 \\} } : \\mathbb { R } ^ { d } \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 508, 555, 542, 572 ], "score": 1.0, "content": "at the", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 568, 541, 583 ], "spans": [ { "bbox": [ 70, 568, 222, 583 ], "score": 1.0, "content": "endpoints instead of i.i.d. samples", "type": "text" }, { "bbox": [ 223, 571, 274, 582 ], "score": 0.94, "content": "X _ { t } \\sim q _ { t } ( x _ { t } )", "type": "inline_equation" }, { "bbox": [ 275, 568, 541, 583 ], "score": 1.0, "content": ", then the objective must be slightly modified. We formulate", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 580, 302, 595 ], "spans": [ { "bbox": [ 69, 580, 302, 595 ], "score": 1.0, "content": "the Energy Conditional Flow Matching Objective as", "type": "text" } ], "index": 31 } ], "index": 30, "bbox_fs": [ 68, 555, 542, 595 ] }, { "type": "interline_equation", "bbox": [ 142, 608, 468, 635 ], "lines": [ { "bbox": [ 142, 608, 468, 635 ], "spans": [ { "bbox": [ 142, 608, 468, 635 ], "score": 0.92, "content": "\\mathcal { L } _ { \\mathrm { E C F M } } = \\mathbb { E } _ { t , \\hat { q } _ { 0 } ( x _ { 0 } ) , \\hat { q } _ { 1 } ( x _ { 1 } ) , p _ { t } ( x | x _ { 0 } , x _ { 1 } ) } \\left[ \\frac { R _ { 0 } ( x _ { 0 } ) R _ { 1 } ( x _ { 1 } ) } { \\hat { q } _ { 0 } ( x _ { 0 } ) \\hat { q } _ { 1 } ( x _ { 1 } ) } \\| v _ { \\theta } ( t , x ) - u _ { t } ( x | x _ { 0 } , x _ { 1 } ) \\| _ { 2 } ^ { 2 } \\right]", "type": "interline_equation", "image_path": "25e9dc760dd42b048a116da829278f4aaed6fad4ed02bf1d66da6905a92125e9.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 142, 608, 468, 635 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 70, 648, 540, 673 ], "lines": [ { "bbox": [ 69, 648, 541, 661 ], "spans": [ { "bbox": [ 69, 648, 541, 661 ], "score": 1.0, "content": "We can use this object to train a flow which matches the energies without access to samples. This is formalized", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 660, 182, 674 ], "spans": [ { "bbox": [ 69, 660, 182, 674 ], "score": 1.0, "content": "in the following theorem.", "type": "text" } ], "index": 34 } ], "index": 33.5, "bbox_fs": [ 69, 648, 541, 674 ] }, { "type": "text", "bbox": [ 69, 679, 538, 704 ], "lines": [ { "bbox": [ 68, 677, 539, 695 ], "spans": [ { "bbox": [ 68, 677, 223, 695 ], "score": 1.0, "content": "Proposition C.1. Assuming that", "type": "text" }, { "bbox": [ 223, 682, 305, 693 ], "score": 0.93, "content": "\\hat { q } _ { \\{ 0 , 1 \\} } ( x ) , p _ { t } ( x ) > 0", "type": "inline_equation" }, { "bbox": [ 306, 677, 338, 695 ], "score": 1.0, "content": "for al l", "type": "text" }, { "bbox": [ 338, 682, 365, 690 ], "score": 0.9, "content": "x \\in \\mathcal { X }", "type": "inline_equation" }, { "bbox": [ 366, 677, 388, 695 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 388, 682, 424, 692 ], "score": 0.94, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 424, 677, 519, 695 ], "score": 1.0, "content": "then the gradients of", "type": "text" }, { "bbox": [ 519, 682, 539, 691 ], "score": 0.9, "content": "\\mathcal { L } _ { \\mathrm { F M } }", "type": "inline_equation" } ], "index": 35 }, { "bbox": [ 70, 692, 403, 705 ], "spans": [ { "bbox": [ 70, 692, 89, 705 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 90, 695, 96, 702 ], "score": 0.61, "content": "\\mathcal { L }", "type": "inline_equation" }, { "bbox": [ 97, 692, 190, 705 ], "score": 1.0, "content": "ECFM with respect to", "type": "text" }, { "bbox": [ 190, 695, 195, 702 ], "score": 0.87, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 196, 692, 393, 705 ], "score": 1.0, "content": "are equal up to some multiplicative constant", "type": "text" }, { "bbox": [ 393, 697, 398, 702 ], "score": 0.84, "content": "c", "type": "inline_equation" }, { "bbox": [ 399, 692, 403, 705 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 36 } ], "index": 35.5, "bbox_fs": [ 68, 677, 539, 705 ] }, { "type": "interline_equation", "bbox": [ 245, 720, 366, 733 ], "lines": [ { "bbox": [ 245, 720, 366, 733 ], "spans": [ { "bbox": [ 245, 720, 366, 733 ], "score": 0.92, "content": "\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { F M } } ( \\boldsymbol { \\theta } ) = c \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathrm { E C F M } } ( \\boldsymbol { \\theta } )", "type": "interline_equation", "image_path": "f6e6f36332c4cb82822b098874005c611e48bafc36b6e4a1f5888c00e1869f15.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 245, 720, 366, 733 ], "spans": [], "index": 37 } ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 71, 82, 344, 95 ], "lines": [ { "bbox": [ 69, 80, 344, 97 ], "spans": [ { "bbox": [ 69, 80, 344, 97 ], "score": 1.0, "content": "Algorithm 2 Simplified Conditional Flow Matching (I-CFM)", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 74, 98, 543, 219 ], "lines": [ { "bbox": [ 79, 97, 504, 112 ], "spans": [ { "bbox": [ 79, 97, 279, 112 ], "score": 1.0, "content": "Input: Empirical or samplable distributions", "type": "text" }, { "bbox": [ 279, 104, 302, 110 ], "score": 0.9, "content": "q _ { 0 } , q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 302, 97, 357, 112 ], "score": 1.0, "content": ", bandwidth", "type": "text" }, { "bbox": [ 357, 104, 363, 108 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 364, 97, 412, 112 ], "score": 1.0, "content": ", batchsize", "type": "text" }, { "bbox": [ 412, 101, 417, 108 ], "score": 0.88, "content": "b", "type": "inline_equation" }, { "bbox": [ 417, 97, 489, 112 ], "score": 1.0, "content": ", initial network", "type": "text" }, { "bbox": [ 489, 104, 499, 110 ], "score": 0.82, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 499, 97, 504, 112 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 1 }, { "bbox": [ 80, 110, 165, 122 ], "spans": [ { "bbox": [ 80, 110, 165, 122 ], "score": 1.0, "content": "while Training do", "type": "text" } ], "index": 2 }, { "bbox": [ 95, 120, 543, 136 ], "spans": [ { "bbox": [ 95, 121, 320, 135 ], "score": 1.0, "content": "/* Sample batches of size b i.i.d. from the datasets", "type": "text" }, { "bbox": [ 528, 120, 543, 136 ], "score": 1.0, "content": "*/", "type": "text" } ], "index": 3 }, { "bbox": [ 96, 136, 216, 147 ], "spans": [ { "bbox": [ 96, 136, 216, 147 ], "score": 0.25, "content": "{ \\pmb x } _ { 0 } \\sim q _ { 0 } ( { \\pmb x } _ { 0 } ) ; \\quad { \\pmb x } _ { 1 } \\sim q _ { 1 } ( { \\pmb x } _ { 1 } )", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 96, 148, 142, 159 ], "spans": [ { "bbox": [ 96, 148, 142, 159 ], "score": 0.79, "content": "t \\sim \\mathcal { U } ( 0 , 1 )", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 96, 160, 188, 171 ], "spans": [ { "bbox": [ 96, 160, 188, 171 ], "score": 0.76, "content": "\\mu _ { t } t { \\pmb x } _ { 1 } + ( 1 - 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With an", "type": "text" } ], "index": 29 }, { "bbox": [ 68, 560, 543, 575 ], "spans": [ { "bbox": [ 68, 560, 400, 575 ], "score": 1.0, "content": "application of Theorem 3.2 the gradients are equivalent up to a factor of", "type": "text" }, { "bbox": [ 400, 566, 419, 573 ], "score": 0.91, "content": "z _ { \\mathrm { 0 } } z _ { \\mathrm { 1 } }", "type": "inline_equation" }, { "bbox": [ 419, 560, 543, 575 ], "score": 1.0, "content": "which does not depend on", "type": "text" } ], "index": 30 }, { "bbox": [ 71, 573, 541, 586 ], "spans": [ { "bbox": [ 71, 578, 77, 583 ], "score": 0.87, "content": "x", "type": "inline_equation" }, { "bbox": [ 77, 574, 83, 586 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 529, 573, 541, 585 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 31 } ], "index": 30 }, { "type": "text", "bbox": [ 72, 597, 541, 622 ], "lines": [ { "bbox": [ 70, 597, 541, 610 ], "spans": [ { "bbox": [ 70, 597, 114, 610 ], "score": 1.0, "content": "Of course", "type": "text" }, { "bbox": [ 115, 600, 146, 609 ], "score": 0.74, "content": "\\mathcal { L } _ { \\mathrm { E C F M } }", "type": "inline_equation" }, { "bbox": [ 146, 597, 541, 610 ], "score": 1.0, "content": "leaves the question of sampling open for high-dimensional spaces. Sampling uniformly does", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 608, 514, 624 ], "spans": [ { "bbox": [ 69, 608, 514, 624 ], "score": 1.0, "content": "not scale well to high dimensions, so for practical reasons we may want a different sampling strategy.", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "text", "bbox": [ 69, 627, 540, 652 ], "lines": [ { "bbox": [ 69, 626, 542, 641 ], "spans": [ { "bbox": [ 69, 626, 542, 641 ], "score": 1.0, "content": "We use this objective in Fig. D.9 with a uniform proposal distribution as a toy example of this type of", "type": "text" } ], "index": 34 }, { "bbox": [ 68, 638, 111, 654 ], "spans": [ { "bbox": [ 68, 638, 111, 654 ], "score": 1.0, "content": "training.", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "title", "bbox": [ 72, 665, 190, 680 ], "lines": [ { "bbox": [ 69, 664, 191, 682 ], "spans": [ { "bbox": [ 69, 664, 191, 682 ], "score": 1.0, "content": "D Additional results", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 70, 691, 412, 704 ], "lines": [ { "bbox": [ 69, 691, 414, 705 ], "spans": [ { "bbox": [ 69, 691, 414, 705 ], "score": 1.0, "content": "We start this section by the definition of the entropy regularized OT problem:", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "interline_equation", "bbox": [ 192, 710, 419, 735 ], "lines": [ { "bbox": [ 192, 710, 419, 735 ], "spans": [ { "bbox": [ 192, 710, 419, 735 ], "score": 0.92, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 , \\lambda } ^ { 2 } = \\operatorname * { i n f } _ { \\pi _ { \\lambda } \\in \\Pi } \\int _ { \\mathcal { X } ^ { 2 } } c ( x , y ) ^ { 2 } \\pi _ { \\lambda } ( d x , d y ) - 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With an", "type": "text" } ], "index": 29 }, { "bbox": [ 68, 560, 543, 575 ], "spans": [ { "bbox": [ 68, 560, 400, 575 ], "score": 1.0, "content": "application of Theorem 3.2 the gradients are equivalent up to a factor of", "type": "text" }, { "bbox": [ 400, 566, 419, 573 ], "score": 0.91, "content": "z _ { \\mathrm { 0 } } z _ { \\mathrm { 1 } }", "type": "inline_equation" }, { "bbox": [ 419, 560, 543, 575 ], "score": 1.0, "content": "which does not depend on", "type": "text" } ], "index": 30 }, { "bbox": [ 71, 573, 541, 586 ], "spans": [ { "bbox": [ 71, 578, 77, 583 ], "score": 0.87, "content": "x", "type": "inline_equation" }, { "bbox": [ 77, 574, 83, 586 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 529, 573, 541, 585 ], "score": 0.999, "content": "□", "type": "text" } ], "index": 31 } ], "index": 30, "bbox_fs": [ 68, 548, 543, 586 ] }, { "type": "text", "bbox": [ 72, 597, 541, 622 ], "lines": [ { "bbox": [ 70, 597, 541, 610 ], "spans": [ { "bbox": [ 70, 597, 114, 610 ], "score": 1.0, "content": "Of course", "type": "text" }, { "bbox": [ 115, 600, 146, 609 ], "score": 0.74, "content": "\\mathcal { L } _ { \\mathrm { E C F M } }", "type": "inline_equation" }, { "bbox": [ 146, 597, 541, 610 ], "score": 1.0, "content": "leaves the question of sampling open for high-dimensional spaces. Sampling uniformly does", "type": "text" } ], "index": 32 }, { "bbox": [ 69, 608, 514, 624 ], "spans": [ { "bbox": [ 69, 608, 514, 624 ], "score": 1.0, "content": "not scale well to high dimensions, so for practical reasons we may want a different sampling strategy.", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 69, 597, 541, 624 ] }, { "type": "text", "bbox": [ 69, 627, 540, 652 ], "lines": [ { "bbox": [ 69, 626, 542, 641 ], "spans": [ { "bbox": [ 69, 626, 542, 641 ], "score": 1.0, "content": "We use this objective in Fig. D.9 with a uniform proposal distribution as a toy example of this type of", "type": "text" } ], "index": 34 }, { "bbox": [ 68, 638, 111, 654 ], "spans": [ { "bbox": [ 68, 638, 111, 654 ], "score": 1.0, "content": "training.", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 68, 626, 542, 654 ] }, { "type": "title", "bbox": [ 72, 665, 190, 680 ], "lines": [ { "bbox": [ 69, 664, 191, 682 ], "spans": [ { "bbox": [ 69, 664, 191, 682 ], "score": 1.0, "content": "D Additional results", "type": "text" } ], "index": 36 } ], "index": 36 }, { "type": "text", "bbox": [ 70, 691, 412, 704 ], "lines": [ { "bbox": [ 69, 691, 414, 705 ], "spans": [ { "bbox": [ 69, 691, 414, 705 ], "score": 1.0, "content": "We start this section by the definition of the entropy regularized OT problem:", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 69, 691, 414, 705 ] }, { "type": "interline_equation", "bbox": [ 192, 710, 419, 735 ], "lines": [ { "bbox": [ 192, 710, 419, 735 ], "spans": [ { "bbox": [ 192, 710, 419, 735 ], "score": 0.92, "content": "W ( q _ { 0 } , q _ { 1 } ) _ { 2 , \\lambda } ^ { 2 } = \\operatorname * { i n f } _ { \\pi _ { \\lambda } \\in \\Pi } \\int _ { \\mathcal { X } ^ { 2 } } c ( x , y ) ^ { 2 } \\pi _ { \\lambda } ( d x , d y ) - \\lambda H ( \\pi ) ,", "type": "interline_equation", "image_path": "504bff277f9d373d0998521ce84e22e398ba8c2c104fc4688b7dbe8d62c87a0e.jpg" } ] } ], "index": 38, "virtual_lines": [ { "bbox": [ 192, 710, 419, 735 ], "spans": [], "index": 38 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 77, 98, 543, 246 ], "lines": [ { "bbox": [ 79, 97, 506, 112 ], "spans": [ { "bbox": [ 79, 97, 279, 112 ], "score": 1.0, "content": "Input: Empirical or samplable distributions", "type": "text" }, { "bbox": [ 279, 104, 302, 110 ], "score": 0.9, "content": "q _ { 0 } , q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 302, 97, 357, 112 ], "score": 1.0, "content": ", bandwidth", "type": "text" }, { "bbox": [ 357, 104, 363, 108 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 364, 97, 415, 112 ], "score": 1.0, "content": ", batch size", "type": "text" }, { "bbox": [ 416, 101, 420, 109 ], "score": 0.87, "content": "b", "type": "inline_equation" }, { "bbox": [ 421, 97, 492, 112 ], "score": 1.0, "content": ", initial network", "type": "text" }, { "bbox": [ 493, 104, 502, 110 ], "score": 0.85, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 503, 97, 506, 112 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 0 }, { "bbox": [ 80, 110, 166, 123 ], "spans": [ { "bbox": [ 80, 110, 166, 123 ], "score": 1.0, "content": "while Training do", "type": "text" } ], "index": 1 }, { "bbox": [ 95, 120, 544, 136 ], "spans": [ { "bbox": [ 95, 120, 320, 135 ], "score": 1.0, "content": "/* Sample batches of size b i.i.d. from the datasets", "type": "text" }, { "bbox": [ 528, 120, 544, 136 ], "score": 1.0, "content": "*/", "type": "text" } ], "index": 2 }, { "bbox": [ 96, 136, 219, 146 ], "spans": [ { "bbox": [ 96, 136, 219, 146 ], "score": 0.26, "content": "{ \\pmb x } _ { 0 } \\sim q _ { 0 } ( { \\pmb x } _ { 0 } ) ; \\quad { \\pmb x } _ { 1 } \\sim q _ { 1 } ( { \\pmb x } _ { 1 } )", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 96, 148, 224, 158 ], "spans": [ { "bbox": [ 96, 148, 224, 158 ], "score": 0.49, "content": "\\pi _ { 2 \\sigma ^ { 2 } } \\gets \\mathrm { S i n k h o r n } ( x _ { 1 } , x _ { 0 } , 2 \\sigma ^ { 2 } )", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 96, 160, 162, 171 ], "spans": [ { "bbox": [ 96, 160, 162, 171 ], "score": 0.58, "content": "( { \\pmb x } _ { 0 } , { \\pmb x } _ { 1 } ) \\sim \\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 96, 172, 143, 182 ], "spans": [ { "bbox": [ 96, 172, 143, 182 ], "score": 0.82, "content": "\\mathbf { \\boldsymbol { t } } \\sim \\mathcal { U } ( 0 , 1 )", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 96, 183, 190, 194 ], "spans": [ { "bbox": [ 96, 183, 190, 194 ], "score": 0.71, "content": "\\mu _ { t } t { \\pmb x } _ { 1 } + ( 1 - t ) { \\pmb x } _ { 0 }", "type": "inline_equation" } ], "index": 7 }, { "bbox": [ 96, 196, 196, 206 ], "spans": [ { "bbox": [ 96, 196, 196, 206 ], "score": 0.44, "content": "\\pmb { x } \\sim \\mathcal { N } ( \\mu _ { t } , \\sigma ^ { 2 } t ( 1 - t ) I )", "type": "inline_equation" } ], "index": 8 }, { "bbox": [ 90, 200, 542, 222 ], "spans": [ { "bbox": [ 90, 200, 335, 222 ], "score": 1.0, "content": "ut(x|z) ← 1−2t2t(1−t) (x − (tx1 + (1 − t)x0)) + (x1 − x0)", "type": "text" }, { "bbox": [ 483, 204, 542, 219 ], "score": 1.0, "content": "▷ From (21)", "type": "text" } ], "index": 9 }, { "bbox": [ 95, 221, 244, 232 ], "spans": [ { "bbox": [ 95, 221, 244, 232 ], "score": 0.34, "content": "\\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) \\gets | | v _ { \\theta } ( t , \\boldsymbol { x } ) - \\boldsymbol { u } _ { t } ( \\boldsymbol { x } | \\boldsymbol { z } ) | | ^ { 2 }", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 96, 233, 217, 245 ], "spans": [ { "bbox": [ 96, 233, 217, 245 ], "score": 0.45, "content": "\\theta \\gets \\mathrm { U p d a t e } ( \\theta , \\nabla _ { \\theta } \\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) )", "type": "inline_equation" } ], "index": 11 } ], "index": 5.5 }, { "type": "text", "bbox": [ 80, 246, 126, 256 ], "lines": [ { "bbox": [ 79, 244, 128, 257 ], "spans": [ { "bbox": [ 79, 244, 128, 257 ], "score": 1.0, "content": "return vθ", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "image", "bbox": [ 80, 277, 537, 549 ], "blocks": [ { "type": "image_body", "bbox": [ 80, 277, 537, 549 ], "group_id": 0, "lines": [ { "bbox": [ 80, 277, 537, 549 ], "spans": [ { "bbox": [ 80, 277, 537, 549 ], "score": 0.972, "type": "image", "image_path": "0912f90607adb9cde0e157c2b68abbd62e2159730aef614261f2d9ee38c617d0.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 80, 277, 537, 367.6666666666667 ], "spans": [], "index": 13 }, { "bbox": [ 80, 367.6666666666667, 537, 458.33333333333337 ], "spans": [], "index": 14 }, { "bbox": [ 80, 458.33333333333337, 537, 549.0 ], "spans": [], "index": 15 } ] }, { "type": "image_caption", "bbox": [ 71, 560, 541, 585 ], "group_id": 0, "lines": [ { "bbox": [ 68, 559, 540, 573 ], "spans": [ { "bbox": [ 68, 559, 306, 573 ], "score": 1.0, "content": "Figure D.1: Evaluation of regularization strength of", "type": "text" }, { "bbox": [ 307, 563, 317, 572 ], "score": 0.89, "content": "\\lambda _ { e }", "type": "inline_equation" }, { "bbox": [ 317, 559, 435, 573 ], "score": 1.0, "content": "over 6 seeds in the range", "type": "text" }, { "bbox": [ 435, 560, 495, 573 ], "score": 0.49, "content": "[ 0 , 1 0 ^ { - 5 } , 1 0 ^ { 2 } ]", "type": "inline_equation" }, { "bbox": [ 496, 559, 502, 573 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 502, 563, 540, 572 ], "score": 0.83, "content": "\\lambda _ { e } = 0 . 1", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 68, 572, 541, 585 ], "spans": [ { "bbox": [ 68, 572, 541, 585 ], "score": 1.0, "content": "performs the best in terms of minimizing path length and test error. We call this model \"Regularized CNF\".", "type": "text" } ], "index": 17 } ], "index": 16.5 } ], "index": 15.25 }, { "type": "text", "bbox": [ 70, 608, 286, 622 ], "lines": [ { "bbox": [ 69, 606, 287, 623 ], "spans": [ { "bbox": [ 69, 606, 99, 623 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 610, 131, 619 ], "score": 0.93, "content": "\\lambda \\in \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 132, 606, 154, 623 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 154, 610, 283, 622 ], "score": 0.92, "content": "\\begin{array} { r } { H ( \\pi ) = \\int \\ln \\pi ( x , y ) d \\pi ( d x , d y ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 283, 606, 287, 623 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 70, 636, 542, 697 ], "lines": [ { "bbox": [ 69, 636, 541, 649 ], "spans": [ { "bbox": [ 69, 636, 541, 649 ], "score": 1.0, "content": "Regularized CNF tuning Continuous normalizing flows with a path length penalty optimize a relaxed", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 649, 541, 661 ], "spans": [ { "bbox": [ 69, 649, 541, 661 ], "score": 1.0, "content": "form of a dynamic optimal transport problem (Tong et al., 2020; Finlay et al., 2020; Onken et al., 2021).", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 659, 542, 674 ], "spans": [ { "bbox": [ 69, 659, 542, 674 ], "score": 1.0, "content": "Where dynamic optimal transport solves for the optimal vector field in terms of average path length where", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 672, 542, 686 ], "spans": [ { "bbox": [ 69, 672, 166, 686 ], "score": 1.0, "content": "the marginals at time", "type": "text" }, { "bbox": [ 167, 675, 189, 682 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 189, 672, 210, 686 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 675, 232, 682 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 233, 672, 430, 686 ], "score": 1.0, "content": "are constrained to equal two input marginals", "type": "text" }, { "bbox": [ 431, 677, 439, 684 ], "score": 0.9, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 440, 672, 461, 686 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 462, 677, 471, 684 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 471, 672, 542, 686 ], "score": 1.0, "content": ". Instead of this", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 684, 518, 696 ], "spans": [ { "bbox": [ 69, 684, 308, 696 ], "score": 1.0, "content": "pair of hard constraints, regularized CNFs instead set", "type": "text" }, { "bbox": [ 308, 686, 379, 696 ], "score": 0.94, "content": "q _ { 0 } : = \\mathcal { N } ( x \\mid 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 379, 684, 518, 696 ], "score": 1.0, "content": "and optimize a loss of the form", "type": "text" } ], "index": 23 } ], "index": 21 }, { "type": "interline_equation", "bbox": [ 203, 708, 408, 735 ], "lines": [ { "bbox": [ 203, 708, 408, 735 ], "spans": [ { "bbox": [ 203, 708, 408, 735 ], "score": 0.93, "content": "L ( x ( t ) ) = - \\log p ( x ( t ) ) + \\lambda _ { e } \\int _ { 0 } ^ { 1 } \\| v _ { \\theta } ( t , x ( t ) ) \\| ^ { 2 } d t", "type": "interline_equation", "image_path": "44fcd5fb7f2f22fe1c8245d48f6c0b2e0b897328312ef692d9ac930b60cef916.jpg" } ] } ], "index": 24.5, "virtual_lines": [ { "bbox": [ 203, 708, 408, 721.5 ], "spans": [], "index": 24 }, { "bbox": [ 203, 721.5, 408, 735.0 ], "spans": [], "index": 25 } ] } ], "page_idx": 24, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 761 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "25", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 77, 98, 543, 246 ], "lines": [ { "bbox": [ 79, 97, 506, 112 ], "spans": [ { "bbox": [ 79, 97, 279, 112 ], "score": 1.0, "content": "Input: Empirical or samplable distributions", "type": "text" }, { "bbox": [ 279, 104, 302, 110 ], "score": 0.9, "content": "q _ { 0 } , q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 302, 97, 357, 112 ], "score": 1.0, "content": ", bandwidth", "type": "text" }, { "bbox": [ 357, 104, 363, 108 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 364, 97, 415, 112 ], "score": 1.0, "content": ", batch size", "type": "text" }, { "bbox": [ 416, 101, 420, 109 ], "score": 0.87, "content": "b", "type": "inline_equation" }, { "bbox": [ 421, 97, 492, 112 ], "score": 1.0, "content": ", initial network", "type": "text" }, { "bbox": [ 493, 104, 502, 110 ], "score": 0.85, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 503, 97, 506, 112 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 0, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 80, 110, 166, 123 ], "spans": [ { "bbox": [ 80, 110, 166, 123 ], "score": 1.0, "content": "while Training do", "type": "text" } ], "index": 1, "is_list_start_line": true, "is_list_end_line": true }, { "bbox": [ 95, 120, 544, 136 ], "spans": [ { "bbox": [ 95, 120, 320, 135 ], "score": 1.0, "content": "/* Sample batches of size b i.i.d. from the datasets", "type": "text" }, { "bbox": [ 528, 120, 544, 136 ], "score": 1.0, "content": "*/", "type": "text" } ], "index": 2 }, { "bbox": [ 96, 136, 219, 146 ], "spans": [ { "bbox": [ 96, 136, 219, 146 ], "score": 0.26, "content": "{ \\pmb x } _ { 0 } \\sim q _ { 0 } ( { \\pmb x } _ { 0 } ) ; \\quad { \\pmb x } _ { 1 } \\sim q _ { 1 } ( { \\pmb x } _ { 1 } )", "type": "inline_equation" } ], "index": 3, "is_list_end_line": true }, { "bbox": [ 96, 148, 224, 158 ], "spans": [ { "bbox": [ 96, 148, 224, 158 ], "score": 0.49, "content": "\\pi _ { 2 \\sigma ^ { 2 } } \\gets \\mathrm { S i n k h o r n } ( x _ { 1 } , x _ { 0 } , 2 \\sigma ^ { 2 } )", "type": "inline_equation" } ], "index": 4, "is_list_end_line": true }, { "bbox": [ 96, 160, 162, 171 ], "spans": [ { "bbox": [ 96, 160, 162, 171 ], "score": 0.58, "content": "( { \\pmb x } _ { 0 } , { \\pmb x } _ { 1 } ) \\sim \\pi _ { 2 \\sigma ^ { 2 } }", "type": "inline_equation" } ], "index": 5, "is_list_end_line": true }, { "bbox": [ 96, 172, 143, 182 ], "spans": [ { "bbox": [ 96, 172, 143, 182 ], "score": 0.82, "content": "\\mathbf { \\boldsymbol { t } } \\sim \\mathcal { U } ( 0 , 1 )", "type": "inline_equation" } ], "index": 6, "is_list_end_line": true }, { "bbox": [ 96, 183, 190, 194 ], "spans": [ { "bbox": [ 96, 183, 190, 194 ], "score": 0.71, "content": "\\mu _ { t } t { \\pmb x } _ { 1 } + ( 1 - t ) { \\pmb x } _ { 0 }", "type": "inline_equation" } ], "index": 7, "is_list_end_line": true }, { "bbox": [ 96, 196, 196, 206 ], "spans": [ { "bbox": [ 96, 196, 196, 206 ], "score": 0.44, "content": "\\pmb { x } \\sim \\mathcal { N } ( \\mu _ { t } , \\sigma ^ { 2 } t ( 1 - t ) I )", "type": "inline_equation" } ], "index": 8, "is_list_end_line": true }, { "bbox": [ 90, 200, 542, 222 ], "spans": [ { "bbox": [ 90, 200, 335, 222 ], "score": 1.0, "content": "ut(x|z) ← 1−2t2t(1−t) (x − (tx1 + (1 − t)x0)) + (x1 − x0)", "type": "text" }, { "bbox": [ 483, 204, 542, 219 ], "score": 1.0, "content": "▷ From (21)", "type": "text" } ], "index": 9 }, { "bbox": [ 95, 221, 244, 232 ], "spans": [ { "bbox": [ 95, 221, 244, 232 ], "score": 0.34, "content": "\\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) \\gets | | v _ { \\theta } ( t , \\boldsymbol { x } ) - \\boldsymbol { u } _ { t } ( \\boldsymbol { x } | \\boldsymbol { z } ) | | ^ { 2 }", "type": "inline_equation" } ], "index": 10, "is_list_end_line": true }, { "bbox": [ 96, 233, 217, 245 ], "spans": [ { "bbox": [ 96, 233, 217, 245 ], "score": 0.45, "content": "\\theta \\gets \\mathrm { U p d a t e } ( \\theta , \\nabla _ { \\theta } \\mathcal { L } _ { \\mathrm { C F M } } ( \\theta ) )", "type": "inline_equation" } ], "index": 11, "is_list_end_line": true } ], "index": 5.5, "bbox_fs": [ 79, 97, 544, 245 ] }, { "type": "text", "bbox": [ 80, 246, 126, 256 ], "lines": [ { "bbox": [ 79, 244, 128, 257 ], "spans": [ { "bbox": [ 79, 244, 128, 257 ], "score": 1.0, "content": "return vθ", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 79, 244, 128, 257 ] }, { "type": "image", "bbox": [ 80, 277, 537, 549 ], "blocks": [ { "type": "image_body", "bbox": [ 80, 277, 537, 549 ], "group_id": 0, "lines": [ { "bbox": [ 80, 277, 537, 549 ], "spans": [ { "bbox": [ 80, 277, 537, 549 ], "score": 0.972, "type": "image", "image_path": "0912f90607adb9cde0e157c2b68abbd62e2159730aef614261f2d9ee38c617d0.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 80, 277, 537, 367.6666666666667 ], "spans": [], "index": 13 }, { "bbox": [ 80, 367.6666666666667, 537, 458.33333333333337 ], "spans": [], "index": 14 }, { "bbox": [ 80, 458.33333333333337, 537, 549.0 ], "spans": [], "index": 15 } ] }, { "type": "image_caption", "bbox": [ 71, 560, 541, 585 ], "group_id": 0, "lines": [ { "bbox": [ 68, 559, 540, 573 ], "spans": [ { "bbox": [ 68, 559, 306, 573 ], "score": 1.0, "content": "Figure D.1: Evaluation of regularization strength of", "type": "text" }, { "bbox": [ 307, 563, 317, 572 ], "score": 0.89, "content": "\\lambda _ { e }", "type": "inline_equation" }, { "bbox": [ 317, 559, 435, 573 ], "score": 1.0, "content": "over 6 seeds in the range", "type": "text" }, { "bbox": [ 435, 560, 495, 573 ], "score": 0.49, "content": "[ 0 , 1 0 ^ { - 5 } , 1 0 ^ { 2 } ]", "type": "inline_equation" }, { "bbox": [ 496, 559, 502, 573 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 502, 563, 540, 572 ], "score": 0.83, "content": "\\lambda _ { e } = 0 . 1", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 68, 572, 541, 585 ], "spans": [ { "bbox": [ 68, 572, 541, 585 ], "score": 1.0, "content": "performs the best in terms of minimizing path length and test error. We call this model \"Regularized CNF\".", "type": "text" } ], "index": 17 } ], "index": 16.5 } ], "index": 15.25 }, { "type": "text", "bbox": [ 70, 608, 286, 622 ], "lines": [ { "bbox": [ 69, 606, 287, 623 ], "spans": [ { "bbox": [ 69, 606, 99, 623 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 610, 131, 619 ], "score": 0.93, "content": "\\lambda \\in \\mathbb { R } ^ { + }", "type": "inline_equation" }, { "bbox": [ 132, 606, 154, 623 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 154, 610, 283, 622 ], "score": 0.92, "content": "\\begin{array} { r } { H ( \\pi ) = \\int \\ln \\pi ( x , y ) d \\pi ( d x , d y ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 283, 606, 287, 623 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 18 } ], "index": 18, "bbox_fs": [ 69, 606, 287, 623 ] }, { "type": "text", "bbox": [ 70, 636, 542, 697 ], "lines": [ { "bbox": [ 69, 636, 541, 649 ], "spans": [ { "bbox": [ 69, 636, 541, 649 ], "score": 1.0, "content": "Regularized CNF tuning Continuous normalizing flows with a path length penalty optimize a relaxed", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 649, 541, 661 ], "spans": [ { "bbox": [ 69, 649, 541, 661 ], "score": 1.0, "content": "form of a dynamic optimal transport problem (Tong et al., 2020; Finlay et al., 2020; Onken et al., 2021).", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 659, 542, 674 ], "spans": [ { "bbox": [ 69, 659, 542, 674 ], "score": 1.0, "content": "Where dynamic optimal transport solves for the optimal vector field in terms of average path length where", "type": "text" } ], "index": 21 }, { "bbox": [ 69, 672, 542, 686 ], "spans": [ { "bbox": [ 69, 672, 166, 686 ], "score": 1.0, "content": "the marginals at time", "type": "text" }, { "bbox": [ 167, 675, 189, 682 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 189, 672, 210, 686 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 210, 675, 232, 682 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 233, 672, 430, 686 ], "score": 1.0, "content": "are constrained to equal two input marginals", "type": "text" }, { "bbox": [ 431, 677, 439, 684 ], "score": 0.9, "content": "q _ { 0 }", "type": "inline_equation" }, { "bbox": [ 440, 672, 461, 686 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 462, 677, 471, 684 ], "score": 0.89, "content": "q _ { 1 }", "type": "inline_equation" }, { "bbox": [ 471, 672, 542, 686 ], "score": 1.0, "content": ". Instead of this", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 684, 518, 696 ], "spans": [ { "bbox": [ 69, 684, 308, 696 ], "score": 1.0, "content": "pair of hard constraints, regularized CNFs instead set", "type": "text" }, { "bbox": [ 308, 686, 379, 696 ], "score": 0.94, "content": "q _ { 0 } : = \\mathcal { N } ( x \\mid 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 379, 684, 518, 696 ], "score": 1.0, "content": "and optimize a loss of the form", "type": "text" } ], "index": 23 } ], "index": 21, "bbox_fs": [ 69, 636, 542, 696 ] }, { "type": "interline_equation", "bbox": [ 203, 708, 408, 735 ], "lines": [ { "bbox": [ 203, 708, 408, 735 ], "spans": [ { "bbox": [ 203, 708, 408, 735 ], "score": 0.93, "content": "L ( x ( t ) ) = - \\log p ( x ( t ) ) + \\lambda _ { e } \\int _ { 0 } ^ { 1 } \\| v _ { \\theta } ( t , x ( t ) ) \\| ^ { 2 } d t", "type": "interline_equation", "image_path": "44fcd5fb7f2f22fe1c8245d48f6c0b2e0b897328312ef692d9ac930b60cef916.jpg" } ] } ], "index": 24.5, "virtual_lines": [ { "bbox": [ 203, 708, 408, 721.5 ], "spans": [], "index": 24 }, { "bbox": [ 203, 721.5, 408, 735.0 ], "spans": [], "index": 25 } ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 73, 117, 538, 239 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 116 ], "group_id": 0, "lines": [ { "bbox": [ 70, 80, 541, 92 ], "spans": [ { "bbox": [ 70, 80, 300, 92 ], "score": 1.0, "content": "Table D.1: Mean training time till convergence in", "type": "text" }, { "bbox": [ 300, 82, 315, 91 ], "score": 0.86, "content": "1 0 ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 315, 80, 541, 92 ], "score": 1.0, "content": "seconds over 5 seeds, with the exception of DSB,", "type": "text" } ], "index": 0 }, { "bbox": [ 69, 91, 542, 106 ], "spans": [ { "bbox": [ 69, 91, 542, 106 ], "score": 1.0, "content": "trained over 1 seed. CFM variants and DSB are trained on a single CPU with 5GB of memory where other", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 104, 541, 118 ], "spans": [ { "bbox": [ 69, 104, 541, 118 ], "score": 1.0, "content": "baselines are given two CPUs and one GPU. CFM, with significantly fewer resources, still trains the fastest.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 73, 117, 538, 239 ], "group_id": 0, "lines": [ { "bbox": [ 73, 117, 538, 239 ], "spans": [ { "bbox": [ 73, 117, 538, 239 ], "score": 0.98, "html": "
N-→8gaussiansmoons-→8gaussiansN→moonsN-→scurvemean
OT-CFM1.284 ± 0.0281.587 ± 0.2041.464 ± 0.1581.499 ± 0.1571.484 ± 0.192
CFM0.993 ± 0.0211.102 ± 0.1711.059 ± 0.1581.008 ± 0.1061.046 ± 0.132
FM0.839 ± 0.0961.076 ± 0.1261.127 ± 0.1231.014 ± 0.170
SB-CFM0.713 ± 0.3860.794 ± 0.2931.143 ± 0.3891.230 ± 0.4240.935 ± 0.397
Reg. CNF2.684 ± 0.0529.154 ± 1.5359.022 ± 3.2078.021 ± 3.288
CNF1.512 ± 0.23417.124 ± 4.39827.416 ± 13.29918.810 ± 12.677
ICNN3.712 ± 0.0913.046 ± 0.4962.558 ± 0.3902.200 ± 0.0342.912 ± 0.626
DSB5.418 ±-5.682 ±-5.428±-5.560 ±-5.522 ±-
", "type": "table", "image_path": "7b519cc93d11ef1a784e13f2a44914aac7d63ead5ad2c42aa17308f58c887e59.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 73, 117, 538, 157.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 73, 157.66666666666666, 538, 198.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 73, 198.33333333333331, 538, 238.99999999999997 ], "spans": [], "index": 5 } ] } ], "index": 2.5 }, { "type": "text", "bbox": [ 70, 258, 295, 271 ], "lines": [ { "bbox": [ 69, 257, 295, 273 ], "spans": [ { "bbox": [ 69, 258, 99, 271 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 259, 166, 272 ], "score": 0.92, "content": "\\begin{array} { r } { \\frac { d x } { d t } = v _ { \\theta } ( t , x ( t ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 167, 257, 188, 273 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 189, 261, 237, 271 ], "score": 0.93, "content": "\\log p ( x ( T ) )", "type": "inline_equation" }, { "bbox": [ 237, 257, 295, 273 ], "score": 1.0, "content": "is defined as", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 141, 280, 470, 308 ], "lines": [ { "bbox": [ 141, 280, 470, 308 ], "spans": [ { "bbox": [ 141, 280, 470, 308 ], "score": 0.91, "content": "\\log p ( x ( T ) ) = p ( x ( 0 ) ) + \\int _ { 0 } ^ { T } { \\frac { \\partial \\log p ( x ( t ) ) } { \\partial t } } d t = p ( x ( 0 ) ) + \\int _ { 0 } ^ { T } - \\operatorname { t r } \\left( { \\frac { d v _ { \\theta } } { d x ( t ) } } \\right) d t", "type": "interline_equation", "image_path": "8a29ff9059564cbcc7fd17f246445db556e2d4fb83c6f499ef65e18168df99fd.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 141, 280, 470, 308 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 315, 541, 399 ], "lines": [ { "bbox": [ 68, 315, 543, 330 ], "spans": [ { "bbox": [ 68, 315, 543, 330 ], "score": 1.0, "content": "where the second equality follows from the instantaneous change of variables theorem (Chen et al., 2018,", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 327, 542, 341 ], "spans": [ { "bbox": [ 69, 327, 273, 341 ], "score": 1.0, "content": "Theorem 1). In practice it is difficult to pick a", "type": "text" }, { "bbox": [ 273, 330, 284, 339 ], "score": 0.92, "content": "\\lambda _ { e }", "type": "inline_equation" }, { "bbox": [ 284, 327, 542, 341 ], "score": 1.0, "content": "which both produces flows with short paths and allows the", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 339, 541, 353 ], "spans": [ { "bbox": [ 69, 339, 541, 353 ], "score": 1.0, "content": "model to fit the data well. We analyze the effect of this parameter over three datasets in Fig. D.1. In this", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 352, 542, 365 ], "spans": [ { "bbox": [ 69, 352, 542, 365 ], "score": 1.0, "content": "figure we analyze the Normalized 2-Wasserstein to the target distribution (which approaches 1 with good fit),", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 363, 541, 377 ], "spans": [ { "bbox": [ 69, 363, 541, 377 ], "score": 1.0, "content": "and the Normalized Path Energy (NPE). We find a tradeoff between short paths (Low NPE) and good fit", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 374, 542, 389 ], "spans": [ { "bbox": [ 69, 374, 211, 389 ], "score": 1.0, "content": "(Low 2-Wasserstein). We choose", "type": "text" }, { "bbox": [ 211, 378, 247, 387 ], "score": 0.92, "content": "\\lambda _ { e } = 0 . 1", "type": "inline_equation" }, { "bbox": [ 248, 374, 542, 389 ], "score": 1.0, "content": "as a good tradeoff across datasets, which has paths that are not too", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 388, 303, 400 ], "spans": [ { "bbox": [ 69, 388, 303, 400 ], "score": 1.0, "content": "much longer than optimal but also fits the data well.", "type": "text" } ], "index": 14 } ], "index": 11 }, { "type": "text", "bbox": [ 70, 411, 540, 496 ], "lines": [ { "bbox": [ 69, 411, 541, 425 ], "spans": [ { "bbox": [ 69, 411, 541, 425 ], "score": 1.0, "content": "Ablation results on batch size. Since we use Minibatch-OT for OT-CFM, when the minibatch size", "type": "text" } ], "index": 15 }, { "bbox": [ 68, 423, 541, 437 ], "spans": [ { "bbox": [ 68, 423, 541, 437 ], "score": 1.0, "content": "is equal to one, then OT-CFM is equivalent to CFM. This effect can be seen in Fig. D.2, where over four", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 435, 541, 449 ], "spans": [ { "bbox": [ 69, 435, 541, 449 ], "score": 1.0, "content": "datasets, OT-CFM starts with equal path length and approximately equal 2-Wasserstein. Then the normalized", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 448, 542, 461 ], "spans": [ { "bbox": [ 69, 448, 406, 461 ], "score": 1.0, "content": "path energy decreases surprisingly quickly plateauing after batchsize reaches", "type": "text" }, { "bbox": [ 406, 451, 424, 458 ], "score": 0.29, "content": "{ \\sim } 6 4", "type": "inline_equation" }, { "bbox": [ 425, 448, 542, 461 ], "score": 1.0, "content": ". While the minibatch size", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 460, 541, 473 ], "spans": [ { "bbox": [ 69, 460, 541, 473 ], "score": 1.0, "content": "needed to approximate the true dynamic optimal transport paths will vary with dataset (for example in the", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 471, 541, 485 ], "spans": [ { "bbox": [ 69, 471, 541, 485 ], "score": 1.0, "content": "moon-8gaussian case we need a larger batch size) it is still somewhat surprising that such small batches are", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 484, 393, 496 ], "spans": [ { "bbox": [ 70, 484, 187, 496 ], "score": 1.0, "content": "needed as this is less than", "type": "text" }, { "bbox": [ 187, 485, 209, 494 ], "score": 0.66, "content": "0 . 5 \\%", "type": "inline_equation" }, { "bbox": [ 209, 484, 393, 496 ], "score": 1.0, "content": "of the entire 10k point dataset per batch.", "type": "text" } ], "index": 21 } ], "index": 18 }, { "type": "text", "bbox": [ 70, 507, 541, 592 ], "lines": [ { "bbox": [ 69, 508, 542, 521 ], "spans": [ { "bbox": [ 69, 508, 137, 521 ], "score": 1.0, "content": "The effect of", "type": "text" }, { "bbox": [ 137, 513, 144, 518 ], "score": 0.82, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 144, 508, 346, 521 ], "score": 1.0, "content": "on fit and path length. Next we consider", "type": "text" }, { "bbox": [ 347, 513, 353, 518 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 353, 508, 542, 521 ], "score": 1.0, "content": ", the bandwidth parameter of the Gaussian", "type": "text" } ], "index": 22 }, { "bbox": [ 68, 518, 542, 535 ], "spans": [ { "bbox": [ 68, 518, 363, 535 ], "score": 1.0, "content": "conditional probability path. In Fig. D.3 we study the effect of", "type": "text" }, { "bbox": [ 364, 525, 370, 530 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 370, 518, 542, 535 ], "score": 1.0, "content": "on the fit (top) and the path energy", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 532, 542, 546 ], "spans": [ { "bbox": [ 69, 532, 146, 546 ], "score": 1.0, "content": "(bottom). With", "type": "text" }, { "bbox": [ 147, 535, 174, 542 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 174, 532, 542, 546 ], "score": 1.0, "content": "methods start to underfit with high 2-Wasserstein error and either very long or", "type": "text" } ], "index": 24 }, { "bbox": [ 68, 544, 542, 556 ], "spans": [ { "bbox": [ 68, 544, 405, 556 ], "score": 1.0, "content": "very short paths. As for specific models, SB-CFM becomes unstable with", "type": "text" }, { "bbox": [ 405, 549, 411, 554 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 411, 544, 542, 556 ], "score": 1.0, "content": "too small due to the lack of", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 555, 541, 569 ], "spans": [ { "bbox": [ 69, 555, 541, 569 ], "score": 1.0, "content": "convergence for the static Sinkhorn optimization with small regularization. FM and CFM follow similar", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 569, 542, 581 ], "spans": [ { "bbox": [ 70, 569, 231, 581 ], "score": 1.0, "content": "trends where they fit fairly well with", "type": "text" }, { "bbox": [ 232, 571, 256, 579 ], "score": 0.92, "content": "\\sigma \\leq 1", "type": "inline_equation" }, { "bbox": [ 257, 569, 542, 581 ], "score": 1.0, "content": "but have paths that are significantly longer than optimal by 2-3x.", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 580, 425, 594 ], "spans": [ { "bbox": [ 70, 580, 395, 594 ], "score": 1.0, "content": "OT-CFM maintains near optimal path energies and near optimal fit until", "type": "text" }, { "bbox": [ 395, 583, 420, 590 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 420, 580, 425, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 } ], "index": 25 }, { "type": "text", "bbox": [ 71, 603, 540, 640 ], "lines": [ { "bbox": [ 70, 602, 541, 618 ], "spans": [ { "bbox": [ 70, 602, 541, 618 ], "score": 1.0, "content": "Schrödinger bridge fit over simulation time. In Fig. D.7 we compare the fit of Diffusion Schrödinger", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 614, 541, 631 ], "spans": [ { "bbox": [ 70, 614, 541, 631 ], "score": 1.0, "content": "Bridge model with SB-CFM conditioned on time. The Diffusion Schrödinger Bridge seems to outperform", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 627, 474, 642 ], "spans": [ { "bbox": [ 69, 627, 474, 642 ], "score": 1.0, "content": "SB-CFM early in the trajectory, however fails to fit the bridge after many integration steps.", "type": "text" } ], "index": 31 } ], "index": 30 }, { "type": "title", "bbox": [ 71, 653, 185, 666 ], "lines": [ { "bbox": [ 69, 651, 187, 668 ], "spans": [ { "bbox": [ 69, 651, 187, 668 ], "score": 1.0, "content": "D.1 Objective variance.", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 70, 674, 541, 711 ], "lines": [ { "bbox": [ 68, 673, 541, 689 ], "spans": [ { "bbox": [ 68, 673, 251, 689 ], "score": 1.0, "content": "We consider the variance of the objective", "type": "text" }, { "bbox": [ 252, 677, 282, 687 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 282, 673, 353, 689 ], "score": 1.0, "content": "with respect to", "type": "text" }, { "bbox": [ 353, 680, 359, 685 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 359, 673, 428, 689 ], "score": 1.0, "content": ". While for any", "type": "text" }, { "bbox": [ 428, 680, 434, 685 ], "score": 0.91, "content": "x", "type": "inline_equation" }, { "bbox": [ 434, 673, 474, 689 ], "score": 1.0, "content": "we have", "type": "text" }, { "bbox": [ 474, 677, 541, 687 ], "score": 0.92, "content": "\\mathbb { E } _ { q } ( z ) u _ { t } ( x | z ) =", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 71, 686, 541, 700 ], "spans": [ { "bbox": [ 71, 689, 94, 699 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 94, 686, 541, 700 ], "score": 1.0, "content": ", we find a lower second moment speeds up training. Specifically, we seek to understand the effect of the", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 699, 346, 712 ], "spans": [ { "bbox": [ 69, 699, 346, 712 ], "score": 1.0, "content": "second moment which we call the objective variance defined as", "type": "text" } ], "index": 35 } ], "index": 34 }, { "type": "interline_equation", "bbox": [ 201, 719, 410, 734 ], "lines": [ { "bbox": [ 201, 719, 410, 734 ], "spans": [ { "bbox": [ 201, 719, 410, 734 ], "score": 0.9, "content": "O V = \\mathbb { E } _ { t \\sim U ( 0 , I ) , x \\sim p _ { t } ( x ) , z \\sim q ( z ) } \\| u _ { t } ( x | z ) - u _ { t } ( x ) \\| ^ { 2 }", "type": "interline_equation", "image_path": "8be63dad3401938f4133a67322d4769b8e7431f7ab6b0dc2a85857e930d4808c.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 201, 719, 410, 734 ], "spans": [], "index": 36 } ] } ], "page_idx": 25, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 26, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "26", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 73, 117, 538, 239 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 116 ], "group_id": 0, "lines": [ { "bbox": [ 70, 80, 541, 92 ], "spans": [ { "bbox": [ 70, 80, 300, 92 ], "score": 1.0, "content": "Table D.1: Mean training time till convergence in", "type": "text" }, { "bbox": [ 300, 82, 315, 91 ], "score": 0.86, "content": "1 0 ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 315, 80, 541, 92 ], "score": 1.0, "content": "seconds over 5 seeds, with the exception of DSB,", "type": "text" } ], "index": 0 }, { "bbox": [ 69, 91, 542, 106 ], "spans": [ { "bbox": [ 69, 91, 542, 106 ], "score": 1.0, "content": "trained over 1 seed. CFM variants and DSB are trained on a single CPU with 5GB of memory where other", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 104, 541, 118 ], "spans": [ { "bbox": [ 69, 104, 541, 118 ], "score": 1.0, "content": "baselines are given two CPUs and one GPU. CFM, with significantly fewer resources, still trains the fastest.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 73, 117, 538, 239 ], "group_id": 0, "lines": [ { "bbox": [ 73, 117, 538, 239 ], "spans": [ { "bbox": [ 73, 117, 538, 239 ], "score": 0.98, "html": "
N-→8gaussiansmoons-→8gaussiansN→moonsN-→scurvemean
OT-CFM1.284 ± 0.0281.587 ± 0.2041.464 ± 0.1581.499 ± 0.1571.484 ± 0.192
CFM0.993 ± 0.0211.102 ± 0.1711.059 ± 0.1581.008 ± 0.1061.046 ± 0.132
FM0.839 ± 0.0961.076 ± 0.1261.127 ± 0.1231.014 ± 0.170
SB-CFM0.713 ± 0.3860.794 ± 0.2931.143 ± 0.3891.230 ± 0.4240.935 ± 0.397
Reg. CNF2.684 ± 0.0529.154 ± 1.5359.022 ± 3.2078.021 ± 3.288
CNF1.512 ± 0.23417.124 ± 4.39827.416 ± 13.29918.810 ± 12.677
ICNN3.712 ± 0.0913.046 ± 0.4962.558 ± 0.3902.200 ± 0.0342.912 ± 0.626
DSB5.418 ±-5.682 ±-5.428±-5.560 ±-5.522 ±-
", "type": "table", "image_path": "7b519cc93d11ef1a784e13f2a44914aac7d63ead5ad2c42aa17308f58c887e59.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 73, 117, 538, 157.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 73, 157.66666666666666, 538, 198.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 73, 198.33333333333331, 538, 238.99999999999997 ], "spans": [], "index": 5 } ] } ], "index": 2.5 }, { "type": "text", "bbox": [ 70, 258, 295, 271 ], "lines": [ { "bbox": [ 69, 257, 295, 273 ], "spans": [ { "bbox": [ 69, 258, 99, 271 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 259, 166, 272 ], "score": 0.92, "content": "\\begin{array} { r } { \\frac { d x } { d t } = v _ { \\theta } ( t , x ( t ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 167, 257, 188, 273 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 189, 261, 237, 271 ], "score": 0.93, "content": "\\log p ( x ( T ) )", "type": "inline_equation" }, { "bbox": [ 237, 257, 295, 273 ], "score": 1.0, "content": "is defined as", "type": "text" } ], "index": 6 } ], "index": 6, "bbox_fs": [ 69, 257, 295, 273 ] }, { "type": "interline_equation", "bbox": [ 141, 280, 470, 308 ], "lines": [ { "bbox": [ 141, 280, 470, 308 ], "spans": [ { "bbox": [ 141, 280, 470, 308 ], "score": 0.91, "content": "\\log p ( x ( T ) ) = p ( x ( 0 ) ) + \\int _ { 0 } ^ { T } { \\frac { \\partial \\log p ( x ( t ) ) } { \\partial t } } d t = p ( x ( 0 ) ) + \\int _ { 0 } ^ { T } - \\operatorname { t r } \\left( { \\frac { d v _ { \\theta } } { d x ( t ) } } \\right) d t", "type": "interline_equation", "image_path": "8a29ff9059564cbcc7fd17f246445db556e2d4fb83c6f499ef65e18168df99fd.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 141, 280, 470, 308 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 315, 541, 399 ], "lines": [ { "bbox": [ 68, 315, 543, 330 ], "spans": [ { "bbox": [ 68, 315, 543, 330 ], "score": 1.0, "content": "where the second equality follows from the instantaneous change of variables theorem (Chen et al., 2018,", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 327, 542, 341 ], "spans": [ { "bbox": [ 69, 327, 273, 341 ], "score": 1.0, "content": "Theorem 1). In practice it is difficult to pick a", "type": "text" }, { "bbox": [ 273, 330, 284, 339 ], "score": 0.92, "content": "\\lambda _ { e }", "type": "inline_equation" }, { "bbox": [ 284, 327, 542, 341 ], "score": 1.0, "content": "which both produces flows with short paths and allows the", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 339, 541, 353 ], "spans": [ { "bbox": [ 69, 339, 541, 353 ], "score": 1.0, "content": "model to fit the data well. We analyze the effect of this parameter over three datasets in Fig. D.1. In this", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 352, 542, 365 ], "spans": [ { "bbox": [ 69, 352, 542, 365 ], "score": 1.0, "content": "figure we analyze the Normalized 2-Wasserstein to the target distribution (which approaches 1 with good fit),", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 363, 541, 377 ], "spans": [ { "bbox": [ 69, 363, 541, 377 ], "score": 1.0, "content": "and the Normalized Path Energy (NPE). We find a tradeoff between short paths (Low NPE) and good fit", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 374, 542, 389 ], "spans": [ { "bbox": [ 69, 374, 211, 389 ], "score": 1.0, "content": "(Low 2-Wasserstein). We choose", "type": "text" }, { "bbox": [ 211, 378, 247, 387 ], "score": 0.92, "content": "\\lambda _ { e } = 0 . 1", "type": "inline_equation" }, { "bbox": [ 248, 374, 542, 389 ], "score": 1.0, "content": "as a good tradeoff across datasets, which has paths that are not too", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 388, 303, 400 ], "spans": [ { "bbox": [ 69, 388, 303, 400 ], "score": 1.0, "content": "much longer than optimal but also fits the data well.", "type": "text" } ], "index": 14 } ], "index": 11, "bbox_fs": [ 68, 315, 543, 400 ] }, { "type": "text", "bbox": [ 70, 411, 540, 496 ], "lines": [ { "bbox": [ 69, 411, 541, 425 ], "spans": [ { "bbox": [ 69, 411, 541, 425 ], "score": 1.0, "content": "Ablation results on batch size. Since we use Minibatch-OT for OT-CFM, when the minibatch size", "type": "text" } ], "index": 15 }, { "bbox": [ 68, 423, 541, 437 ], "spans": [ { "bbox": [ 68, 423, 541, 437 ], "score": 1.0, "content": "is equal to one, then OT-CFM is equivalent to CFM. This effect can be seen in Fig. D.2, where over four", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 435, 541, 449 ], "spans": [ { "bbox": [ 69, 435, 541, 449 ], "score": 1.0, "content": "datasets, OT-CFM starts with equal path length and approximately equal 2-Wasserstein. Then the normalized", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 448, 542, 461 ], "spans": [ { "bbox": [ 69, 448, 406, 461 ], "score": 1.0, "content": "path energy decreases surprisingly quickly plateauing after batchsize reaches", "type": "text" }, { "bbox": [ 406, 451, 424, 458 ], "score": 0.29, "content": "{ \\sim } 6 4", "type": "inline_equation" }, { "bbox": [ 425, 448, 542, 461 ], "score": 1.0, "content": ". While the minibatch size", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 460, 541, 473 ], "spans": [ { "bbox": [ 69, 460, 541, 473 ], "score": 1.0, "content": "needed to approximate the true dynamic optimal transport paths will vary with dataset (for example in the", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 471, 541, 485 ], "spans": [ { "bbox": [ 69, 471, 541, 485 ], "score": 1.0, "content": "moon-8gaussian case we need a larger batch size) it is still somewhat surprising that such small batches are", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 484, 393, 496 ], "spans": [ { "bbox": [ 70, 484, 187, 496 ], "score": 1.0, "content": "needed as this is less than", "type": "text" }, { "bbox": [ 187, 485, 209, 494 ], "score": 0.66, "content": "0 . 5 \\%", "type": "inline_equation" }, { "bbox": [ 209, 484, 393, 496 ], "score": 1.0, "content": "of the entire 10k point dataset per batch.", "type": "text" } ], "index": 21 } ], "index": 18, "bbox_fs": [ 68, 411, 542, 496 ] }, { "type": "text", "bbox": [ 70, 507, 541, 592 ], "lines": [ { "bbox": [ 69, 508, 542, 521 ], "spans": [ { "bbox": [ 69, 508, 137, 521 ], "score": 1.0, "content": "The effect of", "type": "text" }, { "bbox": [ 137, 513, 144, 518 ], "score": 0.82, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 144, 508, 346, 521 ], "score": 1.0, "content": "on fit and path length. Next we consider", "type": "text" }, { "bbox": [ 347, 513, 353, 518 ], "score": 0.88, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 353, 508, 542, 521 ], "score": 1.0, "content": ", the bandwidth parameter of the Gaussian", "type": "text" } ], "index": 22 }, { "bbox": [ 68, 518, 542, 535 ], "spans": [ { "bbox": [ 68, 518, 363, 535 ], "score": 1.0, "content": "conditional probability path. In Fig. D.3 we study the effect of", "type": "text" }, { "bbox": [ 364, 525, 370, 530 ], "score": 0.89, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 370, 518, 542, 535 ], "score": 1.0, "content": "on the fit (top) and the path energy", "type": "text" } ], "index": 23 }, { "bbox": [ 69, 532, 542, 546 ], "spans": [ { "bbox": [ 69, 532, 146, 546 ], "score": 1.0, "content": "(bottom). With", "type": "text" }, { "bbox": [ 147, 535, 174, 542 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 174, 532, 542, 546 ], "score": 1.0, "content": "methods start to underfit with high 2-Wasserstein error and either very long or", "type": "text" } ], "index": 24 }, { "bbox": [ 68, 544, 542, 556 ], "spans": [ { "bbox": [ 68, 544, 405, 556 ], "score": 1.0, "content": "very short paths. As for specific models, SB-CFM becomes unstable with", "type": "text" }, { "bbox": [ 405, 549, 411, 554 ], "score": 0.9, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 411, 544, 542, 556 ], "score": 1.0, "content": "too small due to the lack of", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 555, 541, 569 ], "spans": [ { "bbox": [ 69, 555, 541, 569 ], "score": 1.0, "content": "convergence for the static Sinkhorn optimization with small regularization. FM and CFM follow similar", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 569, 542, 581 ], "spans": [ { "bbox": [ 70, 569, 231, 581 ], "score": 1.0, "content": "trends where they fit fairly well with", "type": "text" }, { "bbox": [ 232, 571, 256, 579 ], "score": 0.92, "content": "\\sigma \\leq 1", "type": "inline_equation" }, { "bbox": [ 257, 569, 542, 581 ], "score": 1.0, "content": "but have paths that are significantly longer than optimal by 2-3x.", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 580, 425, 594 ], "spans": [ { "bbox": [ 70, 580, 395, 594 ], "score": 1.0, "content": "OT-CFM maintains near optimal path energies and near optimal fit until", "type": "text" }, { "bbox": [ 395, 583, 420, 590 ], "score": 0.91, "content": "\\sigma > 1", "type": "inline_equation" }, { "bbox": [ 420, 580, 425, 594 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 } ], "index": 25, "bbox_fs": [ 68, 508, 542, 594 ] }, { "type": "text", "bbox": [ 71, 603, 540, 640 ], "lines": [ { "bbox": [ 70, 602, 541, 618 ], "spans": [ { "bbox": [ 70, 602, 541, 618 ], "score": 1.0, "content": "Schrödinger bridge fit over simulation time. In Fig. D.7 we compare the fit of Diffusion Schrödinger", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 614, 541, 631 ], "spans": [ { "bbox": [ 70, 614, 541, 631 ], "score": 1.0, "content": "Bridge model with SB-CFM conditioned on time. The Diffusion Schrödinger Bridge seems to outperform", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 627, 474, 642 ], "spans": [ { "bbox": [ 69, 627, 474, 642 ], "score": 1.0, "content": "SB-CFM early in the trajectory, however fails to fit the bridge after many integration steps.", "type": "text" } ], "index": 31 } ], "index": 30, "bbox_fs": [ 69, 602, 541, 642 ] }, { "type": "title", "bbox": [ 71, 653, 185, 666 ], "lines": [ { "bbox": [ 69, 651, 187, 668 ], "spans": [ { "bbox": [ 69, 651, 187, 668 ], "score": 1.0, "content": "D.1 Objective variance.", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 70, 674, 541, 711 ], "lines": [ { "bbox": [ 68, 673, 541, 689 ], "spans": [ { "bbox": [ 68, 673, 251, 689 ], "score": 1.0, "content": "We consider the variance of the objective", "type": "text" }, { "bbox": [ 252, 677, 282, 687 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 282, 673, 353, 689 ], "score": 1.0, "content": "with respect to", "type": "text" }, { "bbox": [ 353, 680, 359, 685 ], "score": 0.88, "content": "z", "type": "inline_equation" }, { "bbox": [ 359, 673, 428, 689 ], "score": 1.0, "content": ". While for any", "type": "text" }, { "bbox": [ 428, 680, 434, 685 ], "score": 0.91, "content": "x", "type": "inline_equation" }, { "bbox": [ 434, 673, 474, 689 ], "score": 1.0, "content": "we have", "type": "text" }, { "bbox": [ 474, 677, 541, 687 ], "score": 0.92, "content": "\\mathbb { E } _ { q } ( z ) u _ { t } ( x | z ) =", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 71, 686, 541, 700 ], "spans": [ { "bbox": [ 71, 689, 94, 699 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 94, 686, 541, 700 ], "score": 1.0, "content": ", we find a lower second moment speeds up training. Specifically, we seek to understand the effect of the", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 699, 346, 712 ], "spans": [ { "bbox": [ 69, 699, 346, 712 ], "score": 1.0, "content": "second moment which we call the objective variance defined as", "type": "text" } ], "index": 35 } ], "index": 34, "bbox_fs": [ 68, 673, 541, 712 ] }, { "type": "interline_equation", "bbox": [ 201, 719, 410, 734 ], "lines": [ { "bbox": [ 201, 719, 410, 734 ], "spans": [ { "bbox": [ 201, 719, 410, 734 ], "score": 0.9, "content": "O V = \\mathbb { E } _ { t \\sim U ( 0 , I ) , x \\sim p _ { t } ( x ) , z \\sim q ( z ) } \\| u _ { t } ( x | z ) - u _ { t } ( x ) \\| ^ { 2 }", "type": "interline_equation", "image_path": "8be63dad3401938f4133a67322d4769b8e7431f7ab6b0dc2a85857e930d4808c.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 201, 719, 410, 734 ], "spans": [], "index": 36 } ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 78, 97, 539, 306 ], "blocks": [ { "type": "image_body", "bbox": [ 78, 97, 539, 306 ], "group_id": 0, "lines": [ { "bbox": [ 78, 97, 539, 306 ], "spans": [ { "bbox": [ 78, 97, 539, 306 ], "score": 0.973, "type": "image", "image_path": "6c13371fdd64c6578127bcd3f58570bfee43da54c34998fba88b258ff531bf48.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 78, 97, 539, 166.66666666666669 ], "spans": [], "index": 0 }, { "bbox": [ 78, 166.66666666666669, 539, 236.33333333333337 ], "spans": [], "index": 1 }, { "bbox": [ 78, 236.33333333333337, 539, 306.00000000000006 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 70, 317, 541, 354 ], "group_id": 0, "lines": [ { "bbox": [ 70, 317, 541, 331 ], "spans": [ { "bbox": [ 70, 317, 126, 331 ], "score": 1.0, "content": "Figure D.2:", "type": "text" }, { "bbox": [ 126, 319, 151, 330 ], "score": 0.86, "content": "\\mu \\pm \\sigma", "type": "inline_equation" }, { "bbox": [ 152, 317, 541, 331 ], "score": 1.0, "content": "of mean path length prediction error over 5 seeds. 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Variance reduction", "type": "text" } ], "index": 19 }, { "bbox": [ 70, 509, 541, 523 ], "spans": [ { "bbox": [ 70, 509, 541, 523 ], "score": 1.0, "content": "leads to faster training, especially for CFM where the objective variance is naturally larger than OT-CFM", "type": "text" } ], "index": 20 }, { "bbox": [ 69, 521, 234, 535 ], "spans": [ { "bbox": [ 69, 521, 234, 535 ], "score": 1.0, "content": "which sees a small performance gain.", "type": "text" } ], "index": 21 } ], "index": 19.5 } ], "index": 15.5 }, { "type": "image", "bbox": [ 77, 575, 539, 678 ], "blocks": [ { "type": "image_body", "bbox": [ 77, 575, 539, 678 ], "group_id": 2, "lines": [ { "bbox": [ 77, 575, 539, 678 ], "spans": [ { "bbox": [ 77, 575, 539, 678 ], "score": 0.965, "type": "image", "image_path": "f8e79e1230abf5b1da9ecf9c6dd8f2e8b60c98f50786ef712daa33be42169407.jpg" } ] } ], "index": 23, "virtual_lines": [ { "bbox": [ 77, 575, 539, 609.3333333333334 ], "spans": [], "index": 22 }, { "bbox": [ 77, 609.3333333333334, 539, 643.6666666666667 ], "spans": [], "index": 23 }, { "bbox": [ 77, 643.6666666666667, 539, 678.0000000000001 ], "spans": [], "index": 24 } ] }, { "type": "image_caption", "bbox": [ 68, 688, 539, 713 ], "group_id": 2, "lines": [ { "bbox": [ 70, 688, 541, 702 ], "spans": [ { "bbox": [ 70, 688, 541, 702 ], "score": 1.0, "content": "Figure D.6: Extended results from Fig. 2 (left) over two more datasets. OT-CFM is still consistently the", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 700, 190, 713 ], "spans": [ { "bbox": [ 69, 700, 190, 713 ], "score": 1.0, "content": "fastest converging method.", "type": "text" } ], "index": 26 } ], "index": 25.5 } ], "index": 24.25 } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 77, 83, 538, 185 ], "blocks": [ { "type": "image_body", "bbox": [ 77, 83, 538, 185 ], "group_id": 0, "lines": [ { "bbox": [ 77, 83, 538, 185 ], "spans": [ { "bbox": [ 77, 83, 538, 185 ], "score": 0.966, "type": "image", "image_path": "55b40fdd16153f131a9edc200d91f8bd2ec6bc077bd34dcae000ac4948306b7b.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 77, 83, 538, 117.0 ], "spans": [], "index": 0 }, { "bbox": [ 77, 117.0, 538, 151.0 ], "spans": [], "index": 1 }, { "bbox": [ 77, 151.0, 538, 185.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 71, 196, 537, 220 ], "group_id": 0, "lines": [ { "bbox": [ 69, 194, 540, 210 ], "spans": [ { "bbox": [ 69, 194, 540, 210 ], "score": 1.0, "content": "Figure D.7: 2-Wasserstein Error between trajectories and ground truth Schrödinger Bridge samples over", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 209, 144, 221 ], "spans": [ { "bbox": [ 70, 209, 144, 221 ], "score": 1.0, "content": "simulation time.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "image", "bbox": [ 100, 237, 506, 389 ], "blocks": [ { "type": "image_body", "bbox": [ 100, 237, 506, 389 ], "group_id": 1, "lines": [ { "bbox": [ 100, 237, 506, 389 ], "spans": [ { "bbox": [ 100, 237, 506, 389 ], "score": 0.971, "type": "image", "image_path": "12aa3bc53e7b3c6742a7eb5def0d4eb89ae3b891abf196e1c41a9febc0f5edb6.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 100, 237, 506, 287.6666666666667 ], "spans": [], "index": 5 }, { "bbox": [ 100, 287.6666666666667, 506, 338.33333333333337 ], "spans": [], "index": 6 }, { "bbox": [ 100, 338.33333333333337, 506, 389.00000000000006 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 70, 395, 540, 432 ], "group_id": 1, "lines": [ { "bbox": [ 70, 395, 541, 410 ], "spans": [ { "bbox": [ 70, 395, 541, 410 ], "score": 1.0, "content": "Figure D.8: (left) Variance of the objective for varying batch size. OT-CFM has a lower variance across", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 407, 541, 421 ], "spans": [ { "bbox": [ 69, 407, 433, 421 ], "score": 1.0, "content": "batch sizes. (right) Validation 2-Wasserstein performance with batch averaging as in", "type": "text" }, { "bbox": [ 433, 408, 455, 420 ], "score": 0.86, "content": "\\ S \\mathrm { C . 1 }", "type": "inline_equation" }, { "bbox": [ 455, 407, 541, 421 ], "score": 1.0, "content": ". Reducing variance", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 418, 194, 434 ], "spans": [ { "bbox": [ 69, 418, 194, 434 ], "score": 1.0, "content": "improves training efficiency.", "type": "text" } ], "index": 10 } ], "index": 9 } ], "index": 7.5 }, { "type": "text", "bbox": [ 70, 457, 541, 530 ], "lines": [ { "bbox": [ 68, 457, 542, 471 ], "spans": [ { "bbox": [ 68, 457, 542, 471 ], "score": 1.0, "content": "on training speed for different objectives in Table 1. We estimate the variance on a small data with a known", "type": "text" } ], "index": 11 }, { "bbox": [ 71, 470, 542, 483 ], "spans": [ { "bbox": [ 71, 471, 94, 482 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 470, 542, 483 ], "score": 1.0, "content": ". We examine this estimated objective variance and its effect on training convergence in Fig. D.8,", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 481, 541, 495 ], "spans": [ { "bbox": [ 69, 481, 541, 495 ], "score": 1.0, "content": "showing that either OT-CFM or variance reduced CFM with averaging over the batch results in lower variance", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 494, 541, 506 ], "spans": [ { "bbox": [ 70, 494, 541, 506 ], "score": 1.0, "content": "of the objective. This in turn leads to faster training times as shown on the right. Averaging over a batch of", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 505, 541, 519 ], "spans": [ { "bbox": [ 69, 505, 541, 519 ], "score": 1.0, "content": "data leads to faster training particularly for methods with high objective variance (CFM) and less so for", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 516, 320, 531 ], "spans": [ { "bbox": [ 69, 516, 320, 531 ], "score": 1.0, "content": "those with low (OT-CFM), which already trains quickly.", "type": "text" } ], "index": 16 } ], "index": 13.5 }, { "type": "text", "bbox": [ 69, 535, 538, 559 ], "lines": [ { "bbox": [ 69, 534, 541, 549 ], "spans": [ { "bbox": [ 69, 534, 262, 549 ], "score": 1.0, "content": "Variance in the conditional objective target", "type": "text" }, { "bbox": [ 262, 537, 293, 548 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 293, 534, 541, 549 ], "score": 1.0, "content": "varies across models. In Fig. D.4 we study the objective", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 546, 475, 561 ], "spans": [ { "bbox": [ 68, 546, 475, 561 ], "score": 1.0, "content": "variance across CFM objective functions. Here we estimate the objective variance in (36) as", "type": "text" } ], "index": 18 } ], "index": 17.5 }, { "type": "interline_equation", "bbox": [ 249, 573, 361, 587 ], "lines": [ { "bbox": [ 249, 573, 361, 587 ], "spans": [ { "bbox": [ 249, 573, 361, 587 ], "score": 0.91, "content": "\\mathbb { E } _ { x , t , z } \\Vert u _ { t } ( x | z ) - v _ { \\theta } ( t , x ) \\Vert ^ { 2 }", "type": "interline_equation", "image_path": "737dd7d12609fad952b4e31d9d7434ea341238c157569cd44b101dc8d66b5e62.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 249, 573, 361, 587 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 70, 600, 540, 649 ], "lines": [ { "bbox": [ 70, 600, 542, 614 ], "spans": [ { "bbox": [ 70, 600, 323, 614 ], "score": 1.0, "content": "after training has converged. After training has converged", "type": "text" }, { "bbox": [ 324, 606, 333, 612 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 333, 600, 437, 614 ], "score": 1.0, "content": "should be very close to", "type": "text" }, { "bbox": [ 438, 603, 460, 613 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 461, 600, 542, 614 ], "score": 1.0, "content": "so we use it as an", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 612, 541, 626 ], "spans": [ { "bbox": [ 68, 612, 167, 626 ], "score": 1.0, "content": "empirical estimator of", "type": "text" }, { "bbox": [ 167, 615, 190, 625 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 190, 612, 541, 626 ], "score": 1.0, "content": "to compute the variance. We find that across all datasets OT-CFM and SB-CFM", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 624, 541, 637 ], "spans": [ { "bbox": [ 70, 624, 541, 637 ], "score": 1.0, "content": "have at least an order of magnitude lower variance than CFM and FM objectives. This correlates with faster", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 636, 542, 650 ], "spans": [ { "bbox": [ 70, 636, 542, 650 ], "score": 1.0, "content": "training as measured by lower validation error in fewer steps for lower variance models as seen in Fig. 2 (left).", "type": "text" } ], "index": 23 } ], "index": 21.5 }, { "type": "text", "bbox": [ 70, 654, 540, 703 ], "lines": [ { "bbox": [ 69, 654, 541, 668 ], "spans": [ { "bbox": [ 69, 654, 324, 668 ], "score": 1.0, "content": "We examine the objective variance OV by conditioning", "type": "text" }, { "bbox": [ 324, 657, 356, 667 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 356, 654, 519, 668 ], "score": 1.0, "content": "on a batch of pairs of data points,", "type": "text" }, { "bbox": [ 519, 657, 541, 667 ], "score": 0.8, "content": "\\bar { z } : =", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 72, 663, 543, 683 ], "spans": [ { "bbox": [ 72, 668, 151, 679 ], "score": 0.94, "content": "\\{ z ^ { i } : = ( x _ { 0 } ^ { i } , x _ { 1 } ^ { i } ) \\} _ { i = 1 } ^ { m }", "type": "inline_equation" }, { "bbox": [ 151, 663, 380, 683 ], "score": 1.0, "content": ", we can reduce the variance of the OV objective to", "type": "text" }, { "bbox": [ 381, 669, 386, 676 ], "score": 0.29, "content": "0", "type": "inline_equation" }, { "bbox": [ 386, 663, 543, 683 ], "score": 1.0, "content": "for all models as batchsize goes to", "type": "text" } ], "index": 25 }, { "bbox": [ 68, 677, 540, 693 ], "spans": [ { "bbox": [ 68, 677, 221, 693 ], "score": 1.0, "content": "population size. For the batchsize", "type": "text" }, { "bbox": [ 221, 684, 230, 688 ], "score": 0.89, "content": "m", "type": "inline_equation" }, { "bbox": [ 230, 677, 530, 693 ], "score": 1.0, "content": "range from 1 to the number of the population, we uniformly sample", "type": "text" }, { "bbox": [ 531, 684, 540, 688 ], "score": 0.87, "content": "m", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 69, 690, 510, 704 ], "spans": [ { "bbox": [ 69, 690, 136, 704 ], "score": 1.0, "content": "pairs of points", "type": "text" }, { "bbox": [ 137, 691, 145, 700 ], "score": 0.91, "content": "z ^ { i }", "type": "inline_equation" }, { "bbox": [ 146, 690, 276, 704 ], "score": 1.0, "content": "and compute the probability", "type": "text" }, { "bbox": [ 276, 692, 307, 703 ], "score": 0.95, "content": "p _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 308, 690, 389, 704 ], "score": 1.0, "content": "and the objective", "type": "text" }, { "bbox": [ 389, 692, 421, 703 ], "score": 0.94, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 421, 690, 510, 704 ], "score": 1.0, "content": "from (25) and (26).", "type": "text" } ], "index": 27 } ], "index": 25.5 }, { "type": "text", "bbox": [ 69, 708, 537, 732 ], "lines": [ { "bbox": [ 68, 706, 540, 722 ], "spans": [ { "bbox": [ 68, 706, 540, 722 ], "score": 1.0, "content": "We also find that averaging over batches makes the network acheive a lower validation error in fewer steps", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 719, 208, 732 ], "spans": [ { "bbox": [ 69, 719, 208, 732 ], "score": 1.0, "content": "and in less walltime (Fig. D.5).", "type": "text" } ], "index": 29 } ], "index": 28.5 } ], "page_idx": 28, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "29", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 77, 83, 538, 185 ], "blocks": [ { "type": "image_body", "bbox": [ 77, 83, 538, 185 ], "group_id": 0, "lines": [ { "bbox": [ 77, 83, 538, 185 ], "spans": [ { "bbox": [ 77, 83, 538, 185 ], "score": 0.966, "type": "image", "image_path": "55b40fdd16153f131a9edc200d91f8bd2ec6bc077bd34dcae000ac4948306b7b.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 77, 83, 538, 117.0 ], "spans": [], "index": 0 }, { "bbox": [ 77, 117.0, 538, 151.0 ], "spans": [], "index": 1 }, { "bbox": [ 77, 151.0, 538, 185.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 71, 196, 537, 220 ], "group_id": 0, "lines": [ { "bbox": [ 69, 194, 540, 210 ], "spans": [ { "bbox": [ 69, 194, 540, 210 ], "score": 1.0, "content": "Figure D.7: 2-Wasserstein Error between trajectories and ground truth Schrödinger Bridge samples over", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 209, 144, 221 ], "spans": [ { "bbox": [ 70, 209, 144, 221 ], "score": 1.0, "content": "simulation time.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "image", "bbox": [ 100, 237, 506, 389 ], "blocks": [ { "type": "image_body", "bbox": [ 100, 237, 506, 389 ], "group_id": 1, "lines": [ { "bbox": [ 100, 237, 506, 389 ], "spans": [ { "bbox": [ 100, 237, 506, 389 ], "score": 0.971, "type": "image", "image_path": "12aa3bc53e7b3c6742a7eb5def0d4eb89ae3b891abf196e1c41a9febc0f5edb6.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 100, 237, 506, 287.6666666666667 ], "spans": [], "index": 5 }, { "bbox": [ 100, 287.6666666666667, 506, 338.33333333333337 ], "spans": [], "index": 6 }, { "bbox": [ 100, 338.33333333333337, 506, 389.00000000000006 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 70, 395, 540, 432 ], "group_id": 1, "lines": [ { "bbox": [ 70, 395, 541, 410 ], "spans": [ { "bbox": [ 70, 395, 541, 410 ], "score": 1.0, "content": "Figure D.8: (left) Variance of the objective for varying batch size. OT-CFM has a lower variance across", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 407, 541, 421 ], "spans": [ { "bbox": [ 69, 407, 433, 421 ], "score": 1.0, "content": "batch sizes. (right) Validation 2-Wasserstein performance with batch averaging as in", "type": "text" }, { "bbox": [ 433, 408, 455, 420 ], "score": 0.86, "content": "\\ S \\mathrm { C . 1 }", "type": "inline_equation" }, { "bbox": [ 455, 407, 541, 421 ], "score": 1.0, "content": ". Reducing variance", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 418, 194, 434 ], "spans": [ { "bbox": [ 69, 418, 194, 434 ], "score": 1.0, "content": "improves training efficiency.", "type": "text" } ], "index": 10 } ], "index": 9 } ], "index": 7.5 }, { "type": "text", "bbox": [ 70, 457, 541, 530 ], "lines": [ { "bbox": [ 68, 457, 542, 471 ], "spans": [ { "bbox": [ 68, 457, 542, 471 ], "score": 1.0, "content": "on training speed for different objectives in Table 1. We estimate the variance on a small data with a known", "type": "text" } ], "index": 11 }, { "bbox": [ 71, 470, 542, 483 ], "spans": [ { "bbox": [ 71, 471, 94, 482 ], "score": 0.93, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 95, 470, 542, 483 ], "score": 1.0, "content": ". We examine this estimated objective variance and its effect on training convergence in Fig. D.8,", "type": "text" } ], "index": 12 }, { "bbox": [ 69, 481, 541, 495 ], "spans": [ { "bbox": [ 69, 481, 541, 495 ], "score": 1.0, "content": "showing that either OT-CFM or variance reduced CFM with averaging over the batch results in lower variance", "type": "text" } ], "index": 13 }, { "bbox": [ 70, 494, 541, 506 ], "spans": [ { "bbox": [ 70, 494, 541, 506 ], "score": 1.0, "content": "of the objective. This in turn leads to faster training times as shown on the right. Averaging over a batch of", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 505, 541, 519 ], "spans": [ { "bbox": [ 69, 505, 541, 519 ], "score": 1.0, "content": "data leads to faster training particularly for methods with high objective variance (CFM) and less so for", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 516, 320, 531 ], "spans": [ { "bbox": [ 69, 516, 320, 531 ], "score": 1.0, "content": "those with low (OT-CFM), which already trains quickly.", "type": "text" } ], "index": 16 } ], "index": 13.5, "bbox_fs": [ 68, 457, 542, 531 ] }, { "type": "text", "bbox": [ 69, 535, 538, 559 ], "lines": [ { "bbox": [ 69, 534, 541, 549 ], "spans": [ { "bbox": [ 69, 534, 262, 549 ], "score": 1.0, "content": "Variance in the conditional objective target", "type": "text" }, { "bbox": [ 262, 537, 293, 548 ], "score": 0.94, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 293, 534, 541, 549 ], "score": 1.0, "content": "varies across models. In Fig. D.4 we study the objective", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 546, 475, 561 ], "spans": [ { "bbox": [ 68, 546, 475, 561 ], "score": 1.0, "content": "variance across CFM objective functions. Here we estimate the objective variance in (36) as", "type": "text" } ], "index": 18 } ], "index": 17.5, "bbox_fs": [ 68, 534, 541, 561 ] }, { "type": "interline_equation", "bbox": [ 249, 573, 361, 587 ], "lines": [ { "bbox": [ 249, 573, 361, 587 ], "spans": [ { "bbox": [ 249, 573, 361, 587 ], "score": 0.91, "content": "\\mathbb { E } _ { x , t , z } \\Vert u _ { t } ( x | z ) - v _ { \\theta } ( t , x ) \\Vert ^ { 2 }", "type": "interline_equation", "image_path": "737dd7d12609fad952b4e31d9d7434ea341238c157569cd44b101dc8d66b5e62.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 249, 573, 361, 587 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 70, 600, 540, 649 ], "lines": [ { "bbox": [ 70, 600, 542, 614 ], "spans": [ { "bbox": [ 70, 600, 323, 614 ], "score": 1.0, "content": "after training has converged. After training has converged", "type": "text" }, { "bbox": [ 324, 606, 333, 612 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 333, 600, 437, 614 ], "score": 1.0, "content": "should be very close to", "type": "text" }, { "bbox": [ 438, 603, 460, 613 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 461, 600, 542, 614 ], "score": 1.0, "content": "so we use it as an", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 612, 541, 626 ], "spans": [ { "bbox": [ 68, 612, 167, 626 ], "score": 1.0, "content": "empirical estimator of", "type": "text" }, { "bbox": [ 167, 615, 190, 625 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 190, 612, 541, 626 ], "score": 1.0, "content": "to compute the variance. We find that across all datasets OT-CFM and SB-CFM", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 624, 541, 637 ], "spans": [ { "bbox": [ 70, 624, 541, 637 ], "score": 1.0, "content": "have at least an order of magnitude lower variance than CFM and FM objectives. This correlates with faster", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 636, 542, 650 ], "spans": [ { "bbox": [ 70, 636, 542, 650 ], "score": 1.0, "content": "training as measured by lower validation error in fewer steps for lower variance models as seen in Fig. 2 (left).", "type": "text" } ], "index": 23 } ], "index": 21.5, "bbox_fs": [ 68, 600, 542, 650 ] }, { "type": "text", "bbox": [ 70, 654, 540, 703 ], "lines": [ { "bbox": [ 69, 654, 541, 668 ], "spans": [ { "bbox": [ 69, 654, 324, 668 ], "score": 1.0, "content": "We examine the objective variance OV by conditioning", "type": "text" }, { "bbox": [ 324, 657, 356, 667 ], "score": 0.95, "content": "u _ { t } ( x | z )", "type": "inline_equation" }, { "bbox": [ 356, 654, 519, 668 ], "score": 1.0, "content": "on a batch of pairs of data points,", "type": "text" }, { "bbox": [ 519, 657, 541, 667 ], "score": 0.8, "content": "\\bar { z } : =", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 72, 663, 543, 683 ], "spans": [ { "bbox": [ 72, 668, 151, 679 ], "score": 0.94, "content": "\\{ z ^ { i } : = ( x _ { 0 } ^ { i } , x _ { 1 } ^ { i } ) \\} _ { i = 1 } ^ { m }", "type": "inline_equation" }, { "bbox": [ 151, 663, 380, 683 ], "score": 1.0, "content": ", we can reduce the variance of the OV objective to", "type": "text" }, { "bbox": [ 381, 669, 386, 676 ], "score": 0.29, "content": "0", "type": "inline_equation" }, { "bbox": [ 386, 663, 543, 683 ], "score": 1.0, "content": "for all models as batchsize goes to", "type": "text" } ], "index": 25 }, { "bbox": [ 68, 677, 540, 693 ], "spans": [ { "bbox": [ 68, 677, 221, 693 ], "score": 1.0, "content": "population size. For the batchsize", "type": "text" }, { "bbox": [ 221, 684, 230, 688 ], "score": 0.89, "content": "m", "type": "inline_equation" }, { "bbox": [ 230, 677, 530, 693 ], "score": 1.0, "content": "range from 1 to the number of the population, we uniformly sample", "type": "text" }, { "bbox": [ 531, 684, 540, 688 ], "score": 0.87, "content": "m", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 69, 690, 510, 704 ], "spans": [ { "bbox": [ 69, 690, 136, 704 ], "score": 1.0, "content": "pairs of points", "type": "text" }, { "bbox": [ 137, 691, 145, 700 ], "score": 0.91, "content": "z ^ { i }", "type": "inline_equation" }, { "bbox": [ 146, 690, 276, 704 ], "score": 1.0, "content": "and compute the probability", "type": "text" }, { "bbox": [ 276, 692, 307, 703 ], "score": 0.95, "content": "p _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 308, 690, 389, 704 ], "score": 1.0, "content": "and the objective", "type": "text" }, { "bbox": [ 389, 692, 421, 703 ], "score": 0.94, "content": "u _ { t } ( x | \\bar { z } )", "type": "inline_equation" }, { "bbox": [ 421, 690, 510, 704 ], "score": 1.0, "content": "from (25) and (26).", "type": "text" } ], "index": 27 } ], "index": 25.5, "bbox_fs": [ 68, 654, 543, 704 ] }, { "type": "text", "bbox": [ 69, 708, 537, 732 ], "lines": [ { "bbox": [ 68, 706, 540, 722 ], "spans": [ { "bbox": [ 68, 706, 540, 722 ], "score": 1.0, "content": "We also find that averaging over batches makes the network acheive a lower validation error in fewer steps", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 719, 208, 732 ], "spans": [ { "bbox": [ 69, 719, 208, 732 ], "score": 1.0, "content": "and in less walltime (Fig. D.5).", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 68, 706, 540, 732 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 73, 80, 536, 228 ], "blocks": [ { "type": "image_body", "bbox": [ 73, 80, 536, 228 ], "group_id": 0, "lines": [ { "bbox": [ 73, 80, 536, 228 ], "spans": [ { "bbox": [ 73, 80, 536, 228 ], "score": 0.965, "type": "image", "image_path": "f952c6be9db3a33617c7e34879463176d9960813ace61f1694912a455a7614da.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 73, 80, 536, 129.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 73, 129.33333333333334, 536, 178.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 73, 178.66666666666669, 536, 228.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 70, 239, 538, 264 ], "group_id": 0, "lines": [ { "bbox": [ 69, 237, 541, 254 ], "spans": [ { "bbox": [ 69, 237, 541, 254 ], "score": 1.0, "content": "Figure D.9: Flows (green) from (a) moons to (b) 8-Gaussians unnormalized density function learned using", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 251, 151, 263 ], "spans": [ { "bbox": [ 70, 251, 151, 263 ], "score": 1.0, "content": "CFM with RWIS.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "title", "bbox": [ 72, 284, 186, 296 ], "lines": [ { "bbox": [ 69, 282, 188, 298 ], "spans": [ { "bbox": [ 69, 282, 188, 298 ], "score": 1.0, "content": "D.2 Energy-based CFM", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 70, 305, 541, 365 ], "lines": [ { "bbox": [ 69, 304, 541, 318 ], "spans": [ { "bbox": [ 69, 304, 541, 318 ], "score": 1.0, "content": "We show how CFM and OT-CFM can be adapted to the case where we do not have access to samples from", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 317, 540, 331 ], "spans": [ { "bbox": [ 69, 317, 514, 331 ], "score": 1.0, "content": "the target distribution, but only an unnormalized density (equivalently, energy function) of the target,", "type": "text" }, { "bbox": [ 515, 320, 540, 330 ], "score": 0.94, "content": "R ( x _ { 1 } )", "type": "inline_equation" } ], "index": 7 }, { "bbox": [ 68, 329, 542, 343 ], "spans": [ { "bbox": [ 68, 329, 542, 343 ], "score": 1.0, "content": "(Fig. D.9). We consider the 10-dimensional funnel dataset from Hoffman & Gelman (2011). We aim to learn", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 342, 541, 354 ], "spans": [ { "bbox": [ 69, 342, 541, 354 ], "score": 1.0, "content": "a flow from the 10-dimensional standard Gaussian to the energy function of the funnel. We consider two", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 353, 289, 366 ], "spans": [ { "bbox": [ 69, 353, 289, 366 ], "score": 1.0, "content": "algorithms, each of which has certain advantages:", "type": "text" } ], "index": 10 } ], "index": 8 }, { "type": "text", "bbox": [ 71, 370, 541, 491 ], "lines": [ { "bbox": [ 69, 370, 542, 386 ], "spans": [ { "bbox": [ 69, 370, 474, 386 ], "score": 1.0, "content": "(1) Reweighted importance sampling (RWIS): We construct a weighted batch of target points", "type": "text" }, { "bbox": [ 474, 376, 484, 383 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 485, 370, 542, 386 ], "score": 1.0, "content": "by sampling", "type": "text" } ], "index": 11 }, { "bbox": [ 89, 383, 542, 397 ], "spans": [ { "bbox": [ 89, 385, 147, 396 ], "score": 0.94, "content": "x _ { 1 } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 147, 383, 281, 397 ], "score": 1.0, "content": "and assigning it a weight of", "type": "text" }, { "bbox": [ 281, 385, 359, 396 ], "score": 0.94, "content": "R ( x _ { 1 } ) / \\mathcal { N } ( x _ { 1 } ; \\mathbf { 0 } , \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 360, 383, 542, 397 ], "score": 1.0, "content": "normalized to sum to 1 over the batch.", "type": "text" } ], "index": 12 }, { "bbox": [ 87, 394, 541, 409 ], "spans": [ { "bbox": [ 87, 394, 541, 409 ], "score": 1.0, "content": "The FM and CFM objectives handle weighted samples in a trivial way (by simply using the weights as", "type": "text" } ], "index": 13 }, { "bbox": [ 89, 407, 541, 421 ], "spans": [ { "bbox": [ 89, 409, 112, 420 ], "score": 0.95, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 112, 407, 541, 421 ], "score": 1.0, "content": "in Table 1), while OT-CFM treats the weights as target marginals in constructing the OT plan", "type": "text" } ], "index": 14 }, { "bbox": [ 87, 418, 541, 433 ], "spans": [ { "bbox": [ 87, 418, 126, 433 ], "score": 1.0, "content": "between", "type": "text" }, { "bbox": [ 127, 424, 137, 430 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 137, 418, 158, 433 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 159, 424, 169, 430 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 169, 418, 541, 433 ], "score": 1.0, "content": ". We expect RWIS to perform well when batches are large and the proposal and target", "type": "text" } ], "index": 15 }, { "bbox": [ 87, 430, 541, 445 ], "spans": [ { "bbox": [ 87, 430, 541, 445 ], "score": 1.0, "content": "distributions are sufficiently similar; otherwise, numerical explosion of the importance weights can hinder", "type": "text" } ], "index": 16 }, { "bbox": [ 87, 442, 128, 457 ], "spans": [ { "bbox": [ 87, 442, 128, 457 ], "score": 1.0, "content": "learning.", "type": "text" } ], "index": 17 }, { "bbox": [ 71, 454, 541, 469 ], "spans": [ { "bbox": [ 71, 454, 541, 469 ], "score": 1.0, "content": "(2) MCMC: We use samples from a long-run Metropolis-adjusted Langevin MCMC chain on the target", "type": "text" } ], "index": 18 }, { "bbox": [ 87, 467, 541, 480 ], "spans": [ { "bbox": [ 87, 467, 541, 480 ], "score": 1.0, "content": "density as approximate target samples. We expect this method to perform well when the MCMC mixes", "type": "text" } ], "index": 19 }, { "bbox": [ 86, 479, 349, 492 ], "spans": [ { "bbox": [ 86, 479, 349, 492 ], "score": 1.0, "content": "well; otherwise, modes of the target density may be missed.", "type": "text" } ], "index": 20 } ], "index": 15.5 }, { "type": "text", "bbox": [ 70, 497, 540, 533 ], "lines": [ { "bbox": [ 70, 497, 540, 509 ], "spans": [ { "bbox": [ 70, 497, 540, 509 ], "score": 1.0, "content": "As an evaluation metric, we use the estimation bias of the log-partition function using a reweighted variational", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 508, 541, 522 ], "spans": [ { "bbox": [ 70, 508, 541, 522 ], "score": 1.0, "content": "bound, following prior work that studied the problem using SDE modeling (Zhang & Chen, 2022). The", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 520, 306, 534 ], "spans": [ { "bbox": [ 69, 520, 306, 534 ], "score": 1.0, "content": "computation of this metric for CNFs is given in §E.6.", "type": "text" } ], "index": 23 } ], "index": 22 }, { "type": "text", "bbox": [ 71, 538, 540, 587 ], "lines": [ { "bbox": [ 69, 538, 541, 552 ], "spans": [ { "bbox": [ 69, 538, 541, 552 ], "score": 1.0, "content": "The results are shown in Table D.2. When an adaptive ODE integrator is used, all algorithms achieve similar", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 550, 542, 564 ], "spans": [ { "bbox": [ 69, 550, 471, 564 ], "score": 1.0, "content": "results (no pair of mean log-partition function estimates is statistically distinguishable with", "type": "text" }, { "bbox": [ 472, 554, 503, 563 ], "score": 0.9, "content": "p < 0 . 1", "type": "inline_equation" }, { "bbox": [ 504, 550, 542, 564 ], "score": 1.0, "content": "under a", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 561, 541, 577 ], "spans": [ { "bbox": [ 69, 561, 106, 577 ], "score": 1.0, "content": "Welch’s", "type": "text" }, { "bbox": [ 106, 566, 110, 573 ], "score": 0.85, "content": "t", "type": "inline_equation" }, { "bbox": [ 111, 561, 541, 577 ], "score": 1.0, "content": "-test) but OT-CFM is about twice as efficient as CFM and FM. However, with a fixed computation", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 574, 371, 588 ], "spans": [ { "bbox": [ 70, 574, 371, 588 ], "score": 1.0, "content": "budget for ODE integration, OT-CFM performs significantly better.", "type": "text" } ], "index": 27 } ], "index": 25.5 }, { "type": "title", "bbox": [ 71, 601, 308, 615 ], "lines": [ { "bbox": [ 69, 600, 309, 617 ], "spans": [ { "bbox": [ 69, 600, 309, 617 ], "score": 1.0, "content": "E Experiment and implementation details", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "title", "bbox": [ 72, 627, 225, 639 ], "lines": [ { "bbox": [ 68, 625, 227, 642 ], "spans": [ { "bbox": [ 68, 625, 227, 642 ], "score": 1.0, "content": "E.1 Physical experimental setup", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 71, 648, 540, 732 ], "lines": [ { "bbox": [ 70, 649, 541, 661 ], "spans": [ { "bbox": [ 70, 649, 541, 661 ], "score": 1.0, "content": "All experiments were performed on a shared heterogenous high-performance-computing cluster. This cluster", "type": "text" } ], "index": 30 }, { "bbox": [ 68, 660, 541, 673 ], "spans": [ { "bbox": [ 68, 660, 541, 673 ], "score": 1.0, "content": "is primarily composed of GPU nodes with RTX8000, A100, and V100 Nvidia GPUs. Since the network and", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 672, 541, 686 ], "spans": [ { "bbox": [ 69, 672, 541, 686 ], "score": 1.0, "content": "nodes are shared, other users may cause high variance in the training times of models. However, we believe", "type": "text" } ], "index": 32 }, { "bbox": [ 68, 683, 542, 699 ], "spans": [ { "bbox": [ 68, 683, 542, 699 ], "score": 1.0, "content": "that the striking difference between the convergence times in Table D.1 and combined with the CFM training", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 695, 540, 709 ], "spans": [ { "bbox": [ 69, 695, 540, 709 ], "score": 1.0, "content": "setup with a single CPU and the baseline models trained with two CPUS and a GPU, paints a clear picture", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 707, 541, 722 ], "spans": [ { "bbox": [ 69, 707, 541, 722 ], "score": 1.0, "content": "as to how efficient CFM training is. Qualitatively, we feel that most CFMs converge quite a bit more rapidly", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 720, 528, 733 ], "spans": [ { "bbox": [ 70, 720, 528, 733 ], "score": 1.0, "content": "than these metrics would suggest, often converging to a near optimal validation performance in minutes.", "type": "text" } ], "index": 36 } ], "index": 33 } ], "page_idx": 29, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 368, 39 ], "spans": [ { "bbox": [ 69, 24, 368, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "30", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 73, 80, 536, 228 ], "blocks": [ { "type": "image_body", "bbox": [ 73, 80, 536, 228 ], "group_id": 0, "lines": [ { "bbox": [ 73, 80, 536, 228 ], "spans": [ { "bbox": [ 73, 80, 536, 228 ], "score": 0.965, "type": "image", "image_path": "f952c6be9db3a33617c7e34879463176d9960813ace61f1694912a455a7614da.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 73, 80, 536, 129.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 73, 129.33333333333334, 536, 178.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 73, 178.66666666666669, 536, 228.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 70, 239, 538, 264 ], "group_id": 0, "lines": [ { "bbox": [ 69, 237, 541, 254 ], "spans": [ { "bbox": [ 69, 237, 541, 254 ], "score": 1.0, "content": "Figure D.9: Flows (green) from (a) moons to (b) 8-Gaussians unnormalized density function learned using", "type": "text" } ], "index": 3 }, { "bbox": [ 70, 251, 151, 263 ], "spans": [ { "bbox": [ 70, 251, 151, 263 ], "score": 1.0, "content": "CFM with RWIS.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "title", "bbox": [ 72, 284, 186, 296 ], "lines": [ { "bbox": [ 69, 282, 188, 298 ], "spans": [ { "bbox": [ 69, 282, 188, 298 ], "score": 1.0, "content": "D.2 Energy-based CFM", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 70, 305, 541, 365 ], "lines": [ { "bbox": [ 69, 304, 541, 318 ], "spans": [ { "bbox": [ 69, 304, 541, 318 ], "score": 1.0, "content": "We show how CFM and OT-CFM can be adapted to the case where we do not have access to samples from", "type": "text" } ], "index": 6 }, { "bbox": [ 69, 317, 540, 331 ], "spans": [ { "bbox": [ 69, 317, 514, 331 ], "score": 1.0, "content": "the target distribution, but only an unnormalized density (equivalently, energy function) of the target,", "type": "text" }, { "bbox": [ 515, 320, 540, 330 ], "score": 0.94, "content": "R ( x _ { 1 } )", "type": "inline_equation" } ], "index": 7 }, { "bbox": [ 68, 329, 542, 343 ], "spans": [ { "bbox": [ 68, 329, 542, 343 ], "score": 1.0, "content": "(Fig. D.9). We consider the 10-dimensional funnel dataset from Hoffman & Gelman (2011). We aim to learn", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 342, 541, 354 ], "spans": [ { "bbox": [ 69, 342, 541, 354 ], "score": 1.0, "content": "a flow from the 10-dimensional standard Gaussian to the energy function of the funnel. We consider two", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 353, 289, 366 ], "spans": [ { "bbox": [ 69, 353, 289, 366 ], "score": 1.0, "content": "algorithms, each of which has certain advantages:", "type": "text" } ], "index": 10 } ], "index": 8, "bbox_fs": [ 68, 304, 542, 366 ] }, { "type": "list", "bbox": [ 71, 370, 541, 491 ], "lines": [ { "bbox": [ 69, 370, 542, 386 ], "spans": [ { "bbox": [ 69, 370, 474, 386 ], "score": 1.0, "content": "(1) Reweighted importance sampling (RWIS): We construct a weighted batch of target points", "type": "text" }, { "bbox": [ 474, 376, 484, 383 ], "score": 0.9, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 485, 370, 542, 386 ], "score": 1.0, "content": "by sampling", "type": "text" } ], "index": 11, "is_list_start_line": true }, { "bbox": [ 89, 383, 542, 397 ], "spans": [ { "bbox": [ 89, 385, 147, 396 ], "score": 0.94, "content": "x _ { 1 } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 147, 383, 281, 397 ], "score": 1.0, "content": "and assigning it a weight of", "type": "text" }, { "bbox": [ 281, 385, 359, 396 ], "score": 0.94, "content": "R ( x _ { 1 } ) / \\mathcal { N } ( x _ { 1 } ; \\mathbf { 0 } , \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 360, 383, 542, 397 ], "score": 1.0, "content": "normalized to sum to 1 over the batch.", "type": "text" } ], "index": 12 }, { "bbox": [ 87, 394, 541, 409 ], "spans": [ { "bbox": [ 87, 394, 541, 409 ], "score": 1.0, "content": "The FM and CFM objectives handle weighted samples in a trivial way (by simply using the weights as", "type": "text" } ], "index": 13 }, { "bbox": [ 89, 407, 541, 421 ], "spans": [ { "bbox": [ 89, 409, 112, 420 ], "score": 0.95, "content": "q ( x _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 112, 407, 541, 421 ], "score": 1.0, "content": "in Table 1), while OT-CFM treats the weights as target marginals in constructing the OT plan", "type": "text" } ], "index": 14 }, { "bbox": [ 87, 418, 541, 433 ], "spans": [ { "bbox": [ 87, 418, 126, 433 ], "score": 1.0, "content": "between", "type": "text" }, { "bbox": [ 127, 424, 137, 430 ], "score": 0.91, "content": "x _ { 0 }", "type": "inline_equation" }, { "bbox": [ 137, 418, 158, 433 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 159, 424, 169, 430 ], "score": 0.89, "content": "x _ { 1 }", "type": "inline_equation" }, { "bbox": [ 169, 418, 541, 433 ], "score": 1.0, "content": ". We expect RWIS to perform well when batches are large and the proposal and target", "type": "text" } ], "index": 15 }, { "bbox": [ 87, 430, 541, 445 ], "spans": [ { "bbox": [ 87, 430, 541, 445 ], "score": 1.0, "content": "distributions are sufficiently similar; otherwise, numerical explosion of the importance weights can hinder", "type": "text" } ], "index": 16 }, { "bbox": [ 87, 442, 128, 457 ], "spans": [ { "bbox": [ 87, 442, 128, 457 ], "score": 1.0, "content": "learning.", "type": "text" } ], "index": 17, "is_list_end_line": true }, { "bbox": [ 71, 454, 541, 469 ], "spans": [ { "bbox": [ 71, 454, 541, 469 ], "score": 1.0, "content": "(2) MCMC: We use samples from a long-run Metropolis-adjusted Langevin MCMC chain on the target", "type": "text" } ], "index": 18, "is_list_start_line": true }, { "bbox": [ 87, 467, 541, 480 ], "spans": [ { "bbox": [ 87, 467, 541, 480 ], "score": 1.0, "content": "density as approximate target samples. We expect this method to perform well when the MCMC mixes", "type": "text" } ], "index": 19 }, { "bbox": [ 86, 479, 349, 492 ], "spans": [ { "bbox": [ 86, 479, 349, 492 ], "score": 1.0, "content": "well; otherwise, modes of the target density may be missed.", "type": "text" } ], "index": 20, "is_list_end_line": true } ], "index": 15.5, "bbox_fs": [ 69, 370, 542, 492 ] }, { "type": "text", "bbox": [ 70, 497, 540, 533 ], "lines": [ { "bbox": [ 70, 497, 540, 509 ], "spans": [ { "bbox": [ 70, 497, 540, 509 ], "score": 1.0, "content": "As an evaluation metric, we use the estimation bias of the log-partition function using a reweighted variational", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 508, 541, 522 ], "spans": [ { "bbox": [ 70, 508, 541, 522 ], "score": 1.0, "content": "bound, following prior work that studied the problem using SDE modeling (Zhang & Chen, 2022). The", "type": "text" } ], "index": 22 }, { "bbox": [ 69, 520, 306, 534 ], "spans": [ { "bbox": [ 69, 520, 306, 534 ], "score": 1.0, "content": "computation of this metric for CNFs is given in §E.6.", "type": "text" } ], "index": 23 } ], "index": 22, "bbox_fs": [ 69, 497, 541, 534 ] }, { "type": "text", "bbox": [ 71, 538, 540, 587 ], "lines": [ { "bbox": [ 69, 538, 541, 552 ], "spans": [ { "bbox": [ 69, 538, 541, 552 ], "score": 1.0, "content": "The results are shown in Table D.2. When an adaptive ODE integrator is used, all algorithms achieve similar", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 550, 542, 564 ], "spans": [ { "bbox": [ 69, 550, 471, 564 ], "score": 1.0, "content": "results (no pair of mean log-partition function estimates is statistically distinguishable with", "type": "text" }, { "bbox": [ 472, 554, 503, 563 ], "score": 0.9, "content": "p < 0 . 1", "type": "inline_equation" }, { "bbox": [ 504, 550, 542, 564 ], "score": 1.0, "content": "under a", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 561, 541, 577 ], "spans": [ { "bbox": [ 69, 561, 106, 577 ], "score": 1.0, "content": "Welch’s", "type": "text" }, { "bbox": [ 106, 566, 110, 573 ], "score": 0.85, "content": "t", "type": "inline_equation" }, { "bbox": [ 111, 561, 541, 577 ], "score": 1.0, "content": "-test) but OT-CFM is about twice as efficient as CFM and FM. However, with a fixed computation", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 574, 371, 588 ], "spans": [ { "bbox": [ 70, 574, 371, 588 ], "score": 1.0, "content": "budget for ODE integration, OT-CFM performs significantly better.", "type": "text" } ], "index": 27 } ], "index": 25.5, "bbox_fs": [ 69, 538, 542, 588 ] }, { "type": "title", "bbox": [ 71, 601, 308, 615 ], "lines": [ { "bbox": [ 69, 600, 309, 617 ], "spans": [ { "bbox": [ 69, 600, 309, 617 ], "score": 1.0, "content": "E Experiment and implementation details", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "title", "bbox": [ 72, 627, 225, 639 ], "lines": [ { "bbox": [ 68, 625, 227, 642 ], "spans": [ { "bbox": [ 68, 625, 227, 642 ], "score": 1.0, "content": "E.1 Physical experimental setup", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 71, 648, 540, 732 ], "lines": [ { "bbox": [ 70, 649, 541, 661 ], "spans": [ { "bbox": [ 70, 649, 541, 661 ], "score": 1.0, "content": "All experiments were performed on a shared heterogenous high-performance-computing cluster. This cluster", "type": "text" } ], "index": 30 }, { "bbox": [ 68, 660, 541, 673 ], "spans": [ { "bbox": [ 68, 660, 541, 673 ], "score": 1.0, "content": "is primarily composed of GPU nodes with RTX8000, A100, and V100 Nvidia GPUs. Since the network and", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 672, 541, 686 ], "spans": [ { "bbox": [ 69, 672, 541, 686 ], "score": 1.0, "content": "nodes are shared, other users may cause high variance in the training times of models. However, we believe", "type": "text" } ], "index": 32 }, { "bbox": [ 68, 683, 542, 699 ], "spans": [ { "bbox": [ 68, 683, 542, 699 ], "score": 1.0, "content": "that the striking difference between the convergence times in Table D.1 and combined with the CFM training", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 695, 540, 709 ], "spans": [ { "bbox": [ 69, 695, 540, 709 ], "score": 1.0, "content": "setup with a single CPU and the baseline models trained with two CPUS and a GPU, paints a clear picture", "type": "text" } ], "index": 34 }, { "bbox": [ 69, 707, 541, 722 ], "spans": [ { "bbox": [ 69, 707, 541, 722 ], "score": 1.0, "content": "as to how efficient CFM training is. Qualitatively, we feel that most CFMs converge quite a bit more rapidly", "type": "text" } ], "index": 35 }, { "bbox": [ 70, 720, 528, 733 ], "spans": [ { "bbox": [ 70, 720, 528, 733 ], "score": 1.0, "content": "than these metrics would suggest, often converging to a near optimal validation performance in minutes.", "type": "text" } ], "index": 36 } ], "index": 33, "bbox_fs": [ 68, 649, 542, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 137, 141, 466, 317 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 139 ], "group_id": 0, "lines": [ { "bbox": [ 69, 80, 541, 93 ], "spans": [ { "bbox": [ 69, 80, 541, 93 ], "score": 1.0, "content": "Table D.2: Energy-based CFM results on the 10-dimensional funnel dataset: log-partition function estimation", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 541, 105 ], "spans": [ { "bbox": [ 70, 93, 541, 105 ], "score": 1.0, "content": "bias (mean and standard deviation over 10 runs) and time to generate 6000 samples from the trained", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 103, 542, 118 ], "spans": [ { "bbox": [ 69, 103, 542, 118 ], "score": 1.0, "content": "ODE. With adaptive integration, OT-CFM requires fewer function evaluations. With a fixed-interval solver,", "type": "text" } ], "index": 2 }, { "bbox": [ 68, 115, 542, 129 ], "spans": [ { "bbox": [ 68, 115, 542, 129 ], "score": 1.0, "content": "OT-CFM has lower discretization error, leading to a better estimate. PIS baseline is from Zhang & Chen", "type": "text" } ], "index": 3 }, { "bbox": [ 67, 127, 103, 141 ], "spans": [ { "bbox": [ 67, 127, 103, 141 ], "score": 1.0, "content": "(2022).", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "table_body", "bbox": [ 137, 141, 466, 317 ], "group_id": 0, "lines": [ { "bbox": [ 137, 141, 466, 317 ], "spans": [ { "bbox": [ 137, 141, 466, 317 ], "score": 0.982, "html": "
RWISMCMC
log 2S timelog 2S time
adaptive Dormand-Prince (tolerance O.O1) integration
CFM OT-CFM-0.068±0.041 -0.076 ± 0.09826.6 ± 8.4s 13.3 ± 1.7s0.029 ± 0.037 0.009 ± 0.04534.6 ± 6.0s 12.8 ± 1.2s
FM-0.033±0.05726.5 ± 7.7s0.027 ± 0.03130.9 ± 5.8s
Euler (N=10) integration
CFM0.281 ± 0.2024.0 ± 0.8s0.336 ± 0.0303.7 ± 0.7s
OT-CFM-0.039±0.0304.2 ± 0.6s0.146±0.1074.1 ± 0.8s
FM0.176±0.0444.1 ± 0.7s0.334±0.0663.9 ± 0.6s
PIS (SDE)Euler-Maruyama (N=100) integration -0.018 ± 0.020
", "type": "table", "image_path": "26520aa7300ef2599e56c440a6a05b703374392747ccc2bcb534b36499e891b6.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 137, 141, 466, 199.66666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 137, 199.66666666666666, 466, 258.3333333333333 ], "spans": [], "index": 6 }, { "bbox": [ 137, 258.3333333333333, 466, 317.0 ], "spans": [], "index": 7 } ] } ], "index": 4.0 }, { "type": "title", "bbox": [ 72, 354, 364, 366 ], "lines": [ { "bbox": [ 68, 351, 366, 371 ], "spans": [ { "bbox": [ 68, 351, 366, 371 ], "score": 1.0, "content": "E.2 2D, single-cell, and Schrödinger bridge experimental setup", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 70, 384, 540, 538 ], "lines": [ { "bbox": [ 69, 383, 541, 397 ], "spans": [ { "bbox": [ 69, 383, 541, 397 ], "score": 1.0, "content": "For all experiments we use the same architecture implemented in PyTorch (Paszke et al., 2019). We", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 395, 541, 408 ], "spans": [ { "bbox": [ 69, 395, 213, 408 ], "score": 1.0, "content": "concatenate the flattened input", "type": "text" }, { "bbox": [ 214, 397, 244, 406 ], "score": 0.93, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 244, 395, 307, 408 ], "score": 1.0, "content": "and the time", "type": "text" }, { "bbox": [ 307, 399, 311, 405 ], "score": 0.84, "content": "t", "type": "inline_equation" }, { "bbox": [ 311, 395, 344, 408 ], "score": 1.0, "content": "as the", "type": "text" }, { "bbox": [ 344, 398, 367, 406 ], "score": 0.93, "content": "d + 1", "type": "inline_equation" }, { "bbox": [ 367, 395, 541, 408 ], "score": 1.0, "content": "inputs to a network with three hidden", "type": "text" } ], "index": 10 }, { "bbox": [ 68, 406, 542, 421 ], "spans": [ { "bbox": [ 68, 406, 542, 421 ], "score": 1.0, "content": "layers of width 64 interspersed with SELU activations (Klambauer et al., 2017) followed by a linear output", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 420, 541, 432 ], "spans": [ { "bbox": [ 69, 420, 135, 432 ], "score": 1.0, "content": "layer of width", "type": "text" }, { "bbox": [ 135, 422, 141, 429 ], "score": 0.86, "content": "d", "type": "inline_equation" }, { "bbox": [ 141, 420, 216, 432 ], "score": 1.0, "content": ". This forms our", "type": "text" }, { "bbox": [ 216, 425, 226, 431 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 226, 420, 541, 432 ], "score": 1.0, "content": "for all experiments. For all 2D and single-cell experiments we train for", "type": "text" } ], "index": 12 }, { "bbox": [ 68, 430, 542, 446 ], "spans": [ { "bbox": [ 68, 430, 542, 446 ], "score": 1.0, "content": "1000 epochs and implement early stopping on the validation loss which checks the loss on a validation set every", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 443, 542, 456 ], "spans": [ { "bbox": [ 69, 443, 542, 456 ], "score": 1.0, "content": "10 epochs and stops training if there is no improvement for 30 epochs. We also set a time limit of 100 minutes", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 454, 542, 469 ], "spans": [ { "bbox": [ 69, 454, 436, 469 ], "score": 1.0, "content": "for each CFM model. This is hit almost exclusively for SB-CFM models with small", "type": "text" }, { "bbox": [ 437, 460, 443, 465 ], "score": 0.87, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 443, 454, 542, 469 ], "score": 1.0, "content": "which are unstable to", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 467, 542, 480 ], "spans": [ { "bbox": [ 69, 467, 542, 480 ], "score": 1.0, "content": "train due to instabilities and non-convergence of the Sinkhorn (Cuturi, 2013) transport plan optimization.", "type": "text" } ], "index": 16 }, { "bbox": [ 68, 477, 542, 494 ], "spans": [ { "bbox": [ 68, 477, 419, 494 ], "score": 1.0, "content": "We use the AdamW (Loshchilov & Hutter, 2019) optimizer with weight decay", "type": "text" }, { "bbox": [ 419, 480, 440, 489 ], "score": 0.9, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 441, 477, 542, 494 ], "score": 1.0, "content": "with batchsize 512 by", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 491, 541, 505 ], "spans": [ { "bbox": [ 69, 491, 541, 505 ], "score": 1.0, "content": "default in 2D experiments and 128 in the single cell datasets. For OT-CFM and SB-CFM we use exact linear", "type": "text" } ], "index": 18 }, { "bbox": [ 68, 502, 542, 517 ], "spans": [ { "bbox": [ 68, 502, 542, 517 ], "score": 1.0, "content": "programming EMD and Sinkhorn algorithms from the python optimal transport package (Flamary et al.,", "type": "text" } ], "index": 19 }, { "bbox": [ 70, 515, 540, 527 ], "spans": [ { "bbox": [ 70, 515, 540, 527 ], "score": 1.0, "content": "2021) For evaluation of trajectories unless otherwise noted we use the Runge-Kutta45 (rk4) ODE solver with", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 527, 187, 539 ], "spans": [ { "bbox": [ 70, 527, 187, 539 ], "score": 1.0, "content": "101 timesteps from 0 to 1.", "type": "text" } ], "index": 21 } ], "index": 15 }, { "type": "title", "bbox": [ 72, 572, 244, 584 ], "lines": [ { "bbox": [ 69, 569, 245, 587 ], "spans": [ { "bbox": [ 69, 569, 245, 587 ], "score": 1.0, "content": "E.3 Variance reduction by averaging", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 600, 540, 637 ], "lines": [ { "bbox": [ 70, 601, 541, 613 ], "spans": [ { "bbox": [ 70, 601, 395, 613 ], "score": 1.0, "content": "We tackle the exploration of the effects of reducing variance of the target", "type": "text" }, { "bbox": [ 396, 603, 426, 613 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 427, 601, 541, 613 ], "score": 1.0, "content": "from two directions. The", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 613, 541, 626 ], "spans": [ { "bbox": [ 70, 613, 368, 626 ], "score": 1.0, "content": "first is for small example where we can compute the ground truth", "type": "text" }, { "bbox": [ 369, 615, 392, 625 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 392, 613, 541, 626 ], "score": 1.0, "content": "quickly, and the second is in the", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 625, 468, 638 ], "spans": [ { "bbox": [ 69, 625, 284, 638 ], "score": 1.0, "content": "setting of trained models where we can estimate", "type": "text" }, { "bbox": [ 285, 627, 308, 637 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 308, 625, 333, 638 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 333, 627, 364, 637 ], "score": 0.94, "content": "v _ { \\boldsymbol { \\theta } } ( t , \\boldsymbol { x } )", "type": "inline_equation" }, { "bbox": [ 365, 625, 391, 638 ], "score": 1.0, "content": "after", "type": "text" }, { "bbox": [ 391, 630, 400, 636 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 401, 625, 468, 638 ], "score": 1.0, "content": "has converged.", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "text", "bbox": [ 71, 642, 540, 678 ], "lines": [ { "bbox": [ 69, 641, 542, 657 ], "spans": [ { "bbox": [ 69, 641, 542, 657 ], "score": 1.0, "content": "We first consider the convergence of each flow matching objective (OT-CFM, CFM, FM, SB-CFM) to zero as", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 653, 542, 669 ], "spans": [ { "bbox": [ 69, 653, 444, 669 ], "score": 1.0, "content": "a function of the batch size relative to the dataset size. 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This means that we", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 720, 271, 733 ], "spans": [ { "bbox": [ 69, 720, 271, 733 ], "score": 1.0, "content": "can compare different aggregation sizes fairly.", "type": "text" } ], "index": 32 } ], "index": 30.5 } ], "page_idx": 30, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 367, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "", "type": "text", "height": 13, "width": 14 } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 137, 141, 466, 317 ], "blocks": [ { "type": "table_caption", "bbox": [ 69, 80, 541, 139 ], "group_id": 0, "lines": [ { "bbox": [ 69, 80, 541, 93 ], "spans": [ { "bbox": [ 69, 80, 541, 93 ], "score": 1.0, "content": "Table D.2: Energy-based CFM results on the 10-dimensional funnel dataset: log-partition function estimation", "type": "text" } ], "index": 0 }, { "bbox": [ 70, 93, 541, 105 ], "spans": [ { "bbox": [ 70, 93, 541, 105 ], "score": 1.0, "content": "bias (mean and standard deviation over 10 runs) and time to generate 6000 samples from the trained", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 103, 542, 118 ], "spans": [ { "bbox": [ 69, 103, 542, 118 ], "score": 1.0, "content": "ODE. With adaptive integration, OT-CFM requires fewer function evaluations. With a fixed-interval solver,", "type": "text" } ], "index": 2 }, { "bbox": [ 68, 115, 542, 129 ], "spans": [ { "bbox": [ 68, 115, 542, 129 ], "score": 1.0, "content": "OT-CFM has lower discretization error, leading to a better estimate. PIS baseline is from Zhang & Chen", "type": "text" } ], "index": 3 }, { "bbox": [ 67, 127, 103, 141 ], "spans": [ { "bbox": [ 67, 127, 103, 141 ], "score": 1.0, "content": "(2022).", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "table_body", "bbox": [ 137, 141, 466, 317 ], "group_id": 0, "lines": [ { "bbox": [ 137, 141, 466, 317 ], "spans": [ { "bbox": [ 137, 141, 466, 317 ], "score": 0.982, "html": "
RWISMCMC
log 2S timelog 2S time
adaptive Dormand-Prince (tolerance O.O1) integration
CFM OT-CFM-0.068±0.041 -0.076 ± 0.09826.6 ± 8.4s 13.3 ± 1.7s0.029 ± 0.037 0.009 ± 0.04534.6 ± 6.0s 12.8 ± 1.2s
FM-0.033±0.05726.5 ± 7.7s0.027 ± 0.03130.9 ± 5.8s
Euler (N=10) integration
CFM0.281 ± 0.2024.0 ± 0.8s0.336 ± 0.0303.7 ± 0.7s
OT-CFM-0.039±0.0304.2 ± 0.6s0.146±0.1074.1 ± 0.8s
FM0.176±0.0444.1 ± 0.7s0.334±0.0663.9 ± 0.6s
PIS (SDE)Euler-Maruyama (N=100) integration -0.018 ± 0.020
", "type": "table", "image_path": "26520aa7300ef2599e56c440a6a05b703374392747ccc2bcb534b36499e891b6.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 137, 141, 466, 199.66666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 137, 199.66666666666666, 466, 258.3333333333333 ], "spans": [], "index": 6 }, { "bbox": [ 137, 258.3333333333333, 466, 317.0 ], "spans": [], "index": 7 } ] } ], "index": 4.0 }, { "type": "title", "bbox": [ 72, 354, 364, 366 ], "lines": [ { "bbox": [ 68, 351, 366, 371 ], "spans": [ { "bbox": [ 68, 351, 366, 371 ], "score": 1.0, "content": "E.2 2D, single-cell, and Schrödinger bridge experimental setup", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 70, 384, 540, 538 ], "lines": [ { "bbox": [ 69, 383, 541, 397 ], "spans": [ { "bbox": [ 69, 383, 541, 397 ], "score": 1.0, "content": "For all experiments we use the same architecture implemented in PyTorch (Paszke et al., 2019). We", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 395, 541, 408 ], "spans": [ { "bbox": [ 69, 395, 213, 408 ], "score": 1.0, "content": "concatenate the flattened input", "type": "text" }, { "bbox": [ 214, 397, 244, 406 ], "score": 0.93, "content": "x \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 244, 395, 307, 408 ], "score": 1.0, "content": "and the time", "type": "text" }, { "bbox": [ 307, 399, 311, 405 ], "score": 0.84, "content": "t", "type": "inline_equation" }, { "bbox": [ 311, 395, 344, 408 ], "score": 1.0, "content": "as the", "type": "text" }, { "bbox": [ 344, 398, 367, 406 ], "score": 0.93, "content": "d + 1", "type": "inline_equation" }, { "bbox": [ 367, 395, 541, 408 ], "score": 1.0, "content": "inputs to a network with three hidden", "type": "text" } ], "index": 10 }, { "bbox": [ 68, 406, 542, 421 ], "spans": [ { "bbox": [ 68, 406, 542, 421 ], "score": 1.0, "content": "layers of width 64 interspersed with SELU activations (Klambauer et al., 2017) followed by a linear output", "type": "text" } ], "index": 11 }, { "bbox": [ 69, 420, 541, 432 ], "spans": [ { "bbox": [ 69, 420, 135, 432 ], "score": 1.0, "content": "layer of width", "type": "text" }, { "bbox": [ 135, 422, 141, 429 ], "score": 0.86, "content": "d", "type": "inline_equation" }, { "bbox": [ 141, 420, 216, 432 ], "score": 1.0, "content": ". This forms our", "type": "text" }, { "bbox": [ 216, 425, 226, 431 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 226, 420, 541, 432 ], "score": 1.0, "content": "for all experiments. For all 2D and single-cell experiments we train for", "type": "text" } ], "index": 12 }, { "bbox": [ 68, 430, 542, 446 ], "spans": [ { "bbox": [ 68, 430, 542, 446 ], "score": 1.0, "content": "1000 epochs and implement early stopping on the validation loss which checks the loss on a validation set every", "type": "text" } ], "index": 13 }, { "bbox": [ 69, 443, 542, 456 ], "spans": [ { "bbox": [ 69, 443, 542, 456 ], "score": 1.0, "content": "10 epochs and stops training if there is no improvement for 30 epochs. We also set a time limit of 100 minutes", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 454, 542, 469 ], "spans": [ { "bbox": [ 69, 454, 436, 469 ], "score": 1.0, "content": "for each CFM model. This is hit almost exclusively for SB-CFM models with small", "type": "text" }, { "bbox": [ 437, 460, 443, 465 ], "score": 0.87, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 443, 454, 542, 469 ], "score": 1.0, "content": "which are unstable to", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 467, 542, 480 ], "spans": [ { "bbox": [ 69, 467, 542, 480 ], "score": 1.0, "content": "train due to instabilities and non-convergence of the Sinkhorn (Cuturi, 2013) transport plan optimization.", "type": "text" } ], "index": 16 }, { "bbox": [ 68, 477, 542, 494 ], "spans": [ { "bbox": [ 68, 477, 419, 494 ], "score": 1.0, "content": "We use the AdamW (Loshchilov & Hutter, 2019) optimizer with weight decay", "type": "text" }, { "bbox": [ 419, 480, 440, 489 ], "score": 0.9, "content": "1 0 ^ { - 5 }", "type": "inline_equation" }, { "bbox": [ 441, 477, 542, 494 ], "score": 1.0, "content": "with batchsize 512 by", "type": "text" } ], "index": 17 }, { "bbox": [ 69, 491, 541, 505 ], "spans": [ { "bbox": [ 69, 491, 541, 505 ], "score": 1.0, "content": "default in 2D experiments and 128 in the single cell datasets. For OT-CFM and SB-CFM we use exact linear", "type": "text" } ], "index": 18 }, { "bbox": [ 68, 502, 542, 517 ], "spans": [ { "bbox": [ 68, 502, 542, 517 ], "score": 1.0, "content": "programming EMD and Sinkhorn algorithms from the python optimal transport package (Flamary et al.,", "type": "text" } ], "index": 19 }, { "bbox": [ 70, 515, 540, 527 ], "spans": [ { "bbox": [ 70, 515, 540, 527 ], "score": 1.0, "content": "2021) For evaluation of trajectories unless otherwise noted we use the Runge-Kutta45 (rk4) ODE solver with", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 527, 187, 539 ], "spans": [ { "bbox": [ 70, 527, 187, 539 ], "score": 1.0, "content": "101 timesteps from 0 to 1.", "type": "text" } ], "index": 21 } ], "index": 15, "bbox_fs": [ 68, 383, 542, 539 ] }, { "type": "title", "bbox": [ 72, 572, 244, 584 ], "lines": [ { "bbox": [ 69, 569, 245, 587 ], "spans": [ { "bbox": [ 69, 569, 245, 587 ], "score": 1.0, "content": "E.3 Variance reduction by averaging", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 70, 600, 540, 637 ], "lines": [ { "bbox": [ 70, 601, 541, 613 ], "spans": [ { "bbox": [ 70, 601, 395, 613 ], "score": 1.0, "content": "We tackle the exploration of the effects of reducing variance of the target", "type": "text" }, { "bbox": [ 396, 603, 426, 613 ], "score": 0.94, "content": "\\boldsymbol u _ { t } ( \\boldsymbol x | \\boldsymbol z )", "type": "inline_equation" }, { "bbox": [ 427, 601, 541, 613 ], "score": 1.0, "content": "from two directions. The", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 613, 541, 626 ], "spans": [ { "bbox": [ 70, 613, 368, 626 ], "score": 1.0, "content": "first is for small example where we can compute the ground truth", "type": "text" }, { "bbox": [ 369, 615, 392, 625 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 392, 613, 541, 626 ], "score": 1.0, "content": "quickly, and the second is in the", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 625, 468, 638 ], "spans": [ { "bbox": [ 69, 625, 284, 638 ], "score": 1.0, "content": "setting of trained models where we can estimate", "type": "text" }, { "bbox": [ 285, 627, 308, 637 ], "score": 0.94, "content": "u _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 308, 625, 333, 638 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 333, 627, 364, 637 ], "score": 0.94, "content": "v _ { \\boldsymbol { \\theta } } ( t , \\boldsymbol { x } )", "type": "inline_equation" }, { "bbox": [ 365, 625, 391, 638 ], "score": 1.0, "content": "after", "type": "text" }, { "bbox": [ 391, 630, 400, 636 ], "score": 0.9, "content": "v _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 401, 625, 468, 638 ], "score": 1.0, "content": "has converged.", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 69, 601, 541, 638 ] }, { "type": "text", "bbox": [ 71, 642, 540, 678 ], "lines": [ { "bbox": [ 69, 641, 542, 657 ], "spans": [ { "bbox": [ 69, 641, 542, 657 ], "score": 1.0, "content": "We first consider the convergence of each flow matching objective (OT-CFM, CFM, FM, SB-CFM) to zero as", "type": "text" } ], "index": 26 }, { "bbox": [ 69, 653, 542, 669 ], "spans": [ { "bbox": [ 69, 653, 444, 669 ], "score": 1.0, "content": "a function of the batch size relative to the dataset size. This is done by first sampling", "type": "text" }, { "bbox": [ 444, 658, 468, 667 ], "score": 0.93, "content": "t , x , z", "type": "inline_equation" }, { "bbox": [ 469, 653, 542, 669 ], "score": 1.0, "content": "then computing", "type": "text" } ], "index": 27 }, { "bbox": [ 70, 666, 399, 680 ], "spans": [ { "bbox": [ 70, 666, 399, 680 ], "score": 1.0, "content": "the true objective variance across many samples. This appears in Fig. D.8.", "type": "text" } ], "index": 28 } ], "index": 27, "bbox_fs": [ 69, 641, 542, 680 ] }, { "type": "text", "bbox": [ 70, 684, 540, 732 ], "lines": [ { "bbox": [ 69, 684, 541, 697 ], "spans": [ { "bbox": [ 69, 684, 541, 697 ], "score": 1.0, "content": "We next consider the effect of averaging over a batch to reduce the variance of the objective in Fig. D.8", "type": "text" } ], "index": 29 }, { "bbox": [ 69, 696, 540, 709 ], "spans": [ { "bbox": [ 69, 696, 540, 709 ], "score": 1.0, "content": "(right). Here Batchsize refers to the size of the batch we are averaging over. We aggregate this into a single", "type": "text" } ], "index": 30 }, { "bbox": [ 70, 708, 542, 721 ], "spans": [ { "bbox": [ 70, 708, 237, 721 ], "score": 1.0, "content": "target so that the model sees a single", "type": "text" }, { "bbox": [ 237, 711, 243, 718 ], "score": 0.9, "content": "d", "type": "inline_equation" }, { "bbox": [ 243, 708, 432, 721 ], "score": 1.0, "content": "dimensional target vector for one sampled", "type": "text" }, { "bbox": [ 432, 712, 446, 720 ], "score": 0.91, "content": "x , t", "type": "inline_equation" }, { "bbox": [ 446, 708, 542, 721 ], "score": 1.0, "content": ". This means that we", "type": "text" } ], "index": 31 }, { "bbox": [ 69, 720, 271, 733 ], "spans": [ { "bbox": [ 69, 720, 271, 733 ], "score": 1.0, "content": "can compare different aggregation sizes fairly.", "type": "text" } ], "index": 32 } ], "index": 30.5, "bbox_fs": [ 69, 684, 542, 733 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 72, 82, 262, 94 ], "lines": [ { "bbox": [ 69, 79, 263, 98 ], "spans": [ { "bbox": [ 69, 79, 263, 98 ], "score": 1.0, "content": "E.4 Schrödinger bridge evaluation setup", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 103, 540, 164 ], "lines": [ { "bbox": [ 69, 102, 541, 117 ], "spans": [ { "bbox": [ 69, 102, 541, 117 ], "score": 1.0, "content": "To evaluate how well Schrödinger Bridge models actually model a Schrödinger Bridge, we constrain ourselves", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 115, 541, 129 ], "spans": [ { "bbox": [ 69, 115, 541, 129 ], "score": 1.0, "content": "to a small example with 1000 points. We note that the closed-form Schrödinger marginals are known", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 126, 542, 141 ], "spans": [ { "bbox": [ 69, 126, 542, 141 ], "score": 1.0, "content": "for discrete densities, for Gaussians (Mallasto et al., 2022), and can be constructed for two approximate", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 138, 542, 153 ], "spans": [ { "bbox": [ 69, 138, 542, 153 ], "score": 1.0, "content": "datasets (Korotin et al., 2021), which present other ways of evaluating Schrödinger bridge performance. For", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 151, 443, 165 ], "spans": [ { "bbox": [ 70, 151, 112, 165 ], "score": 1.0, "content": "any time", "type": "text" }, { "bbox": [ 113, 155, 117, 161 ], "score": 0.86, "content": "t", "type": "inline_equation" }, { "bbox": [ 117, 151, 406, 165 ], "score": 1.0, "content": "we can sample from the ground truth Schrödinger bridge density", "type": "text" }, { "bbox": [ 406, 153, 428, 164 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 429, 151, 443, 165 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 5 } ], "index": 3 }, { "type": "interline_equation", "bbox": [ 213, 172, 397, 201 ], "lines": [ { "bbox": [ 213, 172, 397, 201 ], "spans": [ { "bbox": [ 213, 172, 397, 201 ], "score": 0.86, "content": "\\begin{array} { c } { ( x _ { 0 } , x _ { 1 } ) \\sim \\pi _ { 2 \\sigma ^ { 2 } } } \\\\ { X _ { t } \\sim \\mathcal N ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma t ( 1 - t ) ) } \\end{array}", "type": "interline_equation", "image_path": "bd34a851223c99bac8a44549cd545efdd60475bd9b905998a3efe0571189ec95.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 213, 172, 397, 186.5 ], "spans": [], "index": 6 }, { "bbox": [ 213, 186.5, 397, 201.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 207, 540, 256 ], "lines": [ { "bbox": [ 69, 208, 541, 221 ], "spans": [ { "bbox": [ 69, 208, 248, 221 ], "score": 1.0, "content": "We sample trajectories of length 20 from", "type": "text" }, { "bbox": [ 249, 211, 271, 218 ], "score": 0.92, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 271, 208, 285, 221 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 286, 211, 308, 218 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 308, 208, 440, 221 ], "score": 1.0, "content": "by integrating over time from", "type": "text" }, { "bbox": [ 441, 211, 462, 218 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 463, 208, 477, 221 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 477, 211, 499, 218 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 500, 208, 541, 221 ], "score": 1.0, "content": ". At each", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 219, 540, 232 ], "spans": [ { "bbox": [ 69, 219, 540, 232 ], "score": 1.0, "content": "of the 18 intermediate timepoints we compute the 2-Wasserstein distance between a sample of size 1000 from", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 231, 542, 246 ], "spans": [ { "bbox": [ 69, 231, 290, 246 ], "score": 1.0, "content": "the trajectories at that time and the ground truth", "type": "text" }, { "bbox": [ 290, 235, 302, 243 ], "score": 0.91, "content": "X _ { t }", "type": "inline_equation" }, { "bbox": [ 302, 231, 542, 246 ], "score": 1.0, "content": "as above at that time. We reported the average across", "type": "text" } ], "index": 10 }, { "bbox": [ 70, 244, 510, 256 ], "spans": [ { "bbox": [ 70, 244, 510, 256 ], "score": 1.0, "content": "the 18 intermediate timepoints in Table 3 and plot the 2-Wasserstein distance over time in Fig. D.7.", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 70, 267, 538, 292 ], "lines": [ { "bbox": [ 70, 267, 541, 280 ], "spans": [ { "bbox": [ 70, 267, 264, 280 ], "score": 1.0, "content": "SB-CFM Model We train SB-CFM with", "type": "text" }, { "bbox": [ 264, 271, 289, 278 ], "score": 0.91, "content": "\\sigma = 1", "type": "inline_equation" }, { "bbox": [ 289, 267, 541, 280 ], "score": 1.0, "content": "and batchsize=512 for each of the datasets. We save 1000", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 280, 420, 291 ], "spans": [ { "bbox": [ 70, 280, 420, 291 ], "score": 1.0, "content": "trajectories from a test set integrated with the tsit5 solver with atol=rtol=1e-4.", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "text", "bbox": [ 70, 303, 540, 435 ], "lines": [ { "bbox": [ 70, 303, 542, 316 ], "spans": [ { "bbox": [ 70, 303, 542, 316 ], "score": 1.0, "content": "Diffusion Schrödinger bridge model implementation details We use the implementation from De Bor-", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 316, 541, 329 ], "spans": [ { "bbox": [ 69, 316, 541, 329 ], "score": 1.0, "content": "toli et al. (2021). Only the networks were changed for a fair comparison with CFM. The forward and backward", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 328, 542, 341 ], "spans": [ { "bbox": [ 69, 328, 542, 341 ], "score": 1.0, "content": "networks are composed of an MLP with three hidden layers of size 64, with SELU activations in between", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 340, 541, 353 ], "spans": [ { "bbox": [ 69, 340, 541, 353 ], "score": 1.0, "content": "layers. We used a time and a positional encoders composed of two layers of size 16 and 32 with LeakyReLU", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 352, 541, 365 ], "spans": [ { "bbox": [ 68, 352, 541, 365 ], "score": 1.0, "content": "activations has inputs to the score network. The architectures are the same for the 2D examples and the", "type": "text" } ], "index": 18 }, { "bbox": [ 70, 365, 541, 376 ], "spans": [ { "bbox": [ 70, 365, 466, 376 ], "score": 1.0, "content": "single-cell examples (except for the input dimension). During training, we set the variance (", "type": "text" }, { "bbox": [ 466, 369, 473, 376 ], "score": 0.86, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 473, 365, 541, 376 ], "score": 1.0, "content": "in the author’s", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 375, 542, 389 ], "spans": [ { "bbox": [ 69, 375, 542, 389 ], "score": 1.0, "content": "code) to 0.001 and did 20 steps to discretize the Langevin dynamic. We trained for 10k iterations with 10k", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 387, 542, 401 ], "spans": [ { "bbox": [ 68, 387, 542, 401 ], "score": 1.0, "content": "particles and batch size of 512, for 20 iterative proportional fitting steps, and a learning rate set to 0.0001. For", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 401, 541, 412 ], "spans": [ { "bbox": [ 70, 401, 541, 412 ], "score": 1.0, "content": "the interpolation task we used the tenth timepoint from the Langevin dynamic with the backward network", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 411, 541, 425 ], "spans": [ { "bbox": [ 70, 411, 262, 425 ], "score": 1.0, "content": "trained to go from the distribution at time", "type": "text" }, { "bbox": [ 262, 415, 283, 422 ], "score": 0.91, "content": "t - 1", "type": "inline_equation" }, { "bbox": [ 283, 411, 298, 425 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 298, 415, 319, 422 ], "score": 0.92, "content": "t + 1", "type": "inline_equation" }, { "bbox": [ 320, 411, 541, 425 ], "score": 1.0, "content": ". All trajectories are evaluated from the backward", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 425, 263, 436 ], "spans": [ { "bbox": [ 70, 425, 149, 436 ], "score": 1.0, "content": "dynamic. We use", "type": "text" }, { "bbox": [ 149, 427, 174, 434 ], "score": 0.91, "content": "\\sigma = 1", "type": "inline_equation" }, { "bbox": [ 174, 425, 263, 436 ], "score": 1.0, "content": "and batchsize=512.", "type": "text" } ], "index": 24 } ], "index": 19 }, { "type": "title", "bbox": [ 72, 449, 234, 461 ], "lines": [ { "bbox": [ 69, 446, 236, 465 ], "spans": [ { "bbox": [ 69, 446, 236, 465 ], "score": 1.0, "content": "E.5 Single-cell experimental setup", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 71, 470, 541, 542 ], "lines": [ { "bbox": [ 69, 470, 541, 484 ], "spans": [ { "bbox": [ 69, 470, 541, 484 ], "score": 1.0, "content": "We strove to be consistent with the experimental setup of Tong et al. (2020). For the Embryoid body (EB)", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 483, 542, 496 ], "spans": [ { "bbox": [ 70, 483, 542, 496 ], "score": 1.0, "content": "data, we use the same processed artifact which contains the first 100 principal components of the data.", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 494, 542, 508 ], "spans": [ { "bbox": [ 69, 494, 542, 508 ], "score": 1.0, "content": "For our tests we truncate to the first five dimensions, then whiten (subtract mean and divide by standard", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 505, 542, 520 ], "spans": [ { "bbox": [ 69, 505, 542, 520 ], "score": 1.0, "content": "deviation) each dimension. For the Embryoid body dataset which consists of 5 timepoints collected over 30", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 519, 542, 531 ], "spans": [ { "bbox": [ 70, 519, 279, 531 ], "score": 1.0, "content": "days we train separate models leaving out times", "type": "text" }, { "bbox": [ 280, 521, 303, 530 ], "score": 0.92, "content": "1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 304, 519, 542, 531 ], "score": 1.0, "content": "in turn. We train a CFM over the full time scale (0-4).", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 529, 466, 544 ], "spans": [ { "bbox": [ 69, 529, 256, 544 ], "score": 1.0, "content": "During testing we push forward all points", "type": "text" }, { "bbox": [ 256, 533, 279, 542 ], "score": 0.93, "content": "X _ { t - 1 }", "type": "inline_equation" }, { "bbox": [ 279, 529, 316, 544 ], "score": 1.0, "content": "to time", "type": "text" }, { "bbox": [ 316, 534, 320, 540 ], "score": 0.87, "content": "t", "type": "inline_equation" }, { "bbox": [ 321, 529, 466, 544 ], "score": 1.0, "content": "as a distribution to test against.", "type": "text" } ], "index": 31 } ], "index": 28.5 }, { "type": "text", "bbox": [ 70, 547, 540, 655 ], "lines": [ { "bbox": [ 69, 547, 541, 561 ], "spans": [ { "bbox": [ 69, 547, 541, 561 ], "score": 1.0, "content": "For the Cite and Multi datasets these are sourced from the Multimodal Single-cell Integration challenge at", "type": "text" } ], "index": 32 }, { "bbox": [ 70, 560, 541, 573 ], "spans": [ { "bbox": [ 70, 560, 541, 573 ], "score": 1.0, "content": "NeurIPS 2022, a NeurIPS challenge hosted on Kaggle where the task was multi-modal prediction (Burkhardt", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 572, 541, 585 ], "spans": [ { "bbox": [ 69, 572, 541, 585 ], "score": 1.0, "content": "et al., 2022). In this competition they used this data to investigate the predictability of RNA from chromatin", "type": "text" } ], "index": 34 }, { "bbox": [ 70, 585, 541, 597 ], "spans": [ { "bbox": [ 70, 585, 541, 597 ], "score": 1.0, "content": "accessibility and protein expression from RNA. Here, we repurpose this data for the task of time series", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 596, 541, 609 ], "spans": [ { "bbox": [ 69, 596, 541, 609 ], "score": 1.0, "content": "interpolation. Both of these datasets consist of four timepoints from CD34+ hematopoietic stem and", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 608, 541, 621 ], "spans": [ { "bbox": [ 69, 608, 541, 621 ], "score": 1.0, "content": "progenitor cells (HSPCs) collected on days 2, 3, 4, and 7. For more information and the raw data see the", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 621, 541, 633 ], "spans": [ { "bbox": [ 69, 621, 541, 633 ], "score": 1.0, "content": "competition site.5 We preprocess this data slightly to remove patient specific effects by focusing on a single", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 631, 541, 645 ], "spans": [ { "bbox": [ 69, 631, 541, 645 ], "score": 1.0, "content": "donor (donor 13176), then we again compute the first five principal components and again whiten each", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 644, 249, 657 ], "spans": [ { "bbox": [ 69, 644, 249, 657 ], "score": 1.0, "content": "dimension to further normalize the data.", "type": "text" } ], "index": 40 } ], "index": 36 }, { "type": "title", "bbox": [ 72, 669, 185, 681 ], "lines": [ { "bbox": [ 69, 668, 186, 684 ], "spans": [ { "bbox": [ 69, 668, 186, 684 ], "score": 1.0, "content": "E.6 Energy-based CFM", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 71, 690, 543, 714 ], "lines": [ { "bbox": [ 69, 689, 542, 704 ], "spans": [ { "bbox": [ 69, 689, 286, 704 ], "score": 1.0, "content": "The 10-dimensional funnel dataset is defined by", "type": "text" }, { "bbox": [ 286, 693, 344, 703 ], "score": 0.92, "content": "\\mathbf { x } _ { 0 } \\sim \\mathcal { N } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 344, 689, 349, 704 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 349, 693, 456, 703 ], "score": 0.89, "content": "\\mathbf { x } _ { 1 , \\dots , 9 } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\exp ( \\mathbf { x } _ { 0 } ) \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 456, 689, 542, 704 ], "score": 1.0, "content": ". We attempted to", "type": "text" } ], "index": 42 }, { "bbox": [ 70, 702, 541, 715 ], "spans": [ { "bbox": [ 70, 702, 424, 715 ], "score": 1.0, "content": "mimic the SDE model architecture from Zhang & Chen (2022) for the flow model", "type": "text" }, { "bbox": [ 425, 704, 456, 715 ], "score": 0.94, "content": "v _ { \\boldsymbol { \\theta } } ( t , \\boldsymbol { x } )", "type": "inline_equation" }, { "bbox": [ 456, 702, 525, 715 ], "score": 1.0, "content": ". The time step", "type": "text" }, { "bbox": [ 525, 706, 529, 712 ], "score": 0.89, "content": "t", "type": "inline_equation" }, { "bbox": [ 530, 702, 541, 715 ], "score": 1.0, "content": "is", "type": "text" } ], "index": 43 } ], "index": 42.5 } ], "page_idx": 31, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 71, 27, 368, 37 ], "lines": [ { "bbox": [ 69, 24, 369, 39 ], "spans": [ { "bbox": [ 69, 24, 369, 39 ], "score": 1.0, "content": "Published in Transactions on Machine Learning Research (03/2024)", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 81, 722, 363, 732 ], "lines": [ { "bbox": [ 81, 720, 365, 733 ], "spans": [ { "bbox": [ 81, 720, 365, 733 ], "score": 1.0, "content": "5https://www.kaggle.com/competitions/open-problems-multimodal/data", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "32", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 72, 82, 262, 94 ], "lines": [ { "bbox": [ 69, 79, 263, 98 ], "spans": [ { "bbox": [ 69, 79, 263, 98 ], "score": 1.0, "content": "E.4 Schrödinger bridge evaluation setup", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 71, 103, 540, 164 ], "lines": [ { "bbox": [ 69, 102, 541, 117 ], "spans": [ { "bbox": [ 69, 102, 541, 117 ], "score": 1.0, "content": "To evaluate how well Schrödinger Bridge models actually model a Schrödinger Bridge, we constrain ourselves", "type": "text" } ], "index": 1 }, { "bbox": [ 69, 115, 541, 129 ], "spans": [ { "bbox": [ 69, 115, 541, 129 ], "score": 1.0, "content": "to a small example with 1000 points. We note that the closed-form Schrödinger marginals are known", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 126, 542, 141 ], "spans": [ { "bbox": [ 69, 126, 542, 141 ], "score": 1.0, "content": "for discrete densities, for Gaussians (Mallasto et al., 2022), and can be constructed for two approximate", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 138, 542, 153 ], "spans": [ { "bbox": [ 69, 138, 542, 153 ], "score": 1.0, "content": "datasets (Korotin et al., 2021), which present other ways of evaluating Schrödinger bridge performance. For", "type": "text" } ], "index": 4 }, { "bbox": [ 70, 151, 443, 165 ], "spans": [ { "bbox": [ 70, 151, 112, 165 ], "score": 1.0, "content": "any time", "type": "text" }, { "bbox": [ 113, 155, 117, 161 ], "score": 0.86, "content": "t", "type": "inline_equation" }, { "bbox": [ 117, 151, 406, 165 ], "score": 1.0, "content": "we can sample from the ground truth Schrödinger bridge density", "type": "text" }, { "bbox": [ 406, 153, 428, 164 ], "score": 0.94, "content": "p _ { t } ( x )", "type": "inline_equation" }, { "bbox": [ 429, 151, 443, 165 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 5 } ], "index": 3, "bbox_fs": [ 69, 102, 542, 165 ] }, { "type": "interline_equation", "bbox": [ 213, 172, 397, 201 ], "lines": [ { "bbox": [ 213, 172, 397, 201 ], "spans": [ { "bbox": [ 213, 172, 397, 201 ], "score": 0.86, "content": "\\begin{array} { c } { ( x _ { 0 } , x _ { 1 } ) \\sim \\pi _ { 2 \\sigma ^ { 2 } } } \\\\ { X _ { t } \\sim \\mathcal N ( x \\mid t x _ { 1 } + ( 1 - t ) x _ { 0 } , \\sigma t ( 1 - t ) ) } \\end{array}", "type": "interline_equation", "image_path": "bd34a851223c99bac8a44549cd545efdd60475bd9b905998a3efe0571189ec95.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 213, 172, 397, 186.5 ], "spans": [], "index": 6 }, { "bbox": [ 213, 186.5, 397, 201.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 70, 207, 540, 256 ], "lines": [ { "bbox": [ 69, 208, 541, 221 ], "spans": [ { "bbox": [ 69, 208, 248, 221 ], "score": 1.0, "content": "We sample trajectories of length 20 from", "type": "text" }, { "bbox": [ 249, 211, 271, 218 ], "score": 0.92, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 271, 208, 285, 221 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 286, 211, 308, 218 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 308, 208, 440, 221 ], "score": 1.0, "content": "by integrating over time from", "type": "text" }, { "bbox": [ 441, 211, 462, 218 ], "score": 0.91, "content": "t = 0", "type": "inline_equation" }, { "bbox": [ 463, 208, 477, 221 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 477, 211, 499, 218 ], "score": 0.91, "content": "t = 1", "type": "inline_equation" }, { "bbox": [ 500, 208, 541, 221 ], "score": 1.0, "content": ". At each", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 219, 540, 232 ], "spans": [ { "bbox": [ 69, 219, 540, 232 ], "score": 1.0, "content": "of the 18 intermediate timepoints we compute the 2-Wasserstein distance between a sample of size 1000 from", "type": "text" } ], "index": 9 }, { "bbox": [ 69, 231, 542, 246 ], "spans": [ { "bbox": [ 69, 231, 290, 246 ], "score": 1.0, "content": "the trajectories at that time and the ground truth", "type": "text" }, { "bbox": [ 290, 235, 302, 243 ], "score": 0.91, "content": "X _ { t }", "type": "inline_equation" }, { "bbox": [ 302, 231, 542, 246 ], "score": 1.0, "content": "as above at that time. We reported the average across", "type": "text" } ], "index": 10 }, { "bbox": [ 70, 244, 510, 256 ], "spans": [ { "bbox": [ 70, 244, 510, 256 ], "score": 1.0, "content": "the 18 intermediate timepoints in Table 3 and plot the 2-Wasserstein distance over time in Fig. D.7.", "type": "text" } ], "index": 11 } ], "index": 9.5, "bbox_fs": [ 69, 208, 542, 256 ] }, { "type": "text", "bbox": [ 70, 267, 538, 292 ], "lines": [ { "bbox": [ 70, 267, 541, 280 ], "spans": [ { "bbox": [ 70, 267, 264, 280 ], "score": 1.0, "content": "SB-CFM Model We train SB-CFM with", "type": "text" }, { "bbox": [ 264, 271, 289, 278 ], "score": 0.91, "content": "\\sigma = 1", "type": "inline_equation" }, { "bbox": [ 289, 267, 541, 280 ], "score": 1.0, "content": "and batchsize=512 for each of the datasets. We save 1000", "type": "text" } ], "index": 12 }, { "bbox": [ 70, 280, 420, 291 ], "spans": [ { "bbox": [ 70, 280, 420, 291 ], "score": 1.0, "content": "trajectories from a test set integrated with the tsit5 solver with atol=rtol=1e-4.", "type": "text" } ], "index": 13 } ], "index": 12.5, "bbox_fs": [ 70, 267, 541, 291 ] }, { "type": "text", "bbox": [ 70, 303, 540, 435 ], "lines": [ { "bbox": [ 70, 303, 542, 316 ], "spans": [ { "bbox": [ 70, 303, 542, 316 ], "score": 1.0, "content": "Diffusion Schrödinger bridge model implementation details We use the implementation from De Bor-", "type": "text" } ], "index": 14 }, { "bbox": [ 69, 316, 541, 329 ], "spans": [ { "bbox": [ 69, 316, 541, 329 ], "score": 1.0, "content": "toli et al. (2021). Only the networks were changed for a fair comparison with CFM. The forward and backward", "type": "text" } ], "index": 15 }, { "bbox": [ 69, 328, 542, 341 ], "spans": [ { "bbox": [ 69, 328, 542, 341 ], "score": 1.0, "content": "networks are composed of an MLP with three hidden layers of size 64, with SELU activations in between", "type": "text" } ], "index": 16 }, { "bbox": [ 69, 340, 541, 353 ], "spans": [ { "bbox": [ 69, 340, 541, 353 ], "score": 1.0, "content": "layers. We used a time and a positional encoders composed of two layers of size 16 and 32 with LeakyReLU", "type": "text" } ], "index": 17 }, { "bbox": [ 68, 352, 541, 365 ], "spans": [ { "bbox": [ 68, 352, 541, 365 ], "score": 1.0, "content": "activations has inputs to the score network. The architectures are the same for the 2D examples and the", "type": "text" } ], "index": 18 }, { "bbox": [ 70, 365, 541, 376 ], "spans": [ { "bbox": [ 70, 365, 466, 376 ], "score": 1.0, "content": "single-cell examples (except for the input dimension). During training, we set the variance (", "type": "text" }, { "bbox": [ 466, 369, 473, 376 ], "score": 0.86, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 473, 365, 541, 376 ], "score": 1.0, "content": "in the author’s", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 375, 542, 389 ], "spans": [ { "bbox": [ 69, 375, 542, 389 ], "score": 1.0, "content": "code) to 0.001 and did 20 steps to discretize the Langevin dynamic. We trained for 10k iterations with 10k", "type": "text" } ], "index": 20 }, { "bbox": [ 68, 387, 542, 401 ], "spans": [ { "bbox": [ 68, 387, 542, 401 ], "score": 1.0, "content": "particles and batch size of 512, for 20 iterative proportional fitting steps, and a learning rate set to 0.0001. For", "type": "text" } ], "index": 21 }, { "bbox": [ 70, 401, 541, 412 ], "spans": [ { "bbox": [ 70, 401, 541, 412 ], "score": 1.0, "content": "the interpolation task we used the tenth timepoint from the Langevin dynamic with the backward network", "type": "text" } ], "index": 22 }, { "bbox": [ 70, 411, 541, 425 ], "spans": [ { "bbox": [ 70, 411, 262, 425 ], "score": 1.0, "content": "trained to go from the distribution at time", "type": "text" }, { "bbox": [ 262, 415, 283, 422 ], "score": 0.91, "content": "t - 1", "type": "inline_equation" }, { "bbox": [ 283, 411, 298, 425 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 298, 415, 319, 422 ], "score": 0.92, "content": "t + 1", "type": "inline_equation" }, { "bbox": [ 320, 411, 541, 425 ], "score": 1.0, "content": ". All trajectories are evaluated from the backward", "type": "text" } ], "index": 23 }, { "bbox": [ 70, 425, 263, 436 ], "spans": [ { "bbox": [ 70, 425, 149, 436 ], "score": 1.0, "content": "dynamic. We use", "type": "text" }, { "bbox": [ 149, 427, 174, 434 ], "score": 0.91, "content": "\\sigma = 1", "type": "inline_equation" }, { "bbox": [ 174, 425, 263, 436 ], "score": 1.0, "content": "and batchsize=512.", "type": "text" } ], "index": 24 } ], "index": 19, "bbox_fs": [ 68, 303, 542, 436 ] }, { "type": "title", "bbox": [ 72, 449, 234, 461 ], "lines": [ { "bbox": [ 69, 446, 236, 465 ], "spans": [ { "bbox": [ 69, 446, 236, 465 ], "score": 1.0, "content": "E.5 Single-cell experimental setup", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 71, 470, 541, 542 ], "lines": [ { "bbox": [ 69, 470, 541, 484 ], "spans": [ { "bbox": [ 69, 470, 541, 484 ], "score": 1.0, "content": "We strove to be consistent with the experimental setup of Tong et al. (2020). For the Embryoid body (EB)", "type": "text" } ], "index": 26 }, { "bbox": [ 70, 483, 542, 496 ], "spans": [ { "bbox": [ 70, 483, 542, 496 ], "score": 1.0, "content": "data, we use the same processed artifact which contains the first 100 principal components of the data.", "type": "text" } ], "index": 27 }, { "bbox": [ 69, 494, 542, 508 ], "spans": [ { "bbox": [ 69, 494, 542, 508 ], "score": 1.0, "content": "For our tests we truncate to the first five dimensions, then whiten (subtract mean and divide by standard", "type": "text" } ], "index": 28 }, { "bbox": [ 69, 505, 542, 520 ], "spans": [ { "bbox": [ 69, 505, 542, 520 ], "score": 1.0, "content": "deviation) each dimension. For the Embryoid body dataset which consists of 5 timepoints collected over 30", "type": "text" } ], "index": 29 }, { "bbox": [ 70, 519, 542, 531 ], "spans": [ { "bbox": [ 70, 519, 279, 531 ], "score": 1.0, "content": "days we train separate models leaving out times", "type": "text" }, { "bbox": [ 280, 521, 303, 530 ], "score": 0.92, "content": "1 , 2 , 3", "type": "inline_equation" }, { "bbox": [ 304, 519, 542, 531 ], "score": 1.0, "content": "in turn. We train a CFM over the full time scale (0-4).", "type": "text" } ], "index": 30 }, { "bbox": [ 69, 529, 466, 544 ], "spans": [ { "bbox": [ 69, 529, 256, 544 ], "score": 1.0, "content": "During testing we push forward all points", "type": "text" }, { "bbox": [ 256, 533, 279, 542 ], "score": 0.93, "content": "X _ { t - 1 }", "type": "inline_equation" }, { "bbox": [ 279, 529, 316, 544 ], "score": 1.0, "content": "to time", "type": "text" }, { "bbox": [ 316, 534, 320, 540 ], "score": 0.87, "content": "t", "type": "inline_equation" }, { "bbox": [ 321, 529, 466, 544 ], "score": 1.0, "content": "as a distribution to test against.", "type": "text" } ], "index": 31 } ], "index": 28.5, "bbox_fs": [ 69, 470, 542, 544 ] }, { "type": "text", "bbox": [ 70, 547, 540, 655 ], "lines": [ { "bbox": [ 69, 547, 541, 561 ], "spans": [ { "bbox": [ 69, 547, 541, 561 ], "score": 1.0, "content": "For the Cite and Multi datasets these are sourced from the Multimodal Single-cell Integration challenge at", "type": "text" } ], "index": 32 }, { "bbox": [ 70, 560, 541, 573 ], "spans": [ { "bbox": [ 70, 560, 541, 573 ], "score": 1.0, "content": "NeurIPS 2022, a NeurIPS challenge hosted on Kaggle where the task was multi-modal prediction (Burkhardt", "type": "text" } ], "index": 33 }, { "bbox": [ 69, 572, 541, 585 ], "spans": [ { "bbox": [ 69, 572, 541, 585 ], "score": 1.0, "content": "et al., 2022). In this competition they used this data to investigate the predictability of RNA from chromatin", "type": "text" } ], "index": 34 }, { "bbox": [ 70, 585, 541, 597 ], "spans": [ { "bbox": [ 70, 585, 541, 597 ], "score": 1.0, "content": "accessibility and protein expression from RNA. Here, we repurpose this data for the task of time series", "type": "text" } ], "index": 35 }, { "bbox": [ 69, 596, 541, 609 ], "spans": [ { "bbox": [ 69, 596, 541, 609 ], "score": 1.0, "content": "interpolation. Both of these datasets consist of four timepoints from CD34+ hematopoietic stem and", "type": "text" } ], "index": 36 }, { "bbox": [ 69, 608, 541, 621 ], "spans": [ { "bbox": [ 69, 608, 541, 621 ], "score": 1.0, "content": "progenitor cells (HSPCs) collected on days 2, 3, 4, and 7. For more information and the raw data see the", "type": "text" } ], "index": 37 }, { "bbox": [ 69, 621, 541, 633 ], "spans": [ { "bbox": [ 69, 621, 541, 633 ], "score": 1.0, "content": "competition site.5 We preprocess this data slightly to remove patient specific effects by focusing on a single", "type": "text" } ], "index": 38 }, { "bbox": [ 69, 631, 541, 645 ], "spans": [ { "bbox": [ 69, 631, 541, 645 ], "score": 1.0, "content": "donor (donor 13176), then we again compute the first five principal components and again whiten each", "type": "text" } ], "index": 39 }, { "bbox": [ 69, 644, 249, 657 ], "spans": [ { "bbox": [ 69, 644, 249, 657 ], "score": 1.0, "content": "dimension to further normalize the data.", "type": "text" } ], "index": 40 } ], "index": 36, "bbox_fs": [ 69, 547, 541, 657 ] }, { "type": "title", "bbox": [ 72, 669, 185, 681 ], "lines": [ { "bbox": [ 69, 668, 186, 684 ], "spans": [ { "bbox": [ 69, 668, 186, 684 ], "score": 1.0, "content": "E.6 Energy-based CFM", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 71, 690, 543, 714 ], "lines": [ { "bbox": [ 69, 689, 542, 704 ], "spans": [ { "bbox": [ 69, 689, 286, 704 ], "score": 1.0, "content": "The 10-dimensional funnel dataset is defined by", "type": "text" }, { "bbox": [ 286, 693, 344, 703 ], "score": 0.92, "content": "\\mathbf { x } _ { 0 } \\sim \\mathcal { N } ( 0 , I )", "type": "inline_equation" }, { "bbox": [ 344, 689, 349, 704 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 349, 693, 456, 703 ], "score": 0.89, "content": "\\mathbf { x } _ { 1 , \\dots , 9 } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\exp ( \\mathbf { x } _ { 0 } ) \\mathbf { I } )", "type": "inline_equation" }, { "bbox": [ 456, 689, 542, 704 ], "score": 1.0, "content": ". We attempted to", "type": "text" } ], "index": 42 }, { "bbox": [ 70, 702, 541, 715 ], "spans": [ { "bbox": [ 70, 702, 424, 715 ], "score": 1.0, "content": "mimic the SDE model architecture from Zhang & Chen (2022) for the flow model", "type": "text" }, { "bbox": [ 425, 704, 456, 715 ], "score": 0.94, "content": "v _ { \\boldsymbol { \\theta } } ( t , \\boldsymbol { x } )", "type": "inline_equation" }, { "bbox": [ 456, 702, 525, 715 ], "score": 1.0, "content": ". 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The two representations are concatenated and processed through another three-layer MLP to make", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 70, 106, 541, 118 ], "spans": [ { "bbox": [ 70, 106, 541, 118 ], "score": 1.0, "content": "the prediction. All MLPs use GELU activation and have 128 units per hidden layer. 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We trained all models", "type": "text" } ], "index": 2 }, { "bbox": [ 69, 117, 541, 131 ], "spans": [ { "bbox": [ 69, 117, 93, 131 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 93, 121, 131, 128 ], "score": 0.89, "content": "\\sigma = 0 . 0 5", "type": "inline_equation" }, { "bbox": [ 131, 117, 212, 131 ], "score": 1.0, "content": "and learning rate", "type": "text" }, { "bbox": [ 212, 120, 233, 128 ], "score": 0.91, "content": "1 0 ^ { - 2 }", "type": "inline_equation" }, { "bbox": [ 233, 117, 541, 131 ], "score": 1.0, "content": ", the highest at which they were table, for 1500 batches of size 300, to", "type": "text" } ], "index": 3 }, { "bbox": [ 69, 129, 330, 144 ], "spans": [ { "bbox": [ 69, 129, 261, 144 ], "score": 1.0, "content": "be consistent with the settings from Zhang", "type": "text" }, { "bbox": [ 262, 132, 269, 140 ], "score": 0.35, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 270, 129, 330, 144 ], "score": 1.0, "content": "Chen (2022).", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "text", "bbox": [ 70, 147, 393, 160 ], "lines": [ { "bbox": [ 70, 147, 393, 162 ], "spans": [ { "bbox": [ 70, 147, 393, 162 ], "score": 1.0, "content": "The importance-weighted estimate of the log-partition function is defined", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 209, 169, 401, 202 ], "lines": [ { "bbox": [ 209, 169, 401, 202 ], "spans": [ { "bbox": [ 209, 169, 401, 202 ], "score": 0.94, "content": "\\log \\hat { Z } = \\log \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\frac { R ( x _ { 1 } ^ { ( i ) } ) } { \\mathcal { N } ( x _ { 0 } ^ { ( i ) } ; 0 , \\mathbf { I } ) } \\left| \\frac { \\partial x _ { 1 } } { \\partial x _ { 0 } } \\right| _ { x _ { 0 } = x _ { 0 } ^ { ( i ) } } ,", "type": "interline_equation", "image_path": "4b8fffe074e3aec7a6d0a01f94c6a3d0bcff0f4a17c62f7b3271bbb2ee423bde.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 209, 169, 401, 185.5 ], "spans": [], "index": 6 }, { "bbox": [ 209, 185.5, 401, 202.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 71, 212, 541, 237 ], "lines": [ { "bbox": [ 65, 206, 543, 231 ], "spans": [ { "bbox": [ 65, 206, 99, 231 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 212, 115, 226 ], "score": 0.94, "content": "x _ { 0 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 115, 206, 373, 231 ], "score": 1.0, "content": "are independent samples from the source distribution and", "type": "text" }, { "bbox": [ 373, 212, 389, 226 ], "score": 0.93, "content": "x _ { 1 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 389, 206, 402, 231 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 402, 212, 417, 226 ], "score": 0.93, "content": "x _ { 0 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 418, 206, 543, 231 ], "score": 1.0, "content": "pushed forward by the flow", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 224, 542, 237 ], "spans": [ { "bbox": [ 69, 224, 460, 237 ], "score": 1.0, "content": "(note that the Jacobian can be computed by differentiating the ODE integrator). We used", "type": "text" }, { "bbox": [ 460, 228, 502, 235 ], "score": 0.91, "content": "K = 6 0 0 0", "type": "inline_equation" }, { "bbox": [ 503, 224, 542, 237 ], "score": 1.0, "content": "samples.", "type": "text" } ], "index": 9 } ], "index": 8.5 }, { "type": "text", "bbox": [ 70, 242, 541, 279 ], "lines": [ { "bbox": [ 69, 241, 541, 257 ], "spans": [ { "bbox": [ 69, 241, 541, 257 ], "score": 1.0, "content": "For MCMC, to be consistent with Zhang & Chen (2022), we generated 15000 samples, each of which was seen", "type": "text" } ], "index": 10 }, { "bbox": [ 69, 254, 542, 270 ], "spans": [ { "bbox": [ 69, 254, 459, 270 ], "score": 1.0, "content": "30 times in training. We used 1000 steps of Metropolis-adjusted Langevin sampling with", "type": "text" }, { "bbox": [ 459, 260, 463, 265 ], "score": 0.88, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 464, 254, 542, 270 ], "score": 1.0, "content": "linearly decaying", "type": "text" } ], "index": 11 }, { "bbox": [ 70, 267, 132, 279 ], "spans": [ { "bbox": [ 70, 267, 132, 279 ], "score": 1.0, "content": "from 0.1 to 0.", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "text", "bbox": [ 70, 284, 464, 297 ], "lines": [ { "bbox": [ 69, 283, 466, 298 ], "spans": [ { "bbox": [ 69, 283, 466, 298 ], "score": 1.0, "content": "The flow network used to generate Fig. D.9 followed similar settings to those used in §5.1.", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "title", "bbox": [ 72, 310, 210, 322 ], "lines": [ { "bbox": [ 69, 309, 211, 324 ], "spans": [ { "bbox": [ 69, 309, 211, 324 ], "score": 1.0, "content": "E.7 Unsupervised translation", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 71, 331, 537, 356 ], "lines": [ { "bbox": [ 70, 331, 540, 345 ], "spans": [ { "bbox": [ 70, 331, 540, 345 ], "score": 1.0, "content": "We trained a vanilla convolutional VAE, with about 7 million parameters in the encoder, on CelebA faces", "type": "text" } ], "index": 15 }, { "bbox": [ 70, 344, 205, 356 ], "spans": [ { "bbox": [ 70, 344, 112, 356 ], "score": 1.0, "content": "scaled to", "type": "text" }, { "bbox": [ 113, 347, 155, 354 ], "score": 0.89, "content": "1 2 8 \\times 1 2 8", "type": "inline_equation" }, { "bbox": [ 155, 344, 205, 356 ], "score": 1.0, "content": "resolution.", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "text", "bbox": [ 71, 361, 540, 422 ], "lines": [ { "bbox": [ 69, 361, 541, 375 ], "spans": [ { "bbox": [ 69, 361, 162, 375 ], "score": 1.0, "content": "For the flow network", "type": "text" }, { "bbox": [ 162, 364, 193, 374 ], "score": 0.94, "content": "v _ { \\boldsymbol { \\theta } } ( t , \\boldsymbol { x } )", "type": "inline_equation" }, { "bbox": [ 193, 361, 541, 375 ], "score": 1.0, "content": ", we used a MLP with four hidden layers of 512 units and leaky ReLU activations", "type": "text" } ], "index": 17 }, { "bbox": [ 71, 374, 540, 385 ], "spans": [ { "bbox": [ 71, 374, 267, 385 ], "score": 1.0, "content": "taking the 129-dimensional concatenation of", "type": "text" }, { "bbox": [ 267, 379, 273, 384 ], "score": 0.9, "content": "x", "type": "inline_equation" }, { "bbox": [ 273, 374, 295, 385 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 295, 377, 299, 384 ], "score": 0.88, "content": "t", "type": "inline_equation" }, { "bbox": [ 300, 374, 540, 385 ], "score": 1.0, "content": "as input. All models CFM and OT-CFM were trained", "type": "text" } ], "index": 18 }, { "bbox": [ 69, 385, 541, 399 ], "spans": [ { "bbox": [ 69, 385, 393, 399 ], "score": 1.0, "content": "for 5000 batches of size 256 and the Adam optimizer with learning rate", "type": "text" }, { "bbox": [ 393, 387, 414, 396 ], "score": 0.9, "content": "1 0 ^ { - 3 }", "type": "inline_equation" }, { "bbox": [ 415, 385, 541, 399 ], "score": 1.0, "content": ". Integration was performed", "type": "text" } ], "index": 19 }, { "bbox": [ 69, 395, 542, 412 ], "spans": [ { "bbox": [ 69, 395, 306, 412 ], "score": 1.0, "content": "using the Dormand-Prince integrator with tolerance", "type": "text" }, { "bbox": [ 306, 399, 327, 407 ], "score": 0.92, "content": "1 0 ^ { - 3 }", "type": "inline_equation" }, { "bbox": [ 327, 395, 542, 412 ], "score": 1.0, "content": ". For each attribute, 1000 positive and negative", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 410, 268, 423 ], "spans": [ { "bbox": [ 70, 410, 268, 423 ], "score": 1.0, "content": "images each were used as a held-out test set.", "type": "text" } ], "index": 21 } ], "index": 19 }, { "type": "text", "bbox": [ 70, 427, 321, 439 ], "lines": [ { "bbox": [ 69, 426, 322, 441 ], "spans": [ { "bbox": [ 69, 426, 322, 441 ], "score": 1.0, "content": "Fig. E.1 shows some examples of the learned trajectories.", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "title", "bbox": [ 71, 452, 267, 465 ], "lines": [ { "bbox": [ 69, 451, 268, 468 ], "spans": [ { "bbox": [ 69, 451, 268, 468 ], "score": 1.0, "content": "E.8 Unconditional CIFAR-10 experiments", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 70, 474, 540, 522 ], "lines": [ { "bbox": [ 70, 474, 541, 488 ], "spans": [ { "bbox": [ 70, 474, 541, 488 ], "score": 1.0, "content": "For the CIFAR-10 experiments we followed the setup as described in Lipman et al. (2023). All methods", "type": "text" } ], "index": 24 }, { "bbox": [ 69, 486, 540, 499 ], "spans": [ { "bbox": [ 69, 486, 540, 499 ], "score": 1.0, "content": "were trained with the same setup, only differeing in the choice of probability path. Since code has not been", "type": "text" } ], "index": 25 }, { "bbox": [ 69, 498, 541, 512 ], "spans": [ { "bbox": [ 69, 498, 541, 512 ], "score": 1.0, "content": "released for this work, there are a few parameters which may differ. 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0 , \\mathbf { I } ) } \\left| \\frac { \\partial x _ { 1 } } { \\partial x _ { 0 } } \\right| _ { x _ { 0 } = x _ { 0 } ^ { ( i ) } } ,", "type": "interline_equation", "image_path": "4b8fffe074e3aec7a6d0a01f94c6a3d0bcff0f4a17c62f7b3271bbb2ee423bde.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 209, 169, 401, 185.5 ], "spans": [], "index": 6 }, { "bbox": [ 209, 185.5, 401, 202.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 71, 212, 541, 237 ], "lines": [ { "bbox": [ 65, 206, 543, 231 ], "spans": [ { "bbox": [ 65, 206, 99, 231 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 100, 212, 115, 226 ], "score": 0.94, "content": "x _ { 0 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 115, 206, 373, 231 ], "score": 1.0, "content": "are independent samples from the source distribution and", "type": "text" }, { "bbox": [ 373, 212, 389, 226 ], "score": 0.93, "content": "x _ { 1 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 389, 206, 402, 231 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 402, 212, 417, 226 ], "score": 0.93, "content": "x _ { 0 } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 418, 206, 543, 231 ], "score": 1.0, "content": "pushed forward by the flow", "type": "text" } ], "index": 8 }, { "bbox": [ 69, 224, 542, 237 ], "spans": [ { "bbox": [ 69, 224, 460, 237 ], "score": 1.0, "content": "(note that the Jacobian can be computed by differentiating the ODE integrator). 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For each attribute, 1000 positive and negative", "type": "text" } ], "index": 20 }, { "bbox": [ 70, 410, 268, 423 ], "spans": [ { "bbox": [ 70, 410, 268, 423 ], "score": 1.0, "content": "images each were used as a held-out test set.", "type": "text" } ], "index": 21 } ], "index": 19, "bbox_fs": [ 69, 361, 542, 423 ] }, { "type": "text", "bbox": [ 70, 427, 321, 439 ], "lines": [ { "bbox": [ 69, 426, 322, 441 ], "spans": [ { "bbox": [ 69, 426, 322, 441 ], "score": 1.0, "content": "Fig. E.1 shows some examples of the learned trajectories.", "type": "text" } ], "index": 22 } ], "index": 22, "bbox_fs": [ 69, 426, 322, 441 ] }, { "type": "title", "bbox": [ 71, 452, 267, 465 ], "lines": [ { "bbox": [ 69, 451, 268, 468 ], "spans": [ { "bbox": [ 69, 451, 268, 468 ], "score": 1.0, "content": "E.8 Unconditional CIFAR-10 experiments", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 70, 474, 540, 522 ], "lines": [ { "bbox": [ 70, 474, 541, 488 ], "spans": [ { "bbox": [ 70, 474, 541, 488 ], "score": 1.0, "content": "For the CIFAR-10 experiments we followed the setup as described in Lipman et al. 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To reproduce Lipman", "type": "text" } ], "index": 28 }, { "bbox": [ 68, 539, 538, 555 ], "spans": [ { "bbox": [ 68, 539, 329, 555 ], "score": 1.0, "content": "et al. 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