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Computational Optimal Transport
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Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.
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Cited by 9 Pith papers
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Fast Whole-Brain, Geometry-Aware Functional Alignment for Cross-Subject Decoding
SpectralOT regularizes entropic optimal transport with the first three Laplace-Beltrami eigenmodes of cortical geometry to produce fast, parsimonious whole-brain functional alignments that improve cross-subject decoding.
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Representations from Pretrained Machine-Learning Interatomic Potentials as Coarse Coordinates for Material Generation and Evaluation
A dual-featurizer transport distance using MACE and contrastive GNN features jointly measures quality and novelty of generated crystals, and the MACE features can condition a flow-matching generator.
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Safe Inference-Time Alignment via Lagrangian Reward Augmentation
Dualizing Safe RLHF yields a one-dimensional convex calibration of λ that defines a drop-in safety-aware reward for Best-of-N and token-level inference-time decoders.
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Consistent pricing of bivariate interest rate exotics via constrained Schr\"odinger optimal transport
A constrained Schrödinger optimal transport dual is used to build a CMS spread-consistent joint distribution and approximate no-arbitrage bounds for bivariate interest-rate exotics.
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D-CLIPSE: Distributed Consensus-based Localization with Passive Listening on Shared State Exchange
D-CLIPSE reaches near-centralized multi-robot localization accuracy and consistency by consensus on shared states only, plus passive listening of preintegrated odometry exchanges.
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Optimal Transport Event Representation for Anomaly Detection
Adding a few optimal-transport-based features to standard jet observables nearly doubles anomaly-detection significance at 0.5% signal injection on LHC Olympics benchmarks.
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Improved Stochastic Optimization of LogSumExp
A rescaled SoftPlus family approximates LogSumExp with O(ρ) error, enabling stable stochastic optimization in entropic OT and KL-DRO.
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Packet-Level Traffic Modeling with Heavy-Tailed Payload and Inter-Arrival Distributions for Digital Twins
A hybrid HMM + mixture-density-network generator reproduces heavy-tailed packet payloads and inter-arrival times on four traces with about 0.16 MB footprint, beating larger baselines on most metrics.
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Affine calculus for constrained minima of the Kullback-Leibler divergence
A geometric calculus using exponential and mixture coordinates is applied to derive natural gradients for KL-type divergences in mean-field, transport, GAN, and variational Bayes settings.
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