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   https://doi.org/10.1109/CSF.2017.17  

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2. Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

   https://doi.org/10.1109/FOCS54457.2022.00064  

   Chen, Li ; Kyng, Rasmus ; Liu, Yang P. ; Peng, Richard ; Gutenberg, Maximilian Probst ; Sachdeva, Sushant ( October 2022 , 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS))

   We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in m^{1+o(1)} time. Our algorithm builds the flow through a sequence of m^{1+o(1)} approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized m^{o(1)} time using a new dynamic graph data structure. Our framework extends to algorithms running in m^{1+o(1)} time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and p-norm isotonic regression on arbitrary directed acyclic graphs. 

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3. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow

   Xingchao Liu ; Chengyue Gong ; Qiang Liu ( January 2023 , International conference on learning representations (ICLR))

   We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions π0 and π1, hence providing a unified solution to generative modeling and domain transfer, among various other tasks involving distribution transport. The idea of rectified flow is to learn the ODE to follow the straight paths connecting the points drawn from π0 and π1 as much as possible. This is achieved by solving a straightforward nonlinear least squares optimization problem, which can be easily scaled to large models without introducing extra parameters beyond standard supervised learning. The straight paths are special and preferred because they are the shortest paths between two points, and can be simulated exactly without time discretization and hence yield computationally efficient models. We show that the procedure of learning a rectified flow from data, called rectification, turns an arbitrary coupling of π0 and π1 to a new deterministic coupling with provably non-increasing convex transport costs. In addition, recursively applying rectification allows us to obtain a sequence of flows with increasingly straight paths, which can be simulated accurately with coarse time discretization in the inference phase. In empirical studies, we show that rectified flow performs superbly on image generation, image-to-image translation, and domain adaptation. In particular, on image generation and translation, our method yields nearly straight flows that give high quality results even with a single Euler discretization step. 

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4. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow

   Xingchao Liu, Chengyue Gong ( September 2022 , arXivorg)
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