{"id":261,"date":"2025-03-14T20:58:41","date_gmt":"2025-03-14T12:58:41","guid":{"rendered":"https:\/\/www.toothlessos.xyz\/?p=261"},"modified":"2025-03-14T20:58:43","modified_gmt":"2025-03-14T12:58:43","slug":"decoding-diffusion-on-the-perspective-of-rectified-flow","status":"publish","type":"post","link":"https:\/\/www.toothlessos.xyz\/index.php\/2025\/03\/14\/decoding-diffusion-on-the-perspective-of-rectified-flow\/","title":{"rendered":"Decoding Diffusion: On the perspective of Rectified Flow."},"content":{"rendered":"\n

Modern diffusion models, like Stable Diffusion 3 and flux, have changed their basic model structure to flow-based models. Understanding flows is crucial and yet challenging. In this article, we try to decode flow-based diffusion modes with Rectified Flow, one of the most elegant solutions. Note that this article aims to explain the concepts from a relatively high level, and detailed proofs can be found in the original paper<\/a>.<\/p>\n\n\n\n

Problem<\/h1>\n\n\n\n

Given distribution \"\pi_0, \pi_1\", want to find a transfer map \"T\" such that:

  <\/span>   <\/span>\"\[Z_1 = T(Z_0) \sim \pi_1, \quad \text{where } Z_0 \sim \pi_0.\]\"<\/p><\/p>\n\n\n\n

Example: \"\pi_0\" is a Gaussian, \"\pi_1\" is the target distribution.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Math <\/h1>\n\n\n\n

(In understandable language)<\/strong><\/em><\/p>\n\n\n\n

    \n
  • Vector field<\/strong> \"V(x)\": A vector field defined in \"\mathbb{R}^n\" that indicates the direction and magnitude at each point \"x\".<\/li>\n\n\n\n
  • Velocity field<\/strong> \"v(x,t)\": A special type of vector field that indicates the rate of change and tangent direction of each point \"x\". Here, \"t\" represents the time step.<\/li>\n<\/ul>\n\n\n\n

    Method:<\/h1>\n\n\n\n

    Rectified Flow<\/h2>\n\n\n\n

    We learn \"T\" implicitly by constructing an Ordinary Differential Equation(ODE):

      <\/span>   <\/span>\"\[\frac{d}{dt} Z_t = v(Z_t, t), \quad Z_0 \sim \pi_0, \quad \forall t \in [0,1].\]\"<\/p><\/p>\n\n\n\n

    Intuitively, the best way to get \"Z_1\" from \"Z_0\" is to have a “straight” flow.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    So we need to find an ODE to match the linear interpolation of points from \"\pi_0\" and \"\pi_1\".<\/p>\n\n\n\n

    Observe \"X_0 \sim \pi_0\", \"X_1 \sim \pi_1\". Let \"X_t\" for \"t \in [0,1]\" be the linear interpolation of \"X_1\" and \"X_0\":

      <\/span>   <\/span>\"\[X_t = t X_1 + (1-t) X_0, \quad t \in [0,1].\]\"<\/p><\/p>\n\n\n\n

    We can have a trivial ODE:

      <\/span>   <\/span>\"\[\frac{d}{dt} X_t = X_1 - X_0, \quad \forall t \in [0,1].\]\"<\/p><\/p>\n\n\n\n

      \n
    • Note that this ODE is not causal or forward simulatable<\/em>, as we need to know \"X_1\" when \"t < 1\" in order to calculate \"(X_1 - X_0)\".<\/li>\n<\/ul>\n\n\n\n

      So we want our ODE \"\frac{d}{dt} Z_t = v(Z_t, t)\" to be as close as possible to the trivial ODE.<\/p>\n\n\n\n

      A simple way is to optimize \"v\" by minimizing:

        <\/span>   <\/span>\"\[\min_v \mathbb{E}_{X_0 \sim \pi_0, X_1 \sim \pi_1} \int_0^1 | (X_1 - X_0) - v(X_t, t) |^2 dt\]\"<\/p>

        <\/span>   <\/span>\"\[where X_t = t X_1 + (1-t) X_0\]\"<\/p><\/p>\n\n\n\n

      And \"v\" can simply be a neural network.<\/p>\n\n\n\n

      Reflow: Straight Flows<\/h2>\n\n\n\n
      \"\"<\/figure>\n\n\n\n

      The method above gives us the flow (b). In order to get flow (c), we can do:

        <\/span>   <\/span>\"\[\min \int_0^1 \mathbb{E}_{X_0 \sim \pi_0, X_1 \sim \pi_1} | (X_1 - X_0) - \text{Flow}_1 (X_t) |^2 dt\]\"<\/p><\/p>\n\n\n\n

        <\/span>   <\/span>\"\[where X_t = t X_1 + (1-t) X_0\]\"<\/p><\/p>\n\n\n\n

      We call it 2-rectified flow.<\/p>\n\n\n\n

        \n
      • Note that we can do such a process multiple times.<\/li>\n\n\n\n
      • Always remember the Reflow process can speed up the generate process, and will not improve the generation performance.<\/li>\n<\/ul>\n\n\n\n

        <\/p>\n\n\n\n

        <\/p>\n","protected":false},"excerpt":{"rendered":"

        Modern diffusion models, like Stable Diffusion 3 and fl […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[19],"tags":[31,32,20],"class_list":["post-261","post","type-post","status-publish","format-standard","hentry","category-ml","tag-diffusion","tag-flow","tag-machine-learning"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/posts\/261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=261"}],"version-history":[{"count":8,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/posts\/261\/revisions"}],"predecessor-version":[{"id":276,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/posts\/261\/revisions\/276"}],"wp:attachment":[{"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.toothlessos.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}