2024-09-27

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# Rectified Flow
Author: .yellow[Cuiem]

[[Index](../../index.html)]
[[16:9](20241023_16_9.html)]

.footnote[Supported by [Remark](https://github.com/gnab/remark)]

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# Agenda
### 1. Theory
### 2. Flow-Gan
### 3. Latest Work
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# Theory
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## Theory
### - Flow Matching
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Let \$x_1\$ denote a random variable distributed according to some unknown data distribution \$q(x_1)\$. We assume we only have access to data samples from \$q(x_1)\$ but have no access to the density function itself.

Furthermore, we let \$p_t\$ be a probability path such that \$p_0 = p\$ is a simple distribution, e.g., the standard normal distribution \$p(x) = N (x|0, I)\$, and let \$p_1\$ be approximately equal in distribution to \$q\$.

The Flow Matching objective is then designed to match this target probability path, which will allow us to flow from \$p_0\$ to \$p_1\$.

<center>
  <img src="example.png" width="100%">
</center>
]

.footnote[.red[*]Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., & Le, M. (2022). Flow matching for generative modeling.]
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.left-column[
## Theory
### - Flow Matching
### - Rectified Flow
]

.right-column[

The rectified flow is an ODE model that transport distribution \$\pi_0\$ to \$\pi_1\$ by following .red[straight line paths] as much as possible.

Given empirical observations of \$X_0 \sim \pi_0\$, \$X_1 \sim \pi_1\$, the rectified flow induced from \$(X_0, X_1)\$ is an ODE on time \$t \in [0, 1]\$,

$$
      \begin{aligned}
        dZ_t = v(Z_t, t) dt
      \end{aligned}
$$

which converts \$Z_0\$ from \$\pi_0\$ to a \$Z_1\$ following \$\pi_1\$. The drift force \$v: \\mathbb{R}^d \longrightarrow \\mathbb{R}^d\$ is learned by solving the following optimization problem:

$$
      \begin{aligned}
        \min_{v} \int_0^1 \mathbb{E}[||(X_1 - X_0) - v(X_t, t)||^2] dt,\quad \text{with}\quad X_t = tX_1+(1-t)X_0
      \end{aligned}
$$

]

.footnote[.red[*]Liu, X., Gong, C., & Liu, Q. (2022). Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003.]
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.left-column[
## Theory
### - Flow Matching
### - Rectified Flow
### - Algorithm

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# Flow-Gan
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## Flow-Gan
Check My log [here](../../GenerativeModel/GmPaperReading/FlowGan/fg.html).
<center>
  <img src="flow-gan.png" width="80%">
</center>
.footnote[.red[*]Grover, A., Dhar, M., & Ermon, S. (2018, April). Flow-gan: Combining maximum likelihood and adversarial learning in generative models. In Proceedings of the AAAI conference on artificial intelligence (Vol. 32, No. 1).]

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# Latest Work
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## Latest Work

So far results:
<center>
  <img src="MNIST.png" width="100%">
</center>
]
.right-column[
Want to combine Rectified Flow and Gan.
<center>
  <img src="rfgan.png" width="100%">
</center>

]
.footnote[.red[*] Check larger image [here](20241023_16_9.html#10).]
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#Thank You!
[[Index](../../index.html)]