OT in a nutshell

docs/index.rst

@@ -87,6 +87,7 @@ Important resources
     user_guide
     user
     developer
+    ot_background
     contributing
     notebooks/tutorials/index
     notebooks/examples/index

docs/ot_background.md

@@ -0,0 +1,82 @@
+# Optimal Transport (OT) in a nutshell
+
+Optimal transport ({term}`OT`) is a general problem in mathematics that has powerful applications in single-cell genomics analysis, especially in the context of spatial and multi-modal data.\
+The problem that OT aims to solve is minimizing some measure of distance $L$ between two distributions of data points, e.g. sets of cells.
+The solution is encoded using a {term}`transport matrix` $\mathbf{P} \in \mathbb{R}_{+}^{n \times m}$ where $\mathbf{P}_{i,j}$ describes the amount of mass that is transported from data point $x_i$ in row $i$ to data point $y_j$ in column $j$.
+
+The regularized {term}`linear problem` reads:
+
+```{math}
+\begin{align*}
+    \mathbf{L_C^{\varepsilon}(a,b) \overset{\mathrm{def.}}{=} \min_{P\in U(a,b)} \left\langle P, C \right\rangle - \varepsilon H(P).}
+\end{align*}
+```
+
+where $\varepsilon$ is the {term}`entropic regularization`, and $\mathbf{H(P) \overset{\mathrm{def.}}{=} - \sum_\mathnormal{i,j} P_\mathnormal{i,j} \left( \log (P_\mathnormal{i,j}) - 1 \right)}$ is the discrete entropy of a coupling matrix.
+
+## Gromov-Wasserstein (GW)
+
+When the data points (e.g. cells) from source and target distributions lie in different metric spaces,
+we only assume that two matrices $\mathbf{D \in \mathbb{R}^\mathnormal{n \times n}}$ and $\mathbf{D' \in \mathbb{R}^\mathnormal{m \times m}}$
+quantify similarity relationships between data points within the respective distribution.
+
+:::{figure} figures/GWapproach.jpeg
+:align: center
+:alt: Gromov-Wasserstein approach.
+:class: img-fluid
+
+Gromov-Wasserstein approach to comparing two metric measure spaces. Figure from {cite}`peyre:19`.
+:::
+
+The {term}`Gromov-Wasserstein` problem reads
+
+```{math}
+\begin{align*}
+    \mathrm{GW}\mathbf{((a,D), (b,D'))^\mathrm{2} \overset{\mathrm{def.}}{=} \min_{P \in U(a,b)} \mathcal{E}_{D,D'}(P)}
+\\
+    \textrm{where} \quad \mathbf{\mathcal{E}_{D,D'}(P) \overset{\mathrm{def.}}{=} \sum_{\mathnormal{i,j,i',j'}} \left| D_{\mathnormal{i,i'}} - D'_{\mathnormal{j,j'}} \right|^\mathrm{2} P_{\mathnormal{i,i'}}P_{\mathnormal{j,j'}}}.
+\end{align*}
+```
+
+## Fused Gromov-Wasserstein (FGW)
+
+{term}`Fused Gromov-Wasserstein` is needed in cases where a data point, e.g. a cell, of the source distribution
+has both features in the same space as the target distribution ({term}`linear term`) and features in a
+different space than a data point in the target distribution ({term}`quadratic term`).
+
+:::{figure} figures/FGWadapted.jpg
+:align: center
+:alt: Fused Gromov-Wasserstein distance.
+:class: img-fluid
+
+Fused Gromov-Wasserstein distance incorporates both feature and structure aspects of the source and target measures.
+Figure adapted from {cite}`vayer:20`.
+:::
+
+The FGW problem is defined as
+
+```{math}
+\begin{align*}
+    \mathrm{FGW}\mathbf{(a,b,D,D',M) \overset{\mathrm{def.}}{=} \min_{P \in U(a,b)} E_{D,D',M}(P)}
+\\
+    \textrm{where} \quad \mathbf{E_{D,D',M}(P) \overset{\mathrm{def.}}{=} \sum_{\mathnormal{i,j,i',j'}} \left( (1-\alpha)M_\mathnormal{i,j} + \alpha  \left| D_{\mathnormal{i,i'}} - D'_{\mathnormal{j,j'}} \right|^\mathrm{2} \right) P_{\mathnormal{i,i'}}P_{\mathnormal{j,j'}}}
+\end{align*}
+```
+
+Here, $D$ and $D'$ are distances in the structure spaces, $M$ - the distance in the feature space,
+and $\alpha \in [0,1]$ is the tradeoff between the feature and the structure costs.
+
+## Unbalanced OT
+
+In cases that require allowing to ignore any outliers or skip points that don’t have a satisfactory mapping,
+we can add a penalty for the amount of mass variation using some divergence $D_{\varphi}$
+and get the minimization of an OT distance between approximate measures
+
+```{math}
+\begin{align*}
+   \mathbf{L_C^{\tau}(a,b) =  \min_{\tilde{a},\tilde{b}}  L_C(a,b) + \tau_1 D_{\varphi}(a,\tilde{a}) + \tau_2 D_{\varphi}(b,\tilde{b})} \\
+   \mathbf{= \min_{P\in \mathbb{R}_+^\mathnormal{n\times m}} \left\langle C,P \right\rangle + \tau_1 D_{\varphi}(P\mathbb{1}_\mathnormal{m}|a) + \tau_2 D_{\varphi}(P^\top\mathbb{1}_\mathnormal{m}|b)}
+\end{align*}
+```
+
+where $(\tau_1, \tau_2)$ controls how much mass variations are penalized as opposed to transportation of the mass.

docs/references.bib

@@ -478,6 +478,18 @@ @article{srivatsan:20
   publisher={American Association for the Advancement of Science}
 }
 
+@article{vayer:20,
+AUTHOR = {Vayer, Titouan and Chapel, Laetitia and Flamary, Remi and Tavenard, Romain and Courty, Nicolas},
+TITLE = {Fused Gromov-Wasserstein Distance for Structured Objects},
+JOURNAL = {Algorithms},
+VOLUME = {13},
+YEAR = {2020},
+NUMBER = {9},
+ARTICLE-NUMBER = {212},
+URL = {https://www.mdpi.com/1999-4893/13/9/212},
+ISSN = {1999-4893},
+DOI = {10.3390/a13090212}}
+
 @misc{klein2023generative,
       title={Generative Entropic Neural Optimal Transport To Map Within and Across Spaces},
       author={Dominik Klein and Théo Uscidda and Fabian Theis and Marco Cuturi},