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z8fpPCuWBLsr(*lH#7L3M*8tkRF-3i1kPYi7a$9)Jv!gWfbqP*_voMc*6lYz-M@&&k zsxverLAA42nz;J(Rv+c1e4G(~H!cu*9mvZ-GjbD_jh=m@yG$=ysuMBBSKboXZ$%Hv z?PR7kRV(<0iP?@=&~Cce+B&+!uH8&A8+dRR88FTkqE7PeuYnpy=A3jR?U}XGqY~48 zI)Z}HI#9e>cZ{6Uk|yfL&S9r--k?}#gW?Oc95%UC z>?`w9=6OrW`2BA)#pKM-{3p1Ne=Z{Y{r%r7!u%Y+J$tUdlD4&JL}~Pw317d$2CvTY zUL5&G6E5XcC(s7zj5BYjqk@B+z%IWW-(R{GSKC~oEyY502Idkt#)~LVpy~K+#ZJ~~ zhoJv}qcY|ly+P{JDp69fm$KYpcM1^IZF8swp@OL`u%s1XhKKKGPOc?Pc=ur9hhxX- zoHMk+_8P(7Zo8Z3F*(8~AhZ{{Z Date: Thu, 2 Jan 2025 18:41:57 +0000 Subject: [PATCH 2/5] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- docs/ot_background.md | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) diff --git a/docs/ot_background.md b/docs/ot_background.md index a16a4f78e..072738f7f 100644 --- a/docs/ot_background.md +++ b/docs/ot_background.md @@ -4,7 +4,7 @@ Optimal transport ({term}`OT`) is a general problem in mathematics that has powe The problem that OT aims to solve is minimizing some measure of distance $L$ between two distributions, 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 OT problem reads: +The regularized OT problem reads: ```math \begin{align*} @@ -16,7 +16,7 @@ where $\varepsilon$ is the {term}`entropic regularization`, and $\mathbf{H(P) \o ## Gromov-Wasserstein (GW) -When the data points (e.g. cells) from source and target distributions lie in different metric spaces, +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.\ @@ -75,10 +75,12 @@ and $\alpha \in [0,1]$ is the tradeoff between the feature and the structure cos 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 + +```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)} + \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. From 4332d81e1459276ab38bc4dd1b2b66b976cd9fdd Mon Sep 17 00:00:00 2001 From: Arina Danilina Date: Thu, 2 Jan 2025 19:58:39 +0100 Subject: [PATCH 3/5] links --- docs/index.rst | 1 + docs/ot_background.md | 2 +- 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/index.rst b/docs/index.rst index 015f1edbf..a072ebf96 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -61,6 +61,7 @@ well as the publication introducing the model, which can be found in the corresp :hidden: installation + ot_background user_guide user developer diff --git a/docs/ot_background.md b/docs/ot_background.md index a16a4f78e..098ebb916 100644 --- a/docs/ot_background.md +++ b/docs/ot_background.md @@ -4,7 +4,7 @@ Optimal transport ({term}`OT`) is a general problem in mathematics that has powe The problem that OT aims to solve is minimizing some measure of distance $L$ between two distributions, 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 OT problem reads: +The regularized {term}`linear OT` problem reads: ```math \begin{align*} From 6812f4c3471d9891759c51f930791c3825c65058 Mon Sep 17 00:00:00 2001 From: Arina Danilina Date: Sat, 4 Jan 2025 12:21:34 +0100 Subject: [PATCH 4/5] formatting --- docs/ot_background.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/docs/ot_background.md b/docs/ot_background.md index ef506b7e6..ff3faccc3 100644 --- a/docs/ot_background.md +++ b/docs/ot_background.md @@ -1,12 +1,12 @@ # 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, e.g. sets of cells. +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 OT` problem reads: +The regularized {term}`linear problem` reads: -```math +```{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*} @@ -30,7 +30,7 @@ Gromov-Wasserstein approach to comparing two metric measure spaces. Figure credi The {term}`Gromov-Wasserstein` problem reads -```math +```{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)} \end{align*} @@ -57,7 +57,7 @@ Figure adapted from {cite}`vayer:20`. The FGW problem is defined as -```math +```{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)} \end{align*} @@ -76,7 +76,7 @@ In cases that require allowing to ignore any outliers or skip points that don’ 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 +```{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)} From fc279b9fc20f4beb842138cb9383ed9ae3325b12 Mon Sep 17 00:00:00 2001 From: Arina Danilina Date: Mon, 6 Jan 2025 15:45:46 +0100 Subject: [PATCH 5/5] link edits --- docs/index.rst | 2 +- docs/ot_background.md | 8 ++------ 2 files changed, 3 insertions(+), 7 deletions(-) diff --git a/docs/index.rst b/docs/index.rst index 49d6417f0..a97e71a76 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -84,10 +84,10 @@ Important resources :hidden: installation - ot_background user_guide user developer + ot_background contributing notebooks/tutorials/index notebooks/examples/index diff --git a/docs/ot_background.md b/docs/ot_background.md index ff3faccc3..1ab2c63fa 100644 --- a/docs/ot_background.md +++ b/docs/ot_background.md @@ -18,14 +18,14 @@ where $\varepsilon$ is the {term}`entropic regularization`, and $\mathbf{H(P) \o 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.\ +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 credit: {cite}`peyre:19`. +Gromov-Wasserstein approach to comparing two metric measure spaces. Figure from {cite}`peyre:19`. ::: The {term}`Gromov-Wasserstein` problem reads @@ -33,9 +33,7 @@ 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)} -\end{align*} \\ -\begin{align*} \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*} ``` @@ -60,9 +58,7 @@ 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)} -\end{align*} \\ -\begin{align*} \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*} ```