Semidiscrete Optimal Transport: Difference between revisions
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<math display="block"> \max_{\varphi \in \R^m} \Big\{ \mathcal{E}(\varphi) = \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\}. </math> | <math display="block"> \max_{\varphi \in \R^m} \Big\{ \mathcal{E}(\varphi) = \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\}. </math> | ||
Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first | Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first is that <math> \varphi^c </math> denotes the c-transform of <math> \varphi </math>. The c-transform can be defined as <math> \varphi^c(x) := \inf_j \{ c(x,y_j) - \varphi_j \} </math>. <math> \varphi_j </math> is used to denote <math> \varphi(y_j) </math>. Furthermore, we note that our original measure <math> \nu </math> is a sum of Dirac masses evaluated at locations <math> y_j </math> with weights <math> b_j </math>, i.e., <math> \nu = \sum_{j=1}^{N} b_j \delta_{y_j} </math>. | ||
== Voronoi cells | == Voronoi cells decomposition == | ||
Now, we will establish the notion of Voronoi cells. The Voronoi cells refer to a special subset of <math> X </math>, and the reason we are interested in such a subset is because we can use the Voronoi cells to find the regions that are sent to each <math> y_j </math>. In particular, if we denote the set of Voronoi cells as <math> V_{\varphi}(j) </math>, we can find our values of <math> \varphi_j </math> using the fact <math> b_j = \int_{V_{\varphi}(j)} f(x)dx </math>. Recall that <math> f(x) </math> refers to a density of the measure <math> \mu </math>, i.e., <math> \mu = f(x)dx </math>. We define the Voronoi cells with | |||
Now, we will establish the notion of Voronoi cells. The Voronoi cells refer to a special subset of <math> X </math>, and the reason we are interested in such a subset is because we can use the Voronoi cells | |||
<math display="block"> V_{\varphi}(j) = \Big\{ x \in X : \forall j' \neq j, \frac{1}{2}|x-y_j|^2 - \varphi_j \leq \frac{1}{2}|x-y_{j'}|^2 - \varphi_{j'} \Big\}. </math> | <math display="block"> V_{\varphi}(j) = \Big\{ x \in X : \forall j' \neq j, \frac{1}{2}|x-y_j|^2 - \varphi_j \leq \frac{1}{2}|x-y_{j'}|^2 - \varphi_{j'} \Big\}. </math> | ||
We use the specific cost function <math> c(x,y) = \frac{1}{2}|x-y|^2 </math> here. This is a special case and we may generalize to other cost functions if we desire. When we have this special case, the decomposition of our space <math> X </math> is known as a "power diagram."<ref name="Merigot"/> | We use the specific cost function <math> c(x,y) = \frac{1}{2}|x-y|^2 </math> here. This is a special case and we may generalize to other cost functions if we desire. When we have this special case, the decomposition of our space <math> X </math> is known as a "power diagram."<ref name="Merigot"/> | ||
== | == The gradient in finding optimal <math> \varphi </math> == | ||
Finding the weights via the above method is equivalent to maximizing <math> \mathcal{E}(\varphi) </math>, and we may do this by taking the partial derivatives of this function with respect to <math> \varphi_j </math>. This is the same as taking the gradient of <math> \mathcal{E}(\varphi) </math>. In partial derivative form, we have | Finding the weights via the above method is equivalent to maximizing <math> \mathcal{E}(\varphi) </math>, and we may do this by taking the partial derivatives of this function with respect to <math> \varphi_j </math>. This is the same as taking the gradient of <math> \mathcal{E}(\varphi) </math>. In partial derivative form, we have | ||
<math display="block"> \frac{\partial \mathcal{E} }{\partial \varphi_j} = - \int_{V_{\varphi}(j)} f(x)dx + b_j </math> | <math display="block"> \frac{\partial \mathcal{E} }{\partial \varphi_j} = - \int_{V_{\varphi}(j)} f(x)dx + b_j </math><ref name="Merigot"/> | ||
and in gradient form, we have | and in gradient form, we have | ||
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<math display="block"> \nabla \mathcal{E}(\varphi)_j = - \int_{V_{\varphi}(j)} f(x)dx + b_j .</math> | <math display="block"> \nabla \mathcal{E}(\varphi)_j = - \int_{V_{\varphi}(j)} f(x)dx + b_j .</math> | ||
Since <math> \nabla \mathcal{E}(\varphi)_j = 0 </math> when it attains a maximum, we have the relation between the weights and the measure density that we established in the previous section. Note that the maximum is taken and not the minimum because our function <math> \mathcal{E}(\varphi) </math> is a concave function. The | Since <math> \nabla \mathcal{E}(\varphi)_j = 0 </math> when it attains a maximum, we have the relation between the weights and the measure density that we established in the previous section. Note that the maximum is taken and not the minimum because our function <math> \mathcal{E}(\varphi) </math> is a concave function. The mapping <math> \varphi \rightarrow \sum_j \varphi_j b_j </math> is linear, but because we consider an infimum for <math> \varphi^c </math>, the overall function <math> \mathcal{E}(\varphi) </math> is concave.<ref name="Merigot"/> | ||
Once we have discovered our optimal <math> \varphi_j </math>, the optimal transport map is is one in which any <math> x \in V_{\varphi_j} </math> is mapped to <math> y_j </math>. We have the mapping to a constant over each particular region, making the optimal transport map piecewise constant.<ref name="Peyré and Cuturi"/> | |||
== Algorithm discussion == | == Algorithm discussion == | ||
We may search for a maximum of value of <math> \mathcal{E}(\varphi) </math> by means of certain algorithms, the first being gradient ascent. Whether or not such an algorithm is capable of being implemented effectively is contingent on the ability to find our power diagram in a practical way. Certain computational geometry algorithms allow the cells to be found efficiently. A second suitable algorithm is Newton's method. Using Newton's method to find the zeros of <math> \frac{\partial \mathcal{E} }{\partial \varphi_j} </math>, one must compute the second derivative of <math> \mathcal{E} </math>, as well as justify certain constraints are met, such as bounded eigenvalues of the Hessian. | We may search for a maximum of value of <math> \mathcal{E}(\varphi) </math> by means of certain algorithms, the first being gradient ascent. Whether or not such an algorithm is capable of being implemented effectively is contingent on the ability to find our power diagram in a practical way. Certain computational geometry<ref name="Santambrogio"/> algorithms allow the cells to be found efficiently. A second suitable algorithm is Newton's method. Using Newton's method to find the zeros of <math> \frac{\partial \mathcal{E} }{\partial \varphi_j} </math>, one must compute the second derivative of <math> \mathcal{E} </math>, as well as justify certain constraints are met, such as bounded eigenvalues of the Hessian.<ref name="Santambrogio"/> | ||
Latest revision as of 04:38, 28 February 2022
Semidiscrete optimal transport refers to situations in optimal transport where two input measures are considered, and one measure is a discrete measure and the other one is absolutely continuous with respect to Lebesgue measure.[1] Hence, because only one of the two measures is discrete, we arrive at the appropriate name "semidiscrete."
Formulation of the semidiscrete dual problem
In particular, we will examine semidiscrete optimal transport in the case of the dual problem. The general dual problem for continuous measures can be stated as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max_{(\psi, \varphi) \in R(c)} \Big\{ \int_X \psi d\mu + \int_Y \varphi d\nu \Big\} } [2]
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu, \nu } denote probability measures on domains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X, Y } respectively, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x,y) } is a cost function defined over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times Y } . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(c) } denotes the set of possible dual potentials, and the condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(x) + \psi(y) \leq c(x,y) } is satisfied. It should also be noted that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } has a density such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = f(x)dx } . Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max_{\varphi \in \R^m} \Big\{ \mathcal{E}(\varphi) = \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\}. }
Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^c } denotes the c-transform of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi } . The c-transform can be defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^c(x) := \inf_j \{ c(x,y_j) - \varphi_j \} } . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_j } is used to denote Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(y_j) } . Furthermore, we note that our original measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is a sum of Dirac masses evaluated at locations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j } with weights Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_j } , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = \sum_{j=1}^{N} b_j \delta_{y_j} } .
Voronoi cells decomposition
Now, we will establish the notion of Voronoi cells. The Voronoi cells refer to a special subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } , and the reason we are interested in such a subset is because we can use the Voronoi cells to find the regions that are sent to each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j } . In particular, if we denote the set of Voronoi cells as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{\varphi}(j) } , we can find our values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_j } using the fact Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_j = \int_{V_{\varphi}(j)} f(x)dx } . Recall that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } refers to a density of the measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = f(x)dx } . We define the Voronoi cells with
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{\varphi}(j) = \Big\{ x \in X : \forall j' \neq j, \frac{1}{2}|x-y_j|^2 - \varphi_j \leq \frac{1}{2}|x-y_{j'}|^2 - \varphi_{j'} \Big\}. }
We use the specific cost function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x,y) = \frac{1}{2}|x-y|^2 } here. This is a special case and we may generalize to other cost functions if we desire. When we have this special case, the decomposition of our space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } is known as a "power diagram."[3]
The gradient in finding optimal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi }
Finding the weights via the above method is equivalent to maximizing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\varphi) } , and we may do this by taking the partial derivatives of this function with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_j } . This is the same as taking the gradient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\varphi) } . In partial derivative form, we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathcal{E} }{\partial \varphi_j} = - \int_{V_{\varphi}(j)} f(x)dx + b_j } [3]
and in gradient form, we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \mathcal{E}(\varphi)_j = - \int_{V_{\varphi}(j)} f(x)dx + b_j .}
Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \mathcal{E}(\varphi)_j = 0 } when it attains a maximum, we have the relation between the weights and the measure density that we established in the previous section. Note that the maximum is taken and not the minimum because our function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\varphi) } is a concave function. The mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \rightarrow \sum_j \varphi_j b_j } is linear, but because we consider an infimum for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^c } , the overall function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\varphi) } is concave.[3]
Once we have discovered our optimal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_j } , the optimal transport map is is one in which any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V_{\varphi_j} } is mapped to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j } . We have the mapping to a constant over each particular region, making the optimal transport map piecewise constant.[1]
Algorithm discussion
We may search for a maximum of value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\varphi) } by means of certain algorithms, the first being gradient ascent. Whether or not such an algorithm is capable of being implemented effectively is contingent on the ability to find our power diagram in a practical way. Certain computational geometry[2] algorithms allow the cells to be found efficiently. A second suitable algorithm is Newton's method. Using Newton's method to find the zeros of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathcal{E} }{\partial \varphi_j} } , one must compute the second derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E} } , as well as justify certain constraints are met, such as bounded eigenvalues of the Hessian.[2]