Semi-discrete unbalanced optimal transport and quantization

TL;DR

This work develops a rigorous framework for semi-discrete unbalanced optimal transport, where mass can be created or destroyed and one measure is diffuse while the other is discrete. The authors extend classical semi-discrete transport to the unbalanced setting via a tessellation approach built from generalized Laguerre cells, derive dual and primal optimality conditions, and propose practical numerical schemes (gradient-based and quasi-Newton Lloyd-type methods) for solving these problems. They then apply the framework to unbalanced quantization, proving a Voronoi-tessellation representation and extending Lloyd’s algorithm to this context, with thorough numerical demonstrations across several models. In two dimensions they establish crystallization-type results: as the number of Dirac masses grows and a length-scale parameter varies, the optimal local density converges to a triangular lattice with a nonlocal density $D(x)$ determined by the ground-cost and margin function; Zador-type asymptotics are recovered in the balanced limit. Overall, the paper provides a cohesive theory, computational tools, and asymptotic insights for unbalanced transport and quantization, including explicit connections between Voronoi structures, mass distributions, and geometric crystallization phenomena.

Abstract

In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions -- optimality of a locally triangular lattice with spatially varying density -- and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.

Figures (11)

• Figure 1: Two (approximate) centroidal Voronoi tessellations (CVTs) of 10 points for the uniform density $\mu=1$ on a unit square. The polygons are the centroidal Voronoi cells $V_i$ and the circles are the generators $x_i$. The CVTs were computed using Lloyd's algorithm. The CVT on the left has a lower energy $J$ than the CVT on the right. The corresponding quantizer $\nu=\sum_{i=1}^{10} m_i \delta_{x_i}$ of $\mu$ is reconstructed from the CVT by taking $m_i$ as the areas of the centroidal Voronoi cells and $x_i$ as their generators.
• Figure 2: Semi-discrete transport between the Lebesgue measure on $\Omega=[0,L]^2$, $L=5$ and a discrete measure with $M=4$ Dirac masses of locations $(x_1,x_2,x_3,x_4)=L \cdot ((0.375,0.375),(0.75,0.35),(0.65,0.75),(0.25,0.8))$ and weights $(m_1,m_2,m_3,m_4)=|\Omega|\cdot(0.38,0.29,0.19,0.14)$. Top row: optimal cells $\{C_i(w)\}_{i=1}^M$; the residual set $R$ is represented by white; location of the discrete points $(x_1,\ldots,x_M)$ is indicated with red dots. Bottom row: optimal marginal $\rho$ (identical colour scale in all figures; regions with $\tfrac{\textnormal{d} \rho}{\textnormal{d} \mu}(x)=0$ are white for emphasis) and boundaries of cells $\{C_i(w)\}_{i=1}^M$ (red) are shown for models \ref{['item:ModelOT']}--\ref{['item:ModelQuadratic']} from \ref{['exm:models', 'exm:modelComparison']}. Figure (e) shows the same model as \ref{['item:ModelQuadratic']}, only with $c(x,y)=[d(x,y)/2]^2$ instead of $c(x,y)=d(x,y)^2$; on some cells $\mathop{\mathrm{spt}}\nolimits \rho$ is now strictly bounded away from the boundaries of $C_i(w)$.
• Figure 3: One-dimensional slices of computational results from \ref{['fig:SemiDisc_model_comparison']} along $[0,L]\times\{0.375\,L\}$ with $L=5$. Left:$\phi_w$ for optimal $w \in \mathbb{R}^M$. For models \ref{['item:ModelOT']}, \ref{['item:ModelGHK']}, and \ref{['item:ModelQuadratic']}, $\phi_{w}$ is piecewise quadratic; for \ref{['item:ModelHK']} the profile is determined by $c_{\textnormal{HK}}$ and $\phi_w=\infty$ on $R \neq \emptyset$. Right: Optimal density $\tfrac{\textnormal{d} \rho}{\textnormal{d} \mu}$, where $\tfrac{\textnormal{d} \rho}{\textnormal{d} \mu}=(F^\ast)'(-\phi_{w})$ on $\Omega \setminus R$ and $0$ elsewhere by \ref{['eq:UnbalancedOptimalityConditionsDensity']}. For \ref{['item:ModelOT']} the density is constant, for \ref{['item:ModelGHK']} it is piecewise Gaussian, for \ref{['item:ModelHK']} it is piecewise given by $\cos(d(y,x_i))^2$ on $\Omega \setminus R$ and $0$ on $R$, and for \ref{['item:ModelQuadratic']} it is given by truncated paraboloids.
• Figure 4: Semi-discrete Hellinger--Kantorovich transport on $\Omega=[0,1]^2$ (using the same values for $x_i/L$ and $m_i/|\Omega|$ as in \ref{['fig:SemiDisc_model_comparison']}) for different length scales $\varepsilon$. Top row: optimal cells $\{C_i(w)\}_{i=1}^M$; the residual set $R$ is represented by white; location of the discrete points $(x_1,\ldots,x_M)$ is indicated with red dots. Bottom row: optimal marginal $\rho$ (using the same colour scale for all images). For large $\varepsilon$ the behaviour is similar to that of the standard semi-discrete Wasserstein-2 distance. As $\varepsilon$ decreases, the effects of unbalanced transport become increasingly prominent.
• Figure 5: Semi-discrete Hellinger--Kantorovich distance on $\Omega=[0,1]^2$ for different length scales $\varepsilon$, as in \ref{['fig:SemiDisc_HK_base']}, but for $M=128$. The evolution of one cell $C_i(w)$ for fixed $i$ is highlighted in red (top row). For large $\varepsilon$, $C_i(w)$ is essentially the standard Wasserstein-2 Laguerre cell, not necessarily containing $x_i$. For small $\varepsilon$, $C_i(w)$ becomes (a fraction of) the open ball $B_{\varepsilon\pi/2}(x_i)$.

Theorems & Definitions (72)

• Theorem 2.1: Villani-OptimalTransport-09
• Definition 2.2: Marginal discrepancy
• Example 2.3: Kullback--Leibler divergence
• Definition 2.4: Unbalanced optimal transport problem
• Theorem 2.5: LieroMielkeSavare-HellingerKantorovich-2015a
• Remark 2.6
• Remark 2.7
• Definition 2.8: Radial cost
• Example 2.9: Unbalanced transport models
• Lemma 2.10: Properties of $F^\ast$