The computation of a measure of similarity (or dissimilarity) between pairs of objects is a crucial subproblem in several applications in Computer VisionRubner98 ; Rubner2000 ; Pele2009 , Computational Statistic Bickel2001 Bassetti2006 ; Bassetti2006b
, and Machine LearningSolomon2014 ; Cuturi2014 ; Frogner2015 ; Arjovsky2017 . In mathematical terms, in order to compute the similarity between a pair of objects, we want to compute a distance. If the distance is equal to zero the two objects are considered to be equal; the more the two objects are different, the greater is their distance value. For instance, the Euclidean norm is the most used distance function to compare a pair of points in . Note that the Euclidean distance requires only operations to be computed. When computing the distance between complex discrete objects, such as for instance a pair of discrete measures, a pair of images, a pair of -dimensional histograms, or a pair of clouds of points, the Kantorovich-Wasserstein distance Villani2008 ; Vershik2013 has proved to be a relevant distance function Rubner98 , which has both nice mathematical properties and useful practical implications. Unfortunately, computing the Kantorovich-Wasserstein distance requires the solution of an optimization problem. Even if the optimization problem is polynomially solvable, the size of practical instances to be solved is very large, and hence the computation of Kantorovich-Wasserstein distances implies an important computational burden.
The optimization problem that yields the Kantorovich-Wasserstein distance can be solved with different methods. Nowadays, the most popular methods are based on (i) the Sinkhorn’s algorithm Cuturi2013 ; Solomon2015 ; Altschuler2017Flood1953 ; Goldberg1989 ; Orlin , which exactly solve the basic optimal transport problem by formulating and solving an equivalent uncapacitated minimum cost flow problem. For a nice overview of both computational approaches, we refer the reader to Chapters 2 and 3 in Peyre2018 , and the references therein contained.
In this paper, we propose a Linear Programming-based method to speed up the computation of Kantorovich-Wasserstein distances of order 2, which exploits the structure of the ground distance to formulate an uncapacitated minimum cost flow problem. The flow problem is then solved with a state-of-the-art implementation of the well-known Network Simplex algorithm Kovacs2015 .
Our approach is along the line of research initiated in LingOkada2007 , where the authors proposed a very efficient method to compute Kantorovich-Wasserstein distances of order 1 (i.e., the so–called Earth Mover Distance), whenever the ground distance between a pair of points is the norm. In LingOkada2007 , the structure of the ground distance and of regular -dimensional histograms is exploited to define a very small flow network. More recently, this approach has been successfully generalized in Bassetti2018 to the case of and norms, providing both exact and approximations algorithms, which are able to compute distances between pairs of gray scale images. The idea of speeding up the computation of Kantorovich-Wasserstein distances by defining a minimum cost flow on smaller structured flow networks is also used in Pele2009 , where a truncated distance is used as ground distance in place of a norm.
The outline of this paper is as follows. Section 2 reviews the basic notion of discrete optimal transport and fixes the notation. Section 3 contains our main contribution, that is, Theorem 1 and Corollary 2, which permits to speed-up the computation of Kantorovich-Wasserstein distances of order 2 under quite general assumptions. Section 4 presents numerical results of our approaches, compared with the Sinkhorn’s algorithm as implemented in Cuturi2013 and a standard Linear Programming formulation on a complete bipartite graph Rubner98 . Finally, Section 5 concludes the paper.
2 Discrete Optimal Transport: an Overview
be two discrete spaces. Given two probability vectorsand defined on and , respectively, and a cost , the Kantorovich-Rubinshtein functional between and is defined as
where is the set of all the probability measures on with marginals and , i.e. the probability measures such that and for every in . Such probability measures are sometimes called transport plans or couplings for and . An important special case is when and the cost function is a distance on . In this case is a distance on the simplex of probability vectors on , also known as Kantorovich-Wasserstein distance of order .
We remark that the Kantorovich-Wasserstein distance of order can be defined, more in general, for arbitrary probability measures on a metric space by
where now is the set of all probability measures on the Borel sets of that have marginals and , see, e.g., AGS . The infimum in (2) is attained, and any probability which realizes the minimum is called an optimal transport plan.
The Kantorovich-Rubinshtein transport problem in the discrete setting can be seen as a special case of the following Linear Programming problem, where we assume now that and are generic vectors of dimension , with positive components,
If we have the so-called balanced transportation problem, otherwise the transportation problem is said to be unbalanced Liero2018 ; Chizat2016 . For balanced optimal transport problems, constraints (4) and (5) must be satisfied with equality, and the problem reduces to the Kantorovich transport problem (up to normalization of the vectors and ).
Problem (P) is related to the so-called Earth Mover’s distance. In this case, , and are the centers of two data clusters, and and give the number of points in the respective cluster. Finally, is some measure of dissimilarity between the two clusters and . Once the optimal transport is determined, the Earth Mover’s distance between and is defined as (e.g., see Rubner98 )
Problem (P) can be formulated as an uncapacitated minimum cost flow problem on a bipartite graph defined as follows Ahuja . The bipartite graph has two partitions of nodes: the first partition has a node for each point of , and the second partition has a node for each point of . Each node of the first partition has a supply of mass equal to , each node of the second partition has a demand of units of mass. The bipartite graph has an (uncapacitated) arc for each element in the Cartesian product having cost equal to . The minimum cost flow problem defined on this graph yields the optimal transport plan , which indeed is an optimal solution of problem (3)–(6). For instance, in case of a regular 2D dimensional histogram of size , that is, having bins, we get a bipartite graph with nodes and arcs (or nodes and arcs). Figure 1–LABEL: shows an example for a histogram, and Figure 1–LABEL: gives the corresponding complete bipartite graph.
In this paper, we focus on the case in equation (2) and the ground distance function is the Euclidean norm , that is the Kantorovich-Wasserstein distance of order , which is denoted by . We provide, in the next section, an equivalent formulation on a smaller -partite graph.
3 Formulation on -partite Graphs
For the sake of clarity, but without loss of generality, we present first our construction considering 2-dimensional histograms and the Euclidean ground distance. Then, we discuss how our construction can be generalized to any pair of -dimensional histograms.
Let us consider the following flow problem: let and be two probability measures over a regular grid denoted by . In the following paragraphs, we use the notation sketched in Figure 2. In addition, we define the set .
Since we are considering the norm as ground distance, we minimize the functional
among all , with real numbers (i.e., flow variables) satisfying the following constraints
Constraints (8) impose that the mass at the point is moved to the points . Constraints (9) force the point to receive from the points a total mass of . Constraints (10) require that all the mass that goes from the points to the point is moved to the points . We call a pair satisfying the constraints (8)–(10) a feasible flow between and . We denote by the set of all feasible flows between and .
Indeed, we can formulate the minimization problem defined by (7)–(10) as an uncapacitated minimum cost flow problem on a tripartite graph . The set of nodes of is , where and are the nodes corresponding to three regular grids. We denote by the node of coordinates in the grid . We define the two disjoint set of arcs between the successive pairs of node partitions as
and, hence, the arcs of are . Note that in this case the graph has nodes and arcs. Whenever is a feasible flow between and , we can think of the values as the quantity of mass that travels from to or, equivalently, that moves along the arc of the tripartite graph, while the values are the mass moving along the arc (e.g., see Figures 1–LABEL: and 2).
Now we can give an idea of the roles of the sets , and : is the node set where is drawn the initial distribution , while on it is drawn the final configuration of the mass . The node set is an auxiliary grid that hosts an intermediate configuration between and .
We are now ready to state our main contribution.
For each measure on that transports into , we can find a feasible flow such that
(Sketch). We will only show how to build a feasible flow starting from a transport plan, the inverse building uses a more technical lemma (the so–called gluing lemma AGS ; Villani2008 ) and can be found in the Additional Material. Let be a transport plan, if we write explicitly the ground distance we find that
If we set and we find
As a straightforward, yet fundamental, consequence we have the following result.
If we set then, for any discrete measures and , we have that
Indeed, we can compute the Kantorovich-Wasserstein distance of order 2 between a pair of discrete measures , by solving an uncapacitated minimum cost flow problem on the given tripartite graph .
We remark that our approach is very general and it can be directly extended to deal with the following generalizations.
More general cost functions.
The structure that we have exploited of the Euclidean distance is present in any cost function that is separable, i.e., has the form
where both and are positive real valued functions defined over . We remark that the whole class of costs is of that kind, so we can compute any of the Kantorovich-Wasserstein distances related to each .
Higher dimensional grids.
Our approach can handle discrete measures in spaces of any dimension , that is, for instance, any -dimensional histogram. In dimension , we get a tripartite graph because we decomposed the transport along the two main directions. If we have a problem in dimension , we need a -plet of grids connected by arcs oriented as the fundamental directions, yielding a -partite graph. As the dimension grows, our approach gets faster and more memory efficient than the standard formulation given on a bipartite graph.
In the Additional Material, we present a generalization of Theorem 1 to any dimension and to separable cost functions .
4 Computational Results
In this section, we report the results obtained on two different set of instances. The goal of our experiments is to show how our approach scales with the size of the histogram and with the dimension of the histogram . As cost distance , with , we use the squared norm. As problem instances, we use the gray scale images (i.e., 2-dimensional histograms) proposed by the DOTMark benchmark Dotmark , and a set of -dimensional histograms obtained by bio medical data measured by flow cytometer Bernas2008 .
We run our experiments using the Network Simplex as implemented in the Lemon C++ graph library111http://lemon.cs.elte.hu (last visited on October, 26th, 2018), since it provides the fastest implementation of the Network Simplex algorithm to solve uncapacitated minimum cost flow problems Kovacs2015 . We did try other state-of-the-art implementations of combinatorial algorithm for solving min cost flow problems, but the Network Simplex of the Lemon graph library was the fastest by a large margin. The tests are executed on a gaming laptop with Windows 10 (64 bit), equipped with an Intel i7-6700HQ CPU and 16 GB of Ram. The code was compiled with MS Visual Studio 2017, using the ANSI standard C++17. The code execution is single threaded. The Matlab implementation of the Sinkhorn’s algorithm Cuturi2013 runs in parallel on the CPU cores, but we do not use any GPU in our test. The C++ and Matlab code we used for this paper is freely available at http://stegua.github.io/dpartion-nips2018.
Results for the DOTmark benchmark.
The DOTmark benchmark contains 10 classes of gray scale images related to randomly generated images, classical images, and real data from microscopy images of mitochondria Dotmark . In each class there are 10 different images. Every image is given in the data set at the following pixel resolutions: , , , , and . The images in Figure 3 are respectively the ClassicImages, Microscopy, and Shapes images (one class for each row), shown at highest resolution.
In our test, we first compared five approaches to compute the Kantorovich-Wasserstein distances on images of size :
EMD: The implementation of Transportation Simplex provided by Rubner98 , known in the literature as EMD code, that is an exact general method to solve optimal transport problem. We used the implementation in the programming language C, as provided by the authors, and compiled with all the compiler optimization flags active.
Sinkhorn: The Matlab implementation of the Sinkhorn’s algorithm222http://marcocuturi.net/SI.html (last visited on October, 26th, 2018) Cuturi2013 , that is an approximate approach whose performance in terms of speed and numerical accuracy depends on a parameter : for smaller values of , the algorithm is faster, but the solution value has a large gap with respect to the optimal value of the transportation problem; for larger values of , the algorithm is more accurate (i.e., smaller gap), but it becomes slower. Unfortunately, for very large value of the method becomes numerically unstable. The best value of is very problem dependent. In our tests, we used and . The second value, , is the largest value we found for which the algorithm computes the distances for all the instances considered without facing numerical issues.
Improved Sinkhorn: We implemented in Matlab an improved version of the Sinkhorn’s algorithm, specialized to compute distances over regular 2-dimensional grids Solomon2015 ; Solomon2018 . The main idea is to improve the matrix-vector operations that are the true computational bottleneck of Sinkhorn’s algorithm, by exploiting the structure of the cost matrix. Indeed, there is a parallelism with our approach to the method presented in Solomon2015 , since both exploits the geometric cost structure. In Solomon2015 , the authors proposes a general method that exploits a heat kernel to speed up the matrix-vector products. When the discrete measures are defined over a regular 2-dimensional grid, the cost matrix used by the Sinkhorn’s algorithm can be obtained using a Kronecker product of two smaller matrices. Hence, instead of performing a matrix-vector product using a matrix of dimension , we perform two matrix-matrix products over matrices of dimension
, yielding a significant runtime improvement. In addition, since the smaller matrices are Toeplitz matrices, they can be embedded into circulant matrices, and, as consequence, it is possible to employ a Fast Fourier Transform approach to further speed up the computation. Unfortunately, the Fast Fourier Transform makes the approach still more numerical unstable, and we did not used it in our final implementation.
-partite: The -partite formulation proposed in this paper, which for 2-dimensional histograms is represented in 1–LABEL:. Again, we use the Network Simplex of the Lemon Graph Library to solve the corresponding uncapacitated minimum cost flow problem.
Tables 1(a) and 1(b) report the averages of our computational results over different classes of images of the DOTMark benchmark. Each class of gray scale image contains 10 instances, and we compute the distance between every possible pair of images within the same class: the first image plays the role of the source distribution , and the second image gives the target distribution
. Considering all pairs within a class, it gives 45 instances for each class. We report the means and the standard deviations (between brackets) of the runtime, measured in seconds. Table1(a) shows in the second column the runtime for EMD Rubner98 . The third and fourth columns gives the runtime and the optimality gap for the Sinkhorn’s algorithm with ; the 6- and 7- columns for . The percentage gap is computed as , where is the upper bound computed by the Sinkhorn’s algorithm, and is the optimal value computed by EMD. The last two columns report the runtime for the bipartite and -partite approaches presented in this paper.
Table 1(b) compares our -partite formulation with the Improved Sinkhorn’s algorithm Solomon2015 ; Solomon2018 , reporting the same statistics of the previous table. In this case, we run the Improved Sinkhorn using three values of the parameter , that are, 1.0, 1.25, and 1.5. While the Improved Sinkhorn is indeed much faster that the general algorithm as presented in Cuturi2013 , it does suffer of the same numerical stability issues, and, it can yield very poor percentage gap to the optimal solution, as it happens for the GRFrough and the WhiteNoise classes, where the optimality gaps are on average 31.0% and 39.2%, respectively.
As shown in Tables 1(a) and 1(b), the -partite approach is clearly faster than any of the alternatives considered here, despite being an exact method. In addition, we remark that, even on the bipartite formulation, the Network Simplex implementation of the Lemon Graph library is order of magnitude faster than EMD, and hence it should be the best choice in this particular type of instances. We remark that it might be unfair to compare an algorithm implemented in C++ with an algorithm implemented in Matlab, but still, the true comparison is on the solution quality more than on the runtime. Moreover, when implemented on modern GPU that can fully exploit parallel matrix-vector operations, the Sinkhorn’s algorithm can run much faster, but they cannot improve the optimality gap.
|EMD Rubner98||Sinkhorn Cuturi2013||Bipartite||-partite|
|Classic||24.0 (3.3)||6.0 (0.5)||17.3%||8.9 (0.7)||9.1%||0.54 (0.05)||0.07 (0.01)|
|Microscopy||35.0 (3.3)||3.5 (1.0)||2.4%||5.3 (1.4)||1.2%||0.55 (0.03)||0.08 (0.01)|
|Shapes||25.2 (5.3)||1.6 (1.1)||5.6%||2.5 (1.6)||3.0%||0.50 (0.07)||0.05 (0.01)|
|Improved Sinkhorn Solomon2015 ; Solomon2018||-partite|
|CauchyDensity||0.22 (0.15)||2.8%||0.33 (0.23)||2.0%||0.41 (0.28)||1.5%||0.07 (0.01)|
|Classic||0.20 (0.01)||17.3%||0.31 (0.02)||12.4%||0.39 (0.03)||9.1%||0.07 (0.01)|
|GRFmoderate||0.19 (0.01)||12.6%||0.29 (0.02)||9.0%||0.37 (0.03)||6.6%||0.07 (0.01)|
|GRFrough||0.19 (0.01)||58.7%||0.29 (0.01)||42.1%||0.38 (0.02)||31.0%||0.05 (0.01)|
|GRFsmooth||0.20 (0.02)||4.3%||0.30 (0.04)||3.1%||0.38 (0.04)||2.2%||0.08 (0.01)|
|LogGRF||0.22 (0.05)||1.3%||0.32 (0.08)||0.9%||0.40 (0.13)||0.7%||0.08 (0.01)|
|LogitGRF||0.22 (0.02)||4.7%||0.33 (0.03)||3.3%||0.42 (0.04)||2.5%||0.07 (0.02)|
|Microscopy||0.18 (0.03)||2.4%||0.27 (0.04)||1.7%||0.34 (0.05)||1.2%||0.08 (0.02)|
|Shapes||0.11 (0.04)||5.6%||0.16 (0.06)||4.0%||0.20 (0.07)||3.0%||0.05 (0.01)|
|WhiteNoise||0.18 (0.01)||76.3%||0.28 (0.01)||53.8%||0.37 (0.02)||39.2%||0.04 (0.00)|
In order to evaluate how our approach scale with the size of the images, we run additional tests using images of size and . Table 2 reports the results for the bipartite and -partite approaches for increasing size of the 2-dimensional histograms. The table report for each of the two approaches, the number of vertices and of arcs , and the means and standard deviations of the runtime. As before, each row gives the averages over 45 instances. Table 2 shows that the -partite approach is clearly better (i) in terms of memory, since the 3-partite graph has a fraction of the number of arcs, and (ii) of runtime, since it is at least an order of magnitude faster in computation time. Indeed, the 3-partite formulation is better essentially because it exploits the structure of the ground distance used, that is, the squared norm.
|Classic||8 193||16 777 216||16.3 (3.6)||12 288||524 288||2.2 (0.2)|
|Microscopy||11.7 (1.4)||1.0 (0.2)|
|Shape||13.0 (3.9)||1.1 (0.3)|
|Classic||32 768||268 435 456||1 368 (545)||49 152||4 194 304||36.2 (5.4)|
|Microscopy||959 (181)||23.0 (4.8)|
|Shape||983 (230)||17.8 (5.2)|
Flow Cytometry biomedical data.
Flow cytometry is a laser-based biophysical technology used to study human health disorders. Flow cytometry experiments produce huge set of data, which are very hard to analyze with standard statistics methods and algorithms Bernas2008 . Currently, such data is used to study the correlations of only two factors (e.g., biomarkers) at the time, by visualizing 2-dimensional histograms and by measuring the (dis-)similarity between pairs of histograms Orlova2016 . However, during a flow cytometry experiment up to hundreds of factors (biomarkers) are measured and stored in digital format. Hence, we can use such data to build -dimensional histograms that consider up to biomarkers at the time, and then comparing the similarity among different individuals by measuring the distance between the corresponding histograms. In this work, we used the flow cytometry data related to Acute Myeloid Leukemia (AML), available at http://flowrepository.org/id/FR-FCM-ZZYA
, which contains cytometry data for 359 patients, classified as “normal” or affected by AML. This dataset has been used by the bioinformatics community to run clustering algorithms, which should predict whether a new patient is affected by AMLAghaeepour2013 .
Table 3 reports the results of computing the distance between pairs of -dimensional histograms, with ranging in the set , obtained using the AML biomedical data. Again, the first -dimensional histogram plays the role of the source distribution , while the second histogram gives the target distribution . For simplicity, we considered regular histograms of size (i.e., is the total number of bins), using and . Table 3 compares the results obtained by the bipartite and -partite approach, in terms of graph size and runtime. Again, the -partite approach, by exploiting the structure of the ground distance, outperforms the standard formulation of the optimal transport problem. We remark that for and , we pass for going out-of-memory with the bipartite formulation, to compute the distance in around 5 seconds with the -partite formulation.
|Bipartite Graph||-partite Graph|
|2||256||512||65 536||0.024 (0.01)||768||8 192||0.003 (0.00)|
|3||4 096||8 192||16 777 216||38.2 (14.0)||16 384||196 608||0.12 (0.02)|
|4||65 536||out-of-memory||327 680||4 194 304||4.8 (0.84)|
|2||1 024||2 048||1 048 756||0.71 (0.14)||3072||65 536||0.04 (0.01)|
|3||32 768||out-of-memory||131 072||3 145 728||5.23 (0.69)|
In this paper, we have presented a new network flow formulation on -partite graphs that can speed up the optimal solution of transportation problems whenever the ground cost function (see objective function (3)) has a separable structure along the main directions, such as, for instance, the squared norm used in the computation of the Kantorovich-Wasserstein distance of order 2.
Our computational results on two different datasets show how our approach scales with the size of the histograms and with the dimension of the histograms . Indeed, by exploiting the cost structure, the proposed approach is better in term of memory consumption, since it has only arcs instead of . In addition, it is much faster since it has to solve an uncapacitated minimum cost flow problem on a much smaller flow network.
We are deeply indebted to Giuseppe Savaré, for introducing us to optimal transport and for many stimulating discussions and suggestions. We thanks Mattia Tani for a useful discussion concerning the Improved Sinkhorn’s algorithm.
This research was partially supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) - Dept. of Mathematics “F. Casorati”, University of Pavia.
The last author’s research is partially supported by “PRIN 2015. 2015SNS29B-002. Modern Bayesian nonparametric methods”.
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Let us take , two probability measures and a ground distance of the form
We can then define
are a -plet of real values satisfying the two congruence conditions
and the following connection conditions
for . We will call the plet of a flow chart between and .
The set of all possible flow charts between two measures and will be indicate with . We will then define
Let and be two probability measures over the grid , a separable ground distance, of the form (15). Then, for each transport plan between and there exists a flow chart such that
Let us consider a transport plan, then we can write
All of those follow from the definition itself, indeed
Let now be a flow chart. We have that, for each , the define a probability measure over . For we easly find that
If we assume that is a probability measure, then, using condition (19), we get that
Thus, by induction, we get that all the are actually probability measures.
Since we showed that and are both probability measures and relation (19) holds we can apply the gluing lemma and find a probability measure such that
Let us now consider and , we have
so we can apply once again the gluing lemma and find a probability measure such that
We can iterate this process for times and find a probability measure