Stochastic Wasserstein Barycenters

Stochastic Wasserstein Barycenters

Sebastian Claici Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
{sclaici, edchien, jsolomon}@mit.edu
Edward Chien Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
{sclaici, edchien, jsolomon}@mit.edu
Justin Solomon Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
{sclaici, edchien, jsolomon}@mit.edu
Abstract

We present a stochastic algorithm to compute the barycenter of a set of probability distributions under the Wasserstein metric from optimal transport. Unlike previous approaches, our method extends to continuous input distributions and allows the support of the barycenter to be adjusted in each iteration. We tackle the problem without regularization, allowing us to recover a much sharper output. We give examples where our algorithm recovers a more meaningful barycenter than previous work. Our method is versatile and can be extended to applications such as generating super samples from a given distribution and recovering blue noise approximations.

1 Introduction

Several scenarios in machine learning require summarizing a collection of probability distributions with shared structure but individual bias. For instance, multiple sensors might gather data from the same environment with different noise distributions; the samples they collect must be assembled into a single signal. As another example, a dataset might be split among multiple computers, each of which carries out MCMC Bayesian inference for a given model; the resulting “subset posterior” latent variable distributions must be reassembled into a single posterior for the entire dataset. In each case, the summarized whole can be better than the sum of its parts: noise in the input distributions cancels when averaging, while shared structure is reinforced.

The theory of optimal transport (OT) provides a promising and theoretically-justified approach to averaging distributions over a geometric domain. OT equips the space of measures with a distance metric known as the Wasserstein distance; the average, or barycenter, of a collection is then defined as a Fréchet mean minimizing the sum of squared Wasserstein distances to the input distributions Agueh and Carlier (2011). This mean is aware of the geometric structure of the underlying space. For example, the Wasserstein barycenter of two Dirac distributions and supported at points is a single Dirac delta at the center point rather than the bimodal superposition obtained by averaging algebraically.

If the input distributions are discrete, then the Wasserstein barycenter is computable in polynomial time by solving a large linear program Anderes et al. (2016). Adding entropic regularization yields elegant and efficient approximation algorithms Genevay et al. (2016); Cuturi and Peyré (2016); Cuturi and Doucet (2014); Ye et al. (2017). These and other state-of-the-art methods typically suffer from any of a few drawbacks, mainly (1) poor behavior as regularization decreases, (2) required access to the distribution functions rather than sampling machinery, and/or (3) a fixed discretization on which the input or output distribution is supported, chosen without knowledge of the barycenter’s structure.

Given sample access to distributions , we propose an algorithm that iteratively refines an approximation to the true Wasserstein barycenter. The support of our barycenter is adjusted in each iteration, adapting to the geometry of the desired output. Unlike most existing OT algorithms, we tackle the problem without regularization, yielding a sharp result. Experiments show that the support of our barycenter is contained (to tolerance) within the support of the true barycenter even though we use stochastic optimization rather than computational geometry.

Contributions.

We give a straightforward parallelizable stochastic algorithm to approximate and sample from the Wasserstein barycenter of a collection of distributions, which does not rely on regularization to make the problem tractable. We only employ samplers from the input distributions, and our technique is not restricted to input or output distributions supported on a fixed set of points. We verify convergence properties and showcase examples where our approach is inherently more suitable than competing approaches that require a fixed support.

2 Related Work

OT has made significant inroads in computation and machine learning; see Lévy and Schwindt (2017); Peyré and Cuturi (2018); Solomon (2018) for surveys. Although most algorithms we highlight approximate OT distances rather barycenters, they serve as potential starting points for barycenter computation.

Cuturi (2013) renewed interest in OT in machine learning through introduction of entropic regularization. The resulting Sinkhorn algorithm is compact and efficient; it has been extended to barycenter problems through gradient descent Cuturi and Doucet (2014) or iterative projection Benamou et al. (2015). Improvements for structured instances enhance Sinkhorn’s efficiency, e.g. via fast convolution Solomon et al. (2015) or multiscale approximation Schmitzer (2016).

Our technique, however, is influenced more by semidiscrete methods, which compute OT distances to distributions supported on a finite set of points. Semidiscrete OT is equivalent to computing a power diagram Aurenhammer (1987); Aurenhammer et al. (1992), a generalization of a Voronoi diagram whose cells receive the mass from each . Algorithms by Mérigot (2011) in 2D and Lévy (2015) in 3D use computational geometry to extract gradients for the dual semidiscrete problem; Kitagawa et al. (2016a) accelerate convergence via a second-order Newton method. Similar to our technique, De Goes et al. (2012) move the support of a discrete approximation to a distribution to reduce Wasserstein distance.

Recent stochastic techniques target learning applications. Genevay et al. (2016) propose a scalable stochastic algorithm based on the dual of the entropically-regularized problem; they are among the first to consider the setting of sample-based access to distributions but rely on entropic regularization to smooth out the problem and approximate OT distances rather than barycenters. Staib et al. (2017) propose a stochastic barycenter algorithm from samples, but a finite, fixed set of support points must be provided a priori. Arjovsky et al. (2017) incorporate a coarse stochastic approximation of the 1-Wasserstein distance into a generative adversarial network (GAN); the 1-Wasserstein distance typically is not suitable for barycenter computation.

Further machine learning applications range from supervised learning to Bayesian inference. Schmitz et al. (2017) leverage OT theory for dictionary learning. Carrière et al. (2017) apply the Wasserstein distance to point cloud segmentation by developing a notion of distance on topological persistence diagrams. Courty et al. (2016) utilize the optimal transport plan for transfer learning on different domains. Srivastava et al. (2015a, b) use the Wasserstein barycenter to approximate the posterior distribution of a full dataset by the barycenter of the posteriors on smaller subsets; their method provably recovers the full posterior as the number of subsets increases.

3 Background and Preliminaries

Let be a metric space, and let be the space of probability measures on with finite second moment. Given two measures , the squared 2-Wasserstein distance between and is given by

(1)

Here, is the set of measure couplings between and :

where and are the two projections of , and the push-forward of a measure through a measurable map is defined as for any set in a -algebra of . The 2-Wasserstein distance is a metric on .

For measures , we can define the Wasserstein barycenter as the minimizer of the functional

(2)

When the input measures are discrete distributions, (2) is a linear program solvable in polynomial time.

If at least one of the measures is absolutely continuous with respect to the Lebesgue measure, then (2) admits a unique minimizer  Agueh and Carlier (2011); Santambrogio (2015). However, will also be absolutely continuous, implying that computational systems typically can only find an inexact finite approximation.

We study a discretization of this problem. Suppose consists of points , and define the functional

(3)

We define the main problem.

Problem 1 (Semidiscrete approximation).

Find a minimizer of subject to the constraints ,

Solving problem (1) for a single input measure is equivalent to finding the optimal -point approximation to the input measure. We can use the solution as a set of supersamples from the input Chen et al. (2010), or if the input distribution is a grayscale image, the solution yields a blue noise approximation to the image De Goes et al. (2012).

4 Mathematical Formulation

The OT problem (1) admits an equivalent dual problem

(4)

where is the Kantorovich potential and is the -transform of  Santambrogio (2015); Villani (2009).

Following Santambrogio (2015), if is a finite measure supported on , then (4) becomes

(5)

where . The key observation is that the function is replaced with a finite-dimensional .

With this formula in mind, define

(6)

Note that constant shifts in the do not change the value of . has a simple derivative with respect to the ’s:

(7)

where is the power cell of point :

From here on we work with compact subsets of the Euclidean space endowed with the Euclidean metric, . To differentiate with respect to the ’s, notice that the first term in equation (6) does not depend on the positions of the points. We rewrite the second term as

Using Reynolds’ transport theorem to differentiate while accounting for boundary terms shows

(8)

Equation (7) confirms the intuition that each cell contains as much mass as its associated source point. We will leverage (8) to design a fixed-point iteration that moves each point to the center of its power cell.

Each subproblem of (3) admits a different Kantorovich potential , giving the following optimization functional

(9)

Define

With this notation in place, the partial derivatives are

(10)

5 Optimization

With our optimization objective function in place, we now introduce our barycenter algorithm. To simplify nomenclature, from here on we refer to the dual potentials as weights on the generalized Voronoi diagram. Our overall strategy is an alternating optimization of in (9):

  • For fixed point positions, is concave in the weights and is optimized using stochastic gradient ascent.

  • For fixed weights, we apply a single fixed point iteration akin to Lloyd’s algorithm Lloyd (1982).

5.1 Estimating Gradients

Each of and can be expressed as an expectation of a simple function with respect to the . We estimate these quantities by a simple Monte Carlo scheme.

In more detail, we can rewrite and as

Here, indicates the indicator function of a set.

Since we have sample access to each , the expectations can be approximated by drawing points independently and computing

(11)

5.2 Concave Maximization

The first step in our alternating optimization maximizes over the weights while the points are fixed. We call this step of the algorithm an ascent step.

For a fixed set of points, the functional is concave in the weights , since it is the dual of the convex semidiscrete transport problem. To solve for the weights, we perform gradient ascent using the formula in (10) where is approximated using . Note that the gradient for a set of weights only requires computation of the density of a single measure , implying that the ascent steps can be decoupled across different measures.

Write for the initial iterate. The simplest version of our algorithm updates

Here is a -dimensional vector representing a weight for each point. The iterates converge when each point contains equal mass in its associated power cell.

has a known Hessian as a function of the that can be used in Newton’s algorithm Kitagawa et al. (2016b). Computing the Hessian, however, is only possible with access to the density functions of the ’s as it requires computing a density of the measure on the boundary between two power cells. The boundary set is inherently lower dimensional than the problem space, and hence sample access to the is insufficient. Moreover, even had we access to the probability density functions, computing the Hessian would require the Delaunay triangulation of the point set, which is expensive in more than two dimensions.

In any event, choosing the step size is important for convergence. Line search is difficult as we do not have access to true objective value at each iterate. Instead, we rely on Nesterov acceleration to improve performance Nesterov (1983). With acceleration, our iterates are

(12)
(13)

In our experiments, we use and . Convergence of the accelerated gradient method can be shown when where is the Lipschitz constant of ; in §6, we give an estimate of this constant. Our convergence criterion for this step is .

5.3 Fixed Point Iteration

The second step of our optimization is a fixed point iteration on the point positions. This step is similar to the point update in a -means algorithm in that it snaps points to the centers of local cells, and we refer to it as a snap step.

To derive the iteration, we set the second gradient in (10) to zero:

which leads to the point update

(14)

This suggests a fixed point iteration for the ’s that can be decomposed into the following steps:

  1. First find the barycenter of the power cells of each with respect to each .

  2. Then, average the points with weights given by the density of each measure in the cell.

If the concave maximization has converged appropriately, and uniform areas have been achieved, then the update step becomes a uniform average over the barycenters with respect to each measure.

5.4 Global and Local Strategies

0:  Estimate of barycenter support
0:  Optimized barycenter support with lower cost.
1:  for  do
2:     for  do
3:        \hfill{Ascent on weights}
4:        
5:        while  do
6:           Compute according to equation (11)
7:           
8:           
9:        end while
10:        
11:     end for
12:     Compute according to equation (11)
13:     for  do
14:         \hfill{Snap points}
15:     end for
16:  end for
Algorithm 1 Optimizing estimate of barycenter support

The ascent and snap steps can be used to refine a configuration of points . Once the iterates converge, we have an -point approximation to the barycenter that can be used as an initialization for point approximation in two ways. A new point is sampled uniformly from , and then we have a choice between (1) moving all points including the new one or (2) allowing only to move.

These two approaches are codified in Algorithm 1 where the choice on the set dictates which points move. The number of iterations of the outer loop is fixed beforehand. Typically, we see convergence in fewer than steps, and empirically, we observe good performance even with . The two most natural choices for are and . If the barycenter is absolutely continuous with respect to the underlying Lebesgue measure, these two strategies converge at the same rate asymptotically Brancolini et al. (2009). The latter, however, can generate spurious samples that are not in the support of the barycenter. Note that optimizing the weights is regardless a global problem as moving or introducing just one point can change the volumes of the power cells of neighboring points.

Both algorithms are highly parallelizable, since (1) the gradient estimates are expectations computed using Monte Carlo integration and (2) the gradient step in the weights decouples across distributions.

6 Analysis

We justify the use of uniform finitely-supported measures, and then prove that our algorithm converges to a minimum cost under mild assumptions.

We assume in this section that at least one of the distributions is absolutely continuous with respect to the Lebesgue measure, ensuring a unique Wasserstein barycenter.

6.1 Approximation Suitability

The simplest approach for absolutely continuous measures is to sample points from each of the measures and solve for the true barycenter of the empirical distributions Anderes et al. (2016). This approach likely approximates the barycenter as the number of samples increases, but requires solution of a linear program with variables. As an alternative, Staib et al. (2017) propose a stochastic problem for approximating barycenters. They are able to prove a rate of convergence, but the support of their approximate barycenter is fixed to a finite set of points.

Our technique allows the support points to move during the optimization procedure, empirically allowing a better approximation of the barycenter with fewer points. The following theoretical result shows that the use of uniform measures supported on a finite set of points can approximate the barycenter arbitrarily well:

Theorem (Metric convergence, Kloeckner (2012); Brancolini et al. (2009)).

Suppose is a uniform measure supported on points that minimizes , and let denote the true barycenter of the measures . Then where depends on the underlying space , the dimension , and the metric .

Note that this shows convergence in probability since the Wasserstein distance metrizes weak convergence Villani (2009). Brancolini et al. (2009) also show asymptotic equivalence of the local and global algorithms.

While we cannot guarantee that our method converges to , these properties indicate that the global minimizer of our objective provides an effective approximant to the true barycenter as the number of support points .

6.2 Algorithmic Properties

The functional is concave in the weights with fixed point positions, and in fact strictly concave up to constant shifts. We can investigate the convergence properties of the gradient ascent step of the algorithm. However, we will show that the gradient of is not necessarily Lipschitz continuous.

Counterexample.

Assume is a compact subset of . There are measures for which the gradient of is not Lipschitz continuous. A set of weights that satisfies may not exist, and if it does, it may not be unique.

Construction.

We provide a counterexample for . Let with the standard metric and . Let be the fixed positions, and take and for small . Then , but .

Non-existence is shown in Figure 1. To see non-uniqueness, take with as before. Any set of weights in minimizes . ∎

Figure 1: Non-existence of a set of weights. Let be the uniform measure on the line segment, and be the two red points such that the line between them is orthogonal to the support of . There is no set of weights such that the mass of is split evenly between the two red points.
Figure 2: Non-unique minimizer on two points for the uniform measure defined on the unit disk. All antipodal points on the dashed circle at distance from the center are valid minimizers.

For mildly behaved measures the gradient of with respect to is Lipschitz continuous:

Lemma.

Assume is a compact subset of , and is absolutely continuous with respect to the Lebesgue measure, with density function . If the points of are distinct and almost everywhere for some constant , then:

where denotes the surface area of and denotes the minimum pairwise distance between points in .

Proof.

Consider the th component of the gradient difference:

The second inequality follows as the area of a power cell is bounded by and the faces of the cells change at a rate linear in . The rate is dependent on the distance between the points, so the constant is required. The Lipschitz bound follows directly from considering all components of the gradient difference together. ∎

This lemma implies convergence for a step size that is the inverse of the Lipschitz constant. While the above requires absolute continuity of , we have found that our ascent steps and method often converge even when this is not satisfied (see Figures 3 and 4).

We may also show that our algorithm monotonically decreases (defined in Equation (3)) after each pair of snap and then ascent steps for compact domain and absolutely continuous . For this purpose, recall that the transport cost for a map sending measure to is:

Fixing the power cells after an ascent step, we may define to be the transport cost for the map sending the power cells to the point set , and we may define to be the joint (average) transport cost. Letting denote the new positions after a snap step, we may now show:

Lemma.

For compact, and absolutely continuous with respect to the Lebesgue measure for all :

Proof.

By strong duality, we have the following equality for each when the have been optimized after an ascent step:

This implies that as is simply the optimal transport cost. We now argue that . We may split up the integrals for transport cost over the power cells corresponding to each th point. We differentiate with respect to to find the point with lowest joint transport cost to the cells . Setting this to 0 yields the following:

Note this is equivalent to the barycenter update step in Equation (14), and with convergence of the previous ascent step, we should have uniform weights. This demonstrates that snapping to the uniform average of barycenters lowers , and we have that . The last inequality follows as the next ascent step will find the optimal transport and decrease the transport cost. ∎

With joint transportation cost being non-negative, this implies that our objective function decreases at every step. This does not imply that our iterates converge, as there may not be a unique minimizing point configuration (see Figure 2). Empirically, our iterates converge in all of our test cases. We note also that our formula bears some resemblance to the mean-shift algorithm and to Lloyd’s algorithm, both of which which are also known to converge under some assumptions Li et al. (2007); Bottou and Bengio (1995).

7 Experiments

(a) (b) (c)
Figure 3: Barycenter of sharp featured distributions. (a) 50 points from our algorithm yields a barycenter supported on a line. (b) The barycenter from Staib et al. (2017) using a grid of 20000 points. (c) Barycenter from Solomon et al. (2015) using a regularizer value of ; smaller regularizers were numerically unstable.
(a) (b) (c)
Figure 4: Barycenter of randomly generated ellipses. (a) 20 input ellipses (b) The output of our algorithm is a sharp distribution approximating a circle. (c) The output of Solomon et al. (2015) with a regularizer value of .
(a) (b) (c)
Figure 5: Barycenter when tested on two uniform distributions over unit squares. (a) Our output: the input distributions are shown in blue, while the output barycenter points are shown in red, with the limits of the true barycenter in black. (b) A similar example in three dimensions. (c) The output barycenter of Staib et al. (2017): note the output has non-zero measure outside the true barycenter.
Figure 6: The point approximation of a mixture of ten Gaussians. Top row: our method with 10, 50, 100, and 200 points. Bottom row: iid sampling with the same number of points.
Figure 7: Blue noise sampling. Left: 10K samples from our algorithm. Right: Original image (approximately 90K pixels).

We showcase the versatility of our method on several applications. We typically use between 16K and 256K samples per input distribution to approximate the power cell density and barycenter. The variance is due to different problem sizes and dimensionality of the input measures. We stop the gradient ascent step when . The snap step empirically converges in under 20 iterations, and several of our examples use only one step.

7.1 Distributions with Sharp Features

Our algorithm is well-suited to problems where the input distributions have very sharp features. We test against the algorithms in Staib et al. (2017) and Solomon et al. (2015) on two test cases: ten uniform distributions over lines in the 2D plane (Figure 3), and 20 uniform distributions over ellipses (Figure 4).

The results of Figures 3 and 4 show that our barycenter is more sharply supported than the results of competing methods. Our output agrees with that of Solomon et al. (2015), but our results more closely match expected behavior. We strongly suspect that the true barycenter in Figure 3 is also a uniform measure on a line, while that in Figure 4 is a circle centered at the origin.

7.2 The Case

In the case of two input measures and , we expect the barycenter to be McCann’s interpolant Agueh and Carlier (2011); McCann (1997):

where is the optimal map, and is the inverse map, while denotes the pushforward of a measure.

We test this on two uniform distributions on the unit square in Figure 5. The transport map in this case is transport of the entire distribution along a straight line. As expected from McCann’s interpolant, we recover a uniform distribution on the unit square halfway between the two input distributions. We show our results alongside those of Staib et al. (2017). Notice that their output barycenter is not uniform, and that it has non-zero measure outside the true barycenter.

7.3 The Case

The case bears interest as well. There are instances when sampling iid from a distribution yields samples that do not approximate the underlying distribution accurately. We showcase two applications in generating super samples from distributions, as well as approximating grayscale images through blue noise.

7.3.1 Blue Noise

The term blue noise refers to an unstructured but even and isotropic distribution of points. It has been used in image dithering as it captures image intensity via local point density, without the need for varying point sizes as in halftoning.

De Goes et al. (2012) described the link between optimal transport and blue noise generation. We recover a stochastic version of their algorithm by taking a discrete distribution over the image pixels proportional to intensity. As our method is more general, we observe performance loss, but the output is of comparable quality (Figure 7).

7.3.2 Super Samples

Our method can be adapted to generate super samples from complex distributions Chen et al. (2010). Figure 6 details our results on a mixture of ten Gaussians. Our method better approximates the shape of the underlying distribution due to negative autocorrelations: points move away from oversampled regions. The points drawn iid from the mixture tend to oversample around the larger modes and do not approximate density contours as well.

8 Conclusion

We have proposed an algorithm for computing the Wasserstein barycenter of continuous measures using only samples from the input distributions. The algorithm decomposes into a concave maximization and a fixed point iteration similar to the mean-shift and -means algorithms. Our algorithm is easy to implement and parallelize, and it does not rely on a fixed-support grid. This allows us to recover much sharper approximations to the barycenter than previous methods. Our algorithm is general and versatile enough to be applied to other problems beyond barycenter computation.

There are several avenues for future work. Solving the concave maximization problem is currently a bottleneck for our algorithm as we do not have access to the function value or the Hessian, but we believe multiscale methods can be adapted to our approach. The potential applications of this method extend beyond what was covered. One application we highlight is in developing coresets that minimize the distance to the empirical distribution on the input data.

Acknowledgements

The authors acknowledge the generous support of Army Research Office grant W911NF-12-R-0011 (“Smooth Modeling of Flows on Graphs”), from the MIT Research Support Committee, from the MIT–IBM Watson AI Lab, from the Skoltech–MIT Next Generation Program, and from an Amazon Research Award.

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