Barycenters summarize populations of measures, but computing them does not scale to high dimensions with existing methods. We propose a scalable algorithm for estimating barycenters in high dimensions by turning the optimization over measures into a more tractable optimization over a space of generative models.

Barycentric averaging is a principled way of summarizing populations of measures. Existing algorithms for estimating barycenters typically parametrize them as weighted sums of atoms and optimize weights and/or locations. However, these approaches do not scale to high-dimensional settings due to the curse of dimensionality. We propose a scalable and general algorithm for estimating barycenters of measures in high dimensions^{1}.

# Barycenters of Probability Measures

The barycenter of $P$ probability measures $\mu_{1},…,\mu_{P}$ weighted by a vector $\boldsymbol{\beta} \in \Delta_P$ can be expressed as the optimal measure $\mu^{\star}$ solving the optimization problem^{2}
$$
\mu^{\star}=\arg\max_{\mu} \sum_{p=1}^{P}\beta_{p}D(\mu, \mu_{p}),
$$
where $D$ is a distance between probability measures. It can be interpreted as an extension of the Euclidean mean to means of measures.

Importantly, different distances $D$ will induce different averaging properties. For instance, we see in Figure 1 that maximum mean discrepancy (MMD) leads to a mixture behavior, whilst the 2-Wasserstein distance interpolates between measures.

Previous approaches^{3}^{4}^{5} typically parametrized barycenters as a discrete measure and optimized its weights and/or locations. This requires an exponentially increasing number of locations as the dimensionality of the domain increases. As a result of this “curse of dimensionality”, these methods are typically restricted to low-dimensional problems in $\mathbb{R}^{\leq 3}$ and hand-tailored to specific choices of $D$.

# Scalable and General Computation of Barycenters

In our paper^{1}, we propose a scalable algorithm for estimating barycenters of measures.
The main idea is to *turn the optimization over measures into a more tractable optimization over a space of generative models*^{6}^{7}.

**Generative barycenter**: Specifically, we parametrize the barycenter $P_\theta$ as a generative model consisting of a latent measure $\rho$ and a parametric functional $G_{\theta}$. We can sample from it as follows: $$\boldsymbol{z} \sim \rho, \qquad \boldsymbol{x} = G_\theta(\boldsymbol{z}).$$ We illustrate a circular generative model in Figure 2. In that example, the radius and centre of the ellipse are the parameters $\theta$ to be optimized, instead of the locations of individual atoms.

**Optimization**: Computing the parametric barycenter consists of solving the following optimization problem by SGD variants: $$ \theta^\star = \arg\min_{\theta} \sum_{p=1}^P\beta_p D(G_{\theta \star}\rho, \mu_{p}). $$**Inductive Biases**: We can incorporate prior knowledge on the form of the barycenter through the generator $G_\theta$'s structure (e.g., CNNs for barycenters of images). This enables scaling to high-dimensional settings. Note that $G_\theta$ is not restricted to being a neural network, and domain knowledge can enable more efficient learning.**Convergence Guarantees**We study local convergence of our algorithms for different $D$. In particular, we show that under smoothness assumptions on the distance, the barycentric problem converges to a stationary point, which holds for popular choices of discrepancies.^{1}

# Experiments

We now discuss briefly some experiments, and recommend looking into the paper^{1} for more details and comparisons to previous works.

**Nested Ellipses**: Firstly, we consider the computation of the Sinkhorn barycenter of $P=30$ nested ellipses. We consider two approaches to parametrizing the generator $G_\theta$, $(i)$ using a multi-layer perceptron (MLP) as $G_\theta$ and $(ii)$ exploiting inductive biases by parametrizing two ellipses ($\theta$: axis lengths and centers of both ellipses). Figure 2 shows that both approaches recover a sensible barycenter.

**Natural Image Datasets**: We now deal with problems in thousands of dimensions, by computing MMD barycenters of**image datasets**. Each atom is now an image, whilst previous CV approaches computed barycenters of single images, and atoms were**individual pixels**.

# Concluding remarks

- Previous approaches used compact
**local basis functions**to parametrize barycenters, which does not scale well with dimensions due to the curse of dimensionality. - We propose a generative approach relying on
**global basis functions**allowing for scalability and generalizability. - We provide local convergence guarantees for a variety of distances.
- We apply our algorithm to problems of unprecedented scales, and study barycentric behavior from theoretical (advancing previous results) and empirical standpoints.

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