The dynamical formulation of the optimal transport can be extended through
various choices of the underlying geometry ($\textit{kinetic energy}$), and the
regularization of density paths ($\textit{potential energy}$). These
combinations yield different variational problems ($\textit{Lagrangians}$),
encompassing many variations of the optimal transport problem such as the
Schr\"odinger bridge, unbalanced optimal transport, and optimal transport with
physical constraints, among others. In general, the optimal density path is
unknown, and solving these variational problems can be computationally
challenging. Leveraging the dual formulation of the Lagrangians, we propose a
novel deep learning based framework approaching all of these problems from a
unified perspective. Our method does not require simulating or backpropagating
through the trajectories of the learned dynamics, and does not need access to
optimal couplings. We showcase the versatility of the proposed framework by
outperforming previous approaches for the single-cell trajectory inference,
where incorporating prior knowledge into the dynamics is crucial for correct
predictions.

通过动态形式化的最优输运，结合底层几何学（动能）选择和密度路径（势能）的正则化，构造了多种变分问题（Lagrangians），包括 Schrödinger 桥、非平衡最优输运和具有物理约束的最优输运等。采用双重 Lagrangians 的对偶形式提出了一种新颖的基于深度学习的统一框架，无需模拟或反向传播经过学习动力学的轨迹，并且不需要访问最优耦合。通过在单细胞轨迹推断中融入先验知识，展示了所提框架的多功能性能，从而在正确预测方面优于先前方法。

求解 Wasserstein 拉格朗日流的计算框架

A Computational Framework for Solving Wasserstein Lagrangian Flows

In this article, we propose a new variational approach to learn private
and/or fair representations. This approach is based on the Lagrangians of a new
formulation of the privacy and fairness optimization problems that we propose.
In this formulation, we aim to generate representations of the data that keep a
prescribed level of the relevant information that is not shared by the private
or sensitive data, while minimizing the remaining information they keep. The
proposed approach (i) exhibits the similarities of the privacy and fairness
problems, (ii) allows us to control the trade-off between utility and privacy
or fairness through the Lagrange multiplier parameter, and (iii) can be
comfortably incorporated to common representation learning algorithms such as
the VAE, the $\beta$-VAE, the VIB, or the nonlinear IB.

本文提出了一种新的变分方法来学习私有和 / 或公平的表示，该方法基于一个新的隐私和公平优化问题的拉格朗日形式，并展示了保持所需信息水平的数据表示，并在最小化保留信息的同时控制效用与隐私或公平平衡的相似之处和可行性。该方法可舒适地纳入常见的表示学习算法，如 VAE、β-VAE、VIB 或非线性 IB 等。

隐私和公正的变分方法

A Variational Approach to Privacy and Fairness

The information bottleneck (IB) problem tackles the issue of obtaining
relevant compressed representations $T$ of some random variable $X$ for the
task of predicting $Y$. It is defined as a constrained optimization problem
which maximizes the information the representation has about the task,
$I(T;Y)$, while ensuring that a certain level of compression $r$ is achieved
(i.e., $ I(X;T) \leq r$). For practical reasons, the problem is usually solved
by maximizing the IB Lagrangian (i.e., $\mathcal{L}_{\text{IB}}(T;\beta) =
I(T;Y) - \beta I(X;T)$) for many values of $\beta \in [0,1]$. Then, the curve
of maximal $I(T;Y)$ for a given $I(X;T)$ is drawn and a representation with the
desired predictability and compression is selected. It is known when $Y$ is a
deterministic function of $X$, the IB curve cannot be explored and another
Lagrangian has been proposed to tackle this problem: the squared IB Lagrangian:
$\mathcal{L}_{\text{sq-IB}}(T;\beta_{\text{sq}})=I(T;Y)-\beta_{\text{sq}}I(X;T)^2$.
In this paper, we (i) present a general family of Lagrangians which allow for
the exploration of the IB curve in all scenarios; (ii) provide the exact
one-to-one mapping between the Lagrange multiplier and the desired compression
rate $r$ for known IB curve shapes; and (iii) show we can approximately obtain
a specific compression level with the convex IB Lagrangian for both known and
unknown IB curve shapes. This eliminates the burden of solving the optimization
problem for many values of the Lagrange multiplier. That is, we prove that we
can solve the original constrained problem with a single optimization.

本文提出一种通用的拉格朗日算子族，允许在所有情况下探索 IB 曲线，并提供拉格朗日乘子与所需压缩率 r 之间的精确一对一映射，同时证明可以通过凸 IB 拉格朗日以及这些 Lagrangian 对于已知和未知的 IB 曲线形状进行逼近以消除求解优化问题的负担。