We present a flow-based approach to the optimal transport (OT) problem
between two continuous distributions $\pi_0,\pi_1$ on $\mathbb{R}^d$, of
minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings
$(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $\pi_0,\pi_1$,
respectively, where $c$ is a cost function. Our method iteratively constructs a
sequence of neural ordinary differentiable equations (ODE), each learned by
solving a simple unconstrained regression problem, which monotonically reduce
the transport cost while automatically preserving the marginal constraints.
This yields a monotonic interior approach that traverses inside the set of
valid couplings to decrease the transport cost, which distinguishes itself from
most existing approaches that enforce the coupling constraints from the
outside. The main idea of the method draws from rectified flow, a recent
approach that simultaneously decreases the whole family of transport costs
induced by convex functions $c$ (and is hence multi-objective in nature), but
is not tailored to minimize a specific transport cost. Our method is a
single-object variant of rectified flow that guarantees to solve the OT problem
for a fixed, user-specified convex cost function $c$.

研究了一种基于流的方法来解决优化运输问题，该方法可通过神经普通可微方程迭代地减少运输成本，同时自动维护边际约束，相对于已有的方法，该方法针对特定运输成本函数且内部减少可行耦合来减少运输成本，是一种单目标变体。

Rectified Flow：一种保持边际的最优传输方法

Rectified Flow: A Marginal Preserving Approach to Optimal Transport

We prove the Duality Theorems for the stochastic optimal transportation
problems with a convex cost function without a regularity assumption that is
often supposed in the proof of the lower semicontinuity of an action integral.
In our new approach, we prove that the stochastic optimal transportation
problems with a convex cost function are equivalent to a class of variational
problems for the Fokker-Planck equation, which lets us revisit them. It is done
by the so-called superposition principle and by an idea from the mather theory.
The superposition principle is the construction of a semimartingale from the
Fokker-Planck equation and can be considered a class of the so-called marginal
problems that construct stochastic processes from given marginal distributions.
It was first considered in stochastic mechanics by E. Nelson, called Nelson's
problem, and was proved by E. Carlen first. The semimartingale is called the
Nelson process, provided it is Markovian. We also consider the Markov property
of a minimizer of the stochastic optimal transportation problem with a
nonconvex cost in a one-dimensional case. In the proof, the superposition
principle and the minimizer of an optimal transportation problem with a concave
cost function play crucial roles. Lastly, we prove the semiconcavity and the
Lipschitz continuity of Schrodinger's problem that is a typical example of the
stochastic optimal transportation problem.

在本文中，我们证明了具有凸代价函数的随机最优运输问题的对偶定理，该代价函数不需要通常假定的正则性假设。我们的新方法是证明了该问题等同于 Fokker-Planck 方程的一类变分问题，并通过所谓的叠加原理和 mather 理论中的想法来完成。我们还考虑了具有一维非凸代价的随机最优运输问题的最小化器的 Markov 性质，并证明了 Schrodinger 问题的半凸性和 Lipschitz 连续性，这是随机最优运输问题的典型实例。

随机最优传输（再探讨）

Stochastic optimal transport revisited

Deformable parts models show a great potential in tracking by principally
addressing non-rigid object deformations and self occlusions, but according to
recent benchmarks, they often lag behind the holistic approaches. The reason is
that potentially large number of degrees of freedom have to be estimated for
object localization and simplifications of the constellation topology are often
assumed to make the inference tractable. We present a new formulation of the
constellation model with correlation filters that treats the geometric and
visual constraints within a single convex cost function and derive a highly
efficient optimization for MAP inference of a fully-connected constellation. We
propose a tracker that models the object at two levels of detail. The coarse
level corresponds a root correlation filter and a novel color model for
approximate object localization, while the mid-level representation is composed
of the new deformable constellation of correlation filters that refine the
object location. The resulting tracker is rigorously analyzed on a highly
challenging OTB, VOT2014 and VOT2015 benchmarks, exhibits a state-of-the-art
performance and runs in real-time.

我们提出了一种新的 constellation 模型配合相关滤波器的公式，用于追踪目标并处理非刚性物体变形和自遮挡问题。我们设计了两个级别的模型来逼近目标位置，经过高难度测试表明本方法能达到最先进表现且实时运行。