最优质量传输在信号处理与机器学习中的应用解析

Optimal

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"Optimal Mass Transport - Signal processing and machine learning applications" 本文主要探讨了最优质量传输（Optimal Mass Transport, OMT）在信号处理和机器学习领域的应用。OMT是一种数学理论，它允许我们衡量两个概率分布之间的距离，通过找到一个最有效的策略将一个分布转化为另一个分布。这一理论在近年来受到了广泛的关注，因为它可以提供对信号强度和其他数据分布的精确生成模型。文章首先介绍了OMT的基本概念和动机。在信号处理中，理解数据分布的几何特性至关重要，而OMT提供了一种量化这些特性的新方法。它的目标是优化分布间的转换过程，使得转化成本最小化，这在处理非结构化或复杂数据时特别有用。 OMT的应用广泛，包括但不限于内容检索、癌症检测、图像超分辨率以及统计机器学习。在这些领域，OMT已经展示了其在产生最新技术成果中的潜力。例如，在癌症检测中，OMT可以帮助识别异常细胞分布模式；在图像超分辨率中，它可以提升低分辨率图像的细节质量。文章深入讨论了OMT的数学基础，包括相关的数值实现。这部分涵盖了如何通过计算方法来解决实际问题，比如使用Sinkhorn迭代算法求解OT问题。此外，软件包的提供使得读者能够实践和理解OMT的运算过程。接下来，文章回顾了OMT在不同应用中的效果，并通过实例进行了演示。这些应用示例进一步证明了OMT在理解和解释数据分布含义上的创新性。通过这些案例，读者可以了解OMT如何在各种实际场景中发挥作用，如在机器学习中用于特征匹配、聚类或分类任务。 "Optimal Mass Transport - Signal processing and machine learning applications"是一篇综合教程，不仅详细阐述了OMT的理论基础，还提供了实用的数值方法和丰富的应用示例，对于想要深入了解如何利用OMT进行信号处理和机器学习研究的学者与工程师来说，具有很高的参考价值。

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unique optimal transportation map

∗

minimizing (1). In addition, the optimal transport plan

is unique, and given by (5). Moreover if

(

x, y

) = |

x − y

then there exists a (unique up to

adding a constant) convex function

such that

∗

= ∇

. A proof is available in [20, 49].

B. Optimal Mass Transport: Geometric properties

1) Wasserstein metric—Let Ω be a bounded subset of ℝ

on which the signals are

defined. For signals (

= 1) or images (

= 2), this can simply be the space [0, 1]

, for

example. Let

(Ω) be the set of probability densities supported on Ω. The p-Wasserstein

metric,

, for

≥ 1 on

(Ω) is then defined as using the optimal transportation problem (4)

with the cost function

(

x, y

) = |

x − y

. For

and

(Ω),

, I

= inf

γ ∈ M P

∫

Ω × Ω

x − y

dγ x, y

For any

≥ 1,

is a metric on

(Ω). The metric space (

(Ω),

) is referred to as the p-

Wasserstein space. To understand the nature of the optimal transportation distances it is

useful to note that for any

≥ 1, the convergence with respect to

is equivalent to the

weak convergence of measures. That is

(

) → 0 as

→ ∞ if and only if for every

bounded and continuous function

: Ω → ℝ

∫

f x I

x dx

∫

f x I x dx .

For the specific case of

= 1 the p-Wasserstein metric is also known as the Monge–

Rubinstein [49] metric, or the earth mover distance [44]. The p-Wasserstein metric in one-

dimension has a simple characterization. For one-dimensional signals

and

the optimal

transport map has a closed form solution. Let

be the cumulative distribution function of

for

= 0, 1. That is

x =

∫

inf Ω

x dx for i = 0, 1.

Note that this is a nondecreasing function going from 0 to 1. We define the pseudoinverse of

as follows: for

∈ (0, 1),

−1

(

) is the smallest

for which

(

) ≥

, that is

−1

z = inf x ∈ Ω:F

x ≥ z

> 0 then

is continuous and increasing (and thus invertible) and the inverse of the

function

is equal to the pseudoinverse we just defined. In other words the pseudoinverse

is a generalization of the notion of the inverse of a function. The pseudoinverse (i.e. the

inverse if

> 0 and

> 0) provides a closed form solution for the p-Wasserstein distance:

Kolouri et al.

Page 7

IEEE Signal Process Mag

. Author manuscript; available in PMC 2018 June 29.

Author Manuscript Author Manuscript Author Manuscript Author Manuscript

, I

∫

−1

z − F

−1

z |

. (7)

The closed-form solution of the p-Wasserstein distance in one dimension is an attractive

property, as it alleviates the need for optimization. This property was employed in the Sliced

Wasserstein metrics as defined below.

2) Sliced-Wasserstein Metric—The idea behind the Sliced Wasserstein metric is to first

obtain a set of one-dimensional representations for a higher-dimensional probability

distribution through projections (slicing the measure), and then calculate the distance

between two input distributions as a functional on the Wasserstein distance of their one-

dimensional representations. In this sense, the distance is obtained by solving several one-

dimensional optimal transport problems, which have closed-form solutions.

Projection of high-dimensional PDFs is closely related to the well known Radon transform

in the imaging and image processing community [8, 25]. The

-dimensional Radon

transform ℛ maps a function

∈

(ℝ

) where

(ℝ

) : = {I:ℝ

ℝ|

∫

ℝ

|I(x)|dx ≤ ∞}

into

the of its integrals over the hyperplanes of ℝ

and is defined as,

ℛ

I t, θ : =

∫

ℝ

I(tθ + sθ

⊥

)ds, ∀t ∈ ℝ, ∀θ ∈ 𝕊

d − 1

here

⊥

is the subspace orthogonal to

, and 𝕊

d − 1

is the unit sphere in ℝ

. Note that

(ℝ

) L

(ℝ × 𝕊

d − 1

). In other words, Radon transform projects a PDF,

∈

(ℝ

), where

> 1, into an infinite set of one-dimensional PDFs ℛ

(.,

). The Sliced Wasserstein metric for

PDFs

and

on ℝ

is then defined as,

, I

∫

𝕊

d − 1

ℛI

. , θ , ℛI

. , θ dθ

where

≥ 1, and

is the p-Wasserstein metric, which for one dimensional PDFs ℛ

(.,

)

and ℛ

(.,

) has a closed form solution (see Equation (7)). For more details and definitions

of the Sliced Wasserstein metric we refer the reader to [8, 25, 29].

3) Wasserstein spaces, geodesics, and Riemannian structure—In this section we

assume that Ω is convex. Here we highlight that the p-Wasserstein space (

(Ω),

) is not

just a metric space, but has additional geometric structure. In particular for any

≥ 1 and any

∈

(Ω) there exists a continuous path (interpolation) between

and

whose length is

the distance between

and

Furthermore the space with

= 2 is special as it possesses a structure of a formal, infinite

dimensional, Riemannian manifold. That structure was first noted by Otto [38] who

Kolouri et al.

Page 8

IEEE Signal Process Mag

. Author manuscript; available in PMC 2018 June 29.

Author Manuscript Author Manuscript Author Manuscript Author Manuscript

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