### 机器学习代写|流形学习代写manifold data learning代考|The Existence of Density Preserving Maps

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|流形学习代写manifold data learning代考|Moser’s Theorem

Riemannian manifolds. We begin by restricting our attention to data subspaces which are Riemannian submanifolds of $\mathbb{R}^{D}$. Riemannian manifolds provide a generalization of the notion of a smooth surface in $\mathbb{R}^{3}$ to higher dimensions. As first clarified by Gauss in the two-dimensional case and by Riemann in the general case, it turns out that intrinsic features of the geometry of a surface, such as the lengths of its curves or intrinsic distances between its points, etc., can be given in terms of the so-called metric tensor ${ }^{2} \mathrm{~g}$, without referring to the particular way the the surface is embedded in $\mathbb{R}^{3}$. A space whose geometry is defined in terms of a metric tensor is called a Riemannian manifold (for a rigorous definition, see, e.g., $[12,16,2])$.

The Gauss/Riemann result mentioned above states that if the intrinsic curvature of a Riemannian manifold $\left(M, \mathbf{g}_{M}\right)$ is not zero in an open set $U \in M$, it is not possible to find

a map from $M$ into $\mathbb{R}^{d}$ that preserves the distances between the points of $U$. Thus, there exists a local obstruction, namely, the curvature, to the existence of distance-preserving maps. It turns out that no such local obstruction exists for volume-preserving maps. The only invariant is a global one, namely, the total volume. ${ }^{3}$ This is the content of Moser’s theorem on volume-preserving maps, which we state next.

Theorem 3.2.1 (Moser [18]) Let $\left(M, \mathbf{g}{M}\right)$ and $\left(N, \mathbf{g}{N}\right)$ be two closed, connected, orientable, $d$-dimensional differentiable manifolds that are diffeomorphic to each other. Let $\tau_{M}$ and $\tau_{N}$ be volume forms, i.e., nowhere vanishing $d$-forms on these manifolds, satisfying $\int_{M} \tau_{M}=$ $\int_{N} \tau_{N}$. Then, there exists a diffeomorphism $\phi: M \rightarrow N$ such that $\tau_{M}=\phi^{*} \tau_{N}$, i.e., the volume form on $M$ is the same as the pull-back of the volume form on $N$ by $\phi .^{4}$

The meaning of this result is that, if two manifolds with the same “global shape” (i.e., two manifolds that are diffeomorphic) have the same total volume, one can find a map between them that preserves the volume locally. The surfaces of a mug and a torus are the classical examples used for describing global, topological equivalence. Although these objects have the same “global shape” (topology/smooth structure) their intrinsic, local geometries are different. Moser’s theorem states that if their total surface areas are the same, one can find a map between them that preserves the areas locally, as well, i.e., a map that sends all small regions on one surface to regions in the other surface in a way that preserves the areas.

Using this theorem, we now show that it is possible to find density-preserving maps between Riemannian manifolds that have the same total volume. This is due to the fact that if local volumes are preserved under a map, the density of a distribution will also be preserved.

## 机器学习代写|流形学习代写manifold data learning代考|Dimensional Reduction

These results were formulated in terms of so-called closed manifolds, i.e., compact manifolds without boundary. The practical dimensionality reduction problem we would like to address, on the other hand, involves starting with a $d$-dimensional data submanifold $M$ of $\mathbb{R}^{D}$ (where $d<D)$, and dimensionally reducing to $\mathbb{R}^{d}$. In order to be able to do this diffeomorphically, $M$ must be diffeomorphic to a subspace of $\mathbb{R}^{d}$, which is not generally the case for closed manifolds. For instance, although we can find a diffeomorphism from a hemisphere (a manifold with boundary, not a closed manifold) into the plane, we cannot find one from the unit sphere (a closed manifold) into the plane. This is a constraint on all dimensional reduction algorithms that preserve the global topology of the data space, not just density preserving maps. Any algorithm that aims to avoid “tearing” or “folding” the data subspace during the reduction will fail on problems like reducing a sphere to $\mathbb{R}^{2.5}$

Thus, in order to show that density preserving maps into $\mathbb{R}^{d}$ exist for a useful class of $d$-dimensional data manifolds, we have to make sure that the conclusion of Moser’s theorem and our corollary work for certain manifolds with boundary, or for certain non-compact manifolds, as well. Fortunately, this is not so hard, at least for a simple class of manifolds that is enough to be useful. In proving his theorem for closed manifolds, Moser [18] first gives a proof for a single “coordinate patch” in such a manifold, which, basically, defines a compact manifold with boundary minus the boundary itself. Not all $d$-dimensional manifolds with boundary (minus their boundaries) can be given by atlases consisting of a single coordinate patch, but the ones that can be so given cover a wide range of curved Riemannian manifolds, including the hemisphere and the Swiss roll, possibly with punctures. In the following, we will assume that $M$ consists of a single coordinate patch.

## 机器学习代写|流形学习代写manifold data learning代考|Intuition on Non-Uniqueness

Note that the results above claim the existence of volume (or density) preserving maps, but not uniqueness. In fact, the space of volume-preserving maps is very large. An intuitive way to see this is to consider the flow of an incompressible fluid in $\mathbb{R}^{3}$. The fluid may cover the same region in space at two given times, but the fluid particles may have gone through significant shuffling. The map from the original configuration of the fluid to the final one is a volume preserving diffeomorphism, assuming the flow is smooth. The infinity of ways a fluid can move shows the infinity of ways of preserving volume.

Distance-preserving maps may also have some non-uniqueness, but this is parametrized by a finite-dimensional group, namely, the isometry group of the Riemannian manifold under consideration. ${ }^{6}$ The case of volume-preserving maps is much worse, the space of volumepreserving diffeomorphisms being infinite-dimensional. Since the aim of this chapter is to describe a manifold-learning method that preserves volumes/densities, we are faced with the following question: Given a data manifold with intrinsic dimension $d$ that is diffeomorphic to a subset of $\mathbb{R}^{d}$, which map, in the infinite-dimensional space of volume-preserving maps from this manifold to $\mathbb{R}^{d}$, is the “best”? In Section 3.4, we will describe an approach to this problem by setting up a specific optimization procedure. But first, let us describe a method for estimating densities on submanifolds.

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