### 机器学习代写|流形学习代写manifold data learning代考| Density Estimation on Submanifolds

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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|Introduction

Kernel density estimation (KDE) [21] is one of the most popular methods of estimating the underlying probability density function (PDF) of a data set. Roughly speaking, KDE consists of having the data points contribute to the estimate at a given point according to their distances from that point – closer the point, the bigger the contribution. More precisely, in the simplest multi-dimensional KDE [5], the estimate $\hat{f}{m}\left(\mathbf{y}{0}\right)$ of a PDF $f\left(\mathbf{y}{0}\right)$ at a point $\mathbf{y}{0} \in \mathbb{R}^{D}$ is given in terms of a sample $\left{\mathbf{y}{1}, \ldots, \mathbf{y}{m}\right}$ as,
$$\hat{f}{m}\left(\mathbf{y}{0}\right)=\frac{1}{m} \sum_{i=1}^{m} \frac{1}{h_{m}^{D}} K\left(\frac{\left|\mathbf{y}{i}-\mathbf{y}{0}\right|}{h_{m}}\right),$$
where $h_{m}>0$, the bandwidth, is chosen to approach to zero in a suitable manner as the number $m$ of data points increases, and $K:[0, \infty) \rightarrow[0, \infty)$ is a kernel function that satisfies certain properties such as boundedness. Various theorems exist on the different types and rates of convergence of the estimator to the correct result. The earliest result on the pointwise convergence rate in the multivariable case seems to be given in [5], where it is stated that under certain conditions for $f$ and $K$, assuming $h_{m} \rightarrow 0$ and $m h_{m}^{D} \rightarrow \infty$ as $m \rightarrow \infty$, the mean squared error in the estimate $\hat{f}\left(\mathbf{y}{0}\right)$ of the density at a point goes to zero with the rate, $$\operatorname{MSE}\left[\hat{f}{m}\left(\mathbf{y}{0}\right)\right]=\mathrm{E}\left[\left(\hat{f}{m}\left(\mathbf{y}{0}\right)-f\left(\mathbf{y}{0}\right)\right)^{2}\right]=O\left(h_{m}^{4}+\frac{1}{m h_{m}^{D}}\right)$$
as $m \rightarrow \infty$. If $h_{m}$ is chosen to be proportional to $m^{-1 /(D+4)}$, one gets,
$$\operatorname{MSE}\left[\hat{f}{m}(p)\right]=O\left(\frac{1}{m^{4 /(D+4)}}\right),$$ as $m \rightarrow \infty$. The two conditions $h{m} \rightarrow 0$ and $m h_{m}^{D} \rightarrow \infty$ ensure that, as the number of data points increases, the density estimate at a point is determined by the values of the density in a smaller and smaller region around that point, but the number of data points contributing to the estimate (which is roughly proportional to the volume of a region of size $h_{m}$ ) grows unboundedly, respectively.

## 机器学习代写|流形学习代写manifold data learning代考|Motivation for the Submanifold Estimator

We would like to estimate the values of a PDF that lives on an (unknown) $d$-dimensional Riemannian submanifold $M$ of $\mathbb{R}^{D}$, where $d<D$. Usually, $D$-dimensional KDE does not work for such a distribution. This can be intuitively understood by considering a distribution on a line in the plane: 1-dimensional KDE performed on the line (with a bandwidth $h_{m}$ satisfying the asymptotics given above) would converge to the correct density on the line, but 2-dimensional KDE, differing from the former only by a normalization factor that blows up as the bandwidth $h_{m} \rightarrow 0$ (compare (3.1) for the cases $D=2$ and $D=1$ ), diverges. This behavior is due to the fact that, similar to a “delta function” distribution on $\mathbb{R}$, the $D$-dimensional density of a distribution on a $d$-dimensional submanifold of $\mathbb{R}^{D}$ is, strictly speaking, undefined – the density is zero outside the submanifold, and in order to have proper normalization, it has to be infinite on the submanifold. More formally, the $D$ dimensional probability measure for a $d$-dimensional PDF supported on $M$ is not absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{D}$, and does not have a probability

density function on $\mathbb{R}^{D}$. If one attempts to use $D$-dimensional KDE for data drawn from such a probability measure, the estimator will “attempt to converge” to a singular PDF; one that is infinite on $M$, zero outside.

For a distribution with support on a line in the plane, we can resort to 1-dimensional KDE to get the correct density on the line, but how could one estimate the density on an unknown, possibly curved submanifold of dimension $d<D$ ? Essentially the same approach works: even for data that lives on an unknown, curved $d$-dimensional submanifold of $\mathbb{R}^{D}$, it suffices to use the $d$-dimensional kernel density estimator with the Euclidean distance on $\mathbb{R}^{D}$ to get a consistent estimator of the submanifold density. Furthermore, the convergence rate of this estimator can be bounded as in (3.3), with $D$ being replaced by $d$, the intrinsic dimension of the submanifold. [20]

The intuition behind this approach is based on three facts: 1) For small bandwidths, the main contribution to the density estimate at a point comes from data points that are nearby; 2) For small distances, a $d$-dimensional Riemannian manifold “looks like” $\mathbb{R}^{d}$, and densities in $\mathbb{R}^{d}$ should be estimated by a $d$-dimensional kernel, instead of a $D$-dimensional one; and 3) For points of $M$ that are close to each other, the intrinsic distances as measured on $M$ are close to Euclidean distances as measured in the surrounding $\mathbb{R}^{D}$. Thus, as the number of data points increases and the bandwidth is taken to be smaller and smaller, estimating the density by using a kernel normalized for $d$ dimensions and distances as measured in $\mathbb{R}^{D}$ should give a result closer and closer to the correct value.

We will next give the formal definition of the estimator motivated by these considerations, and state the theorem on its asymptotics. As in the original work of Parzen [21], the pointwise consistence of the estimator can be proven by using a bias-variance decomposition. The asymptotic unbiasedness of the estimator follows from the fact that as the bandwidth converges to zero, the kernel function becomes a “delta function.” Using this fact, it is possible to show that with an appropriate choice for the vanishing rate of the bandwidth, the variance also vanishes asymptotically, completing the proof of the pointwise consistency of the estimator.

## 机器学习代写|流形学习代写manifold data learning代考|Statement of the Theorem

Let $(M, \mathbf{g})$ be a $d$-dimensional, embedded, complete, compact Riemannian submanifold of $\mathbb{R}^{D}(d0 .^{7}$ Let $d(p, q)=d_{p}(q)$ be the length of a length-minimizing geodesic in $M$ between $p, q \in M$, and let $u(p, q)=u_{p}(q)$ be the geodesic distance between $p$ and $q$ as measured in $\mathbb{R}^{D}$ (thus, $u(p, q)$ is simply the Euclidean distance between $p$ and $q$ in $\left.\mathbb{R}^{D}\right)$. Note that $u(p, q) \leq d(p, q)$. We will denote the Riemannian volume measure on $M$ by $V$, and the volume form by $d V .^{8}$
Theorem 3.3.1 Let $f: M \rightarrow[0, \infty)$ be a probability density function defined on $M$ (so that the related probability measure is $f V)$, and $K:[0, \infty) \rightarrow[0, \infty)$ be a continuous function that vanishes outside $[0,1)$, is differentiable with a bounded derivative in $[0,1)$, and satisfies the normalization condition, $\int_{|\mathbf{z}| \leq 1} K(|\mathbf{z}|) d^{d} \mathbf{z}=1$. Assume $f$ is differentiable to second order in a neighborhood of $p \in M$, and for a sample $q_{1}, \ldots, q_{m}$ of size $m$ drawn from the

density $f$, define an estimator $\hat{f}{m}(p)$ of $f(p)$ as, $$\hat{f}{m}(p)=\frac{1}{m} \sum_{j=1}^{m} \frac{1}{h_{m}^{d}} K\left(\frac{u_{p}\left(q_{j}\right)}{h_{m}}\right)$$
where $h_{m}>0$. If $h_{m}$ satisfies $\lim {m \rightarrow \infty} h{m}=0$ and $\lim {m \rightarrow \infty} m h{m}^{d}=\infty$, then, there exist non-negative numbers $m_{}, C_{b}$, and $C_{V}$ such that for all $m>m_{}$ the mean squared error of the estimator (3.4) satisfies,
$$\operatorname{MSE}\left[\hat{f}{m}(p)\right]=\mathrm{E}\left[\left(\hat{f}{m}(p)-f(p)\right)^{2}\right]<C_{b} h_{m}^{4}+\frac{C_{V}}{m h_{m}^{d}}$$
If $h_{m}$ is chosen to be proportional to $m^{-1 /(d+4)}$, this gives,
$$\mathrm{E}\left[\left(f_{m}(p)-f(p)\right)^{2}\right]=O\left(\frac{1}{m^{4 /(d+4)}}\right)$$
as $m \rightarrow \infty$.
Thus, the bound on the convergence rate of the submanifold density estimator is as in (3.2), (3.3), with the dimensionality $D$ replaced by the intrinsic dimension $d$ of $M$. As mentioned above, the proof of this theorem follows from two lemmas on the convergence rates of the bias and the variance; the $h_{m}^{4}$ term in the bound corresponds to the bias, and the $1 / m h_{m}^{d}$ term corresponds to the variance; see $[20]$ for details. This approach to submanifold density estimation was previously mentioned in [11], and the thesis [10] contains the details, although in a more technical and general approach than the elementary one followed in [20].

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

F^米(是0)=1米∑一世=1米1H米Dķ(|是一世−是0|H米),

MSE⁡[F^米(p)]=这(1米4/(D+4)),作为米→∞. 两个条件H米→0和米H米D→∞确保随着数据点数量的增加，一个点的密度估计值由该点周围越来越小的区域中的密度值决定，但对估计值有贡献的数据点数量（大致成正比）到一个大小区域的体积H米) 分别无限增长。

## 机器学习代写|流形学习代写manifold data learning代考|Statement of the Theorem

MSE⁡[F^米(p)]=和[(F^米(p)−F(p))2]<CbH米4+C在米H米d

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