Convergence Rates for Empirical Estimation of Binary Classification Bounds

Abstract

Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze–Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman–Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman–Rafsky estimator of the Henze–Penrose divergence. We validate our results experimentally and illustrate their application to real datasets.

1. Introduction

Divergence measures between probability density functions are used in many signal processing applications including classification, segmentation, source separation, and clustering (see [1,2,3]). For more applications of divergence measures, we refer to [4].

In classification problems, the Bayes error rate is the expected risk for the Bayes classifier, which assigns a given feature vector $\mathbf{x}$ to the class with the highest posterior probability. The Bayes error rate is the lowest possible error rate of any classifier for a particular joint distribution. Mathematically, let ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N\in {\mathbb{R}}^d$ be realizations of random vector $\mathbf{X}$ and class labels $S\in \{0,1\}$, with prior probabilities $p=P(S=0)$ and $q=P(S=1)$, such that $p+q=1$. Given conditional probability densities $f_0\left(\mathbf{x}\right)$ and $f_1\left(\mathbf{x}\right)$, the Bayes error rate is given by

$${\displaystyle ϵ={\int }_{{\mathbb{R}}^d}min\left\{p{f}_0\left(\mathbf{x}\right),q{f}_1\left(\mathbf{x}\right)\right\}\mathrm{d}\mathbf{x}.}$$

(1)

The Bayes error rate provides a measure of classification difficulty. Thus, when known, the Bayes error rate can be used to guide the user in the choice of classifier and tuning parameter selection. In practice, the Bayes error is rarely known and must be estimated from data. Estimation of the Bayes error rate is difficult due to the nonsmooth min function within the integral in (1). Thus, research has focused on deriving tight bounds on the Bayes error rate based on smooth relaxations of the min function. Many of these bounds can be expressed in terms of divergence measures such as the Bhattacharyya [5] and Jensen–Shannon [6]. Tighter bounds on the Bayes error rate can be obtained using an important divergence measure known as the Henze–Penrose (HP) divergence [7,8].

Many techniques have been developed for estimating divergence measures. These methods can be broadly classified into two categories: (i) plug-in estimators in which we estimate the probability densities and then plug them in the divergence function [9,10,11,12], (ii) entropic graph approaches, in which the relationship between the divergence function and a graph functional in Euclidean space is derived [8,13]. Examples of plug-in methods include k-nearest neighbor (K-NN) and Kernel density estimator (KDE) divergence estimators. Examples of entropic graph approaches include methods based on minimal spanning trees (MST), K-nearest neighbors graphs (K-NNG), minimal matching graphs (MMG), traveling salesman problem (TSP), and their power-weighted variants.

Disadvantages of plug-in estimators are that these methods often require assumptions on the support set boundary and are more computationally complex than direct graph-based approaches. Thus, for practical and computational reasons, the asymptotic behavior of entropic graph approaches has been of great interest. Asymptotic analysis has been used to justify graph based approaches. For instance, in [14], the authors showed that a cross match statistic based on optimal weighted matching converges to the the HP-divergence. In [15], a more complex approach based on the K-NNG was proposed that also converges to the HP-divergence.

The first contribution of our paper is that we obtain a bound on the convergence rates for the Friedman and Rafsky (FR) estimator of the HP-divergence, which is based on a multivariate extension of the non-parametric run length test of equality of distributions. This estimator is constructed using a multicolored MST on the labeled training set where MST edges connecting samples with dichotomous labels are colored differently from edges connecting identically labeled samples. While previous works have investigated the FR test statistic in the context of estimating the HP-divergence (see [8,16]), to the best of our knowledge, its minimax MSE convergence rate has not been previously derived. The bound on convergence rate is established by using the umbrella theorem of [17], for which we define a dual version of the multicolor MST. The proposed dual MST in this work is different than the standard dual MST introduced by Yukich in [17]. We show that the bias rate of the FR estimator is bounded by a function of N, $\eta$ and d, as $O\left({\left(N\right)}^{-{\eta }^2/\left(d(\eta +1)\right)}\right)$, where N is the total sample size, d is the dimension of the data samples $d\ge 2$, and $\eta$ is the Hölder smoothness parameter $0<\eta \le 1$. We also obtain the variance rate bound as ${\displaystyle O\left({\left(N\right)}^{-1}\right)}$.

The second contribution of our paper is a new concentration bound for the FR test statistic. The bound is obtained by establishing a growth bound and a smoothness condition for the multicolored MST. Since the FR test statistic is not a Euclidean functional, we cannot use the standard subadditivity and superadditivity approaches of [17,18,19]. Our concentration inequality is derived using a different Hamming distance approach and a dual graph to the multicolored MST.

We experimentally validate our theoretic results. We compare the MSE theory and simulation in three experiments with various dimensions $d=2,4,8$. We observe that, in all three experiments, as sample size increases, the MSE rate decreases and, for higher dimensions, the rate is slower. In all sets of experiments, our theory matches the experimental results. Furthermore, we illustrate the application of our results on estimation of the Bayes error rate on three real datasets.

1.1. Related Work

Much research on minimal graphs has focused on the use of Euclidean functionals for signal processing and statistics applications such as image registration [20,21], pattern matching [22], and non-parametric divergence estimation [23]. A K-NNG-based estimator of Rényi and f-divergence measures has been proposed in [13]. Additional examples of direct estimators of divergence measures include statistic based on the nonparametric two sample problem, the Smirnov maximum deviation test [24], and the Wald–Wolfowitz [25] runs test, which have been studied in [26].

Many entropic graph estimators such as MST, K-NNG, MMG, and TSP have been considered for multivariate data from a single probability density f. In particular, the normalized weight function of graph constructions all converge almost surely to the Rényi entropy of f [17,27]. For N uniformly distributed points, the MSE is $O\left(N^{-1/d}\right)$ [28,29]. Later, Hero et al. [30,31] reported bounds on $L_{\gamma }$-norm bias convergence rates of power-weighted Euclidean weight functionals of order $\gamma$ for densities f belonging to the space of Hölder continuous functions ${\Sigma }_d(\eta ,K)$ as $O\left(N^{-\alpha \eta /(\alpha \eta +1)1/d}\right)$, where $0<\eta \le 1$, $d\ge 1$, $\gamma \in (1,d)$, and $\alpha =(d-\gamma )/d$. In this work, we derive a bound on convergence rate of FR estimator for the HP-divergence when the density functions belong to the Hölder class, ${\Sigma }_d(\eta ,K)$, for $0<\eta \le 1$, $d\ge 2$ [32]. Note that throughout the paper we assume the density functions are absolutely continuous and bounded with support on the unit cube ${[0,1]}^d$.

In [28], Yukich introduced the general framework of continuous and quasi-additive Euclidean functionals. This has led to many convergence rate bounds of entropic graph divergence estimators.

The framework of [28] is as follows: Let F be finite subset of points in ${[0,1]}^d$, $d\ge 2$, drawn from an underlying density. A real-valued function $L_{\gamma }$ defined on F is called a Euclidean functional of order $\gamma$ if it is of the form $L_{\gamma }\left(F\right)=\underset{E\in \mathcal{E}}{min}{\displaystyle \sum _{e\in E}}{|e\left(F\right)|}^{\gamma }$, where $\mathcal{E}$ is a set of graphs, e is an edge in the graph E, $|e|$ is the Euclidean length of e, and $\gamma$ is called the edge exponent or power-weighting constant. The MST, TSP, and MMG are some examples for which $\gamma =1$.

Following this framework, we show that the FR test statistic satisfies the required continuity and quasi-additivity properties to obtain similar convergence rates to those predicted in [28]. What distinguishes our work from previous work is that the count of dichotomous edges in the multicolored MST is not Euclidean. Therefore, the results in [17,27,30,31] are not directly applicable.

Using the isoperimetric approach, Talagrand [33] showed that, when the Euclidean functional $L_{\gamma }$ is based on the MST or TSP, then the functional $L_{\gamma }$ for derived random vertices uniformly distributed in a hypercube ${[0,1]}^d$ is concentrated around its mean. Namely, with high probability, the functional $L_{\gamma }$ and its mean do not differ by more than $C{(NlogN)}^{(d-\gamma )/2d}$. In this paper, we establish concentration bounds for the FR statistic: with high probability $1-\delta$, the FR statistic differs from its mean by not more than $O\left({\left(N\right)}^{(d-1)/d}{\left(log(C/\delta )\right)}^{(d-1)/d}\right)$, where C is a function of N and d.

1.2. Organization

This paper is organized as follows. In Section 2, we first introduce the HP-divergence and the FR multivariate test statistic. We then present the bias and variance rates of the FR-based estimator of HP-divergence followed by the concentration bounds and the minimax MSE convergence rate. Section 3 provides simulations that validate the theory. All proofs and relevant lemmas are given in the Appendix A, Appendix B, Appendix C, Appendix D and Appendix E.

Throughout the paper, we denote expectation by $\mathbb{E}$ and variance by abbreviation $\mathrm{Var}$. Bold face type indicates random variables. In this paper, when we say number of samples we mean number of observations.

2. The Henze–Penrose Divergence Measure

Consider parameters $p\in (0,1)$ and $q=1-p$. We focus on estimating the HP-divergence measure between distributions $f_0$ and $f_1$ with domain ${\mathbb{R}}^d$ defined by

$${\displaystyle D_p(f_0,f_1)=\frac{1}{4pq}\left[\int \frac{{\left(p{f}_0\left(\mathbf{x}\right)-q{f}_1\left(\mathbf{x}\right)\right)}^2}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}-{(p-q)}^2\right].}$$

(2)

It can be verified that this measure is bounded between 0 and 1 and, if $f_0\left(\mathbf{x}\right)=f_1\left(\mathbf{x}\right)$, then $D_p=0$. In contrast with some other divergences such as the Kullback–Liebler [34] and Rényi divergences [35], the HP-divergence is symmetrical, i.e., $D_p(f_0,f_1)=D_q(f_1,f_0)$. By invoking relation (3) in [8],

$$\begin{array}{c}\hfill {\displaystyle \int \frac{{\left(p{f}_0\left(\mathbf{x}\right)-q{f}_1\left(\mathbf{x}\right)\right)}^2}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}=1-4pq{A}_p(f_0,f_1),}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{ccc}{A}_p(f_0,f_1)& =& {\displaystyle \int \frac{f_0\left(\mathbf{x}\right)f_1\left(\mathbf{x}\right)}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}={\mathbb{E}}_{f_0}\left[{\left(p\frac{f_0\left(\mathbf{X}\right)}{f_1\left(\mathbf{X}\right)}+q\right)}^{-1}\right],}\hfill \\ {\displaystyle u_p(f_0,f_1)}& =& 1-4pq{A}_p(f_0,f_1),\hfill \end{array}\end{array}$$

one can rewrite $D_p$ in the alternative form:

$$\begin{array}{c}\hfill {\displaystyle D_p(f_0,f_1)=1-A_p(f_0,f_1)=\frac{u_p(f_0,f_1)}{4pq}-\frac{{(p-q)}^2}{4pq}.}\end{array}$$

Throughout the paper, we refer to $A_p(f_0,f_1)$ as the HP-integral. The HP-divergence measure belongs to the class of $\varphi$-divergences [36]. For the special case $p=0.5$, the divergence (2) becomes the symmetric ${\chi }^2$-divergence and is similar to the Rukhin f-divergence. See [37,38].

2.1. The Multivariate Runs Test Statistic

The MST is a graph of minimum weight among all graphs $\mathcal{E}$ that span n vertices. The MST has many applications including pattern recognition [39], clustering [40], nonparametric regression [41], and testing of randomness [42]. In this section, we focus on the FR multivariate two sample test statistic constructed from the MST.

Assume that sample realizations from $f_0$ and $f_1$, denoted by ${\mathfrak{X}}_m\in {\mathbb{R}}^{m\times d}$ and ${\mathfrak{Y}}_n\in {\mathbb{R}}^{n\times d}$, respectively, are available. Construct an MST spanning the samples from both $f_0$ and $f_1$ and color the edges in the MST that connect dichotomous samples green and color the remaining edges black. The FR test statistic ${\mathfrak{R}}_{m,n}:={\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ is the number of green edges in the MST. Note that the test assumes a unique MST, therefore all inter point distances between data points must be distinct. We recall the following theorem from [7,8]:

Theorem 1.

As $m\to \infty$ and $n\to \infty$ such that ${\displaystyle \frac{m}{n+m}\to p}$ and ${\displaystyle \frac{n}{n+m}\to q}$,

$$\begin{array}{c}\hfill {\displaystyle 1-{\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\frac{m+n}{2mn}\to D_p(f_0,f_1),a.s.}\end{array}$$

(3)

In the next section, we obtain bounds on the MSE convergence rates of the FR approximation for HP-divergence between densities that belong to ${\Sigma }_d(\eta ,K)$, the class of Hölder continuous functions with Lipschitz constant K and smoothness parameter $0<\eta \le 1$ [32]:

Definition 1

(Hölder class). Let $\mathcal{X}\subset {\mathbb{R}}^d$ be a compact space. The Hölder class ${\Sigma }_d(\eta ,K)$, with η-Hölder parameter, of functions with the $L_d$-norm, consists of the functions g that satisfy

$$\begin{array}{c}\hfill \begin{array}{c}\left\{g:\parallel g\left(\mathbf{z}\right)-p_{\mathbf{x}}^{\lfloor \eta \rfloor }\left(\mathbf{z}\right){\parallel }_d\le K\parallel \mathbf{x}-\mathbf{z}{\parallel }_d^{\eta },\mathbf{x},\mathbf{z}\in \mathcal{X}\right\},\hfill \end{array}\end{array}$$

(4)

where $p_{\mathbf{x}}^k\left(\mathbf{z}\right)$ is the Taylor polynomial (multinomial) of g of order k expanded about the point $\mathbf{x}$ and $\lfloor \eta \rfloor$ is defined as the greatest integer strictly less than η.

In what follows, we will use both notations ${\mathfrak{R}}_{m,n}$ and ${\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ for the FR statistic over the combined samples.

2.2. Convergence Rates

In this subsection, we obtain the mean convergence rate bounds for general non-uniform Lebesgue densities $f_0$ and $f_1$ belonging to the Hölder class ${\Sigma }_d(\eta ,K)$. Since the expectation of ${\mathfrak{R}}_{m,n}$ can be closely approximated by the sum of the expectation of the FR statistic constructed on a dense partition of ${[0,1]}^d$, ${\mathfrak{R}}_{m,n}$ is a quasi-additive functional in mean. The family of bounds (A16) in Appendix B enables us to achieve the minimax convergence rate for the mean under the Hölder class assumption with smoothness parameter $0<\eta \le 1$, $d\ge 2$:

Theorem 2

(Convergence Rate of the Mean). Let $d\ge 2$, and ${\mathfrak{R}}_{m,n}$ be the FR statistic for samples drawn from Hölder continuous and bounded density functions $f_0$ and $f_1$ in ${\Sigma }_d(\eta ,K)$. Then, for $d\ge 2$,

$$\begin{array}{c}{\displaystyle \left|\frac{\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]}{m+n}-2pq\int \frac{f_0\left(\mathbf{x}\right)f_1\left(\mathbf{x}\right)}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}\right|\le O\left({(m+n)}^{-{\eta }^2/\left(d(\eta +1)\right)}\right).}\hfill \end{array}$$

(5)

This bound holds over the class of Lebesgue densities $f_0,f_1\in {\Sigma }_d(\eta ,K)$, $0<\eta \le 1$. Note that this assumption can be relaxed to $f_0\in {\Sigma }_d^s(\eta ,K_0)$ and $f_1\in {\Sigma }_d^s(\eta ,K_1)$ that is Lebesgue densities $f_0$ and $f_1$ belong to the Strong Hölder class with the same Hölder parameter $\eta$ and different constants $K_0$ and $K_1$, respectively.

The following variance bound uses the Efron–Stein inequality [43]. Note that in Theorem 3 we do not impose any strict assumptions. We only assume that the density functions are absolutely continuous and bounded with support on the unit cube ${[0,1]}^d$. Appendix C contains the proof.

Theorem 3.

The variance of the HP-integral estimator based on the FR statistic, ${\displaystyle {\mathfrak{R}}_{m,n}/(m+n)}$ is bounded by

$$\begin{array}{c}\hfill {\displaystyle Var\left(\frac{{\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)}{m+n}\right)\le \frac{32c_d^{2}q}{(m+n)},}\end{array}$$

(6)

where the constant $c_d$ depends only on d.

By combining Theorems 2 and 3, we obtain the MSE rate of the form $O\left({m+n)}^{-{\eta }^2/\left(d(\eta +1)\right)}\right)+O\left({(m+n)}^{-1}\right)$. Figure 1 indicates a heat map showing the MSE rate as a function of d and $N=m=n$. The heat map shows that the MSE rate of the FR test statistic-based estimator given in (3) is small for large sample size N.

Figure 1. Heat map of the theoretical MSE rate of the FR estimator of the HP-divergence based on Theorems 2 and 3 as a function of dimension and sample size when $N=m=n$. Note the color transition (MSE) as sample size increases for high dimension. For fixed sample size N, the MSE rate degrades in higher dimensions.

2.3. Proof Sketch of Theorem 2

In this subsection, we first establish subadditivity and superadditivity properties of the FR statistic, which will be employed to derive the MSE convergence rate bound. This will establish that the mean of the FR test statistic is a quasi-additive functional:

Theorem 4.

Let ${\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ be the number of edges that link nodes from differently labeled samples ${\mathfrak{X}}_m=\{{\mathbf{X}}_1,\cdots ,{\mathbf{X}}_m\}$ and ${\mathfrak{Y}}_n=\{{\mathbf{Y}}_1,\cdots ,{\mathbf{Y}}_n\}$ in ${[0,1]}^d$. Partition ${[0,1]}^d$ into $l^d$ equal volume subcubes $Q_i$ such that $m_i$ and $n_i$ are the number of samples from $\{{\mathbf{X}}_1,\cdots ,{\mathbf{X}}_m\}$ and $\{{\mathbf{Y}}_1,\cdots ,{\mathbf{Y}}_n\}$, respectively, that fall into the partition $Q_i$. Then, there exists a constant $c_1$ such that

$$\begin{array}{c}{\displaystyle \mathbb{E}\left[{\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\right]\le \sum _{i=1}^{l^d}\mathbb{E}\left[{\mathfrak{R}}_{m_i,n_i}\left(({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\cap Q_i\right)\right]+2c_1{l}^{d-1}{(m+n)}^{1/d}.}\hfill \end{array}$$

(7)

Here, ${\mathfrak{R}}_{m_i,n_i}$ is the number of dichotomous edges in partition $Q_i$. Conversely, for the same conditions as above on partitions $Q_i$, there exists a constant $c_2$ such that

$$\begin{array}{c}{\displaystyle \mathbb{E}\left[{\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\right]\ge \sum _{i=1}^{l^d}\mathbb{E}\left[{\mathfrak{R}}_{m_i,n_i}\left(({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\cap Q_i\right)\right]-2c_2{l}^{d-1}{(m+n)}^{1/d}.}\hfill \end{array}$$

(8)

The inequalities (7) and (8) are inspired by corresponding inequalities in [30,31]. The full proof is given in Appendix A. The key result in the proof is the inequality:

$${\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\le \sum _{i=1}^{l^d}{\mathfrak{R}}_{m_i,n_i}\left(({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\cap Q_i\right)+2|D|,$$

where $|D|$ indicates the number of all edges of the MST which intersect two different partitions.

Furthermore, we adapt the theory developed in [17,30] to derive the MSE convergence rate of the FR statistic-based estimator by defining a dual MST and dual FR statistic, denoted by ${\mathrm{MST}}^{*}$ and ${\mathfrak{R}}_{m,n}^{*}$ respectively (see Figure 2):

Figure 2. The dual MST spanning the merged set ${\mathfrak{X}}_m$ (blue points) and ${\mathfrak{Y}}_n$ (red points) drawn from two Gaussian distributions. The dual FR statistic (${\mathfrak{R}}_{m,n}^{*}$) is the number of edges in the ${\mathrm{MST}}^{*}$ (contains nodes in ${\mathfrak{X}}_m\cup {\mathfrak{Y}}_n\cup \left\{2\mathrm{corner}\mathrm{points}\right\}$) that connect samples from different color nodes and corners (denoted in green). Black edges are the non-dichotomous edges in the ${\mathrm{MST}}^{*}$.

Definition 2

(Dual MST, MST${}^{*}$ and dual FR statistic ${\mathfrak{R}}_{m,n}^{*}$). Let ${\mathbb{F}}_i$ be the set of corner points of the subsection $Q_i$ for $1\le i\le l^d$. Then, we define ${\mathrm{MST}}^{*}({\mathfrak{X}}_m\cup {\mathfrak{Y}}_n\cap Q_i)$ as the boundary MST graph of partition $Q_i$ [17], which contains ${\mathfrak{X}}_m$ and ${\mathfrak{Y}}_n$ points falling inside the section $Q_i$ and those corner points in ${\mathbb{F}}_i$ which minimize total MST length. Notice it is allowed to connect the MSTs in $Q_i$ and $Q_j$ through points strictly contained in $Q_i$ and $Q_j$ and corner points are taken into account under condition of minimizing total MST length. Another word, the dual MST can connect the points in $Q_i\cup Q_j$ by direct edges to pair to another point in $Q_i\cup Q_j$ or the corner the corner points (we assume that all corner points are connected) in order to minimize the total length. To clarify this, assume that there are two points in $Q_i\cup Q_j$, then the dual MST consists of the two edges connecting these points to the corner if they are closed to a corner point; otherwise, dual MST consists of an edge connecting one to another. Furthermore, we define ${\mathfrak{R}}_{m,n}^{*}({\mathfrak{X}}_m,{\mathfrak{Y}}_n\cap Q_i)$ as the number of edges in an ${\mathrm{MST}}^{*}$ graph connecting nodes from different samples and number of edges connecting to the corner points. Note that the edges connected to the corner nodes (regardless of the type of points) are always counted in dual FR test statistic ${\mathfrak{R}}_{m,n}^{*}$.

In Appendix B, we show that the dual FR test statistic is a quasi-additive functional in mean and ${\mathfrak{R}}_{m,n}^{*}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)\ge {\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$. This property holds true since $\mathrm{MST}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ and ${\mathrm{MST}}^{*}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ graphs can only be different in the edges connected to the corner nodes, and in ${\mathfrak{R}}^{*}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)$ we take all of the edges between these nodes and corner nodes into account.

To prove Theorem 2, we partition ${[0,1]}^d$ into $l^d$ subcubes. Then, by applying Theorem 4 and the dual MST, we derive the bias rate in terms of partition parameter l (see (A16) in Theorem A1). See Appendix B and Appendix E for details. According to (A16), for $d\ge 2$, and $l=1,2,\cdots$, the slowest rates as a function of l are $l^d{(m+n)}^{\eta /d}$ and $l^{-\eta d}$. Therefore, we obtain an l-independent bound by letting l be a function of $m+n$ that minimizes the maximum of these rates i.e.,

$$l(m+n)=arg\underset{l}{min}max\left\{l^d{(m+n)}^{-\eta /d},l^{-\eta d}\right\}.$$

The full proof of the bound in (2) is given in Appendix B.

2.4. Concentration Bounds

Another main contribution of our work in this part is to provide an exponential inequality convergence bound derived for the FR estimator of the HP-divergence. The error of this estimator can be decomposed into a bias term and a variance-like term via the triangle inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\left|{\displaystyle {\mathfrak{R}}_{m,n}-\int \frac{f_0\left(\mathbf{x}\right)f_1\left(\mathbf{x}\right)}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}}\right|\le \underset{\mathrm{variance}-\mathrm{like}\mathrm{term}}{\underbrace{\left|{\mathfrak{R}}_{m,n}-\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]\right|}}\hfill \\ +\underset{\mathrm{bias}\mathrm{term}}{\underbrace{\left|{\displaystyle \mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]-\int \frac{f_0\left(\mathbf{x}\right)f_1\left(\mathbf{x}\right)}{p{f}_0\left(\mathbf{x}\right)+q{f}_1\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}}\right|}}.\hfill \end{array}\end{array}$$

The bias bound was given in Theorem 2. Therefore, we focus on an exponential concentration bound for the variance-like term. One application of concentration bounds is to employ these bounds to compare confidence intervals on the HP-divergence measure in terms of the FR estimator. In [44,45], the authors provided an exponential inequality convergence bound for an estimator of Rény divergence for a smooth Hölder class of densities on the d-dimensional unite cube ${[0,1]}^d$. We show that if ${\mathfrak{X}}_m$ and ${\mathfrak{Y}}_n$ are the set of m and n points drawn from any two distributions $f_0$ and $f_1$, respectively, the FR criteria ${\mathfrak{R}}_{m,n}$ is tightly concentrated. Namely, we establish that, with high probability, ${\mathfrak{R}}_{m,n}$ is within

$$1-O\left({(m+n)}^{-2/d}{{ϵ}^{*}}^2\right)$$

of its expected value, where ${ϵ}^{*}$ is the solution of the following convex optimization problem:

$$\begin{array}{cccc}\hfill & \underset{ϵ\ge 0}{\min }\hfill & \hfill & {\displaystyle C_{m,n}^{\prime }\left(ϵ\right)exp\left(\frac{-{(t/\left(2ϵ\right))}^{d/(d-1)}}{(m+n)\tilde{C}}\right)}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & ϵ\ge O\left(7^{d+1}{(m+n)}^{1/d}\right),\hfill \end{array}$$

(9)

where $\tilde{C}=8{\left(4\right)}^{d/(d-1)}$ and

$$\begin{array}{c}{C}_{m,n}^{\prime }\left(ϵ\right)=8{\left(1-O\left({(m+n)}^{-2/d}{ϵ}^2\right)\right)}^{-2}.\hfill \end{array}$$

(10)

Note that, under the assumption ${(m+n)}^{1/d}\simeq 1$, $C_{m,n}^{\prime }\left(ϵ\right)$ becomes a constant depending only on $ϵ$ by $8{\left(1-(c{ϵ}^2\right)}^{-2}$, where c is a constant. This is inferred from Theorems 5 and 6 below as ${(m+n)}^{1/d}\simeq 1$. See Appendix D, specifically Lemmas A8–A12 for more detail. Indeed, we first show the concentration around the median. A median is by definition any real number $M_e$ that satisfies the inequalities ${\displaystyle P(X\le M_e)\ge 1/2}$ and ${\displaystyle P(X\ge M_e)\ge 1/2}$. To derive the concentration results, the properties of growth bounds and smoothness for ${\mathfrak{R}}_{m,n}$, given in Appendix D, are exploited.

Theorem 5

(Concentration around the median). Let $M_e$ be a median of ${\mathfrak{R}}_{m,n}$ which implies that ${\displaystyle P\left({\mathfrak{R}}_{m,n}\le M_e\right)\ge 1/2}$. Recall ${ϵ}^{*}$ from (9) then we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle P\left(|{\mathfrak{R}}_{m,n}({\mathfrak{X}}_m,{\mathfrak{Y}}_n)-M_e|\ge t\right)\le C_{m,n}^{\prime }\left({ϵ}^{*}\right)exp\left(\frac{-{(t/{ϵ}^{*})}^{d/(d-1)}}{(m+n)\tilde{C}}\right),}\hfill \end{array}\end{array}$$

(11)

where $\tilde{C}=8{\left(4\right)}^{d/(d-1)}$.

Theorem 6

(Concentration of ${\mathfrak{R}}_{m,n}$ around the mean). Let ${\mathfrak{R}}_{m,n}$ be the FR statistic. Then,

$$\begin{array}{c}P\left(|{\mathfrak{R}}_{m,n}-\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]|\ge t\right)\le C_{m,n}^{\prime }\left({ϵ}^{*}\right)exp\left({\displaystyle \frac{-{(t/\left(2{ϵ}^{*}\right))}^{d/(d-1)}}{(m+n)\tilde{C}}}\right).\hfill \end{array}$$

(12)

Here, $\tilde{C}=8{\left(4\right)}^{d/(d-1)}$ and the explicit form for $C_{m,n}^{\prime }\left({ϵ}^{*}\right)$ is given by (10) when $ϵ={ϵ}^{*}$.

See Appendix D for full proofs of Theorems 5 and 6. Here, we sketch the proofs. The proof of the concentration inequality for ${\mathfrak{R}}_{m,n}$, Theorem 6, requires involving the median $M_e$, where $P({\mathfrak{R}}_{m,n}\le M_e)\ge 1/2$, inside the probability term by using

$$|{\mathfrak{R}}_{m,n}-\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]|\le |{\mathfrak{R}}_{m,n}-M_e|+|\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]-M_e|.$$

To prove the expressions for the concentration around the median, Theorem 5, we first consider the $h^d$ uniform partitions of ${[0,1]}^d$, with edges parallel to the coordinate axes having edge lengths $h^{-1}$ and volumes $h^{-d}$. Then, by applying the Markov inequality, we show that with at least probability ${\displaystyle 1-\left({\delta }_{m,n}^h/ϵ\right)}$, where ${\delta }_{m,n}^h=O\left(h^{d-1}{(m+n)}^{1/d}\right)$, the FR statistic ${\mathfrak{R}}_{m,n}$ is subadditive with $2ϵ$ threshold. Afterward, owing to the induction method [17], the growth bound can be derived with at least probability ${\displaystyle 1-\left(h{\delta }_{m,n}^h/ϵ\right)}$. The growth bound explains that with high probability there exists a constant depending on $ϵ$ and h, $C_{ϵ,h}$, such that ${\mathfrak{R}}_{m,n}\le C_{ϵ,h}{\left(mn\right)}^{1-1/d}$. Applying the law of total probability and semi-isoperimetric inequality (A108) in Lemma A11 gives us (A35). By considering the solution to convex optimization problem (9), i.e., ${ϵ}^{*}$ and optimal $h=7$ the claimed results (11) and (12) are derived. The only constraint here is that $ϵ$ is lower bounded by a function of ${\delta }_{m,n}^h=O\left(h^{d-1}{(m+n)}^{1/d}\right)$.

Next, we provide a bound for the variance-like term with high probability at least $1-\delta$. According to the previous results, we expect that this bound depends on ${ϵ}^{*}$, d, m and n. The proof is short and is given in Appendix D.

Theorem 7

(Variance-like bound for ${\mathfrak{R}}_{m,n}$). Let ${\mathfrak{R}}_{m,n}$ be the FR statistic. With at least probability $1-\delta$, we have

$$\begin{array}{c}{\displaystyle \left|{\mathfrak{R}}_{m,n}-\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]\right|\le O\left({ϵ}^{*}{(m+n)}^{(d-1)/d}{\left(log\left(C_{m,n}^{\prime }\left({ϵ}^{*}\right)/\delta \right)\right)}^{(d-1)/d}\right).}\hfill \end{array}$$

(13)

or, equivalently,

$$\begin{array}{c}{\displaystyle \left|{\displaystyle \frac{{\mathfrak{R}}_{m,n}}{m+n}-\frac{\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]}{m+n}}\right|\le O\left({ϵ}^{*}{(m+n)}^{-1/d}{\left(log\left(C_{m,n}^{\prime }\left({ϵ}^{*}\right)/\delta \right)\right)}^{(d-1)/d}\right),}\hfill \end{array}$$

(14)

where $C_{m,n}^{\prime }\left({ϵ}^{*}\right)$ depends on $m,n$, and d is given in (10) when $ϵ={ϵ}^{*}$.

3. Numerical Experiments

3.1. Simulation Study

In this section, we apply the FR statistic estimate of the HP-divergence to both simulated and real data sets. We present results of a simulation study that evaluates the proposed bound on the MSE. We numerically validate the theory stated in Section 2.2 and Section 2.4 using multiple simulations. In the first set of simulations, we consider two multivariate Normal random vectors $\mathbf{X}$, $\mathbf{Y}$ and perform three experiments $d=2,4,8$, to analyze the FR test statistic-based estimator performance as the sample sizes m, n increase. For the three dimensions $d=2,4,8$, we generate samples from two normal distributions with identity covariance and shifted means: ${\mu }_1=[0,0]$, ${\mu }_2=[1,0]$ and ${\mu }_1=[0,0,0,0]$, ${\mu }_2=[1,0,0,0]$ and ${\mu }_1=[0,0,\dots ,0]$, ${\mu }_2=[1,0,\dots ,0]$ when $d=2$, $d=4$ and $d=8$, respectively. For all of the following experiments, the sample sizes for each class are equal ($m=n$).

We vary $N=m=n$ up to 800. From Figure 3, we deduce that, when the sample size increases, the MSE decreases such that for higher dimensions the rate is slower. Furthermore, we compare the experiments with the theory in Figure 3. Our theory generally matches the experimental results. However, the MSE for the experiments tends to decrease to zero faster than the theoretical bound. Since the Gaussian distribution has a smooth density, this suggests that a tighter bound on the MSE may be possible by imposing stricter assumptions on the density smoothness as in [12].

Figure 3. Comparison of the bound on the MSE theory and experiments for $d=2,4,8$ standard Gaussian random vectors versus sample size from 100 trials.

In our next simulation, we compare three bivariate cases: first, we generate samples from a standard Normal distribution. Second, we consider a distinct smooth class of distributions i.e., binomial Gamma density with standard parameters and dependency coefficient $\rho =0.5$. Third, we generate samples from Standard t-student distributions. Our goal in this experiment is to compare the MSE of the HP-divergence estimator between two identical distributions, $f_0=f_1$, when $f_0$ is one of the Gamma, Normal, and t-student density function. In Figure 4, we observe that the MSE decreases as N increases for all three distributions.

Figure 4. Comparison of experimentally predicted MSE of the FR-statistic as a function of sample size $m=n$ in various distributions Standard Normal, Gamma (${\alpha }_1={\alpha }_2=1,{\beta }_1={\beta }_2=1,\rho =0.5$) and Standard t-Student.

3.2. Real Datasets

We now show the results of applying the FR test statistic to estimate the HP-divergence using three different real datasets [46]:

• Human Activity Recognition (HAR), Wearable Computing, Classification of Body Postures and Movements (PUC-Rio): This dataset contains five classes (sitting-down, standing-up, standing, walking, and sitting) collected on eight hours of activities of four healthy subjects.
• Skin Segmentation dataset (SKIN): The skin dataset is collected by randomly sampling B,G,R values from face images of various age groups (young, middle, and old), race groups (white, black, and asian), and genders obtained from the FERET and PAL databases [47].
• Sensorless Drive Diagnosis (ENGIN) dataset: In this dataset, features are extracted from electric current drive signals. The drive has intact and defective components. The dataset contains 11 different classes with different conditions. Each condition has been measured several times under 12 different operating conditions, e.g., different speeds, load moments, and load forces.

We focus on two classes from each of the HAR, SKIN, and ENGIN datasets, specifically, for HAR dataset two classes “sitting” and “standing” and for SKIN dataset the classes “Skin” and “Non-skin” are considered. In the ENGIN dataset, the drive has intact and defective components, which results in 11 different classes with different conditions. We choose conditions 1 and 2.

In the first experiment, we computed the HP-divergence using KDE plug-in estimator and then the MSE for the FR test statistic estimator is derived as the sample size $N=m=n$ increases. We used 95% confidence interval as the error bars. We observe in Figure 5 that the estimated HP-divergence ranges in $[0,1]$, which is one of the HP-divergence properties [8]. Interestingly, when N increases the HP-divergence tends to 1 for all HAR, SKIN, and ENGIN datasets. Note that in this set of experiments we have repeated the experiments on independent parts of the datasets to obtain the error bars. Figure 6 shows that the MSE expectedly decreases as the sample size grows for all three datasets. Here, we have used the KDE plug-in estimator [12], implemented on the all available samples, to determine the true HP-divergence. Furthermore, according to Figure 6, the FR test statistic-based estimator suggests that the Bayes error rate is larger for the SKIN dataset compared to the HAR and ENGIN datasets.

Figure 5. HP-divergence vs. sample size for three real datasets HAR, SKIN, and ENGIN.

Figure 6. The empirical MSE vs. sample size. The empirical MSE of the FR estimator for all three datasets HAR, SKIN, and ENGIN decreases for larger sample size N.

In our next experiment, we add the first six features (dimensions) in order to our datasets and evaluate the FR test statistic’s performance as the HP-divergence estimator. Surprisingly, the estimated HP-divergence doesn’t change for the HAR sample; however, big changes are observed for the SKIN and ENGIN samples (see Figure 7).

Figure 7. HP-divergence vs. dimension for three datasets HAR, SKIN, and ENGIN.

Finally, we apply the concentration bounds on the FR test statistic (i.e., Theorems 6 and 7) and compute theoretical implicit variance-like bound for the FR criteria with $\delta =0.05$ error for the real datasets ENGIN, HAR, and SKIN. Since datasets ENGIN, HAR, and SKIN have the equal total sample size $N=m+n=1200$ and different dimensions $d=14,12,4$, respectively; here, we first intend to compare the concentration bound (13) on the FR statistic in terms of dimension d when $\delta =0.05$. For real datasets ENGIN, HAR, and SKIN, we obtain

$$P\left(|{\mathfrak{R}}_{m,n}-\mathbb{E}\left[{\mathfrak{R}}_{m,n}\right]|\le \xi \right)\ge 0.95,$$

where $\xi ={\xi }^{\prime }[0.257,0.005,0.6\times {10}^{-11}]$, respectively, and ${\xi }^{\prime }$ is a constant not dependent on d. One observes that as the dimension decreases the interval becomes significantly tighter. However, this could not be generally correct and computing bound (13) precisely requires the knowledge of distributions and unknown constants. In Table 1, we compute the standard variance-like bound by applying the percentiles technique and observe that the bound threshold is not monotonic in terms of dimension d. Table 1 shows the FR test statistic, HP-divergence estimate (denoted by ${\mathfrak{R}}_{m,n}$, ${\widehat{D}}_p$, respectively), and standard variance-like interval for the FR statistic using the three real datasets HAR, SKIN, and ENGIN.

Table 1. ${\mathfrak{R}}_{m,n}$, ${\widehat{D}}_p$, m, and n are the FR test statistic, HP-divergence estimates using ${\mathfrak{R}}_{m,n}$, and sample sizes for two classes, respectively.

FR Test Statistic
Dataset	$\mathbb{E}\left[{\mathfrak{R}}_{\mathit{m},\mathit{n}}\right]$	${\widehat{\mathit{D}}}_{\mathit{p}}$	$\mathit{m}$	$\mathit{n}$	Variance-Like Interval
HAR	3	0.995	600	600	(2.994,3.006)
SKIN	4.2	0.993	600	600	(4.196,4.204)
ENGIN	1.8	0.997	600	600	(1.798,1.802)

4. Conclusions

We derived a bound on the MSE convergence rate for the Friedman–Rafsky estimator of the Henze–Penrose divergence assuming the densities are sufficiently smooth. We employed a partitioning strategy to derive the bias rate which depends on the number of partitions, the sample size $m+n$, the Hölder smoothness parameter $\eta$, and the dimension d. However, by using the optimal partition number, we derived the MSE convergence rate only in terms of $m+n$, $\eta$, and d. We validated our proposed MSE convergence rate using simulations and illustrated the approach for the meta-learning problem of estimating the HP-divergence for three real-world data sets. We also provided concentration bounds around the median and mean of the estimator. These bounds explicitly provide the rate that the FR statistic approaches its median/mean with high probability, not only as a function of the number of samples, m, n, but also in terms of the dimension of the space d. By using these results, we explored the asymptotic behavior of a variance-like rate in terms of m, n, and d.