# Weighted Riesz Particles

Xiongming Dai
Division of Computer Science and Engineering
Louisiana State University
Baton Rouge,LA70803, USA
{xdai2}@email
&Gerald Baumgartner
Division of Computer Science and Engineering
Louisiana State University
Baton Rouge,LA70803, USA
{gb}@email

###### Abstract

Markov chain Monte Carlo (MCMC) methods are simulated by local exploration of
complex statistical distributions, and while bypassing the cumbersome
requirement of a specific analytical expression for the target, this
stochastic exploration of an uncertain parameter space comes at the expense of
a large number of samples, and this computational complexity increases with
parameter dimensionality. Although at the exploration level, some methods are
proposed to accelerate the convergence of the algorithm, such as tempering,
Hamiltonian Monte Carlo, Rao-redwellization, and scalable methods for better
performance, it cannot avoid the stochastic nature of this exploration. We
consider the target distribution as a mapping where the infinite-dimensional
Eulerian space of the parameters consists of a number of deterministic
submanifolds and propose a generalized energy metric, termed weighted Riesz
energy, where a number of points is generated through pairwise interactions,
to discretize rectifiable submanifolds. We study the properties of the point,
called Riesz particle, and embed it into sequential MCMC, and we find that
there will be higher acceptance rates with fewer evaluations, we validate it
through experimental comparative analysis from a linear Gaussian state-space
model with synthetic data and a non-linear stochastic volatility model with
real-world data.

_K_ eywords Markov chain Monte Carlo $\cdot$ Riesz

##  1 Introduction

Markov chain Monte Carlo integration methods provide a feasible way for
Bayesian analysis [1, 2, 3, 4], which requires the evaluation of complex and
often high-dimensional integrals to obtain posterior distributions for the
unobserved latent quantities of interest in the model. The advantage of this
method is that it guarantees convergence to the target distribution after a
"burn-in" period [5]. This robustness may, however, lead to very low
convergence rates in that the exploration of the uncertain space, meaning some
part of the space supporting the target distribution that has a great
probability mass under that distribution may take a long time, as the
simulation usually traverses by local jumps in the vicinity of the current
position [6].

Many discretization techniques in numerical analysis are based on finite
samples that adequately cover the underlying space [7]. Quasi-Monte Carlo
design was studied for compact smooth Riemannian manifolds [8]. Points on a
design space that derives from energy-based functions often have desirable
separation, the worse-case error of integration bounds the covering radius and
it provides asymptotically optimal covering radii[9, 10, 11, 12, 13]. This
yields equilibrium points that are useful in a variety of applications,
especially in high-dimensional sampling. We consider the target distribution
as a mapping where the infinite-dimensional Eulerian space of parameters
consists of several deterministic submanifolds. We propose a new flexible
minimum energy metric to discretize rectifiable submanifolds via particle
interaction. We study the characterization of deterministic points (called
weighted Riesz particles) and embed them into sequential MCMC, and we find
that there will be higher acceptance rates with fewer evaluations, we validate
our assertion through experimental comparative analysis from a linear Gaussian
state-space model with synthetic data and a non-linear stochastic volatility
model with real-world data.

In this paper, we focus on analyzing our novel energy criterion and the
characterization of particles generated based on this criterion, as well as on
how to improve the acceptance rate of the MCMC with fewer evaluations. We
present an efficient algorithm for deterministically sampling the target
distribution with weighted Riesz energy minimization, from which particles
sparsely represent integrable geometric manifolds, with a small number of
samples to approximate the target posterior distribution.

In Section 2, we briefly introduced the minimum energy model. Here, we propose
a new generalized energy criterion and focus on asymptotic behavior,
separation and coverage radius. In Section 3, we present a novel sampler with
weighted Riesz particles, where the discretized deterministic submanifold
inherits a special representation of the sampled space. We then describe the
procedure for sequentially sampling weighted Riesz particles and embedding
them in the Metropolis-Hastings algorithm for the hidden Markov model. In
Section 4, we validate the algorithm through tracking the stochastic
volatility and present its performance and error analysis. Section 5
summarizes our contributions.

##  2 Weighted Riesz Energy Criterion

In this section, we introduce the main idea of discrete minimum energies on
rectifiable high-dimensional manifolds and propose a generalized energy
criterion. Then, we study the asymptotic behavior of the corresponding
configurations and representations of weighted Ritz particles in terms of
separation and covering radii.

###  2.1 Discrete Weighted Riesz Energy

Let $\mathfrak{C}$ denote a compact set in $\mathbb{R}^{d}$ whose
$d$-dimensional Borel measure,
$\mathbb{B}_{d}(\mathfrak{C})\subset(\mathfrak{C},\mathbb{R}^{d})$, is finite,
and $h$ denote a bi-Lipschitz mapping from $\mathfrak{C}\times\mathfrak{C}$ to
$\mathbb{R}^{d}$, for a collection of $n(\geq 2)$ distinct points of
configuration in $\mathfrak{C}$, let $X_{1:n}=\\{x_{1},...,x_{n}\\}$, we
define the energy of $X_{1:n}$ to be

${}E(X_{1:n}):=\sum_{i=1}^{n}\sum_{j=1,j\neq i}^{n}h(x_{i},x_{j})=\sum_{i\neq
j}h(x_{i},x_{j}),$ (1)

and let

${}\mathcal{E}(\mathfrak{C},n):=\text{inf}\\{E(X_{1:n}):X_{1:n}\subset\mathfrak{C},\left|X_{1:n}\right|=n\\}$
(2)

be the minimal discrete $n$-point energy of the configuration in
$\mathfrak{C}$, where $\left|X_{1:n}\right|$ represents the cardinality of the
set $X_{1:n}$. (I) For $h(x_{i},x_{j})=-\text{log}\parallel
x_{i}-x_{j}\parallel$, it was first proposed by M.Fekete who explored the
connection between discretized manifolds and polynomial interpolation[14]. In
computational complexity theory, Smale [15] proposed the $7th$ problem in his
list of "Mathematical problems for the next century", which is how to design a
polynomial time algorithm for generating “nearly" optimal logarithmic energy
points $X_{1:n}^{*}$, also called Fekete points, on the unit sphere in
$\mathbb{R}^{3}$ that satisfy
$E(X_{1:n}^{*})-\mathcal{E}(\mathbb{S}^{2},n)\leq{{C_{1}}}\cdot\log n$111In
this paper $C_{i}$ $\in\mathbb{R}^{+},i=1,2,...$denote different constants.
for some universal constant
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{1}}$ ;
(II) when $h(x_{i},x_{j})=\frac{1}{\parallel
x_{i}-x_{j}\parallel^{s}},s\in\mathbb{R}^{+}$, let
$\mathcal{E}_{s}(\mathfrak{C},n)$ denote the Riez $s$-energy, by Taylor’s
formula, for any $s\in(0,+\infty)$, we have

${}\begin{split}\lim_{s\rightarrow
0^{+}}\mathcal{E}_{s}(\mathfrak{C},n)=\lim_{s\rightarrow
0^{+}}\frac{n(n-1)+s\mathcal{E}_{\text{log}}(\mathfrak{C},n)+\mathcal{O}(s)}{s}=\mathcal{E}_{\text{log}}(\mathfrak{C},n).\end{split}$
(3)

Consequently, the Fekete points set $X_{1:n}^{(s)}$ can be considered as
limiting cases of point sets that minimize the discrete Riez $s$-energy, which
is widely used to discretize manifolds via particle interactions in Euclidean
space [12, 16].

From the point of view of statistical high-dimensional sampling, we consider a
sufficiently large $d$ and propose the minimally weighted Riesz energy
criterion of

$\displaystyle\begin{split}&\mathcal{E}_{\beta}(\mathfrak{C},n)=\min_{x_{i},x_{j}}\left\\{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{\omega(x_{i},x_{j})}{\parallel
x_{i}-x_{j}\parallel^{s}}\right\\}^{\frac{1}{s}},\omega(x_{i},x_{j})\propto
e^{\left[\alpha\cdot\kappa(x_{i})\kappa(x_{j})+\beta\cdot\parallel
x_{i}-x_{j}\parallel\right]^{-\frac{s}{2d}}}.\\\ \end{split}$ (4)

When $s\rightarrow\infty$, the formulation (4) is convex, the denominator for
$\mathcal{E}_{\beta}(\mathfrak{C},n)$ approximates $\parallel
x_{i}-x_{j}\parallel$, thus, it inherits the properties of Riesz energy,
termed as weighted Riesz energy criterion. To obtain a finite collection of
point sets distributed according to a specified non-uniform density, e.g.
points that can be used as weighted integrals or for the design of complex
surfaces, where more points are needed in regions of higher curvature, we
introduce $\omega(x_{i},x_{j})$ in (4), where $\kappa(x)\propto-\ln{f(x)}$,
$\parallel x_{i}-x_{j}\parallel$ is involved to ensure that is locally bounded
for $\alpha=-1$, $\beta$ is a local discrepancy coefficient to balance off the
local conflict with the distributed points when short-range interactions
between points are the dominator. Here, $\beta>0$. Thus, given a proper
distribution $f(x)$, we can use $\mathcal{E}_{\beta}(\mathfrak{C},n)$ to
generate a sequence of $n$-point configurations that are "well-separated" and
have asymptotic distribution $f(x)$.

Our weighted Riesz energy $\mathcal{E}_{\beta}(\mathfrak{C},n)$ is continuous
and derivable with respect to the parameter $\beta\subset\mathbb{R}$ from (4),
it provides a more flexible and versatile framework when we discretize the
submanifolds via particle interactions.

###  2.2 Asymptotics for Extremal Weighted Riesz Energy Criterion

Properties of $\mathfrak{C}(x_{i},x_{j})$ . (I) $\omega(x_{i},x_{j})$ is
continuous as a function of $\kappa(x)\propto-\ln{f(x)}$ when
$\exists\beta_{0}>0$ satisfying $\beta\leq\beta_{0}$; it is a positive
constant when $\exists\beta_{1}>0$ satisfying $\beta\geq\beta_{1}$; (II) There
exists a neighborhood set $\mathfrak{C}^{\prime}$, where
$x_{i}^{\prime},x_{j}^{\prime}\in\mathfrak{C}^{\prime},\omega(x_{i}^{\prime},x_{j}^{\prime})$
is bounded and larger than zero; (III) $\omega(x_{i},x_{j})$ is bounded on any
closed and compact metric space $\mathfrak{C}$.

Assume the compact set $\mathfrak{C}\subset\mathbb{R}^{d}$, for high dimension
$s>d$, we define the generalized Borel measure on sets
$\mathbb{S}\subset\mathfrak{C}$ with
$\mathcal{U}_{d}^{s}(\mathbb{S}):=\int_{\mathbb{S}}\omega(x_{i},x_{j})d\mathcal{U}_{d}(x).$
It is bounded and the corresponding normalized form:
$u_{d}^{s}(\mathbb{S}):=\mathcal{U}_{d}^{s}(\mathbb{S})/\mathcal{U}_{d}^{s}({\mathfrak{C}}).$

Measure Metric. Consider a Euclidean space $\mathbb{R}^{d},d\geq 2$, for
$s>d$, let $\mu(\sigma\text{-algebra}):=\cup_{d=2}^{\infty}\\{\parallel
x_{i}-x_{j}\parallel^{d}\\}$, represent a Borel measure from the
$\sigma$-algebra on $\mathfrak{C}$, a measure $\phi$ in $\mathfrak{C}_{i}$ is
a non-negative $\sigma$-algebra set function defined on
$\mu(\sigma\text{-algebra})$ and finite on all compact sets
$\mathfrak{C}_{i}\subset\mathfrak{C},i\in[1,n]$. If $\phi<\infty$, then the
measure $\phi$ is called finite. Generally, for the smallest $\sigma$-algebra,
containing all compact subsets of $\mathfrak{C}_{i}$.

According to the measure theoretics [17], we have the following novel version
of the Poppy-Seed Bagel Theorem [18] for weighted Riesz energy.
Theorem 2.2.1.  Given a distribution $f(x)$ with respect to $d$-rectifiable
set $\mathfrak{C}$ embedded in Euclidean space, $\omega(x_{i},x_{j})>0$ is
bounded and continuous on the closed Borel sets
$\mathbb{S}\subset\mathfrak{C}\times\mathfrak{C}$, for $s>d$, the minimal
weighted Riesz energy configuration on $\mathfrak{C}$ from
$\mathcal{E}_{\beta}(\mathfrak{C},n)$ where the $n$-point interacts via the
$h_{\beta}(x_{i},x_{j})$ potential, have

$\lim_{n\rightarrow\infty}\frac{\mathcal{E}_{\beta}(\mathfrak{C},n)}{n^{\frac{2}{s}+\frac{1}{d}}}=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{2}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}}.$
(5)

Moreover, if $\mathcal{U}_{d}^{s}(\mathbb{S})>0$, any configuration
$X_{1:n},n>1$ generated by asymptotically minimizing weighted Riesz energy
$\mathcal{E}_{\beta}(\mathfrak{C},n)$ is uniformly distributed with respect to
$\mathcal{U}_{d}$, that is,

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1,i\neq j}^{n}\parallel
x_{i}-x_{j}\parallel=u_{d}^{s}(\mathbb{S}).$ (6)

Proof of Theorem 2.2.1. We divide the proof of Theorem 2.2.1 into two parts;
The proof works via induction with Lemma 2.2.2 for (5)  in the main text, and
Lemmas 2.2.3, 2.2.4, 2.2.5, 2.2.6 and 2.2.7 for (6)  in the main text.

Lemma 2.2.2. Given a distribution $f(x)$ with respect to $d$-rectifiable set
$\mathfrak{C}$ embedded in Euclidean space, $\omega(x_{i},x_{j})>0$ is bounded
and continuous on the closed Borel sets
$\mathbb{S}\subset\mathfrak{C}\times\mathfrak{C}$, for $s>d$ and
$\beta\in(0,\beta_{0}]\cup[\beta_{1},+\infty),\beta_{0},\beta_{1},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{3}}\in\mathbb{R}$,
the minimal weighted Riesz energy configuration on $\mathfrak{C}$ from
$\mathcal{E}_{\beta}(\mathfrak{C},n)$ where the $n$-point interacts via the
$h_{\beta}(x_{i},x_{j})$ potential, have

$\lim_{\begin{subarray}{c}\mathllap{n}\to\infty\\\
\mathllap{\beta^{-}}\to\beta_{0}\end{subarray}}\frac{\mathcal{E}_{\beta}(\mathfrak{C},n)}{n^{\frac{2}{s}+\frac{1}{d}}}=\frac{{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{3}}}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}},\
\ \lim_{\begin{subarray}{c}\mathllap{n}\to\infty\\\
\mathllap{\beta^{+}}\to\beta_{1}\end{subarray}}\frac{\mathcal{E}_{\beta}(\mathfrak{C},n)}{n^{\frac{2}{s}+\frac{1}{d}}}=\frac{{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{3}}}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}}.$
(7)

Proof $\mathcal{E}_{\beta}(\mathfrak{C},n)$ is strictly decreasing as $\beta$
increases, this monotonicity makes it possible to analyze the asymptotics and
extend it into high-dimensional sampling on the compact set $\mathfrak{C}$
under mild assumptions. Let
$J(\beta^{\prime}):=\lim_{n\rightarrow\infty}\lim_{\beta\rightarrow\beta^{\prime}}\mathcal{E}_{\beta}(\mathfrak{C},n)$,

${}J(\beta):=\sum_{i=1}^{n}\sum_{j=1,j\neq
i}^{n}\left[h_{\beta}(x_{i},x_{j})\right]^{\frac{1}{s}}.$

$h_{\beta}(x_{i},x_{j})$ is also strictly decreasing as $\beta$ increases, we
firstly focus on
$\beta\in(0,\beta_{0}]\cup[\beta_{1},+\infty),\beta_{0},\beta_{1}\in\mathbb{R}^{+}$
, then relax this assumption later, define

$h_{\beta^{\prime}}(x_{i},x_{j}):=\lim_{\beta\rightarrow\beta^{\prime}}\frac{\omega(x_{i},x_{j})}{\parallel
x_{i}-x_{j}\parallel^{s}},\omega(x_{i},x_{j})>0,$

if $\beta\leq\beta_{0}$ is sufficiently small such that

$\kappa(x_{i})\kappa(x_{j})\gg\beta_{0}\cdot\parallel x_{i}-x_{j}\parallel,$
(8)

then,

$h_{\beta_{0}^{-}}(x_{i},x_{j}):=\lim_{\beta\rightarrow\beta_{0}^{-}}h_{\beta}(x_{i},x_{j})=\frac{e^{\left[-\kappa(x_{i})\kappa(x_{j})\right]^{-\frac{s}{2d}}}}{\parallel
x_{i}-x_{j}\parallel^{s}}.$ (9)

From Taylor’s theorem

$e^{z}=1+z+\frac{z^{2}}{2!}+\cdots+\frac{z^{k^{\prime}}}{k^{\prime}!},k^{\prime}\to\infty,z\in\mathbb{R}.$

Let $z={\left[-\kappa(x_{i})\kappa(x_{j})\right]^{-\frac{s}{2d}}}$, substitute
(8) into (9),

$\begin{split}h_{\beta_{0}^{-}}(x_{i},x_{j})&=\frac{1+z+\frac{z^{2}}{2!}+\cdots+\frac{z^{k^{\prime}}}{k^{\prime}!}}{\parallel
x_{i}-x_{j}\parallel^{s}}\geq\frac{1}{\parallel
x_{i}-x_{j}\parallel^{s}}+\frac{\left[-\beta_{0}\cdot\parallel
x_{i}-x_{j}\parallel\right]^{-\frac{s}{2d}}}{\parallel
x_{i}-x_{j}\parallel^{s}}+\\\ &...+\frac{\left[-\beta_{0}\cdot\parallel
x_{i}-x_{j}\parallel\right]^{\frac{-{s{k^{\prime}}}}{2d}}}{{k^{\prime}}!\parallel
x_{i}-x_{j}\parallel^{s}}.\end{split}$

For $s>d$, the right-hand side terms belong to the classical Riesz-kernel
model, from the Poppy-Seed Bagel Theorem [18], there exists a
${{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{4}}}$,

$h_{\beta_{0}^{-}}(x_{i},x_{j})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{4}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{s}{d}}}\cdot
n^{1+\frac{s}{d}}\cdot n.$

Thus,

$J(\beta_{0}^{-}):=\sum_{i=1}^{n}\sum_{j=1,j\neq
i}^{n}\left[h_{\beta_{0}^{-}}(x_{i},x_{j})\right]^{\frac{1}{s}}=\frac{{{C_{8}}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}}\cdot
n^{\frac{1}{s}+\frac{1}{d}}\cdot n^{\frac{1}{s}}.$

Similarly, if $\beta\geq\beta_{1}$ is sufficiently large such that
$\kappa(x_{i})\kappa(x_{j})\ll\beta_{1}\cdot\parallel x_{i}-x_{j}\parallel$,
then

$\displaystyle\begin{split}h_{\beta_{1}^{+}}(x_{i},x_{j}):=&\lim_{\beta\rightarrow\beta_{1}^{+}}h_{\beta}(x_{i},x_{j})=\frac{e^{\left[\beta_{1}\parallel
x_{i}-x_{j}\parallel\right]^{-\frac{s}{2d}}}}{\parallel
x_{i}-x_{j}\parallel^{s}}=\frac{1}{\parallel x_{i}-x_{j}\parallel^{s}}+\\\
&\frac{\left[\beta_{1}\cdot\parallel
x_{i}-x_{j}\parallel\right]^{-\frac{s}{2d}}}{\parallel
x_{i}-x_{j}\parallel^{s}}+...+\frac{\left[\beta_{1}\cdot\parallel
x_{i}-x_{j}\parallel\right]^{\frac{-{s{k^{\prime}}}}{2d}}}{{k^{\prime}}!\parallel
x_{i}-x_{j}\parallel^{s}}.\end{split}$ (10)

It provides a flexible framework to prove the asymptotics of the proposed
weighted Riesz energy criterion for (10) that we will frequently refer to it
for the following lemma and related proof.

For $s>d$, the right-hand side terms belong to the classical Riesz-kernel
model, from the Poppy-Seed Bagel Theorem [18], there exists a
${{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{5}}}$,

$h_{\beta_{1}^{+}}(x_{i},x_{j})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{5}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{s}{d}}}\cdot
n^{1+\frac{s}{d}}\cdot n.$

Thus,

$J(\beta_{1}^{+}):=\sum_{i=1}^{n}\sum_{j=1,j\neq
i}^{n}\left[h_{\beta_{1}^{+}}(x_{i},x_{j})\right]^{\frac{1}{s}}=\frac{{{C_{7}}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}}\cdot
n^{\frac{1}{s}+\frac{1}{d}}\cdot n^{\frac{1}{s}}.$

As $J({\beta})$ is strictly decreasing, and continuous and derivative for
$\beta\in\mathbb{R}^{+}$, Consequently, There exists a
${{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{3}}}$,

$\lim_{n\rightarrow\infty}\frac{\mathcal{E}_{\beta}(\mathfrak{C},n)}{n^{\frac{2}{s}+\frac{1}{d}}}=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{3}}}{[\mathcal{U}_{d}^{s}(\mathbb{S})]^{\frac{1}{d}}}.$

Thus, (7) holds.

From Lemma 2.2.2, as $n\to\infty$, the approximation of
$\mathcal{E}_{\beta}(\mathfrak{C},n)$ is not correlated with $\beta$. That is,
we are assuming that $\beta$ approximates a specific real value, and for the
convenience of introducing Taylor’s theorem to derive, it does not affect the
final limit value of $\mathcal{E}_{\beta}(\mathfrak{C},n)$ for $n\to\infty$.

Analogous to the proof of classical Poppy-Seed Bagel Theorem [18], we define

$\mathcal{T}(d):=2+\frac{s}{d},n\geq 2.$

$\lambda(n):=n^{\mathcal{T}(d)}$, for $n\geq 2$, $\lambda(1):=1$. And define

$\psi_{s,d}(\mathfrak{C}):=\lim_{n\rightarrow\infty}\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C},n)}{\lambda(n)},$
(11)

let
$\psi_{s,d}^{\text{inf}}(\mathfrak{C})={\text{inf}}(\psi_{s,d}(\mathfrak{C}))$,
$\psi_{s,d}^{\text{sup}}(\mathfrak{C})={\text{sup}}(\psi_{s,d}(\mathfrak{C}))$
and decompose the $d$-rectifiable set $\mathfrak{C}$ into different subsets
$\mathfrak{C}_{i},i\in\mathbb{R}^{+}$, satisfying
$\cup_{i=1}^{\infty}{\mathfrak{C}_{i}}=\mathfrak{C}$.
Lemma 2.2.3. [18] $\exists\alpha_{1},\alpha_{2}\in\mathbb{R}^{+}$,
$\mathcal{T}(d)$ is continuous and derivative for $d\in\mathbb{R}^{+}$, the
function
$U(t)=\alpha_{1}t^{\mathcal{T}(d)-1}+\alpha_{2}(1-t)^{\mathcal{T}(d)-1}$ has
the minimum for $t\in[0,1]$ where occurs at the points
$t^{*}:=\frac{1}{1+(\frac{\alpha_{1}}{\alpha_{2}})^{\frac{1}{\mathcal{T}(d)-2}}}$
with
$U(t^{*})=\left[\alpha_{2}^{\frac{-1}{\mathcal{T}(d)-2}}+\alpha_{1}^{\frac{-1}{\mathcal{T}(d)-2}}\right]^{2-\mathcal{T}(d)}$.
The proof is straightforward from the first order derivative of the function
$\frac{dU(t)}{dt}=0$.

We will introduce the subadditivity and superadditivity properties as follows.

Lemma 2.2.4. $\exists\mathfrak{C}_{j},\mathfrak{C}_{k}\subset\mathfrak{C}$,
and $\mathfrak{C}_{j},\mathfrak{C}_{k}\not\subset\emptyset$, $j\neq k$,
$\mathcal{T}(d)$ is continuous and derivative for $d\in\mathbb{R}^{+}$, let
${X_{1:n}^{(i)}},i,j$ and $k\in\mathbb{R}^{+}$ be an infinite sequence of
$n$-point configurations on $\mathfrak{C}_{j}\cup\mathfrak{C}_{k}$ and define
the unit $x_{1:n}^{(i)}$ that belongs to the compact and closed subset
$\mathfrak{C}_{j}$ with probability

${p_{j}}:=P(x_{1:n}^{(i)}\in\mathfrak{C}_{j}).$

Assume that both $\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j})$ and
$\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{k})$ are bounded, for $s>d$, we have

$\lim_{n\rightarrow\infty}{\text{inf}}\left[\frac{\mathcal{E}^{s}_{\beta}({X_{1:n}})}{n^{\mathcal{T}(d)}}\right]\geq
U(p_{j}),$ (12)

where
$U(p_{j}):=\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j})p_{j}^{\mathcal{T}(d)-1}+\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{k})(1-p_{j})^{\mathcal{T}(d)-1}$.
Proof we follow an argument in [18] and define $N_{j}:=\left\lfloor p_{j}\cdot
n\right\rfloor$ the units that belong to $\mathfrak{C}_{j}$, for
$\mathfrak{C}_{k}$, $N_{k}:=\left\lfloor(1-p_{j})\cdot n\right\rfloor$. Here,
we assume $n\rightarrow\infty$,

$\begin{split}E^{s}_{\beta}({X_{1:n}},n)&\geq
E^{s}_{\beta}({X_{1:n}^{(i)}}\cap\mathfrak{C}_{j},N_{j})+E^{s}_{\beta}({X_{1:n}^{(i)}}\setminus{\mathfrak{C}_{j}},N_{k})\geq\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j},N_{j})+\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k}).\end{split}$

We introduce the decomposed units into $\mathfrak{C}_{j}$ and
$\mathfrak{C}_{k}$, and combine (11), let
$\psi_{s,d}({X_{1:n}}):=\mathcal{E}^{s}_{\beta}({X_{1:n}},n)/n^{\mathcal{T}(d)}$,
then

$\begin{split}\lim_{n\rightarrow\infty}{\text{inf}}\left[\psi_{s,d}({X_{1:n}})\right]&\geq\lim_{n\rightarrow\infty}{\text{inf}}\left[\psi_{s,d}(\mathfrak{C}_{j})\cdot(\frac{N_{j}}{n})^{\mathcal{T}(d)}\right]+\lim_{n\rightarrow\infty}{\text{inf}}\left[\psi_{s,d}(\mathfrak{C}_{k})\cdot(\frac{N_{k}}{n})^{\mathcal{T}(d)}\right]\\\
&\geq\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j})p_{j}^{\mathcal{T}(d)-1}+\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{k})(1-p_{j})^{\mathcal{T}(d)-1}=U(p_{j}).\end{split}$

If $p_{j}=0$, we have $N_{k}\rightarrow n$. Since

$E^{s}_{\beta}({X_{1:n}},n)\geq
E^{s}_{\beta}({X_{1:n}}\setminus\mathfrak{C}_{j})\geq\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k}),$

motivated by Lemma 2.2.3, we have

$\begin{split}\lim_{n\rightarrow\infty}{\text{inf}}\left[\frac{E^{s}_{\beta}(X_{1:n})}{n^{\mathcal{T}(d)}}\right]&\geq\lim_{n\rightarrow\infty}{\text{inf}}\left[\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k})}{n^{\mathcal{T}(d)}}\cdot(\frac{N_{k}}{n})^{\mathcal{T}(d)}\right]\geq\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{k})=U(0).\end{split}$

Similarly, for $p_{j}=1$,

$\lim_{n\rightarrow\infty}{\text{inf}}\left[\frac{E^{s}_{\beta}(X_{1:n})}{n^{\mathcal{T}(d)}}\right]\geq
U(1).$

Thus, (12) holds.
Lemma 2.2.5. $\exists\mathfrak{C}_{j},\mathfrak{C}_{k}\subset\mathfrak{C}$,
and $\mathfrak{C}_{j},\mathfrak{C}_{k}\not\subset\emptyset$, $j\neq k$,
$\mathcal{T}(d)>2$ is continuous and derivative for $d\in\mathbb{R}^{+}$, let
$\alpha_{3}=\frac{1}{2-\mathcal{T}(d)}$, for $s>d$,

$\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k})^{\alpha_{3}}\geq\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})^{\alpha_{3}}.$
(13)

Proof If $\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})$ or
$\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})$ equals zero, or one of the
quantities $\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})$ or
$\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})$ approximates infinite, as the size
of set increase, $\mathcal{E}_{\beta}^{s}(\mathfrak{C},n)$ will increase, the
lemma holds.

Hereafter, we follow an argument in [18] and consider the general case of
$\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})\in(0,\infty),\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})\in(0,\infty)$,
the distance of two set is defined with
$r:=\left\|a_{i}-b_{j}\right\|,a_{i}\in\mathfrak{C}_{j},b_{j}\in\mathfrak{C}_{k},i,j\in\mathbb{R}^{+}$.
Motivated by Lemma 2.2.4 with
$\alpha_{1}=\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})$ and
$\alpha_{2}=\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})$, for a given $n$ units,
${X_{1:n}^{(i)}}\cap\mathfrak{C}_{j}$ and
${X_{1:n}^{(i)}}\setminus{\mathfrak{C}_{j}}$ be configurations of
$N_{j}:=\left\lfloor\tilde{p}\cdot n\right\rfloor$ and $N_{k}:=n-N_{j}$ points
respectively such that
$E^{s}_{\beta}({X_{1:n}^{\mathfrak{C}_{j}}})<\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j},N_{j})+{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{6}}}}$
and
$E^{s}_{\beta}({X_{1:n}^{\mathfrak{C}_{k}}})<\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k})+{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{6}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{6}}}}\in\mathbb{R}^{+}$,
where

$\tilde{p}=\frac{\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})^{\alpha_{3}}}{\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})^{\alpha_{3}}}.$

Then

$\begin{split}\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k},n)&\leq
E^{s}_{\beta}(X_{1:n}^{\mathfrak{C}_{j}}\cup
X_{1:n}^{\mathfrak{C}_{k}})=E^{s}_{\beta}(X_{1:n}^{\mathfrak{C}_{j}})+E^{s}_{\beta}(X_{1:n}^{\mathfrak{C}_{k}})+2\sum_{\begin{subarray}{c}x\in
X_{1:n}^{\mathfrak{C}_{j}}\\\ y\in
X_{1:n}^{\mathfrak{C}_{k}}\end{subarray}}\frac{\omega(x,y)}{\left\|x-y\right\|}\\\
&\leq\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j},N_{j})+\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k})+{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{7}}}}+2\cdot\frac{n^{2}}{r}\cdot\left\|\omega\right\|_{\mathfrak{C}_{j}\times\mathfrak{C}_{k}},\end{split}$

where $\left\|\omega\right\|_{\mathfrak{C}_{j}\times\mathfrak{C}_{k}}$ denotes
the supremum of $\mathfrak{C}$ over $\mathfrak{C}_{j}\times\mathfrak{C}_{k}$.
Dividing by $\lambda(n)$ and taking into account that
$\lambda(N_{j})/\lambda(n)\rightarrow{(\tilde{p})^{\mathcal{T}(d)}}$ as
$n\rightarrow\infty$, we obtain

$\displaystyle\begin{split}\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k})&=\lim_{n\rightarrow\infty}{\text{sup}}\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j}\cup\mathfrak{C}_{j},n)}{n^{\mathcal{T}(d)}}\leq\lim_{n\rightarrow\infty}{\text{sup}}\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j},N_{j})}{n^{\mathcal{T}(d)}}+\lim_{n\rightarrow\infty}{\text{sup}}\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{k},N_{k})}{n^{\mathcal{T}(d)}}\\\
&=\lim_{n\rightarrow\infty}{\text{sup}}\frac{\mathcal{E}^{s}(\mathfrak{C}_{j},N_{j})}{N_{j}^{\mathcal{T}(d)}}{(\tilde{p})^{\mathcal{T}(d)}}+\lim_{n\rightarrow\infty}{\text{sup}}\frac{\mathcal{E}^{s}(\mathfrak{C}_{k},N_{k})}{N_{k}^{\mathcal{T}(d)}}{(1-\tilde{p})^{\mathcal{T}(d)}}\\\
&\leq\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})\cdot(\tilde{p})^{\mathcal{T}(d)}+\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})\cdot(1-\tilde{p})^{\mathcal{T}(d)}\\\
&=\left[\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{sup}}(\mathfrak{C}_{k})^{\alpha_{3}}\right]^{\frac{1}{\alpha_{3}}}.\end{split}$

Thus, Lemma 2.2.5 holds.
Lemma 2.2.6. $\exists\mathfrak{C}_{j},\mathfrak{C}_{k}\subset\mathfrak{C}$,
and $\mathfrak{C}_{j},\mathfrak{C}_{k}\not\subset\emptyset$, $j\neq k$,
$\mathcal{T}(d)>2$ is continuous and derivative for $d\in\mathbb{R}^{+}$, let
$\alpha_{3}=\frac{1}{2-\mathcal{T}(d)}$ for $s>d$,

$\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k})^{\alpha_{3}}\leq\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{k})^{\alpha_{3}}$
(14)

Furthermore, if
$\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j}),\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{k})\geq
0$ and at least one of these oracles is finite, then for any infinite subset
$N^{\prime}$ of $n$ and any sequence $\\{X_{1:n}\\}_{n\in N^{\prime}}$ of
$n$-point configurations in $\mathfrak{C}_{j}\cup\mathfrak{C}_{k}$ such that

$\lim_{n\rightarrow\infty}\frac{\mathcal{E}^{s}_{\beta}(X_{1:n},n)}{\lambda(n)}=(\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{k})^{\alpha_{3}})^{\frac{1}{\alpha_{3}}}$
(15)

holds, we have

$p_{j}=\frac{\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})^{\alpha_{3}}}{\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{k})^{\alpha_{3}}}.$
(16)

Proof We assume $\psi_{s,d}^{\text{inf}}(\mathfrak{C})$ is bounded on the
compact set $\mathfrak{C}_{j}$ and $\mathfrak{C}_{k}$. Let $\hat{N}$ be an
infinite subsequence such that

$\lim_{n\rightarrow\hat{N}}\frac{\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k},n)}{n^{\mathcal{T}(d)}}:=\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k}).$

If $X_{1:n}^{(i)}$ is a sequence of $n$-point configurations on
$\mathfrak{C}_{j}\cup\mathfrak{C}_{k}$ such that
$E^{s}_{\beta}(X_{1:n})\leq\mathcal{E}^{s}_{\beta}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k},n)+{{C_{11},C_{11}}}\in\mathbb{R}^{+}$.
Similar to the proof of Lemma 2.2.4, if at least one of the oracles
$\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})$ or
$\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{k})$ is infinite, where $p_{j}=0$ or
$1$. Generally, for $p_{j}\in(0,1)$, by Lemma 2.2.4, we obtain

$\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k})=\lim_{n\rightarrow\hat{N}}\frac{E^{s}_{\beta}(X_{1:n})}{n^{\mathcal{T}(d)}}\geq
U(p_{j}).$

We follow an argument in [18] and let

$\alpha^{\prime}=\frac{\psi_{s,d}^{\text{inf}}(A)^{\alpha_{3}}}{\psi_{s,d}^{\text{inf}}(A)^{\alpha_{3}}+\psi_{s,d}^{\text{inf}}(B)^{\alpha_{3}}}$

and $\\{X_{1:n}\\}_{n\in N^{\prime}}$ be any sequence of $n$-point sets in
$A\cup B$ satisfying (15). Let $n\subset\hat{N}$ be any infinite subsequence
on $\mathfrak{C}_{j}\cup\mathfrak{C}_{k}$ such that the quantity $p_{j}\geq 0$
as $n\rightarrow\infty$, $n\in\hat{N}$. From Lemma 2.2.5 and (15), we have

$U(\alpha^{\prime})=\lim_{n\rightarrow\infty}\frac{\mathcal{E}^{s}_{\beta}(X_{1:n},n)}{\lambda(n)}\geq
U(p_{j}),$

where $U(\alpha^{\prime})$ is equal to the right-hand side of (15), obviously,
$\alpha^{\prime}$ is the only minimum point of $U(\cdot)$. Consequently,
$p_{j}=\alpha^{\prime}$, (16) holds. If $X_{1:n}$ is a subsequence on
$N^{\prime}:=\mathfrak{C}_{j}\cup\mathfrak{C}_{k}\setminus\hat{N}$, we denote
the corresponding units that belong to the compact subset with probability
$p_{j}^{\prime}$, then

$\displaystyle\begin{split}\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{j}\cup\mathfrak{C}_{k})&=\lim_{n\rightarrow
N^{\prime}}\frac{E^{s}_{\beta}(X_{1:n})}{n^{\mathcal{T}(d)}}\geq
U(p^{\prime}_{j})\geq U(\alpha^{\prime})\\\
&=\left[\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{j})^{\alpha_{3}}+\psi_{s,d}^{\text{inf}}(\mathfrak{C}_{k})^{\alpha_{3}}\right]^{\frac{1}{\alpha_{3}}}.\end{split}$

Thus, (14) holds.
Lemma 2.2.7. Suppose that $s>d$, and $\mathfrak{C}\subset\mathbb{R}^{d}$ is a
compact set with $0<\mu(\mathfrak{C})<\infty$, $\mathcal{T}(d)>2$ is
continuous and derivative for $d\in\mathbb{R}^{+}$. Furthermore, suppose that
for any compact subset $\mathfrak{C}_{i}\subset\mathfrak{C}$, the limit
$\psi_{s,d}(\mathfrak{C}_{i}),i\in\mathbb{R}^{+}$ exists and is given by

$\psi_{s,d}(\mathfrak{C}_{i})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}{\mathcal{U}_{d}^{s}(\mathfrak{C}_{i})^{\mathcal{T}(d)-2}}.$

Then, $\psi_{s,d}(\mathfrak{C})$ exists and is given by

$\psi_{s,d}(\mathfrak{C})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}{\mathcal{U}_{d}^{s}(\mathfrak{C})^{\mathcal{T}(d)-2}}.$
(17)

Moreover, if a sequence of $n$-point configurations $X_{1:n}$ is
asymptotically weighted Riesz energy minimizing on the set $\mathfrak{C}$ and
$\mu(\mathfrak{C})>0$, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1,i\neq j}^{n}\parallel
x_{i}-x_{j}\parallel\rightarrow u_{d}^{s}(\mathbb{S}).$ (18)

Proof To prove (17), we firstly decompose the entire metric space
$\mathfrak{C}$ into extremely small disconnected parts with diameter less than
$\epsilon>0$, according to the property of Borel metrics, then

$\sum_{P\in\mathfrak{C}_{i}}\mathcal{U}_{d}(P)\leq\mathcal{U}_{d}(\mathfrak{C}).$
(19)

Hereafter we follow an argument in [18] and define a sufficiently small space
$\mathfrak{C}_{i}$ as follows, the rule refers to [12]. We consider the
hyperplane $\mathfrak{C}^{\prime}$ consisting of all points, $(-l,l)$ is a
cube embedded in $\mathfrak{C}^{\prime}$, we discretize the cube with tiny
intervals for $j$-th ordinate, $-l=h_{0}^{j}<h_{1}^{j}\cdots<h_{k}^{j}=l,j\in
i^{d^{\prime}},d^{\prime}\in(1,d),i=(i_{1},i_{2}\cdots,i_{n})$, $k$ is
sufficiently large, $\exists\left\|h_{k}^{j}-h_{k-1}^{j}\right\|<\epsilon$
such that (19) holds. $\mathfrak{C}_{i}$ can be written as

$\mathfrak{C}_{i}:=[h_{i_{1}^{1}}^{1},h_{i_{1}^{1}+1}^{1})\times\cdots\times[h_{i_{n}^{d^{\prime}}-1}^{d^{\prime}},h_{i_{n}^{d^{\prime}}}^{d^{\prime}}),$

For $\mathfrak{C}_{i}\subset\mathfrak{C}$, if $\omega(x_{i},x_{j})$ is
bounded, let

$\overline{\omega}_{{\mathfrak{C}}_{i}}=\sup_{x_{i},x_{j}\in\mathfrak{C}_{i}}\omega(x_{i},x_{j}),\text{
and}\
\underline{\omega}_{{\mathfrak{C}}_{i}}=\inf_{x_{i},x_{j}\in\mathfrak{C}_{i}}\omega(x_{i},x_{j}),$

we introduce the radial basis functions $\varphi(\cdot)$ to approximate the
corresponding bounded $\omega(x_{i},x_{j})$:

$\overline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})=\sum_{P\in\mathfrak{C}_{i}}\overline{\omega}_{P}\varphi(\left\|x_{i}-x_{j}\right\|),\
\
\underline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})=\sum_{P\in\mathfrak{C}_{i}}\underline{\omega}_{P}\varphi(\left\|x_{i}-x_{j}\right\|).$
(20)

From Lemma 2.2.5, and (13),

$\displaystyle\begin{split}\psi_{s,d}^{{\text{sup}}}(\mathfrak{C})^{\alpha
3}&\geq\sum_{i=1}^{n}\psi_{s,d}^{{\text{sup}}}(\mathfrak{C}_{i})^{\alpha
3}\geq\sum_{i=1}^{n}\left[\overline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})\cdot\psi_{s,d}^{{\text{sup}}}(\mathfrak{C}_{i})\right]^{\alpha
3}={{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}\sum_{x_{i},x_{j}\in\mathfrak{C}_{i}}\overline{\omega}_{{\mathfrak{C}}_{i}}^{\alpha
3}\cdot\mathcal{U}_{d}(\mathfrak{C}_{i})\\\
&\geq{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}\int_{x_{i},x_{j}\in\mathfrak{C}_{i}}\overline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})^{\alpha
3}d\mathcal{U}_{d}(\mathfrak{C}_{i}).\end{split}$

From Lemma 2.2.6 and (14), similarly, we have

$\displaystyle\begin{split}\psi_{s,d}^{{\text{inf}}}(\mathfrak{C})^{\alpha
3}&\leq\sum_{i=1}^{n}\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{i})^{\alpha
3}\leq\sum_{i=1}^{n}\left[\underline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})\cdot\psi_{s,d}^{{\text{inf}}}(\mathfrak{C}_{i})\right]^{\alpha
3}={{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}\sum_{x_{i},x_{j}\in\mathfrak{C}_{i}}\underline{\omega}_{{\mathfrak{C}}_{i}}^{\alpha
3}\cdot\mathcal{U}_{d}(\mathfrak{C}_{i})\\\
&\leq{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}\int_{x_{i},x_{j}\in\mathfrak{C}_{i}}\underline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})^{\alpha
3}d\mathcal{U}_{d}(\mathfrak{C}_{i}).\end{split}$

Given a sufficiently small $P$, for (20), use the equation limit, we have

$\displaystyle\begin{split}&\overline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})=\sum_{P\in\mathfrak{C}_{i}}\overline{\omega}_{P}\varphi(\left\|x_{i}-x_{j}\right\|)={\omega}_{{\mathfrak{C}}}(x_{i},x_{j}),\\\
&\underline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})=\sum_{P\in\mathfrak{C}_{i}}\underline{\omega}_{P}\varphi(\left\|x_{i}-x_{j}\right\|)={\omega}_{{\mathfrak{C}}}(x_{i},x_{j}).\end{split}$

Since $\omega(x_{i},x_{j})$ is continuous on $\mathfrak{C}$, both
$\int_{x_{i},x_{j}\in\mathfrak{C}_{i}}\overline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})^{\alpha
3}d\mathcal{U}_{d}(\mathfrak{C}_{i})$ and
$\int_{x_{i},x_{j}\in\mathfrak{C}_{i}}\underline{\omega}_{{\mathfrak{C}}_{i}}(x_{i},x_{j})^{\alpha
3}d\mathcal{U}_{d}(\mathfrak{C}_{i})$ converge to
$\mathcal{U}_{d}^{s}(\mathfrak{C})$. Consequently, the limit
$\psi_{s,d}(\mathfrak{C}_{i}),i\in\mathbb{R}^{+}$ exists and can be given by

$\psi_{s,d}(\mathfrak{C}_{i})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{8}}}{\mathcal{U}_{d}^{s}(\mathfrak{C}_{i})^{\mathcal{T}(d)-2}}.$

By the Fatou’s Lemma and Monotone Convergence Theorem, Thus, (17) holds on
$\mathfrak{C}$.

To prove (18), suppose that $X_{1:n}$ is an asymptotically weighted Riesz
energy minimizing sequence of $n$-point configuration on $\mathfrak{C}$, the
corresponding signed finite Borel measures
$\cup_{i=1}^{n}{\mu_{d}^{s}(\mathfrak{C}_{i})}$ in $\mathbb{R}^{d}$ converges
weak∗ to a signed finite Borel measure $\mu_{d}(\mathfrak{C})$, as
$n\rightarrow\infty$. Consequently, (18) is equivalent to the assertion that

$\lim_{n\rightarrow\infty}\sum_{j=1}^{n}p_{j}=\cup_{i=1}^{n}\mu_{d}^{s}(\mathfrak{C}_{j})=\mu_{d}(\mathbb{S})$

holds for any almost $\sigma\text{-algebra}$ subset on $\mathfrak{C}$, let
$\mathfrak{C}_{\sigma}=\cup_{i=1}^{n}\mathfrak{C}_{i}$ be a subset of
$\sigma\text{-algebra}$ on $\mathfrak{C}$, for any Borel subset
$\mathfrak{C}_{\sigma}\subset\mathfrak{C}$. Since $\mathfrak{C}_{\sigma}$ and
$\mathfrak{C}/\mathfrak{C}_{\sigma}$ are the compact subsets of
$\mathfrak{C}$, suppose
$\psi_{s,d}(\mathfrak{C}_{\sigma})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{9}}}{\mu(\mathfrak{C}_{\sigma})^{-\frac{1}{\alpha_{3}}}}$
and
$\psi_{s,d}(\mathfrak{C}/\mathfrak{C}_{\sigma})=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{10}}}{\mu(\mathfrak{C}/\mathfrak{C}_{\sigma})^{-\frac{1}{\alpha_{3}}}}$,
for the asymptotically weighted Riesz energy minimal sequence $X_{1:n}$,

$\begin{split}\lim_{n\rightarrow\infty}\frac{E^{s}(X_{1:n})}{\lambda(n)}&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}C_{s,d}}\cdot(\mu(\mathfrak{C}))^{\frac{1}{\alpha_{3}}}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}C_{s,d}}\cdot(\mu(\mathfrak{C}_{\sigma})+\mu(\mathfrak{C}/\mathfrak{C}_{\sigma}))^{\frac{1}{\alpha_{3}}}\\\
&=\left[\psi_{s,d}(\mathfrak{C}_{\sigma})^{\alpha_{3}}+\psi_{s,d}(\mathfrak{C}/\mathfrak{C}_{\sigma})^{\alpha_{3}})\right]^{\frac{1}{\alpha_{3}}}.\end{split}$

Using (16) in Lemma 2.2.6 and (17) which holds for $\mathfrak{C}_{\sigma}$ and
$\mathfrak{C}/\mathfrak{C}_{\sigma}$, we have

$\begin{split}\lim_{n\rightarrow\infty}\sum_{j=1}^{n}p_{j}&=\frac{\psi_{s,d}(\mathfrak{C}/\mathfrak{C}_{\sigma})^{-\alpha_{3}}}{\psi_{s,d}(\mathfrak{C}_{\sigma})^{-\alpha_{3}}+\psi_{s,d}(\mathfrak{C}/\mathfrak{C}_{\sigma})^{-\alpha_{3}}}=\frac{\mathcal{U}_{d}^{s}(\mathfrak{C}_{\sigma})}{\mathcal{U}_{d}^{s}(\mathfrak{C}_{\sigma})+\mathcal{U}_{d}^{s}(\mathfrak{C}/\mathfrak{C}_{\sigma})}=\mu_{d}(\mathbb{S}).\end{split}$

Thus, (18) holds.

###  2.3 Separation Distance

In this section, we focus on the features of points generated by minimizing
the weighted Riesz energy criterion from the point of discrete geometry. We
state and prove the relationship between asymptotically best-packing and
weighted Riesz energy-minimizing configuration first. We are devoted to
deriving the lower bounds for the minimal pairwise separation in optimal
configurations. Suppose that $\mathfrak{C}$ is a compact infinite space with
Euclidean distance $r$, $\mathfrak{C}\times\mathfrak{C}\rightarrow[0,\infty)$,
and define the separation distance of an $n$-point configuration
$X_{1:n}=\\{x_{1},x_{2},...,x_{n}\\}$ on $(r,\mathfrak{C})$ as
$r_{\text{min}}(X_{1:n}):=\min_{1\leq i\neq j\leq n}\parallel
x_{i}-x_{j}\parallel,$ the $n$-point best-packing configuration on
$\mathfrak{C}$ is $X_{1:n}^{*}$ satisfying

$\displaystyle\begin{split}&\lim_{m\rightarrow\infty}{(\cup
X_{1:n}^{(s)})}=\mathfrak{C},r_{\text{min}}^{\text{max}}(\mathfrak{C}):=\max\\{r_{\text{min}}(X_{1:n}^{*}),X_{1:n}^{*}\subset\mathfrak{C}\\}.\\\
\end{split}$ (21)

For the weighted Riesz energy criterion, the $n$-point best-packing
configuration is the limiting case of $s$ approximating infinity. To derive
the well-separateness property, we will first provide the upper estimate of
the Borel measure in a restricted compact space for the potential and then
prove the estimate for our weighted Riesz energy configurations next.
Lemma 2.3.1. Let $\mu$ be a closed Borel measure in $\mathbb{R}^{d}$,
$\epsilon>0$, suppose $B(y,\epsilon)\subset\sigma\text{-algebra}$, there exist
$\mu(\sigma\text{-algebra})\leq\epsilon$, $y\in\mathfrak{C}_{i}^{\infty}$,
then the potential

$\begin{split}G_{s,\mu}(y,\epsilon)&:=\left[\int_{\mathfrak{C}\setminus
B(y,\epsilon)}\frac{\omega(x_{i},y)}{\parallel
x_{i}-y\parallel^{s}}d\mu(x_{i})\right]^{\frac{1}{s}}\leq\sum_{k^{\prime}=1}^{\infty}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{11}}}{\epsilon}^{\frac{2d}{s(2d+k^{\prime}!)}-1}.\end{split}$
(22)

Proof of Lemma 2.3.1 For every $y\in\mathfrak{C}$ and $\epsilon>0$, the point
$x_{i}=y$ is enclosed in an infinite dimensional space (denoted by sphere) of
radius $\epsilon$, in order to make the energy function (7) in the main text
integrable, this space will be excluded with $\mathfrak{C}\setminus
B(y,\epsilon)$. Substitute (10) into (7) in the main text, then

$\displaystyle\begin{split}G_{s,\mu}^{s}(y,\epsilon):&=\int_{\mathfrak{C}\setminus
B(y,\epsilon)}h_{\beta}(x_{i},y)d\mu(x_{i})=\int_{0}^{{\epsilon}^{-s}}\mu\\{x\in\mathfrak{C}\setminus
B(y,\epsilon):h_{\beta_{1}^{+}}(x_{i},y)>t\\}dt\\\
&=\int_{0}^{{\epsilon}^{-s}}\mu\\{x\in\mathfrak{C}\setminus
B(y,\epsilon):\frac{e^{\left[\beta_{1}\parallel
x_{i}-y\parallel\right]^{-\frac{s}{2d}}}}{\parallel
x_{i}-y\parallel^{s}}>t\\}dt\\\
&=\sum_{k^{\prime}=1}^{\infty}\int_{0}^{{\epsilon}^{-s}}\mu\\{x\in\mathfrak{C}\setminus
B(y,\epsilon):\frac{\left[\beta_{1}\cdot\parallel
x_{i}-y\parallel\right]^{\frac{-{s{k^{\prime}}}}{2d}}}{{k^{\prime}}!\parallel
x_{i}-y\parallel^{s}}>t\\}dt\\\
&\leq\sum_{k^{\prime}=1}^{\infty}\int_{0}^{{\epsilon}^{-s}}\mu\left[B(y,\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}t^{\frac{-2d}{2sd+sk^{\prime}!}})\right]dt\\\
&\leq\sum_{k^{\prime}=1}^{\infty}\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}\int_{0}^{{\epsilon}^{-s}}t^{\frac{-2d}{2sd+sk^{\prime}!}}dt\\\
&\leq\sum_{k^{\prime}=1}^{\infty}\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}\frac{2sd+sk^{\prime}!}{2sd+sk^{\prime}!-2d}{\epsilon}^{\frac{2d}{2d+k^{\prime}!}-s}\\\
&\leq\sum_{k^{\prime}=1}^{\infty}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{10}}}{\epsilon}^{\frac{2d}{2d+k^{\prime}!}-s}.\end{split}$
(23)

Thus, Lemma 2.3.1 holds.

Let $v$ the county measure of the intersection of compact subsets
$\mathfrak{C}_{i}$ and $\mathfrak{C}_{j}$, there exists an infinitely small
value $\zeta>0$ such that
$v=\lim_{i,j\in[1,\infty)}\\#(\mathfrak{C}_{i}\cap\mathfrak{C}_{j})>\zeta$,
and
$\lim_{i,j\in[1,\infty)}\left\|x_{i}-x_{j}\right\|\subset\mu(\mathfrak{C}_{i}^{\infty}\cap\mathfrak{C}_{j}^{\infty})\subset\sigma\text{-algebra}$.
Suppose that $\overline{W}$ be the bounded value of $\omega(x_{i},x_{j})$ in
(4), we can estimate the upper bound for extremal weighted Riesz energy
criterion.
Lemma 2.3.2. Let $s\geq d>0$, there exist constant
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{12}}$
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{13}}$
such that for any compact set $\mathfrak{C}\subset\mathbb{R}^{d}$ with the
measure $\mu(\sigma\text{-algebra})>0$ and any $\omega(x_{i},x_{j})$ that is
bounded and lower semicontinuous on $\mathfrak{C}\times\mathfrak{C}$, then

$\mathcal{E}_{\beta}(\mathfrak{C},n)\leq\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{12}}\overline{W}}{\mu(\sigma\text{-algebra})}n^{\frac{1}{d}+\frac{2}{s}}.$

holds for any weighted Riesz energy minimizing configuration
$X_{1:n}^{*},s\geq 2$, and

$r_{\text{min}}(X_{1:n}^{*})\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{13}}\cdot\mu(\sigma\text{-algebra})^{\frac{2d}{2d+1}}\cdot
n^{\frac{-1}{d}+\frac{-2}{s}}.$ (24)

Proof of Lemma 2.3.2 Let
$E_{i}(x)=\left\\{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{\omega(x,x_{j})}{\parallel
x-x_{j}\parallel^{s}}\right\\}^{\frac{1}{s}},x\in\mathfrak{C}$. If we define
$x_{i}$ is from the unit of minimum energy configuration, $E_{i}(x_{i})\leq
E_{i}(x),x\in\mathfrak{C},i=1,...,n$. If $\mu$ is a measure on $\mathfrak{C}$
where $x_{j}$ is excluded by
$\mathfrak{C}_{\setminus{x_{j}}}:=\mathfrak{C}\setminus\sum_{j\neq i}(\cup
B(x_{j},\epsilon))$, as $\epsilon$ is extremely small, there exists a constant
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{14}}$
that is close to 1,
$0<{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{14}}<1$,
$\mu(\mathfrak{C}_{\setminus{x_{j}}})\geq\mu(\mathfrak{C})-\mu(\sum_{j\neq
i}(\cup
B(x_{j},\epsilon)))\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{14}}\cdot\mu(\mathfrak{C})$.
When $s>d$, substitute (7) in the main text into $E_{i}^{s}(x_{i})$, we have

$\displaystyle\begin{split}E_{i}^{s}(x_{i})&\leq\frac{2}{\mu(\mathfrak{C}_{\setminus{x_{j}}})}\int_{\mathfrak{C}_{\setminus{x_{j}}}}\int_{\mathfrak{C}_{\setminus{x_{j}}}}h_{\beta}(x_{i},x_{j})d\mu(x_{i})d\mu(x_{j})\\\
&=\frac{2}{\mu(\mathfrak{C})}\int_{\mathfrak{C}_{\setminus{x_{j}}}}\int_{\mathfrak{C}_{\setminus{x_{j}}}}\frac{\omega(x,x_{j})}{\parallel
x-x_{j}\parallel^{s}}d\mu(x)d\mu(x_{j})=\frac{2}{\mu(\mathfrak{C}_{\setminus{x_{j}}})}\sum_{j:j\neq
i}G_{s,\mu}^{s}(x_{j},\epsilon)\\\
&\leq\frac{1}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{15}}\cdot\mu(\mathfrak{C}_{\setminus{x_{j}}})}\sum_{j:j\neq
i}\sum_{k^{\prime}=1}^{\infty}\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}\frac{2sd+sk^{\prime}!}{2sd+sk^{\prime}!-2d}{\epsilon}^{\frac{2d}{2d+k^{\prime}!}-s}\\\
&\leq{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{15}}}\left(\frac{n}{\mu(\mathfrak{C})}\right)^{\frac{s}{d}+2}\leq\left[\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{16}}\overline{W}}{\mu(\sigma\text{-algebra})}n\right]^{\frac{s}{d}+2}.\end{split}$
(25)

Let $X_{1:n}^{*},n\geq 2$, be a weighted Riesz energy minimizing configuration
on $\mathfrak{C}$ and $i_{n}$ and $j_{n}$ be such that
$r_{\text{min}}(X_{1:n}^{*})=\left\|x_{i}^{*}-x_{j}^{*}\right\|$ and for the
bound $\left\|\omega(x_{i},x_{j})\right\|\leq\overline{W}$, then

$\mathcal{E}_{\beta}(\mathfrak{C},n)\geq\sum_{j=1,j\neq
i}^{n}\left[h_{\beta}(x_{i},x_{j})\right]^{\frac{1}{s}}\geq\frac{1}{\left\|x_{i}^{*}-x_{j}^{*}\right\|}.$
(26)

Combine (25) and (26), we have

$\frac{1}{\left\|x_{i}^{*}-x_{j}^{*}\right\|}\leq\mathcal{E}_{\beta_{1}}(\mathfrak{C},n)\leq\left[\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{12}}\overline{W}}{\mu(\sigma\text{-algebra})}n\right]^{\frac{1}{d}+\frac{2}{s}},$

then,
$r_{\text{min}}(X_{1:n}^{*})\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{13}}\cdot\mu(\sigma\text{-algebra})^{\frac{2d}{2d+1}}\cdot
n^{\frac{-1}{d}+\frac{-2}{s}}$. Consequently, Lemma 2.3.2 holds.

###  2.4 Covering Radius for journal

In this section, we state and prove the bound of the covering radius. And
extend to deal with the weak* limit distribution of best-covering $n$-point
configurations on rectifiable sets $\mathfrak{C}$. Suppose that $\mathfrak{C}$
is bounded in a compact infinite metric space with Euclidean metric $r$,
$\mathfrak{C}\times\mathfrak{C}\rightarrow[0,\infty)$, we generalize
$r_{\text{min}}^{\text{max}}(\mathfrak{C})$ in (21) and define the covering
radius of an $n$-point configuration $X_{1:n}$ in a metric space
$(\mathfrak{C},r)$ as
$\rho(X_{1:n},\mathfrak{C}):=\max_{x\in\mathfrak{C}}\min_{i=1,...,n}r(x,x_{i}).$
From the geometrical perspective, the covering radius of $X_{1:n}$ can be
considered as the minimal radius of $n$ adjacent closed balls centered at
$X_{1:n}$ whose union contains the entire $\mathfrak{C}$. Among finite element
analysis and approximation theory, this quantity is known as the best
approximation of the set $\mathfrak{C}$ by the configuration $X_{1:n}$ [18].
The optimal values of this quantity are also of interest and we define the
minimal $N$-point covering radius of a set $\mathfrak{C}$ as

$\rho_{n}(\mathfrak{C}):=\min\\{\rho(X_{1:n},\mathfrak{C}):X_{1:n}\subset\mathfrak{C}\\}.$

$\rho_{n}(\mathfrak{C})$ is also called an $n$-point best-covering
configuration for $\mathfrak{C}$ [10].
Theorem 2.4.1. Assume that $s>d$ and $\mathfrak{C}\subset\mathbb{R}^{d}$, if
the compact support of $\mu$ is contained in $\mathfrak{C}$, with respect to
some finite Borel measure for a compact Euclidean metrical and bounded set,
there exists positive constant
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{17}}$
such that every sequence $X_{1:n}^{*}$ of weighted Riesz energy minimizing
configurations on $\mathfrak{C}$ satisfies

$\rho{(X_{1:n}^{*})}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{17}}n^{(1-\frac{s}{d})\cdot\frac{2d+1}{s+2d+2ds}}$
(27)

Proof of Theorem 2.4.1 Let $X_{1:n}^{*}=\\{x_{1}^{*}...,x_{n}^{*}\\}$ be an
$n$-point energy minimizing configuration for the compact and measurable space
$\mathfrak{C}$, we consider the function

$\overline{H}(y):=\frac{1}{n}\sum_{i=1}^{n}\frac{\omega(x_{i},y)}{\parallel
x_{i}-y\parallel^{s}}.$

Given a specified $1\leq j\leq n$ and $d>0$ in $\mathfrak{C}$,

$\overline{H}(y):=\frac{1}{n}\frac{\omega(x_{j},y)}{\parallel
x_{j}-y\parallel^{s}}+\frac{1}{n}\sum_{i=1,i\neq
j}^{n}\frac{\omega(x_{i},y)}{\parallel x_{i}-y\parallel^{s}}.$ (28)

since $X_{1:n}^{*}$ is the energy minimizing configuration on $\mathfrak{C}$,
the point $x_{j}$ minimizes (28), consequently, for each fixed $j$ and
$y\in\mathfrak{C}$,

$\overline{H}(y)\geq\frac{1}{n}\frac{\omega(x_{j},y)}{\parallel
x_{j}-y\parallel^{s}}+\frac{1}{N}\sum_{i=1,i\neq
j}^{n}\frac{\omega(x_{j},y)}{\parallel x_{j}-y\parallel^{s}}.$ (29)

Summing over (29) with different $j$, from Jensen’s inequality, it gives

$\displaystyle\begin{split}n\cdot\overline{H}(y)&\geq
H(y)+\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1,j\neq
i}^{n}\frac{\omega(x_{j},x_{i})}{\parallel
x_{i}-x_{j}\parallel}\geq\overline{H}(y)+\frac{1}{n}E^{s}(x_{i},x_{j}),\end{split}$

thus,

$\overline{H}(y)\geq\frac{1}{n(n-1)}E^{s}(x_{i},x_{j})\geq\frac{E^{s}(x_{i},x_{j})}{n^{2}},(y\in\mathfrak{C}).$
(30)

Since $\mathfrak{C}$ is compact, there exists a point $y^{*}\in\mathfrak{C}$
such that

$\min_{x\leq i\leq n}r(y^{*},x_{i})=\rho(X^{*}_{1:n},\mathfrak{C}).$ (31)

$\omega$ is bounded on $\mathfrak{C}$, by Theorem 2.2.1, there are constants
$N_{0}$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{18}}>0$
such that

$E^{s}(x_{i},x_{j})\geq\mathcal{E}_{\beta}\left[(\mathfrak{C},n)\right]^{s}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{18}}n^{2+\frac{s}{d}},n\geq
N_{0}.$

Since (30) holds for the point $y^{*}$ of (31), substitute (31) into (30),

$\overline{H}(y)\geq\frac{\mathcal{E}_{\beta}\left[(\mathfrak{C},n)\right]^{s}}{n^{2}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{18}}n^{\frac{s}{d}},(y\in\mathfrak{C}).$
(32)

Next we determine an upper bound for $\overline{H}(y)$ on $\mathfrak{C}$ by
introducing (31). Lemma 2.3.3 applies that there exist some
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{19}}>0$
such that
$r_{\text{min}}(X_{1:n}^{*})\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{19}}\cdot\mu(\sigma\text{-algebra})^{\frac{2d}{2d+1}}\cdot
n^{\frac{-1}{d}+\frac{-2}{s}}$ for $n\geq 2$. Thus, we have

$\rho(X_{1:n}^{*},\mathfrak{C})\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{20}}\cdot\mu(\sigma\text{-algebra})^{\frac{2d}{2d+1}}\cdot
n^{\frac{-1}{d}+\frac{-2}{s}}$ (33)

Hereafter we follow an argument in [18] and define

$\rho_{0}:=\zeta\cdot\rho(X_{1:n}^{*},\mathfrak{C})$

where $\zeta$ is sufficiently small, the particles are well-separated that
have no intersection with the neighbor within the Ball
$B(x_{i},r_{0})\subset\mathfrak{C}$. For any $x\in B(x,x_{i})$, combine (31)
and (33)

$\displaystyle\begin{split}r(x,y^{*})&\leq
r(x,x_{i})+r(x_{i},y^{*})\leq\rho_{0}+r(x_{i},y^{*})\leq\zeta\rho(X_{1:N}^{*})+r(x_{i},y^{*})\leq(1+\zeta)r(x_{i},y^{*}),\end{split}$
(34)

the lower bound for $r(x,y^{*})$

$\displaystyle\begin{split}r(x,y^{*})&\geq r(x,x_{i})-r(x,x_{i})\geq
r(x_{i},y^{*})-\rho_{0}\geq
r(x_{i},y^{*})-\zeta\rho(X_{1:n}^{*})\geq(1-\zeta)\rho(X_{1:n}^{*}),\end{split}$
(35)

implies

${\bigcup_{i=1}^{n}}B(x_{i},\rho_{0})\subseteq\mathfrak{C}(y^{*},(1-\zeta)\rho(X_{1:n}^{*})).$
(36)

For fixed $n$ points, using (34) and taking the average value on
$B(x_{i},\rho_{0})$, we obtain

$\displaystyle\begin{split}\overline{K}(y)&\leq\frac{\parallel\omega\parallel(1+\zeta)^{s}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{21}}}{n\rho_{0}}\sum_{i=1}^{n}\int_{B(y,\epsilon)}\frac{d\mu(x_{i})}{\parallel
x-y^{*}\parallel^{s}}\\\
&\leq\frac{(1+\zeta)^{s}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{21}}}{n\rho_{0}}\int_{\mathfrak{C}\setminus
B(y^{*},(1-\zeta)\rho(X_{1:n}^{*}))}\frac{\omega(x,y^{*})d\mu(x_{i})}{\parallel
x-y^{*}\parallel^{s}}\\\
&=\frac{(1+\zeta)^{s}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{21}}}{n\rho_{0}}\int_{\mathfrak{C}\setminus
B^{\prime}}h_{\beta}(x,y^{*})d\mu(x_{i}),\end{split}$

where $B^{\prime}=B(y^{*},(1-\zeta)\rho(X_{1:n}^{*}))$, let
$c(k^{\prime})={\frac{-2d}{2sd+sk^{\prime}!}}$, we decompose the right-hand
side of (36) into two terms and proceed as in (23) to obtain

$\displaystyle\begin{split}\int_{\mathfrak{C}\setminus
B^{\prime}}h_{\beta}(x,y^{*})d\mu(x_{i})&=\int_{B^{\prime}}h_{\beta}(x,y^{*})d\mu(x_{i})+\int_{\mathfrak{C}\setminus
B(y^{*},\rho_{0})}h_{\beta}(x,y^{*})d\mu(x_{i})\\\
&\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{22}}\int_{0}^{\rho_{0}^{-s}}t^{c(k^{\prime})}dt+\sum_{k^{\prime}=1}^{\infty}\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}\int_{\rho_{0}^{-s}}^{{[(1-\zeta)\rho(X_{1:n}^{*})]}^{-s}}t^{c(k^{\prime})}dt\\\
&={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{22}}\cdot
c^{\prime}(k^{\prime})+\sum_{k^{\prime}=1}^{\infty}\frac{k^{\prime}}{{\beta_{1}}^{\frac{-sk^{\prime}}{2d}}}\left[\frac{[(1-\zeta){\rho{(X_{1:n}^{*})}}]^{c(k^{\prime})s-s}}{c(k^{\prime})s-s}-c(\rho_{0})\right]\\\
&\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{23}}n\cdot[(1-\zeta){\rho{(X_{1:n}^{*})}}]^{\frac{-2ds-2d-s}{2d+1}},\end{split}$
(37)

where
$c^{\prime}(k^{\prime})=\frac{\rho_{0}^{c(k^{\prime})s-s}}{c(k^{\prime})s-s}$.
Combine (32) with (37)

$\frac{\mathcal{E}_{\beta}\left[(\mathfrak{C},n)\right]^{s}}{n^{2}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{18}}n^{\frac{s}{d}},(y\in\mathfrak{C})\leq\overline{H}(y)\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{23}}n\cdot[(1-\zeta){\rho{(X_{1:n}^{*})}}]^{\frac{-2ds-2d-s}{2d+1}}.$

Thus, we have

$\rho{(X_{1:n}^{*})}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}C_{17}}n^{(\frac{s}{d}-1)\cdot\frac{2d+1}{-s-2d-2ds}}$

(27) in the main text holds.

##  3 Weighted Riesz Particles MCMC

In this section, we develop a new sampler in which the propagation of
particles originates from weighted Riesz energy minimization, called weighted
Riesz particles, which inherits the special properties introduced in Section 2
when traversing through discrete deterministic subdomains of the parameter
space via particle interactions. We further extend this to sequential sampling
in the particle Metropolis-Hastings framework for the inference of hidden
Markov models, where the acceptance rate is approximated by the pseudo-
marginal Metropolis-Hastings algorithm.

###  3.1 Sequential Weighted Riesz Particles Sampling

Finding the optimal design of a configuration is non-deterministic, especially
for high dimensions, where point-by-point traversal leads to exponential
growth of the computational load. Many optimization algorithms have been
proposed for the optimal design of different configurations. Park [19]
proposed a two-stage exchange and Newton-type optimal design that minimizes
the integrated mean square error and maximizes the entropy, respectively. Ye
[20] further extended the column-pair algorithm. Morris and Mitchell [21]
adapted simulated annealing [22] to explore cells in the reachable domain.
Inspired by [21] and [23], we propose a constrained one-point-at-a-time greedy
algorithm for developing sequential designs of weighted Riesz particles as
follows.

(I) The selection of the initial point is critical because it is closely
related to the sampling of subsequent points. For the sake of stability of the
initial point, we use the particle with the largest mean value as the initial
point. Our desired value is
$\mathbb{E}(x)=\int_{0}^{x}xf(x)dx,x\in\mathfrak{C}$. The maximum point
$x_{0}$ can be obtained by $x_{0}=\text{arg}_{x}[\max\mathbb{E}(x)]$.
(II) Given an initial point $x_{0}$, we can sequentially generate
$x_{2},x_{3}...,x_{n}$. Suppose we have $n$ points using (4). Then the
$(n+1)$th point can be obtained by

$\displaystyle\begin{split}x_{n+1}&=\underset{x}{\text{arg}}\mathcal{E}_{\beta}(\mathfrak{C},n)=\text{arg}\min_{x}\left\\{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{\omega(x_{i},x_{j})}{\parallel
x_{i}-x_{j}\parallel^{s}}\right\\}^{\frac{1}{s}}.\end{split}$ (38)

(III) If $\left|x_{n+1}-x_{n}\right|\geq r_{\text{min}}(x_{1:n}^{*})$, we
further develop an acceptance criterion for $x_{n+1}$: Given $u\sim U(u\mid
0,1)$, if $\frac{\left|x_{n+1}-x_{n}\right|}{\left|x_{n}\right|}\geq u$, we
accept $x_{n+1}$; otherwise, we reject it.
(IV) From $n$ points, we can use some statistical methods such as regression
or kriging to estimate the latent manifold, where the density can be updated
with the ancestors of the samples, and then recursively generated for
different configurations of discrete manifolds.

###  3.2 Pseudo-marginal Metropolis-Hastings Sampling

Consider a hidden Markov model, where $x_{t}\sim f_{\theta}(x_{t}\mid
x_{t-1}),y_{t}\mid x_{t}\sim g_{\theta}(y_{t}\mid x_{t}),$ given
$\hat{x_{0}}$, $x_{t}(t=1,2,...n)$ is a latent variable to be observed, and
assuming that the measurements ${y_{t}}$ are conditionally independent of the
given ${x_{t}}$, the objective is to estimate $\\{x_{1:t},\theta\\}$. The
Particle Metropolis-Hastings [24], proposed an MCMC method to randomly explore
in the assumed measurable $\theta$ space and generate samples from the
approximated posterior $\hat{p}(x_{1:t},\theta\mid y_{1:t})$, whose closed-
form ${p}(x_{1:t},\theta\mid y_{1:t})={p}(\theta\mid
y_{1:t})\cdot{p}(x_{1:t}\mid y_{1:t},\theta)$ is not reachable for exact
pointwise evaluation.

We will introduce how weighted Riesz particles are used for the following
steps: In the parameter space, given $\\{\theta,x_{1:t}\\}$, a new status
$\\{\theta^{\prime},x_{1:t}^{{}^{\prime}}\\}$ is obtained from a proposal
$q(\theta^{\prime},x_{1:t}^{{}^{\prime}}\mid\theta,x_{1:t})$ with the
probability of acceptance

$\displaystyle\begin{split}\alpha&=\min\left\\{1,\frac{p(x_{1:t}^{{}^{\prime}},\theta^{\prime}\mid
y_{1:t})q(\theta,x_{1:t}\mid\theta^{\prime},x_{1:t}^{{}^{\prime}})}{p(x_{1:t},\theta\mid
y_{1:t})q(\theta^{\prime},x_{1:t}^{{}^{\prime}}\mid\theta,x_{1:t})}\right\\}=\min\left\\{1,\frac{p(y_{1:t}\mid\theta^{\prime})p(\theta^{{}^{\prime}})q(\theta\mid\theta^{\prime})}{p(y_{1:t}\mid\theta)p(\theta)q(\theta^{\prime}\mid\theta)}\right\\}.\end{split}$
(39)

The optimal importance density function that minimizes the variance of
importance weights, conditioned upon $x_{t-1}^{i}$ and ${y_{t}}$ has been
shown [25] to be

$\displaystyle\begin{split}q(x_{t}\mid x_{t-1}^{i},y_{t})_{opt}&=p(x_{t}\mid
x_{t-1}^{i},y_{t})=\frac{p(y_{t}\mid x_{t},x_{t-1}^{i})p(x_{t}\mid
x_{t-1}^{i})}{p(y_{t}\mid x_{t-1}^{i})}.\end{split}$

While sampling from $p(y_{t}\mid x_{t},x_{t-1}^{i})$ may not be
straightforward, since $x_{1:t}$ belongs to the "deterministic" part of the
discrete manifolds of the space, $x_{1:t}\in\mathfrak{C}$, the selection of
importance density $q(x_{t}|y_{t},x_{t-1}^{a_{t-1}^{(i)}})$ is from the
configuration of the minimum energy, where $a_{t}^{(i)}$ denotes the ancestor
of particle $x_{t}^{i}$. If $n\rightarrow\infty$, we have
$\lim_{n\rightarrow\infty}q(x_{t}|y_{t},x_{t-1}^{a_{t-1}^{(i)}})=p(x_{t}|x_{t-1}^{i},y_{t})$.
Thus, our proposal converges to the optimal importance density. We can obtain
a stochastic estimator of $p(y_{1:T}\mid\theta)$ and this likelihood can be
estimated by the weights

$\hat{p}_{\theta}(y_{1:T})=\prod_{t=1}^{T}(\frac{1}{N_{x}}\sum_{i=1}^{N_{x}}\frac{p(x_{t}|x_{t-1}^{i},y_{t})}{q(x_{t}|y_{t},x_{t-1}^{a_{t-1}^{(i)}})}).$
(40)

It can be shown that
$\mathbb{E}[\hat{p}_{\theta}(y_{1:T})]=p_{\theta}(y_{1:T})$ [26]. The variance
of the weights will approximate to $0$ when $N_{x}$ goes to infinity, this
would be verified by the following experiments.

From (39) and (40), we can get the estimated acceptance ratio

$\hat{\alpha}=\min\left\\{1,\frac{\hat{p}(y_{1:t}\mid\theta^{\prime})p(\theta^{{}^{\prime}})q(\theta\mid\theta^{\prime})}{\hat{p}(y_{1:t}\mid\theta)p(\theta)q(\theta^{\prime}\mid\theta)}\right\\}.$

##  4 Experiments

In this section, we present simulations embedding Riesz particles into
sequential Monte Carlo, as well as extending it to Bayesian analysis of linear
and nonlinear models. We ran the experiments on an HP Z200 workstation with an
Intel Core i5 and an $\\#82-18.04.1-$ Ubuntu SMP kernel. The code is available
at https://github.com/986876245/Weighted-Riesz-Particles.

###  4.1 Linear Gaussian State Space Model

The expression for the linear model is

$X_{t}\mid X_{t-1}\sim g(X_{t}|X_{t-1}),\ \ \ Y_{t}\mid X_{t}\sim
f(Y_{t}|X_{t})+e_{o}.$

Where $g(X_{t}|X_{t-1})=\phi X_{t-1}+e_{v}$, the noise from tracking
$e_{v}\sim\mathbb{N}(0,\delta_{v}^{2})$, the noise from observations
$e_{o}\sim\mathbb{N}(0,\delta_{o}^{2})$. Here we use (38), to compute
$\widehat{x}^{n}_{0:T}$, and $\widehat{p}^{n}_{\theta}(y_{1:T})$ with Riesz
particles by

$\displaystyle\begin{split}g(\widehat{x}_{t}|\widehat{x}_{t-1})&=\text{arg}\min_{x}\mathcal{E}_{\beta}(\mathfrak{C},N^{\prime})=\text{arg}\min_{x}\left\\{\sum_{x\neq\widehat{x}_{i},i=1}^{t-1}\frac{\omega(\widehat{x}_{i},x)}{\parallel\widehat{x}_{i}-x\parallel^{s}}\right\\}^{\frac{1}{s}},\\\
\omega(\widehat{x}_{i},x)&\propto
e^{\left[\kappa(\widehat{x}_{i})\kappa(x)+\beta\parallel\widehat{x}_{i}-x\parallel\right]^{-\frac{s}{2d}}}.\end{split}$

For a linear Gaussian state-space model, the optimal proposed distribution of
propagating particles $X^{i}_{t},i=1,...n$ is derived [27] from

$\begin{split}p_{\theta}^{\text{opt}}(X_{t}^{i}\mid X_{t-1}^{i},Y_{t})&\propto
g_{\theta}(Y_{t}\mid Y_{t}^{i})f_{\theta}(X_{t}^{i}\mid
X_{t-1}^{i})=\mathbb{N}(X^{i}_{t};\sigma^{2}[\sigma_{o}^{-2}Y_{t}+\sigma^{-2}_{v}\phi
X^{i}_{t-1}],\sigma^{2})\end{split}$

with $\sigma^{-2}=\sigma_{v}^{-2}+\sigma_{o}^{-2}$. To ensure the stability of
the algorithm and to minimize the variance of the incremental particle weights
for the current time step, we set $\kappa(x_{t})\propto
p_{\theta}^{\text{opt}}(X_{t}^{i}\mid X_{t-1}^{i},Y_{t})$. The latent state
$x_{t}$ can be represented with an unbiased quantity
$\widehat{x}^{n}_{t}=\frac{1}{n}\sum_{i=1}^{n}x^{i}_{t}$, here, $n$ is the
number of particle to estimate the current state, $N^{\prime}$ is the quantity
of Riesz particles to discretize the submanifolds. Ensuring a uniform
embedding of the particles with labels, we allocate the indices one by one
with the remainder of $n$ divided by $N^{\prime},(n>N^{\prime})$ for the
specific particles $x^{i}_{t}$.

We first study the generation of sets of particles with different cardinality
in the same space under the same conditions to test and verify the geometric
characteristics. When these particles are mapped to a particular low-
dimensional space shown in Figure 1, they will approach different straight
lines with the target parameters of the Ritz particles as
$\\{\beta=2,m=40,d=1\\}$, it satisfies the uniform distribution, which is
verified Theorem 2.2.1.

Figure 1: Theoretical Quantiles for 20, 100 and 200 Riesz particles.

We then embed these particles in a sequential Monte Carlo, where these states
interact recursively in the set of Riesz particles set $\mathbb{P}$, and for
comparison with the ground truth, we provide a simulated data record of the
model, with observations of $T=250$, initial values of
${\phi=0.75,\delta_{v}=1.00,\delta_{o}=0.10}$, $\widehat{x}_{0}=0$. We
recorded the estimated logarithmic bias and logarithmic MSE for Riesz
particles embedded in sequential Monte Carlo at different particle sets with
different cardinality, respectively. It is shown in Table 1.

Figure 2: Posterior estimate, burning process and ACF for different step size:$h_{1}=0.05,h_{2}=0.1,h_{3}=0.5$. Number of particles(n) | 10 | 20 | 50 | 100 | 200 | 500 | 1000
---|---|---|---|---|---|---|---
log-bias | -3.67 | -4.05 | -4.46 | -4.87 | -5.24 | -5.64 | -5.97
log-MSE | -6.94 | -7.64 | -8.43 | -9.27 | -9.98 | -10.81 | -11.44
Table 1: The log-bias and the log-MSE of the filte red states under 100 Riesz
particles for varying n.

Here, we extend the weighted Riesz particles to the pseudo-marginal
Metropolis-Hastings algorithm for Bayesian parameter inference of hidden
Markov models provided in Section 3.2. We estimate $\phi$, $\phi\in(-1,1)$
describing the persistence of the state and keeping
${delta_{v}=1.00,\delta_{e}=0.10}$ fixed, and $\phi_{0}=0.75$ a prior, and
specify the number of Riesz particles in the set $\mathbb{P}$ to be $100$,
which is much smaller than the number of iterations ($\geq 2000$), and from
this point on we have essentially scaled the particle set of the model, which
then requires very little evaluation for us to infer. We performed iterations
with different step sizes, $h_{1}=0.05,h_{2}=0.1,h_{3}=0.5$, and the lagged
autocorrelation plots of the posterior estimates, the combustion process, are
shown in Figure 2. The corresponding table is shown in Table 2.

Number of observations(T) | 10 | 20 | 50 | 100 | 200 | 500
---|---|---|---|---|---|---
Estimated posterior mean | 0.587 | 0.775 | 0.745 | 0.712 | 0.689 | 0.722
Estimated posterior variance | 0.031 | 0.016 | 0.011 | 0.009 | 0.005 | 0.002
Table 2: The estimated posterior mean and variance when varying T.

###  4.2 Nonlinear State Space Model

We continue by presenting a practical application of our proposal for tracking
stochastic volatility, a nonlinear state-space model with Gaussian noise, in
which log volatility is considered as a latent variable and an important
factor in financial risk management analysis. The stochastic volatility is
given by

$\displaystyle\begin{split}&x_{0}\sim\mathbb{N}(\mu,\frac{\sigma_{v}^{2}}{1-\rho^{2}}),\
\ x_{t}\mid x_{t-1}\sim\mathbb{N}(\mu+\rho(x_{t-1}-\mu),\sigma_{v}^{2}),\ \
y_{t}\mid x_{t}\sim\mathbb{N}(0,exp(x_{t})\tau),\end{split}$ (41)

where the parameters $\theta=\left\\{\mu,\rho,\sigma_{v},\tau\right\\}$,
$\mu\in\mathbb{R},\rho\in[-1,1]$, $\sigma_{v}$ and $\tau\in\mathbb{R}_{+}$,
denotes the mean value, persistence of volatility, standard deviation of the
state process, and instantaneous volatility, respectively.

The observation $y_{t}=\log(p_{t}/p_{t-1})$, also called log-returns,
represents the logarithm of the daily difference in the exchange rate $p_{t}$,
here, $\\{{p_{t}}\\}_{t=1}^{T}$ is the daily closing prices of the NASDAQ
OMXS30 index (a weighted average of the 30 most traded stocks on the Stockholm
stock exchange) [27]. We extract the data from  Quandl for the period between
January 2, 2015 and January 2, 2016. The resulting logarithmic returns are
shown in Figure 3. Large fluctuations are frequent, which is known as
volatility clustering in finance, and from equation (41), volatility
clustering effect is more likely to occur when $|\phi|$ is close to $1$ and
the standardized variance is very small. Here, the parameters of the objective
for Riesz particles: $\\{\beta=2,m=40,d=1\\}$, the size of Riesz particles is
$180$. The initial values are
$\mu_{0}=0,\sigma_{0}=1,\phi_{0}=0.95,\sigma_{\phi}=0.05,\delta_{v_{0}}=0.2,\sigma_{v}=0.03$
We obtain good performance, inferring a posteriori estimates after only a few
evaluations, greatly reducing the computational load of high-dimensional
sampling, as shown in Figure 3.

Figure 3: Top: The daily log-returns and estimated log-volatility with $95\%$
confidence intervals of the NASDAQ OMXS30 index for the period between
February 4, 2015 and February 4, 2016. Bottom: the posterior estimate(left),
the trace of the Markov chain(middle) and the corresponding ACF(right) of
$\mu$(purple), $\phi$(magenta) and $\sigma_{v}$(green) obtained from Riesz
particles embedded PMH. The dotted and solid gray lines in the left and middle
plots indicate the parameter posterior mean and the parameter priors,
respectively.

##  5 Conclusion

Markov Chain Monte Carlo (MCMC) provides a viable method for the inference of
hidden Markov models, but it tends to be computationally overloaded,
especially by the curse of dimensionality, as the Monte Carlo sampler
traverses the uncertain region of the parameter space in small random steps.
In the process, a large number of duplicate samples are burned, and these
duplicate samples increase the computational load significantly. We introduce
a deterministic sampling mechanism where all generated samples are produced by
particle interactions under a weighted Riesz energy minimization criterion.
All samples inherit the properties of good separation and bounded coverage
radius. We embed it into MCMC, obtain high performance through experiments
with Hidden Markov Models, require only a small number of evaluations, and we
can extend the method to high-dimensional sampling. For future research, we
will propose some kernel for the Riesz particles and scale the model with low
complexity of computations from the perspective of equilibrium states on high-
dimensional sampling.

## Acknowledgments

This was supported in part by BRBytes project.

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