$L^p$-norm inequality using q-moment and its applications

TL;DR

This paper establishes new inequalities relating Lp-norms in Euclidean space using q-moments of escort distributions, with applications to entropy bounds and probability estimates.

Contribution

It introduces Lp-norm inequalities based on q-moments for Euclidean spaces, extending previous finite measure results and applying them to entropy and probability bounds.

Findings

Derived upper bounds for Renyi and Tsallis entropies using q-moments.

Established inequalities between two Renyi entropies.

Provided bounds for probabilities of subsets in Euclidean space.

Abstract

For a measurable function on a set which has a finite measure, an inequality holds between two Lp-norms. In this paper, we show similar inequalities for the Euclidean space and the Lebesgue measure by using a q-moment which is a moment of an escort distribution. As applications of these inequalities, we first derive upper bounds for the Renyi and the Tsallis entropies with given q-moment and derive an inequality between two Renyi entropies. Second, we derive an upper bound for the probability of a subset in the Euclidean space with given Lp-norm on the same set.

Equations

$$\Vert f{\Vert }_q\le \mu (S{)}^{\frac{1}{q}-\frac{1}{r}}\Vert f{\Vert }_r,$$

$$\displaystyle\|f\|_{p}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}{\bigl{(}\int_{S}|f(x)|^{p}\mathrm{d}^{n}x\bigr{)}}^{\frac{1}{p}}.$$

$$\Vert f{\Vert }_{\infty }\stackrel{\mathrm{def}}{=}\underset{x\in S}{\mathrm{esssup}}|f|.$$

$$\Vert f{\Vert }_q\le {(C{{\mu }_{q,\alpha }}^{\frac{1}{\alpha }})}^{\frac{1}{q}-\frac{1}{r}}\Vert f{\Vert }_r,$$

$$E_q[X]\stackrel{\mathrm{def}}{=}\frac{{\int }_{{\mathbb{R}}^n}x|f(x){|}^q{\mathrm{d}}^{n}x}{{\int }_{{\mathbb{R}}^n}|f(x){|}^q{\mathrm{d}}^{n}x}$$

$${\mu }_{q,\alpha }\stackrel{\mathrm{def}}{=}{E}_q[|X-b{|}^{\alpha }]$$

$$\Vert f{\Vert }_q\le {(C{{\mu }_{q,\alpha }}^{\frac{1}{\alpha }})}^{\frac{1}{q}-\frac{1}{r}}\Vert f{\Vert }_r,$$

$$V={\int }_{\mathbb{R}}|f(x){|}^q{\varphi }_t({\mu }_{q,\alpha }^{-1}|x-b{|}^{\alpha })\mathrm{d}x,$$

$$V={\int }_{\mathbb{R}}|f(x){|}^q\mathrm{d}x\times E_q[{\varphi }_t({\mu }_{q,\alpha }^{-1}|X-b{|}^{\alpha })]=\Vert f{\Vert }_q^q{E}_q[{\varphi }_t({\mu }_{q,\alpha }^{-1}|X-b{|}^{\alpha })]$$

$$V\ge \Vert f{\Vert }_q^q{\varphi }_t({\mu }_{q,\alpha }^{-1}{E}_q[|X-b{|}^{\alpha }])=\Vert f{\Vert }_q^q{\varphi }_t(1)=\Vert f{\Vert }_q^q.$$

$$V\le \Vert |f{|}^q{\Vert }_s{({\int }_{\mathbb{R}}{\varphi }_t({\mu }_{q,\alpha }^{-1}|x-b{|}^{\alpha }{)}^t\mathrm{d}x)}^{\frac{1}{t}},$$

$$V\le \Vert f{\Vert }_r^q{({\int }_{\mathbb{R}}{\varphi }_t({\mu }_{q,\alpha }^{-1}|x-b{|}^{\alpha }{)}^t\mathrm{d}x)}^{\frac{1}{t}},$$

$$\displaystyle\int_{\mathbb{R}}\phi_{t}(\mu_{q,\alpha}^{-1}|x-b|^{\alpha})^{t}\mathrm{d}x=\int_{\mathbb{R}}\exp\bigl{(}-\beta(\mu_{q,\alpha}^{-1}|x-b|^{\alpha}-1)\bigr{)}\mathrm{d}x,$$

$${\int }_{\mathbb{R}}{\varphi }_t({\mu }_{q,\alpha }^{-1}|x-b{|}^{\alpha }{)}^t\mathrm{d}x$$

$$V\le \Vert f{\Vert }_r^q{(C{{\mu }_{q,\alpha }}^{\frac{1}{\alpha }})}^{\frac{1}{t}},$$

$$\Vert f{\Vert }_q\le \Vert f{\Vert }_r{(C{{\mu }_{q,\alpha }}^{\frac{1}{\alpha }})}^{\frac{1}{qt}}.$$

$${\Sigma }_{q,b}\stackrel{\mathrm{def}}{=}{E}_q[(X-b)(X-b{)}^T],$$

$$\Vert f{\Vert }_q\le {(C{(\det {\Sigma }_{q,b})}^{\frac{1}{2}})}^{\frac{1}{q}-\frac{1}{r}}\Vert f{\Vert }_r.$$

$$V={\int }_{{\mathbb{R}}^n}|f(x){|}^q{\varphi }_t((x-b{)}^T{\Sigma }_{q,b}^{-1}(x-b)){\mathrm{d}}^{n}x,$$

$$V\ge \Vert f{\Vert }_q^q{\varphi }_t(n)=\Vert f{\Vert }_q^q.$$

$$\displaystyle V\leq\|f\|_{r}^{q}{\biggl{(}\int_{\mathbb{R}^{n}}\phi_{t}\bigl{(}(x-b)^{T}\Sigma_{q,b}^{-1}(x-b)\bigr{)}^{t}\mathrm{d}x\biggr{)}}^{\frac{1}{t}}.$$

$$\displaystyle\int_{\mathbb{R}^{n}}\phi_{t}\bigl{(}(x-b)^{T}\Sigma_{q,b}^{-1}(x-b)\bigr{)}^{t}\mathrm{d}x=\exp(\beta n)\int_{\mathbb{R}^{n}}\exp\bigl{(}-\beta(x-b)^{T}\Sigma_{q,b}^{-1}(x-b)\bigr{)}\mathrm{d}^{n}x.$$

$$\displaystyle\exp(\beta n)\int_{\mathbb{R}^{n}}\exp\bigl{(}-\beta(x-b)^{T}\Sigma_{q,b}^{-1}(x-b)\bigr{)}\mathrm{d}^{n}x=\exp(\beta n){(\det\Sigma_{q,b})}^{\frac{1}{2}}{\biggl{(}\frac{\pi}{\beta}\biggr{)}}^{\frac{n}{2}}.$$

$$V\le \Vert f{\Vert }_r^q{(C{(\det {\Sigma }_{q,b})}^{\frac{1}{2}})}^{\frac{1}{t}},$$

$$\Vert f{\Vert }_q\le \Vert f{\Vert }_r{(C{(\det {\Sigma }_{q,b})}^{\frac{1}{2}})}^{\frac{1}{qt}}.$$

$$h_p(X)\le \frac{1}{2}\log (\det \Sigma )+\log C.$$

$$h_p(X)\le \frac{1}{2}\log (\det {\Sigma }_p)+\log C.$$

$$h(X)\le \frac{1}{2}\log (\det \Sigma )+\log C,$$

$$\displaystyle\frac{1-q}{q}h_{q}(X)\leq\frac{1-r}{r}h_{r}(X)+\biggl{(}\frac{1}{q}-\frac{1}{r}\biggr{)}\bigl{(}\frac{1}{2}\log(\det\Sigma_{q})+\log C\bigr{)}.$$

$${\exp }_q(S_q(X))\le C{(\det \Sigma )}^{\frac{1}{2}},$$

Full text

Keywords: Lp-norm, q-moment, q-expectation value, Tsallis entropy, Renyi entropy, maximum entropy, escort distribution.

1. Introduction

We consider the measure space $(\mathbb{R}^{n},\mathcal{M},\mu)$, where $\mathcal{M}$ is a $\sigma$-algebra and $\mu$ is the Lebesgue measure. When $S\in\mathcal{M}$ has a finite measure, for $0<q<r$ and measurable function $f$ on $S$, the following inequality holds.

[TABLE]

where $\|f\|_{p}$ is a $L^{p}$-norm[7] defined as follows.

Definition 1**.**

Let $S\in\mathcal{M}$. For $0<p<\infty$,

[TABLE]

For $p=\infty$,

[TABLE]

In this paper, we show a similar inequality between two $L^{p}$-norms by using a $q$-moment when $S=\mathbb{R}^{n}$ (See Theorem 1 and 2 in section 2).

For $0<q<r\leq\infty$ and $n=1$,

[TABLE]

where $C$ is a constant and $\mu_{q,\alpha}$ is the $\alpha$-th order $q$-moment defined as follows. .

Definition 2**.**

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a measurable function which satisfies $0<\|f\|_{q}<\infty$. Let $X\in\mathbb{R}^{n}$. We define a q-expectation value [9, 1].

[TABLE]

Especially, when $q=1$, we write $E_{1}[X]$ as $E[X]$.

$E_{q}[X]$ is also a expected value of a escort distribution $\frac{|f(x)|^{q}}{\int_{\mathbb{R}^{n}}|f(x)|^{q}\mathrm{d}^{n}x}$[2].

Definition 3**.**

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function and $X\in\mathbb{R}$. For $b\in\mathbb{R}$ and $\alpha>0$, we define the $\alpha$-th order $q$-moment as follows.

[TABLE]

When $b=E_{q}[X]$, $\mu_{q,\alpha}$ is the central $q$-moment.

$\mu_{q,\alpha}$ is also the $\alpha$-th order moment of a escort distribution.

In (2), ${\mu_{q,\alpha}}^{\frac{1}{\alpha}}$ corresponds to $\mu(S)$ in (1) and we can interpret ${\mu_{q,\alpha}}^{\frac{1}{\alpha}}$ as the “range” of the region function $f$ spreads.

As applications of the inequality (2) and the multivariate version of (2), we derive an inequality between two Rényi entropies[6] and derive upper bounds for the Rényi and the Tsallis entropies with given $q$-moment. We also obtain an upper bound for the Shannon differential entropy [3] as a limit of the Rényi entropy.

Furthermore, we derive an upper bound for the probability of a subset in $\mathbb{R}^{n}$ with given $L^{p}$-norm on the same set. This is a generalization of the result in[5].

2. Main Results

Theorem 1**.**

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function with finite $q$-moment. Let $\|f\|_{q},\|f\|_{r}<\infty$.

For $0<q<r\leq\infty$,

[TABLE]

where $C$ is a constant which only depends on $\alpha$. The example value of $C$ is $C=\frac{2}{\alpha}\Gamma(\frac{1}{\alpha})(\alpha e)^{\frac{1}{\alpha}}$.

Proof. For a non-negative convex function $\phi_{t}$, we consider the following value.

[TABLE]

where the function $\phi_{t}(x)$ satisfies $\phi_{t}(1)=1$. We transform this equation as follows.

[TABLE]

Applying the Jensen’s inequality to this equation and using Definition 3 give

[TABLE]

Furthermore, for $1\leq s,t\leq\infty$, by applying the Hölder’s inequality to (4), we have

[TABLE]

where $\frac{1}{s}+\frac{1}{t}=1$. By assumption $q<r$, we can put $r=qs$ and $1\leq t<\infty$. Then, we have

[TABLE]

where we use $\||f|^{q}\|_{s}=\|f\|_{qs}^{q}=\|f\|_{r}^{q}$.

Since \phi_{t}(x)=\exp\bigl{(}-\frac{\beta}{t}(x-1)\bigr{)} is a convex function and satisfies $\phi_{t}(1)=1$, substituting \phi_{t}(x)=\exp\bigl{(}-\frac{\beta}{t}(x-1)\bigr{)} into RHS of this inequality, we have

[TABLE]

where $\beta>0$. Changing from the variable $x$ to $y=\mu_{q,\alpha}^{-\frac{1}{\alpha}}(x-b)$ gives

[TABLE]

$\exp(\beta)\beta^{-\frac{1}{\alpha}}$ has a minimum value at $\beta=\frac{1}{\alpha}$. Substituting this condition into (10) and (8) gives

[TABLE]

where $C=\frac{2}{\alpha}\Gamma(\frac{1}{\alpha})(\alpha e)^{\frac{1}{\alpha}}$. Combining (6) and (11) gives

[TABLE]

Combining $\frac{1}{s}+\frac{1}{t}=1$ and $r=qs$, we have $\frac{1}{qt}=\frac{1}{q}-\frac{1}{r}$. Substituting this equation into (12), the result follows.

Next, we prove the multivariate version.

Definition 4**.**

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a measurable function.

For $b\in\mathbb{R}^{n}$ and $X\in\mathbb{R}^{n}$, we define a multivariate $q$-moment as follows.

[TABLE]

where $T$ denotes the transpose of a vector.

When $b=E_{q}[X]$, we write $\Sigma_{q,b}$ as $\Sigma_{q}$ and $\Sigma_{q}$ is equal to a $q$-covariance matrix.

When $q=1$, we write $\Sigma_{q,b}$ as $\Sigma_{b}$ and $\Sigma$ denotes a covariance matrix.

Theorem 2**.**

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a measurable function with finite multivariate $q$-moment. Let $\|f\|_{q},\|f\|_{r}<\infty$.

For $0<q<r\leq\infty$ and $n\geq 1$

[TABLE]

$C$ is a constant which only depends on $n$. The example value of $C$ is $C={(2\pi e)}^{\frac{n}{2}}$.

Proof. We can prove this theorem in the same way as the theorem 1.

First, we consider the following value.

[TABLE]

where $\phi_{t}$ is a non-negative convex function which satisfies $\phi_{t}(n)=1$. By applying the Jensen’s inequality to this equation in the same way as Theorem 1, we get

[TABLE]

Next, for $1\leq s,t\leq\infty$, by applying the Hölder’s inequality to (14) and putting $r=qs$, we have

[TABLE]

Substituting \phi_{t}(x)=\exp\bigl{(}-\frac{\beta}{t}(x-n)\bigr{)} into this inequality gives

[TABLE]

Changing the variable from $x$ to $y=\Sigma_{q,b}^{-\frac{1}{2}}(x-b)$ gives

[TABLE]

$\exp(\beta)\beta^{-\frac{1}{2}}$ has a minimum value at $\beta=\frac{1}{2}$. Substituting this condition into (18) and combining (16) and (17) give

[TABLE]

where $C={(2\pi e)}^{\frac{n}{2}}$. Combining (15) and (19) gives

[TABLE]

Combining $\frac{1}{s}+\frac{1}{t}=1$ and $r=qs$, we have $\frac{1}{qt}=\frac{1}{q}-\frac{1}{r}$. Substituting this equation into (20), the result follows.

3. Applications

In this section, $C$ denotes a constant which only depends on the dimension $n$. The example value of $C$ is $(2\pi e)^{\frac{n}{2}}$.

3.1. Application for the Rényi and the Tsallis entropies

We derive upper bounds for the Rényi and the Tsallis entropies with given $q$-covariance matrix and derive an inequality between two Rényi entropies by using Theorem 2.

Corollary 1**.**

Let $f$ be a probability density function on $\mathbb{R}^{n}$ with finite $q$-covariance matrix and $h_{p}(X)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{p}{1-p}\log\|f\|_{p}$ be the Rényi entropy.

For $p>1$,

[TABLE]

For $0<p<1$,

[TABLE]

Proof. For $p>1$, by putting $q=1$, $r=p$, $b=E_{q}[X]$ and using $\|f\|_{1}=1$ in Theorem 2, the result follows.

For $0<p<1$, by putting $r=1$, $q=p$, $b=E_{q}[X]$ and using $\|f\|_{1}=1$ in Theorem 2, the result follows.

We can derive the optimal constant $C$ by using the distribution that maximizes the Rényi entropy[4].

In the limit $p\rightarrow 1$, the Rényi entropy is the Shannon differential entropy and we have

[TABLE]

where $h(X)$ is the Shannon differential entropy. When $C=(2\pi e)^{\frac{n}{2}}$, this inequality is consistent with the well-known upper bound of the Shannon entropy.

Corollary 2**.**

Let $f$ be a probability density function on $\mathbb{R}^{n}$ with finite $q$-covariance matrix and $h_{p}(X)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{p}{1-p}\log\|f\|_{p}$ be the Rényi entropy.

For $0<q<r\leq\infty$,

[TABLE]

Proof. Taking the logarithm of Theorem 2, the result follows.

Corollary 3**.**

Let $f$ be a probability density function on $\mathbb{R}^{n}$ with finite $q$-covariance matrix and $S_{q}(X)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{q-1}(1-\|f\|_{q}^{q})$ be the Tsallis entropy[8].

For $q>1$,

[TABLE]

where $\exp_{q}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[1+(1-q)x]^{\frac{1}{1-q}}$.

For $0<q<1$,

[TABLE]

Proof. From the definition of the Tsallis entropy and $\exp_{q}(x)$, we have $\exp_{q}(S_{q}(X))=\|f\|_{q}^{\frac{q}{1-q}}$. The rest part of the proof is almost the same as Corollary 1.

3.2. Application for the upper bound of a probability

We derive an upper bound for the probability of a subset with given $L^{p}$-norm on the same set. Since we use some functions in this subsection, we use the following notation.

Notation.

• •

For a non-negative measurable function $F$,

[TABLE]

When $b=E_{q}[X]$, we write $\Sigma_{F,b}$ as $\Sigma_{F}$.

• •

For $\Omega\subseteq\mathbb{R}^{n}$,

[TABLE]

Proposition 1**.**

Let $f$ be a probability density function on $\mathbb{R}^{n}$ with finite covariance matrix $\Sigma_{f}$. Let $\Omega\subseteq\mathbb{R}^{n}$ and $P(\Omega)$ be a probability of $\Omega$.

For $r>1$ and $n\geq 1$,

[TABLE]

Proof. First, we put

[TABLE]

Since ${(\det\Sigma_{f})}^{\frac{1}{2}}\mathrm{d}^{n}y=\mathrm{d}^{n}x$, we have

[TABLE]

From this equation, we have

[TABLE]

Especially, when $r=1$, $\|\hat{g}\|_{1}=\|g\|_{1}=P(\Omega)$.

Furthermore, using the GM-AM inequality ${(\det A)}^{\frac{1}{n}}\leq\frac{1}{n}\mathrm{Tr}A$, we have

[TABLE]

and

[TABLE]

(31) and (32) give

[TABLE]

Applying Theorem 2 for $\hat{g}$, $q=1$ and $b=0$ gives

[TABLE]

By using (30) and (33), we have

[TABLE]

By transforming this equation and using $P(\Omega)=\|g\|_{1}$, the result follows.

Corollary 4**.**

Let $f$ be a probability density function on $\mathbb{R}^{n}$ with finite covariance matrix $\Sigma_{f}$. Let $\Omega\subseteq\mathbb{R}^{n}$ and $P(\Omega)$ be a probability of $\Omega$.

For $n\geq 1$,

[TABLE]

Proof. By substituting $r=\infty$ into (27), the result follows.

Since $\|fI_{\Omega}\|_{\infty}=\operatorname*{ess\,sup}_{x\in\Omega}f(x)$, when the supremum of $f(x)$ in $\Omega$ is given, we can derive the probability upper bound by using Corollary 4.

4. Conclusion

In the first half, we have shown inequalities between two $L^{p}$-norms by using the $q$-moment for the Euclidean space and the Lebesgue measure.

In the latter half, by applying these inequalities to probability theory, we have derived the inequality that holds between two Rényi entropies, and have derived upper bounds for the Rényi and the Tsallis entropies with given $q$-moment. In particular, by using the result of the Rényi entropy, we have shown an upper bound for the Shannon entropy in the limit $q\rightarrow 1$.

Furthermore, we have derived the upper bound for the probability of the subset in $\mathbb{R}^{n}$ with given $L^{p}$-norm on the same set.

We hope we will find the optimal constants $C$ for each inequality.