# Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures

## Abstract

**:**

## 1. Introduction

## 2. Rate-Distortion Function and Shannon Lower Bound

#### 2.1. Rate-Distortion Function

#### 2.2. Shannon Lower Bound

**Lemma**

**1.**

**Proof.**

#### 2.3. Probability Density Achieving Tight SLB for All D

**Theorem**

**1.**

**Proof.**

## 3. Generalized Gaussian Source and Power Distortion Measure

#### 3.1. $\beta $-th Power Distortion Measure

#### 3.2. Tightness of the SLB

**Corollary**

**1.**

**Corollary**

**2.**

## 4. Rate-Distortion Bounds for Mismatching Pairs

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Example**

**1.**

**Example**

**2.**

**Corollary**

**5.**

**Proof.**

## 5. Distortion-Rate Bounds for $\mathit{\u03f5}$-Insensitive Loss

**Theorem**

**4.**

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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Watanabe, K. Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures . *Entropy* **2017**, *19*, 262.
https://doi.org/10.3390/e19060262

**AMA Style**

Watanabe K. Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures . *Entropy*. 2017; 19(6):262.
https://doi.org/10.3390/e19060262

**Chicago/Turabian Style**

Watanabe, Kazuho. 2017. "Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures " *Entropy* 19, no. 6: 262.
https://doi.org/10.3390/e19060262