# Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Frame Fundamentals

**Definition**

**1**

**Definition**

**2**

- 1.
- ${\u2225{f}_{j}\u2225}_{2}=1$,
- 2.
- $|\u2329{f}_{j},{f}_{k}\u232a|\text{}=\sqrt{\frac{N-m}{m(N-1)}}$, for $j\ne k$ where $\forall j,k\in J$.

#### 2.2. Mutual Coherence

#### 2.3. Kronecker Product

#### 2.4. Other Notations

## 3. Problem Formulation for Kronecker-Product-Based Frames

#### 3.1. Problem Formulation

#### 3.2. Kronecker Product with Unitary Matrix

**Theorem**

**1.**

**Proof of Theorem 1.**

#### 3.3. Kronecker Product with Special Unitary Matrix

**Corollary**

**1.**

#### 3.4. Computational Complexity of Objective Function

## 4. Algorithm for Solving Optimization Problem

## 5. Results

#### 5.1. Mutual Coherence

#### 5.2. CS Recovery Performance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The optimization process for obtaining the Kronecker-product-based frame. In this process, eliminating redundant elements in $\tilde{\mathbf{B}}$ and implementing local brute-force search are carried out by Steps 5 and 6 of Algorithm 2 in [41], respectively.

**Figure 4.**Performance of sparse signal recovery for various frames of $N=256$ and $m=64$. (

**a**) Success rate of support set recovery, (

**b**) mean squared error (MSE).

**Figure 5.**The distribution of average magnitude of inner products for 30-frame vectors randomly selected from the optimized harmonic and Kronecker-product-based frames, where $N=256$ and $m=64$.

**Table 1.**The computational complexities of several objective functions for the frames from Kronecker product, where $N=pq$.

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**MDPI and ACS Style**

Kwon, J.; Yu, N.Y.
Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames. *Appl. Sci.* **2022**, *12*, 11055.
https://doi.org/10.3390/app122111055

**AMA Style**

Kwon J, Yu NY.
Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames. *Applied Sciences*. 2022; 12(21):11055.
https://doi.org/10.3390/app122111055

**Chicago/Turabian Style**

Kwon, Jooeun, and Nam Yul Yu.
2022. "Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames" *Applied Sciences* 12, no. 21: 11055.
https://doi.org/10.3390/app122111055