# Quantum Minimum Distance Classifier

## Abstract

**:**

## 1. Introduction

## 2. Minimum Distance Classification

- Computation of the centroid (i.e., the sample mean [26]) associated to each class, whose corresponding feature vector is given by:$${\overrightarrow{\mu}}_{l}=\frac{1}{{N}_{l}}\sum _{n=1}^{{N}_{l}}{\overrightarrow{x}}_{n},\phantom{\rule{1.em}{0ex}}l=1,2,\dots ,L,$$
- Classification of the object $\overrightarrow{x}$, provided by:$$argmi{n}_{l=1,\dots L}{d}_{E}(\overrightarrow{x},{\overrightarrow{\mu}}_{l}),\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{1.em}{0ex}}{d}_{E}(\overrightarrow{x},{\overrightarrow{\mu}}_{l})={\parallel \overrightarrow{x}-{\overrightarrow{\mu}}_{l}\parallel}_{2},$$

- True Positive Rate (TPR): $\mathrm{TPR}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}$;
- True Negative Rate (TNR): $\mathrm{TNR}=\frac{\mathrm{TN}}{\mathrm{TN}+\mathrm{FP}}$;
- False Positive Rate (FPR): $\mathrm{FPR}=\frac{\mathrm{FP}}{\mathrm{FP}+\mathrm{TN}}=1-\mathrm{TPN}$;
- False Negative Rate (FNR): $\mathrm{FNR}=\frac{\mathrm{FN}}{\mathrm{FN}+\mathrm{TP}}=1-\mathrm{TPR}$.

- Classification error (E): $\mathrm{E}=1-\frac{\mathrm{TP}}{{N}^{\prime}-N}$;
- Precision (P): $\mathrm{P}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}$;
- Cohen’s Kappa (K): $\mathrm{K}=\frac{\mathrm{Pr}\left(\mathrm{a}\right)-\mathrm{Pr}\left(\mathrm{e}\right)}{1-\mathrm{Pr}\left(\mathrm{e}\right)}$, where$\mathrm{Pr}(\mathrm{a})=\frac{\mathrm{TP}+\mathrm{TN}}{{N}^{\prime}-N}$, $\mathrm{Pr}(\mathrm{e})=\frac{(\mathrm{TP}+\mathrm{FP})(\mathrm{TP}+\mathrm{FN})+(\mathrm{FP}+\mathrm{TN})(\mathrm{TN}+\mathrm{FN})}{{({N}^{\prime}-N)}^{2}}$.

## 3. Mapping Real Patterns into Quantum States

- We map the vector $\overrightarrow{x}\in {\mathbb{R}}^{d}$ into a vector ${\overrightarrow{x}}^{\prime}\in {\mathbb{R}}^{d+1}$, whose first d features are the components of the vector $\overrightarrow{x}$ and the $(d+1)$-th feature is the norm of $\overrightarrow{x}$. Formally:$$\overrightarrow{x}=[{x}^{\left(1\right)},\dots ,{x}^{\left(d\right)}]\phantom{\rule{4pt}{0ex}}\mapsto \phantom{\rule{4pt}{0ex}}{\overrightarrow{x}}^{\prime}=[{x}^{\left(1\right)},\dots ,{x}^{\left(d\right)},\parallel \overrightarrow{x}\parallel ].$$
- We obtain the vector ${\overrightarrow{x}}^{\u2033}$ by dividing the first d components of the vector ${\overrightarrow{x}}^{\prime}$ for $\parallel \overrightarrow{x}\parallel $:$${\overrightarrow{x}}^{\prime}\phantom{\rule{4pt}{0ex}}\mapsto \phantom{\rule{4pt}{0ex}}{\overrightarrow{x}}^{\u2033}=\left(\right)open="["\; close="]">\frac{{x}^{\left(1\right)}}{\parallel \overrightarrow{x}\parallel},\dots ,\frac{{x}^{\left(d\right)}}{\parallel \overrightarrow{x}\parallel},\parallel \overrightarrow{x}\parallel $$
- We compute the norm of the vector ${\overrightarrow{x}}^{\u2033}$, i.e., $\parallel {\overrightarrow{x}}^{\u2033}\parallel =\sqrt{\parallel \overrightarrow{x}{\parallel}^{2}+1}$ and we map the vector ${\overrightarrow{x}}^{\u2033}$ into the normalized vector ${\overrightarrow{x}}^{\u2034}$ as follows:$${\overrightarrow{x}}^{\u2033}\phantom{\rule{4pt}{0ex}}\mapsto \phantom{\rule{4pt}{0ex}}{\overrightarrow{x}}^{\u2034}=\frac{{\overrightarrow{x}}^{\u2033}}{\parallel {\overrightarrow{x}}^{\u2033}\parallel}=\left(\right)open="["\; close="]">\frac{{x}^{\left(1\right)}}{\parallel \overrightarrow{x}\parallel \sqrt{\left|\right|\overrightarrow{x}{\left|\right|}^{2}+1}},\dots ,\frac{{x}^{\left(d\right)}}{\parallel \overrightarrow{x}\parallel \sqrt{\left|\right|\overrightarrow{x}{\left|\right|}^{2}+1}},\frac{\parallel \overrightarrow{x}\parallel}{\sqrt{\left|\right|\overrightarrow{x}{\left|\right|}^{2}+1}}$$

**Definition**

**1**(Density Pattern).

## 4. Density Pattern Classification

**Definition**

**2**(Quantum Centroid).

**Definition**

**3**(Trace Distance).

- Constructing the sets ${\mathcal{S}}_{\mathrm{tr}}^{q}$, ${\mathcal{S}}_{\mathrm{ts}}^{q}$ by mapping each pattern of the sets ${\mathcal{S}}_{\mathrm{tr}}$, ${\mathcal{S}}_{\mathrm{ts}}$ via the encoding introduced in Definition 1;
- Calculating the quantum centroids ${\rho}_{l}$ ($\forall l\in \{1,\dots L\}$), by using the quantum training set ${\mathcal{S}}_{\mathrm{tr}}^{q}$, in accordance with Definition 2;
- Classifying a density pattern ${\rho}_{\overrightarrow{x}}\in {S}_{\mathrm{ts}}^{q}$ by means of the optimization problem:$$argmi{n}_{l=1,\dots ,L}{d}_{T}({\rho}_{\overrightarrow{x}},{\rho}_{l}),$$

## 5. Experimental Results

#### 5.1. Comparison between QNMC and NMC

#### 5.2. Non-Invariance Under Rescaling

## 6. Conclusions and Future Developments

## Supplementary Materials

## Conflicts of Interest

## References

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**Figure 1.**Comparison between NMC (Nearest Mean Classifier) and QNMC (Quantum Nearest Mean Classifier) performance in terms of the classification error for the datasets (

**a**–

**c**) Appendicitis, (

**d**–

**f**) Monk, (

**g**–

**i**) Moon. In all the subfigures, the simple dashed line represents the QNMC classification error without rescaling, the dashed line with points represents the NMC classification error (which does not depend on the rescaling parameter), points with related error bars (red for Appendicitis, blue for Monk and green for Moon) represent the QNMC classification error for increasing values of the parameter $\gamma $.

**Table 1.**Characteristics of the datasets used in our experiments. The number of each class is shown between brackets.

Data Set | Class Size | Features (d) |
---|---|---|

Appendicitis | 106 (85 + 21) | 7 |

Balance | 625 (49 + 288 + 288) | 4 |

Banana | 5300 (2376 + 2924) | 2 |

Bands | 365 (135 + 230) | 19 |

Breast Cancer (I) | 683 (444 + 239) | 10 |

Breast Cancer (II) | 699 (458 + 241) | 9 |

Bupa | 345 (145 + 200) | 6 |

Chess | 3196 (1669 + 1527) | 36 |

Gaussian (I) | 400 (200 + 200) | 30 |

Gaussian (II) | 1000 (100 + 900) | 8 |

Gaussian (III) | 2050 (50 + 500 + 1500) | 8 |

Hayes-Roth | 132 (51 + 51 + 30) | 5 |

Ilpd | 583 (416 + 167) | 9 |

Ionosphere | 351 (225 + 126) | 34 |

Iris | 150 (50 + 50 + 50) | 4 |

Iris0 | 150 (100 + 50) | 4 |

Liver | 578 (413 + 165) | 10 |

Monk | 432 (204 + 228) | 6 |

Moon | 200 (100 + 100) | 2 |

Mutagenesis-Bond | 3995 (1040 + 2955) | 17 |

Page | 5472 (4913 + 559) | 10 |

Pima | 768 (500 + 268) | 8 |

Ring | 7400 (3664 + 3736) | 20 |

Segment | 2308 (1979 + 329) | 19 |

Thyroid (I) | 215 (180 + 35) | 5 |

Thyroid (II) | 215 (35 + 180) | 5 |

TicTac | 958 (626 + 332) | 9 |

QNMC | |||||

Dataset | E | TPR | TNR | P | K |

Appendicitis | 0.124 ± 0.058 | 0.876 ± 0.058 | 0.708 ± 0.219 | 0.886 ± 0.068 | 0.553 ± 0.223 |

Balance | 0.148 ± 0.018 | 0.852 ± 0.018 | 0.915 ± 0.014 | 0.862 ± 0.022 | 0.767 ± 0.029 |

Banana | 0.316 ± 0.017 | 0.684 ± 0.017 | 0.660 ± 0.017 | 0.684 ± 0.018 | 0.350 ± 0.034 |

Bands | 0.394 ± 0.053 | 0.606 ± 0.053 | 0.528 ± 0.071 | 0.606 ± 0.058 | 0.133 ± 0.112 |

Breast Cancer (I) | 0.386 ± 0.038 | 0.614 ± 0.038 | 0.444 ± 0.045 | 0.583 ± 0.044 | 0.062 ± 0.069 |

Breast Cancer (II) | 0.040 ± 0.015 | 0.946 ± 0.023 | 0.986 ± 0.016 | 0.993 ± 0.009 | 0.912 ± 0.033 |

Bupa | 0.389 ± 0.044 | 0.610 ± 0.044 | 0.641 ± 0.052 | 0.359 ± 0.052 | 0.066 ± 0.044 |

Chess | 0.256 ± 0.017 | 0.744 ± 0.017 | 0.747 ± 0.016 | 0.748 ± 0.016 | 0.488 ± 0.033 |

Gaussian (I) | 0.274 ± 0.051 | 0.726 ± 0.051 | 0.728 ± 0.049 | 0.745 ± 0.048 | 0.452 ± 0.099 |

Gaussian (II) | 0.210 ± 0.025 | 0.790 ± 0.025 | 0.744 ± 0.061 | 0.900 ± 0.019 | 0.308 ± 0.058 |

Gaussian (III) | 0.401 ± 0.036 | 0.599 ± 0.036 | 0.558 ± 0.026 | 0.654 ± 0.041 | 0152 ± 0.043 |

Hayes-Roth | 0.413 ± 0.039 | 0.588 ± 0.039 | 0.780 ± 0.025 | 0.602 ± 0.063 | 0.339 ± 0.060 |

Ilpd | 0.351 ± 0.037 | 0.649 ± 0.037 | 0.705 ± 0.056 | 0.734 ± 0.041 | 0.292 ± 0.073 |

Ionosphere | 0.165 ± 0.049 | 0.835 ± 0.049 | 0.764 ± 0.059 | 0.842 ± 0.051 | 0.624 ± 0.105 |

Iris | 0.047 ± 0.031 | 0.953 ± 0.031 | 0.977 ± 0.014 | 0.957 ± 0.028 | 0.929 ± 0.045 |

Iris0 | 0 ± 0 | 1 ± 0 | 1 ± 0 | 1 ± 0 | 1 ± 0 |

Liver | 0.342 ± 0.037 | 0.607 ± 0.057 | 0.783 ± 0.059 | 0.870 ± 0.039 | 0.318 ± 0.061 |

Monk | 0.132 ± 0.034 | 0.869 ± 0.034 | 0.885 ± 0.030 | 0.891 ± 0.025 | 0.738 ± 0.065 |

Moon | 0.156 ± 0.042 | 0.857 ± 0.063 | 0.831 ± 0.066 | 0.841 ± 0.066 | 0.683 ± 0.085 |

Mutagenesis-Bond | 0.266 ± 0.021 | 0.734 ± 0.021 | 0.281 ± 0.017 | 0.662 ± 0.040 | 0.023 ± 0.021 |

Page | 0.154 ± 0.009 | 0.846 ± 0.009 | 0.471 ± 0.039 | 0.869 ± 0.010 | 0.274 ± 0.035 |

Pima | 0.304 ± 0.030 | 0.696 ± 0.030 | 0.690 ± 0.044 | 0.720 ± 0.030 | 0.365 ± 0.066 |

Ring | 0.098 ± 0.006 | 0.902 ± 0.006 | 0.903 ± 0.006 | 0.905 ± 0.006 | 0.805 ± 0.012 |

Segment | 0.194 ± 0.017 | 0.807 ± 0.017 | 0.718 ± 0.045 | 0.864 ± 0.015 | 0.401 ± 0.041 |

Thyroid (I) | 0.078 ± 0.040 | 0.922 ± 0.040 | 0.747 ± 0.148 | 0.923 ± 0.043 | 0.695 ± 0.153 |

Thyroid (II) | 0.081 ± 0.034 | 0.919 ± 0.034 | 0.754 ± 0.122 | 0.923 ± 0.035 | 0.684 ± 0.121 |

Tic Tac | 0.410 ± 0.032 | 0.590 ± 0.032 | 0.597 ± 0.039 | 0.629 ± 0.036 | 0.172 ± 0.061 |

NMC | |||||

Dataset | E | TPR | TNR | P | K |

Appendicitis | 0.218 ± 0.086 | 0.782 ± 0.086 | 0.724 ± 0.167 | 0.835 ± 0.070 | 0.423 ± 0.201 |

Balance | 0.267 ± 0.038 | 0.733 ± 0.038 | 0.969 ± 0.014 | 0.925 ± 0.025 | 0.686 ± 0.034 |

Banana | 0.453 ± 0.019 | 0.548 ± 0.019 | 0.552 ± 0.020 | 0.556 ± 0.020 | 0.098 ± 0.038 |

Bands | 0.435 ± 0.048 | 0.565 ± 0.048 | 0.582 ± 0.055 | 0.605 ± 0.054 | 0.135 ± 0.092 |

Breast Cancer (I) | 0.442 ± 0.037 | 0.558 ± 0.037 | 0.464 ± 0.046 | 0.551 ± 0.039 | 0.022 ± 0.076 |

Breast Cancer (II) | 0.042 ± 0.015 | 0.973 ± 0.015 | 0.931 ± 0.032 | 0.963 ± 0.017 | 0.908 ± 0.033 |

Bupa | 0.530 ± 0.029 | 0.470 ± 0.029 | 0.625 ± 0.030 | 0.620 ± 0.036 | 0.066 ± 0.044 |

Chess | 0.307 ± 0.018 | 0.693 ± 0.018 | 0.707 ± 0.016 | 0.714 ± 0.016 | 0.393 ± 0.033 |

Gaussian (I) | 0.322 ± 0.042 | 0.679 ± 0.042 | 0.680 ± 0.043 | 0.685 ± 0.042 | 0.355 ± 0.085 |

Gaussian (II) | 0.320 ± 0.032 | 0.680 ± 0.032 | 0.588 ± 0.102 | 0.860 ± 0.032 | 0.129 ± 0.055 |

Gaussian (III) | 0.530 ± 0.029 | 0.470 ± 0.029 | 0.625 ± 0.030 | 0.620 ± 0.036 | 0.066 ± 0.044 |

Hayes-Roth | 0.503 ± 0.066 | 0.497 ± 0.066 | 0.689 ± 0.063 | 0.514 ± 0.075 | 0.180 ± 0.121 |

Ilpd | 0.470 ± 0.037 | 0.530 ± 0.037 | 0.757 ± 0.041 | 0.761 ± 0.037 | 0.193 ± 0.051 |

Ionosphere | 0.323 ± 0.051 | 0.677 ± 0.051 | 0.676 ± 0.051 | 0.680 ± 0.051 | 0.351 ± 0.102 |

Iris | 0.110 ± 0.052 | 0.890 ± 0.052 | 0.946 ± 0.033 | 0.904 ± 0.041 | 0.831 ± 0.087 |

Iris0 | 0.023 ± 0.021 | 0.977 ± 0.021 | 0.990 ± 0.009 | 0.980 ± 0.018 | 0.946 ± 0.050 |

Liver | 0.472 ± 0.048 | 0.388 ± 0.057 | 0.891 ± 0.055 | 0.905 ± 0.045 | 0.193 ± 0.060 |

Monk | 0.224 ± 0.022 | 0.776 ± 0.022 | 0.775 ± 0.022 | 0.779 ± 0.022 | 0.550 ± 0.043 |

Moon | 0.234 ± 0.065 | 0.772 ± 0.089 | 0.762 ± 0.085 | 0.771 ± 0.091 | 0.528 ± 0.130 |

Mutagenesis-Bond | 0.481 ± 0.013 | 0.519 ± 0.013 | 0.525 ± 0.029 | 0.630 ± 0.020 | 0.034 ± 0.029 |

Page | 0.215 ± 0.013 | 0.785 ± 0.013 | 0.205 ± 0.028 | 0.809 ± 0.014 | -0.010 ± 0.024 |

Pima | 0.375 ± 0.033 | 0.625 ± 0.033 | 0.546 ± 0.045 | 0.622 ± 0.037 | 0.173 ± 0.075 |

Ring | 0.238 ± 0.011 | 0.763 ± 0.011 | 0.761 ± 0.011 | 0.768 ± 0.011 | 0.524 ± 0.022 |

Segment | 0.311 ± 0.022 | 0.689 ± 0.022 | 0.824 ± 0.041 | 0.870 ± 0.014 | 0.286 ± 0.038 |

Thyroid (I) | 0.134 ± 0.042 | 0.867 ± 0.042 | 0.739 ± 0.150 | 0.887 ± 0.040 | 0.545 ± 0.139 |

Thyroid (II) | 0.134 ± 0.048 | 0.866 ± 0.048 | 0.777 ± 0.159 | 0.897 ± 0.046 | 0.542 ± 0.157 |

Tic Tac | 0.439 ± 0.031 | 0.561 ± 0.031 | 0.571 ± 0.042 | 0.606 ± 0.036 | 0.119 ± 0.063 |

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**MDPI and ACS Style**

Santucci, E.
Quantum Minimum Distance Classifier. *Entropy* **2017**, *19*, 659.
https://doi.org/10.3390/e19120659

**AMA Style**

Santucci E.
Quantum Minimum Distance Classifier. *Entropy*. 2017; 19(12):659.
https://doi.org/10.3390/e19120659

**Chicago/Turabian Style**

Santucci, Enrica.
2017. "Quantum Minimum Distance Classifier" *Entropy* 19, no. 12: 659.
https://doi.org/10.3390/e19120659