- the $\texttt{i.d.}$ of representation found by internal layers of deep neural netwoks trained to recognize object and faces.
- the $\texttt{i.d.}$ of data taken from a variety of sensors and used to model human cognitive/emotional states.

$\textbf{Dataset}$ | $d$ | $\texttt{MLE}$ | $\texttt{kNNG}_{1}$ | $\texttt{kNNG}_{2}$ | $\texttt{BPCA}$ | $\texttt{Hein}$ | $\texttt{CD}$ | $\texttt{MiND}_{\texttt{KL}}$ | $\texttt{DANCo}$ | $\texttt{MLSVD}$ |
---|---|---|---|---|---|---|---|---|---|---|

$\mathcal{M}_1$ | $\textit{10.00}$ | 9.10 | 9.16 | 9.89 | 5.45 | 9.45 | 9.12 | 10.30 | 10.09 | $\textbf{10.00}$ |

$\mathcal{M}_2$ | $\textit{3.00}$ | 2.88 | 2.95 | 3.03 | $\textbf{3.00}$ | $\textbf{3.00}$ | 2.88 | $\textbf{3.00}$ | $\textbf{3.00}$ | $\textbf{3.00}$ |

$\mathcal{M}_3$ | $\textit{4.00}$ | 3.83 | 3.75 | 3.82 | $\textbf{4.00}$ | $\textbf{4.00}$ | 3.23 | $\textbf{4.00}$ | $\textbf{4.00}$ | 2.08 |

$\mathcal{M}_4$ | $\textit{4.00}$ | 3.95 | 4.05 | 4.76 | 4.25 | $\textbf{4.00}$ | 3.88 | 4.15 | $\textbf{4.00}$ | 8.00 |

$\mathcal{M}_5$ | $\textit{2.00}$ | 1.97 | 1.96 | 2.06 | $\textbf{2.00}$ | $\textbf{2.00}$ | 1.98 | $\textbf{2.00}$ | $\textbf{2.00}$ | $\textbf{2.00}$ |

$\mathcal{M}_6$ | $\textit{6.00}$ | 6.39 | 6.46 | 11.24 | 12.00 | $\textbf{5.95}$ | 5.91 | 6.50 | 7.00 | 12.00 |

$\mathcal{M}_7$ | $\textit{2.00}$ | 1.96 | 1.97 | 2.09 | $\textbf{2.00}$ | $\textbf{2.00}$ | 1.93 | 2.07 | $\textbf{2.00}$ | 2.35 |

$\mathcal{M}_9$ | $\textit{20.00}$ | 14.64 | 15.25 | 10.59 | 13.55 | 15.50 | 13.75 | 19.15 | 19.71 | $\textbf{20.00}$ |

$\mathcal{M}_{10a}$ | $\textit{10.00}$ | 8.26 | 8.62 | 10.21 | 5.20 | 8.90 | 8.09 | 9.85 | 9.86 | $\textbf{10.00}$ |

$\mathcal{M}_{10b}$ | $\textit{17.00}$ | 12.87 | 13.69 | 15.38 | 9.46 | 13.85 | 12.30 | 16.25 | 16.62 | $\textbf{17.00}$ |

$\mathcal{M}_{10c}$ | $\textit{24.00}$ | 16.96 | 17.67 | 21.42 | 13.3 | 17.95 | 15.58 | 22.55 | 24.28 | $\textbf{24.00}$ |

$\mathcal{M}_{10d}$ | $\textit{70.00}$ | 36.49 | 39.67 | 40.31 | 71.00 | 38.69 | 31.4 | 64.38 | 70.52 | $\textbf{70.00}$ |

$\mathcal{M}_{11}$ | $\textit{2.00}$ | 2.21 | 1.95 | 2.03 | 1.55 | 2.00 | 2.19 | $\textbf{2.00}$ | $\textbf{2.00}$ | 1.00 |

$\mathcal{M}_{12}$ | $\textit{20.00}$ | 15.82 | 16.40 | 24.89 | 13.7 | 15.00 | 11.26 | 19.35 | 19.90 | $\textbf{20.00}$ |

$\mathcal{M}_{13}$ | $\textit{1.00}$ | $\textbf{1.00}$ | 0.97 | 1.07 | 5.70 | $\textbf{1.00}$ | 1.14 | $\textbf{1.00}$ | $\textbf{1.00}$ | $\textbf{1.00}$ |

$\mathcal{M}_{N1}$ | $\textit{18.00}$ | 12.25 | 14.26 | 19.8 | 36.00 | 14.10 | 10.40 | 17.76 | 18.76 | $\textbf{18.00}$ |

$\mathcal{M}_{N2}$ | $\textit{24.00}$ | 14.72 | 17.62 | 26.87 | 48.00 | 17.76 | 12.43 | 23.76 | 25.76 | $\textbf{24.00}$ |

$\mathcal{M}_{beta}$ | $\textit{10.00}$ | 6.36 | 6.45 | 14.77 | 19.7 | 4.00 | 3.05 | 7.00 | 7.00 | $\textbf{10.00}$ |

$\mathcal{M}_{P3}$ | $\textit{3.00}$ | 2.89 | 2.93 | 3.12 | 7.00 | 2.00 | 2.43 | $\textbf{3.00}$ | $\textbf{3.00}$ | 1.00 |

$\mathcal{M}_{P6}$ | $\textit{6.00}$ | 4.96 | 4.98 | 5.82 | 7.00 | 2.66 | 3.58 | 5.04 | $\textbf{6.00}$ | 1.00 |

$\mathcal{M}_{P9}$ | $\textit{9.00}$ | 6.35 | 6.89 | 8.04 | 10.95 | 2.85 | 4.55 | 7.00 | $\textbf{8.00}$ | 1.00 |

$\texttt{MPE}$ | 17.29 | 14.50 | 16.79 | 62.62 | 19.92 | 25.96 | 5.55 | $\textbf{3.70}$ | 26.34 |

[1] P Campadelli, E Casiraghi, C Ceruti, and A Rozza. Intrinsic dimension
estimation: Relevant techniques and a benchmark framework. 2015.

[2] C. Ceruti, S. Bassis, A Rozza, G. Lombardi, E. Casiraghi, and P. Campadelli.
DANCo: an intrinsic Dimensionalty estimator exploiting Angle and Norm
Concentration. Pattern recognition, 2014.

[3] G. Lombardi, A. Rozza, C. Ceruti, E. Casiraghi, and P. Campadelli. Mini-
mum neighbor distance estimators of intrinsic dimension. Proc. of ECML-
PKDD, 6912:374–389, 2011.