consecutive eigenvalues is decreasing. Then, $$s_1<s_i, \;\;\forall i\ge 2.$$

The proofs of Propositions and Lemmas in this section are presented in the supplementary material. Moreover, the results of Proposition \[pr:rom\] and \[pr:sens2\] are also verified in supplementary material.

  Average samples per class                         2           3           4           5           6           7           8           9          10
  -------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
  Random                                          59.87       65.15       68.25       71.16       71.42       73.80       75.41       76.36       76.65
  Spectral Clustering                             61.11       66.66       68.01       69.31       70.97       71.60       71.58       73.88       74.14
  K-medoids                                       62.51       66.29       69.36       69.28       71.55       73.06       73.96       75.25       75.20
  DS3 [@Elhamifar2016DissimilaritySelection]      63.94       67.19       68.64       68.78       70.86       72.37       72.66       72.87       73.19
  Uncertainty [@Gal2017DeepData]                  59.50       67.27       69.89       72.64       74.01       74.86       74.67       77.08       76.79
  IPM                                           **64.93**     68.86       71.02       72.74       73.71       73.93       75.89       76.36       77.18
  IPM + Uncertainty                             **64.93**   7**0.23**   **73.85**   **73.46**   **76.97**   **76.52**   **77.66**   **78.19**   **78.11**

Applications of IPM {#sec:experiments}
===================

To validate our theoretical investigation and to empirically demonstrate the behavior and effectiveness of the proposed selection technique, we have performed extensive sets of experiments considering several different scenarios. We divide our experiments into three different subsections. In Section \[subsec:active-learning\], we use our algorithm in the *active learning* setting and show that IPM is able to reduce the sampling 