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# `sklearn.manifold`.trustworthiness

sklearn.manifold.trustworthiness(X, X_embedded, *, n_neighbors=5, metric='euclidean')

  Indicate to what extent the local structure is retained.

  The trustworthiness is within [0, 1]. It is defined as

  \[T(k) = 1 - \frac{2}{nk (2n - 3k - 1)} \sum^n_{i=1} \sum_{j \in \mathcal{N}_{i}^{k}} \max(0, (r(i, j) - k))\]

  where for each sample i, \(\mathcal{N}_{i}^{k}\) are its k nearest neighbors in the output space, and every sample j is its \(r(i, j)\)-th nearest neighbor in the input space. In other words, any unexpected nearest neighbors in the output space are penalised in proportion to their rank in the input space.

  Parameters:
    X{array-like, sparse matrix} of shape (n_samples, n_features) or

      (n_samples, n_samples) If the metric is ‘precomputed’ X must be a square distance matrix. Otherwise it contains a sample per row.

    X_embedded{array-like, sparse matrix} of shape (n_samples, n_components)

      Embedding of the training data in low-dimensional space.

    n_neighbors int, default=5

      The number of neighbors that will be considered. Should be fewer than `n_samples / 2` to ensure the trustworthiness to lies within [0, 1], as mentioned in [1]. An error will be raised otherwise.

    metric str or callable, default=’euclidean’

      Which metric to use for computing pairwise distances between samples from the original input space. If metric is ‘precomputed’, X must be a matrix of pairwise distances or squared distances. Otherwise, for a list of available metrics, see the documentation of argument metric in `sklearn.pairwise.pairwise_distances` and metrics listed in `sklearn.metrics.pairwise.PAIRWISE_DISTANCE_FUNCTIONS`. Note that the “cosine” metric uses `cosine_distances`.

      New in version 0.20.

  Returns:
    trustworthiness float

      Trustworthiness of the low-dimensional embedding.

  References

  [1]

  Jarkko Venna and Samuel Kaski. 2001. Neighborhood Preservation in Nonlinear Projection Methods: An Experimental Study. In Proceedings of the International Conference on Artificial Neural Networks (ICANN ‘01). Springer-Verlag, Berlin, Heidelberg, 485-491.

  [2]

  Laurens van der Maaten. Learning a Parametric Embedding by Preserving Local Structure. Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics, PMLR 5:384-391, 2009.