Blurb::
Use the Bing Li "ladle" diagnostic to truncate subspace
Description::
Uses a trade-off criterion to determine where to truncate the active subspace.
The criterion is a function of the eigenvalues
and eigenvectors of the active subspace gradient matrix. This function compares
the decrease in eigenvalue amplitude with the increase in eigenvector variability
under bootstrap sampling of the gradient matrix. The active subspace size is taken to
be the index of the first minimum of this quantity.

*Usage Tips*

If this automated diagnostic does not yield desirable results, consider
using the explicit :dakkw:`model-active_subspace-dimension` truncation option or one of the other
truncation methods.
Topics::

Examples::

Theory::
Below is a brief outline of the Bing Li method of active subspace
identification. The first two steps are common to all active subspace
truncation methods.

1. Compute the gradient of the quantity of interest, :math:`q = f(\mathbf{x})` ,
    at several locations sampled from the input space,
    
.. math:: \nabla_{\mathbf{x}} f_i = \nabla f(\mathbf{x}_i).

2. Compute the eigendecomposition of the matrix :math:`\hat{\mathbf{C}}` ,
    
.. math:: \hat{\mathbf{C}} = \frac{1}{M}\sum_{i=1}^{M}\nabla_{\mathbf{x}} f_i\nabla_{\mathbf{x}} f_i^T = \hat{\mathbf{W}}\hat{\mathbf{\Lambda}}\hat{\mathbf{W}}^T,

where :math:`\hat{\mathbf{W}}`  has eigenvectors as columns,
    :math:`\hat{\mathbf{\Lambda}} = \text{diag}(\hat{\lambda}_1,\:\ldots\:,\hat{\lambda}_N)` 
    contains eigenvalues, and :math:`N`  is the total number of parameters.

3. Normalize the eigenvalues,
    
.. math:: \lambda_i = \frac{\hat{\lambda}_i}{\sum_j^N \hat{\lambda}_j}.

4. Use bootstrap sampling of the gradients found in step 1 to compute replicate
    eigendecompositions,
    
.. math:: \hat{\mathbf{C}}_j^* = \hat{\mathbf{W}}_j^*\hat{\mathbf{\Lambda}}_j^*\left(\hat{\mathbf{W}}_j^*\right)^T.

5. Compute variability of eigenvectors,
    
.. math:: f_i^0 = \frac{1}{M_{boot}}\sum_j^{M_{boot}}\left\lbrace 1 - \left\vert\text{det}\left(\hat{\mathbf{W}}_i^T\hat{\mathbf{W}}_{j,i}^*\right)\right\vert\right\rbrace ,

where :math:`\hat{\mathbf{W}}_i`  and :math:`\hat{\mathbf{W}}_{j,i}^*`  both
    contain only the first :math:`i`  eigenvectors and :math:`M_{boot}`  is the
    number of bootstrap samples. The value of the variability at the first index,
    :math:`f_1^0` , is defined as zero.

6. Normalize the eigenvector variability,
    
.. math:: f_i = \frac{f_i^0}{\sum_j^N f_j^0}.

7. The criterion, :math:`g_i` , is defined as,
    
.. math:: g_i = \lambda_i + f_i.

8. The index of first minimum of :math:`g_i`  is then the estimated active
    subspace rank.

For additional information, see Luo, Wei, and Bing Li. "Combining eigenvalues and variation of eigenvectors for order determination." SIAM, 2015.
Faq::

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