Facility Location

Facility location functions are general purpose submodular functions that, when maximized, choose examples that represent the space of the data well. In many ways, optimizing a facility location function is simply a greedy version of k-medoids, where after the first few examples are selected, the subsequent ones are at the center of clusters. The function, like most graph-based functions, operates on a pairwise similarity matrix, and successively chooses examples that are similar to examples whose current most-similar example is still very dissimilar. Phrased another way, successively chosen examples are representative of underrepresented examples.

The general form of a facility location function is

\[f(X, Y) = \sum\limits_{y in Y} \max_{x in X} \phi(x, y)\]

where \(f\) indicates the function, \(X\) is a subset, \(Y\) is the ground set, and \(\phi\) is the similarity measure between two examples. Like most graph-based functons, the facility location function requires access to the full ground set.

API Reference

This file contains code that implements facility location submodular selection algorithms.

class apricot.functions.facilityLocation.FacilityLocationSelection(n_samples=10, metric='euclidean', initial_subset=None, optimizer='two-stage', optimizer_kwds={}, n_neighbors=None, n_jobs=1, random_state=None, verbose=False)

A selector based off a facility location submodular function.

NOTE: All ~pairwise~ values in your data must be non-negative for this selection to work.

This selector uses a facility location submodular function to perform selection. The facility location function is based on maximizing the pairwise similarities between the points in the data set and their nearest chosen point. The similarity function can be species by the user but must be non-negative where a higher value indicates more similar.

This implementation allows users to pass in either their own symmetric square matrix of similarity values, or a data matrix as normal and a function that calculates these pairwise values.

For more details, see https://las.inf.ethz.ch/files/krause12survey.pdf page 4.

Parameters:

n_samples : int

The number of samples to return.

metric : str, optional

The method for converting a data matrix into a square symmetric matrix of pairwise similarities. If a string, can be any of the metrics implemented in sklearn (see https://scikit-learn.org/stable/modules/ generated/sklearn.metrics.pairwise_distances.html), including “precomputed” if one has already generated a similarity matrix. Note that sklearn calculates distance matrices whereas apricot operates on similarity matrices, and so a distances.max() - distances transformation is performed on the resulting distances. For backcompatibility, ‘corr’ will be read as ‘correlation’. Default is ‘euclidean’.

initial_subset : list, numpy.ndarray or None, optional

If provided, this should be a list of indices into the data matrix to use as the initial subset, or a group of examples that may not be in the provided data should beused as the initial subset. If indices, the provided array should be one-dimensional. If a group of examples, the data should be 2 dimensional. Default is None.

optimizer : string or optimizers.BaseOptimizer, optional

The optimization approach to use for the selection. Default is ‘two-stage’, which makes selections using the naive greedy algorithm initially and then switches to the lazy greedy algorithm. Must be one of

‘random’ : randomly select elements (dummy optimizer) ‘modular’ : approximate the function using its modular upper bound ‘naive’ : the naive greedy algorithm ‘lazy’ : the lazy (or accelerated) greedy algorithm ‘approximate-lazy’ : the approximate lazy greedy algorithm ‘two-stage’ : starts with naive and switches to lazy ‘stochastic’ : the stochastic greedy algorithm ‘sample’ : randomly take a subset and perform selection on that ‘greedi’ : the GreeDi distributed algorithm ‘bidirectional’ : the bidirectional greedy algorithm

Default is ‘two-stage’.

optimizer_kwds : dict, optional

Arguments to pass into the optimizer object upon initialization. Default is {}.

n_neighbors : int or None, optional

The number of nearest neighbors to keep in the KNN graph, discarding all other neighbors. This process can result in a speedup but is an approximation.

n_jobs : int, optional

The number of cores to use for processing. This value is multiplied by 2 when used to set the number of threads. If set to -1, use all cores and threads. Default is -1.

random_state : int or RandomState or None, optional

The random seed to use for the random selection process. Only used for stochastic greedy.

verbose : bool

Whether to print output during the selection process.

Attributes:

n_samples : int

The number of samples to select.

ranking : numpy.array int

The selected samples in the order of their gain.

gains : numpy.array float

The gain of each sample in the returned set when it was added to the growing subset. The first number corresponds to the gain of the first added sample, the second corresponds to the gain of the second added sample, and so forth.

fit(self, X, y=None, sample_weight=None, sample_cost=None)

Run submodular optimization to select the examples.

This method is a wrapper for the full submodular optimization process. It takes in some data set (and optionally labels that are ignored during this process) and selects n_samples from it in the greedy manner specified by the optimizer.

This method will return the selector object itself, not the transformed data set. The transform method will then transform a data set to the selected points, or alternatively one can use the ranking stored in the self.ranking attribute. The fit_transform method will perform both optimization and selection and return the selected items.

Parameters:

X : list or numpy.ndarray, shape=(n, d)

The data set to transform. Must be numeric.

y : list or numpy.ndarray or None, shape=(n,), optional

The labels to transform. If passed in this function will return both the data and th corresponding labels for the rows that have been selected.

sample_weight : list or numpy.ndarray or None, shape=(n,), optional

The weight of each example. Currently ignored in apricot but included to maintain compatibility with sklearn pipelines.

sample_cost : list or numpy.ndarray or None, shape=(n,), optional

The cost of each item. If set, indicates that optimization should be performed with respect to a knapsack constraint.

Returns:

self : FacilityLocationSelection

The fit step returns this selector object.

fit_transform(self, X, y=None, sample_weight=None, sample_cost=None)

Run optimization and select a subset of examples.

This method will first perform the fit step and then perform the transform step, returning a transformed data set.

Parameters:

X : list or numpy.ndarray, shape=(n, d)

The data set to transform. Must be numeric.

y : list or numpy.ndarray or None, shape=(n,), optional

The labels to transform. If passed in this function will return both the data and the corresponding labels for the rows that have been selected. Default is None.

sample_weight : list or numpy.ndarray or None, shape=(n,), optional

The sample weights to transform. If passed in this function will return the selected labels (y) and the selected samples, even if no labels were passed in. Default is None.

sample_cost : list or numpy.ndarray or None, shape=(n,), optional

The cost of each item. If set, indicates that optimization should be performed with respect to a knapsack constraint.

Returns:

X_subset : numpy.ndarray, shape=(n_samples, d)

A subset of the data such that n_samples < n and n_samples is the integer provided at initialization.

y_subset : numpy.ndarray, shape=(n_samples,), optional

The labels that match with the indices of the samples if y is passed in. Only returned if passed in.

sample_weight_subset : numpy.ndarray, shape=(n_samples,), optional

The weight of each example.

transform(self, X, y=None, sample_weight=None)

Transform a data set to include only the selected examples.

This method will return a selection of X and optionally selections of y and sample_weight. The default setting is to select items based on the ranking determined in the fit step with examples in the same order as that ranking. Optionally, the whole data set can be returned, with the weights corresponding to samples that were not selected set to 0. This setting can be controlled by setting pipeline=True.

Parameters:

X : list or numpy.ndarray, shape=(n, d)

The data set to transform. Must be numeric.

y : list or numpy.ndarray or None, shape=(n,), optional

The labels to transform. If passed in this function will return both the data and the corresponding labels for the rows that have been selected. Default is None.

sample_weight : list or numpy.ndarray or None, shape=(n,), optional

The sample weights to transform. If passed in this function will return the selected labels (y) and the selected samples, even if no labels were passed in. Default is None.

Returns:

X_subset : numpy.ndarray, shape=(n_samples, d)

A subset of the data such that n_samples < n and n_samples is the integer provided at initialization.

y_subset : numpy.ndarray, shape=(n_samples,), optional

The labels that match with the indices of the samples if y is passed in. Only returned if passed in.

sample_weight_subset : numpy.ndarray, shape=(n_samples,), optional

The weight of each example.