# Mixtures¶

A convenient property of submodular functions is that the addition of two of them is still a submodular function. More generally, the linear combination of any number of submodular functions (assuming non-negative weights) is still a submodular function. Because of this, a mixture of submodular functions can be optimized using the same algorithms as an individual submodular function. Mixtures can be useful in situations where there are different important aspects of data that are each submodular.

The general form of a mixture function is

$f(X) = \sum\limits_{i=1}^{M} \alpha_{i} g_{i}(X)$

where $$f$$ indicates the mixture function, $$M$$ is the number of functions in the mixture, $$X$$ is a subset, $$\alpha_{i}$$ is the weight of the $$i$$-th function and $$g_{i}$$ is the $$i$$-th function.

## API Reference¶

This file contains code that implements mixtures of submodular functions.

class apricot.functions.mixture.MixtureSelection(n_samples, functions, weights=None, metric='ignore', initial_subset=None, optimizer='two-stage', optimizer_kwds={}, n_jobs=1, random_state=None, verbose=False)[source]

A selection approach based on a mixture of submodular functions.

This class implements a simple mixture of submodular functions for the purpose of selecting a representative subset of the data. The user passes in a list of instantiated submodular functions and their respective weights to the initialization. At each iteration in the selection procedure the gains from each submodular functions will be scaled by their respective weight and added together.

This class can also be used to add regularizers to the selection procedure. If a submodular function is mixed with another submodular function that acts as a regularizer, such as feature based selection mixed with a custom function measuring some property of the selected subset.

Parameters: n_samples : int The number of samples to return. submodular_functions : list The list of submodular functions to mix together. The submodular functions should be instantiated. weights : list, numpy.ndarray or None, optional The relative weight of each submodular function. This is the value that the gain from each submodular function is multiplied by before being added together. The default is equal weight for each function. 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. 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_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, optional Whether to print output during the selection process. pq : PriorityQueue The priority queue used to implement the lazy greedy algorithm. n_samples : int The number of samples to select. submodular_functions : list A concave function for transforming feature values, often referred to as phi in the literature. weights : numpy.ndarray The weights of each submodular function. ranking : numpy.array int The selected samples in the order of their gain with the first number in the ranking corresponding to the index of the first sample that was selected by the greedy procedure. 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)[source]

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. self : MixtureSelection 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. 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. 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.