fit

Fit simple model of local interpretable model-agnostic explanations (LIME)

Syntax

newresults = fit(results,queryPoint,numImportantPredictors)

newresults = fit(results,queryPoint,numImportantPredictors,Name,Value)

Description

newresults = fit(results,queryPoint,numImportantPredictors) fits a new simple model for the specified query point (queryPoint) by using the specified number or predictors (numImportantPredictors). The function returns a lime object newresults that contains the new simple model.

fit uses the simple model options that you specify when you create the lime object results. You can change the options using the name-value pair arguments of the fit function.

newresults = fit(results,queryPoint,numImportantPredictors,Name,Value) specifies additional options using one or more name-value pair arguments. For example, you can specify 'SimpleModelType','tree' to fit a decision tree model.

Examples

Explain Prediction with Linear Simple Model

Train a regression model and create a lime object that uses a linear simple model. When you create a lime object, if you do not specify a query point and the number of important predictors, then the software generates samples of a synthetic data set but does not fit a simple model. Use the object function fit to fit a simple model for a query point. Then display the coefficients of the fitted linear simple model by using the object function plot.

Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

load carbig

Create a table containing the predictor variables Acceleration, Cylinders, and so on, as well as the response variable MPG.

tbl = table(Acceleration,Cylinders,Displacement,Horsepower,Model_Year,Weight,MPG);

Removing missing values in a training set can help reduce memory consumption and speed up training for the fitrkernel function. Remove missing values in tbl.

tbl = rmmissing(tbl);

Create a table of predictor variables by removing the response variable from tbl.

tblX = removevars(tbl,'MPG');

Train a blackbox model of MPG by using the fitrkernel function.

rng('default') % For reproducibility
mdl = fitrkernel(tblX,tbl.MPG,'CategoricalPredictors',[2 5]);

Create a lime object. Specify a predictor data set because mdl does not contain predictor data.

results = lime(mdl,tblX)

results = 
  lime with properties:

             BlackboxModel: [1x1 RegressionKernel]
              DataLocality: 'global'
     CategoricalPredictors: [2 5]
                      Type: 'regression'
                         X: [392x6 table]
                QueryPoint: []
    NumImportantPredictors: []
          NumSyntheticData: 5000
             SyntheticData: [5000x6 table]
                    Fitted: [5000x1 double]
               SimpleModel: []
       ImportantPredictors: []
            BlackboxFitted: []
         SimpleModelFitted: []

results contains the generated synthetic data set. The SimpleModel property is empty ([]).

Fit a linear simple model for the first observation in tblX. Specify the number of important predictors to find as 3.

queryPoint = tblX(1,:)

queryPoint=1×6 table
    Acceleration    Cylinders    Displacement    Horsepower    Model_Year    Weight
    ____________    _________    ____________    __________    __________    ______

         12             8            307            130            70         3504 

results = fit(results,queryPoint,3);

Plot the lime object results by using the object function plot. To display an existing underscore in any predictor name, change the TickLabelInterpreter value of the axes to 'none'.

f = plot(results);
f.CurrentAxes.TickLabelInterpreter = 'none';

The plot displays two predictions for the query point, which correspond to the BlackboxFitted property and the SimpleModelFitted property of results.

The horizontal bar graph shows the coefficient values of the simple model, sorted by their absolute values. LIME finds Horsepower, Model_Year, and Cylinders as important predictors for the query point.

Model_Year and Cylinders are categorical predictors that have multiple categories. For a linear simple model, the software creates one less dummy variable than the number of categories for each categorical predictor. The bar graph displays only the most important dummy variable. You can check the coefficients of the other dummy variables using the SimpleModel property of results. Display the sorted coefficient values, including all categorical dummy variables.

[~,I] = sort(abs(results.SimpleModel.Beta),'descend');
table(results.SimpleModel.ExpandedPredictorNames(I)',results.SimpleModel.Beta(I), ...
    'VariableNames',{'Exteded Predictor Name','Coefficient'})

ans=17×2 table
      Exteded Predictor Name      Coefficient
    __________________________    ___________

    {'Horsepower'            }    -3.4485e-05
    {'Model_Year (74 vs. 70)'}    -6.1279e-07
    {'Model_Year (80 vs. 70)'}     -4.015e-07
    {'Model_Year (81 vs. 70)'}     3.4176e-07
    {'Model_Year (82 vs. 70)'}    -2.2483e-07
    {'Cylinders (6 vs. 8)'   }    -1.9024e-07
    {'Model_Year (76 vs. 70)'}     1.8136e-07
    {'Cylinders (5 vs. 8)'   }     1.7461e-07
    {'Model_Year (71 vs. 70)'}      1.558e-07
    {'Model_Year (75 vs. 70)'}     1.5456e-07
    {'Model_Year (77 vs. 70)'}      1.521e-07
    {'Model_Year (78 vs. 70)'}     1.4272e-07
    {'Model_Year (72 vs. 70)'}     6.7001e-08
    {'Model_Year (73 vs. 70)'}     4.7214e-08
    {'Cylinders (4 vs. 8)'   }     4.5118e-08
    {'Model_Year (79 vs. 70)'}    -2.2598e-08
      ⋮

Fit Simple Models for Multiple Query Points

Train a classification model and create a lime object that uses a decision tree simple model. Fit multiple models for multiple query points.

Load the CreditRating_Historical data set. The data set contains customer IDs and their financial ratios, industry labels, and credit ratings.

tbl = readtable('CreditRating_Historical.dat');

Create a table of predictor variables by removing the columns of customer IDs and ratings from tbl.

tblX = removevars(tbl,["ID","Rating"]);

Train a blackbox model of credit ratings by using the fitcecoc function.

blackbox = fitcecoc(tblX,tbl.Rating,'CategoricalPredictors','Industry')

blackbox = 
  ClassificationECOC
           PredictorNames: {1x6 cell}
             ResponseName: 'Y'
    CategoricalPredictors: 6
               ClassNames: {'A'  'AA'  'AAA'  'B'  'BB'  'BBB'  'CCC'}
           ScoreTransform: 'none'
           BinaryLearners: {21x1 cell}
               CodingName: 'onevsone'


  Properties, Methods

Create a lime object with the blackbox model.

rng('default') % For reproducibility
results = lime(blackbox);

Find two query points whose true rating values are AAA and B, respectively.

queryPoint(1,:) = tblX(find(strcmp(tbl.Rating,'AAA'),1),:);
queryPoint(2,:) = tblX(find(strcmp(tbl.Rating,'B'),1),:)

queryPoint=2×6 table
    WC_TA    RE_TA    EBIT_TA    MVE_BVTD    S_TA     Industry
    _____    _____    _______    ________    _____    ________

    0.121    0.413     0.057      3.647      0.466       12   
    0.019    0.009     0.042      0.257      0.119        1   

Fit a linear simple model for the first query point. Set the number of important predictors to 4.

newresults1 = fit(results,queryPoint(1,:),4);

Plot the LIME results newresults1 for the first query point. To display an existing underscore in any predictor name, change the TickLabelInterpreter value of the axes to 'none'.

f1 = plot(newresults1);
f1.CurrentAxes.TickLabelInterpreter = 'none';

Fit a linear decision tree model for the first query point.

newresults2 = fit(results,queryPoint(1,:),6,'SimpleModelType','tree');
f2 = plot(newresults2);
f2.CurrentAxes.TickLabelInterpreter = 'none';

The simple models in newresults1 and newresults2 both find MVE_BVTD and RE_TA as important predictors.

Fit a linear simple model for the second query point, and plot the LIME results for the second query point.

newresults3 = fit(results,queryPoint(2,:),4);
f3 = plot(newresults3);
f3.CurrentAxes.TickLabelInterpreter = 'none';

The prediction from the blackbox model is B, but the prediction from the simple model is not B. When the two predictions are not the same, you can specify a smaller 'KernelWidth' value. The software fits a simple model using weights that are more focused on the samples near the query point. If a query point is an outlier or is located near a decision boundary, then the two prediction values can be different, even if you specify a small 'KernelWidth' value. In such a case, you can change other name-value pair arguments. For example, you can generate a local synthetic data set (specify 'DataLocality' of lime as 'local') for the query point and increase the number of samples ('NumSyntheticData' of lime or fit) in the synthetic data set. You can also use a different distance metric ('Distance' of lime or fit).

Fit a linear simple model with a small 'KernelWidth' value.

newresults4 = fit(results,queryPoint(2,:),4,'KernelWidth',0.01);
f4 = plot(newresults4);
f4.CurrentAxes.TickLabelInterpreter = 'none';

The credit ratings for the first and second query points are AAA and B, respectively. The simple models in newresults1 and newresults4 both find MVE_BVTD, RE_TA, and WC_TA as important predictors. However, their coefficient values are different. The plots show that these predictors act differently depending on the credit ratings.

Input Arguments

results — LIME results

lime object

LIME results, specified as a lime object.

queryPoint — Query point

row vector of numeric values | single-row table

Query point around which the fit function fits the simple model, specified as a row vector of numeric values or a single-row table. The queryPoint value must have the same data type and the same number of columns as the predictor data (results.X or results.SyntheticData) in the lime object results.

queryPoint must not contain missing values.

Data Types: single | double | table

numImportantPredictors — Number of important predictors to use in simple model

positive integer scalar value

Number of important predictors to use in the simple model, specified as a positive integer scalar value.

• If 'SimpleModelType' is 'linear', then the software selects the specified number of important predictors and fits a linear model of the selected predictors.

• If 'SimpleModelType' is 'tree', then the software specifies the maximum number of decision splits (or branch nodes) as the number of important predictors so that the fitted decision tree uses at most the specified number of predictors.

Data Types: single | double

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'NumSyntheticData',2000,'SimpleModelType','tree' sets the number of samples to generate for the synthetic data set to 2000 and specifies the simple model type as a decision tree.

Cov — Covariance matrix for Mahalanobis distance metric

positive definite matrix

Covariance matrix for the Mahalanobis distance metric, specified as the comma-separated pair consisting of 'Cov' and a K-by-K positive definite matrix, where K is the number of predictors.

This argument is valid only if 'Distance' is 'mahalanobis'.

The default value is the 'Cov' value that you specify when creating the lime object results. The default 'Cov' value of lime is cov(PD,'omitrows'), where PD is the predictor data or synthetic predictor data. If you do not specify the 'Cov' value, then the software uses different covariance matrices when computing the distances for both the predictor data and the synthetic predictor data.

Example: 'Cov',eye(3)

Data Types: single | double

Distance — Distance metric

character vector | string scalar | function handle

Distance metric, specified as the comma-separated pair consisting of 'Distance' and a character vector, string scalar, or function handle.

• If the predictor data includes only continuous variables, then fit supports these distance metrics.

  Value	Description
  'euclidean'	Euclidean distance.
  'seuclidean'	Standardized Euclidean distance. Each coordinate difference between observations is scaled by dividing by the corresponding element of the standard deviation, S = std(PD,'omitnan'), where PD is the predictor data or synthetic predictor data. To specify different scaling, use the 'Scale' name-value argument.
  'mahalanobis'	Mahalanobis distance using the sample covariance of PD, C = cov(PD,'omitrows'). To change the value of the covariance matrix, use the 'Cov' name-value argument.
  'cityblock'	City block distance.
  'minkowski'	Minkowski distance. The default exponent is 2. To specify a different exponent, use the 'P' name-value argument.
  'chebychev'	Chebychev distance (maximum coordinate difference).
  'cosine'	One minus the cosine of the included angle between points (treated as vectors).
  'correlation'	One minus the sample correlation between points (treated as sequences of values).
  'spearman'	One minus the sample Spearman's rank correlation between observations (treated as sequences of values).
  @distfun	Custom distance function handle. A distance function has the form function D2 = distfun(ZI,ZJ) % calculation of distance ... where ZI is a 1-by-t vector containing a single observation. ZJ is an s-by-t matrix containing multiple observations. distfun must accept a matrix ZJ with an arbitrary number of observations. D2 is an s-by-1 vector of distances, and D2(k) is the distance between observations ZI and ZJ(k,:). If your data is not sparse, you can generally compute distance more quickly by using a built-in distance metric instead of a function handle.

• If the predictor data includes both continuous and categorical variables, then fit supports these distance metrics.

  Value	Description
  'goodall3'	Modified Goodall distance
  'ofd'	Occurrence frequency distance

For definitions, see Distance Metrics.

The default value is the 'Distance' value that you specify when creating the lime object results. The default 'Distance' value of lime is 'euclidean' if the predictor data includes only continuous variables, or 'goodall3' if the predictor data includes both continuous and categorical variables.

Example: 'Distance','ofd'

Data Types: char | string | function_handle

KernelWidth — Kernel width

numeric scalar value

Kernel width of the squared exponential (or Gaussian) kernel function, specified as the comma-separated pair consisting of 'KernelWidth' and a numeric scalar value.

The fit function computes distances between the query point and the samples in the synthetic predictor data set, and then converts the distances to weights by using the squared exponential kernel function. If you lower the 'KernelWidth' value, then fit uses weights that are more focused on the samples near the query point. For details, see LIME.

The default value is the 'KernelWidth' value that you specify when creating the lime object results. The default 'KernelWidth' value of lime is 0.75.

Example: 'KernelWidth',0.5

Data Types: single | double

NumNeighbors — Number of neighbors of query point

positive integer scalar value

Number of neighbors of the query point, specified as the comma-separated pair consisting of 'NumNeighbors' and a positive integer scalar value. This argument is valid only when the DataLocality property of results is 'local'.

The fit function estimates the distribution parameters of the predictor data using the specified number of nearest neighbors of the query point. Then the function generates synthetic predictor data using the estimated distribution.

If you specify a value larger than the number of observations in the predictor data set (results.X) in the lime object results, then fit uses all observations.

The default value is the 'NumNeighbors' value that you specify when creating the lime object results. The default 'NumNeighbors' value of lime is 1500.

Example: 'NumNeighbors',2000

Data Types: single | double

NumSyntheticData — Number of samples to generate for synthetic data set

results.NumSyntheticData (default) | positive integer scalar value

Number of samples to generate for the synthetic data set, specified as the comma-separated pair consisting of 'NumSyntheticData' and a positive integer scalar value.

The default value is the NumSyntheticData property value of results.

Example: 'NumSyntheticData',2500

Data Types: single | double

P — Exponent for Minkowski distance metric

positive scalar

Exponent for the Minkowski distance metric, specified as the comma-separated pair consisting of 'P' and a positive scalar.

This argument is valid only if 'Distance' is 'minkowski'.

The default value is the 'P' value that you specify when creating the lime object results. The default 'P' value of lime is 2.

Example: 'P',3

Data Types: single | double

Scale — Scale parameter value for standardized Euclidean distance metric

nonnegative numeric vector

Scale parameter value for the standardized Euclidean distance metric, specified as the comma-separated pair consisting of 'Scale' and a nonnegative numeric vector of length K, where K is the number of predictors.

This argument is valid only if 'Distance' is 'seuclidean'.

The default value is the 'Scale' value that you specify when creating the lime object results. The default 'Scale' value of lime is std(PD,'omitnan'), where PD is the predictor data or synthetic predictor data. If you do not specify the 'Scale' value, then the software uses different scale parameters when computing the distances for both the predictor data and the synthetic predictor data.

Example: 'Scale',quantile(X,0.75) - quantile(X,0.25)

Data Types: single | double

SimpleModelType — Type of simple model

'linear' | 'tree'

Type of the simple model, specified as the comma-separated pair consisting of 'SimpleModelType' and 'linear' or 'tree'.

• 'linear' — The software fits a linear model by using fitrlinear for regression or fitclinear for classification.

• 'tree' — The software fits a decision tree model by using fitrtree for regression or fitctree for classification.

The default value is the 'SimpleModelType' value that you specify when creating the lime object results. The default 'SimpleModelType' value of lime is 'linear'.

Example: 'SimpleModelType','tree'

Data Types: char | string

Output Arguments

newresults — LIME results

lime object

LIME results, returned as a lime object. newresults contains the new simple model.

To overwrite the input argument results, assign the output of fit to results:

results = fit(results,queryPoint,numImportantPredictors);

More About

Distance Metrics

A distance metric is a function that defines a distance between two observations. fit supports various distance metrics for continuous variables and a mix of continuous and categorical variables.

• Distance metrics for continuous variables

  Given an mx-by-n data matrix X, which is treated as mx (1-by-n) row vectors x₁, x₂, ..., x_mx, and an my-by-n data matrix Y, which is treated as my (1-by-n) row vectors y₁, y₂, ...,y_my, the various distances between the vector x_s and y_t are defined as follows:

  • Euclidean distance

    $$d_{st}^2=(x_s-y_t)(x_s-y_t{)}^{\prime }.$$

    The Euclidean distance is a special case of the Minkowski distance, where p = 2.

  • Standardized Euclidean distance

    $$d_{st}^2=(x_s-y_t)V^{-1}(x_s-y_t{)}^{\prime },$$

    where V is the n-by-n diagonal matrix whose jth diagonal element is (S(j))², where S is a vector of scaling factors for each dimension.

  • Mahalanobis distance

    $$d_{st}^2=(x_s-y_t)C^{-1}(x_s-y_t{)}^{\prime },$$

    where C is the covariance matrix.

  • City block distance

    $$d_{st}={\displaystyle \sum _{j=1}^n\left|x_{sj}-y_{tj}\right|}.$$

    The city block distance is a special case of the Minkowski distance, where p = 1.

  • Minkowski distance

    $$d_{st}=\sqrt[p]{{\displaystyle \sum _{j=1}^n{\left|x_{sj}-y_{tj}\right|}^p}}.$$

    For the special case of p = 1, the Minkowski distance gives the city block distance. For the special case of p = 2, the Minkowski distance gives the Euclidean distance. For the special case of p = ∞, the Minkowski distance gives the Chebychev distance.

  • Chebychev distance

    $$d_{st}={\max }_j\left\{\left|x_{sj}-y_{tj}\right|\right\}.$$

    The Chebychev distance is a special case of the Minkowski distance, where p = ∞.

  • Cosine distance

    $$d_{st}=\left(1-\frac{x_s{y^{\prime }}_t}{\sqrt{\left(x_s{x^{\prime }}_s\right)\left(y_t{y^{\prime }}_t\right)}}\right).$$

  • Correlation distance

    $$d_{st}=1-\frac{\left(x_s-{\overline{x}}_s\right){\left(y_t-{\overline{y}}_t\right)}^{\prime }}{\sqrt{\left(x_s-{\overline{x}}_s\right){\left(x_s-{\overline{x}}_s\right)}^{\prime }}\sqrt{\left(y_t-{\overline{y}}_t\right){\left(y_t-{\overline{y}}_t\right)}^{\prime }}},$$

    where

    $${\overline{x}}_s=\frac{1}{n}{\displaystyle \sum _j{x}_{sj}}$$

    and

    $${\overline{y}}_t=\frac{1}{n}{\displaystyle \sum _j{y}_{tj}}.$$

  • Spearman distance

    $$d_{st}=1-\frac{\left(r_s-{\overline{r}}_s\right){\left(r_t-{\overline{r}}_t\right)}^{\prime }}{\sqrt{\left(r_s-{\overline{r}}_s\right){\left(r_s-{\overline{r}}_s\right)}^{\prime }}\sqrt{\left(r_t-{\overline{r}}_t\right){\left(r_t-{\overline{r}}_t\right)}^{\prime }}},$$

    where

    • r_sj is the rank of x_sj taken over x_1j, x_2j, ...x_mx,j, as computed by tiedrank.

    • r_tj is the rank of y_tj taken over y_1j, y_2j, ...y_my,j, as computed by tiedrank.

    • r_s and r_t are the coordinate-wise rank vectors of x_s and y_t, that is, r_s = (r_s1, r_s2, ... r_sn) and r_t = (r_t1, r_t2, ... r_tn).

    • $${\overline{r}}_s=\frac{1}{n}{\displaystyle \sum _j{r}_{sj}}=\frac{\left(n+1\right)}{2}$$.

    • $${\overline{r}}_t=\frac{1}{n}{\displaystyle \sum _j{r}_{tj}}=\frac{\left(n+1\right)}{2}$$.

• Distance metrics for a mix of continuous and categorical variables

  • Modified Goodall distance

    This distance is a variant of the Goodall distance, which assigns a small distance if the matching values are infrequent regardless of the frequencies of the other values. For mismatches, the distance contribution of the predictor is 1/(number of variables).

  • Occurrence frequency distance

    For a match, the occurrence frequency distance assigns zero distance. For a mismatch, the occurrence frequency distance assigns a higher distance on a less frequent value and a lower distance on a more frequent value.

Algorithms

LIME

To explain a prediction of a machine learning model using LIME [1], the software generates a synthetic data set and fits a simple interpretable model to the synthetic data set by using lime and fit, as described in steps 1–5.

• If you specify the queryPoint and numImportantPredictors values of lime, then the lime function performs all steps.

• If you do not specify queryPoint and numImportantPredictors and specify 'DataLocality' as 'global' (default), then the lime function generates a synthetic data set (steps 1–2), and the fit function fits a simple model (steps 3–5).

• If you do not specify queryPoint and numImportantPredictors and specify 'DataLocality' as 'local', then the fit function performs all steps.

The lime and fit functions perform these steps:

1. Generate a synthetic predictor data set X_s using a multivariate normal distribution for continuous variables and a multinomial distribution for each categorical variable. You can specify the number of samples to generate by using the 'NumSyntheticData' name-value argument.

   • If 'DataLocality' is 'global' (default), then the software estimates the distribution parameters from the whole predictor data set (X or predictor data in blackbox).

   • If 'DataLocality' is 'local', then the software estimates the distribution parameters using the k-nearest neighbors of the query point, where k is the 'NumNeighbors' value. You can specify a distance metric to find the nearest neighbors by using the 'Distance' name-value argument.

   The software ignores missing values in the predictor data set when estimating the distribution parameters.

   Alternatively, you can provide a pregenerated, custom synthetic predictor data set by using the customSyntheticData input argument of lime.

2. Compute the predictions Y_s for the synthetic data set X_s. The predictions are predicted responses for regression or classified labels for classification. The software uses the predict function of the blackbox model to compute the predictions. If you specify blackbox as a function handle, then the software computes the predictions by using the function handle.

3. Compute the distances d between the query point and the samples in the synthetic predictor data set using the distance metric specified by 'Distance'.

4. Compute the weight values w_q of the samples in the synthetic predictor data set with respect to the query point q using the squared exponential (or Gaussian) kernel function

   $$w_q(x_s)=\exp \left(-\frac{1}{2}{\left(\frac{d(x_s,q)}{\sqrt{p}\sigma }\right)}^2\right).$$

   • x_s is a sample in the synthetic predictor data set X_s.

   • d(x_s,q) is the distance between the sample x_s and the query point q.

   • p is the number of predictors in X_s.

   • σ is the kernel width, which you can specify by using the 'KernelWidth' name-value argument. The default 'KernelWidth' value is 0.75.

   The weight value at the query point is 1, and then it converges to zero as the distance value increases. The 'KernelWidth' value controls how fast the weight value converges to zero. The lower the 'KernelWidth' value, the faster the weight value converges to zero. Therefore, the algorithm gives more weight to samples near the query point. Because this algorithm uses such weight values, the selected important predictors and fitted simple model effectively explain the predictions for the synthetic data locally, around the query point.

5. Fit a simple model.

   • If 'SimpleModelType' is 'linear' (default), then the software selects important predictors and fits a linear model of the selected important predictors.

     • Select n important predictors ($${\tilde{X}}_s$$) by using the group orthogonal matching pursuit (OMP) algorithm [2][3], where n is the numImportantPredictors value. This algorithm uses the synthetic predictor data set (X_s), predictions (Y_s), and weight values (w_q).

     • Fit a linear model of the selected important predictors ($${\tilde{X}}_s$$) to the predictions (Y_s) using the weight values (w_q). The software uses fitrlinear for regression or fitclinear for classification. For a multiclass model, the software uses the one-versus-all scheme to construct a binary classification problem. The positive class is the predicted class for the query point from the blackbox model, and the negative class refers to the other classes.

   • If 'SimpleModelType' is 'tree', then the software fits a decision tree model by using fitrtree for regression or fitctree for classification. The software specifies the maximum number of decision splits (or branch nodes) as the number of important predictors so that the fitted decision tree uses at most the specified number of predictors.

References

[1] Ribeiro, Marco Tulio, S. Singh, and C. Guestrin. "'Why Should I Trust You?': Explaining the Predictions of Any Classifier." In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 1135–44. San Francisco, California: ACM, 2016.

[2] Świrszcz, Grzegorz, Naoki Abe, and Aurélie C. Lozano. "Grouped Orthogonal Matching Pursuit for Variable Selection and Prediction." Advances in Neural Information Processing Systems (2009): 1150–58.

[3] Lozano, Aurélie C., Grzegorz Świrszcz, and Naoki Abe. "Group Orthogonal Matching Pursuit for Logistic Regression." Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (2011): 452–60.