# fitrkernel

Fit Gaussian kernel regression model using random feature expansion

## Syntax

``Mdl = fitrkernel(X,Y)``
``Mdl = fitrkernel(Tbl,ResponseVarName)``
``Mdl = fitrkernel(Tbl,formula)``
``Mdl = fitrkernel(Tbl,Y)``
``Mdl = fitrkernel(___,Name,Value)``
``[Mdl,FitInfo] = fitrkernel(___)``
``[Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(___)``

## Description

`fitrkernel` trains or cross-validates a Gaussian kernel regression model for nonlinear regression. `fitrkernel` is more practical to use for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.

`fitrkernel` maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. Obtaining the linear model in the high-dimensional space is equivalent to applying the Gaussian kernel to the model in the low-dimensional space. Available linear regression models include regularized support vector machine (SVM) and least-squares regression models.

To train a nonlinear SVM regression model on in-memory data, see `fitrsvm`.

example

````Mdl = fitrkernel(X,Y)` returns a compact Gaussian kernel regression model trained using the predictor data in `X` and the corresponding responses in `Y`.```
````Mdl = fitrkernel(Tbl,ResponseVarName)` returns a kernel regression model `Mdl` trained using the predictor variables contained in the table `Tbl` and the response values in `Tbl.ResponseVarName`.```
````Mdl = fitrkernel(Tbl,formula)` returns a kernel regression model trained using the sample data in the table `Tbl`. The input argument `formula` is an explanatory model of the response and a subset of predictor variables in `Tbl` used to fit `Mdl`.```
````Mdl = fitrkernel(Tbl,Y)` returns a kernel regression model using the predictor variables in the table `Tbl` and the response values in vector `Y`.```

example

````Mdl = fitrkernel(___,Name,Value)` specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can implement least-squares regression, specify the number of dimension of the expanded space, or specify cross-validation options.```

example

````[Mdl,FitInfo] = fitrkernel(___)` also returns the fit information in the structure array `FitInfo` using any of the input arguments in the previous syntaxes. You cannot request `FitInfo` for cross-validated models.```

example

````[Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(___)` also returns the hyperparameter optimization results when you optimize hyperparameters by using the `'OptimizeHyperparameters'` name-value pair argument. ```

## Examples

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Train a kernel regression model for a tall array by using SVM.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer` function.

`mapreducer(0)`

Create a datastore that references the folder location with the data. The data can be contained in a single file, a collection of files, or an entire folder. Treat `'NA'` values as missing data so that `datastore` replaces them with `NaN` values. Select a subset of the variables to use. Create a tall table on top of the datastore.

```varnames = {'ArrTime','DepTime','ActualElapsedTime'}; ds = datastore('airlinesmall.csv','TreatAsMissing','NA',... 'SelectedVariableNames',varnames); t = tall(ds);```

Specify `DepTime` and `ArrTime` as the predictor variables (`X`) and `ActualElapsedTime` as the response variable (`Y`). Select the observations for which `ArrTime` is later than `DepTime`.

```daytime = t.ArrTime>t.DepTime; Y = t.ActualElapsedTime(daytime); % Response data X = t{daytime,{'DepTime' 'ArrTime'}}; % Predictor data```

Standardize the predictor variables.

`Z = zscore(X); % Standardize the data`

Train a default Gaussian kernel regression model with the standardized predictors. Extract a fit summary to determine how well the optimization algorithm fits the model to the data.

`[Mdl,FitInfo] = fitrkernel(Z,Y)`
```Found 6 chunks. |========================================================================= | Solver | Iteration / | Objective | Gradient | Beta relative | | | Data Pass | | magnitude | change | |========================================================================= | INIT | 0 / 1 | 4.307833e+01 | 4.345788e-02 | NaN | | LBFGS | 0 / 2 | 3.705713e+01 | 1.577301e-02 | 9.988252e-01 | | LBFGS | 1 / 3 | 3.704022e+01 | 3.082836e-02 | 1.338410e-03 | | LBFGS | 2 / 4 | 3.701398e+01 | 3.006488e-02 | 1.116070e-03 | | LBFGS | 2 / 5 | 3.698797e+01 | 2.870642e-02 | 2.234599e-03 | | LBFGS | 2 / 6 | 3.693687e+01 | 2.625581e-02 | 4.479069e-03 | | LBFGS | 2 / 7 | 3.683757e+01 | 2.239620e-02 | 8.997877e-03 | | LBFGS | 2 / 8 | 3.665038e+01 | 1.782358e-02 | 1.815682e-02 | | LBFGS | 3 / 9 | 3.473411e+01 | 4.074480e-02 | 1.778166e-01 | | LBFGS | 4 / 10 | 3.684246e+01 | 1.608942e-01 | 3.294968e-01 | | LBFGS | 4 / 11 | 3.441595e+01 | 8.587703e-02 | 1.420892e-01 | | LBFGS | 5 / 12 | 3.377755e+01 | 3.760006e-02 | 4.640134e-02 | | LBFGS | 6 / 13 | 3.357732e+01 | 1.912644e-02 | 3.842057e-02 | | LBFGS | 7 / 14 | 3.334081e+01 | 3.046709e-02 | 6.211243e-02 | | LBFGS | 8 / 15 | 3.309239e+01 | 3.858085e-02 | 6.411356e-02 | | LBFGS | 9 / 16 | 3.276577e+01 | 3.612292e-02 | 6.938579e-02 | | LBFGS | 10 / 17 | 3.234029e+01 | 2.734959e-02 | 1.144307e-01 | | LBFGS | 11 / 18 | 3.205763e+01 | 2.545990e-02 | 7.323180e-02 | | LBFGS | 12 / 19 | 3.183341e+01 | 2.472411e-02 | 3.689625e-02 | | LBFGS | 13 / 20 | 3.169307e+01 | 2.064613e-02 | 2.998555e-02 | |========================================================================= | Solver | Iteration / | Objective | Gradient | Beta relative | | | Data Pass | | magnitude | change | |========================================================================= | LBFGS | 14 / 21 | 3.146896e+01 | 1.788395e-02 | 5.967293e-02 | | LBFGS | 15 / 22 | 3.118171e+01 | 1.660696e-02 | 1.124062e-01 | | LBFGS | 16 / 23 | 3.106224e+01 | 1.506147e-02 | 7.947037e-02 | | LBFGS | 17 / 24 | 3.098395e+01 | 1.564561e-02 | 2.678370e-02 | | LBFGS | 18 / 25 | 3.096029e+01 | 4.464104e-02 | 4.547148e-02 | | LBFGS | 19 / 26 | 3.085475e+01 | 1.442800e-02 | 1.677268e-02 | | LBFGS | 20 / 27 | 3.078140e+01 | 1.906548e-02 | 2.275185e-02 | |========================================================================| ```
```Mdl = RegressionKernel PredictorNames: {'x1' 'x2'} ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 64 KernelScale: 1 Lambda: 8.5385e-06 BoxConstraint: 1 Epsilon: 5.9303 Properties, Methods ```
```FitInfo = struct with fields: Solver: 'LBFGS-tall' LossFunction: 'epsiloninsensitive' Lambda: 8.5385e-06 BetaTolerance: 1.0000e-03 GradientTolerance: 1.0000e-05 ObjectiveValue: 30.7814 GradientMagnitude: 0.0191 RelativeChangeInBeta: 0.0228 FitTime: 50.0477 History: [1x1 struct] ```

`Mdl` is a `RegressionKernel` model. To inspect the regression error, you can pass `Mdl` and the training data or new data to the `loss` function. Or, you can pass `Mdl` and new predictor data to the `predict` function to predict responses for new observations. You can also pass `Mdl` and the training data to the `resume` function to continue training.

`FitInfo` is a structure array containing optimization information. Use `FitInfo` to determine whether optimization termination measurements are satisfactory.

For improved accuracy, you can increase the maximum number of optimization iterations (`'IterationLimit'`) and decrease the tolerance values (`'BetaTolerance'` and `'GradientTolerance'`) by using the name-value pair arguments of `fitrkernel`. Doing so can improve measures like `ObjectiveValue` and `RelativeChangeInBeta` in `FitInfo`. You can also optimize model parameters by using the `'OptimizeHyperparameters'` name-value pair argument.

Load the `carbig` data set.

`load carbig`

Specify the predictor variables (`X`) and the response variable (`Y`).

```X = [Acceleration,Cylinders,Displacement,Horsepower,Weight]; Y = MPG;```

Delete rows of `X` and `Y` where either array has `NaN` values. Removing rows with `NaN` values before passing data to `fitrkernel` can speed up training and reduce memory usage.

```R = rmmissing([X Y]); % Data with missing entries removed X = R(:,1:5); Y = R(:,end); ```

Standardize the predictor variables.

`Z = zscore(X);`

Cross-validate a kernel regression model using 5-fold cross-validation.

`Mdl = fitrkernel(Z,Y,'Kfold',5)`
```Mdl = RegressionPartitionedKernel CrossValidatedModel: 'Kernel' ResponseName: 'Y' NumObservations: 392 KFold: 5 Partition: [1x1 cvpartition] ResponseTransform: 'none' Properties, Methods ```
`numel(Mdl.Trained)`
```ans = 5 ```

`Mdl` is a `RegressionPartitionedKernel` model. Because `fitrkernel` implements five-fold cross-validation, `Mdl` contains five `RegressionKernel` models that the software trains on training-fold (in-fold) observations.

Examine the cross-validation loss (mean squared error) for each fold.

`kfoldLoss(Mdl,'mode','individual')`
```ans = 5×1 13.0610 14.0975 24.0104 21.1223 24.3979 ```

Optimize hyperparameters automatically using the `'OptimizeHyperparameters'` name-value pair argument.

Load the `carbig` data set.

`load carbig`

Specify the predictor variables (`X`) and the response variable (`Y`).

```X = [Acceleration,Cylinders,Displacement,Horsepower,Weight]; Y = MPG;```

Delete rows of `X` and `Y` where either array has `NaN` values. Removing rows with `NaN` values before passing data to `fitrkernel` can speed up training and reduce memory usage.

```R = rmmissing([X Y]); % Data with missing entries removed X = R(:,1:5); Y = R(:,end); ```

Standardize the predictor variables.

`Z = zscore(X);`

Find hyperparameters that minimize five-fold cross-validation loss by using automatic hyperparameter optimization. Specify `'OptimizeHyperparameters'` as `'auto'` so that `fitrkernel` finds the optimal values of the `'KernelScale'`, `'Lambda'`, and `'Epsilon'` name-value pair arguments. For reproducibility, set the random seed and use the `'expected-improvement-plus'` acquisition function.

```rng('default') [Mdl,FitInfo,HyperparameterOptimizationResults] = fitrkernel(Z,Y,'OptimizeHyperparameters','auto',... 'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus'))```
```|====================================================================================================================| | Iter | Eval | Objective: | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Epsilon | | | result | log(1+loss) | runtime | (observed) | (estim.) | | | | |====================================================================================================================| | 1 | Best | 4.8295 | 1.0202 | 4.8295 | 4.8295 | 0.011518 | 6.8068e-05 | 0.95918 | | 2 | Best | 4.1488 | 0.20075 | 4.1488 | 4.1855 | 477.57 | 0.066115 | 0.091828 | | 3 | Accept | 4.1521 | 0.23448 | 4.1488 | 4.1747 | 0.0080478 | 0.0052867 | 520.84 | | 4 | Accept | 4.1506 | 0.19343 | 4.1488 | 4.1488 | 0.10935 | 0.35931 | 0.013372 | | 5 | Best | 4.1446 | 0.22183 | 4.1446 | 4.1446 | 326.29 | 2.5457 | 0.22475 | | 6 | Accept | 4.1521 | 0.20642 | 4.1446 | 4.1447 | 932.16 | 0.19667 | 873.68 | | 7 | Accept | 4.1501 | 0.44117 | 4.1446 | 4.1461 | 0.052426 | 2.5402 | 0.051319 | | 8 | Best | 4.1408 | 0.2327 | 4.1408 | 4.1402 | 850.91 | 0.01462 | 0.37284 | | 9 | Accept | 4.1521 | 0.35173 | 4.1408 | 4.1427 | 0.019352 | 0.012035 | 63.493 | | 10 | Accept | 4.1521 | 0.21123 | 4.1408 | 4.1452 | 853.22 | 1.0698 | 55.679 | | 11 | Accept | 4.1521 | 0.34269 | 4.1408 | 4.1416 | 1.4548 | 0.022234 | 26.275 | | 12 | Accept | 4.1509 | 0.19879 | 4.1408 | 4.1469 | 877.82 | 0.0071133 | 0.012021 | | 13 | Accept | 4.1422 | 0.85467 | 4.1408 | 4.1455 | 944.08 | 0.011177 | 0.31055 | | 14 | Accept | 4.2032 | 0.29126 | 4.1408 | 4.1405 | 979.21 | 0.010842 | 13.776 | | 15 | Accept | 4.1438 | 0.20706 | 4.1408 | 4.1509 | 0.001234 | 0.018449 | 0.044225 | | 16 | Best | 4.1372 | 0.17378 | 4.1372 | 4.1511 | 1.7802 | 2.5477 | 0.014737 | | 17 | Accept | 4.1521 | 0.24019 | 4.1372 | 4.1466 | 0.0015946 | 2.5474 | 590.35 | | 18 | Accept | 4.1452 | 0.20963 | 4.1372 | 4.1464 | 0.058846 | 1.0766 | 0.20569 | | 19 | Accept | 4.1521 | 0.32159 | 4.1372 | 4.1461 | 2.187 | 2.5531e-06 | 278.92 | | 20 | Accept | 4.1451 | 0.32948 | 4.1372 | 4.1461 | 0.0050283 | 0.039894 | 0.14402 | |====================================================================================================================| | Iter | Eval | Objective: | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Epsilon | | | result | log(1+loss) | runtime | (observed) | (estim.) | | | | |====================================================================================================================| | 21 | Best | 4.1362 | 0.26969 | 4.1362 | 4.1426 | 0.0029885 | 0.039099 | 6.3938 | | 22 | Accept | 4.1521 | 0.20719 | 4.1362 | 4.1449 | 0.035949 | 0.038533 | 80.585 | | 23 | Accept | 4.1399 | 0.36116 | 4.1362 | 4.1446 | 50.001 | 0.095432 | 0.19954 | | 24 | Accept | 4.1487 | 0.52381 | 4.1362 | 4.1374 | 0.012199 | 0.089894 | 0.034773 | | 25 | Accept | 4.1521 | 0.22703 | 4.1362 | 4.1447 | 0.0011871 | 0.30153 | 425.89 | | 26 | Accept | 4.1466 | 0.45165 | 4.1362 | 4.145 | 0.0011773 | 0.052213 | 0.017592 | | 27 | Accept | 4.1418 | 0.17754 | 4.1362 | 4.145 | 7.556 | 1.655 | 0.016225 | | 28 | Accept | 4.1407 | 0.4172 | 4.1362 | 4.145 | 0.01201 | 1.6696 | 0.38806 | | 29 | Accept | 5.4153 | 3.9701 | 4.1362 | 4.1365 | 0.0010531 | 1.1032e-05 | 0.034083 | | 30 | Accept | 4.1521 | 0.34684 | 4.1362 | 4.1364 | 652.19 | 2.6286e-06 | 882.02 | ```

```__________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 26.6629 seconds Total objective function evaluation time: 13.4353 Best observed feasible point: KernelScale Lambda Epsilon ___________ ________ _______ 0.0029885 0.039099 6.3938 Observed objective function value = 4.1362 Estimated objective function value = 4.1364 Function evaluation time = 0.26969 Best estimated feasible point (according to models): KernelScale Lambda Epsilon ___________ ________ _______ 0.0029885 0.039099 6.3938 Estimated objective function value = 4.1364 Estimated function evaluation time = 0.31488 ```
```Mdl = RegressionKernel ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 256 KernelScale: 0.0030 Lambda: 0.0391 BoxConstraint: 0.0652 Epsilon: 6.3938 Properties, Methods ```
```FitInfo = struct with fields: Solver: 'LBFGS-fast' LossFunction: 'epsiloninsensitive' Lambda: 0.0391 BetaTolerance: 1.0000e-04 GradientTolerance: 1.0000e-06 ObjectiveValue: 1.7716 GradientMagnitude: 0.0051 RelativeChangeInBeta: 8.5572e-05 FitTime: 0.0237 History: [] ```
```HyperparameterOptimizationResults = BayesianOptimization with properties: ObjectiveFcn: @createObjFcn/inMemoryObjFcn VariableDescriptions: [5x1 optimizableVariable] Options: [1x1 struct] MinObjective: 4.1362 XAtMinObjective: [1x3 table] MinEstimatedObjective: 4.1364 XAtMinEstimatedObjective: [1x3 table] NumObjectiveEvaluations: 30 TotalElapsedTime: 26.6629 NextPoint: [1x3 table] XTrace: [30x3 table] ObjectiveTrace: [30x1 double] ConstraintsTrace: [] UserDataTrace: {30x1 cell} ObjectiveEvaluationTimeTrace: [30x1 double] IterationTimeTrace: [30x1 double] ErrorTrace: [30x1 double] FeasibilityTrace: [30x1 logical] FeasibilityProbabilityTrace: [30x1 double] IndexOfMinimumTrace: [30x1 double] ObjectiveMinimumTrace: [30x1 double] EstimatedObjectiveMinimumTrace: [30x1 double] ```

For big data, the optimization procedure can take a long time. If the data set is too large to run the optimization procedure, you can try to optimize the parameters using only partial data. Use the `datasample` function and specify `'Replace','false'` to sample data without replacement.

## Input Arguments

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Predictor data to which the regression model is fit, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictor variables.

The length of `Y` and the number of observations in `X` must be equal.

Data Types: `single` | `double`

Response data, specified as an n-dimensional numeric vector. The length of `Y` must be equal to the number of observations in `X` or `Tbl`.

Data Types: `single` | `double`

Sample data used to train the model, specified as a table. Each row of `Tbl` corresponds to one observation, and each column corresponds to one predictor variable. Optionally, `Tbl` can contain one additional column for the response variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

• If `Tbl` contains the response variable, and you want to use all remaining variables in `Tbl` as predictors, then specify the response variable by using `ResponseVarName`.

• If `Tbl` contains the response variable, and you want to use only a subset of the remaining variables in `Tbl` as predictors, then specify a formula by using `formula`.

• If `Tbl` does not contain the response variable, then specify a response variable by using `Y`. The length of the response variable and the number of rows in `Tbl` must be equal.

Data Types: `table`

Response variable name, specified as the name of a variable in `Tbl`. The response variable must be a numeric vector.

You must specify `ResponseVarName` as a character vector or string scalar. For example, if `Tbl` stores the response variable `Y` as `Tbl.Y`, then specify it as `'Y'`. Otherwise, the software treats all columns of `Tbl`, including `Y`, as predictors when training the model.

Data Types: `char` | `string`

Explanatory model of the response variable and a subset of the predictor variables, specified as a character vector or string scalar in the form `"Y~x1+x2+x3"`. In this form, `Y` represents the response variable, and `x1`, `x2`, and `x3` represent the predictor variables.

To specify a subset of variables in `Tbl` as predictors for training the model, use a formula. If you specify a formula, then the software does not use any variables in `Tbl` that do not appear in `formula`.

The variable names in the formula must be both variable names in `Tbl` (`Tbl.Properties.VariableNames`) and valid MATLAB® identifiers. You can verify the variable names in `Tbl` by using the `isvarname` function. If the variable names are not valid, then you can convert them by using the `matlab.lang.makeValidName` function.

Data Types: `char` | `string`

Note

The software treats `NaN`, empty character vector (`''`), empty string (`""`), `<missing>`, and `<undefined>` elements as missing values, and removes observations with any of these characteristics:

• Missing value in the response variable

• At least one missing value in a predictor observation (row in `X` or `Tbl`)

• `NaN` value or `0` weight (`'Weights'`)

### 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: ```Mdl = fitrkernel(X,Y,'Learner','leastsquares','NumExpansionDimensions',2^15,'KernelScale','auto')``` implements least-squares regression after mapping the predictor data to the `2^15` dimensional space using feature expansion with a kernel scale parameter selected by a heuristic procedure.

Note

You cannot use any cross-validation name-value argument together with the `'OptimizeHyperparameters'` name-value argument. You can modify the cross-validation for `'OptimizeHyperparameters'` only by using the `'HyperparameterOptimizationOptions'` name-value argument.

Kernel Regression Options

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Box constraint, specified as the comma-separated pair consisting of `'BoxConstraint'` and a positive scalar.

This argument is valid only when `'Learner'` is `'svm'`(default) and you do not specify a value for the regularization term strength `'Lambda'`. You can specify either `'BoxConstraint'` or `'Lambda'` because the box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations (rows in `X`).

Example: `'BoxConstraint',100`

Data Types: `single` | `double`

Half the width of the epsilon-insensitive band, specified as the comma-separated pair consisting of `'Epsilon'` and `'auto'` or a nonnegative scalar value.

For `'auto'`, the `fitrkernel` function determines the value of `Epsilon` as `iqr(Y)/13.49`, which is an estimate of a tenth of the standard deviation using the interquartile range of the response variable `Y`. If `iqr(Y)` is equal to zero, then `fitrkernel` sets the value of `Epsilon` to 0.1.

`'Epsilon'` is valid only when `Learner` is `svm`.

Example: `'Epsilon',0.3`

Data Types: `single` | `double`

Number of dimensions of the expanded space, specified as the comma-separated pair consisting of `'NumExpansionDimensions'` and `'auto'` or a positive integer. For `'auto'`, the `fitrkernel` function selects the number of dimensions using `2.^ceil(min(log2(p)+5,15))`, where `p` is the number of predictors.

Example: `'NumExpansionDimensions',2^15`

Data Types: `char` | `string` | `single` | `double`

Kernel scale parameter, specified as the comma-separated pair consisting of `'KernelScale'` and `'auto'` or a positive scalar. MATLAB obtains the random basis for random feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

If you specify `'auto'`, then MATLAB selects an appropriate kernel scale parameter using a heuristic procedure. This heuristic procedure uses subsampling, so estimates can vary from one call to another. Therefore, to reproduce results, set a random number seed by using `rng` before training.

Example: `'KernelScale','auto'`

Data Types: `char` | `string` | `single` | `double`

Regularization term strength, specified as the comma-separated pair consisting of `'Lambda'` and `'auto'` or a nonnegative scalar.

For `'auto'`, the value of `'Lambda'` is 1/n, where n is the number of observations (rows in `X`).

You can specify either `'BoxConstraint'` or `'Lambda'` because the box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn).

Example: `'Lambda',0.01`

Data Types: `char` | `string` | `single` | `double`

Linear regression model type, specified as the comma-separated pair consisting of `'Learner'` and `'svm'` or `'leastsquares'`.

In the following table, $f\left(x\right)=T\left(x\right)\beta +b.$

• x is an observation (row vector) from p predictor variables.

• $T\left(·\right)$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$).

• β is a vector of m coefficients.

• b is the scalar bias.

ValueAlgorithmResponse rangeLoss function
`'leastsquares'`Linear regression via ordinary least squaresy ∊ (-∞,∞)Mean squared error (MSE): $\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^{2}$
`'svm'`Support vector machine regressionSame as `'leastsquares'`Epsilon-insensitive: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$

Example: `'Learner','leastsquares'`

Verbosity level, specified as the comma-separated pair consisting of `'Verbose'` and either `0` or `1`. `Verbose` controls the amount of diagnostic information `fitrkernel` displays at the command line.

ValueDescription
`0``fitrkernel` does not display diagnostic information.
`1``fitrkernel` displays and stores the value of the objective function, gradient magnitude, and other diagnostic information. `FitInfo.History` contains the diagnostic information.

Example: `'Verbose',1`

Data Types: `single` | `double`

Maximum amount of allocated memory (in megabytes), specified as the comma-separated pair consisting of `'BlockSize'` and a positive scalar.

If `fitrkernel` requires more memory than the value of `BlockSize` to hold the transformed predictor data, then MATLAB uses a block-wise strategy. For details about the block-wise strategy, see Algorithms.

Example: `'BlockSize',1e4`

Data Types: `single` | `double`

Random number stream for reproducibility of data transformation, specified as the comma-separated pair consisting of `'RandomStream'` and a random stream object. For details, see Random Feature Expansion.

Use `'RandomStream'` to reproduce the random basis functions that `fitrkernel` uses to transform the data in `X` to a high-dimensional space. For details, see Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.

Example: `'RandomStream',RandStream('mlfg6331_64')`

Other Regression Options

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Categorical predictors list, specified as one of the values in this table.

ValueDescription
Vector of positive integers

Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and `p`, where `p` is the number of predictors used to train the model.

If `fitrkernel` uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The `CategoricalPredictors` values do not count the response variable, observation weight variable, or any other variables that the function does not use.

Logical vector

A `true` entry means that the corresponding predictor is categorical. The length of the vector is `p`.

Character matrixEach row of the matrix is the name of a predictor variable. The names must match the entries in `PredictorNames`. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the entries in `PredictorNames`.
`"all"`All predictors are categorical.

By default, if the predictor data is in a table (`Tbl`), `fitrkernel` assumes that a variable is categorical if it is a logical vector, categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (`X`), `fitrkernel` assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the `'CategoricalPredictors'` name-value argument.

For the identified categorical predictors, `fitrkernel` creates dummy variables using two different schemes, depending on whether a categorical variable is unordered or ordered. For an unordered categorical variable, `fitrkernel` creates one dummy variable for each level of the categorical variable. For an ordered categorical variable, `fitrkernel` creates one less dummy variable than the number of categories. For details, see Automatic Creation of Dummy Variables.

Example: `'CategoricalPredictors','all'`

Data Types: `single` | `double` | `logical` | `char` | `string` | `cell`

Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of `PredictorNames` depends on the way you supply the training data.

• If you supply `X` and `Y`, then you can use `PredictorNames` to assign names to the predictor variables in `X`.

• The order of the names in `PredictorNames` must correspond to the column order of `X`. That is, `PredictorNames{1}` is the name of `X(:,1)`, `PredictorNames{2}` is the name of `X(:,2)`, and so on. Also, `size(X,2)` and `numel(PredictorNames)` must be equal.

• By default, `PredictorNames` is `{'x1','x2',...}`.

• If you supply `Tbl`, then you can use `PredictorNames` to choose which predictor variables to use in training. That is, `fitrkernel` uses only the predictor variables in `PredictorNames` and the response variable during training.

• `PredictorNames` must be a subset of `Tbl.Properties.VariableNames` and cannot include the name of the response variable.

• By default, `PredictorNames` contains the names of all predictor variables.

• A good practice is to specify the predictors for training using either `PredictorNames` or `formula`, but not both.

Example: `"PredictorNames",["SepalLength","SepalWidth","PetalLength","PetalWidth"]`

Data Types: `string` | `cell`

Response variable name, specified as a character vector or string scalar.

Example: `"ResponseName","response"`

Data Types: `char` | `string`

Response transformation, specified as either `'none'` or a function handle. The default is `'none'`, which means `@(y)y`, or no transformation. For a MATLAB function or a function you define, use its function handle for the response transformation. The function handle must accept a vector (the original response values) and return a vector of the same size (the transformed response values).

Example: Suppose you create a function handle that applies an exponential transformation to an input vector by using `myfunction = @(y)exp(y)`. Then, you can specify the response transformation as `'ResponseTransform',myfunction`.

Data Types: `char` | `string` | `function_handle`

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and a vector of scalar values or the name of a variable in `Tbl`. The software weights each observation (or row) in `X` or `Tbl` with the corresponding value in `Weights`. The length of `Weights` must equal the number of rows in `X` or `Tbl`.

If you specify the input data as a table `Tbl`, then `Weights` can be the name of a variable in `Tbl` that contains a numeric vector. In this case, you must specify `Weights` as a character vector or string scalar. For example, if weights vector `W` is stored as `Tbl.W`, then specify it as `'W'`. Otherwise, the software treats all columns of `Tbl`, including `W`, as predictors when training the model.

By default, `Weights` is `ones(n,1)`, where `n` is the number of observations in `X` or `Tbl`.

`fitrkernel` normalizes the weights to sum to 1.

Data Types: `single` | `double` | `char` | `string`

Cross-Validation Options

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Cross-validation flag, specified as the comma-separated pair consisting of `'Crossval'` and `'on'` or `'off'`.

If you specify `'on'`, then the software implements 10-fold cross-validation.

You can override this cross-validation setting using the `CVPartition`, `Holdout`, `KFold`, or `Leaveout` name-value pair argument. You can use only one cross-validation name-value pair argument at a time to create a cross-validated model.

Example: `'Crossval','on'`

Cross-validation partition, specified as a `cvpartition` partition object created by `cvpartition`. The partition object specifies the type of cross-validation and the indexing for the training and validation sets.

To create a cross-validated model, you can specify only one of these four name-value arguments: `CVPartition`, `Holdout`, `KFold`, or `Leaveout`.

Example: Suppose you create a random partition for 5-fold cross-validation on 500 observations by using `cvp = cvpartition(500,'KFold',5)`. Then, you can specify the cross-validated model by using `'CVPartition',cvp`.

Fraction of the data used for holdout validation, specified as a scalar value in the range (0,1). If you specify `'Holdout',p`, then the software completes these steps:

1. Randomly select and reserve `p*100`% of the data as validation data, and train the model using the rest of the data.

2. Store the compact, trained model in the `Trained` property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: `CVPartition`, `Holdout`, `KFold`, or `Leaveout`.

Example: `'Holdout',0.1`

Data Types: `double` | `single`

Number of folds to use in a cross-validated model, specified as a positive integer value greater than 1. If you specify `'KFold',k`, then the software completes these steps:

1. Randomly partition the data into `k` sets.

2. For each set, reserve the set as validation data, and train the model using the other `k` – 1 sets.

3. Store the `k` compact, trained models in a `k`-by-1 cell vector in the `Trained` property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: `CVPartition`, `Holdout`, `KFold`, or `Leaveout`.

Example: `'KFold',5`

Data Types: `single` | `double`

Leave-one-out cross-validation flag, specified as the comma-separated pair consisting of `'Leaveout'` and `'on'` or `'off'`. If you specify `'Leaveout','on'`, then, for each of the n observations (where n is the number of observations excluding missing observations), the software completes these steps:

1. Reserve the observation as validation data, and train the model using the other n – 1 observations.

2. Store the n compact, trained models in the cells of an n-by-1 cell vector in the `Trained` property of the cross-validated model.

To create a cross-validated model, you can use one of these four name-value pair arguments only: `CVPartition`, `Holdout`, `KFold`, or `Leaveout`.

Example: `'Leaveout','on'`

Convergence Controls

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Relative tolerance on the linear coefficients and the bias term (intercept), specified as the comma-separated pair consisting of `'BetaTolerance'` and a nonnegative scalar.

Let ${B}_{t}=\left[{\beta }_{t}{}^{\prime }\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$, that is, the vector of the coefficients and the bias term at optimization iteration t. If ${‖\frac{{B}_{t}-{B}_{t-1}}{{B}_{t}}‖}_{2}<\text{BetaTolerance}$, then optimization terminates.

If you also specify `GradientTolerance`, then optimization terminates when the software satisfies either stopping criterion.

Example: `'BetaTolerance',1e-6`

Data Types: `single` | `double`

Absolute gradient tolerance, specified as the comma-separated pair consisting of `'GradientTolerance'` and a nonnegative scalar.

Let $\nabla {ℒ}_{t}$ be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If ${‖\nabla {ℒ}_{t}‖}_{\infty }=\mathrm{max}|\nabla {ℒ}_{t}|<\text{GradientTolerance}$, then optimization terminates.

If you also specify `BetaTolerance`, then optimization terminates when the software satisfies either stopping criterion.

Example: `'GradientTolerance',1e-5`

Data Types: `single` | `double`

Size of the history buffer for Hessian approximation, specified as the comma-separated pair consisting of `'HessianHistorySize'` and a positive integer. At each iteration, `fitrkernel` composes the Hessian by using statistics from the latest `HessianHistorySize` iterations.

Example: `'HessianHistorySize',10`

Data Types: `single` | `double`

Maximum number of optimization iterations, specified as the comma-separated pair consisting of `'IterationLimit'` and a positive integer.

The default value is 1000 if the transformed data fits in memory, as specified by `BlockSize`. Otherwise, the default value is 100.

Example: `'IterationLimit',500`

Data Types: `single` | `double`

Hyperparameter Optimization Options

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Parameters to optimize, specified as the comma-separated pair consisting of `'OptimizeHyperparameters'` and one of these values:

• `'none'` — Do not optimize.

• `'auto'` — Use `{'KernelScale','Lambda','Epsilon'}`.

• `'all'` — Optimize all eligible parameters.

• Cell array of eligible parameter names.

• Vector of `optimizableVariable` objects, typically the output of `hyperparameters`.

The optimization attempts to minimize the cross-validation loss (error) for `fitrkernel` by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the `HyperparameterOptimizationOptions` name-value pair argument.

Note

The values of `'OptimizeHyperparameters'` override any values you specify using other name-value arguments. For example, setting `'OptimizeHyperparameters'` to `'auto'` causes `fitrkernel` to optimize hyperparameters corresponding to the `'auto'` option and to ignore any specified values for the hyperparameters.

The eligible parameters for `fitrkernel` are:

• `Epsilon``fitrkernel` searches among positive values, by default log-scaled in the range `[1e-3,1e2]*iqr(Y)/1.349`.

• `KernelScale``fitrkernel` searches among positive values, by default log-scaled in the range `[1e-3,1e3]`.

• `Lambda``fitrkernel` searches among positive values, by default log-scaled in the range `[1e-3,1e3]/n`, where `n` is the number of observations.

• `Learner``fitrkernel` searches among `'svm'` and `'leastsquares'`.

• `NumExpansionDimensions``fitrkernel` searches among positive integers, by default log-scaled in the range `[100,10000]`.

Set nondefault parameters by passing a vector of `optimizableVariable` objects that have nondefault values. For example:

```load carsmall params = hyperparameters('fitrkernel',[Horsepower,Weight],MPG); params(2).Range = [1e-4,1e6];```

Pass `params` as the value of `'OptimizeHyperparameters'`.

By default, the iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function is log(1 + cross-validation loss). To control the iterative display, set the `Verbose` field of the `'HyperparameterOptimizationOptions'` name-value argument. To control the plots, set the `ShowPlots` field of the `'HyperparameterOptimizationOptions'` name-value argument.

For an example, see Optimize Kernel Regression.

Example: `'OptimizeHyperparameters','auto'`

Options for optimization, specified as a structure. This argument modifies the effect of the `OptimizeHyperparameters` name-value argument. All fields in the structure are optional.

Field NameValuesDefault
`Optimizer`
• `'bayesopt'` — Use Bayesian optimization. Internally, this setting calls `bayesopt`.

• `'gridsearch'` — Use grid search with `NumGridDivisions` values per dimension.

• `'randomsearch'` — Search at random among `MaxObjectiveEvaluations` points.

`'gridsearch'` searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command `sortrows(Mdl.HyperparameterOptimizationResults)`.

`'bayesopt'`
`AcquisitionFunctionName`

• `'expected-improvement-per-second-plus'`

• `'expected-improvement'`

• `'expected-improvement-plus'`

• `'expected-improvement-per-second'`

• `'lower-confidence-bound'`

• `'probability-of-improvement'`

Acquisition functions whose names include `per-second` do not yield reproducible results because the optimization depends on the runtime of the objective function. Acquisition functions whose names include `plus` modify their behavior when they are overexploiting an area. For more details, see Acquisition Function Types.

`'expected-improvement-per-second-plus'`
`MaxObjectiveEvaluations`Maximum number of objective function evaluations.`30` for `'bayesopt'` and `'randomsearch'`, and the entire grid for `'gridsearch'`
`MaxTime`

Time limit, specified as a positive real scalar. The time limit is in seconds, as measured by `tic` and `toc`. The run time can exceed `MaxTime` because `MaxTime` does not interrupt function evaluations.

`Inf`
`NumGridDivisions`For `'gridsearch'`, the number of values in each dimension. The value can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. This field is ignored for categorical variables.`10`
`ShowPlots`Logical value indicating whether to show plots. If `true`, this field plots the best observed objective function value against the iteration number. If you use Bayesian optimization (`Optimizer` is `'bayesopt'`), then this field also plots the best estimated objective function value. The best observed objective function values and best estimated objective function values correspond to the values in the `BestSoFar (observed)` and ```BestSoFar (estim.)``` columns of the iterative display, respectively. You can find these values in the properties `ObjectiveMinimumTrace` and `EstimatedObjectiveMinimumTrace` of `Mdl.HyperparameterOptimizationResults`. If the problem includes one or two optimization parameters for Bayesian optimization, then `ShowPlots` also plots a model of the objective function against the parameters.`true`
`SaveIntermediateResults`Logical value indicating whether to save results when `Optimizer` is `'bayesopt'`. If `true`, this field overwrites a workspace variable named `'BayesoptResults'` at each iteration. The variable is a `BayesianOptimization` object.`false`
`Verbose`

Display at the command line:

• `0` — No iterative display

• `1` — Iterative display

• `2` — Iterative display with extra information

For details, see the `bayesopt` `Verbose` name-value argument and the example Optimize Classifier Fit Using Bayesian Optimization.

`1`
`UseParallel`Logical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization.`false`
`Repartition`

Logical value indicating whether to repartition the cross-validation at every iteration. If this field is `false`, the optimizer uses a single partition for the optimization.

The setting `true` usually gives the most robust results because it takes partitioning noise into account. However, for good results, `true` requires at least twice as many function evaluations.

`false`
Use no more than one of the following three options.
`CVPartition`A `cvpartition` object, as created by `cvpartition``'Kfold',5` if you do not specify a cross-validation field
`Holdout`A scalar in the range `(0,1)` representing the holdout fraction
`Kfold`An integer greater than 1

Example: `'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)`

Data Types: `struct`

## Output Arguments

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Trained kernel regression model, returned as a `RegressionKernel` model object or `RegressionPartitionedKernel` cross-validated model object.

If you set any of the name-value pair arguments `CrossVal`, `CVPartition`, `Holdout`, `KFold`, or `Leaveout`, then `Mdl` is a `RegressionPartitionedKernel` cross-validated model. Otherwise, `Mdl` is a `RegressionKernel` model.

To reference properties of `Mdl`, use dot notation. For example, enter `Mdl.NumExpansionDimensions` in the Command Window to display the number of dimensions of the expanded space.

Note

Unlike other regression models, and for economical memory usage, a `RegressionKernel` model object does not store the training data or training process details (for example, convergence history).

Optimization details, returned as a structure array including fields described in this table. The fields contain final values or name-value pair argument specifications.

FieldDescription
`Solver`

Objective function minimization technique: `'LBFGS-fast'`, `'LBFGS-blockwise'`, or `'LBFGS-tall'`. For details, see Algorithms.

`LossFunction`Loss function. Either mean squared error (MSE) or epsilon-insensitive, depending on the type of linear regression model. See `Learner`.
`Lambda`Regularization term strength. See `Lambda`.
`BetaTolerance`Relative tolerance on the linear coefficients and the bias term. See `BetaTolerance`.
`GradientTolerance`Absolute gradient tolerance. See `GradientTolerance`.
`ObjectiveValue`Value of the objective function when optimization terminates. The regression loss plus the regularization term compose the objective function.
`GradientMagnitude`Infinite norm of the gradient vector of the objective function when optimization terminates. See `GradientTolerance`.
`RelativeChangeInBeta`Relative changes in the linear coefficients and the bias term when optimization terminates. See `BetaTolerance`.
`FitTime`Elapsed, wall-clock time (in seconds) required to fit the model to the data.
`History`History of optimization information. This field also includes the optimization information from training `Mdl`. This field is empty (`[]`) if you specify `'Verbose',0`. For details, see `Verbose` and Algorithms.

To access fields, use dot notation. For example, to access the vector of objective function values for each iteration, enter `FitInfo.ObjectiveValue` in the Command Window.

Examine the information provided by `FitInfo` to assess whether convergence is satisfactory.

Cross-validation optimization of hyperparameters, returned as a `BayesianOptimization` object or a table of hyperparameters and associated values. The output is nonempty when the value of `'OptimizeHyperparameters'` is not `'none'`. The output value depends on the `Optimizer` field value of the `'HyperparameterOptimizationOptions'` name-value pair argument:

Value of `Optimizer` FieldValue of `HyperparameterOptimizationResults`
`'bayesopt'` (default)Object of class `BayesianOptimization`
`'gridsearch'` or `'randomsearch'`Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

## Limitations

• `fitrkernel` does not accept initial conditions for the linear coefficients beta (β) and bias term (b) used to determine the decision function, $f\left(x\right)=T\left(x\right)\beta +b.$

• `fitrkernel` does not support standardization.

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### Random Feature Expansion

Random feature expansion, such as Random Kitchen Sinks[1] and Fastfood[2], is a scheme to approximate Gaussian kernels of the kernel regression algorithm for big data in a computationally efficient way. Random feature expansion is more practical for big data applications that have large training sets but can also be applied to smaller data sets that fit in memory.

The kernel regression algorithm searches for an optimal function that deviates from each response data point (yi) by values no greater than the epsilon margin (ε) after mapping the predictor data into a high-dimensional space.

Some regression problems cannot be described adequately using a linear model. In such cases, obtain a nonlinear regression model by replacing the dot product x1x2 with a nonlinear kernel function $G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉$, where xi is the ith observation (row vector) and φ(xi) is a transformation that maps xi to a high-dimensional space (called the “kernel trick”). However, evaluating G(x1,x2) , the Gram matrix, for each pair of observations is computationally expensive for a large data set (large n).

The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,

`$G\left({x}_{1},{x}_{2}\right)=〈\phi \left({x}_{1}\right),\phi \left({x}_{2}\right)〉\approx T\left({x}_{1}\right)T\left({x}_{2}\right)\text{'},$`

where T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$). The Random Kitchen Sink[1] scheme uses the random transformation

`$T\left(x\right)={m}^{-1/2}\mathrm{exp}\left(iZx\text{'}\right)\text{'},$`

where $Z\in {ℝ}^{m×p}$ is a sample drawn from $N\left(0,{\sigma }^{-2}\right)$ and σ2 is a kernel scale. This scheme requires O(mp) computation and storage. The Fastfood[2] scheme introduces another random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces computation cost to O(m`log`p) and reduces storage to O(m).

You can specify values for m and σ2, using the `NumExpansionDimensions` and `KernelScale` name-value pair arguments of `fitrkernel`, respectively.

The `fitrkernel` function uses the Fastfood scheme for random feature expansion and uses linear regression to train a Gaussian kernel regression model. Unlike solvers in the `fitrsvm` function, which require computation of the n-by-n Gram matrix, the solver in `fitrkernel` only needs to form a matrix of size n-by-m, with m typically much less than n for big data.

### Box Constraint

A box constraint is a parameter that controls the maximum penalty imposed on observations that lie outside the epsilon margin (ε), and helps to prevent overfitting (regularization). Increasing the box constraint can lead to longer training times.

The box constraint (C) and the regularization term strength (λ) are related by C = 1/(λn), where n is the number of observations.

## Algorithms

`fitrkernel` minimizes the regularized objective function using a Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver with ridge (L2) regularization. To find the type of LBFGS solver used for training, type `FitInfo.Solver` in the Command Window.

• `'LBFGS-fast'` — LBFGS solver.

• `'LBFGS-blockwise'` — LBFGS solver with a block-wise strategy. If `fitrkernel` requires more memory than the value of `BlockSize` to hold the transformed predictor data, then it uses a block-wise strategy.

• `'LBFGS-tall'` — LBFGS solver with a block-wise strategy for tall arrays.

When `fitrkernel` uses a block-wise strategy, `fitrkernel` implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, `fitrkernel` refines the initial estimates of the linear coefficients and the bias term by fitting the model locally to parts of the data and combining the coefficients by averaging. If you specify `'Verbose',1`, then `fitrkernel` displays diagnostic information for each data pass and stores the information in the `History` field of `FitInfo`.

When `fitrkernel` does not use a block-wise strategy, the initial estimates are zeros. If you specify `'Verbose',1`, then `fitrkernel` displays diagnostic information for each iteration and stores the information in the `History` field of `FitInfo`.

## References

[1] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.

[2] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.

[3] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, pp. 205–209.