# fscnca

Feature selection using neighborhood component analysis for classification

## Syntax

## Description

`fscnca`

performs feature selection using neighborhood
component analysis (NCA) for classification.

To perform NCA-based feature selection for regression, see `fsrnca`

.

returns the NCA feature selection model for classification using the sample data
contained in the table `mdl`

= fscnca(`Tbl`

,`ResponseVarName`

)`Tbl`

.
`ResponseVarName`

is the name of the variable in
`Tbl`

that contains the class labels.

`fscnca`

learns the feature weights by using a diagonal
adaptation of NCA with regularization.

specifies additional options using one or more name-value arguments. For example,
you can specify the method for fitting the model, the regularization parameter, and
the initial feature weights.`mdl`

= fscnca(`X`

,`Y`

,`Name,Value`

)

## Examples

### Detect Relevant Features in Data Using NCA for Classification

Generate toy data where the response variable depends on the 3rd, 9th, and 15th predictors.

rng(0,'twister'); % For reproducibility N = 100; X = rand(N,20); y = -ones(N,1); y(X(:,3).*X(:,9)./X(:,15) < 0.4) = 1;

Fit the neighborhood component analysis model for classification.

mdl = fscnca(X,y,'Solver','sgd','Verbose',1);

o Tuning initial learning rate: NumTuningIterations = 20, TuningSubsetSize = 100 |===============================================| | TUNING | TUNING SUBSET | LEARNING | | ITER | FUN VALUE | RATE | |===============================================| | 1 | -3.755936e-01 | 2.000000e-01 | | 2 | -3.950971e-01 | 4.000000e-01 | | 3 | -4.311848e-01 | 8.000000e-01 | | 4 | -4.903195e-01 | 1.600000e+00 | | 5 | -5.630190e-01 | 3.200000e+00 | | 6 | -6.166993e-01 | 6.400000e+00 | | 7 | -6.255669e-01 | 1.280000e+01 | | 8 | -6.255669e-01 | 1.280000e+01 | | 9 | -6.255669e-01 | 1.280000e+01 | | 10 | -6.255669e-01 | 1.280000e+01 | | 11 | -6.255669e-01 | 1.280000e+01 | | 12 | -6.255669e-01 | 1.280000e+01 | | 13 | -6.255669e-01 | 1.280000e+01 | | 14 | -6.279210e-01 | 2.560000e+01 | | 15 | -6.279210e-01 | 2.560000e+01 | | 16 | -6.279210e-01 | 2.560000e+01 | | 17 | -6.279210e-01 | 2.560000e+01 | | 18 | -6.279210e-01 | 2.560000e+01 | | 19 | -6.279210e-01 | 2.560000e+01 | | 20 | -6.279210e-01 | 2.560000e+01 | o Solver = SGD, MiniBatchSize = 10, PassLimit = 5 |==========================================================================================| | PASS | ITER | AVG MINIBATCH | AVG MINIBATCH | NORM STEP | LEARNING | | | | FUN VALUE | NORM GRAD | | RATE | |==========================================================================================| | 0 | 9 | -5.658450e-01 | 4.492407e-02 | 9.290605e-01 | 2.560000e+01 | | 1 | 19 | -6.131382e-01 | 4.923625e-02 | 7.421541e-01 | 1.280000e+01 | | 2 | 29 | -6.225056e-01 | 3.738784e-02 | 3.277588e-01 | 8.533333e+00 | | 3 | 39 | -6.233366e-01 | 4.947901e-02 | 5.431133e-01 | 6.400000e+00 | | 4 | 49 | -6.238576e-01 | 3.445763e-02 | 2.946188e-01 | 5.120000e+00 | Two norm of the final step = 2.946e-01 Relative two norm of the final step = 6.588e-02, TolX = 1.000e-06 EXIT: Iteration or pass limit reached.

Plot the selected features. The weights of the irrelevant features should be close to zero.

figure() plot(mdl.FeatureWeights,'ro') grid on xlabel('Feature index') ylabel('Feature weight')

`fscnca`

correctly detects the relevant features.

### Identify Relevant Features for Classification

Load and partition the ovarian cancer data set, and determine if feature selection is necessary. Fit the model, plot the feature weights, and then classify observations using the selected features.

```
load ovariancancer;
whos
```

Name Size Bytes Class Attributes grp 216x1 28512 cell obs 216x4000 3456000 single

The `obs`

variable consists of 216 observations with 4000 features. Each element in `grp`

defines the group to which the corresponding row of `obs`

belongs.

Use `cvpartition`

to divide the data into a training set of size 160 and a test set of size 56. Both the training set and the test set have roughly the same group proportions as in `grp`

.

rng(1,"twister"); % For reproducibility cvp = cvpartition(grp,Holdout=56)

cvp = Hold-out cross validation partition NumObservations: 216 NumTestSets: 1 TrainSize: 160 TestSize: 56 IsCustom: 0

Xtrain = obs(cvp.training,:); ytrain = grp(cvp.training,:); Xtest = obs(cvp.test,:); ytest = grp(cvp.test,:);

To determine if feature selection is necessary, first compute the generalization error without fitting.

```
nca = fscnca(Xtrain,ytrain,FitMethod="none");
loss(nca,Xtest,ytest)
```

ans = 0.0893

The software computes the generalization error of the neighborhood component analysis (NCA) feature selection model using the initial feature weights (in this case, the default feature weights) provided by `fscnca`

.

Fit the NCA model without the regularization parameter (that is, `Lambda`

= 0).

nca = fscnca(Xtrain,ytrain,FitMethod="exact",Lambda=0,... Solver="sgd",Standardize=true); loss(nca,Xtest,ytest)

ans = 0.0714

The improvement in the loss value suggests that feature selection is worthwhile. Tuning the regularization parameter (`Lambda`

value) usually improves the results.

Tuning the regularization parameter for the NCA model means finding the `Lambda`

value that produces the minimum classification loss. To tune the parameter using five-fold cross-validation:

1. Partition the training data into five folds and extract the number of validation (test) sets. For each fold, `cvpartition`

assigns four-fifths of the data as a training set, and one-fifth of the data as a test set.

cvp = cvpartition(ytrain,KFold=5); numvalidsets = cvp.NumTestSets;

Assign `Lambda`

values and create an array to store the loss function values.

n = length(ytrain); lambdavals = linspace(0,20,20)/n; lossvals = zeros(length(lambdavals),numvalidsets);

2. Train the NCA model for each `Lambda`

value, using the training set in each fold.

3. Compute the classification loss for the corresponding test set in the fold using the NCA model. Record the loss value.

4. Repeat this process for all folds and all `Lambda`

values.

for i = 1:length(lambdavals) for k = 1:numvalidsets X = Xtrain(cvp.training(k),:); y = ytrain(cvp.training(k),:); Xvalid = Xtrain(cvp.test(k),:); yvalid = ytrain(cvp.test(k),:); nca = fscnca(X,y,FitMethod="exact", ... Solver="sgd",Lambda=lambdavals(i), ... IterationLimit=30,GradientTolerance=1e-4, ... Standardize=true); lossvals(i,k) = loss(nca,Xvalid,yvalid,LossFunction="classiferror"); end end

Compute the average loss obtained from the folds for each `Lambda`

value.

meanloss = mean(lossvals,2);

Plot the average loss values versus the `Lambda`

values.

figure() plot(lambdavals,meanloss,"ro-") xlabel("Lambda") ylabel("Loss (MSE)") grid on

Find the best `Lambda`

value that corresponds to the minimum average loss.

`[~,idx] = min(meanloss) % Find the index`

idx = 2

`bestlambda = lambdavals(idx) % Find the best Lambda value`

bestlambda = 0.0066

bestloss = meanloss(idx)

bestloss = 0.0312

Fit the NCA model on all the data using the best `Lambda`

value. Use the solver `sgd`

and standardize the predictor values.

nca = fscnca(Xtrain,ytrain,FitMethod="exact",Solver="sgd",... Lambda=bestlambda,Standardize=true,Verbose=1);

o Tuning initial learning rate: NumTuningIterations = 20, TuningSubsetSize = 100 |===============================================| | TUNING | TUNING SUBSET | LEARNING | | ITER | FUN VALUE | RATE | |===============================================| | 1 | 2.403497e+01 | 2.000000e-01 | | 2 | 2.275050e+01 | 4.000000e-01 | | 3 | 2.036845e+01 | 8.000000e-01 | | 4 | 1.627647e+01 | 1.600000e+00 | | 5 | 1.023512e+01 | 3.200000e+00 | | 6 | 3.864283e+00 | 6.400000e+00 | | 7 | 4.743816e-01 | 1.280000e+01 | | 8 | -7.260138e-01 | 2.560000e+01 | | 9 | -7.260138e-01 | 2.560000e+01 | | 10 | -7.260138e-01 | 2.560000e+01 | | 11 | -7.260138e-01 | 2.560000e+01 | | 12 | -7.260138e-01 | 2.560000e+01 | | 13 | -7.260138e-01 | 2.560000e+01 | | 14 | -7.260138e-01 | 2.560000e+01 | | 15 | -7.260138e-01 | 2.560000e+01 | | 16 | -7.260138e-01 | 2.560000e+01 | | 17 | -7.260138e-01 | 2.560000e+01 | | 18 | -7.260138e-01 | 2.560000e+01 | | 19 | -7.260138e-01 | 2.560000e+01 | | 20 | -7.260138e-01 | 2.560000e+01 | o Solver = SGD, MiniBatchSize = 10, PassLimit = 5 |==========================================================================================| | PASS | ITER | AVG MINIBATCH | AVG MINIBATCH | NORM STEP | LEARNING | | | | FUN VALUE | NORM GRAD | | RATE | |==========================================================================================| | 0 | 9 | 4.016078e+00 | 2.835465e-02 | 5.395984e+00 | 2.560000e+01 | | 1 | 19 | -6.726156e-01 | 6.111354e-02 | 5.021138e-01 | 1.280000e+01 | | 1 | 29 | -8.316555e-01 | 4.024186e-02 | 1.196031e+00 | 1.280000e+01 | | 2 | 39 | -8.838656e-01 | 2.333416e-02 | 1.225834e-01 | 8.533333e+00 | | 3 | 49 | -8.669034e-01 | 3.413162e-02 | 3.421902e-01 | 6.400000e+00 | | 3 | 59 | -8.906936e-01 | 1.946295e-02 | 2.232511e-01 | 6.400000e+00 | | 4 | 69 | -8.778630e-01 | 3.561290e-02 | 3.290645e-01 | 5.120000e+00 | | 4 | 79 | -8.857135e-01 | 2.516638e-02 | 3.902979e-01 | 5.120000e+00 | Two norm of the final step = 3.903e-01 Relative two norm of the final step = 6.171e-03, TolX = 1.000e-06 EXIT: Iteration or pass limit reached.

Plot the feature weights.

figure() plot(nca.FeatureWeights,"ro") xlabel("Feature Index") ylabel("Feature Weight") grid on

Most of the feature weights are very close to zero, which means that they are irrelevant. Some features have much higher feature weight values. In this case, to select a reasonable number of predictors, specify a threshold of 0.02 times the maximum feature weight value.

selidx = find(nca.FeatureWeights > 0.02*max(1,max(nca.FeatureWeights)))

`selidx = `*72×1*
565
611
654
681
737
743
744
750
754
839
⋮

Compute the classification loss using the test set.

loss(nca,Xtest,ytest)

ans = 0.0179

Extract the features with feature weights greater than the specified threshold value from the training data.

features = Xtrain(:,selidx);

Apply a support vector machine classifier to the reduced training set using the selected features.

svmMdl = fitcsvm(features,ytrain);

Evaluate the accuracy of the trained classifier on the test data, which has not been used for feature selection.

loss(svmMdl,Xtest(:,selidx),ytest)

`ans = `*single*
0

## Input Arguments

`Tbl`

— Sample data

table

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.

**Data Types: **`table`

`ResponseVarName`

— Response variable name

name of variable in `Tbl`

Response variable name, specified as the name of a variable in `Tbl`

. The
remaining variables in the table are predictors.

**Data Types: **`char`

| `string`

`X`

— Predictor variable values

*n*-by-*p* matrix

Predictor variable values, specified as an *n*-by-*p*
matrix, where *n* is the number of observations and
*p* is the number of predictor variables.

**Data Types: **`single`

| `double`

`formula`

— Explanatory model of response variable and subset of predictor variables

string | character vector

Explanatory model of the response variable and a subset of the predictor variables, specified
as a string or a character vector 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`

`Y`

— Class labels

categorical array | logical vector | numeric vector | string array | cell array of character vectors of length *n* | character matrix with *n* rows

Class labels, specified as a categorical array, logical vector, numeric vector, string
array, cell array of character vectors of length *n*, or character
matrix with *n* rows. *n* is the number of
observations. Element *i* or row *i* of
`Y`

is the class label corresponding to row *i*
of `X`

(observation *i*).

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

| `categorical`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'Solver','sgd','Weights',W,'Lambda',0.0003`

specifies
the solver as the stochastic gradient descent, the observation weights
as the values in the vector `W`

, and sets the regularization
parameter at 0.0003.

**Fitting Options**

`FitMethod`

— Method for fitting the model

`'exact'`

(default) | `'none'`

| `'average'`

Method for fitting the model, specified as the comma-separated
pair consisting of `'FitMethod'`

and one of the following:

`'exact'`

— Performs fitting using all of the data.`'none'`

— No fitting. Use this option to evaluate the generalization error of the NCA model using the initial feature weights supplied in the call to fscnca.`'average'`

— Divides the data into partitions (subsets), fits each partition using the`exact`

method, and returns the average of the feature weights. You can specify the number of partitions using the`NumPartitions`

name-value pair argument.

**Example: **`'FitMethod','none'`

`NumPartitions`

— Number of partitions

`max(2,min(10,`*n*))

(default) | integer between 2 and *n*

*n*))

Number of partitions to split the data for using with `'FitMethod','average'`

option,
specified as the comma-separated pair consisting of `'NumPartitions'`

and
an integer value between 2 and *n*, where *n* is
the number of observations.

**Example: **`'NumPartitions',15`

**Data Types: **`double`

| `single`

`Lambda`

— Regularization parameter

1/*n* (default) | nonnegative scalar

Regularization parameter to prevent overfitting, specified as the
comma-separated pair consisting of `'Lambda'`

and a
nonnegative scalar.

As the number of observations *n* increases, the
chance of overfitting decreases and the required amount of
regularization also decreases. See Identify Relevant Features for Classification and Tune Regularization Parameter to Detect Features Using NCA for Classification to learn how to tune the regularization parameter.

**Example: **`'Lambda',0.002`

**Data Types: **`double`

| `single`

`LengthScale`

— Width of the kernel

`1`

(default) | positive real scalar

Width of the kernel, specified as the comma-separated pair consisting
of `'LengthScale'`

and a positive real scalar.

A length scale value of 1 is sensible when all predictors are
on the same scale. If the predictors in `X`

are
of very different magnitudes, then consider standardizing the predictor
values using `'Standardize',true`

and setting `'LengthScale',1`

.

**Example: **`'LengthScale',1.5`

**Data Types: **`double`

| `single`

`CategoricalPredictors`

— Categorical predictors list

vector of positive integers | logical vector | character matrix | string array | cell array of character vectors | `"all"`

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

Value | Description |
---|---|

Vector of positive integers | Each entry in the vector is an index value corresponding to the column of the predictor data (`X` ) that contains a categorical variable. |

Logical vector | A `true` entry means that the corresponding column of predictor data (`X` ) is a categorical variable. |

Character matrix | Each row of the matrix is the name of a predictor variable in the table `X` . 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 vectors | Each element in the array is the name of a predictor variable in the table `X` . The names must match the entries in `PredictorNames` . |

`"all"` | All predictors are categorical. |

By default, if the predictor data is in a table,
`fscnca`

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, `fscnca`

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, `fscnca`

creates dummy variables using two different schemes, depending on whether a categorical variable is unordered or ordered:

For an unordered categorical variable,

`fscnca`

creates one dummy variable for each level of the categorical variable.For an ordered categorical variable,

`fscnca`

creates one less dummy variable than the number of categories. For details, see Automatic Creation of Dummy Variables.

For the table `X`

, categorical predictors can be ordered and unordered. For the matrix `X`

, `fscnca`

treats categorical predictors as unordered.

**Example: **`CategoricalPredictors="all"`

**Data Types: **`double`

| `logical`

| `char`

| `string`

`PredictorNames`

— Predictor variable names

string array of unique names | cell array of unique character vectors

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`

as a matrix, 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 predictor order in`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

`X`

as a table, then you can use`PredictorNames`

to specify which predictor variables to use in training. That is,`fscnca`

uses only the predictor variables in`PredictorNames`

and the response variable during training.`PredictorNames`

must be a subset of`X.Properties.VariableNames`

and cannot include the name of the response variable.By default,

`PredictorNames`

contains the names of all predictor variables.Specify the predictors for training using either

`PredictorNames`

or a formula string in`Y`

(such as`'y ~ x1 + x2 + x3'`

), but not both.

**Example: **`"PredictorNames={"SepalLength","SepalWidth","PetalLength","PetalWidth"}`

**Data Types: **`string`

| `cell`

`ResponseName`

— Response variable name

`"Y"`

(default) | character vector | string scalar

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

If you supply

`Y`

, then you can use`ResponseName`

to specify a name for the response variable.If you supply

`ResponseVarName`

or`formula`

, then you cannot use`ResponseName`

.

**Example: **`ResponseName="response"`

**Data Types: **`char`

| `string`

`InitialFeatureWeights`

— Initial feature weights

`ones(M,1)`

(default) | *M*-by-1 vector of positive numbers

Initial feature weights, specified as an *M*-by-1 vector of positive numbers,
where *M* is the number of predictor variables after dummy variables
are created for categorical variables (for details, see
`CategoricalPredictors`

).

The regularized objective function for optimizing feature weights is nonconvex. As a result,
using different initial feature weights might give different results. Setting all
initial feature weights to 1 generally works well, but in some cases, random
initialization using `rand(M,1)`

might give better quality
solutions.

For more information about feature weights, see Neighborhood Component Analysis (NCA) Feature Selection.

**Data Types: **`double`

| `single`

`Weights`

— Observation weights

*n*-by-1 vector of 1s (default) | *n*-by-1 vector of real positive scalars

Observation weights, specified as the comma-separated pair consisting of
`'Weights'`

and an *n*-by-1 vector of real
positive scalars. Use observation weights to specify higher importance of some
observations compared to others. The default weights assign equal importance to all
observations.

**Data Types: **`double`

| `single`

`Prior`

— Prior probabilities for each class

`'empirical'`

(default) | `'uniform'`

| structure

Prior probabilities for each class, specified as the comma-separated pair consisting of
`'Prior'`

and one of the following:

`'empirical'`

—`fscnca`

obtains the prior class probabilities from class frequencies.`'uniform'`

—`fscnca`

sets all class probabilities equal.Structure with two fields:

`ClassProbs`

— Vector of class probabilities. If these are numeric values with a total greater than 1,`fsnca`

normalizes them to add up to 1.`ClassNames`

— Class names corresponding to the class probabilities in`ClassProbs`

.

**Example: **`'Prior','uniform'`

`Standardize`

— Indicator for standardizing predictor data

`false`

(default) | `true`

Indicator for standardizing the predictor data, specified as the comma-separated pair
consisting of `'Standardize'`

and either `false`

or
`true`

. For more information, see Impact of Standardization.

**Example: **`'Standardize',true`

**Data Types: **`logical`

`Verbose`

— Verbosity level indicator

0 (default) | 1 | >1

Verbosity level indicator for the convergence summary display,
specified as the comma-separated pair consisting of `'Verbose'`

and
one of the following:

0 — No convergence summary

1 — Convergence summary, including norm of gradient and objective function values

> 1 — More convergence information, depending on the fitting algorithm

When using

`'minibatch-lbfgs'`

solver and verbosity level > 1, the convergence information includes iteration the log from intermediate mini-batch LBFGS fits.

**Example: **`'Verbose',1`

**Data Types: **`double`

| `single`

`Solver`

— Solver type

`'lbfgs'`

| `'sgd'`

| `'minibatch-lbfgs'`

Solver type for estimating feature weights, specified as the
comma-separated pair consisting of `'Solver'`

and
one of the following:

`'lbfgs'`

— Limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm`'sgd'`

— Stochastic gradient descent (SGD) algorithm`'minibatch-lbfgs'`

— Stochastic gradient descent with LBFGS algorithm applied to mini-batches

Default is `'lbfgs'`

for *n* ≤
1000, and `'sgd'`

for *n* > 1000.

**Example: **`'solver','minibatch-lbfgs'`

`LossFunction`

— Loss function

`'classiferror'`

(default) | function handle

Loss function, specified as the comma-separated pair consisting
of `'LossFunction'`

and one of the following.

`'classiferror'`

— Misclassification error$$l\left({y}_{i},{y}_{j}\right)=\{\begin{array}{cc}1& \text{if}\text{\hspace{0.17em}}{y}_{i}\ne {y}_{j,}\\ 0& \text{otherwise}\text{.}\end{array}$$

`@`

— Custom loss function handle. A loss function has this form.`lossfun`

function L = lossfun(Yu,Yv) % calculation of loss ...

`Yu`

is a*u*-by-1 vector and`Yv`

is a*v*-by-1 vector.`L`

is a*u*-by-*v*matrix of loss values such that`L(i,j)`

is the loss value for`Yu(i)`

and`Yv(j)`

.

The objective function for minimization includes the loss function *l*(*y*_{i},*y*_{j}) as follows:

$$f\left(w\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\displaystyle \sum _{j=1,j\ne i}^{n}{p}_{ij}l\left({y}_{i},{y}_{j}\right)}+\lambda {\displaystyle \sum _{r=1}^{p}{w}_{r}^{2}}},$$

where *w* is the feature weight
vector, *n* is the number of observations, and
*p* is the number of predictor variables. *p*_{ij} is the probability that *x*_{j} is the reference point for *x*_{i}. For details, see NCA Feature Selection for Classification.

**Example: **`'LossFunction',@lossfun`

`CacheSize`

— Memory size

`1000MB`

(default) | integer

Memory size, in MB, to use for objective function and gradient
computation, specified as the comma-separated pair consisting of `'CacheSize'`

and
an integer.

**Example: **`'CacheSize',1500MB`

**Data Types: **`double`

| `single`

**LBFGS Options**

`HessianHistorySize`

— Size of history buffer for Hessian approximation

`15`

(default) | positive integer

Size of history buffer for Hessian approximation for the `'lbfgs'`

solver,
specified as the comma-separated pair consisting of `'HessianHistorySize'`

and
a positive integer. At each iteration the function uses the most recent `HessianHistorySize`

iterations
to build an approximation to the inverse Hessian.

**Example: **`'HessianHistorySize',20`

**Data Types: **`double`

| `single`

`InitialStepSize`

— Initial step size

`'auto'`

(default) | positive real scalar

Initial step size for the `'lbfgs'`

solver,
specified as the comma-separated pair consisting of `'InitialStepSize'`

and
a positive real scalar. By default, the function determines the initial
step size automatically.

**Data Types: **`double`

| `single`

`LineSearchMethod`

— Line search method

`'weakwolfe'`

(default) | `'strongwolfe'`

| `'backtracking'`

Line search method, specified as the comma-separated pair consisting
of `'LineSearchMethod'`

and one of the following:

`'weakwolfe'`

— Weak Wolfe line search`'strongwolfe'`

— Strong Wolfe line search`'backtracking'`

— Backtracking line search

**Example: **`'LineSearchMethod','backtracking'`

`MaxLineSearchIterations`

— Maximum number of line search iterations

`20`

(default) | positive integer

Maximum number of line search iterations, specified as the comma-separated
pair consisting of `'MaxLineSearchIterations'`

and
a positive integer.

**Example: **`'MaxLineSearchIterations',25`

**Data Types: **`double`

| `single`

`GradientTolerance`

— Relative convergence tolerance

`1e-6`

(default) | positive real scalar

Relative convergence tolerance on the gradient norm for solver `lbfgs`

,
specified as the comma-separated pair consisting of `'GradientTolerance'`

and
a positive real scalar.

**Example: **`'GradientTolerance',0.000002`

**Data Types: **`double`

| `single`

**SGD Options**

`InitialLearningRate`

— Initial learning rate for `'sgd'`

solver

`'auto'`

(default) | positive real scalar

Initial learning rate for the `'sgd'`

solver,
specified as the comma-separated pair consisting of `'InitialLearningRate'`

and
a positive real scalar.

When using solver type `'sgd'`

, the learning
rate decays over iterations starting with the value specified for `'InitialLearningRate'`

.

The default `'auto'`

means that the initial
learning rate is determined using experiments on small subsets of
data. Use the `NumTuningIterations`

name-value
pair argument to specify the number of iterations for automatically
tuning the initial learning rate. Use the `TuningSubsetSize`

name-value
pair argument to specify the number of observations to use for automatically
tuning the initial learning rate.

For solver type `'minibatch-lbfgs'`

, you can
set `'InitialLearningRate'`

to a very high value.
In this case, the function applies LBFGS to each mini-batch separately
with initial feature weights from the previous mini-batch.

To make sure the chosen initial learning rate decreases the
objective value with each iteration, plot the `Iteration`

versus
the `Objective`

values saved in the `mdl.FitInfo`

property.

You can use the `refit`

method with `'InitialFeatureWeights'`

equal
to `mdl.FeatureWeights`

to start from the current
solution and run additional iterations

**Example: **`'InitialLearningRate',0.9`

**Data Types: **`double`

| `single`

`MiniBatchSize`

— Number of observations to use in each batch for the `'sgd'`

solver

min(10,*n*) (default) | positive integer value from 1 to *n*

Number of observations to use in each batch for the `'sgd'`

solver,
specified as the comma-separated pair consisting of `'MiniBatchSize'`

and
a positive integer from 1 to *n*.

**Example: **`'MiniBatchSize',25`

**Data Types: **`double`

| `single`

`PassLimit`

— Maximum number of passes for solver `'sgd'`

`5`

(default) | positive integer

Maximum number of passes through all *n* observations
for solver `'sgd'`

, specified as the comma-separated
pair consisting of `'PassLimit'`

and a positive integer.
Each pass through all of the data is called an epoch.

**Example: **`'PassLimit',10`

**Data Types: **`double`

| `single`

`NumPrint`

— Frequency of batches for displaying convergence summary

10 (default) | positive integer value

Frequency of batches for displaying convergence summary for
the `'sgd'`

solver , specified as the comma-separated
pair consisting of `'NumPrint'`

and a positive integer.
This argument applies when the `'Verbose'`

value
is greater than 0. `NumPrint`

mini-batches are
processed for each line of the convergence summary that is displayed
on the command line.

**Example: **`'NumPrint',5`

**Data Types: **`double`

| `single`

`NumTuningIterations`

— Number of tuning iterations

20 (default) | positive integer

Number of tuning iterations for the `'sgd'`

solver,
specified as the comma-separated pair consisting of `'NumTuningIterations'`

and
a positive integer. This option is valid only for `'InitialLearningRate','auto'`

.

**Example: **`'NumTuningIterations',15`

**Data Types: **`double`

| `single`

`TuningSubsetSize`

— Number of observations to use for tuning initial learning rate

min(100,*n*) (default) | positive integer value from 1 to *n*

Number of observations to use for tuning the initial learning
rate, specified as the comma-separated pair consisting of `'TuningSubsetSize'`

and
a positive integer value from 1 to *n*. This option
is valid only for `'InitialLearningRate','auto'`

.

**Example: **`'TuningSubsetSize',25`

**Data Types: **`double`

| `single`

**SGD or LBFGS Options**

`IterationLimit`

— Maximum number of iterations

positive integer

Maximum number of iterations, specified as the comma-separated
pair consisting of `'IterationLimit'`

and a positive
integer. The default is 10000 for SGD and 1000 for LBFGS and mini-batch
LBFGS.

Each pass through a batch is an iteration. Each pass through
all of the data is an epoch. If the data is divided into *k* mini-batches,
then every epoch is equivalent to *k* iterations.

**Example: **`'IterationLimit',250`

**Data Types: **`double`

| `single`

`StepTolerance`

— Convergence tolerance on the step size

1e-6 (default) | positive real scalar

Convergence tolerance on the step size, specified as the comma-separated
pair consisting of `'StepTolerance'`

and a positive
real scalar. The `'lbfgs'`

solver uses an absolute
step tolerance, and the `'sgd'`

solver uses a relative
step tolerance.

**Example: **`'StepTolerance',0.000005`

**Data Types: **`double`

| `single`

**Mini-Batch LBFGS Options**

`MiniBatchLBFGSIterations`

— Maximum number of iterations per mini-batch LBFGS step

10 (default) | positive integer

Maximum number of iterations per mini-batch LBFGS step, specified
as the comma-separated pair consisting of `'MiniBatchLBFGSIterations'`

and
a positive integer.

**Example: **`'MiniBatchLBFGSIterations',15`

**Data Types: **`double`

| `single`

**Note**

The mini-batch LBFGS algorithm is a combination of SGD and LBFGS methods. Therefore, all of the name-value pair arguments that apply to SGD and LBFGS solvers also apply to the mini-batch LBFGS algorithm.

## Output Arguments

`mdl`

— Neighborhood component analysis model for classification

`FeatureSelectionNCAClassification`

object

Neighborhood component analysis model for classification, returned
as a `FeatureSelectionNCAClassification`

object.

## Version History

**Introduced in R2016b**

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