Nystrom (Java Statistical Analysis Tool 0.0.8 API)

java.lang.Object
- jsat.datatransform.DataTransformBase
- - jsat.datatransform.kernel.Nystrom

All Implemented Interfaces:

Serializable, Cloneable, DataTransform, Parameterized
```
public class Nystrom
extends DataTransformBase
```
An implementation of the Nystrom approximation for any Kernel Trick. The full rank kernel is approximated by a basis set of a subset of the data points that make up the original data set. Instead of explicitly forming the approximately decomposed matrix, this transform projects the original numeric features of a data set into a new feature space where the dot product in the linear space approximates the dot product in the given kernel space.

See:
- Williams, C.,&Seeger, M. (2001). Using the Nyström Method to Speed Up Kernel Machines. Advances in Neural Information Processing Systems 13 (pp. 682–688). MIT Press. Retrieved from here
- Yang, T., Li, Y.-F., Mahdavi, M., Jin, R.,&Zhou, Z.-H. (2012). Nystrom Method vs Random Fourier Features A Theoretical and Empirical Comparison. Advances in Neural Information Processing Systems (pp. 485–493). Retrieved from here
- Kumar, S., Mohri, M.,&Talwalkar, A. (2012). Sampling methods for the Nyström method. The Journal of Machine Learning Research, 5, 981–1006. Retrieved from here
Author:

Edward Raff

See Also:

Serialized Form

Nested Class Summary

Nested Classes
Modifier and Type	Class and Description
`static class`	`Nystrom.SamplingMethod` Different sample methods may be used to select a better and more representative set of vectors to form the basis vectors at increased cost, where `n` is the number of data points in the full data set and `b` is the number of basis vectors to obtain.

Constructor Summary

Constructors
Modifier	Constructor and Description
	`Nystrom()` Creates a new Nystrom approximation object using the `RBF Kernel` with 500 basis vectors
	`Nystrom(KernelTrick k, DataSet dataset, int basisSize, Nystrom.SamplingMethod method)` Creates a new Nystrom approximation object
	`Nystrom(KernelTrick k, DataSet dataset, int basisSize, Nystrom.SamplingMethod method, double ridge, boolean sampleWithReplacment)` Creates a new Nystrom approximation object
	`Nystrom(KernelTrick k, int basisSize)` Creates a new Nystrom approximation object
	`Nystrom(KernelTrick k, int basisSize, Nystrom.SamplingMethod method, double ridge, boolean sampleWithReplacment)` Creates a new Nystrom approximation object
`protected`	`Nystrom(Nystrom toCopy)` Copy constructor

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`Nystrom`	`clone()`
`void`	`fit(DataSet dataset)` Fits this transform to the given dataset.
`Nystrom.SamplingMethod`	`getBasisSamplingMethod()` Returns the method of selecting the basis vectors
`int`	`getBasisSize()` Returns the number of basis vectors to use
`int`	`getDimension()` Returns the number of dimensions to project down too
`KernelTrick`	`getKernel()`
`double`	`getRidge()` Returns the regularization value added to each eigen value
`static List<Vec>`	`sampleBasisVectors(KernelTrick k, DataSet dataset, List<Vec> X, Nystrom.SamplingMethod method, int basisSize, boolean sampleWithReplacment, Random rand)` Performs sampling of a data set for a subset of the vectors that make a good set of basis vectors for forming an approximation of a full kernel space.
`void`	`setBasisSamplingMethod(Nystrom.SamplingMethod method)` Sets the method of selecting the basis vectors
`void`	`setBasisSize(int basisSize)` Sets the basis size for the Kernel PCA to be learned from.
`void`	`setDimension(int dimension)` Sets the dimension of the new feature space, which is the number of principal components to select from the kernelized feature space.
`void`	`setKernel(KernelTrick k)`
`void`	`setRidge(double ridge)` Sets the regularization parameter to add to the eigen values of the gram matrix.
`DataPoint`	`transform(DataPoint dp)` Returns a new data point that is a transformation of the original data point.

Methods inherited from class jsat.datatransform.DataTransformBase
getParameter, getParameters

Methods inherited from class java.lang.Object
equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - Nystrom
```
public Nystrom(KernelTrick k,
               DataSet dataset,
               int basisSize,
               Nystrom.SamplingMethod method)
```
    Creates a new Nystrom approximation object
    
    Parameters:
    
    k - the kernel trick to form an approximation of
    
    dataset - the data set to form the approximate feature space from
    
    basisSize - the number of basis vectors to use, this is the output dimension size.
    
    method - what sampling method should be used to select the basis vectors from the full data set.
  - Nystrom
```
public Nystrom()
```
    Creates a new Nystrom approximation object using the RBF Kernel with 500 basis vectors
  - Nystrom
```
public Nystrom(KernelTrick k,
               int basisSize)
```
    Creates a new Nystrom approximation object
    
    Parameters:
    
    k - the kernel trick to form an approximation of
    
    basisSize - the number of basis vectors to use, this is the output dimension size.
  - Nystrom
```
public Nystrom(KernelTrick k,
               int basisSize,
               Nystrom.SamplingMethod method,
               double ridge,
               boolean sampleWithReplacment)
```
    Creates a new Nystrom approximation object
    
    Parameters:
    
    k - the kernel trick to form an approximation of
    
    basisSize - the number of basis vectors to use, this is the output dimension size.
    
    method - what sampling method should be used to select the basis vectors from the full data set.
    
    ridge - a non negative additive term to regularize the eigen values of the decomposition.
    
    sampleWithReplacment - true if the basis vectors should be sampled with replacement, false if they should not.
  - Nystrom
```
public Nystrom(KernelTrick k,
               DataSet dataset,
               int basisSize,
               Nystrom.SamplingMethod method,
               double ridge,
               boolean sampleWithReplacment)
```
    Creates a new Nystrom approximation object
    
    Parameters:
    
    k - the kernel trick to form an approximation of
    
    dataset - the data set to form the approximate feature space from
    
    basisSize - the number of basis vectors to use, this is the output dimension size.
    
    method - what sampling method should be used to select the basis vectors from the full data set.
    
    ridge - a non negative additive term to regularize the eigen values of the decomposition.
    
    sampleWithReplacment - true if the basis vectors should be sampled with replacement, false if they should not.
  - Nystrom
```
protected Nystrom(Nystrom toCopy)
```
    Copy constructor
    
    Parameters:
    
    toCopy - the object to copy
- Method Detail
  - fit
```
public void fit(DataSet dataset)
```
    Description copied from interface: DataTransform
    
    Fits this transform to the given dataset. Some transforms can only be learned from classification or regression datasets. If an incompatible dataset type is given, a FailedToFitException exception may be thrown.
    
    Parameters:
    
    dataset - the dataset to fir this transform to
  - sampleBasisVectors
```
public static List<Vec> sampleBasisVectors(KernelTrick k,
                                           DataSet dataset,
                                           List<Vec> X,
                                           Nystrom.SamplingMethod method,
                                           int basisSize,
                                           boolean sampleWithReplacment,
                                           Random rand)
```
    Performs sampling of a data set for a subset of the vectors that make a good set of basis vectors for forming an approximation of a full kernel space. While these methods are motivated from Nystrom's algorithm, they are also useful for others.
    
    Parameters:
    
    k - the kernel trick to form the basis for
    
    dataset - the data set to sample from
    
    X - the list of vectors from the data set
    
    method - the sampling method to use
    
    basisSize - the number of basis vectors to select
    
    sampleWithReplacment - whether or not the sample with replacement
    
    rand - the source of randomness for the sampling
    
    Returns:
    
    a list of basis vectors sampled from the data set.
    
    See Also:
    
    Nystrom.SamplingMethod
  - transform
```
public DataPoint transform(DataPoint dp)
```
    Description copied from interface: DataTransform
    
    Returns a new data point that is a transformation of the original data point. This new data point is a different object, but may contain the same references as the original data point. It is not guaranteed that you can mutate the transformed point without having a side effect on the original point.
    
    Parameters:
    
    dp - the data point to apply a transformation to
    
    Returns:
    
    a transformed data point
  - clone
```
public Nystrom clone()
```
    Specified by:
    
    clone in interface DataTransform
    
    Specified by:
    
    clone in class DataTransformBase
  - setRidge
```
public void setRidge(double ridge)
```
    Sets the regularization parameter to add to the eigen values of the gram matrix. This can be particularly useful when using a large (500+) number of components.
    
    Parameters:
    
    ridge - the non-negative value in [0, ∞) to add to each eigen value
  - getRidge
```
public double getRidge()
```
    Returns the regularization value added to each eigen value
    
    Returns:
    
    the regularization value added to each eigen value
  - setDimension
```
public void setDimension(int dimension)
```
    Sets the dimension of the new feature space, which is the number of principal components to select from the kernelized feature space.
    
    Parameters:
    
    dimension - the number of dimensions to project down too
  - getDimension
```
public int getDimension()
```
    Returns the number of dimensions to project down too
    
    Returns:
    
    the number of dimensions to project down too
  - setBasisSamplingMethod
```
public void setBasisSamplingMethod(Nystrom.SamplingMethod method)
```
    Sets the method of selecting the basis vectors
    
    Parameters:
    
    method - the method of selecting the basis vectors
  - getBasisSamplingMethod
```
public Nystrom.SamplingMethod getBasisSamplingMethod()
```
    Returns the method of selecting the basis vectors
    
    Returns:
    
    the method of selecting the basis vectors
  - setBasisSize
```
public void setBasisSize(int basisSize)
```
    Sets the basis size for the Kernel PCA to be learned from. Increasing the basis increase the accuracy of the transform, but increased the training time at a cubic rate.
    
    Parameters:
    
    basisSize - the number of basis vectors to build Kernel PCA from
  - getBasisSize
```
public int getBasisSize()
```
    Returns the number of basis vectors to use
    
    Returns:
    
    the number of basis vectors to use
  - setKernel
```
public void setKernel(KernelTrick k)
```
    Parameters:
    
    k - the kernel trick to use
  - getKernel
```
public KernelTrick getKernel()
```
    Returns:
    
    the kernel trick to use

Class Nystrom

Nested Class Summary

Constructor Summary

Method Summary

Methods inherited from class jsat.datatransform.DataTransformBase

Methods inherited from class java.lang.Object

Constructor Detail

Nystrom

Nystrom

Nystrom

Nystrom

Nystrom

Nystrom

Method Detail

fit

sampleBasisVectors

transform

clone

setRidge

getRidge

setDimension

getDimension

setBasisSamplingMethod

getBasisSamplingMethod

setBasisSize

getBasisSize

setKernel

getKernel