Package smile.feature.extraction

Class RandomProjection

java.lang.Object

smile.feature.extraction.Projection

smile.feature.extraction.RandomProjection

All Implemented Interfaces:

Serializable, Function<Tuple,Tuple>, Transform

public class RandomProjection
extends Projection

Random projection is a promising dimensionality reduction technique for learning mixtures of Gaussians. According to Johnson-Lindenstrauss lemma, any n data points in high dimension can be mapped down to d = O(log n / ε²) dimension without distorting their pairwise distances by more than (1 + ε). However, this reduced dimension is still far too high. Let ε = 1, we need 2^d data points, and this usually exceeds n by many orders of magnitude.

Fortunately, we can reduce the dimension of the data far more drastically for the particular case of mixtures of Gaussians. In fact, we can map the data into just d = O(log k) dimensions, where k is the number of Gaussians. Therefore, the amount of data we will need is only polynomial in k. Note that this projected dimension is independent of the number of data points and of their original dimension. Experiments show that a value of log k works nicely.

Besides, even if the original clusters are highly eccentric (that is, far from spherical), random projection will make them more spherical. Note that eccentric clusters are problematic for the EM algorithm because intermediate covariance matrices may become singular or close to singular. Note that for high enough dimension, almost the entire Gaussian distribution lies in a thin shell.

References

1. S. Dasgupta. Experiments with random projection. UAI, 2000.
2. D. Achlioptas. Database-friendly random projections. 2001.
3. Chinmay Hegde, Michael Wakin, and Richard Baraniuk. Random projections for manifold learning. NIPS, 2007.

See Also:

• Serialized Form

Field Summary

Fields inherited from class smile.feature.extraction.Projection

columns, projection, schema

Constructor Summary

Constructors

Constructor	Description
RandomProjection(Matrix projection, String... columns)	Constructor.

Method Summary

Modifier and Type	Method	Description
static RandomProjection	of(int n, int p, String... columns)	Generates a non-sparse random projection.
static RandomProjection	sparse(int n, int p, String... columns)	Generates a sparse random projection.

Methods inherited from class smile.feature.extraction.Projection

apply, apply, apply, apply, postprocess, preprocess

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface java.util.function.Function

andThen, compose

Methods inherited from interface smile.data.transform.Transform

andThen, compose

Constructor Details

RandomProjection

public RandomProjection(Matrix projection,
 String... columns)

Constructor.

Parameters:

projection - the projection matrix.

columns - the columns to transform when applied on Tuple/DataFrame.

Method Details

of

public static RandomProjection of(int n,
 int p,
 String... columns)

Generates a non-sparse random projection.

Parameters:

n - the dimension of input space.

p - the dimension of feature space.

columns - the columns to transform when applied on Tuple/DataFrame.

Returns:

the model.

sparse

public static RandomProjection sparse(int n,
 int p,
 String... columns)

Generates a sparse random projection.

Parameters:

n - the dimension of input space.

p - the dimension of feature space.

columns - the columns to transform when applied on Tuple/DataFrame.

Returns:

the model.