de.citec.ml.rng

Class RelationalDistances

java.lang.Object

de.citec.ml.rng.RelationalDistances

public final class RelationalDistances
extends Object

This is a helper class for the computation of distances between data points and convex combinations. As demonstrated in Hasenfuss and Hammer (2009), if a prototype w_k is given in terms of a convex combination

w_k = Σ_i=1,...,m α_{k, i} · x_i

were all convex coefficients α_{k, i} are non-negative and

Σ_i=1,...,m α_{k, i} = 1

we obtain for all i:

d(w_k, x_i)² = α_k · D(:, i)² - 0.5 · α_k · D² · α_k^T

Author:

Benjamin Paassen - bpaassen(at)techfak.uni-bielefeld.de

Method Summary

Modifier and Type	Method and Description
protected static double[][]	getDistancesToPrototypes(double[][] D, double[][] Alpha, double[] Z) Calculates the squared distances of n data points to all prototypes, based on the distances from the test to the training data D, the normalization terms for each prototype Z, and the convex coefficients representing the prototypes Alpha.
static double[][]	getDistancesToPrototypes(double[][] D, RNGModel model) Calculates the squared distances of n data points to all prototypes, based on the distances from the test to the training data D and a relational neural gas model.
protected static double[]	getDistancesToPrototypes(double[] d, double[][] Alpha, double[] Z) Calculates the squared distances of data point to all prototypes, based on the distances to the training data d, the normalization terms for each prototype Z, and the convex coefficients representing the prototypes Alpha.
static double[]	getDistancesToPrototypes(double[] d, RNGModel model) Calculates the squared distances of a data point to all prototypes, based on the distances to the training data d and a relational neural gas model.
protected static double[]	getNormalizationTerms(double[][] D, double[][] Alpha) Computes the normalization terms

Methods inherited from class java.lang.Object

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

Method Detail

getDistancesToPrototypes

public static double[][] getDistancesToPrototypes(double[][] D,
                                                  RNGModel model)

Calculates the squared distances of n data points to all prototypes, based on the distances from the test to the training data D and a relational neural gas model.

Parameters:

D - The n x m distance matrix from test data points to training data points.

model - a RNGModel.

Returns:

a n x K matrix Dp where Dp[j][k] contains the squared distance between data point j and prototype k.

getDistancesToPrototypes

public static double[] getDistancesToPrototypes(double[] d,
                                                RNGModel model)

Calculates the squared distances of a data point to all prototypes, based on the distances to the training data d and a relational neural gas model.

Parameters:

d - The 1 x m vector of distances from the data point to all training data points.

model - a RNGModel.

Returns:

a 1 x K vector dp where dp[k] contains the squared distance between the data point and prototype k.

getDistancesToPrototypes

protected static double[][] getDistancesToPrototypes(double[][] D,
                                                     double[][] Alpha,
                                                     double[] Z)

Calculates the squared distances of n data points to all prototypes, based on the distances from the test to the training data D, the normalization terms for each prototype Z, and the convex coefficients representing the prototypes Alpha. In particular, let m be the number of training data points and K be the number of prototypes. Then, the output is a n x K matrix Dp, where

Dp[j][k] = Σ_i=1,...,m Alpha[k][i] · D[j][i]² + Z[k] - 0.5 * Alpha[k] · D_train² · Alpha[k]^T

Parameters:

D - The n x m distance matrix from test data points to training data points.

Alpha - The prototypes given as a K x m matrix of convex coefficients.

Z - The K x 1 vector of normalization terms for each prototype, meaning prototype k contains - 0.5 * Alpha[k] · D_train² · Alpha[k]^T where D_train is the matrix of pairwise distances for the training data points.

Returns:

a n x K matrix Dp where Dp[j][k] contains the squared distance between data point j and prototype k.

getDistancesToPrototypes

protected static double[] getDistancesToPrototypes(double[] d,
                                                   double[][] Alpha,
                                                   double[] Z)

Calculates the squared distances of data point to all prototypes, based on the distances to the training data d, the normalization terms for each prototype Z, and the convex coefficients representing the prototypes Alpha. In particular, let m be the number of training data points and K be the number of prototypes. Then, the output is a 1 x K vector dp, where

dp[k] = Σ_i=1,...,m Alpha[k][i] · d[i]² + Z[k] - 0.5 * Alpha[k] · D_train² · Alpha[k]^T

Parameters:

d - The 1 x m vector of distances from the data point to all training data points.

Alpha - The prototypes given as a K x m matrix of convex coefficients.

Z - The K x 1 vector of normalization terms for each prototype, meaning prototype k contains - 0.5 * Alpha[k] · D_train² · Alpha[k]^T where D_train is the matrix of pairwise distances for the training data points.

Returns:

a 1 x K vector dp where dp[k] contains the squared distance between the data point and prototype k.

getNormalizationTerms

protected static double[] getNormalizationTerms(double[][] D,
                                                double[][] Alpha)

Computes the normalization terms

-0.5 * Alpha[k] · D² · Alpha[k]^T

for all prototypes k, where Alpha[k] are the convex coefficients representing prototype k and D² is the matrix of squared distances for all training data points.

Parameters:

D - a m x m matrix of distances for all training data points.

Alpha - The prototypes given as a K x m matrix of convex coefficients.

Returns:

a K x 1 vector containing the entries -0.5 * Alpha[k] · D² · Alpha[k]^T