In []:

```
import numpy
x = numpy.array([[0,0],[-1,0.1],[0.3,-0.05],[0.7,0.3],[-0.2,-0.6],[-0.15,-0.63],[-0.25,0.55],[-0.28,0.67]])
y = numpy.array([0,0,0,0,1,1,2,2])
```

In []:

```
import matplotlib.pyplot as pyplot
%matplotlib inline
def plot_data(features,labels,axis,alpha=1.0):
# separate features according to their class
X0,X1,X2 = features[labels==0], features[labels==1], features[labels==2]
# class 0 data
axis.plot(X0[:,0], X0[:,1], 'o', color='green', markersize=12, alpha=alpha)
# class 1 data
axis.plot(X1[:,0], X1[:,1], 'o', color='red', markersize=12, alpha=alpha)
# class 2 data
axis.plot(X2[:,0], X2[:,1], 'o', color='blue', markersize=12, alpha=alpha)
# set axes limits
axis.set_xlim(-1.5,1.5)
axis.set_ylim(-1.5,1.5)
axis.set_aspect('equal')
axis.set_xlabel('x')
axis.set_ylabel('y')
figure,axis = pyplot.subplots(1,1)
plot_data(x,y,axis)
axis.set_title('Toy data set')
pyplot.show()
```

*k-nearest neighbors*) in a data set like this, we would expect quite some errors using the standard Euclidean distance. This is due to the fact that the spread of the data is not similar amongst the feature dimensions. The following piece of code plots an ellipse on top of the data set. The ellipse in this case is in fact a circunference that helps to visualize how the Euclidean distance weights equally both feature dimensions.

In []:

```
def make_covariance_ellipse(covariance):
import matplotlib.patches as patches
import scipy.linalg as linalg
# the ellipse is centered at (0,0)
mean = numpy.array([0,0])
# eigenvalue decomposition of the covariance matrix (w are eigenvalues and v eigenvectors),
# keeping only the real part
w,v = linalg.eigh(covariance)
# normalize the eigenvector corresponding to the largest eigenvalue
u = v[0]/linalg.norm(v[0])
# angle in degrees
angle = 180.0/numpy.pi*numpy.arctan(u[1]/u[0])
# fill Gaussian ellipse at 2 standard deviation
ellipse = patches.Ellipse(mean, 2*w[0]**0.5, 2*w[1]**0.5, 180+angle, color='orange', alpha=0.3)
return ellipse
# represent the Euclidean distance
figure,axis = pyplot.subplots(1,1)
plot_data(x,y,axis)
ellipse = make_covariance_ellipse(numpy.eye(2))
axis.add_artist(ellipse)
axis.set_title('Euclidean distance')
pyplot.show()
```

*k*, find the Mahalanobis distance measure which maximizes kNN accuracy (using the given value for *k*) in a training data set. As we usually do in machine learning, under the assumption that the training data is an accurate enough representation of the underlying process, the distance learnt will not only perform well in the training data, but also have good generalization properties.

First, we need to wrap the data into Shogun's feature and label objects:

In []:

```
from modshogun import RealFeatures, MulticlassLabels
features = RealFeatures(x.T)
labels = MulticlassLabels(y.astype(numpy.float64))
```

Secondly, perform LMNN training:

In []:

```
from modshogun import LMNN
# number of target neighbours per example
k = 1
lmnn = LMNN(features,labels,k)
# set an initial transform as a start point of the optimization
init_transform = numpy.eye(2)
lmnn.set_maxiter(2000)
lmnn.train(init_transform)
```

In []:

```
# get the linear transform from LMNN
L = lmnn.get_linear_transform()
# square the linear transform to obtain the Mahalanobis distance matrix
M = numpy.matrix(numpy.dot(L.T,L))
# represent the distance given by LMNN
figure,axis = pyplot.subplots(1,1)
plot_data(x,y,axis)
ellipse = make_covariance_ellipse(M.I)
axis.add_artist(ellipse)
axis.set_title('LMNN distance')
pyplot.show()
```

LMNN is one of the so-called linear metric learning methods. What this means is that we can understand LMNN's output in two different ways: on the one hand, as a distance measure, this was explained above; on the other hand, as a linear transformation of the input data. Like any other linear transformation, LMNN's output can be written as a matrix, that we will call $L$. In other words, if the input data is represented by the matrix $X$, then LMNN can be understood as the data transformation expressed by $X'=L X$. We use the convention that each column is a feature vector; thus, the number of rows of $X$ is equal to the input dimension of the data, and the number of columns is equal to the number of vectors.

So far, so good. But, if the output of the same method can be interpreted in two different ways, then there must be a relation between them! And that is precisely the case! As mentioned above, the ellipses that were plotted in the previous section represent a distance measure. This distance measure can be thought of as a matrix $M$, being the distance between two vectors $\vec{x_i}$ and $\vec{x_j}$ equal to $d(\vec{x_i},\vec{x_j})=(\vec{x_i}-\vec{x_j})^T M (\vec{x_i}-\vec{x_j})$. In general, this type of matrices are known as *Mahalanobis* matrices. In LMNN, the matrix $M$ is precisely the 'square' of the linear transformation $L$, i.e. $M=L^T L$. Note that a direct consequence of this is that $M$ is guaranteed to be positive semi-definite (PSD), and therefore define a valid metric.

This distance measure/linear transform duality in LMNN has its own advantages. An important one is that the optimization problem can go back and forth between the $L$ and the $M$ representations, giving raise to a very efficient solution.

In []:

```
# project original data using L
lx = numpy.dot(L,x.T)
# represent the data in the projected space
figure,axis = pyplot.subplots(1,1)
plot_data(lx.T,y,axis)
plot_data(x,y,axis,0.3)
ellipse = make_covariance_ellipse(numpy.eye(2))
axis.add_artist(ellipse)
axis.set_title('LMNN\'s linear transform')
pyplot.show()
```

**both** the projection **and** the learnt Mahalanobis distance.

In []:

```
import numpy
import matplotlib.pyplot as pyplot
%matplotlib inline
def sandwich_data():
from numpy.random import normal
# number of distinct classes
num_classes = 6
# number of points per class
num_points = 9
# distance between layers, the points of each class are in a layer
dist = 0.7
# memory pre-allocation
x = numpy.zeros((num_classes*num_points, 2))
y = numpy.zeros(num_classes*num_points)
for i,j in zip(xrange(num_classes), xrange(-num_classes//2, num_classes//2 + 1)):
for k,l in zip(xrange(num_points), xrange(-num_points//2, num_points//2 + 1)):
x[i*num_points + k, :] = numpy.array([normal(l, 0.1), normal(dist*j, 0.1)])
y[i*num_points:i*num_points + num_points] = i
return x,y
def plot_sandwich_data(x, y, axis=pyplot, cols=['r', 'b', 'g', 'm', 'k', 'y']):
for idx,val in enumerate(numpy.unique(y)):
xi = x[y==val]
axis.scatter(xi[:,0], xi[:,1], s=50, facecolors='none', edgecolors=cols[idx])
x, y = sandwich_data()
figure, axis = pyplot.subplots(1, 1, figsize=(5,5))
plot_sandwich_data(x, y, axis)
axis.set_aspect('equal')
axis.set_title('"Sandwich" toy data set')
axis.set_xlabel('x')
axis.set_ylabel('y')
pyplot.show()
```

In []:

```
from modshogun import KNN, EuclideanDistance, LMNN, RealFeatures, MulticlassLabels
def plot_neighborhood_graph(x, nn, axis=pyplot, cols=['r', 'b', 'g', 'm', 'k', 'y']):
for i in xrange(x.shape[0]):
xs = [x[i,0], x[nn[1,i], 0]]
ys = [x[i,1], x[nn[1,i], 1]]
axis.plot(xs, ys, cols[int(y[i])])
features = RealFeatures(x.T)
labels = MulticlassLabels(y)
fig, axes = pyplot.subplots(1, 3, figsize=(15, 10))
# use k = 2 instead of 1 because otherwise the method nearest_neighbors just returns the same
# points as their own 1-nearest neighbours
k = 2
knn = KNN(k, EuclideanDistance(features, features), labels)
plot_sandwich_data(x, y, axes[0])
plot_neighborhood_graph(x, knn.nearest_neighbors(), axes[0])
axes[0].set_title('Euclidean neighbourhood in the input space')
lmnn = LMNN(features, labels, k)
# set a large number of iterations. The data set is small so it does not cost a lot, and this way
# we ensure a robust solution
lmnn.set_maxiter(3000)
lmnn.train()
knn.set_distance(lmnn.get_distance())
plot_sandwich_data(x, y, axes[1])
plot_neighborhood_graph(x, knn.nearest_neighbors(), axes[1])
axes[1].set_title('LMNN neighbourhood in the input space')
# plot features in the transformed space, with the neighbourhood graph computed using the Euclidean distance
L = lmnn.get_linear_transform()
xl = numpy.dot(x, L.T)
features = RealFeatures(xl.T)
knn.set_distance(EuclideanDistance(features, features))
plot_sandwich_data(xl, y, axes[2])
plot_neighborhood_graph(xl, knn.nearest_neighbors(), axes[2])
axes[2].set_ylim(-3, 2.5)
axes[2].set_title('Euclidean neighbourhood in the transformed space')
[axes[i].set_xlabel('x') for i in xrange(len(axes))]
[axes[i].set_ylabel('y') for i in xrange(len(axes))]
[axes[i].set_aspect('equal') for i in xrange(len(axes))]
pyplot.show()
```

In []:

```
from modshogun import CSVFile, RealFeatures, MulticlassLabels
ape_features = RealFeatures(CSVFile('../../../data/multiclass/fm_ape_gut.dat'))
ape_labels = MulticlassLabels(CSVFile('../../../data/multiclass/label_ape_gut.dat'))
```

In []:

```
print('Number of examples = %d, number of features = %d.' % (ape_features.get_num_vectors(), ape_features.get_num_features()))
```

In []:

```
def visualize_tdsne(features, labels):
from modshogun import TDistributedStochasticNeighborEmbedding
converter = TDistributedStochasticNeighborEmbedding()
converter.set_target_dim(2)
converter.set_perplexity(25)
embedding = converter.embed(features)
import matplotlib.pyplot as pyplot
% matplotlib inline
x = embedding.get_feature_matrix()
y = labels.get_labels()
pyplot.scatter(x[0, y==0], x[1, y==0], color='green')
pyplot.scatter(x[0, y==1], x[1, y==1], color='red')
pyplot.scatter(x[0, y==2], x[1, y==2], color='blue')
pyplot.show()
visualize_tdsne(ape_features, ape_labels)
```

In []:

```
from modshogun import KNN, EuclideanDistance
from modshogun import StratifiedCrossValidationSplitting, CrossValidation
from modshogun import CrossValidationResult, MulticlassAccuracy
# set up the classifier
knn = KNN()
knn.set_k(3)
knn.set_distance(EuclideanDistance())
# set up 5-fold cross-validation
splitting = StratifiedCrossValidationSplitting(ape_labels, 5)
# evaluation method
evaluator = MulticlassAccuracy()
cross_validation = CrossValidation(knn, ape_features, ape_labels, splitting, evaluator)
# locking is not supported for kNN, deactivate it to avoid an inoffensive warning
cross_validation.set_autolock(False)
# number of experiments, the more we do, the less variance in the result
num_runs = 200
cross_validation.set_num_runs(num_runs)
# perform cross-validation and print the result!
result = cross_validation.evaluate()
result = CrossValidationResult.obtain_from_generic(result)
print('kNN mean accuracy in a total of %d runs is %.4f.' % (num_runs, result.mean))
```

*skewed* data sets, where the number of examples among classes varies significantly.

*feature selection*. Using Shogun, it is extremely easy to switch to this so-called *diagonal* mode for LMNN: just call the method `set_diagonal(use_diagonal)`

with `use_diagonal`

set to `True`

.

In []:

```
from modshogun import LMNN
import numpy
# to make training faster, use a portion of the features
fm = ape_features.get_feature_matrix()
ape_features_subset = RealFeatures(fm[:150, :])
# number of targer neighbours in LMNN, here we just use the same value that was used for KNN before
k = 3
lmnn = LMNN(ape_features_subset, ape_labels, k)
lmnn.set_diagonal(True)
lmnn.set_maxiter(1000)
init_transform = numpy.eye(ape_features_subset.get_num_features())
lmnn.train(init_transform)
diagonal = numpy.diag(lmnn.get_linear_transform())
print('%d out of %d elements are non-zero.' % (numpy.sum(diagonal != 0), diagonal.size))
```

It is a fair question to ask how did we know that the maximum number of iterations in this experiment should be around 1200 iterations. Well, the truth is that we know this only because we have run this experiment with this same data beforehand, and we know that after this number of iterations the algorithm has converged. This is not something nice, and the ideal case would be if one could completely forget about this parameter, so that LMNN uses as many iterations as it needs until it converges. Nevertheless, this is not practical at least because of two reasons:

- If you are dealing with many examples or with very high dimensional feature vectors, you might not want to wait until the algorithm converges and have a look at what LMNN has found before it has completely converged.
- As with any other algorithm based on gradient descent, the termination criteria can be tricky. Let us illustrate this further:

In []:

```
import matplotlib.pyplot as pyplot
%matplotlib inline
statistics = lmnn.get_statistics()
pyplot.plot(statistics.obj.get())
pyplot.grid(True)
pyplot.xlabel('Number of iterations')
pyplot.ylabel('LMNN objective')
pyplot.show()
```

`set_stepsize_threshold`

and `set_obj_threshold`

. These methods can be used to modify the lower bound required in the step size and the increment in the objective (relative to its absolute value), respectively, to stop training. Also, it is possible to set a hard upper bound on the number of iterations using `set_maxiter`

as we have done above. In case the internal termination criteria did not fire before the maximum number of iterations was reached, you will receive a warning message, similar to the one shown above. This is not a synonym that the training went wrong; but it is strongly recommended at this event to have a look at the objective plot as we have done in the previous block of code.

In []:

```
from modshogun import CSVFile, RealFeatures, MulticlassLabels
wine_features = RealFeatures(CSVFile('../../../data/uci/wine/fm_wine.dat'))
wine_labels = MulticlassLabels(CSVFile('../../../data/uci/wine/label_wine.dat'))
assert(wine_features.get_num_vectors() == wine_labels.get_num_labels())
print('%d feature vectors with %d features from %d different classes.' % (wine_features.get_num_vectors(), \
wine_features.get_num_features(), wine_labels.get_num_classes()))
```

In []:

```
from modshogun import KNN, EuclideanDistance
from modshogun import StratifiedCrossValidationSplitting, CrossValidation
from modshogun import CrossValidationResult, MulticlassAccuracy
import numpy
# kNN classifier
k = 5
knn = KNN()
knn.set_k(k)
knn.set_distance(EuclideanDistance())
splitting = StratifiedCrossValidationSplitting(wine_labels, 5)
evaluator = MulticlassAccuracy()
cross_validation = CrossValidation(knn, wine_features, wine_labels, splitting, evaluator)
cross_validation.set_autolock(False)
num_runs = 200
cross_validation.set_num_runs(num_runs)
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
euclidean_means = numpy.zeros(3)
euclidean_means[0] = result.mean
print('kNN accuracy with the Euclidean distance %.4f.' % result.mean)
```

Seconly, we will use LMNN to find a distance measure and use it with kNN:

In []:

```
from modshogun import LMNN
# train LMNN
lmnn = LMNN(wine_features, wine_labels, k)
lmnn.set_maxiter(1500)
lmnn.train()
# evaluate kNN using the distance learnt by LMNN
knn.set_distance(lmnn.get_distance())
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
lmnn_means = numpy.zeros(3)
lmnn_means[0] = result.mean
print('kNN accuracy with the distance obtained by LMNN %.4f.' % result.mean)
```

In []:

```
print('minima = ' + str(numpy.min(wine_features, axis=1)))
print('maxima = ' + str(numpy.max(wine_features, axis=1)))
```

In []:

```
from modshogun import RescaleFeatures
# preprocess features so that all of them vary within [0,1]
preprocessor = RescaleFeatures()
preprocessor.init(wine_features)
wine_features.add_preprocessor(preprocessor)
wine_features.apply_preprocessor()
# sanity check
assert(numpy.min(wine_features) >= 0.0 and numpy.max(wine_features) <= 1.0)
# perform kNN classification after the feature rescaling
knn.set_distance(EuclideanDistance())
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
euclidean_means[1] = result.mean
print('kNN accuracy with the Euclidean distance after feature rescaling %.4f.' % result.mean)
# train kNN in the new features and classify with kNN
lmnn.train()
knn.set_distance(lmnn.get_distance())
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
lmnn_means[1] = result.mean
print('kNN accuracy with the distance obtained by LMNN after feature rescaling %.4f.' % result.mean)
```

*whitening*. Whitening, which is explained in an article in wikipedia, transforms the covariance matrix of the data into the identity matrix.

In []:

```
import scipy.linalg as linalg
# shorthand for the feature matrix -- this makes a copy of the feature matrix
data = wine_features.get_feature_matrix()
# remove mean
data = data.T
data-= numpy.mean(data, axis=0)
# compute the square of the covariance matrix and its inverse
M = linalg.sqrtm(numpy.cov(data.T))
# keep only the real part, although the imaginary that pops up in the sqrtm operation should be equal to zero
N = linalg.inv(M).real
# apply whitening transform
white_data = numpy.dot(N, data.T)
wine_white_features = RealFeatures(white_data)
```

In []:

```
import matplotlib.pyplot as pyplot
%matplotlib inline
fig, axarr = pyplot.subplots(1,2)
axarr[0].matshow(numpy.cov(wine_features))
axarr[1].matshow(numpy.cov(wine_white_features))
pyplot.show()
```

In []:

```
wine_features = wine_white_features
# perform kNN classification after whitening
knn.set_distance(EuclideanDistance())
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
euclidean_means[2] = result.mean
print('kNN accuracy with the Euclidean distance after whitening %.4f.' % result.mean)
# train kNN in the new features and classify with kNN
lmnn.train()
knn.set_distance(lmnn.get_distance())
result = CrossValidationResult.obtain_from_generic(cross_validation.evaluate())
lmnn_means[2] = result.mean
print('kNN accuracy with the distance obtained by LMNN after whitening %.4f.' % result.mean)
```

In []:

```
assert(euclidean_means.shape[0] == lmnn_means.shape[0])
N = euclidean_means.shape[0]
# the x locations for the groups
ind = 0.5*numpy.arange(N)
# bar width
width = 0.15
figure, axes = pyplot.subplots()
figure.set_size_inches(6, 5)
euclidean_rects = axes.bar(ind, euclidean_means, width, color='y')
lmnn_rects = axes.bar(ind+width, lmnn_means, width, color='r')
# attach information to chart
axes.set_ylabel('Accuracies')
axes.set_ylim(top=1.4)
axes.set_title('kNN accuracy by distance and feature preprocessing')
axes.set_xticks(ind+width)
axes.set_xticklabels(('Raw', 'Rescaling', 'Whitening'))
axes.legend(( euclidean_rects[0], lmnn_rects[0]), ('Euclidean', 'LMNN'), loc='upper right')
def autolabel(rects):
# attach text labels to bars
for rect in rects:
height = rect.get_height()
axes.text(rect.get_x()+rect.get_width()/2., 1.05*height, '%.3f' % height,
ha='center', va='bottom')
autolabel(euclidean_rects)
autolabel(lmnn_rects)
pyplot.show()
```