Nearest NeighbourCondensing and Editing

David Claus

February 27, 2004

Computer Vision Reading Group

Oxford

Nearest Neighbour Rule

Non-parametric pattern classification.

Consider a two class problem where each sample consists of two measurements (x,y).

k = 1

k = 3

For a given query point q, assign the class of the nearest neighbour.

Compute the k nearest neighbours and assign the class by majority vote.

Example: Digit Recognition

• Yann LeCunn – MNIST Digit Recognition

– Handwritten digits– 28x28 pixel images: d = 784– 60,000 training samples– 10,000 test samples

• Nearest neighbour is competitive

Test Error Rate (%)

Linear classifier (1-layer NN) 12.0

K-nearest-neighbors, Euclidean 5.0

K-nearest-neighbors, Euclidean, deskewed 2.4

K-NN, Tangent Distance, 16x16 1.1

K-NN, shape context matching 0.67

1000 RBF + linear classifier 3.6

SVM deg 4 polynomial 1.1

2-layer NN, 300 hidden units 4.7

2-layer NN, 300 HU, [deskewing] 1.6

LeNet-5, [distortions] 0.8

Boosted LeNet-4, [distortions] 0.7

Nearest Neighbour Issues

• Expensive– To determine the nearest neighbour of a query point q, must compute

the distance to all N training examples+ Pre-sort training examples into fast data structures (kd-trees)+ Compute only an approximate distance (LSH)+ Remove redundant data (condensing)

• Storage Requirements– Must store all training data P

+ Remove redundant data (condensing)- Pre-sorting often increases the storage requirements

• High Dimensional Data– “Curse of Dimensionality”

• Required amount of training data increases exponentially with dimension• Computational cost also increases dramatically• Partitioning techniques degrade to linear search in high dimension

Questions

• What distance measure to use?

– Often Euclidean distance is used

– Locally adaptive metrics

– More complicated with non-numeric data, or when different dimensions have different scales

• Choice of k?

– Cross-validation

– 1-NN often performs well in practice

– k-NN needed for overlapping classes

– Re-label all data according to k-NN, then classify with 1-NN

– Reduce k-NN problem to 1-NN through dataset editing

Exact Nearest Neighbour

• Asymptotic error (infinite sample size) is less than twice the Bayes classification error

– Requires a lot of training data

• Expensive for high dimensional data (d>20?)

• O(Nd) complexity for both storage and query time

– N is the number of training examples, d is the dimension of each sample

– This can be reduced through dataset editing/condensing

Decision Regions

Each cell contains one sample, and every location within the cell is closer to that sample than to any other sample.

A Voronoi diagram divides the space into such cells.

Every query point will be assigned the classification of the sample within that cell. The decision boundary separates the class regions based on the 1-NN decision rule.

Knowledge of this boundary is sufficient to classify new points.

The boundary itself is rarely computed; many algorithms seek to retain only those points necessary to generate an identical boundary.

Condensing

• Aim is to reduce the number of training samples• Retain only the samples that are needed to define the decision boundary• This is reminiscent of a Support Vector Machine

• Decision Boundary Consistent – a subset whose nearest neighbour decision boundary is identical to the boundary of the entire training set

• Minimum Consistent Set – the smallest subset of the training data that correctly classifies all of the original training data

Original data Condensed data Minimum Consistent Set

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Condensing• Condensed Nearest Neighbour (CNN)

Hart 1968– Incremental– Order dependent– Neither minimal nor decision

boundary consistent– O(n3) for brute-force method– Can follow up with reduced NN

[Gates72] • Remove a sample if doing so

does not cause any incorrect classifications

1. Initialize subset with a single training example

2. Classify all remaining samples using the subset, and transfer any incorrectly classified samples to the subset

3. Return to 2 until no transfers occurred or the subset is full

Proximity Graphs

• Condensing aims to retain points along the decision boundary• How to identify such points?

– Neighbouring points of different classes

• Proximity graphs provide various definitions of “neighbour”

NNG MST RNG GG DT

NNG = Nearest Neighbour Graph

MST = Minimum Spanning Tree

RNG = Relative Neighbourhood Graph

GG = Gabriel Graph

DT = Delaunay Triangulation

Proximity Graphs: Delaunay

• The Delaunay Triangulation is the dual of the Voronoi diagram

• Three points are each others neighbours if their tangent sphere contains no other points

• Voronoi editing: retain those points whose neighbours (as defined by the Delaunay Triangulation) are of the opposite class

• The decision boundary is identical

• Conservative subset

• Retains extra points

• Expensive to compute in high dimensions

Proximity Graphs: Gabriel

• The Gabriel graph is a subset of the Delaunay Triangulation

• Points are neighbours only if their (diametral) sphere of influence is empty

• Does not preserve the identical decision boundary, but most changes occur outside the convex hull of the data points

• Can be computed more efficiently

Green lines denote “Tomek links”

Proximity Graphs: RNG

• The Relative Neighbourhood Graph (RNG) is a subset of the Gabriel graph

• Two points are neighbours if the “lune” defined by the intersection of their radial spheres is empty

• Further reduces the number of neighbours• Decision boundary changes are often

drastic, and not guaranteed to be training set consistent

Gabriel edited RNG edited – not consistent

Matlab demo

Dataset Reduction: Editing

• Training data may contain noise, overlapping classes

– starting to make assumptions about the underlying distributions

• Editing seeks to remove noisy points and produce smooth decision boundaries – often by retaining points far from the decision boundaries

• Results in homogenous clusters of points

Wilson Editing• Wilson 1972• Remove points that do not agree with the majority of their k nearest neighbours

Wilson editing with k=7

Original data

Earlier example

Wilson editing with k=7

Original data

Overlapping classes

Multi-edit

• Multi-edit [Devijer & Kittler ’79]– Repeatedly apply Wilson editing

to random partitions– Classify with the 1-NN rule

• Approximates the error rate of the Bayes decision rule

1. Diffusion: divide data into N ≥ 3 random subsets

2. Classification: Classify Si using 1-NN with S(i+1)Mod N as the training set (i = 1..N)

3. Editing: Discard all samples incorrectly classified in (2)

4. Confusion: Pool all remaining samples into a new set

5. Termination: If the last I iterations produced no editing then end; otherwise go to (1)

Multi-edit, 8 iterations – last 3 sameVoronoi editing

Combined Editing/Condensing

• First edit the data to remove noise and smooth the boundary• Then condense to obtain a smaller subset

Where are we?

• Simple method, pretty powerful rule

• Can be made to run fast

• Requires a lot of training data

• Edit to reduce noise, class overlap

• Condense to remove redundant data

Questions

• What distance measure to use?

– Often Euclidean distance is used

– Locally adaptive metrics

– More complicated with non-numeric data, or when different dimensions have different scales

• Choice of k?

– Cross-validation

– 1-NN often performs well in practice

– k-NN needed for overlapping classes

– Re-label all data according to k-NN, then classify with 1-NN

– Reduce k-NN problem to 1-NN through dataset editing