Developer Guide and Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

k-Nearest Neighbors (kNN) Classifier

k-Nearest Neighbors (kNN) classification is a non-parametric classification algorithm. The model of the kNN classifier is based on feature vectors and class labels from the training data set. This classifier induces the class of the query vector from the labels of the feature vectors in the training data set to which the query vector is similar. A similarity between feature vectors is determined by the type of distance (for example, Euclidian) in a multidimensional feature space.

Details

Given n feature vectors LaTex Math image. of size
p
and a vector of class labels LaTex Math image. , where LaTex Math image. and
C
is the number of classes, describes the class to which the feature vector LaTex Math image. belongs, the problem is to build a kNN classifier.
Given a positive integer parameter
k
and a test observation LaTex Math image. , the kNN classifier does the following:
  1. Identifies the set LaTex Math image. of the k feature vectors in the training data that are closest to LaTex Math image. according to the distance metric
  2. Estimates the conditional probability for the class
    j
    as the fraction of vectors in LaTex Math image. whose labels y are equal to
    j
  3. Assigns the class with the largest probability to the test observation LaTex Math image.
On CPU, kNN classification might use K-D tree, a space-partitioning data structure, or Brute Force search to find nearest neighbors, while on GPU only Brute Force search is available.
K-D tree
On CPU, the library provides kNN classification based on multidimensional binary search tree (K-D tree, where D means the dimension and K means the number of dimensions in the feature space). For more details, see [James2013], [Patwary2016].
oneDAL version of the kNN algorithm with K-D trees uses the PANDA algorithm [Patwary2016].
Each non-leaf node of a tree contains the identifier of a feature along which to split the feature space and an appropriate feature value (a cut-point) that defines the splitting hyperplane to partition the feature space into two parts. Each leaf node of the tree has an associated subset (a bucket) of elements of the training data set. Feature vectors from any bucket belong to the region of the space defined by tree nodes on the path from the root node to the respective leaf.
Brute Force
Brute Force kNN algorithm calculates the squared distances from each query feature vector to each reference feature vector in the training data set. Then, for each query feature vector it selects
k
objects from the training set that are closest to that query feature vector. For details, see [Li2015], [Verma2014].
Training Stage
Training using K-D Tree
For each non-leaf node, the process of building a K-D tree involves the choice of the feature (that is, dimension in the feature space) and the value for this feature (a cut-point) to split the feature space. This procedure starts with the entire feature space for the root node of the tree, and for every next level of the tree deals with ever smaller part of the feature space.
The PANDA algorithm constructs the K-D tree by choosing the dimension with the maximum variance for splitting [Patwary2016].
Therefore, for each new non-leaf node of the tree, the algorithm computes the variance of values that belong to the respective region of the space for each of the features and chooses the feature with the largest variance. Due to high computational cost of this operation, PANDA uses a subset of feature values to compute the variance.
PANDA uses a sampling heuristic to estimate the data distribution for the chosen feature and chooses the median estimate as the cut-point.
PANDA generates new K-D tree levels until the number of feature vectors in a leaf node gets less or equal to a predefined threshold. Once the threshold is reached, PANDA stops growing the tree and associates the feature vectors with the bucket of the respective leaf node.
Training using Brute Force
During training with the Brute Force approach, the algorithm stores all feature vectors from the training data set to calculate their distances to the query feature vectors.
Prediction Stage
Given kNN classifier and query vectors LaTex Math image. , the problem is to calculate the labels for those vectors.
Prediction using K-D Tree
To solve the problem for each given query vector LaTex Math image. , the algorithm traverses the K-D tree to find feature vectors associated with a leaf node that are closest to LaTex Math image. . During the search, the algorithm limits exploration of the nodes for which the distance between the query vector and respective part of the feature space is not less than the distance from the LaTex Math image. neighbor. This distance is progressively updated during the tree traverse.
Prediction using Brute Force
To solve the problem, the algorithm computes distances between vectors from training and testing sets: LaTex Math image. . For example, if Euclidean distance is used, LaTex Math image. would be the following:
LaTex Math image.
K training vectors with minimal distance to the testing vector are the nearest neighbors the algorithms searches for.

Batch Processing

kNN classification follows the general workflow described in Classification Usage Model.
Training
For a description of the input and output, refer to Usage Model: Training and Prediction.
At the training stage, both Brute Force and K-D tree based kNN classifier have the following parameters:
Parameter
Default Value
Description
algorithmFPType
float
The floating-point type that the algorithm uses for intermediate computations. Can be
float
or
double
.
method
defaultDense
The computation method used by kNN classification. The only training method supported so far is the default dense method.
nClasses
2
The number of classes.
dataUseInModel
doNotUse
A parameter to enable/disable use of the input data set in the kNN model. Possible values:
  • doNotUse
    - the algorithm does not include the input data and labels in the trained kNN model but creates a copy of the input data set.
  • doUse
    - the algorithm includes the input data and labels in the trained kNN model.
K-D tree based kNN reorders feature vectors and corresponding labels in the input data set or its copy to improve performance at the prediction stage.
If the value is
doUse
, do not deallocate the memory for input data and labels.
engine
SharePtr< engines:: mt19937:: Batch>()
Pointer to the random number generator engine that is used internally to perform sampling needed to choose dimensions and cut-points for the K-D tree.
Prediction
For a description of the input and output, refer to Usage Model: Training and Prediction.
At the prediction stage, both Brute Force and K-D tree based kNN classifier have the following parameters:
Parameter
Default Value
Description
algorithmFPType
float
The floating-point type that the algorithm uses for intermediate computations. Can be
float
or
double
.
method
defaultDense
The computation method used kNN classification. The only prediction method supported so far is the default dense method.
nClasses
2
The number of classes.
k
1
The number of neighbors.
resultsToCompute
0
The 64-bit integer flag that specifies which extra characteristics of the kNN algorithm to compute. Provide one of the following values to request a single characteristic or use bitwise OR to request a combination of the characteristics:
  • computeIndicesOfNeighbors
  • computeDistances
voteWeights
voteUniform
The voting method for prediction:
  • voteUniform
    – Uniform weighting is used. All neighbors weight equally.
  • voteDistance
    – Inverse-distance weighting is used. The closer to the query point the neighbor is, the more it weights.
Output
In addition to classifier output, kNN calculates the results described below. Pass the
Result ID
as a parameter to the methods that access the result of your algorithm.
Result ID
Result
indices
A numeric table LaTex Math image. containing indices of rows from training dataset that are nearest neighbors computed when the
computeIndicesOfNeigtbors
option is on.
By default, this result is an object of the
HomogenNumericTable
class, but you can define the result as an object of any class derived from
NumericTable
.
distances
A numeric table LaTex Math image. containing distances to nearest neighbors computed when the
computeDistances
option is on.
By default, this result is an object of the
HomogenNumericTable
class, but you can define the result as an object of any class derived from
NumericTable
.

Examples

oneAPI DPC++
oneAPI C++
Java*
There is no support for Java on GPU.
Batch Processing:
Python* with DPC++ support

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.