Finding k-nearest neighbors for a given vector?

Question

Given that I have the following in my knowledge-database:

1  0  6 20  0  0  6 20
1  0  3  6  0  0  3  6
1  0 15 45  0  0 15 45
1  0 17 44  0  0 17 44
1  0  2  5  0  0  2  5

I want to be able to find the nearest neighbors of the following vector:

1  0  5 16  0  0  5 16

according to a distance metric. So in this case, given a particular threshold, I should find that the first vector listed is a near-neighbor to the given vector. Currently, the size of my knowledge database is in the order of millions so calculating the distance metric for each and every point and then comparing is proving expensive. Are there any alternatives on how to achieve this with a significant speedup?

I am open to pretty much any approach including using spatial indexes in MySQL (except that I am not entirely sure on how this problem can be solved) or some kind of hashing (this would be great but again, I am not entirely sure).

Answer 1

In Python (from www.comp.mq.edu.au/):

def count_different_values(k_v1s, k_v2s):
    """kv1s and kv2s should be dictionaries mapping keys to 
    values.  count_different_values() returns the number of keys in 
    k_v1s and k_v2s that don't have the same value"""
    ks = set(k_v1s.iterkeys()) | set(k_v2s.iterkeys())
    return sum(1 for k in ks if k_v1s.get(k) != k_v2s.get(k))


def sum_square_diffs(x0s, x1s):
    """x1s and x2s should be equal-lengthed sequences of numbers.
    sum_square_differences() returns the sum of the squared differences 
    of x1s and x2s."""
    sum((pow(x1-x2,2) for x1,x2 in zip(x1s,x2s)))

def incr(x_c, x, inc=1):
    """increments the value associated with key x in dictionary x_c
    by inc, or sets it to inc if key x is not in dictionary x_c."""
    x_c[x] = x_c.get(x, 0) + inc

def count_items(xs, x_c=None):
    """returns a dictionary x_c whose keys are the items in xs, and 
    whose values are the number of times each item occurs in xs."""
    if x_c == None:
        x_c = {}
    for x in xs:
        incr(x_c, x)
    return x_c

def second(xy):
    """returns the second element in a sequence"""
    return xy[1]

def most_frequent(xs):
    """returns the most frequent item in xs"""
    x_c = count_items(xs)
    return sorted(x_c.iteritems(), key=second, reverse=True)[0][0]


class kNN_classifier:
    """This is a k-nearest-neighbour classifer."""
    def __init__(self, train_data, k, distf):
        self.train_data = train_data
        self.k = min(k, len(train_data))
        self.distf = distf

    def classify(self, x):
        Ns = sorted(self.train_data, 
                    key=lambda xy: self.distf(xy[0], x))
        return most_frequent((y for x,y in Ns[:self.k]))

    def batch_classify(self, xs):
        return [self.classify(x) for x in xs]

def train(train_data, k=1, distf=count_different_values):
    """Returns a kNN_classifer that contains the data, the number of
    nearest neighbours k and the distance function"""
    return kNN_classifier(train_data, k, distf)

also another implementation of www.umanitoba.ca/

#!/usr/bin/env python
# This code is part of the Biopython distribution and governed by its
# license.  Please see the LICENSE file that should have been included
# as part of this package.
"""
This module provides code for doing k-nearest-neighbors classification.

k Nearest Neighbors is a supervised learning algorithm that classifies
a new observation based the classes in its surrounding neighborhood.

Glossary:
distance   The distance between two points in the feature space.
weight     The importance given to each point for classification. 


Classes:
kNN           Holds information for a nearest neighbors classifier.


Functions:
train        Train a new kNN classifier.
calculate    Calculate the probabilities of each class, given an observation.
classify     Classify an observation into a class.

    Weighting Functions:
equal_weight    Every example is given a weight of 1.

"""

import numpy

class kNN:
    """Holds information necessary to do nearest neighbors classification.

    Members:
    classes  Set of the possible classes.
    xs       List of the neighbors.
    ys       List of the classes that the neighbors belong to.
    k        Number of neighbors to look at.

    """
    def __init__(self):
        """kNN()"""
        self.classes = set()
        self.xs = []
        self.ys = []
        self.k = None

def equal_weight(x, y):
    """equal_weight(x, y) -> 1"""
    # everything gets 1 vote
    return 1

def train(xs, ys, k, typecode=None):
    """train(xs, ys, k) -> kNN

    Train a k nearest neighbors classifier on a training set.  xs is a
    list of observations and ys is a list of the class assignments.
    Thus, xs and ys should contain the same number of elements.  k is
    the number of neighbors that should be examined when doing the
    classification.

    """
    knn = kNN()
    knn.classes = set(ys)
    knn.xs = numpy.asarray(xs, typecode)
    knn.ys = ys
    knn.k = k
    return knn

def calculate(knn, x, weight_fn=equal_weight, distance_fn=None):
    """calculate(knn, x[, weight_fn][, distance_fn]) -> weight dict

    Calculate the probability for each class.  knn is a kNN object.  x
    is the observed data.  weight_fn is an optional function that
    takes x and a training example, and returns a weight.  distance_fn
    is an optional function that takes two points and returns the
    distance between them.  If distance_fn is None (the default), the
    Euclidean distance is used.  Returns a dictionary of the class to
    the weight given to the class.

    """
    x = numpy.asarray(x)

    order = []  # list of (distance, index)
    if distance_fn:
        for i in range(len(knn.xs)):
            dist = distance_fn(x, knn.xs[i])
            order.append((dist, i))
    else:
        # Default: Use a fast implementation of the Euclidean distance
        temp = numpy.zeros(len(x))
        # Predefining temp allows reuse of this array, making this
        # function about twice as fast.
        for i in range(len(knn.xs)):
            temp[:] = x - knn.xs[i]
            dist = numpy.sqrt(numpy.dot(temp,temp))
            order.append((dist, i))
    order.sort()

    # first 'k' are the ones I want.
    weights = {}  # class -> number of votes
    for k in knn.classes:
        weights[k] = 0.0
    for dist, i in order[:knn.k]:
        klass = knn.ys[i]
        weights[klass] = weights[klass] + weight_fn(x, knn.xs[i])

    return weights

def classify(knn, x, weight_fn=equal_weight, distance_fn=None):
    """classify(knn, x[, weight_fn][, distance_fn]) -> class

    Classify an observation into a class.  If not specified, weight_fn will
    give all neighbors equal weight.  distance_fn is an optional function
    that takes two points and returns the distance between them.  If
    distance_fn is None (the default), the Euclidean distance is used.
    """
    weights = calculate(
        knn, x, weight_fn=weight_fn, distance_fn=distance_fn)

    most_class = None
    most_weight = None
    for klass, weight in weights.items():
        if most_class is None or weight > most_weight:
            most_class = klass
            most_weight = weight
    return most_class