Nearest-Neighbour Search in High Dimension and Molecular Clustering

Frdric Cazals

Algorithms project, INRIA Rocquencourt

Algorithms Seminar

June 30, 1997

[summary by Frdric Cazals]

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1   Introduction and prerequisites

1.1   Problem statement

Given a set of points P={p1,...,pn} Rd, the nearest-neighbour (NN) and k nearest neighbours (k-NN) problems can be stated as follows: pre-process P in order to return as fast as possible the nearest or k nearest neighbour(s) of an arbitrary point q according to any Euclidian metric d(p,q) = (i=1d (pi - qi)2)1/2. A weakened version of the NN problem consists in returning a point p' which e-approximates the NN p of q in the sense d(p',q) d(p,q) 1+e for any e > 0. If one denotes pi1,...,pin the points of P sorted by increasing distance to q, an equivalent formulation for the k-NN problem consists in returning a subset S={s1,...,sk} with d(q,sj) (1+e)d(q,pij) for j=1,...,k.

The naive algorithm to compute the NN of a point q consists in checking all the points of P and returning the closest, which has complexity O(dn). On the other hand, the most sophisticated algorithms known until recently had complexities in O(exp(d) log n) with exp(d) a function growing at least as quickly as 2d---see e.g., [1]. So that whenever d log n nothing better than the brute force method was known!

Kleinberg's break-through [4] has been to get around the exponential difficulty by an heavy use of random sampling aiming at ``comparing'' the points of P through their projections on random lines passing through the origin rather than decomposing the d-dimensional space containing them. The first result is an algorithm returning an approximation of the k-NN in a deterministic way but with an exponential time/space pre-processing. The second algorithm returns an approximation of the NN in a randomised way but with a polynomial pre-processing only. This talk presents these two algorithms and discusses their potential use to a clustering problem arising in chemistry---see section 4.

1.2   Prerequisites

1.2.1   A geometric lemma

The core idea of Kleinberg's method lies in the following property:
Lemma 1   Let x and y be two vectors of Rd such that ||y||/||x|| 1+g with g 1/2. Then, if v is a random vector on the unit sphere Sd-1 we have Pr[|x v| |y v|] 1/2 - g/3.
Intuitively, short vectors ``should not defeat too often'' longer ones when comparing their projection on a random line determined by a vector on Sd-1. In order to compare two vectors from their projections, the key point is therefore to use a large enough set of lines to capture the probabilistic property contained in the above theorem.

1.2.2   Empirical measures and Vapnik-Chervonenkis bounds

Let (W, F,) be a probability space, S F a set of events, and X1,X2,...,Xn n random variables following the law . If one calls the empirical measure of an event s the fraction of Xi's falling into s, the quantity
DSn =
 
sup
s S



1s(X1)+...+1s(Xn)
n
-(s)


measures the maximum difference over the class S between the empirical measure and the probability. It is a random variable and Vapnik-Chervonenkis's contribution [5] has been to elucidate the conditions under which it converges in probability to zero, that is the conditions under which limn Pr[DSn > e] = 0 for any e. To sketch this contribution, let a range-space be a couple ( P, R)= ((W, F,), R F ). We shall say that a set A of finite cardilality is shattered by R if " a 2A  $ r R such that a=r A. The dimension of Vapnik-Chervonenkis of ( P, R) is the cardinality of the biggest A W shattered by R.
Definition 1   A g-sample for ( P, R) is a finite set A W such that |(r)-| r A |/| A || g, " r R.
Theorem 1  [[5]]   For a range space of dimension k, a random sample of size
l
16
g2
(k log
16k
g2
+ log
4
d
)
is a g-sample with a probability at least 1-d.

1.2.3   Exceptional and r-distinguishing sets

As pointed out above, we are interested in comparing points with respect to their projections on vectors from Sd-1. For two vectors x and y with ||y|| / ||x|| 1+g we call their exceptional set
W
 
x,y
= { v Sd-1 such that | x v| | y v | }.
And a random set of vectors V from Sd-1 is called r-distinguishing if
" W
 
x,y
,  (W
 
x,y
)<r | V W
 
x,y
| / | V | < 1/2.
More prosaically, a set V is r-distinguishing if a majority of its points do not fall into some exceptional set of size smaller than r.

1.2.4   Hyper-planes arrangements

An arrangement of n hyper-planes in Rd is said to be in general position if any d hyper-planes have a unique point in common, and any d+1 hyper-planes do not share a point. Given a hyper-plane h and a point p, p is either above, on, or below h, which is called its position. A face on an arrangement is the set of points having the same position with respect to all the hyper-planes. The dimension of a face is its affine dimension. It is known from [2] that
Theorem 2   The number of d-faces of an arrangement of n hyper-planes in general position is fd(n)=i=0d (
n
i
).

1.2.5   Digraphs

A complete digraph G on n vertices 1,2,...,n is a directed graph which contains for any pair of vertices {i,j} either the edge (i,j) or (i,j). An apex of G is a vertex with a directed path of length at most two to any vertex. At least, an apex ordering of G is an ordering i1,...,in of its vertices such that ik is an apex for the sub-digraph G[ik,ik+1,...,in]. The following is straightforward:
Theorem 3   Every n-node complete digraph has an apex ordering computable in O(n2).

1.3   First results

The following theorems can be proved:
Lemma 2   Let be the uniform measure on Sd-1. The dimension of the range-space
((Sd-1, F , ), { W
 
x,y
|(W
 
x,y
) r })
is less than d' = 8(d+1)log (4d+4).

Lemma 3   There exists c0 such that a random sample of Sd-1 of size f(d,g) is a g/2-sample for the range-space ((Sd-1, F, ), { Wx,y|(Wx,y) r }) with f(d,g)= c0/g2 (d' log d'/g2 + log 1/d)= q(d log2 d).

Corollary 1   A set V of f(g,d) vectors from Sd-1 is (1/2-g)-distinguishing with a probability at least 1-d.

2   First algorithm

2.1   Construction of the data structure

This algorithm returns an approximation of the k-NN of a point q. To build the data structure from which it does so, we first draw uniformly at random a set V of L = f(e/3,d) = q(d log2 d) vectors from Sd-1. Then for each vector vl V, the following is done: 1. compute vl pij with pij=(pi+pj)/2, 1 i,j n; 2. sort the pij according to the values of vl pij and denote Sl the list obtained. The list of lists S1,...,SL is denoted S.

For each such list, a pair of consecutive entries is called a primitive interval, and a sequence of primitive intervals is called a trace---see Figure 1. The maximum number of traces is upper-bounded by (n2)L = nO(d log2 d). But a trace is realizable if
$ q Rd, " k=1,...,L, vk p
(k)
 
i1i2
< vk q < vk p
(k)
 
i3i4
.
So that realizable traces are defined with respect to the Ln2 hyper-planes vk (pi1i2(k) - x). And from theorem 2, the number of such traces is i=0d (
Ln2
i
) = O(n log d)2d.

Definition 2   For a realizable trace s=s1 sk sL, pi is said to s-dominate pj in Sk if pij lies in the interval (sk,...,pjj).

For each realizable trace s, the construction is as follows: (1.) build a complete digraph Gs on {1,2,...,n} with the edge (i,j) if pi s-dominates pj in half of the lists of S; (2.) build an apex ordering (s,Gs) of the nodes of Gs.

The idea behind the domination definition is depicted on Figure 2: if pij falls in the desired interval denoted I1, then pi I2 and we have |vk (pi-q)| < |vk (pj-q)|.



2.2   Algorithm

To process a query associated to a point q: 1. compute s(q)=s1(q) sL(q) with sk(q) the primitive interval from Sl containing vl q; 2. retrieve the apex ordering associated to s(q) and return the k first entries.

This algorithm therefore returns an e-approximation of the k nearest neighbours of a point q in a deterministic fashion, so that the answer is guaranteed to be correct if the random sample V is actually (1/2-e/3)-distinguishing---see 1.2.3 and Corollary 1. Roughly speaking, the correctness of the algorithm comes from the fact that if a non-desired point p1 has been returned instead of a desired point p2, then p1 dominated p2 in more than half of the lists of S, and the sample V was not distinguishing enough with respect to the exceptional set Wq-p1,q-p2.

The pre-processing requires O (Ln2 (n log d)(2d) ) time and O (n (n log d)(2d) ) space. The cost of a query is O(k+L (d+log n))=O(k + (d log2 d) (d + log n)).

3   Second algorithm

As opposed to the first algorithm, the second one does not try to compute a partition of the requests' space and proceeds in two steps. The first one consists in drawing a random sample from P of the appropriate size and returning the closest point to q. The second one compares iteratively q to pairs of points of P in a tournament way depicted on Figure 3. Assuming that n=2m, the overview of this second stage is the following:
  1. a random sample V of Q(d log2 n (log2 d + log d log log n)) vectors is drawn uniformly on Sd-1, and a multi-set Gv of V is assigned to each internal node v of the binary tree whose leaves are a permutation of the points of P. For a query point q, two points pi and pj of P, and a multi-set Gv, pi is said to dominate pj if |vk (pi-q)| < |vk (pj-q)| holds for a majority of vectors vk in Gv. Otherwise pj dominates pi;
  2. each internal node v of the tree is assigned its dominating child for the multi-set Gv.
The point eventually returned is the best candidate from the two points returned by the two steps. It can be shown that an e-approximation is returned with a probability greater than 1-d in O(n + d log3 n) time with a space requirement of n| V |.



Figure 1: Tournament tree


Figure 2: Molecule and fragments




4   Application to molecular clustering

Suppose we are given a set of d molecular fragments, say F={f1,...,fd} and a set M={m1,...,mNm} of Nm molecules, each described by a set of fragments of F---see Figure 4. We shall represent a molecule mi by a sequence of d boolean values mi=bi, 1 bi, 2 bi,d with bi, j=1 if mi contains fj and bi, j=0 otherwise. This formulation fails to report multiple occurrences of a given fragment in a molecule, but it has the nice property that a molecule is represented by a point on the hyper-cube Hd={0,1}d. Given two molecules, we call their similarity the number of common fragments that is the quantity sim(mi,mj) = k=1d bi, k bj, k.

Given the set M we are interested in the following problem: find a partition of M into subsets of neighbours or clusters. To do so, one way to proceed see [3] consists in first building a Minimum Spanning Tree on the input data set, second removing those ``too long'' edges from the MST, and third computing the connected components we are left with. The key point lies in the MST computation, and it is shown in [6] that
Theorem 4   A MST on n points in dimension d can be found in
O(2dn2-1/2d+1 (log n)1-1/2d+1) time.

Unfortunately for our concern where d can range from 500 to 2000, the 2d constant is prohibitive. Kleinberg's algorithms customised to the hyper-cube setting could make Yao's algorithm interesting in practice.

References

[1]
Arya (S.), Mount (D. M.), Netanyahu (N. S.), Silverman (R.), and Wu (A. Y.). -- An optimal algorithm for approximate nearest neighbour searching. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 573--582. -- New York, 1994.

[2]
Edelsbrunner (Herbert). -- Algorithms in combinatorial geometry. -- Springer-Verlag, Berlin, 1987, EATCS Monographs on Theoretical Computer Science, vol. 10, xvi+423p.

[3]
Jain (Anil K.) and Dubes (Richard C.). -- Algorithms for clustering data. -- Prentice-Hall Inc., Englewood Cliffs, NJ, 1988, Prentice-Hall Advanced Reference Series, xiv+320p.

[4]
Kleinberg (J.). -- Two algorithms for nearest-neighbour search in high dimension. In ACM STOC. -- El Paso, Texas, USA, 1997.

[5]
Vapnik (N.) and Chervonenkis (A.). -- On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, vol. 16, n°2, 1971, pp. 264--280.

[6]
Yao (Andrew Chi Chih). -- On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, vol. 11, n°4, 1982, pp. 721--736.

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