Random projection trees and low dimensional manifolds

2013. 01.07( 월 )Jeonbuk National Univ.

DBLAB 김태훈

Yoav Freund, Sanjoy Dasgupta

University of California, San Diego 2008

Contents

1. Introduction

2. Detailed overview

3. An RP-Tree-MAX adapts to assouad dimension.

4. An RP-Tree-MEAN adapts to local covariance dimension.

Introduction

A k-d Tree is spatial data structure that partitions into hyperrect-angular cells.• k-d Tree 는 hyperrectangular cells 속 파티션들의 공간적인 데이터 구조

It is built in a recursive manner, splitting along one coordinate di-rection at a time.• k-d Tree 는 한 방향을 따라서 한번에 분리되는 재귀적인 방법을 이용

Introduction

The succession of splits corresponds to a binary tree whose leave contain the individual cells in .• 이 분리의 연속은 각 셀들을 포함하고 있는 잎의 이진 트리와 부합함 .

suppose that

*. The dots are points in a database. *. The cross is a query point q.

Introduction

K-d Tree requires D level in order to halve the cell diameter.• K-d 트리는 각 반경을 나누기 위해서 D level 을 요구

If the data lie in , it could take 1000 levels of the tree to bring the diameter of cells down to half that of the entire data set.• 만약 data 가 주어 졌을 경우 1000 level 을 내려가야 함 .

This would require data points!

Introduction

Thus k-d trees are susceptible to the same curse of dimensional-ity.• 그래서 k-d tree 는 차원의 저주를 받을 정도로 민감 .

However, a recent positive development in machine learning has been realization that a lot of data which superficially lie in a very high-dimensional space , actually have low intrinsic dimension.• 하지만 최근 machine learning 에서 깨닫게 되었는데 많은 데이터들이 주어졌을 때 실제로는 매우 높은 는 낮은 고유한 차원을 가짐 .

d << D• d(nonparameter 실제 주어지는 데이터 ) 보다 D 차원에 더 민감함

Introduction

In this paper, we are interested in techniques that automaticallyadapt to intrinsic low dimensional structure without having to explic-itly learn this structure.

• 이 논문에서는 명시적으로 이 구조에 배울 필요 없이 관심 있는 테크닉인 자동적으로 적응하는 고유의 저차원 구조에 대해서 서술 하고자 함 .

Detailed overview

Both k-d trees and RP trees are built by recursive binary splits.• K-d tree 와 RP tree 는 재귀적으로 이진으로 분리되서 만듬 .

The core tree-building algorithm is called MakeTree, and takesas input a data set S

• 이 코어 트리 빌딩 알고리즘은 MakeTree 라고 불리는데 이것은 어떤 집합셋인 S가 Rd 에 속하는 input 데이터를 가짐 .

MakeTree algorithmprocedure MakeTree(S)

if |S| < MinSize return (Leaf)

Rule ← ChooseRule(S)LeftTree ← MakeTree({x ∈ S : Rule(x) = true})RightTree ← MakeTree({x ∈ S : Rule(x) = false})

return ([Rule, LeftTree, RightTree])

K-d tree version

procedure ChooseRule(S)comment: k-d tree version

choose a coordinate direction Rule() := ≤ median({ : ∈ S})

return (Rule)

procedure MakeTree(S)

if |S| < MinSize return (Leaf)

Rule ← ChooseRule(S)LeftTree ← MakeTree({x ∈ S : Rule(x) = true})RightTree ← MakeTree({x ∈ S : Rule(x) = false})

return ([Rule, LeftTree, RightTree])

RP-tree version PCA

• 임의의 방향을 선정해서 중점을 기준으로 방향을 선택 .

Principal Component Analysis( 주성분 분석 )

RP-tree Max version

procedure ChooseRule(S)comment: RPTree-Max version

choose a random unit direction v ∈ pick any x ∈ S; let y ∈ S be the farthest point from itchoose δ uniformly at random in [−1, 1] · /

Rule() := ≤ (

return (Rule

procedure MakeTree(S)

if |S| < MinSize return (Leaf)

Rule ← ChooseRule(S)LeftTree ← MakeTree({x ∈ S : Rule(x) = true})RightTree ← MakeTree({x ∈ S : Rule(x) = false})

return ([Rule, LeftTree, RightTree])

RP-tree Mean versionprocedure MakeTree(S)

if |S| < MinSize return (Leaf)

Rule ← ChooseRule(S)LeftTree ← MakeTree({x ∈ S : Rule(x) = true})RightTree ← MakeTree({x ∈ S : Rule(x) = false})

return ([Rule, LeftTree, RightTree])