1.
Seminar@ochanomizu university , October 14, 2010(Thu.)
Yasuo Tabei(JST Minato ERATO Project)
Joint work with Takeaki Uno(NII),
Masashi Sugiyama (TITECH), Koji Tsuda (AIST)

2.
 Motivation
- Large-scale data
- Needs for all pairs similarity search method
- Single sorting sethod, and its drawbacks
 Method
- Multiple sorting method
- Locality sensitive hashing
- SketchSort
 Experiments
- Comparison of other state-of-the art methods
- Use large-scale image datasets

3.
 Image Chemical Compounds
- 80 million tiny images - NCBI PubChem
(Torralba et al., (2008)) - 28 million chemical
- size: 32×32 pixes compounds
Genome Sequences
- NCBI Sequence Read Archive
- A large-scale genome sequences
from various organisms

4.
Image
sift Vector
x=(0.3, -0.3, 0.5, 1.2, …)
Chemical Compound
fingerprint
x=(1, 0, 1, 0, 0, 0, 1, …)
Text, Protein, DNA/RNA etc

5.
 Mapping vector to binary string (sketch)
- Conserve the distance in the original space
x=(0.3, 0.1, 0.5, 0.6, 0.7, 1.2, -0.2,…)
Mapping
s=1010010001110001010…
 Advantages
- Can keep giga-scale data in main memory
- Accelerate various algorithms

6.
 Finding all neighbor pairs from vector data
- Given a set of data-points
- Find all pairs within a distance ,
xi , x j Δ( xi , x j ε
s.t. )
ε

7.
 Can build a neighborhood graph
- Vertex: a data-point
- Edge : a neighbor pair
Applications: semi-supervised learning, spectral
clustering, ROI detection in images, retrieval of
protein sequences, etc
ε

8.
 Finding neighbor pairs by sorting
 Map vector data to skeches
(a)Input (b) Sort (c) Scan neighbors
1:101111 7:000000 7:000000
2:110101 4:010000 4:010000
3:110010 8:010110 8:010110 w
4:010000 10:100100 10:100100
5:101000 5:101000 5:101000
6:111100 1:101111 1:101111
7:000000 3:110010 3:110010 w
8:010110 2:110101 2:110101
9:110110 9:110110 9:110110
10:100100 6:111100 6:111100

9.
 Need a large number of distance calcuration for
achieving reasonable accuracy
 Can not derive an analytical estimation of the
fraction of missing neighbors

10.
 Motivation
- Large-scale data
- Needs for all pairs similarity search method
- Single sorting sethod, and its drawbacks
 Method
- Multiple sorting method
- Locality sensitive hashing
- SketchSort
 Experiments
- Comparison of other state-of-the art methods
- Use large-scale image datasets

11.
 Input: set of fixed-length strings S={s1,…,sn }
 Output: all pairs of strings within a Hamming
distance d
 By appling radixsort, enumerate all pairs in O(n+m)
- n: number of strings, m: output pairs
 Introduce block-wise masking technique for acceleration

12.
 Sort strings by radixsort, divide strings into equivalence
classes O(n)
 Draw edges within all strings in an equivalece class O(m)
 Computational Complexity: O(n+m)
EMILY ALICE
DAVID ALICE
CHRIS BOBBY
ALICE Sort CHRIS
Equivalence
DAVID DAVID Classes
BOBBY DAVID
DAVID DAVID
ALICE EMILY

13.
 Mask d characters in all possible ways
l 
 Performe radixsort  d  times
 
 
 Linear time to the number of strings
 Time exponential to d, polynomial to the length of
strings l
 Ex)d=2 7:0000 0001 0011 1110 7:0000 0001 0011 1110
4:0100 0001 1101 1100 4:0100 0001 1101 1100
8:0101 1001 0111 1000 8:0101 1001 0111 1000
10:1001 0011 1001 0111 5:1010 0010 1110 1010
5:1010 0010 1110 1010 3:1100 1000 1101 1100
1:1011 1111 0011 1110 6:1111 0011 1001 0111
2:1101 0111 0111 0001 10:1001 0011 1001 0111
3:1100 1000 1101 1100 2:1101 0111 0111 0001
9:1101 1000 1101 1110 9:1101 1000 1101 1110
6:1111 0011 1001 0111 1:1011 1111 0011 1110

14.
 Mask d blocks in all possible ways
 Can reduce the number of sorting operations
 Non-neighbor might be detected
 Filter out by calcurating actual Hamming distance
 Ex)d=2
7:0000 0001 0011 1110 7:0000 0001 0011 1110 7:0000 0000 0000 1110
4:0100 0001 1101 1100 4:0100 0001 1101 1100 4:0100 0000 0000 1100
8:0101 1001 0111 1000 8:0101 1001 0111 1000 8:0101 0000 0000 1000
10:1001 0011 1001 0111 10:1001 0011 1001 0111 10:1001 0000 0000 0111
5:1010 0010 1110 1010 5:1010 0000 1110 0000 5:1010 0000 0000 1010
1:1011 1111 0011 1110 1:1011 0000 0011 0000 1:1011 0000 0000 1110
3:1100 1000 0000 0000 3:1100 0000 1101 0000 3:1100 0000 0000 1100
2:1101 0111 0000 0000 2:1101 0000 0111 0000 2:1101 0000 0000 0001
9:1101 1000 0000 0000 9:1101 0000 1101 0000 9:1101 0000 0000 1110
6:1111 0011 0000 0000 6:1111 0000 1001 0000 6:1111 0000 0000 0111
7:0000 0001 0011 0000 4:0000 0001 0000 1100 1:0000 0000 0011 1110
4:0000 0001 1101 0000 7:0000 0001 0000 1110 7:0000 0000 0011 1110
5:0000 0010 1110 0000 5:0000 0010 0000 1010 2:0000 0000 0111 0001
6:0000 0011 1001 0000 6:0000 0011 0000 0111 8:0000 0000 0111 1000
10:0000 0011 1001 0000 10:0000 0011 0000 0111 4:0000 0000 0111 1000
2:0000 0111 0111 0000 2:0000 0111 0000 0001 6:0000 0000 1001 0111
3:0000 1000 1101 0000 3:0000 1000 0000 1100 10:0000 0000 1001 0111
9:0000 1000 1101 0000 9:0000 1000 0000 1110 3:0000 0000 1101 1100
8:0000 1001 0111 0000 8:0000 1001 0000 1000 9:0000 0000 1101 1110
1:0000 1111 0011 0000 1:0000 1111 0000 1110 5:0000 0000 1110 1010

15.
Step1. Perform radixsort in a block, and detect
equivalence classes
Step2. For each equivalence class, perform radixsort
the next block
Recursive
7:0000 0001 0011 0000
7:0000 0001 0011 0000 7:0000 0001 0011 0000 4:0000 0001 1101 0000
4:0000 0001 1101 0000 4:0000 0001 1101 0000
5:0000 0010 1110 0000 5:0000 0010 1110 0000
6:0000 0011 1001 0000 6:0000 0011 1001 0000 6:0000 0011 1001 0000
10:0000 0011 1001 0000 10:0000 0011 1001 0000 10:0000 0011 1001 0000
2:0000 0111 0111 0000 2:0000 0111 0111 0000
3:0000 1000 1101 0000 3:0000 1000 1101 0000
9:0000 1000 1101 0000 9:0000 1000 1101 0000
8:0000 1001 0111 0000 8:0000 1001 0111 0000 3:0000 1000 1101 0000
1:0000 1111 0011 0000 1:0000 1111 0011 0000 9:0000 1000 1101 0000

16.
 All neighbor pairs can be
7:0000 0001 0011 1110
enumerated 4:0100 0001 1101 1100 2:1101 0111 0000 0000
8:0101 1001 0111 1000 9:1101 1000 0000 0000
10:1001 0011 1001 0111
5:1010 0010 1110 1010 2:1101 0111 0011 0000
1:1011 1111 0011 1110 9:1101 1000 0000 0000
3:1100 1000 0000 0000
2:1101 0111 0000 0000 2:1101 0111 0000 0001
9:1101 1000 0000 0000 9:1101 1000 0000 1110
6:1111 0011 0000 0000
7:0000 0001 0011 0000
4:0000 0001 1101 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
7:0000 0001 0011 0000 1:0000 0000 0011 1110
1:0000 0000 0011 1110 7:0000 0000 0011 1110
4:0000 0001 1101 0000 3:0000 1000 1101 0000
7:0000 0000 0011 1110
5:0000 0010 1110 0000 9:0000 1000 1101 0000
2:0000 0000 0111 0001 2:0000 0000 0111 0001
6:0000 0011 1001 0000
8:0000 0000 0111 1000 8:0000 0000 0111 1000
10:0000 0011 1001 0000 4:0000 0001 0011 1000
4:0000 0000 0111 1000 4:0000 0000 0111 1000
2:0000 0111 0111 0000 7:0000 0001 1101 1110
6:0000 0000 1001 0111
3:0000 1000 1101 0000
6:0000 0011 1001 0111 10:0000 0000 1001 0111
9:0000 1000 1101 0000 3:0000 0000 1101 1100
10:0000 0011 1001 0111 3:0000 0000 1101 1100
8:0000 1001 0111 0000 9:0000 0000 1101 1110
9:0000 0000 1101 1110
1:0000 1111 0011 0000 3:0000 1000 1101 1100 5:0000 0000 1110 1010
9:0000 1000 1101 1110

17.
 The same pair may be
7:0000 0001 0011 1110
detcted in different block 4:0100 0001 1101 1100 2:1101 0111 0000 0000
8:0101 1001 0111 1000 9:1101 1000 0000 0000
combinations 10:1001 0011 1001 0111
5:1010 0010 1110 1010 2:1101 0111 0011 0000
Ex) (3,9), (6,10) 1:1011 1111 0011 1110 9:1101 1000 0000 0000
3:1100 1000 0000 0000
 Naïve method takes n2 memory 2:1101 0111 0000 0000 2:1101 0111 0000 0001
9:1101 1000 0000 0000 9:1101 1000 0000 1110
6:1111 0011 0000 0000
7:0000 0001 0011 0000
4:0000 0001 1101 0000
6:0000 0011 1001 0000 1:0000 0000 0011 1110
10:0000 0011 1001 0000 1:0000 0000 0011 1110 7:0000 0000 0011 1110
7:0000 0001 0011 0000 7:0000 0000 0011 1110
4:0000 0001 1101 0000 3:0000 1000 1101 0000 2:0000 0000 0111 0001 2:0000 0000 0111 0001
5:0000 0010 1110 0000 9:0000 1000 1101 0000 8:0000 0000 0111 1000 8:0000 0000 0111 1000
6:0000 0011 1001 0000 4:0000 0000 0111 1000 4:0000 0000 0111 1000
10:0000 0011 1001 0000 4:0000 0001 0011 1000 6:0000 0000 1001 0111
2:0000 0111 0111 0000 7:0000 0001 1101 1110 6:1111 0011 1001 0111
10:0000 0000 1001 0111
3:0000 1000 1101 0000 10:1001 0011 1001 0111
6:0000 0011 1001 0111 3:0000 0000 1101 1100
9:0000 1000 1101 0000 9:0000 0000 1101 1110
10:0000 0011 1001 0111 3:0000 0000 1101 1100
8:0000 1001 0111 0000 5:0000 0000 1110 1010
1:0000 1111 0011 0000 9:0000 0000 1101 1110
3:0000 1000 1101 1100
9:0000 1000 1101 1110

18.
1 2 3 4
Step1: Make a total order among
6:1111 0011 1001 0111
blocks from left to right 10:1001 0011 1001 0111
Step2: Make a total order among
block combinations (1,2)<(1,3)<(1,4)
<(2,3)<(2,4)<(3,4)
Step3: Take the minimum among
matched block combinations
Combination 1 Combination 2
1 2 3 4 1 2 3 4
6:1111 0011 1001 0111 6:1111 0011 1001 0111
10:1001 0011 1001 0111 10:1001 0011 1001 0111
(1,2) < (1,4)

19.
If the number of blocks is k-d,
Eliminate duplicate pairs, and
Calculate Hamming distance
Call function to equivalence
classes

20.
 Enumerates all neighbor pairs within a distance
- (xi, xj), i < j, Δ(xi,xj) ≦ε,
 Basic idea
- Map vector data to sketches by LSH
- Enumerate all neighbor pairs by MSM
 SketchSort with cosine LSH
- Enumerate all neighbor pairs within a consine
distance threshold ε xiT x j
- (xi, xj), i < j,
Δ( xi , x j )  1  ≦ε
|| xi |||| x j ||
 Applicable to Euclidean distance (Raginsky,10),
Jaccard-coffecients

21.
 Basic idea
 Generate a random hyperplain centered at 0
 Map vector data to ‘1 if a
data-points is above the hyperplain,
or else ‘0’
 Repeat l times 10….
l
1
1
1 0….
0 0

22.
 Basic idea: Map vector data to sketches and apply MSM
 Not good: create long sketches and apply MSM at once
 Divide long sketches to Q short sketches of length l
(chunks)
 Apply MSM to each chunk, obtain neighbor pairs w.r.t
Hamming distance
l l l
1100101010010101 0101010101001010 101010101010011
0101010101010101 0101000101010101 010101010010100
1001010101011110 1010101010101010 101010101010100
1000001010101010 1010101010101010 101010010101011
1111001010111110 1010101011111010 010101010101010 ......
1010101010010101 0100111101010100 001111010001011
1000010000100001 0011111101000100 010010010001111
1111000001110101 0101001010101001 001000111100101
0001010010101001 0100101011100100 101001010001000
1111000100100010 0011010100010010 010010100010001
MSM MSM MSM
 Report neighbor pairs no more than a cosine distance
threshold ε

23.
 A neighbor pair of sketches can be detected in
several chunks within Hamming distance d
Si1100101010010101 0101010101001010 101010101010011 10101111010011 111010101001001 …
Sj0101010101010101 0101000101010101 010101010010100 11110101010011 111111111010011 …
HamDist(si1,sj1)>d HamDist(si3,sj3)>d HamDist(si5,sj5)≦d
HamDist(si2,sj2)≦d HamDist(si4,sj4)>d
Duplication!!
 The same pair is outputted several times
(Duplication)

24.
Step1: Order chunks from left to right.
1 2 3 4 5
1100101010010101 0101010101001010 101010101010011 10101111010011 111010101001001 …
0101010101010101 0101000101010101 010101010010100 11110101010011 111111111010011 …a
Step2: Check whether left chunks are no more than
Hamming distance d
1 2 3 4 5
1100101010010101 0101010101001010 101010101010011 10101111010011 111010101001001 …
0101010101010101 0101000101010101 010101010010100 11110101010011 111111111010011 …a
HamDist(si1,sj1)>d? HamDist(si3,sj3)>d? HamDist(si5,sj5)≦d
HamDist(si2,sj2)>d? HamDist(si4,sj4)>d?
- If such chunk is found, trash the pair,
Or else check cosine distance

25.
Block level Yes
Trash
duplication?
No
Check Hamming >d
Distance? Trash
≦d
chunk level Yes
Trash
duplication
No
Check Cosine >ε
Distance Trash
≦ε
Output

26.
 Call function for
each chunk
 Check duplication
four times
 Divide sketches into
equivalence classes
 Call function recursively

27.
 True edges E*, Our results E
 Type-I error (false positive): A non-neighbor pair has
a Hamming distance within d in at least one chunk
 Type II-error (false negative): A neighbor pair has a
Hamming distance larger than d in all chunks

28.
 Basically, type-II error is more crucial
- type-I errors are filtered out by distance calculations
 Missing edge ratio (type-II error) is bounded as
where p is an upper bound of the non-collision
probability of neighbors

29.
 Motivation
- Large-scale data
- Needs for All Pairs Similarity Search Method
- Single Sorting Method, and its drawbacks
 Method
- Multiple Sorting Method
- Cosine Locality Sensitive Hashing
- SketchSort
 Experiments
- Comparison of other state-of-the art methods
- Use large-scale image datasets

30.
 Two image datasets
- MNIST (60,000 data, 748 dimension)
- Tiny Image (100,000 data, 960 dimension)
 Use missing edge ratio as an evaluation measure
 Set cosine distance threshold of 0.15π
 Length of each chunk to 32bit
 Hamming distance and number of blocks are set
to (2,5) and (3,6).
 Make number of chunks vary from 2, 4, 6, …, 50
 Compare our method to Lanczos bisection method
(JMLR, 2009)

31.
 K-nearest neighbor graph construction by
SketchSort
- Keep k-nearest neighbor pairs by priority queue
 Compare SketchSort to
- Cover Tree (Beygelzimer et al., ICML 2006)
- AllKNN (Ram et al., NIPS 2009),
- Lanczos-bisection (JMLR, 2009)

32.
 Set parameters so as to keep missing edge ratio
no more than 1.0×10-6
 Enable to detect similar pairs nearly exactly
 Take only 4.3 hours for 1.6 million images
0.05π 0.1π 0.15π

33.
 Fast all pairs similarity search method
 Applicable to large-scale vector data
 Various applications
 Software
- http://code.google.com/p/sketchsort/