- The document describes a multiple sorting method called SketchSort for efficiently finding all pairs of similar items in large-scale datasets. - SketchSort maps high-dimensional vector data to binary sketches while preserving distances. It then performs multiple sorting on the sketches to enumerate similar item pairs. - Experiments show SketchSort can efficiently find neighbor pairs in large image and genetic datasets, outperforming other state-of-the-art methods. It enables applications like clustering and information retrieval in big data domains.

1. Yasuo Tabei(JST Minato ERATO Project)
Joint work with Takeaki Uno(NII),
Masashi Sugiyama (TITECH), Koji Tsuda (AIST)
Seminar@Sugiyamalab, October 26, 2010(Tue.)

2.  Motivation
- Development of large-scale data
- Needs for all pairs similarity search method
- Single sorting method 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.  Images
- flicker, image search etc
- 80 million tiny images
(Torralba et al., (2008))
Genome Sequences
- NCBI Sequence Read Archive
- A large-scale genome sequences
from various organisms
Chemical Compounds
- NCBI PubChem
- 28 million chemical
compounds

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

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,…)
s=1010010001110001010…
 Advantage
- Can keep giga-scale data in main memory
- Accelerate various algorithms
Mapping

6.  Finding all neighbor pairs from vector data
- Given a set of data-points
- Find all pairs within a distance ,
s.t. ji xx , ε Δ ),( ji xx
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
1:101111
2:110101
3:110010
4:010000
5:101000
6:111100
7:000000
8:010110
9:110110
10:100100
(b) Sort
7:000000
4:010000
8:010110
10:100100
5:101000
1:101111
3:110010
2:110101
9:110110
6:111100
(c) Scan neighbors
7:000000
4:010000
8:010110
10:100100
5:101000
1:101111
3:110010
2:110101
9:110110
6:111100
w
w

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 method, 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. Block level
duplication?
Check Hamming
Distance?
Trash
Trash
Yes
>d
No
≦d
Output
Enumerate all pairs
which share matching
block combinations by
MSM
Input fixed length strings
s1=01001001001010100
s2=11010010010101010
s3=00010100100100100
…
s6:1111 0011 1001 0111
s8:1001 0011 1001 0111

13.  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
DAVID
CHRIS
ALICE
DAVID
BOBBY
DAVID
ALICE
Sort
ALICE
ALICE
BOBBY
CHRIS
DAVID
DAVID
DAVID
EMILY
Equivalence
Classes

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

15.  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
4:0100 0001 1101 1100
8:0101 1001 0111 1000
10:1001 0011 1001 0111
5:1010 0010 1110 1010
1:1011 1111 0011 1110
3:1100 1000 0000 0000
2:1101 0111 0000 0000
9:1101 1000 0000 0000
6:1111 0011 0000 0000
7:0000 0001 0011 1110
4:0100 0001 1101 1100
8:0101 1001 0111 1000
10:1001 0011 1001 0111
5:1010 0000 1110 0000
1:1011 0000 0011 0000
3:1100 0000 1101 0000
2:1101 0000 0111 0000
9:1101 0000 1101 0000
6:1111 0000 1001 0000
7:0000 0000 0000 1110
4:0100 0000 0000 1100
8:0101 0000 0000 1000
10:1001 0000 0000 0111
5:1010 0000 0000 1010
1:1011 0000 0000 1110
3:1100 0000 0000 1100
2:1101 0000 0000 0001
9:1101 0000 0000 1110
6:1111 0000 0000 0111
7:0000 0001 0011 0000
4:0000 0001 1101 0000
5:0000 0010 1110 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
2:0000 0111 0111 0000
3:0000 1000 1101 0000
9:0000 1000 1101 0000
8:0000 1001 0111 0000
1:0000 1111 0011 0000
4:0000 0001 0000 1100
7:0000 0001 0000 1110
5:0000 0010 0000 1010
6:0000 0011 0000 0111
10:0000 0011 0000 0111
2:0000 0111 0000 0001
3:0000 1000 0000 1100
9:0000 1000 0000 1110
8:0000 1001 0000 1000
1:0000 1111 0000 1110
1:0000 0000 0011 1110
7:0000 0000 0011 1110
2:0000 0000 0111 0001
8:0000 0000 0111 1000
4:0000 0000 0111 1000
6:0000 0000 1001 0111
10:0000 0000 1001 0111
3:0000 0000 1101 1100
9:0000 0000 1101 1110
5:0000 0000 1110 1010

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

17. 7:0000 0001 0011 1110
4:0100 0001 1101 1100
8:0101 1001 0111 1000
10:1001 0011 1001 0111
5:1010 0010 1110 1010
1:1011 1111 0011 1110
3:1100 1000 0000 0000
2:1101 0111 0000 0000
9:1101 1000 0000 0000
6:1111 0011 0000 0000
2:1101 0111 0000 0000
9:1101 1000 0000 0000
2:1101 0111 0011 0000
9:1101 1000 0000 0000
2:1101 0111 0000 0001
9:1101 1000 0000 1110
7:0000 0001 0011 0000
4:0000 0001 1101 0000
5:0000 0010 1110 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
2:0000 0111 0111 0000
3:0000 1000 1101 0000
9:0000 1000 1101 0000
8:0000 1001 0111 0000
1:0000 1111 0011 0000
7:0000 0001 0011 0000
4:0000 0001 1101 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
3:0000 1000 1101 0000
9:0000 1000 1101 0000
4:0000 0001 0011 1000
7:0000 0001 1101 1110
6:0000 0011 1001 0111
10:0000 0011 1001 0111
3:0000 1000 1101 1100
9:0000 1000 1101 1110
1:0000 0000 0011 1110
7:0000 0000 0011 1110
2:0000 0000 0111 0001
8:0000 0000 0111 1000
4:0000 0000 0111 1000
6:0000 0000 1001 0111
10:0000 0000 1001 0111
3:0000 0000 1101 1100
9:0000 0000 1101 1110
5:0000 0000 1110 1010
1:0000 0000 0011 1110
7:0000 0000 0011 1110
2:0000 0000 0111 0001
8:0000 0000 0111 1000
4:0000 0000 0111 1000
3:0000 0000 1101 1100
9:0000 0000 1101 1110
 All neighbor pairs can be
enumerated

18. 7:0000 0001 0011 1110
4:0100 0001 1101 1100
8:0101 1001 0111 1000
10:1001 0011 1001 0111
5:1010 0010 1110 1010
1:1011 1111 0011 1110
3:1100 1000 0000 0000
2:1101 0111 0000 0000
9:1101 1000 0000 0000
6:1111 0011 0000 0000
2:1101 0111 0000 0000
9:1101 1000 0000 0000
2:1101 0111 0011 0000
9:1101 1000 0000 0000
2:1101 0111 0000 0001
9:1101 1000 0000 1110
7:0000 0001 0011 0000
4:0000 0001 1101 0000
5:0000 0010 1110 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
2:0000 0111 0111 0000
3:0000 1000 1101 0000
9:0000 1000 1101 0000
8:0000 1001 0111 0000
1:0000 1111 0011 0000
7:0000 0001 0011 0000
4:0000 0001 1101 0000
6:0000 0011 1001 0000
10:0000 0011 1001 0000
3:0000 1000 1101 0000
9:0000 1000 1101 0000
4:0000 0001 0011 1000
7:0000 0001 1101 1110
6:0000 0011 1001 0111
10:0000 0011 1001 0111
3:0000 1000 1101 1100
9:0000 1000 1101 1110
1:0000 0000 0011 1110
7:0000 0000 0011 1110
2:0000 0000 0111 0001
8:0000 0000 0111 1000
4:0000 0000 0111 1000
6:0000 0000 1001 0111
10:0000 0000 1001 0111
3:0000 0000 1101 1100
9:0000 0000 1101 1110
5:0000 0000 1110 1010
1:0000 0000 0011 1110
7:0000 0000 0011 1110
2:0000 0000 0111 0001
8:0000 0000 0111 1000
4:0000 0000 0111 1000
3:0000 0000 1101 1100
9:0000 0000 1101 1110
 The same pair can be
detected in different block
combinations
Ex) (3,9), (6,10)
 Naïve method takes n2 memory
6:1111 0011 1001 0111
10:1001 0011 1001 0111

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

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

21.  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
- (xi, xj), i < j, ε≦Δ
||||||||
1),(
ji
j
T
i
ji
xx
xx
xx 

22. Block level
duplication?
Check Hamming
Distance?
Chunk Level
Duplication
Check Cosine
Distance
Trash
Trash
Trash
Trash
Yes
>d
Yes
>ε
No
≦d
No
≦ε
Output
Mapping to
sketches by LSH
s1=1000101001...
s2=0110000101…
s3=0101001000…
Enumerating matching
block combinations by
MSM
Input vector data
x1=(0.1,1.2,-0.9,2.3,…)
x2=(1.5,-0.1,-1.2,-1.2,…)
x3=(-1.6,1.9,0.5,-0.6,…)
…
s6:1111 0011 1001 0111
s8:1001 0011 1001 0111

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

24.  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
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
l l
MSM MSM MSM
 Report neighbor pairs no more than a cosine distance
threshold ε
l

25.  A neighbor pair of sketches can be detected in
several chunks within Hamming distance d
1100101010010101 0101010101001010 101010101010011 10101111010011 111010101001001 …
0101010101010101 0101000101010101 010101010010100 11110101010011 111111111010011 …
HamDist(si
1,sj
1)>d
Si
Sj
HamDist(si
2,sj
2)≦d
HamDist(si
3,sj
3)>d
HamDist(si
4,sj
4)>d
HamDist(si
5,sj
5)≦d
Duplication!!
 The same pair is outputted several times
(Duplication)

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

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

28.  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

29.  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

30.  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

31.  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).
 Number of chunks vary from 2, 4, 6, …, 50
 Compare our method to Lanczos bisection method
(JMLR, 2009)

34.  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)

37.  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

38.  Fast all pairs similarity search method
 Applicable to large-scale vector data
 Applicable to Euclidean distance (Raginsky,10),
Jaccard-coffecients (Broder,00)
 Various applications
 Software
- http://code.google.com/p/sketchsort/