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GRAPH REGULARISED HASHING
SEAN MORAN†, VICTOR LAVRENKO
† SEAN.MORAN@ED.AC.UK
RESEARCH QUESTION
• Locality sensitive hashing (LSH) [1] fractures the input feature
space space with randomly placed hyperplanes.
• Can we do better by using supervision to adjust the hyperplanes?
INTRODUCTION
• Problem: Constant time nearest-neighbour search in large datasets.
• Hashing-based approximate nearest neighbour (NN) search:
– Index image/documents into the buckets of hashtable(s)
– Encourage collisions between similar images/documents
110101
010111
111101
H
H
Content Based IR
Image: Imense Ltd
Image: Doersch et al.
Image: Xu et al.
Location Recognition
Near duplicate detection
010101
111101
.....
H
QUERY
DATABASE
QUERY
NEAREST
NEIGHBOURS
HASH TABLE
COMPUTE
SIMILARITY
• Advantages:
– O(1) lookup per query rather than O(N) (brute-force)
– Memory/storage saving due to compact binary codes
GRAPH REGULARISED HASHING (GRH)
• We propose a two step iterative hashing model, Graph Regularised
Hashing (GRH) [5]. GRH uses supervision in the form of an adja-
cency matrix that specifies whether or not data-points are related.
– Step A: Graph Regularisation: the K-bit hashcode of a data-
point is set to the average of the data-points of its nearest
neighbours as specified by the adjacency graph:
Lm ← sgn α SD−1
Lm−1 + (1−α)L0
∗ S: Affinity (adjacency) matrix
∗ D: Diagonal degree matrix
∗ L: Binary bits at iteration m
∗ α ∈ {0, 1}: Linear interpolation parameter
– Step A is a simple sparse-sparse matrix multiplication, and can
be implemented very efficiently. Any existing hash function
e.g. LSH [1] can be used to initialise the bits in L0
– Step B: Data-Space Partitioning: the hashcodes produced in
Step A are used as the labels to learn K binary classifiers. This
is the out-of-sample extension step, allowing the encoding of
data-points not seen before:
for k = 1. . .K : min ||wk||2
+ C
Ntrd
i=1 ξik
s.t. Lik(wk xi + tk) ≥ 1 − ξik for i = 1. . .Ntrd
∗ wk: Hyperplane k tk: bias k
∗ xi: data-point i Lik: bit k of data-point i
∗ ξik: slack variable K: # bits Ntrd: # data-points
• Steps A-B are repeated for a set number of iterations (M) (e.g. < 10).
The learnt hyperplanes wk can then be used to encode unseen data-
points (via a simple dot-product).
STEP A: GRAPH REGULARISATION
• Toy example: nodes are images with 3-bit LSH encoding. Arcs
indicate nearest neighbour relationships. We show two images
(c,e) having their hashcodes updated in Step A:
-1 1 1
1 1 1
-1 -1 -1
1 1 -1
ba
c
e
f
g
h
d
-1 1 1
1 -1 -1
1 -1 -1
1 1 1 -1 1 1
1 1 -1
STEP B: DATA-SPACE PARTITIONING
• Here we show a hyperplane being learnt using the first bit as
(highlighted with bold box) as label. One hyperplane is learnt
per bit.
-1 1 1
1 1 1
-1 -1 -1
1 1 -1
ba
c
e
f
g
h
d
-1 1 1
w1. x−t1=0
w1
Negative
(-1)
half space
Positive
(+1)
half space
1 1 -1
1 -1 -1
-1 1 1
QUANTITATIVE RESULTS (CIFAR-10) (MORE DATASETS IN PAPER)
• Mean average precision (mAP) image retrieval results using
GIST features on CIFAR-10 ( : is significant at p < 0.01):
CIFAR-10
16 bits 32 bits 48 bits 64 bits
ITQ+CCA [2] 0.2015 0.2130 0.2208 0.2237
STH [3] 0.2352 0.2072 0.2118 0.2000
KSH [4] 0.2496 0.2785 0.2849 0.2905
GRH [5] 0.2991 0.3122 0.3252 0.3350
• Timings (seconds) averaged over 10 runs. GRH is 1) faster to
train and 2) is faster to encode unseen data-points:
Training Testing Total
GRH [5] 8.01 0.03 8.04
KSH [4] 74.02 0.10 74.12
BRE [6] 227.84 0.37 228.21
SUMMARY OF KEY FINDINGS
• First both accurate and scalable supervised hashing model
• Future work will extend GRH to streaming data sources
• Code online: https://github.com/sjmoran/grh
References:
[1] P. Indyk, R. Motwani: Approximate nearest neighbors: Towards removing the curse of dimensionality. In: STOC (1998).
[2] Y. Gong, S. Lazebnik: Iterative Quantisation. In: CVPR (2011). , [3] D. Zhang et al. Self-Taught Hashing. In: SIGIR (2010)., [4] W. Liu et
al. Supervised Hashing with Kernels. In: CVPR (2012)., [5] S. Moran, V. Lavrenko. Graph Regularised Hashing. In: ECIR (2015).,
[6] B. Kulis, T. Darrell et al. Binary Reconstructive Embedding. In: NIPS (2009).

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Graph Regularised Hashing

  • 1. GRAPH REGULARISED HASHING SEAN MORAN†, VICTOR LAVRENKO † SEAN.MORAN@ED.AC.UK RESEARCH QUESTION • Locality sensitive hashing (LSH) [1] fractures the input feature space space with randomly placed hyperplanes. • Can we do better by using supervision to adjust the hyperplanes? INTRODUCTION • Problem: Constant time nearest-neighbour search in large datasets. • Hashing-based approximate nearest neighbour (NN) search: – Index image/documents into the buckets of hashtable(s) – Encourage collisions between similar images/documents 110101 010111 111101 H H Content Based IR Image: Imense Ltd Image: Doersch et al. Image: Xu et al. Location Recognition Near duplicate detection 010101 111101 ..... H QUERY DATABASE QUERY NEAREST NEIGHBOURS HASH TABLE COMPUTE SIMILARITY • Advantages: – O(1) lookup per query rather than O(N) (brute-force) – Memory/storage saving due to compact binary codes GRAPH REGULARISED HASHING (GRH) • We propose a two step iterative hashing model, Graph Regularised Hashing (GRH) [5]. GRH uses supervision in the form of an adja- cency matrix that specifies whether or not data-points are related. – Step A: Graph Regularisation: the K-bit hashcode of a data- point is set to the average of the data-points of its nearest neighbours as specified by the adjacency graph: Lm ← sgn α SD−1 Lm−1 + (1−α)L0 ∗ S: Affinity (adjacency) matrix ∗ D: Diagonal degree matrix ∗ L: Binary bits at iteration m ∗ α ∈ {0, 1}: Linear interpolation parameter – Step A is a simple sparse-sparse matrix multiplication, and can be implemented very efficiently. Any existing hash function e.g. LSH [1] can be used to initialise the bits in L0 – Step B: Data-Space Partitioning: the hashcodes produced in Step A are used as the labels to learn K binary classifiers. This is the out-of-sample extension step, allowing the encoding of data-points not seen before: for k = 1. . .K : min ||wk||2 + C Ntrd i=1 ξik s.t. Lik(wk xi + tk) ≥ 1 − ξik for i = 1. . .Ntrd ∗ wk: Hyperplane k tk: bias k ∗ xi: data-point i Lik: bit k of data-point i ∗ ξik: slack variable K: # bits Ntrd: # data-points • Steps A-B are repeated for a set number of iterations (M) (e.g. < 10). The learnt hyperplanes wk can then be used to encode unseen data- points (via a simple dot-product). STEP A: GRAPH REGULARISATION • Toy example: nodes are images with 3-bit LSH encoding. Arcs indicate nearest neighbour relationships. We show two images (c,e) having their hashcodes updated in Step A: -1 1 1 1 1 1 -1 -1 -1 1 1 -1 ba c e f g h d -1 1 1 1 -1 -1 1 -1 -1 1 1 1 -1 1 1 1 1 -1 STEP B: DATA-SPACE PARTITIONING • Here we show a hyperplane being learnt using the first bit as (highlighted with bold box) as label. One hyperplane is learnt per bit. -1 1 1 1 1 1 -1 -1 -1 1 1 -1 ba c e f g h d -1 1 1 w1. x−t1=0 w1 Negative (-1) half space Positive (+1) half space 1 1 -1 1 -1 -1 -1 1 1 QUANTITATIVE RESULTS (CIFAR-10) (MORE DATASETS IN PAPER) • Mean average precision (mAP) image retrieval results using GIST features on CIFAR-10 ( : is significant at p < 0.01): CIFAR-10 16 bits 32 bits 48 bits 64 bits ITQ+CCA [2] 0.2015 0.2130 0.2208 0.2237 STH [3] 0.2352 0.2072 0.2118 0.2000 KSH [4] 0.2496 0.2785 0.2849 0.2905 GRH [5] 0.2991 0.3122 0.3252 0.3350 • Timings (seconds) averaged over 10 runs. GRH is 1) faster to train and 2) is faster to encode unseen data-points: Training Testing Total GRH [5] 8.01 0.03 8.04 KSH [4] 74.02 0.10 74.12 BRE [6] 227.84 0.37 228.21 SUMMARY OF KEY FINDINGS • First both accurate and scalable supervised hashing model • Future work will extend GRH to streaming data sources • Code online: https://github.com/sjmoran/grh References: [1] P. Indyk, R. Motwani: Approximate nearest neighbors: Towards removing the curse of dimensionality. In: STOC (1998). [2] Y. Gong, S. Lazebnik: Iterative Quantisation. In: CVPR (2011). , [3] D. Zhang et al. Self-Taught Hashing. In: SIGIR (2010)., [4] W. Liu et al. Supervised Hashing with Kernels. In: CVPR (2012)., [5] S. Moran, V. Lavrenko. Graph Regularised Hashing. In: ECIR (2015)., [6] B. Kulis, T. Darrell et al. Binary Reconstructive Embedding. In: NIPS (2009).