S UPERVISED D ISCRETE H ASHING
F UMIN S HEN , C HUNHUA S HEN , W EI L IU , H ENG TAO S HEN
A major difficulty of learning to hash lies in
handling the discrete constraints imposed on the
pursued hash codes,
In the proposed framework, the label information is exploited such that these binary feature
vectors are easy to be classified. To implement
this idea, we propose a joint optimization problem which jointly learns the binary embeddings
and the linear classifier. We aim to learn a set of
binary codes B = {bi }ni=1 ∈ {−1, 1}L×n for X =
{xi }ni=1 to well preserve the semantic similarities.
We adopt the multi-class classification formula>
>
> >
tion y = G(b) = W b = [w1 b, · · · , wC b] where
wk ∈ RL×1 , k = 1, · · · , C is the classification vector for class k and y is the label vector, of which
the maximum item indicates the assigned class of
x. We then formulate the hashing problem as follows,
r
b ∈ {−1, 1} ,
which typically makes hash optimizations very
challenging (NPhard in general).
Relaxation: (1) discard the discrete constraints, (2) threshold (or quantize) the solved
continuous solution.
Such approximation typically leads to lowquality hash codes due to the accumulated quantization error, especially when with long code
lengths.
min
C ONTRIBUTIONS
We named the proposed supervised hashing
approach employing discrete cyclic coordinate
descent as Supervised Discrete Hashing (SDH).
Our contributions included
1. We proposed a novel supervised hashing
framework based on the assumption that
good hash codes are optimal for linear classification.
2. We proposed a supervised algorithm to
learn binary codes without continuous relaxations.
• Efficient: By means of discrete cyclic
coordinate descent (DCC), at each
step SDH solves the associated binary
optimization and obtains an analytical solution
• Locally optimal: Optimal solution for
each bit, with all other bits fixed.
B,W,F
n
X
L(yi , W> bi ) + λ||W||2 + ν
i=1
[1] F. Shen, C. Shen, W. Liu, H. T. Shen. Supervised Discrete
Hashing. In CVPR’15
||bi − F (xi )||2
(2)
L
s.t. bi ∈ {−1, 1} .
The hash function F (xi ) = P> φ(xi ), where φ(x)
is obtained by the RBF kernel mapping, is simultaneously learned to fit the generated binary bits.
The overall optimization problem is then solved
in an iterative fashion with three corresponding
sub-problems: each sub-problem solves for one
variable while keeping other two variables fixed.
With the `2 loss, it is trivial to solve for W and F .
The most important B sub-problem writes
>
2
min ||W B|| − 2Tr(B Q)
B
L×n
B ∈ {−1, 1}
s.t.
(3)
.
where Q = WY + νF (X).
We choose to learn the binary codes B by the
discrete cyclic coordinate descent (DCC) method. In
other words, We learn B bit by bit. Let z> be the
lth row of B, l = 1, · · · , L and B0 the matrix of B
excluding z. Then z is one bit for all n samples.
Similarly, let q> be the lth row of Q, Q0 the matrix
of Q excluding q, v> the lth row of W and W0 the
matrix of W excluding v. Then we have w.r.t. z:
min
R EFERENCES
n
X
i=1
>
z
s.t.
>
R ESULTS
F ORMULATION C ONT.
0>
0
>
(v W B − q )z
n
z ∈ {−1, 1} .
(4)
This problem has the optimal solution
z = sgn(q − B0> W0 v).
Table 2: Results on CIFAR with 64 bits.
Method
Precision MAP Training time Test time
BRE
0.1299
0.1156
12042.0
6.4e-5
MLH
0.2251
0.1730
2297.5
3.2e-5
KSH
0.1656
0.3822
2625.0
3.1e-6
SSH
0.2860
0.2091
96.9
3.6e-6
CCA-ITQ
0.3524
0.3379
4.3
1.7e-6
FastHash
0.1880
0.4187
1340.7
7.1e-4
SDH
0.4229
0.4555
62.6
2.6e-6
(1)
It is easy to see form (1) that, each bit z is computed based on the pre-learned L − 1 bits B0 . This
is expected because, one can iteratively update
each bit till the procedure converges with a set of
better codes B. In our experiments, the whole L
bits for X can be iteratively learned in tL times by
(1), where usually t = 2 ∼ 5.
By carefully choosing loss functions of the
classifier, the DCC method produces the optimal hash bits in a closed form, which consequently makes the entire optimization procedure
very efficient. Therefore, the proposed binary
code learning method can easily deal with largescale datasets. We name the proposed supervised
hashing method employing discrete cyclic coordinate as Supervised Discrete Hashing (SDH).
Fig. 1: Results on NUS-WIDE.
NUSWIDE
0.6
0.5
Precision
F ORMULATION
0.4
0.3
0.1
0
D ISCRETE OR N OT ?
To see how much the discrete optimization
will benefit the hash code learning, we perform
a comparison of our hash formulation (2) with or
without the discrete constraints. The comparative
results on CIFAR are shown in Table 1. As can be
seen, our discrete hashing framework SDH consistently yields better hash codes than the relaxed
one by removing the sign function. In particular for precision, the performance gaps between
these two methods are increased with longer hash
bits.
Table 1: Discrete vs. relaxed solution
Constraint 32 bits 64 bits 96 bits
Discrete
0.5090 0.4229 0.3515
Precision
Relaxed
0.4718 0.3354 0.1685
Discrete
0.4307 0.4555 0.4582
MAP
Relaxed
0.3777 0.4150 0.4244
S OURCE CODE
Available
soon
at
https://sites.
google.com/site/shenfumin or https:
//github.com/bd622.
SDH
KSH
CCA−ITQ
BRE
MLH
0.2
16 32
64
96
Code length
NUSWIDE
128
0.55
0.5
MAP
P ROBLEM
0.45
0.4
0.35
16 32
64
96
Code length
128
Table 3: Classification accuracy on CIFAR
with the learned binary codes by linear SVM
Method
64 bits 128 bits
BRE
0.332
0.335
CCA-ITQ 0.564
0.570
SSH
0.432
0.460
KSH
0.590
0.598
FastHash
0.578
0.603
SDH
0.636
0.651
GIST
0.580