# Deep Unsupervised Hashing by Distilled Smooth Guidance

###### Abstract

Hashing has been widely used in approximate nearest neighbor search for its
storage and computational efficiency. Deep supervised hashing methods are not
widely used because of the lack of labeled data, especially when the domain is
transferred. Meanwhile, unsupervised deep hashing models can hardly achieve
satisfactory performance due to the lack of reliable similarity signals. To
tackle this problem, we propose a novel deep unsupervised hashing method,
namely Distilled Smooth Guidance (DSG), which can learn a distilled dataset
consisting of similarity signals as well as smooth confidence signals. To be
specific, we obtain the similarity confidence weights based on the initial
noisy similarity signals learned from local structures and construct a
priority loss function for smooth similarity-preserving learning. Besides,
global information based on clustering is utilized to distill the image pairs
by removing contradictory similarity signals. Extensive experiments on three
widely used benchmark datasets show that the proposed DSG consistently
outperforms the state-of-the-art search methods.

Index Terms—  Learning to hash, Unsupervised learning, Deep learning

##  1 Introduction

Deep learning-based hashing methods can be divided into supervised hashing and
unsupervised hashing[1]. At the early stage, many researchers mainly focused
on the supervised hashing methods, which utilize semantic labels to greatly
improve the performance of image retrieval[2]. However, supervised hashing
methods are difficult to be applied in practice when there is not enough
labeled data, especially when the domain is transferred. To solve this
problem, several deep learning-based unsupervised methods were proposed,
including deep binary descriptors (DeepBit)[3], semantic structure-based
unsupervised deep hashing (SSDH)[4] and unsupervised deep hashing by
distilling data pairs (DistillHash)[5].

Although deep learning-based unsupervised hashing methods can be applied on
unlabelled data, they still have evident limitations. [6] takes clustering
information to generate pairwise pseudo-labels. SSDH further studies the deep
feature statistics empirically from a pre-trained model and captures the
semantic relationships across different data points. Specifically, it selects
image pairs with confident pseudo-labels to guide the training of the model.
Nevertheless, SSDH discards most image pairs which are hard to be decided
whether they are semantically similar or dissimilar, which causes much
information loss and thus limits the performance of the model in further image
retrieval. In general, these methods only consider either the local
information of similarity signals or the global information such as clustering
labels but do not consider this task comprehensively.

To tackle the above issues, we propose a novel method, which comprehensively
explores correlations among image pairs. First of all, for the correlation
among different samples, we adopt the pre-trained deep convolutional neural
network (CNN) to generate features for the input images. Then we compute the
pairwise cosine distance and construct the similarity pseudo graph. We take
all the image pairs into consideration. To alleviate the effect of wrong
pseudo labels, different weights are assigned to image pairs according to the
confidence of the pseudo labels. Furthermore, for the global robustness, we
adopt clustering on the deep image features and then obtain another similarity
graph. According to these two similarity graphs, image pairs with different
correlation identification are considered contradictory and thus distilled.
Finally, we design a deep neural network based on the distilled data pair set
and adopt a deep learning framework to perform the deep representation and
hash code learning simultaneously. Our main contributions can be summarized as
follows:

  * •

We introduce two similarity graphs based on the local (i.e., pairwise cosine
similarity) and global information (i.e., clustering) respectively and then
obtain the ensemble similarity graph by distilling the contradictory pairs.

  * •

We construct a priority loss function for similarity-preserving learning,
which prioritizes confident image pairs over fuzzy image pairs based on their
pairwise distance to learn deep representations smoothly.

  * •

Experiments on three popular benchmark datasets show that our method DSG
outperforms current state-of-the-art unsupervised hashing methods by a large
margin.

Fig. 1: The framework of our model. First, deep features of the dataset are
extracted through the pre-trained VGG-F network. Second, the pseudo-graph $S$
and the smooth weight matrix $W_{1}$ are constructed based on the local
information of the cosine distance distribution of deep features. Third, the
relationship matrix $C$ is constructed based on the global information of the
K-means clustering on deep features and the distilling matrix $W_{2}$ is
obtained by comparing the relationship matrix $C$ with the pseudo-graph $S$.
Lastly, iterations are done aiming to minimize the weighted $L_{2}$ loss for
preserving similarity relationship of image pairs (i.e., Equation 6) by using
the mini-batch stochastic gradient descent method.

##  2 Related Work

Deep Supervised Hashing. Deep supervised hashing methods usually map the data
points into Hamming space where the semantic similarities can be preserved by
learning a deep neural network[1]. Typically, as the first supervised deep
hashing method, CNNH[2] splits the hash learning into two stages based on a
convolutional neural network. Hash codes are learned on the first stage, a
specific deep network is learned on the second stage to map the input samples
to the learned hash codes. Deep Supervised Hashing[7] uses a loss function
with product form based on the pairwise similarities and Hamming distances,
which can be trained by the end-to-end back propagation algorithm.

Deep Unsupervised Hashing. Unsupervised deep hashing methods aim to turn
unsupervised problems into supervised problems by constructing pseudo labels
based on deep features. Semantic Structure-based Unsupervised Deep Hashing
(SSDH)[4] uses a specific truncated function on the pairwise distances and
constructs the similarity matrices. DistillHash[5] improves the performance of
SSDH by distilling the data pairs for confident similarity signals. MLS3RDUH
[8] utilizes the intrinsic manifold structure in feature space to reconstruct
the local semantic similarity structure, and achieves the state-of-the-art
performance.

##  3 Method

In the problem of deep unsupervised hashing,
$\mathcal{X}=\\{x_{i}\\}_{i=1}^{N}$ denotes the training set with $N$ samples
without label annotations, it aims to learn a hash function

$\mathcal{H}:x\rightarrow b\in\\{-1,1\\}^{L},$

where $x$ is the input sample and $b$ is a compact $L$-bit hash code. It is
noticed that $x_{i}\in\mathbb{R}^{d}$ is the normalized extracted deep feature
for the $i$-th image through the pre-trained neural network by removing the
last fully-connected layer. Here we use VGG-F[9] to be consistent with other
articles.

###  3.1 Pseduo-Similarity Graph

In our model, the pseudo-similarity graph is constructed firstly. The pseudo-
similarity graph is used to capture pairwise similarity information from a
local perspective. Based on the pre-trained deep feature $x_{i}$, the cosine
distance between the i-th and the j-th samples can be computed by
$d_{ij}=1-\frac{x_{i}\cdot x_{j}}{||x_{i}||_{2}||x_{j}||_{2}}$. We set a large
threshold $t$, and consider data points with the cosine distance smaller than
$t$ as potential similar and data points with the cosine distance larger than
$t$ as potential dissimilar. Based on the threshold $t$, we construct the
pseudo-graph $S$ as:

$S_{ij}=\left\\{\begin{array}[]{ll}1&d_{ij}\leq t,\\\
-1&d_{ij}>t\end{array}\right.$ (1)

Where $S_{ij}$ is set to 1 if points $x_{i}$ and $x_{j}$ are potential
similar, and -1 if points $x_{i}$ and $x_{j}$ are potential dissimilar.

###  3.2 Smooth Weight Matrix

Although pseudo-similarity graph is constructed, the semantic confidence of
pseudo-label for each pair is different. In this section, we construct the
weight matrix for the pseudo-graph based on the semantic confidence.

By observing the distribution of cosine distance for deep feature pairs, [4]
finds that each distance histogram is similar to two half Gaussian
distributions, where $m_{l}$ and $\sigma_{l}$ denote the mean and the standard
deviation of the first (left half) distribution and $m_{r}$ and $\sigma_{r}$
denote the mean and the standard deviation of the second (right half)
distribution. Accordingly, we obtain two distance thresholds
$d_{l}=m_{l}-\alpha\sigma_{l}$ and $d_{r}=m_{r}+\beta\sigma_{r}$, where the
hyper-parameters $\alpha$ and $\beta$ control the values of the distance
thresholds $d_{l}$ and $d_{r}$ respectively and dictate the percentage of
similar points and dissimilar points from all data points respectively as
well. According to the theory of confidence interval, the pairs with distances
smaller than $d_{l}$ or larger than $d_{r}$ have confident semantic similarity
information. As a result, we set the weights for confident pairs to 1. Since
the pairs with distance in the interval $[d_{l},d_{r}]$ probably have no
certain semantic information, we obtain the weight as:

$W_{1,ij}=\left\\{\begin{array}[]{ll}\frac{(t-d_{ij})^{2}}{(t-d_{l})^{2}}&d_{l}<d_{ij}\leq
t,\\\ \frac{(d_{ij}-t)^{2}}{(d_{r}-t)^{2}}&t<d_{ij}<d_{r},\\\ 1&d_{ij}\leq
d_{l}\ or\ d_{ij}\geq d_{r}\end{array}\right.$ (2)

From the equation, the weight for each image pair with the distance out of the
interval is set to $1$ and the weight for each image pair with the distance in
the interval is smaller if their distance is closer to the threshold in a
quadratic form. In this way, all the image pairs are taken into consideration
with the guidance of the smooth weight matrix. Accordingly, the confidence of
their similarity relationship is guided by the smooth weight $W_{1}$.

###  3.3 Pair Distilling based on clustering

As we know, the obtained pseudo-label is very coarse. In this section, we try
to use the clustering method to distill the pairs by removing contradictory
results. To be specific, we first use the extracted
features$\\{x_{i}\\}_{i=1}^{N}$ to construct a K-means clustering model. Based
on the K-means clustering result, we construct a relationship matrix $C$:

$C_{ij}=\left\\{\begin{array}[]{ll}1&c_{i}=c_{j},\\\ -1&c_{i}\neq
c_{j}\end{array}\right.$ (3)

where $c_{i}$ represents the label of the cluster that the data point $x_{i}$
belongs to. $c_{i}\in\\{0,1,...,K\\}$ and $K$ is the parameter that represents
the number of clusters. By comparing the relationship matrix $C$ with the
pseudo-graph $S$, we obtain the distilling matrix $W_{2}$ as:

$W_{2,ij}=\left\\{\begin{array}[]{ll}1&S_{ij}=C_{ij},\\\ 0&S_{ij}\neq
C_{ij}\end{array}\right.$ (4)

in which $S_{ij}=C_{ij}$ means that the result of pseudo-graph and clustering
for the pair $x_{i}$ and $x_{j}$ is consistent.

After setting the weight of contradictory pairs to zero, the final distilled
weight matrix is obtained as

$W=W_{1}\odot W_{2},$

in which $\odot$ denotes the element-wise product of vectors.

###  3.4 Hash code learning

For the purpose of preserving the similarity relationship of data points,
similar data points are expected to be mapped into similar hash codes and
dissimilar data points are expected to be mapped into dissimilar hash codes.
The similarity output $H_{N\times N}$ of hash codes is formulated as

$H_{ij}=\frac{1}{L}\boldsymbol{b}_{i}\top\boldsymbol{b}_{j},\quad\boldsymbol{b}_{i}\in\\{-1,+1\\}^{L}$
(5)

To preserve the obtained semantic structures, we minimize the weighted $L_{2}$
loss between the hash code similarity and the pseudo-graph. In formulation,

$\mathcal{L}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}W_{ij}(H_{ij}-S_{ij})^{2}$

in which $b_{i}=sign(F(x_{i};\Theta))$ and $F(x_{i};\Theta)$ denotes the
output of the neural network, and $\Theta$ is the parameters. In this way, we
can integrate the above loss function into the deep architecture. However, it
is infeasible to train the neural network with binary outputs by the standard
Back Propagation algorithm because of the ill-posed gradient problem. As a
result, $tanh(\cdot)$ is utilized to relieve the binary constraint.

Algorithm 1 Training algorithm for our model

0: Training images $\mathcal{I}=\\{I_{1},\dots,I_{N}\\}$; Number of clusters:
$K$; Cosine distance threshold: $t$

0: Parameters $\Theta$ for the neural network; Hash codes $B$ for training
images.

1: Get deep features of $\mathcal{I}$ through VGG-F:
$\mathcal{X}=\\{x_{1},\dots,x_{N}\\}$.

2: Construct the pseudo-graph $S$ by Equation 1

3: Construct the weight matrix $W_{1}$ by Equation 2

4: Cluster $\mathcal{X}$ into $K$ different groups by $K$-means. and construct
the distilling matrix $W_{2}$ by Equation 3

5: Calculate distilled weight matrix $W$ by Equation 4

6: repeat

7: Sample N points from $\mathcal{X}$ and then construct a mini-batch.

8: Calculate the outputs by forward-propagating through the network.

9: Update parameters of the VGG-F network by Back propagation by Equation 6

10: until convergence

Thus we adopt the following objective function:

$\begin{array}[]{c}\min_{\Theta}\mathcal{L}(\Theta)=\frac{1}{m^{2}}\sum_{i=1}^{m}\sum_{j=1}^{m}W_{ij}\left(H_{ij}-S_{ij}\right)^{2}\\\
\\\ \text{s.t.}\quad
H_{ij}=\frac{1}{L}\boldsymbol{v}_{i}\top\boldsymbol{v}_{j},\quad\boldsymbol{v}_{i}=\tanh\left(F\left(\boldsymbol{x}_{i};\Theta\right)\right)\end{array}$
(6)

in which $\boldsymbol{v}_{i}$ denotes the relaxed binary representation.

For the point $q_{i}$ not in the training set, its hash code
$\boldsymbol{b}_{i}$ is obtained by directly forward propagating it through
the learned neural network.

$\boldsymbol{b}_{i}=\operatorname{sign}\left(F\left(\boldsymbol{q}_{i};\Theta\right)\right)$
(7)

###  3.5 Optimization

To optimize the problem, we construct the pseudo-graph $S$ from the pre-
trained neural network by using Equation 1. Then the smooth weight matrix
$W_{1}$ and the distilling matrix $W_{2}$ are constructed to get the distilled
weight matrix $W$. Lastly, we minimize Equation 6 by using the standard
stochastic gradient descent (SGD) method. The whole learning procedure is
summarized in Algorithm 1.

##  4 Experiments

Table 1: MAP for different methods on FLICKR25K, CIFAR-10 and NUS-WIDE datasets. | FLICKR25K | CIFAR-10 | NUS-WIDE
---|---|---|---
Method | 16bits | 32bits | 64bits | 128bits | 16bits | 32bits | 64bits | 128bits | 16bits | 32bits | 64bits | 128bits
SH | 0.6091 | 0.6105 | 0.6033 | 0.6014 | 0.1605 | 0.1583 | 0.1509 | 0.1538 | 0.4350 | 0.4129 | 0.4062 | 0.4100
SpH | 0.6119 | 0.6315 | 0.6381 | 0.6451 | 0.1439 | 0.1665 | 0.1783 | 0.1840 | 0.4458 | 0.4537 | 0.4926 | 0.5000
SGH | 0.6362 | 0.6283 | 0.6253 | 0.6206 | 0.1795 | 0.1827 | 0.1889 | 0.1904 | 0.4994 | 0.4869 | 0.4851 | 0.4945
DeepBit | 0.5934 | 0.5933 | 0.6199 | 0.6349 | 0.2204 | 0.2410 | 0.2521 | 0.2530 | 0.3844 | 0.4341 | 0.4461 | 0.4917
SSDH | 0.7240 | 0.7276 | 0.7377 | 0.7343 | 0.2568 | 0.2560 | 0.2587 | 0.2601 | 0.6374 | 0.6768 | 0.6829 | 0.6831
DistillHash | 0.6964 | 0.7056 | 0.7075 | 0.6995 | 0.2844 | 0.2853 | 0.2867 | 0.2895 | 0.6667 | 0.6752 | 0.6769 | 0.6747
CUDH | 0.7332 | 0.7426 | 0.7549 | 0.7561 | 0.2856 | 0.2903 | 0.3025 | 0.3000 | 0.6996 | 0.7222 | 0.7451 | 0.7418
MLS3RDUH | 0.7587 | 0.7754 | 0.7870 | 0.7927 | 0.2876 | 0.2962 | 0.3139 | 0.3117 | 0.7056 | 0.7384 | 0.7629 | 0.7818
DSG | 0.7994 | 0.8172 | 0.8197 | 0.8245 | 0.3225 | 0.3241 | 0.3453 | 0.3450 | 0.7795 | 0.7981 | 0.8098 | 0.8187
Table 2: Ablation analysis on datasets FLICKR25K, CIFAR-10 and NUS-WIDE. | FLICKR25K | CIFAR-10 | NUS-WIDE
---|---|---|---
Method | 16bits | 32bits | 64bits | 128bits | 16bits | 32bits | 64bits | 128bits | 16bits | 32bits | 64bits | 128bits
DSG-v2 | 0.6896 | 0.6570 | 0.6228 | 0.6178 | 0.2356 | 0.2099 | 0.1704 | 0.1567 | 0.6967 | 0.6736 | 0.6257 | 0.6062
DSG-v1 | 0.7584 | 0.7432 | 0.7355 | 0.7240 | 0.2612 | 0.2496 | 0.2441 | 0.1988 | 0.7632 | 0.7718 | 0.7719 | 0.7703
DSG | 0.7994 | 0.8172 | 0.8197 | 0.8245 | 0.3225 | 0.3241 | 0.3453 | 0.3450 | 0.7795 | 0.7981 | 0.8098 | 0.8187

We implement extensive experiments on three datasets to evaluate our DSG by
comparing with several state-of-the-art unsupervised hashing methods.

###  4.1 Datasets and Baselines

CIFAR-10[10] is a dataset for image classification and retrieval, containing
60K images from 10 different categories. For each class, we randomly select
1,000 images as queries and 500 as training images, resulting in a query set
containing 10,000 images and a training set made up of 5,000 images. All
images except for the query set are used as the retrieval set. NUS-WIDE[11]
contains 269,648 images, each of the images is annotated with multiple labels
referring to 81 concepts. The subset containing the 10 most popular concepts
is used here. We randomly select 5,000 images as a test set; the remaining
images are used as a retrieval set, and 5000 images are randomly selected from
the retrieval set as the training set. FLICKR25K[12] contains 25,000 images
collected from the Flickr website. Each image is manually annotated with at
least one of the 24 unique labels provided. We randomly select 2,000 images as
a test set; the remaining images are used as a retrieval set, from which we
randomly select 10,000 images as a training set.

Our method is compared with both traditional hashing methods and state-of-the-
art unsupervised deep learning methods. Traditional methods includes SpH[13],
DSH[14] and SGH[15]. Deep unsupervised hashing methods includes DeepBits[3],
SSDH[4], DistillHash [5], CUDH [16], and MLS3RUDH[8].

###  4.2 Implementation Details

The framework is implemented by Pytorch V1.4. The mini-batch size is set to 24
and the momentum to 0.9. The learning rate is fixed at 0.001. The initial
weights of the first seven layers of the neural network are from the model
pre-trained with ImageNet, and the last fully-connected layer is learnt from
scratch. The parameter $\alpha$ and $\beta$ are all set following [4] and the
threshold $t$ to $0.1$. The number of clusters is $70$.

###  4.3 Evaluation

The ground-truth similarity information for evaluation is constructed from the
ground-truth image labels: two data points are considered similar if they
share the same label (for CIFAR-10) or share at least one common label (for
FLICKR25K and NUSWIDE).

The retrieval quality are evaluated by the following three evaluation metrics:
Mean Average Precision (MAP), Precision-Recall curve and Top N precision
curve.

MAP is a widely-used criteria to evaluate retrieval accuracy. Given a query
and a list of $R$ ranked retrieval results, the average precision (AP) for the
given query can be computed. MAP is defined as the average of APs for all
queries. For datasets FLICKR25K and NUSWIDE, we set $R$ as $5000$ for the
experiments. For CIFAR-10, $R$ is set as $50000$. Precision-recall curve
reveals the precision at different recall levels and is a good indicator of
overall performance. In addition, Top N precision curve, which is the
precision curve with respect to the top $K$ retrieved instances, also
visualizes the performance from a different perspective .

###  4.4 Overall Performance and Ablation Study

Table 1 shows the MAPs for different methods on three datasets with hash code
lengths varying from 16 to 128. It can be seen that the performances of deep
learning-based algorithms are overall better than traditional methods. For our
proposed method, we find that DSG has a significant improvement over the
state-of-the-art deep learning-based methods in all cases, which implies the
superiority of our model. By comparing with MLS3RUDH, which has the best
performance of MAP among the deep hashing methods, DSG improves the MAP by
3.18%, 3.33% and 3.69% for 128 bit length on datasets FLICKER25K, CIFAR-10 and
NUS-WIDE respectively. We also find that the larger the bit length, the
greater the improvement of DSG over other methods, which implies that DSG can
generate more independent hash bits.

(a) FLICKR25K

(b) CIFAR-10

(c) NUS-WIDE

(d) FLICKR25K

(e) CIFAR-10

(f) NUS-WIDE

Fig. 2: (a), (b) and (c) are the Top N precision curves with code length 128
on FLICKR25K, CIFAR-10 and NUS-WIDE. (d) and (e) and (f) are the precision-
recall curves with code length 128 on FLICKR25K, CIFAR-10 and NUS-WIDE.

We also compare the performance of the DSG full model, the DSG model without
the weight matrix $W_{1}$ , which is denoted as DSG-v1 and the DSG model
without the weight matrix $W_{1}$ and the distilling matrix $W_{2}$, which is
denoted as DSG-v2. The results are shown in Table 2. It is easy to find that
DSG always achieves the highest MAPs, which implies that both the confidence
information provided by the weight matrix $W_{1}$ and the global consistency
that is ensured by the distilling matrix $W_{2}$ are necessary in our model.
We can also find that DSG-v1 always outperforms DSG-v2, which proves that the
clustering alone can help improve performance by removing contradictory
similarity relationships between the pseudo-graph $S$ and the relationship
matrix $C$.

For a more comprehensive comparison, we draw precision-recall curves and Top N
precision curves for our method DSG and three state-of-art methods CUDH, SSDH
and MLS3RUDH with the hash code length of 128. Figure 2 (a), (b) and (c) show
the Top N precision curves of CUDH, DSG, SSDH and MLS3RUDH on datasets
FLICKR25K, CIFAR-10 and NUS-WIDE. It can be seen that DSG always has the
highest precision among these four models and MLS3RUDH always has the second-
highest precision. Since the precision curves are based on the ranks of
Hamming distance, DSG is able to achieve the highest recall if we directly use
Hamming distance for retrieval. It’s known that hash codes can also be used
for coarse filtering in the form of hash table lookup; we also plot the
precision-recall curves for these four models on the same datasets, which are
shown in Figure 2 (d), (e) and (f). It can be clearly seen that the curves of
DSG are always on top of the other three models’ curves, which implies that
the hash codes obtained by DSG are also more suitable for the hash table
lookup search strategy, which further demonstrates the superiority of our
method.

Fig. 3: MAP w.r.t different numbers of clusters and threshold $t$ values with
code length 128 on CIFAR-10.

###  4.5 Parameter Sensitivity

We further study the number of clusters $K$ and threshold $t$. Figure 3 shows
the MAPs of different $K$ values ranging $[50,150]$ on the dataset CIFAR-10
with code length 128 and the MAPs of different threshold $t$ values ranging
$[0.05,0.20]$ on the dataset CIFAR-10 with code length 128 as well. We find
the MAP with the $K$ value of 100 is slightly lower than other $K$ values’. In
general, DSG’s performance with different $K$ values ranging $[50,150]$ is
relatively stable, indicating that the model performance is not sensitive to
different $K$ values ranging $[50,150]$. Furthermore, we show that the MAP
decreases as the threshold $t$ is over $0.1$ with K fixed to $70$. In
addition, the performance of DSG with different $t$ ranging $[0.05,0.1]$ is
relatively stable, which indicates that the suitable interval for $t$ value is
$[0.05,0.1]$. Accordingly, we set $K$ as $70$ and $t$ as $0.1$ in our other
experiments as default.

##  5 Conclusion

In this paper, we proposed Distilled Smooth Guidance (DSG) for deep
unsupervised hashing. DSG not only considers local similarity signals but also
considers the confidence of similarity signals from local structure for
smoothness. What’s more, global information is also explored and the image
pairs are distilled by removing contradictory image pairs from two views for
the purpose of accuracy. Numeric experiments demonstrates that DSG outperforms
the existing state-of-the-art methods.

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