A Lower Bound of Hash Codes' Performance
Authors: Xiaosu Zhu
Jingkuan Song
Yu Lei
Lianli Gao
Hengtao Shen
Venue: NeurIPS 2022 Accept (2022)
Keywords: Learning to hash
Compact representation
Fast retrieval
Hamming space metric learning

Abstract: As a crucial approach for compact representation learning, hashing has achieved great success in effectiveness and efficiency. Numerous heuristic Hamming space metric learning objectives are designed to obtain high-quality hash codes. Nevertheless, a theoretical analysis of criteria for learning good hash codes remains largely unexploited. In this paper, we prove that inter-class distinctiveness and intra-class compactness among hash codes determine the lower bound of hash codes' performance. Promoting these two characteristics could lift the bound and improve hash learning. We then propose a surrogate model to fully exploit the above objective by estimating the posterior of hash codes and controlling it, which results in a low-bias optimization. Extensive experiments reveal the effectiveness of the proposed method. By testing on a series of hash-models, we obtain performance improvements among all of them, with an up to $26.5\%$ increase in mean Average Precision and an up to $20.5\%$ increase in accuracy. Our code is publicly available at https://github.com/VL-Group/LBHash.

A Lower Bound of Hash Codes' Performance
Xiaosu Zhu' xi xiaosu. www.oztlok..tooccoooooo com
Jingkuan Song j jingkunn. song@gmail com
Lei'
1eiyu6969@gail..comm
Lianli Gaot
Heng Tao Shen 12
lianli. wwwplagggguuueeeeee..e edu.cn
shenhengtaoGhhommaill com
Center for Future Media, University of Electronic Science and Technology of China
?engg Cheng Laboratory
Abstract
As a crucial approach for compact representation learning, hashing has achieved great success in effectiveness and efficiency. Numerous heuristic Hamming space metric learning objectives are designed to obtain high-quality hash codes. Nev- ertheless, a theoretical analysis of criteria for learning good hash codes remains largely unexploited. In this paper, we prove that inter-class distinctiveness and intra-class compactness among hash codes determine the lower bound of hash codes' performance. Promoting these two characteristics could lift the bound and improve hash learning. We then propose a surrogate model to fully exploit the above objective by estimating the posterior of hash codes and controlling it, which results in a low-bias optimization. Extensive experiments reveal the effectiveness of the proposed method. By testing series of model,, obtain performance improvements among all of them, with an up to 26.5% increase in mean Average Precision and an up 20.5% increase in accuracy. Our code is publicly available at https::// //github.
Introduction
The explosive increase of multimedia digital content requires people to develop large-scale processing techniques for effective and efficient multimedia understanding [47, 50, 52, 53]. Meanwhile, growing focus on carbon neutrality initiatives urges researchers to handle above issues with low power and memory consumption [30, 31j. Fortunately, hashing, as a key role compact representation learning, has demonstrated efficiency and effectiveness over the past few decades towards above demands [22]. By digesting multimedia contents into short binary codes, hashing able to significantly reduce storage, boost speed, and preserve key information [15, 21, 25, 27, 56].
In recent years, learning to hash has shown its advantages alleviating information loss by performing data-dependent Hamming space projection with a non-linear model [24, 32, 46] compared to hand- crafted hashing functions. The purpose of optimizing hash-models meaningful hash codes for downstream tasks. A practical convention is to increase similarities among codes of the same category, otherwise the opposite. Such convention is validated to effective in many applications [8, 26, 33, 40, 41, 48).
to generate semantically
Practices in current advances still leave two questions which remain primarily unexploited: When should we determine hash-model for producing good codes, and by what means to achieve this goal? Taking comprehensive study on the above questions essential since it not only guides to
CCoresponding author.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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develop -egh-peeformacce algorithms in typical basse applications but also provides potential possibilities to inspire future works in hash learning.
Although some works [1, 46] try to answer the above questions by building connections between hash codes and common descriptors with domain knowledge, still seek theoretical guarantees for hash codes' performance. Similar situations occur in related works [30, 44, 45], where objectives to optimize hash-models are designed by intuitions or heuristics.
By formulating correlations among hash codes as inter-class distinctiveness and intra-class com- pactness, we try to address the above issues and provide a lower bound of hash codes' performance. Based on this, we could further derive from them as objectives to lift the lower bound. Moreover, to effectively train model towards such targets, estimating and further controlling the posterior of hash codes is necessary since bit-level dependence exists in posterior distribution. If we neglect this, bias will be introduced during hash-model optimization, hindering performance. Summarizing both of them, Our main contributions are summarized as follows:
1) To the best of our knowledge, we are the first to formulate a lower bound of hash codes' per- formance, which is directly proportional to inter-class distinctiveness and intra-class compactness among codes. Based on this, objective is further proposed to lift this lower bound and thus enhance performance.
2) A novel posterior estimation over hash codes is proposed to perform low-bias optimization towards the above objective. As a non-linear function, could be utilized for optimizing hash-models via gradient descent. Such plug-and-play component is able to integrate into current hash-models to boost performance.
3) Extensive experiments on three benchmark datasets reveal the effectiveness of the proposed method in two typical tasks. Specifically, we obtain an up to 26.5% increase on mean Average Precision for fast retrieval. We also observe an to 20.5% increase on accuracy for hash-based recognition.
The rest of paper is organized follows: We first briefly review current hashing methods and give preliminaries (Secs. 2 and 3). Then, lower bound on hash codes' performance by inter-class distinctiveness and intra-class compactness is proposed (Sec. 4). To fully exploit such guidance, posterior estimation over multivariate Bernoulli distribution is designed for code control (Sec. 5). Experiments (Sec. 6) are then conducted for evaluation.
2 Related Works
This paper will focus on data-dependent learning-to-hast methods. Recent advances in building these models are validated to be effective in hash-based recognition and fast retrieval [1, 17, 18, 26, 40]. The typical approach utilizes a projection function to convert raw inputs into compact binary codes and conduct classification downstream tasks [8, 41, 48]. As mentioned in introduction, to obtain such a projector, current practices suggest that we formulate it as a Hamming space metric learning problem under binary constraints. Pointwise [21, 29, 49], pairwise [4, 5, 25, 26, 37, 55), triplet [10, 28), listwise [43, 54] and prototype- based based [2, 6, 17, 51] objectives are all studied for model training, where the core idea is to adjust similarities among samples from same / different categories.
approximate nearest neighbour search on codes for
Besides the above objectives, we should also note that hash-model optimization is NP-hard [I, 29, 40, 46). That is caused by the discrete, binarized hash code formulation. To tackle the issue, previous works propose many methods that target on following purposes: a) Surrogate functions that approximate hashing by continuous functions, such as tanh [5], Stochastic neuron [9], affine [38] or Straight-Through [3]. By adopting these, the discrete hash-modes optimization is transformed into approximated but easier one via gradient descent. b)
optimization algorithm design, which focuses solutions that directly handle discrete inputs, including Coordinate descent [36], Discrete gradient estimation [13], etc. There also bag of tricks which help for stable training. For example, minimizing quantization error reduces the gap between raw outputs of the model and final hash codes [26], and code- aade-balancing alleviates the problem of producing trivial codes [12].
Please refer to literature reviews on above works [30, 44, 45] for comprehensive studies learning to hash. We could see two issues exist and remain primarily unexploited in above works. The first is objective design, which is commonly proposed by intuitive and with few theoretical guarantees.

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The second is model optimization, where bit-level dependence is essential in code control but most works neglect To tackle these, we would firstly give lower bound hash codes' performance and utilize it as guidance for hash-model training. Then, novel model optimization approach is proposed in a multivariate perspective for low-bias code control.
Preliminaries
Learning to hash performs data-dependent compact representation learning. Inputs E X C R are firstly transformed into h-dim real-valued vector by Fe. projection parameterized by 0:
Rd Fo, R And l is then binarized to h bits binary code b E B H":
+1, (+) 2 0,
Fo
hin
b, where bin (.) 1, (-) <.
Typical usage of hash codes. Common hashing-based tasks include fast retrieval [2, 19, 27, 39] and recognition [26, 32, 36, 40]. For fast retrieval, given a query under the same distribution of X. convert it into query hash code q and conduct fast approximate nearest neighbour search in to produce rank list R. In R, samples are organized from nearest to furthest to query. Correspondingly among all results in R, true positives tp' TP indicate samples that match with query while others are false positives fy' FP indicates rank). As a criterion, Average Precision (AP) is commonly adopted to determine how well the retrieved results are. It encourages true positives to have higher ranks than false positives: AP [TP] Extp,. ETP P@i, where P is the rank-i precision. As for hash-based recognition, since current works formulate it as a variant of retrieval, we could still adopt AP criterion to indicate performance. The detailed explanation is placed in Supp. Sec. C.
4 Lower Bound by Inter-Class Distinctiveness and Intra-Class Compactness
We start at an arbitrary rank list for studying the characteristics of codes in the above scenarios. To calculate AP easily, first introduce mis-rank that indicates how many false positives take higher places than true positives. We refer readers to Supp. Secs. A and B for detailed description and proof. Definition. Mis-rank indicates how many false positives have higher ranks than a true positive, egg.. tp- is at rank i, meanwhile m=| {d (q.fp) < d (q, tn) ) }., where d (q, +) is distance between and a sample, and | is number of elements set.
Remark. The rank-i precision P@i is derived be (i-mi/i.
Then, average precision could be derived as:
Σ P@i = TP
Σ
i m
AP
TP
EIP
Vtpi. E TP
for all true positives in rank list. This is because the rank-i precision Pûi for any true positive tp equals to (f-m)/s.
From the derivation, we could immediately obtain that AP increases iff m decreases. Therefore, now we could focus on how to reduce m to thereby increase average precision. Noticed that the value of m is highly related to two distances, d (g) and d (g, (p), the following proposition is raised: Proposition.
m max d (g. tp)
Vtp TP, fp FP min d (q, fp)
where denotes upper bound. Correspondingly.
min d (q. Vtp E TP, fp E FP AP x maxd (q. tp)
where denotes lower bound.
If inputs æn have associated labels y² the above studies are able to be further generalized. Class-wise centers, inter-class distinetiveness and intra-class compactness are introduced to formulate our final lower-bound of hash codes' performance.
3

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Definition. Center cf E C' is a representative hash code of a specific class
c' = arg min Σ d (b) 4 3d = =c}
b
where b is the j-th sample in set B and y is the label of b².
Definition. Inter-class distinctiveness is min D Pinter where DD inter is set that measures distances between all centers over C: {d (c), d Ve c² C.. In contrast, intra-class compactness is 7/max Disros where D intrro = fd (b", c Vy = c} measures the distances among samples of the same class to their corresponding center.
Combining above propositions and definitions, we could reach: Proposition.
min d (q. fp) AP x max d (q, tp)
min D Dinter
2 const
(I)
max Dintro
This proposition reveals how AP is affected by the above two inter- and intra- factors. Actually, could cover common scenarios criterion of hash codes' performance, e.g., fast retrieval and recognition, which will be explained in supplementary materials. Intuitively, lifting this lower bound would enhance performance, formulating the following objective:
marimize min Dinters
(2)
minimize max D Dintra
(3)
s.t. X C RR
To. bin B BC H
Inspired by [51], we give a specific solution to implement Eqs. (2) and (3) under supervised hashing by firstly defining distance maximized class-specifir centers and then shrinking hash codes their corresponding centers. We will validate the effectiveness of such solution in experiments. It is worth noting that achieving above goal an open topic, including unsupervised or other complex scenarios, which we leave for future study. Next, to optimize hash-models with the above supervision, we introduce posterior estimation over hash codes and then control
5 Posterior Estimation for Code Control
is essentially hard to minimize the distance between hash code and a target to achieve Eq. (3) for two reasons. Firstly, the posterior distribution of B given inputs X' is formulated to be under multivariate Bernoulli distribution: p PB X)) ~ MVB. If we do not take correlations among different variables into account, optimization on B3 is biased. Secondly, dealing with hash codes is a a (|-1,+1) discrete optimization, which is not trivial to handle. Therefore, we try to organize Eq. (3) as Maximum Likelihood Estimation (MLE) tackle the above issues. Considering the definition of Hamming distance between an arbitrary hash code and its target t:
h
d (b, t)
Σ 1 [t)
Σ 1 (est, < 0}
where 1 is the characteristic function and (-); indicates value on the i-th dimension. Since Hamming distance measures how many bits are different between two codes, the probability of d (b, t) å can be formulated as:
p
= =))
Σ vie (te
(b, + t, b, = =t,)]
Therefore, b and & have distance 8 iff p (d (b, t) =) 5) is maximized. However, to precisely calculate it difficult, since it involves the joint probability of b. Therefore, try estimate all joint probabilities by adopting a surrogate model Pn parameterized by ग to perform estimation:
L Pr where E O OCCR
(4)

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where o is the probabilities of a Categorical distribution (0) X) = p (B X), which contains 2
entries. Each entry of o estimates one of specific joint probability by feeding into Pr::
o, = p (bb. () (bx 0)
IskSh. 05152"-1
where is the h bits binary representation of i. For example, 03 pb (+1-1+1+1) when h 4. To train Pr, we perform Maximum Likelihood Estimation (MLE) over p (0 x)):
L (P:: 0) -log Oi. h where i= Σ (bx > >0 00-22-1
(5)
k=1
If ready for estimation, could directly adopt it nice non-linea function maximize p (b t) = k). Note that to realize Eq. (3), we want b and its corresponding center с are closest, i.e.
maximize p (d, c) = 0). So, we could reformulate this maximization another MLE:
Le (Fo; b) pog (d (b, c) 0), where (b, c) = 0) = =p (b, c, S h)),
(6)
which could be optimized by calculating surrogate gradients of Eq. (6) through Px:
log 0;; i= i=Et=1" (cr > 0} 2
oL.
dn de OLg
00
Ôπ
de 00
00
Unfortunately, such joint probability estimation on MVB requires O (2*) complexity, which not flexible when h is large. By adopting the idea of block code [42], the above estimation is able to perform on long hash bits by separating hash codes into series of blocks. Specifically, any bits hash codes can be split into blocks while each block consumes A/u bits. Correspondinglyy independent surrogate networks Pr.,7 1 I S u are adopted to perform the above estimation and back- sockprroaggation simultaneously. With such decomposition, the whole problem is transformed into sub-problems with reduced complexity O (2*) O 2%/").
As summarization all of things, overall optimization via gradient descent placed in Alg. 1. We first perform Eq. (2) step by using the pre-defined centers and then perform Eq. (3) step by back- propagation via Eq. (7). Two models Fe, Pr are optimized with learning rate 7/, 72 respectively.
Algorithm 1 One of implementations under supervised circumstance.
Training procedure of two models Fo, Pr. 2: Generate class-specific centers e E [Cl Class-num;
1:
procedure TRAIN(Fo, Px)
(Eq. (2)).
3: repeat
Main training loop.
4:
Sample æ from with label y/:
5:
L = Fo (a);
6:
o Pa (E);
7:
π +- π
7)
0Lg. tm
(Eq. (5)).
8:
<- 1/2 80
BLe with corresponding center e;
Eq. (3), Eq. (7).
9:
until Total epoch exceeds;
return Fe:
Optimized hash-model Fg
11:
end procedure
Experiments
We conduct extensive experiments on three benchmark datasets to confirm the effectiveness of our proposed method. To make fair comparisons, first provide experiments setup.
5

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80
Methods
Clunchy
80
60
BCE
Oure
70
Eрoch
40
mAP
16
32
80
64
16 bits
32 bits
bits
0.2
0.4
0.8
10
20
300
50
70
80
Piner
Epoch
per epoch on ImageNet 64 bits during training. These two values have positive relationship.
Figure mAP w.l. max P oster
Figure 2: Convergence curve of three variants on ImageNet 64 bits. Our Pa helps for achieving higher performance than other two.
6.1 Setup
Our experiments focus on performance comparisons on two typical hash-based tasks, fast retrieval and recognition, with and without integrating our proposed method. The general evaluation pipeline to train models in the training split, then hash all samples the base split. Then for retrieval, any hashed queries in query split are used to get rank lists of the base split from nearest to farthest according Hamming distance. As for recognition, we adopt a &NN classifier linear model to produce queries' predictions.
6.1.1 Datasets
Our experiments are conducted on three datasets, varying scale, variety and perplexity vali- date methods' performance in different scenar- ios. Both single-label and multi-label datasets are included. We follow previous works to gen- erate three splits, which are detailed below:
Table 1: Total training time with different variants,
where we substitute our surrogate model with BCE loss from [51] or Cauchy loss from [4].
Training time per epoch (s)
Method
CIFAR- 10 [20] is a single-label 1 10-class dataset.
16 bits
32 bits
64 bits
The whole dataset contains 6, 000 images for each class. Following [5], we split the dataset
BCE
38.13
38.20
38.52
Cauchy
38.21
38.30
38.42
into 500 5, 400 100 for each class ran-
domly as train, base and query splits, respec- tively.
Ours
38.95
39.78
40.84
NUS-WIDE [7] consists of 81 labels and im- ages may have
more labels. We follow previous works [5] pick the most frequent 21 labels and their associated images (195, 834) for experiments. Specifically, 193, 734 images are randomly picked to form the base split while remaining 2, 100 images adopted for queries. 10, 500 images are randomly sampled from the base split for training models.
ImageNet [11] is a large-scale dataset consists of 1, 000 classes. To conduct experiments, follow [5] to pick a subset 100 classes where all images these classes in training set base split query split respectively (128, 503 4, 988 images). We then
validation set
randomly sample 100 images per class in the base split for training.
6.1.2 Implementation Details
Our method able to integrate into common hash-models. Due the limitation of computation resources, we choose series of representative deep hashing methods for comparison, including HashNet [5], DBDH [55], DSDH [26], DCH [4], GreedHash [40] and CSQ [51]. When our method is integrated, we append a subscript (-), to the original methods' name. All methods adopted for

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experiments are implemented from a public benchmark with PyTorch [34]. For fair comparisons, we conduct experiments with the same backbone (ResNet-50 [16]) and hyper-parameters for all tested methods. We adopt Adam [35] optimizer with default configuration and learning rate of our method ?/1 = ?22 1e for training. For multi-label datasets, we simply modify Alg. I Line with the sum of multiple losses. Block number of P is set to bits/s. For example, if the length of hash code is 64, there will be sub-models Pm, Pr, trained in parallel. To mitigate randomness, we report average performance for 5 runs on all experiments. The evaluation metric adopted for retrieval is mean Average Precision (mAP@/ Rt where = 54, 000 5, 000 | 1, 000 on CIFAR-10, NUS- WIDE, ImageNet, respectively. For recognition performance, if methods have auxiliary classification branch, we directly use its predictions from it. Otherwise, we adopt a kNN classifier built on base split and vote with 100 nearest neighbours.
6.2 Ablation Study
In the ablation study, we try reveal thecorrect- ness of our proposed objectives Eqs. (2) and (3), and the effectiveness of our proposed posterior estimation model Pæ. We answer the following questions by conducting experiments on Ima- geNet to demystify the above concerns. We focus on retrieval performance in the ablation study, while we observe similar results on recog- nition.
X10
3.5
Real
3.0
Eai.
2.0
Is hash-model's performance lower-bounded by Eq. (1)? Validating correctness of Eq. (1) is important, but it is essentially hard to conduct ex- periments to confirm it. Nevertheless, we still re- veal the correlation between model performance and inter-class distinctiveness / intra-class com- min Putter dur- pactness by tracking mAP w.r.t. max intre ing training. To calculate Dinter and DD Dintra we first hash all samples from the base split and cal- culate their centers of them over all classes. The 99.9 percentile of min Dinter and max D inton is picked to avoid outliers. We conduct tracking on CSQp with 16, 32, 64 bits per epoch to draw Fig. 1. As the figure shows, lines go from lower left to upper right with small fluctuations, which indicates the linear relationship between mAP and confirm our theoretical analysis in Sec.
00
(Ed, Real) = 7.88c
Method
KLreal
Naïve
0.538
Ours
0.008
Figure 3:
Multivariate Bernoulli estimation by our proposed Pa,, and KL-divergence comparisons with naïve independent estimation.
min Dester on all bit-lengths. This may partially max Destra
Convergence speed and efficiency of Par. To verify the convergence speed and efficiency of our posterior estimation model Par test it by substituting with two variants: BCE from [51] and Cauchy loss from [4]. mAP is evaluated per epoch for three variants with three bit-lengths 16, 32, 64 and the convergence curve is plotted in Fig. 2. From the figure, we could see that ours and BCE's convergence speed are similar, while Cauchy is slightly slow. Meanwhile, our method has continuously higher performance than other two after 10 epochs for all bit-lengths, which is potentially due to our method could perform low-bias optimization. Note that when training with Cauchy loss, performance is instead lower when codes are longer. This may be caused by the inner-product based Cauchy loss could not handle long bits. We also measure averaged training time per epoch of all variants on single NVIDIA RTX 3090, which is placed in Tab. 1. In this table, our model does not consume significant longer time than others to train. The above observations reveal the efficiency of our proposed method. could be a drop-in replacement without introducing significant overhead. Moreover, convergence speed and efficiency will not drop when the bit-length becomes long, which reveals the flexibility of the blocked code design.
Could Pr estimate multivariate Bernoulli distribution? We design an extra toy experiments to validate the ability of estimating multivariate Bernoulli distribution by our surrogate model P,.
tattts::/gittaa wttpecwwww.oooowoy//eeeeeeee ppttoh
7

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Table 2: mAP comparisons three benchmark datasets for 16,32, 64 bits codes. 1() indicates
performance enhancement with our method integrated.
CIFAR-10
NUS-WIDE
ImageNet
Method
16 bits
32 bits
64 bits
16 bits
32 bits
64 bits
16 bits
32 bits
64 bits
HashNet
63.0
81.5
84.7
77.8
83.3
85.0
50.6
63.1
68.4
HashNet
79.1
f2.3
85.3
78.6
84.0
86.2
72.6
022..84... 84.3
DBDH 83.1
85.0
85.6
82.8
84.6
85.7
55.7
63.8
76.9
DBDHd 84.611.5
86.0
11.0
86.5
1o.9
83.8
85.6
86.5
10.8
82.2
26.
85.8
12.00
86. 86.1 19.2
DSDH 75.6
83.1
84.5
83.3
84.5
85.6
57.2
72.1
75.3
DSDHd 18.3
84.6
87.3 22.8
83.6 10.3
15.3
86.4
81.7
r24.587.3.
+24.
21..88.99
(12.6
DCH 83.4
84.4
85.3
80.7
81.7
80.9
85.5
86.2
86.4
DCHD 83.6
84.6
87.1
82.9
84.2
84.9 4.00
86.1
87.5
88. 88.1
GreedHash
83.3
84.3
86.9
78.6
80.3
82.0
83.1
85.9
86.4
GreedHashp
85.2
85.5
87.6
79.4
83.1
f3.7
83.8
86.6
86.8
CSQ
83.2
83.4
84.7
82.0
83.5
84.6
83.4
86.9
87.9
CSQD 88.
88.7
15.55
89.2 15.8
90.3
83.3
85.3
85.8
88.5
89.5
90.2
Specifically, 8 bits MVB is generated with randomly 256 joint probabilities. We then take 10, 000 samples from it as inputs to train Pr. The model will further estimate distribution by feeding another 100 samples and taking the mean of os (Eq. (4)) as result. The estimated distribution evaluated by Kullback-Leibler divergence [23] (KL) with real distribution, as well as bar plot, demonstrated in Fig. 3. As the figure shows, our predicted joint probabilities almost cover real probabilities and KL low. As a comparison, if we estimate real distribution with the product of edge probabilities directly (w/ o correlations between variables), KL will be significantly increased (row 1 in table). Such toy experiment reveals that our method is better for estimating MVBs than the naïve one.
6.3 Performance Enhancements when Integrating into State-of-the-Ar
To evaluate performance gain when integrating proposed method into hash-models, con- duct experiments on three datasets. Specifically, first report the original performance of tested models and then re-run with our method incor- porated make a comparison. Both 16, 32, 64 bits results are reported to show performance from short codes to long codes.
95
NachNNt
DRDH
90
DSDH
DCH
GroedHHall
85
CSQ
mAP
80
Retrieval Performance. Retrieval performance comparisons under mAP is shown in Tab. 2. With our integrated method, all tested hashing methods have performance enhancement. We observe an up to 26.5% increase and 5.02% on average, which is significant improvement. Specifically, performance increase on HashNet, DBDH and DSDH is higher than other methods (first 3 rows), especially on ImageNet dataset. potential reason is that all of these three methods use pairwise metric learning objectives and inner-product to approximate Hamming dis- tance, which may not handle optimization well when dataset size and class number is large. Meanwhile, our method works on multi-label datasets where methods also obtain perfor- mance gain, indicating that multi-labe data may also benefit from our objective. Furthermore,
75
65
1.0
min Dastes
manac D Pintra
Figure 4: mAP W.E.f. min Dinter,
max Pintey for different
methods. By integrating our auxiliary objective, we could observe significant performance and min Dutte increase in most cases (from solid dots max Danre translucent dots). Regression line is also plotted with 95% confidence interval to reveal linear rela- tionship between two metrics.
8

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(a) HashNet 64 bits, Pinise. 0.8562
(b) HashNet 64 bits, min miter.
DRDH 64 hits, min Jutor max D botro
(d) DBDHa 64 bits, max 2 Puctr
0.9334
0.9447
Pister 0.9761
Daa Pindes
max 27 Inpees
Figure 5: I-SNE visualization for HashNet and DBDH w/ and w /o our integration, where the (+), variants have more compact structures and larger margin among different classes than original ones.
also observe 3. 3.46% average mAP increase CSQ. It adopts similar objective but different optimization approach with ours. This increase shows the effectiveness of our proposed posterior estimation approach, which could perform better optimization than theirs.
Recognition Performance. Similarly, we report the recognition performance of these methods by measuring classification accuracy on Ima- geNet with 16, 32, 64 bits codes. Results are reported Tab. 3. Generally, we observe sim- ilar performance gains. When methods have our integration, their accuracy is significantly increased, up to 20.5% and 5.3% on average. Specifically, some of them utilize an auxiliary classification branch to produce predictions. i.e.. GresdyHash and DSDH. While for others, we adopt &NN classifier to classify samples by voting with the base split. We obtain increase for both approaches. This indicates that our approach also helps for hash-based recognition in different scenarios, by leveraging inter-class distinctiveness and intra-class compactness.
Table 3: Recognition performance on ImageNet.
ImageNet
Method
16 bits
32 bits
64 bits
HashNet HashNetp
69.1
74.1
82.4
83.2 1...1
08..-114.09..50
DBDH 80.9
82.9
88.2
DBDHd 86.8
89.1
88.4
DSDH 40.6
49.9
54.1
DSDH, 61.1 120.5
62.
122.
63.2 03.2.99 19.1
DCH 88.5
89.2
88.1
DCH 88.9
89.5
0.33
88.7 10.6
GreedHash GreedHash
79.1
86.8
87.0
From the above comparisons, we confirm the effectiveness of our proposed method. Over- all, our method is validated to be flexible and promising to deploy in the above scenarios with various bits.
80.2
87.1-0.3
88.4
CSQ
87.7
89.0
89.4
CSQp
87.9
89.1-0.1
91.0
6.4 Qualitative Comparisons
t-SNE visualization. To further analyze how our objective affects hash codes by adjusting Dinter and D Dintra. we conduct codes visualization by t-SNE [14] HashNet and DBDH. Specifically, randomly sample 5, 000 64 bits codes from CIFAR-10 base split. t-SNE is then performed on these codes shown in Fig. 5, where points' color indicates class. From these figures, we give our humble explanation. Firstly, please look at Fig. 5(a), codes extracted from the original HashNet are mixed among different classes. While in Fig. 5(c), clusters are loose. These representations result low min D inter and high max D intra When they are trained along with our method (Figs, 5(b) and 5(d),, we get more separable and distinct clusters compared to the original ones. Meanwhile, codes in cluster are more compact than the original methods. Quantitative results of Dunter figures also reveal this phenomenon.
max intre placed under
mAP w.r.f. min Datie visualization. To further confirm the relationship between hashing performance Dasto
and max D pitra
D petter as in Sec. 6.2, we plot mAP w.t.f max Dinss
Dener for all tested methods on 64 bits ImageNet min Dinler
base split (Fig, 4). As the figure shows, most of the methods obtain higher mAP with larger
max Panre

---

and move from lower-left region to upper-right region (solid dots to translucent dots indicate original methods to integrated methods). Meanwhile, according to the positions of these dots. mAP and min Dister are supposed to be under linear relationship for different methods, i.e. higher min Disie. max D estro
max Patees leads to higher mAP. Therefore, we give a regression line with 95% confidence interval on the plot to confirm our observations. Extended from this, our proposed lower bound could also be criterion of hash codes' performance.
7 Conclusion
In this paper, we conduct comprehensive study the characteristics of hash codes, As result, prove that hash codes' performance is lower-bounded by inter-class distinctiveness and intra-class compactness, Formulating such lower bound as an objective, we could further lift in hash-model training. Meanwhile, our proposed surrogate model for posterior estimation over hash codes' fully exploits the above guidance to perform low-bias model optimization and finally produce good codes. Extensive experiments conducted on three benchmark datasets confirm the effectiveness of our proposed method. We are able to boost current
performance with flexible integration.
Limitation and Broader Impacts
Hashing and related compact representation learning are able significantly reduce computational requirements while improving memory efficiency when deploying to real scenarios. However, the main challenge for these techniques is information loss which results in performance drop. To tackle this, our work on on hash codes' performance gives lower bound, validated in two specific tasks. However, generalizing such lower bounds to various scenarios, e.g. semi-supervised, unsupervised hash learning, are left for future study. Meanwhile, our posterior estimation approach is not verified for extremely long bits. Nevertheless, to improve not only hash learning, but also broader areas that adopt metric learning. Our study on posterior estimation may also help for precise discrete optimizations.
work may still provide the potential to inspire researchers
Acknowledgments and Disclosure of Funding
This work is supported by National Key Research and Development Program of China (No. 2018AAA0102200), the National Natural Science Foundation China (Grant No. 62122018, No. 61772116, No. 62020106008, No. 61872064).
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Checklist
1. For all authors...
(a) Do the main claims made the abstract and introduction accurately reflect the paper's contribu- tions and scope? [Yes]
(b) Did you describe the limitations your work? [Yes] Sec. 7.
(c) Did you discuss any potential negative societal impacts of your work? [No] No explicit negative
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(d) Have you read the ethics review guidelines and ensured that your paper conforms them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full of assumptions of all theoretical results? [Yes] See Sec. 4.
(b) Did you include complete proofs of all theoretical results? [Yes] See Sec. 4.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as URL)? [Yes] See supplemental material.
(h) Did you specify all the training details (e.g., data splits, hyperparamcters, how they were chosen)? [Yes] See Sec. 6.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Sec. 6.
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13


Reviews:
[{"review_id": "jZQ7bKv66K9", "reviewer": "Reviewer_N6zv", "summary": "The paper on hand addresses the problem if hashing indicating that interclass distinctiveness and intraclass compactness determine the lower bound of hand code performance. Based on these assumptions a model is proposed, showing beneficial properties for different applications.", "questions": "I would argue with benefits in terms of speed but not on \"carbon neutrality\"\n\nAs seen above from the weaknesses, there are several points that would need to be addressed in a revised version to increase the clarity and the reading flow. In particular, the technical contribution needs to discussed in more clear way.\n\nOverall, there are too many flaws for accepting the paper for NeurIPS.", "limitations": "The paper on hand is a theoretical contribution, thus there are no limitations in this context.\n\n", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nIn general, using hashing could be beneficial for many applications.\n\nThe presented results indicate that the approach is beneficial for at least three different applications.\n\n\nWeaknesses:\n\nThere is no clear description of related work. Even though there is a large number of of references, these are quite general and not directly related to the problem on hand.\n\nTo fully understand the problem, the preliminary section need to be extended. Neither the overall problem, nor the required technical details are given on an adequate level. \n\nIn general, the reading flow is hampered by inadequate structure and mathematical writing. The (wrong) overuse of definitions, propositions, and corollary prevents a fluently reading the paper. \n\nIn addition, the mathematical writing needs to be improved. This includes the correct embedding of of equations into the text, missing or insufficiently defined mathematical terms, and a proper argumentation flow.\n\nIn general, the paper would benefit from a careful proofreading. There are countless typographical and grammatical errors, hampering the reading flow.\n\nThe structure of the paper needs to improved. In particular, the introduction, the conclusion, and the discussion need to be structured differently. The argumentation is not straight forward and somehow redundant.\n\nThe caption of Fig. 4 needs to be extended. Even though described in the text, from the figure it is not clear what is shown.\n\nThe meaning of Fig. 5 is not fully clear. In this case a more thorough discussion in the text is necessarily required.The ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "IdTpMPK4lA_", "reviewer": "Reviewer_HCrQ", "summary": "This paper first proves that inter-class distinctiveness and intra-class compactness among hash codes determine the lower bound of hash codes’ performance. And it shows that promoting these two characteristics could lift the bound and improve hash learning. Then it proposes a surrogate model to fully exploit such objective by estimating posterior of hash codes. Extensive experiments reveal effectiveness of the proposed method.", "questions": "Is the performance improvement still significant when we increase the size of training set? ", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The studied problem is interesting and important because a theoretical analysis on criteria of learning good hash codes remains largely unexplored.\n2. The proposed method seems to be reasonable and effective.\n3. Experiment seems to be extensive. \n\nWeaknesses:\n1. There exist some typos and grammatical errors in the paper. \n2. The training sets on all datasets are relatively small.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "8PwhIJl2sn", "reviewer": "Reviewer_jH1U", "summary": "This paper studies the learn-to-hash problem, where we need to transform images into hash codes for fast retrieval.  The major research highlight of this paper is the lower bound for hash codes' performance. This paper uses inter-class distinctiveness and intra-class compactness to present a lower bound for the average precision, an evaluation metric for learn-to-hash. Moreover, this paper uses this lower bound as an objective for learn-to-hash. As a result, this paper presents a significant increase in average precision and accuracy.", "questions": "1. This paper introduces a lower bound for AP using inter-class distinctiveness and intra-class compactness. Is this lower bound tight? On which situation the lower bound is equal to the AP? Is there any intuition on the edge cases that the lower bound is far way from the AP?\n\n2. This paper introduces the strength of the proposed lower bound as an objective function. Is there any more potential use case for this lower bound other than the objective? For instance, can we use this lower bound in the parameter search of hash code length?\n\n3. This paper presents improvements in both mAP and accuracy using the proposed lower bound as an objective. Since the lower bound is designed for AP. Is there a connection between it and the accuracy so that we observe this improvement. In fact, this leads to a more general question: how does this lower bound connect to other metrics such as recall, and precision? \n", "limitations": "In general, this paper has a nice presentation and organization. Both theoretical analysis and experimental evaluation are presented for better illustration. However, it would be better to provide a deeper analysis of the functionality of the lower bound. I would like to see the answers to the raised questions in the previous section.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths;\n\n1. The proposed lower bound is useful. It connects the evaluation metric with the current state of the learn-to-hash model. As a result, we can use it as a guide in the model training.\n\n2. The experimental evaluation is extensive. The authors present a comparison with solid baselines. Moreover, the authors study the training efficiency of the proposed objective with Cauchy and BCE.\n\n3.  The paper provides a theoretical analysis of the proposed lower-bound and its gradient when we use it as a loss. In the supplementary material, this paper also includes necessary implementation details.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept