Original: A Lower Bound of Hash Codes’ Performance
Revised: A Lower Bound of Hash Codes’ Performance

Original: Abstract
Revised: Abstract

Original: As a crucial approach for compact representation learning, hashing has achievedgreat success in effectiveness and efficiency.
Revised: As a crucial approach for compact representation learning, hashing has achievedgreat success in effectiveness and efficiency.

Original: Numerous heuristic Hamming spacemetric learning objectives are designed to obtain high quality hash codes.
Revised: Numerous heuristic Hamming spacemetric learning objectives are designed to obtain high quality hash codes.

Original: Nevertheless, a theoretical analysis on criteria of learning good hash codes remains largelyunexploited.
Revised: Nevertheless, a theoretical analysis on criteria of learning good hash codes remains largelyunexploited.

Original: In this paper, we prove that inter-class distinctiveness and intra-classcompactness among hash codes determine the lower bound of hash codes’ performance.
Revised: In this paper, we prove that inter-class distinctiveness and intra-classcompactness among hash codes determine the lower bound of hash codes’ performance.

Original: Promoting these two characteristics could lift the bound and improvehash learning.
Revised: Promoting these two characteristics could lift the bound and improvehash learning.

Original: We propose a surrogate model to fully exploit such objective byestimating posterior of hash codes and further control it, which results in low-biasoptimization.
Revised: We propose a surrogate model to fully exploit such objective byestimating posterior of hash codes and further control it, which results in low-biasoptimization.

Original: Extensive experiments reveal effectiveness of the proposed method.
Revised: Extensive experiments reveal effectiveness of the proposed method.

Original: By testing on a series of hashing methods, we obtain performance improvementsamong all of them, with an up to 26 .
Revised: By testing on a series of hashing methods, we obtain performance improvementsamong all of them, with an up to 26 .

Original: 5% increase in mean Average Precision andan up to 20 .
Revised: 5% increase in mean Average Precision andan up to 20 .

Original: 5% increase in accuracy.
Revised: 5% increase in accuracy.

Original:  Introduction
Revised:  Introduction

Original: Explosive increase of multimedia digital contents requires people to develop large-scale processingtechniques for effective and efficient multimedia understanding [36, 49, 52, 11, 32, 29].
Revised: Explosive increase of multimedia digital contents requires people to develop large-scale processingtechniques for effective and efficient multimedia understanding [30, 44, 47, 8, 26, 22].

Original: Meanwhile,growing focus on carbon neutrality initiatives urge researchers to handle above issues with lowpower and memory consumption.
Revised: Meanwhile,growing concerns on carbon neutrality initiatives urge researchers to handle above issues with lowpower and memory consumption.

Original: Fortunately, hashing, as a key role in compact representationlearning, has been demonstrating its efficiency and effectiveness over past few decades towards abovedemands [18, 22, 43, 15, 27, 25, 21].
Revised: Fortunately, hashing, as a key role in compact representationlearning, has been demonstrating its efficiency and effectiveness over past few decades towards abovedemands [12, 16, 38, 10, 21, 19, 15].

Original: By digesting multimedia contents into short binary codes,hashing is able to significantly reduce storage, boost speed, and preserve key information.
Revised: By digesting multimedia contents into short binary codes,hashing is able to significantly reduce storage, boost speed, and preserve key information.

Original: Recent years,compared to hand-crafted hashing functions, learning to hash shows its advantagesto alleviate information loss by performing data-dependent Hamming space projection with a nonlinear model [33, 48, 46, 47, 27, 25, 21, 24].
Revised: Recent years, learning to hash shows its advantages to alleviate information loss by performingdata-dependent projection [27, 43, 41, 42, 21, 19, 15, 18].

Original: The purpose of optimizing hash models is to generatesemantically meaningful hash codes for downstream tasks.
Revised: People design Hamming space metriclearning objectives to optimize hash models, with the purpose of generating meaningful hash codesfor downstream tasks.

Original: A practical convention is to increasesimilarities among codes of same category, otherwise the opposite.
Revised: Specifically, a practical convention is to increase similarities among codesof same category, otherwise the opposite.

Original: Such convention is validated tobe effective in many applications [8, 42, 26, 41, 50, 34].
Revised: Such convention is validated to be effective in manyapplications [7, 37, 20, 36, 45, 28], e.g ., fast retrieval and hash-based recognition.

Original: Practices in current advances still leave two questions which remain largely unexploited:
Revised: Practices in current advances still leave two questions which remains largely unexploited:

Original: Howcould we determine a hash modelcould produce good codes, and by what means to achieve thisgoal?
Revised: Howcould we determine a hashing model produce good codes, and by what means to achieve abovegoal?

Original: Taking comprehensive study on above questions is essential, since it not only guides us todevelop high performance algorithms in typical hash-based applications, but also provides potentialpossibilities to inspire future works in hash learning.
Revised: Taking comprehensive study on above questions is essential, since it not only guides us todevelop high performance algorithms in typical hash-based applications, but also provides potentialpossibilities to inspire future works in hash learning.

Original: Although some works [48, 1] try to answer above questions by building connections between hashcodes and common descriptors with domain knowledge, we still seeks for theoretical guarantees forhash codes’ performance.
Revised: Although some works [43, 1] try to answer above questions by building connections between hashcodes and common descriptors with domain knowledge, we still seeks for theoretical guarantees forhash codes’ performance.

Original: Similar situations occur in related works [47, 46, 31], where objectives tooptimize hashing models are designed by intuitions or heuristics.
Revised: Similar situations occur in related works [42, 41, 25], where objectives tooptimize hashing models are designed by intuitions or heuristics.

Original: By formulating correlations among hash codes as inter-class distinctiveness and intra-class compactness, we try to address above issues and provide a lower bound of hash codes’ performance.
Revised: By formulating correlations among hash codes as inter-class distinctiveness and intra-class compactness, we try to address above issues and provide a lower bound of hash codes’ performance.

Original: Based on this, we could further derive from them as objectives to lift lower bound.
Revised: Based on this, we could further adopt them as objectives to lift lower bound.

Original: Moreover, toeffectively train model towards such targets, estimating and further controlling posterior of hash codesis necessary, since bit-level dependence exists in posterior distribution.
Revised: However, estimationon posterior of codes is necessary, since bit-level dependence exists in posterior distribution.

Original: If we neglect this, bias will beintroduced during hashing model optimization and therefore hinders performance.
Revised: Ifwe neglect this, bias will be introduced during hashing model optimization and therefore hindersperformance.

Original: Summarizing bothof them, our contribution is presented below:
Revised: Summarizing both of them, our contribution is presented below:

Original: 1) To the best of our knowledge, we are the first to formulate lower bound of hash codes’ performance,which is directly proportional to inter-class distinctiveness and intra-class compactness among codes.
Revised: 1) To the best of our knowledge, we are the first to formulate lower bound of hash codes’ performance,which is directly proportional to inter-class distinctiveness and intra-class compactness among codes.

Original: Base on this, an objective is further proposed to lift this lower bound and thus enhance performance.
Revised: Base on this, an objective is further proposed to lift this lower bound and thus enhance performance.2)

Original: 2) To perform low-bias optimization towards above objective, a novel posterior estimation over hashcodes is proposed.
Revised: To perform low-bias optimization towards above objective a novel posterior estimation over hash codes is proposed.

Original: As a non-linear function, it could be utilized for optimizing hashing modelsviagradient descent.
Revised: As a non-linear function, it could be utilized for optimizing hashing models

Revised: to

Revised: produce good codes.

Original: Such plug-and-play component is able to integrate into current hashing models toboost performance.3)
Revised: Such plug-and-play component is able to integrate into current hashing modelsto boost performance.3)

Original: Extensive experiments on three benchmark datasets reveal effectiveness of proposed method in  two typical tasks.
Revised: Extensive experiments on three benchmark datasets reveal effectiveness of proposed method intwo typical tasks.

Original: Specifically we obtain an up to 26 .
Revised: Specifically, we obtain an up to 26 .

Original:  increase on mean Average Precision for fast  retrieval.
Revised: 5% increase on mean Average Precision for fastretrieval.

Original: We also observe an up to 20 .
Revised: We also observe an up to 20 .

Original: 5% increase on accuracy for hash-based recognition.
Revised: 5% increase on accuracy for hash-based recognition.

Original: Rest of paper is organized as follows: We first givea short review on current hashing methodsand preliminaries (Secs. 2 and 3).
Revised: Rest of paper is organized as follows: We first give preliminaries on hash models and applications(Sec. 2).

Original: Then, lower bound on hash codes’ performance by inter-classdistinctiveness and intra-class compactness is proposed (Sec. 4).
Revised: Then, lower bound on hash codes’ performance by inter-class distinctiveness and intra-classcompactness is proposed (Sec. 3).

Original: To fully exploit such guidance,posterior estimation over multivariate Bernoulli distributionis designed for code control (Sec. 5).
Revised: To fully exploit such guidance, we propose posterior estimationover multivariate Bernoulli distribution for code control (Sec. 4).

Original: Experiments (Sec.  6) arethen conducted for evaluation.
Revised: Discussions (Sec. 5) and experiments(Sec. 6) are conducted in next few sections.

Original:  Related Works
Revised:  Preliminaries

Original: This paper will focus on data-dependent learning-to-hash methods.
Revised: Learning to hash performs data-dependent compact representation learning.

Original: Recent advances on buildingthese models are validated to be effective on hash-based recognition and fast retrieval [18, 1, 26, 41,17].

Original: The typical approach utilizes a projection function to convert raw inputs into compact binarycodes and conduct classification or approximate nearest neighbor search on codes for downstreamtasks [8, 42, 50]. As aforementioned in introduction, to obtain such projector, current practicessuggest us to formulate it as a Hamming space metric learning problem under binary constraints. Pointwise [51, 21, 30], pairwise [26, 55, 4, 25, 5], triplet [28, 10], listwise [54, 45] and prototypebased [6, 53, 17, 2] objectives are all studied for model training, where the core idea is to adjustsimilarities among samples from same / different categories. Besides above objectives, we should also notice that hashing model optimization is NP-hard [1, 48,30, 41].
Revised: Generally, inputsx ∈ X ⊆

Original: This is caused by the discrete, binarized hash codes formulation.

Original: To tackle this, previousworks propose a lot of methods that target on following purposes: a) Surrogate functions thatapproximate hashing by continuous functions, such as tanh [5], Stochastic neuron [9], affine [39] or

Original: Straight-Through [3]. By adopting these, the discrete hashing model optimization is transformed intoapproximated but easier one via gradient descent. b) Non-differentiable optimization algorithm design,which focus on solutions that directly handle discrete inputs, including Coordinate descent [38], Discrete gradient estimation [13], etc .
Revised: R d are firstly transformed into h -dim real-valued vector ℓ and then binarized to h bitsbinary code b ∈ B ⊆

Original: There are also bag of tricks which help for stable training. Forexample, minimizing quantization error reduces gap between raw outputs of model and final hashcodes [26], and code-balancing alleviates the problem of producing trivial codes [12]. Please refer to literature reviews on above works [47, 46, 31] for comprehensive studies on learningto hash. We could see two issues exist and remain largely unexploited in above works.
Revised: H h :x

Original: The firstis objective design, which is commonly proposed by intuitive and with few theoretical guarantees. The second is model optimization, where bit-level dependence is important and involved in codecontrol but most of works neglect it. To tackle these, we would firstly give a lower bound of hashcodes’ performance and utilize it as a guidance for hashing model training. Then, a novel modeloptimization approach is proposed in a multivariate perspective for low-bias code control.
Revised: [Equation]

Original: Learning to hash performs data-dependent compact representation learning. Inputs x ∈ X ⊆ R dare firstly transformed into h -dim real-valued vector ℓ by F θ , a projection parameterized by θ :
Revised: bin −−→ b , where bin is the {− 1 , +1 } binary function, and F θ is a projection parameterized by θ to map d -diminput real space to h

Original:  R d F θ −−→ R h .
Revised: -dim: R d F θ −−→ R h .

Original: And ℓ is then binarized to h bits binary code b ∈ B ⊆ H h :x F θ −−→ ℓ bin −−→ b , where bin ( · ) = (cid:26) +1 , ( · ) ≥ 0 , − 1 , ( · ) < 0 .
Revised: The posterior distribution of B given inputs X is formulatedunder multivariate Bernoulli distribution: p ( B | X ) ∼ MVB .

Original: Typical usage of hash codes.
Revised: Hash codes’ performance.

Original: Common hashing-based tasks include fast retrieval [19, 40, 27, 2] andrecognition [26, 38, 41, 33]. For fast retrieval, given a  queryunder same distribution of X , we convertit into query hash code q and conduct fast approximate nearest neighbor search in B to produce ranklist R . In R  , samples are organized from nearest to furthest to query.
Revised: Generally, Hamming space metric learning matters in hash code learning,and neighborhood structure between codes needs to be regarded essentially [27, 28, 24]. We couldorganize above as a ranking problem: Given an arbitrary query code q , rank list R with total lengthr is obtained by sorting distances d ( q , b ) among all codes of X in ascending order.

Original: Correspondingly among allresults in R , true positives tp i ∈ TP indicate samples that match with query while others are falsepositives fp i ∈ FP ( i indicates rank).
Revised: For eachresult in rank list, true positives tp i ∈ TP is a sample matched with query while others are falsepositives fp i ∈ FP ( i indicates rank).

Original: As a criterion, Average Precision (AP) is commonly adoptedto determine how well the retrieved results are. It encourages true positives to have higher ranks thanfalse positives: AP = 1 | TP | (cid:80) ∀ tp i,m ∈ TP P @ i , where P @
Revised: As a criterion, Average Precision (AP) could be adoptedto determine hash codes’ performance, which encourages true positives to have higher ranks thanfalse positives.

Original: i is the rank- i precision. As for hash-basedrecognition, since current works formulate it as a variant of retrieval, we could still adopt AP as acriterion to indicate performance. Detailed explanation is placed in Supp. Sec. C.
Revised: Such criterion could cover common hash tasks, e.g ., fast retrieval [13, 35, 21, 2],recognition [20, 32, 36, 27], which is explained in supplementary materials.

Original:  Lower Bound by Inter-Class Distinctiveness and Intra-Class Compactness
Revised:  Lower Bound by Inter-Class Distinctiveness and Intra-Class Compactness

Original: We start at an arbitrary rank list for studying characteristics of codes in above scenarios.
Revised: To study characteristics of codes under above scenarios, we start at an arbitrary rank list.

Original: To calculate
Revised: Forsimplicity, query and all samples are assumed to be under same distribution, and size of TP and FPis fixed. As a good model, we expect true positives are placed near queries in order to be on top ofrank list.

Original: AP easily, we first introduce mis-rank that indicates how many false positives take higher places thantrue positives.We refer readers to Supp. Secs. A and B for detailed description and proof.
Revised: To easily calculate AP, mis-rank is defined to indicate how many false positives take higherplaces than true positives.

Original: Definition.
Revised: Definition.

Original: Mis-rank m indicates how many false positives have higher ranks than a true positive i.e .,tp i,m is at rank i , meanwhile m = | (cid:8) d ( q , fp ) <
Revised: Mis-rank m indicates how many false positives have higher ranks than a true positive i.e .,tp i,m is at rank i , meanwhile m = | (cid:8) d ( q , fp ) <

Original: d (cid:0) q , tp i,m (cid:1)(cid:9) | , whered ( q , · ) is distance between qand a sample, and | · | is number of elements in set.
Revised: d (cid:0) q , tp i,m (cid:1)(cid:9) | , where | · | is number of elements. Remark. For any true positive tp i,m , the rank- i precision P @ i is derived to be ( i − m ) / i .

Original: Then, average precision could be derived as:
Revised: Then, average precision could be derived as:

Original: [Equation]
Revised: [Equation]

Original:  for all true positives in rank list.
Revised: i − m i , for all true positives in rank list.

Original: This is because the rank- i precision P @ i for any true positive tp i,mequals to ( i − m ) / i .
Revised: Corollary.

Original: From the derivation, we could immediately obtain that AP increases iff m decreases.
Revised: Average precision increases iff m decreases.

Original: Therefore,now we could focus on how to reduce m to thereby increase average precision.
Revised: Now we could focus on how to reduce m to thereby increase average precision.

Original: Noticed that value ofm is highly related to two distances, d ( q , fp ) and d ( q , tp ) , following proposition is raised:
Revised: Noticed that value ofm is highly related to two distances, d ( q , fp ) and d ( q , tp ) , following proposition is raised:

Original: Proposition.
Revised: Proposition.

Original: [Equation]
Revised: [Equation]

Original: where · denotes upper bound.
Revised: where · denotes upper bound.

Original: Correspondingly,
Revised: Correspondingly,

Original: [Equation]
Revised: [Equation]

Original: where · denotes lower bound.
Revised: where · denotes lower bound.

Original: If inputs x j have associated labels y j , above studies are able to befurther generalized.
Revised: If inputs x j have associated labels y j , above studies are able to be generalized.

Original: Class-wisecenters, inter-class distinctiveness and intra-class compactness are introduced to formulate our finallower-bound of hash codes’ performance.
Revised: We firstly defineclass-wise centers to help for introducing inter-class distinctiveness and intra-class compactness.

Original: Definition.
Revised: Definition.

Original: Center c c ∈ C is a representative hash code of a specific class c :
Revised: Center c c ∈ C is a representative hash code of a specific class c :

Original: [Equation]
Revised: [Equation]

Original: where b j is the j th sample in setB and y j is the label of b j .
Revised: where b j is the j th sample in set and y j is the label of b j , and | C | is number of total classes.

Original: Definition.
Revised: Definition.

Original: Inter-class distinctiveness is min D inter where D inter is a set measures distances betweenall centers over C : (cid:8) d (cid:0) c
Revised: Inter-class distinctiveness is min D inter where D inter is a set measures distances betweenall centers over C : (cid:8) d (cid:0) c

Original: j , c k (cid:1) | ∀ c j , c k ∈ C (cid:9) .
Revised: j , c k (cid:1) | ∀ c j , c k ∈ C (cid:9) .

Original: In contrast, intra-class compactness is 1 / max D
Revised: In contrast, intra-class compactness is 1 / max D

Original: intrawhere D intra = (cid:8) d (cid:0) b
Revised: intrawhere D intra = (cid:8) d (cid:0) b

Original: j , c c (cid:1) | ∀ y j
Revised: j , c c (cid:1) | ∀ y j

Original: = c (cid:9) measures distances among samples of same class to theircorresponding center.
Revised: = c (cid:9) measures distance among samples of same class to theircorresponding center.

Original: Combining above propositions and definitions, we could reach:
Revised: Combining above propositions and definitions, we could reach:

Original: Proposition.
Revised: Proposition.

Original: [Equation]
Revised: [Equation]

Original: This proposition reveals how AP is affected byabove twointer- and intra- factors.
Revised: This proposition reveals how AP is affected by two factors.

Original: Actually, it couldcover common scenarios as a criterion of hash codes’ performance, e.g ., fast retrieval and recognition,which will be explained in supplementary materials.
Revised: Actually, it could cover common scenariosas a criterion of hash codes’ performance, e.g ., fast retrieval and recognition, which will be explainedin supplementary materials.

Original: Intuitively, lifting this lower bound wouldenhance performance, formulating thefollowing objective:
Revised: Lifting this lower bound would further enhance performance, whichformulates the objective:

Original: [Equation]
Revised: [Equation]

Original: Inspired by [53], we give a specific solution to implement Eqs.
Revised: Inspired by [48], we give a specific solution to implement Eqs.

Original: (2) and (3) under supervised hashingby firstly defining distance maximized class-specific centers and then shrinking hash codes to theircorresponding centers.
Revised: (2) and (3) under supervised hashing by firstly defining distance maximized class-specific centers and then shrinking hash codes to their corresponding centers.

Original: We will validate effectiveness with such solution in experiments.
Revised: We will validate effectiveness with such solution in experiments.

Original: It is worthnoting that achieving above goal is an open topic, including unsupervised or other complex scenarios,which we leave for future study.
Revised: It is worth noting that achieving above goal is an open topic, including unsupervised or other complex scenarios, which we leave for future study.

Original: Next, to optimize hashing models with above supervision, weintroduce posterior estimation over hash codes andthen control it.
Revised: Next, to optimize hashing models with above supervision, weintroduce posterior estimation over MVB to model and control codes’ distribution.

Original:  Posterior Estimation for Code Control
Revised:  Posterior Estimation for Code Control

Original: It is essentially hard to minimize distance between a hash code and a target to achieve Eq.
Revised: It is essentially hard to minimize distance between a hash code and a target to achieve Eq.

Original: (3) due totwo reasons.
Revised: (3) dueto two reasons.

Original: Firstly,  posterior distribution of B given inputs X is formulated to be under multivariate
Revised: Firstly, as mentioned before, posterior distribution p ( B | X ) are under multivariate

Original: Bernoulli distribution: p ( B | X ) ∼ MVB . If we do not take correlations among different variablesinto account, optimization on B is biased.
Revised: Bernoulli distribution, resulting in biases if we do not take correlations among different variablesinto account.

Original: Secondly, dealing with hash codes is a {− 1 , +1 } discreteoptimization, which is not trivial to handle.
Revised: Secondly, dealing with hash codes is a {− 1 , +1 } discrete optimization, which is nottrivial to handle.

Original: Therefore, we try to organize Eq.
Revised: Therefore, we try to organize Eq.

Original: (3) as a Maximum Likelihood Estimation (MLE) to tackle above issues.
Revised: (3) as a Maximum Likelihood Estimation (MLE)to tackle above issues.

Original: Considering definition of Hamming distancebetween an arbitrary hash code b and its target t :
Revised: Considering definition of Hamming distance between an arbitrary hash code band its target t :

Original: [Equation]
Revised: [Equation]

Original: < 0 } where 1 is the characteristic function and ( · ) i indicates value on the i th dimension.
Revised: where 1 is the characteristic function and ( · ) i indicates value on the i th dimension.

Original: Since Hammingdistance measures how many bits are different between two codes, probability of d ( b , t )
Revised: Since Hammingdistance measures how many bits are different between two codes, probability of d ( b , t )

Original: = δ can beformulated as:
Revised: = δ can beformulated as:

Original: [Equation]
Revised: [Equation]

Original: Therefore, b and t have distance δ iff p ( d ( b , t ) = δ ) is maximized.
Revised: Therefore, b and t have distance δ iff p ( d ( b , t ) = δ ) is maximized.

Original: However, to precisely calculateit is difficult, since it involves joint probability of b .
Revised: However, to precisely calculateit is not trivial, since it involves joint probability of b .

Original: Therefore, we try to estimate all joint probabilitiesby adopting a surrogate model P π parameterized by π to perform estimation:
Revised: Therefore, we try to estimate all jointprobabilities by adopting a surrogate model P π parameterized by π to produce estimation:

Original: [Equation]
Revised: [Equation]

Original: where  o is probabilities of a Categorical distribution p ( O | X ) (cid:98) = p ( B | X ) , which contains 2 hentries.
Revised: where [Equation] where o is probabilities of a Categorical distribution p ( O | X ) to estimate p ( B | X ) , which contains2 h entries.

Original: Each entry of o estimates one of a specific joint probability by feeding ℓ into P π :
Revised: Each entry of o estimates one of a specific joint probability by feeding ℓ into P π :

Original: [Equation]
Revised: [Equation]

Original: where ι is the h bits binary representation of i .
Revised: where ι is the h bits binary representation of i .

Original: For example, o 3 = p ( b = ( -1-1+1+1 )) when h = 4 .
Revised: For example, o 3 = p ( b = ( -1-1+1+1 )) when h = 4 .

Original: To train P π , we perform Maximum Likelihood Estimation (MLE) over p ( O | X ) :
Revised: To train P π , we perform Maximum Likelihood Estimation (MLE) over p ( O | X ) :

Original: [Equation]
Revised: [Equation]

Original: where i =
Revised: where i =

Original: [Equation]
Revised: [Equation]

Original: If P π is ready for estimation, we could directly adopt it as a nice non-linear function to maximizep ( d ( b , t ) = k ) .
Revised: If P π is ready for estimation, we could directly adopt it as a nice non-linear function to maximizep ( d ( b , t ) = k ) .

Original: Note that to realize Eq.
Revised: Note that to realize Eq.

Original: (3), we want b and its corresponding center c are closest i.e .maximize p ( d ( b , c ) = 0) .
Revised: (3), we want b and its corresponding center c are closest i.e .maximize p ( d ( b , c ) = 0) .

Original: So, we could reformulate this maximization as another MLE:
Revised: So, we could reformulate this maximization as another MLE:

Original: [Equation]
Revised: [Equation]

Original: where p ( d ( b , c ) = 0) =
Revised: where p ( d ( b , c ) = 0) =

Original: p ( b i = c
Revised: p ( b i = c

Original: i , 1 ≤ i ≤ h ) ,
Revised: i , 1 ≤ i ≤ h ) ,

Original: [Equation]
Revised: [Equation]

Original: 6) which could be optimized by calculating surrogate gradients of Eq.
Revised: 6) which could be optimized by calculating surrogate gradients of Eq.

Original: (6) through P π :
Revised: (6) through P π :

Original: [Equation]
Revised: [Equation]

Original: Unfortunately, such joint probability estimation on MVB requires O (cid:0) 2 h (cid:1) complexity.
Revised: However, such joint probability estimation on MVB requires O (cid:0) 2 h (cid:1) complexity.

Original: This is notflexible when h is large.
Revised: This is not flexiblewhen h is large.

Original: By adopting the idea of block code [44], above estimation is able to performon long hash bits by separating hash codes into a series of blocks.
Revised: By adopting the idea of block code [39], above estimation is able to perform onlong hash bits by separating hash codes into a series of blocks.

Original: Specifically, any h bits hashcodes can be split into u blocks while each block consumes h / u bits.
Revised: Specifically, any h bits hash codescan be split into m blocks while each block consumes h / m bits.

Original: Correspondingly, u independentsurrogate networks P π i , 1 ≤ i ≤ u are adopted to perform above estimation and back-propagationsimultaneously.
Revised: Correspondingly, m independentsurrogate networks P π i , 1 ≤ i ≤ m are adopted to perform above estimation and back-propagationsimultaneously.

Original: With such decomposition, the whole problem is transformed into u sub-problemswith a reduced complexity O (cid:0) 2 h (cid:1) → O (cid:0) u · 2 h / u (cid:1) .
Revised: With such decomposition, the whole problem is transformed into m sub-problemswith a reduced complexity O (cid:0) 2 h (cid:1) → O (cid:0) m · 2 h / m (cid:1) .

Original: Summarizing all of things, overall optimization via gradient descent is placed in Alg.
Revised: Summarizing all of things, overall optimization via gradient descent is placed in Alg.

Original: We firstperform Eq.
Revised: We firstperform Eq.

Original: (2) step by using pre-defined centers and then perform Eq.
Revised: (2) step by using pre-defined centers and then perform Eq.

Original: (3) step by back-propagationvia Eq.
Revised: (3) step by back-propagationvia Eq.

Original: Two models F θ , P π are optimized with learning rate η 1 , η 2 respectively.
Revised: Two models F θ , P π are optimized with learning rate η 1 , η 2 respectively.

Original: Algorithm 1 One of

Original: 1: procedure T RAIN ( F θ , P π ) ▷ Training procedure of two models F θ , P π .
Revised: 1: procedure T RAIN ( F θ , P π ) ▷ Training procedure of two models F θ , P π .

Original: Generate class-specific centers c ∈ C , | C | =
Revised: Generate class-specific centers c ∈ C , | C | =

Original: Class-num; ▷ (Eq.
Revised: Class-num; ▷ (Eq.

Original: 3: repeat ▷ Main training loop.
Revised: 3: repeat ▷ Main training loop.

Original: Sample x from X with label y ; 5: ℓ = F θ ( x ) ; 6: o = P π ( ℓ ) ; 7: π ← π − η 1 ∂L π ∂ π ; ▷ (Eq.
Revised: Sample x from X with label y ; 5: ℓ = F θ ( x ) ; 6: o = P π ( ℓ ) ; 7: π ← π − η 1 ∂L π ∂ π ; ▷ (Eq.

Original: 8: θ ← θ − η 2 ˆ ∂L θ ∂ θ with corresponding center c ; ▷ Eq.
Revised: 8: θ ← θ − η 2 ˆ ∂L θ ∂ θ with corresponding center c ; ▷ Eq.

Original: (3), Eq.
Revised: (3), Eq.

Original: 9: until Total epoch exceeds; 10: return F θ ; ▷ Optimized hashing model F θ 11: end procedure
Revised: 9: until Total epoch exceeds; 10: return F θ ; ▷ Optimized hashing model F θ 11: end procedure

Original:  Experiments
Revised:  Discussions

Revised: In this paper, we provide analysis to guarantee the lower-bound of hashing performance.

Revised: By maximizing inter-class distinctiveness and intra-class compactness, such lower-bound is lifted.

Revised: Relatedworks include CSQ [48], which uses centroids and binary cross entropy to achieve above goal, and[36] constrains inter-class relation by auxiliary classification branch.

Revised: Similar ideas present in a seriesof works [34, 15, 40, 33], but comprehensive study on effectiveness of them still remains largelyunexploited.

Revised: Our work unifies these ideas by Eq.

Revised: (1), and try to bridge connection between aboveintuitions and theory behind it.

Revised: To perform discrete optimization, [48] uses binary cross entropy:

Revised: [Equation]

Revised: where σ ( · ) is sigmoid function.

Revised: Adopting such objective results in processing hash codes amongdifferent dimensions separately, which reveals their assumption that hash codes are under singlevariate Bernoulli distribution.

Revised: [3] and similar works [23, 4, 46, 5] rather take approximation of

Revised: Hamming distance by cosine similarity:

Revised: [Equation]

Revised: or inner product with tanh ( ℓ ) .

Revised: But such practices are validated to be not flexible when code lengthis large [24].

Revised: Both of two approaches will be evaluated in experiments.

Revised:  Experiments

Original: We conduct extensive experiments on three benchmark datasets to confirm effectiveness of ourproposed method.
Revised: To confirm effectiveness of our proposed method, we conduct extensive experiments on threebenchmark datasets.

Original: To make fair comparisons, we first provide experiments setup.
Revised: To make fair comparisons, we first provide experiments setup.

Original: 6.1 Setup
Revised: 6.1 Setup

Original: Our experiments focus on performance evaluation on two typical hash-based tasks, fast retrieval andrecognition.
Revised: Our experiments focus on performance evaluation on two typical hash-based tasks, fast retrieval andrecognition.

Original: The pipeline is to firstly train models on training split, then hash all samples in basesplit.
Revised: The pipeline is to firstly train models on training split, then hash all samples in basesplit.

Original: Then, hashed queries in query split get rank lists from base split for retrieval, while queries’predictions from classifier are used for recognition.
Revised: Then, hashed queries in query split get rank lists from base split for retrieval, while queries’predictions from classifier are used for recognition.

Original: 6.1.1 Datasets
Revised: 6.1.1 Datasets

Original: Our experiments are conducted on three datasets, varying from scale, variety and perplexity to validatemethods’ performance in different scenarios.
Revised: Our experiments are conducted on three datasets, varying from scale, variety and perplexity to validatemethods’ performance in different scenarios.

Original: Both single-label and multi-label datasets are included.
Revised: Both single-label and multi-label datasets are included.

Original: We follow previous works to generate three splits, which is detailed below: 
Revised: We follow previous works to generate three splits, which is detailed below: [Figure] [Figure] Method

Revised: [Table]

Revised: BCE 38 .38 .38 .

Original:  CIFAR-10 [20] is a single-label 10 -class dataset.
Revised: Cauchy 38 .38 .38 . Ours 38 .39 .40 . CIFAR-10 [14] is a single-label 10 -class dataset.

Original: The whole dataset contains 6 , 000 images for eachclass.
Revised: The whole dataset contains 6 , 000 images foreach class.

Original: Following [5], we form the dataset by 500 | 5 , 400 | 100 for each class randomly as train,base, query splits, respectively.[Figure] [Figure] [Table]
Revised: Following [4], we form the datasetby 500 | 5 , 400 | 100 for each class randomlyas train, base, query splits, respectively.

Original: NUS-WIDE [7] consists of 81 labels and images may have one or more labels.
Revised: NUS-WIDE [6] consists of 81 labels and images may have one or more labels.

Original: We followprevious works [5] to pick the most frequent21 labels and their associated images ( 195 , 834 )for experiments.
Revised: We followprevious works [4] to pick the most frequent21 labels and their associated images ( 195 , 834 )for experiments.

Original: Specifically, 193 , 734 imagesare randomly picked to form base split whileremaining 2 , 100 images are adopted for queries.10 , 500 images are randomly sampled from basesplit for training models.
Revised: Specifically, 193 , 734 imagesare randomly picked to form base split whileremaining 2 , 100 images are adopted for queries. 10 , 500 images are randomly sampled from basesplit for training models.

Original: ImageNet [11] is a large-scale dataset consistsof 1 , 000 classes.
Revised: ImageNet [8] is a large-scale dataset consists of 1 , 000 classes.

Original: To conduct experiments, wefollow [5] to pick a subset of 100 classes where all images of these classes in training set / validationset are as base split / query split respectively ( 128 , 503 | 4 , 983 images).
Revised: To conduct experiments, we follow [4]to pick a subset of 100 classes where all images of these classes in training set / validation set areas base split / query split respectively ( 128 , 503 | 4 , 983 images).

Original: We then randomly sample100 images per class in base split for training.
Revised: We then randomly sample 100images per class in base split for training.

Original: 6.1.2 Implementation Details
Revised: 6.1.2 Implementation Details

Original: Our method is able to integrate into common hashing models.
Revised: Our method is able to integrate into common hashing models.

Original: Due to limitation of computationresources, we choose a series of representative deep hashing methods for comparison, including
Revised: Due to limitation of computationresources, we choose a series of representative deep hashing methods for comparison, including

Original: HashNet [5], DBDH [55], DSDH [26], DCH [4], GreedHash [41] and CSQ [53] and denotemethods with our integration as ( · ) D .
Revised: HashNet [4], DBDH [49], DSDH [20], DCH [3], GreedHash [36] and CSQ [48] and denotemethods with our integration as ( · ) D .

Original: All methods adopted for experiments are from a publicbenchmark implemented with PyTorch [35].
Revised: All methods adopted for experiments are from a publicbenchmark implemented with PyTorch [29].

Original: For fair comparisons, we conduct experiments withsame backbone (ResNet-50 [16]) and hyper-parameters for all tested methods.
Revised: For fair comparisons, we conduct experiments withsame backbone (ResNet-50 [11]) and hyper-parameters for all tested methods.

Original: We adopt Adam [37]optimizer with default configuration and learning rate of our method η 1 = η 2 = 1 e − 3 for training.
Revised: We adopt Adam [31]optimizer with default configuration and learning rate of our method η 1 = η 2 = 1 e − 3 for training.

Original: For multi-label datasets, we simply modify Alg.
Revised: For multi-label datasets, we simply modify Alg.

Original: 1 Line 8 with sum of multiple losses.
Revised: 1 Line 8 with sum of multiple losses.

Original: Blocknumber u of P is set to bits / 8 .
Revised: Blocknumber m of P is set to bits / 8 .

Original: For example, if hash code length is 64 , there will be 8 sub models
Revised: For example, if hash code length is 64 , there will be 8 sub models

Original: P π 1 ∼ P π 8 trained in parallel.
Revised: P π 1 ∼ P π 8 trained in parallel.

Original: To mitigate randomness, we report average performance for 5 runson all experiments.
Revised: To mitigate randomness, we report average performance for 5 runson all experiments.

Original: Evaluation metric adopted for retrieval is mean Average Precision (mAP@ K )where K = 54 , 000 | 5 , 000 | 1 , 000 on CIFAR-10, NUS-WIDE, ImageNet, respectively.
Revised: Evaluation metric adopted for retrieval is mean Average Precision (mAP@ K )where K = 54 , 000 | 5 , 000 | 1 , 000 on CIFAR-10, NUS-WIDE, ImageNet, respectively.

Original: Forrecognition performance, if methods have auxiliary classification branch, we directly use predictionsfrom it.
Revised: Forrecognition performance, if methods have auxiliary classification branch, we directly use predictionsfrom it.

Original: Otherwise, we adopt a k NN classifier built on base split and vote with 100 nearest neighbors.
Revised: Otherwise, we adopt a k NN classifier built on base split and vote with 100 nearest neighbors.

Original: 6.2 Ablation Study
Revised: 6.2 Ablation Study

Revised: DeepHash-

Original: In ablation study, we try to reveal correctness of our proposed objectives Eqs.
Revised: In ablation study, we try to reveal correctness of our proposed objectives Eqs.

Original: (2) and (3), andeffectiveness of our proposed posterior estimation model P π .
Revised: (2) and (3), andeffectiveness of our proposed posterior estimation model P π .

Original: We answer following questions byconducting experiments on ImageNet to demystify above concerns.
Revised: We answer following questions byconducting experiments on ImageNet to demystify above concerns.

Original: We focus on retrieval performancein ablation study, while we observe similar results on recognition.
Revised: We focus on retrieval performancein ablation study, while we observe similar results on recognition.

Original: KL ( Est ., Real )
Revised: KL ( Est ., Real )

Original: = 7 . 88 e −0 .0 .1 .1 .2 .2 .3 .3 .p
Revised: = 7 . 88 e −0 .0 .1 .1 .2 .2 .3 .3 .p

Original: × 10 −
Revised: × 10 −

Original: Real Est.
Revised: Real Est.

Original: Method
Revised: Method

Original: KL real
Revised: KL real

Original: Naïve 0.
Revised: Naïve 0.

Original: Ours 0.
Revised: Ours 0.

Original: Is hash model
Revised: Is hash model

Original: ’s performance lower-boundedby Eq.
Revised: ’s performance lower-boundedby Eq.

Original: Validating correctness of Eq.
Revised: Validating correctness of Eq.

Original: (1) isimportant, but it is essentially hard to conductexperiments to confirm it.
Revised: (1) isimportant, but it is essentially hard to conductexperiments to confirm it.

Original: Nevertheless, we stillreveal correlation between model performanceand inter-class distinctiveness / intra-class compactness by tracking mAP w.r.t .
Revised: Nevertheless, we stillreveal correlation between model performanceand inter-class distinctiveness / intra-class compactness by tracking mAP w.r.t .

Original: min D inter max D intra during training.
Revised: min D inter max D intra during training.

Original: To calculate D inter and D intra ,we first hash all samples from base split andcalculate centers of them over all classes.
Revised: To calculate D inter and D intra ,we first hash all samples from base split andcalculate centers of them over all classes.

Original: The99 .
Revised: The99 .

Original: 9 percentile of min D inter and max D intrais picked to avoid outliers.
Revised: 9 percentile of min D inter and max D intrais picked to avoid outliers.

Original: We conduct trackingon CSQ D with 16 , 32 , 64 bits per epoch to draw
Revised: We conduct trackingon CSQ D with 16 , 32 , 64 bits per epoch to draw

Original: Fig.
Revised: Fig.

Original: As figure shows, lines goes from lowerleft to upper right with small fluctuations, whichindicates linear relationship between mAP andmin D inter max D intra on all bit-lengths.
Revised: As figure shows, lines goes from lowerleft to upper right with small fluctuations, whichindicates linear relationship between mAP andmin D inter max D intra on all bit-lengths.

Original: This may partiallyconfirm our theoretical analysis in Sec. 4.
Revised: This may partiallyconfirm our theoretical analysis in Sec. 3.

Original: Convergence speed and efficiency of P π .
Revised: Convergence speed and efficiency of P π .

Original: Toverify convergence speed and efficiency of ourposterior estimation model P π , we test it bysubstituting with two variants: BCE from [53] and Cauchy loss from [4].
Revised: Toverify convergence speed and efficiency of ourposterior estimation model P π , we test it bysubstituting with two variants: BCE from [48]and Cauchy loss from [3].

Original: mAP is evaluated perepoch for three variants with three bit-lengths 16 , 32 , 64 .
Revised: mAP is evaluated per epoch for three variants with three bit-lengths16 , 32 , 64 .

Original: The convergence curve is plotted in Fig.
Revised: The convergence curve is plotted in Fig.

Original: From the figure, we could see convergence speed of ours and BCE are similar, while Cauchy isslightly slow.
Revised: From the figure, we could see convergencespeed of ours and BCE are similar, while Cauchy is slightly slow.

Original: Meanwhile, our method has continuously higher performance than other two after ∼ 10epochs for all bit-lengths, which is potentially due to our method could perform low-bias optimization.
Revised: Meanwhile, our method hascontinuously higher performance than other two after ∼ 10 epochs for all bit-lengths, which ispotentially due to our method could perform low-bias optimization.

Original: Note that when training with Cauchy loss, performance is instead lower when codes are longer.
Revised: Note that when training with Cauchy loss, performance is instead lower when codes are longer.

Original: This may because inner-product based Cauchy loss could not handle long bits.
Revised: This may because inner-productbased Cauchy loss could not handle long bits.

Original: We also measureaveraged training time per epoch of all variants on a single NVIDIA RTX 3090, which is placed in Tab. 1.
Revised: We also measure averaged training time per epochof all variants on a single NVIDIA RTX 3090, which is placed in Tab. 1.

Original: In this table, our model does not consume significant longer time than others to train.
Revised: In this table, our modeldoes not consume significant longer time than others to train.

Original: Aboveobservations reveal efficiency of our proposed method.
Revised: Above observations reveal efficiency ofour proposed method.

Original: It could be a drop-in replacement withoutintroducing significant overhead.
Revised: It could be a drop-in replacement without introducing significant overhead.

Original: Moreover, convergence speed and efficiency will not drop whenbit-length becomes long, which reveals flexibility of the blocked design.
Revised: Moreover, convergence speed and efficiency will not drop when bit-length becomes long, whichreveals flexibility of the blocked design.

Original: Could P π estimate multivariate Bernoulli distribution?
Revised: Could P π estimate multivariate Bernoulli distribution?

Original: We design an extra toy experimentsto validate the ability of estimating multivariate Bernoulli distribution by our surrogate model P π .
Revised: We design an extra toy experimentsto validate the ability of estimating multivariate Bernoulli distribution by our surrogate model P π .

Original: Specifically, 8 bits MVB is generated with randomly 256 joint probabilities.
Revised: Specifically, 8 bits MVB is generated with randomly 256 joint probabilities.

Original: We then take 10 , 000samples from it as inputs to train P π .
Revised: We then take 10 , 000samples from it as inputs to train P π .

Original: The model will further estimate distribution by feeding another100 samples and taking mean of all o s (Eq.
Revised: The model will further estimate distribution by feeding another100 samples and taking mean of all o s (Eq.

Original: (4)) as result.
Revised: (4)) as result.

Original: The estimated distribution is evaluated by
Revised: The estimated distribution is evaluated by

Original: Kullback–Leibler divergence [23] ( KL ) with real distribution, as well as bar plot, demonstrated in
Revised: Kullback–Leibler divergence [17] ( KL ) with real distribution, as well as bar plot, demonstrated in

Original: Fig.
Revised: Fig.

Original: As figure shows, our predicted joint probabilities almost cover real probabilities and KL islow.
Revised: As figure shows, our predicted joint probabilities almost cover real probabilities and KL islow.

Original: As a comparison, if we estimate real distribution with product of edge probabilities directly( w / o correlations between variables), KL will be significantly increased (row 1 in table).
Revised: As a comparison, if we estimate real distribution with product of edge probabilities directly( w / o correlations between variables), KL will be significantly increased (row 1 in table).

Original: Such toyexperiment reveals our method is better to estimate MVB s than the naïve one.
Revised: Such toyexperiment reveals our method is better to estimate MVB s than the naïve one.

Original: 6.3 Performance Enhancements when Integrating into State-of-the-Art
Revised: 6.3 Performance Enhancements when Integrating into State-of-the-Art

Original: To evaluate performance gain when integrating our proposed method into hashing models, we conductexperiments on three datasets.
Revised: To evaluate performance gain when integrating our proposed method into hashing models, we conductexperiments on three datasets.

Original: Specifically, we first report original performance of tested modelsand then re-run with our method incorporated to make comparison.
Revised: Specifically, we first report original performance of tested modelsand then re-run with our method incorporated to make comparison.

Original: Both 16 , 32 , 64 bits results arereported to show performance from short codes to long codes.
Revised: Both 16 , 32 , 64 bits results arereported to show performance from short codes to long codes.

Original: Retrieval Performance.
Revised: Retrieval Performance.

Original: Retrieval performance comparisons under mAP is shown in Tab. 2.
Revised: Retrieval performance comparisons under mAP is shown in Tab. 2.

Original: Withour method integrated, all of tested hashing methods have performance enhancement.
Revised: Withour method integrated, all of tested hashing methods have performance enhancement.

Original: We observean up to 26 .
Revised: We observe [Table] an up to 26 .

Original: 5% increase and 5 .
Revised: 5% increase and 5 .

Original: 02% in average, which is a significant improvement.
Revised: 02% in average, which is a significant improvement.

Original: Specifically,performance increase on HashNet, DBDH and DSDH is higher than other methods (first 3 rows),especially on ImageNet dataset.
Revised: Specifically,performance increase on HashNet, DBDH and DSDH is higher than other methods (first 3 rows),especially on ImageNet dataset.

Original: A potential reason is all of these three methods use pairwise metriclearning objectives and inner-product to approximate Hamming distance, which may not handleoptimization well when dataset size and class number is large.
Revised: A potential reason is all of these three methods use pairwise metriclearning objectives and inner-product to approximate Hamming distance, which may not handleoptimization well when dataset size and class number is large.

Original: Meanwhile, our method works on [Table] multi-label datasets where all methods also obtain performance gain, indicating multi-label datamay also benefit from our objective.
Revised: Meanwhile, our method works onmulti-label datasets where all methods also obtain performance gain, indicating multi-label datamay also benefit from our objective.

Original: Furthermore, we also observe 3 .
Revised: Furthermore, we also observe 3 .

Original: 46% average mAP increase on
Revised: 46% average mAP increase on

Original: CSQ.
Revised: CSQ.

Original: It adopts similar objective but different optimization approach with ours.
Revised: It adopts similar objective but different optimization approach with ours.

Original: This increase showseffectiveness of our proposed posterior estimation approach, which could perform better optimizationthan theirs.
Revised: This increase showseffectiveness of our proposed posterior estimation approach, which could perform better optimizationthan theirs.

Original: [Table]
Revised: [Table]

Original: CSQ 87 .89 .89 .

Original: CSQ D 87 .

Original: [Table]

Original: [Table]

Original: [Table]

Original: Recognition Performance.
Revised: Recognition Performance.

Original: Similarly, we report recognition performance of these methodsby measuring classification accuracy on ImageNet with 16 , 32 , 64 bits codes.
Revised: Similarly, we report recognition performance of these methodsby measuring classification accuracy on ImageNet with 16 , 32 , 64 bits codes.

Original: Results arereported in Tab. 3.
Revised: Results arereported in Tab. 3.

Original: Generally, we observe similar performance gain.
Revised: Generally, we observe similar performance gain.

Original: When methods haveour integration, their accuracy is significantlyincreased, with an up to 20 .
Revised: When methods haveour integration, their accuracy is significantlyincreased, with an up to 20 .

Original: 5% and 5 .
Revised: 5% and 5 .

Original: 3% inaverage.
Revised: 3% inaverage.

Original: Specifically, some of them utilize anauxiliary classification branch to produce predictions, i.e ., GreedyHash and DSDH.
Revised: Specifically, some of them utilize anauxiliary classification branch to produce predictions, i.e ., GreedyHash and DSDH.

Original: Whilefor others, we adopt a k NN classifier to classifysamples by voting with base split.
Revised: Whilefor others, we adopt a k NN classifier to classifysamples by voting with base split.

Original: We obtainincrease for both approaches.
Revised: We obtainincrease for both approaches.

Original: This indicates ourapproach also helps for hash-based recognitionin different scenarios, by leveraging inter-classdistinctiveness and intra-class compactness.
Revised: This indicates ourapproach also helps for hash-based recognitionin different scenarios, by leveraging inter-classdistinctiveness and intra-class compactness.

Original: From above comparisons, we confirm effectiveness of our proposed method.
Revised: From above comparisons, we confirm effectiveness of our proposed method.

Original: Overall, ourmethod is validated to be flexible and promising to deploy in above scenarios with various bits.
Revised: Overall, ourmethod is validated to be flexible and promising to deploy in above scenarios with various bits.

Original: 6.4 Qualitative Comparisons
Revised: 6.4 Qualitative Comparisons

Original: t-SNE visualization.
Revised: t-SNE visualization.

Original: To further analyze how our objective affects hash codes by adjusting D interand D intra , we conduct codes visualization by t-SNE [14] on HashNet and DBDH.
Revised: To further analyze how our objective affects hash codes by adjusting D interand D intra , we conduct codes visualization by t-SNE [9] on HashNet and DBDH.

Original: Specifically,we randomly sample 5 , 000 64 bits codes from CIFAR-10 base split.
Revised: Specifically, werandomly sample 5 , 000 64 bits codes from CIFAR-10 base split.

Original: t-SNE is then performed onthese codes as shown in Fig.
Revised: t-SNE is then performed on these [Figure] codes as shown in Fig.

Original: 4, where points’ color indicates class.
Revised: 4, where points’ color indicates class.

Original: From these figures, we give ourhumble explanation.
Revised: From these figures, we give our humbleexplanation.

Original: Firstly, please look at Fig. 4(a), codes extracted from original HashNet aremixed among different classes.
Revised: Firstly, please look at Fig. 4(a), codes extracted from original HashNet are mixed amongdifferent classes.

Original: While in Fig. 4(c), clusters are loose.
Revised: While in Fig. 4(c), clusters are loose.

Original: These representations resultin low min D inter and high max D intra .
Revised: These representations result in low min D interand high max D intra .

Original: When they are trained along with our method (Figs. 4(b)and 4(d)), we get more separable and distinct clusters compared to original ones.
Revised: When they are trained along with our method (Figs. 4(b) and 4(d)), we getmore separable and distinct clusters compared to original ones.

Original: Meanwhile, codes 
Revised: Meanwhile, codes in a cluster aremore compact than original methods.

Original: [Figure]
Revised: Quantitative results of min D inter max D intra placed under figures alsoreveal this phenomenon.

Original: [Figure]
Revised: [Figure]

Original: different methods,
Revised: mAP w.r.t .

Original: i
Revised: min max D intra visualization.

Original: e . 
Revised: To furtherconfirm relationship between hashing performance and min D inter max D intra , as in Sec. 6.2, we plotmAP w.r.t . min D inter max D intra for all tested methods on64 bits ImageNet base split (Fig. 5). As figureshows, most of methods obtain higher mAP withlarger min D inter max D intra (solid dots to translucent dotsindicate original methods to integrated methods).

Original: higher min D inter max D intra leads tohigher mAP.
Revised: Meanwhile, mAP and min D inter max D intra are nearlyunder linear relationship for different methods.

Original: Therefore, we give regression linewith 95% confidence interval on plot to confirmour observations.
Revised: Therefore, we give regression line with 95%confidence interval on plot to confirm our observations.

Original: Extended from this, our proposed lower bound could also be a criterion ofhash codes’ performance.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we conduct comprehensive study on characteristics of hash codes.
Revised: In this paper, we conduct comprehensive study on characteristics of hash codes.

Original: As a result, we provehash codes’ performance is lower-bounded by inter-class distinctiveness and intra-class compactness. Formulating such lower bound as an objective, we could further lift it in hash model training.
Revised: As a result, we pointout two key factors that benefit for hash learning. Specifically, by formulating hash performancelower-bound by inter-class distinctiveness and intra-class compactness among hash codes, we areable to further lift it with derived objective.

Original: Meanwhile, our proposed surrogate model for posterior estimation over hash codes’  fully exploitsabove guidance toperform low-bias model optimization and finally produce good codes.
Revised: Meanwhile, our proposed surrogate model for posteriorestimation are able to model codes’ distribution, and fully exploit above guidance to produce goodcodes.

Original: Extensiveexperiments conducted on three benchmark datasets confirm effectiveness of our proposed method. We are able to boost current hashing models’ performance with a flexible integration.
Revised: Extensive experiments conducted on three benchmark datasets confirm effectiveness of ourproposed method to boost current hashing models’ performance with a flexible integration.

Original: Limitation and Broader Impacts
Revised: Limitation and Broader Impacts

Original: Hashing and related compact representation learning are able to significantly reduce computationalrequirements while improving memory efficiency when deploying to real scenarios.
Revised: Hashing and related compact representation learning are able to significantly reduce computationalrequirements while improving memory efficiency when deploying to real scenarios.

Original: However, amain challenge for these techniques is information loss which results in performance drop.
Revised: However, amain challenge for these techniques is information loss which results in performance drop.

Original: To tacklethis, our work studies on hash codes’ performance gives a lower-bound, which is validated in twospecific tasks.
Revised: To tacklethis, our work studies on hash codes’ performance gives a lower-bound, which is validated in twospecific tasks.

Original: However, generalizing such lower-bounds to various scenarios, e.g . semi-supervised,unsupervised hash learning, are leaved for future study.
Revised: However, generalizing such lower-bounds to various scenarios, e.g . semi-supervised,unsupervised hash learning, are leaved for future study.

Original: Meanwhile, our posterior estimation approachis not verified for extremely long bits.
Revised: Meanwhile, our posterior estimation approachis not verified for extremely long bits.

Original: Nevertheless, our work may still provide potentials of inspiringresearchers to improve not only hash learning, but also broader areas that adopt metric learning.
Revised: Nevertheless, our work may still provide potentials of inspiringresearchers to improve not only hash learning, but also broader areas that adopt metric learning.

Original: Ourstudy on posterior estimation may also help for precise discrete optimizations.
Revised: Ourstudy on posterior estimation may also help for precise discrete optimizations.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontribu-
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: tions and scope?
Revised: contributions and scope?

Original: [Yes] 
Revised: [Yes] See Sec. 5.(b)

Original: (b) Did you describe the limitations of your work?
Revised: Did you describe the limitations of your work?

Original: [Yes] See Sec. 7.(c)
Revised: [Yes] See Sec. 7.

Original:  Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No] No explicit negative impacts.
Revised: [No] No explicitnegative impacts.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] See Sec. 4.(b)
Revised: [Yes] See Sec. 3.(b)

Original: Did you include complete proofs of all theoretical results?
Revised: Did you include complete proofs of all theoretical results?

Original: [Yes] See Sec. 4.
Revised: [Yes] See Sec. 3.3.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experimental
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original:  results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [Yes] See supplemental material.
Revised: [Yes] See supplemen- tal material.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See Sec. 6.
Revised: [Yes] See Sec. 6.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] See Sec. 6.
Revised: [Yes] See Sec. 6.

Original: (d) Did you include the total amount of compute and the type of resources used (e
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: g., type of GPUs, internal cluster, or cloud provider)?
Revised: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Sec. 6.
Revised: [Yes] See Sec. 6.4.

Original: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] See Sec. 6.
Revised: [Yes] See Sec. 6.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] See Sec. 6.
Revised: [Yes] See Sec. 6.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes] Codes appear in supplementary materials.
Revised: [Yes]Codes appear in supplementary materials.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: informa- tion or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]5.

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of 
Revised: (a) Did you include the full text of instructions given to participants and screenshots, ifapplicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [Table]
Revised: [N/A]

Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Revised: spent on participant compensation?

Revised: [N/A]