Unified Optimal Transport Framework for Universal Domain Adaptation
s
Abstract
Universal Domain Adaptation (UniDA) aims to transfer knowledge from a sourcedomain to a target domain without any constraints on label sets.
Since both domainsmay hold private classes, identifying target common samples for domain alignmentis an essential issue in UniDA.
Most existing methods require manually specifiedor hand-tuned threshold values to detect common samples thus they are hard toextend to more realistic UniDA because of the diverse ratios of common classes.
Moreover, they cannot recognize different categories among target-private samplesas these private samples are treated as a whole.
In this paper, we propose to use
Optimal Transport (OT) to handle these issues under a unified framework, namely
UniOT.
First, an OT-based partial alignment with adaptive filling is designedto detect common classes without any predefined threshold values for realistic
UniDA.
It can automatically discover the intrinsic difference between commonand private classes based on the statistical information of the assignment matrixobtained from OT.
Second, we propose an OT-based target representation learningthat encourages both global discrimination and local consistency of samples toavoid the over-reliance on the source.
Notably, UniOT is the first method withthe capability to automatically discover and recognize private categories in thetarget domain for UniDA.
Accordingly, we introduce a new metric H 3 -score toevaluate the performance in terms of both accuracy of common samples andclustering performance of private ones.
Extensive experiments clearly demonstratethe advantages of UniOT over a wide range of state-of-the-art methods in UniDA.
Introduction
Deep neural networks have boosted performance in extensive computer vision tasks but still struggleto generalize well in cross-domain tasks that source and target domain data are drawn from thedifferent data distributions.
Unsupervised Domain Adaptation (UDA)[27] aims to transfer knowledgefrom fully labeled source to unlabeled target domain by minimizing the domain gap between sourceand target.
However, existing UDA methods tackle the domain gap under a strong closed-setassumption that two domains share identical label sets, limiting their applications to real-worldscenarios.
Recently, Partial Domain Adaptation (PDA)[4] and Open set domain adaptation (OSDA)[2]are proposed to relax the closed-set assumption, allowing the existence of private classes in the sourceand target domain respectively.
However, all the above-mentioned settings heavily rely on priorknowledge where the common classes lie in the target domain.
It may be not reasonable and realisticfor UDA since the target domain is unsupervised.
To address the above problem, a generalized setting, termed as Universal Domain Adaptation(UniDA)[37], is proposed to allow both domains to own private classes but without knowing prior
[Figure]
information, e.g. matched common classes and classes numbers in the target domain.
Detectingcommon and target-private samples in the target domain is an essential issue in UniDA.
Existing UniDA methods [37, 16, 31] detect common and target-private samples by using somemanually specified or hand-tuned threshold values.
Therefore, these methods are not applicable tomore realistic UniDA due to the diverse ratios of common categories as shown in Fig. 1(a).
Moreover,most existing UniDA methods treat all target-private samples as a single class and cannot recognizedifferent categories among them, as shown in Fig. 1(b).
This paper is the first to emphasize that
UniDA methods should have the capability to automatically discover and recognize private categoriesin the target domain.
Essentially, the common class detection and private class discovery problems can be both viewed asdistribution transportation problems.
Therefore, these issues can be formulated within the frameworkof Optimal Transport (OT), which is a promising optimization problem to seek an efficient solutionfor transporting one distribution to another.
Even though many OT-based methods [12, 23, 36, 13]have been proposed for Unsupervised Domain Adaptation, most of them consider cross-domainsample-to-sample mapping under the closed-set condition and are not specialized in UniDA problemwith unaligned label sets.
Specifically, OT encourages a global mapping to mine domain statisticsproperty for discovering intrinsic differences among common and target private samples.
Additionally,the OT constraints can also avoid degenerate solutions in clustering representation problem [1, 7].
Inspired by this, OT can be a proper formulation with respect to inter-domain common class detectionand intra-domain private class discovery.
In this paper, we propose a unified optimal transport framework, namely UniOT, to solve Universal
Domain Adaptation from the perspectives of (1) partial alignment for common class detection and (2)target representation learning for private class discovery.
With regards to common class detection,we propose an OT-based partial alignment to detect common samples in the target domain anddevelop an adaptive filling method to handle the diverse ratios of common categories.
Without anypredefined threshold values, it can automatically discover the intrinsic difference between commonand private classes based on the global statistical information of the assignment matrix obtainedfrom OT.
For private class discovery, we propose an OT-based target representation learning thatencourages both global discrimination and local consistency of samples to avoid the over-reliance onsource supervision.
In addition, UniOT has the capability to automatically discover and recognizecategories in the target domain, benefiting from the representation learning with OT.
We believe acompetitive UniDA method should achieve high classification accuracy for the common class andalso learn a discriminative representation for the target private class.
However, existing methodsdo not provide any quantitative metric to evaluate the target representation performance for thoseunknown samples.
For target-private class discovery purpose, we introduce a new evaluation metric,
H 3 -score, considering not only common class accuracy but also the clustering performance of privateclasses.
Our contributions are summarized as follows: (1) We propose UniOT to handle two essential issuesin UniDA, including common class detection and private class discovery.
To our best knowledge, thisis the first attempt to jointly consider common class detection and private class discovery in a unifiedframework by optimal transport.
(2) We propose an OT-based partial alignment with adaptive fillingfor common class detection without any predefined threshold values.
It can automatically adapt to more realistic UniDA scenarios, where the ratios of common classes are diverse as shown in Fig.
1(a).(3)
We design an OT-based representation learning technique for private class discovery, consideringboth global discrimination of clusters and local consistency of samples.
Unlike most existing methodsthat treat all target-private samples as a whole, our UniOT can automatically discover and recognizeprivate categories in the target domain.
Related Work
Universal Domain Adaptation As a more generalized UDA setting, UniDA [37] is more challengingand realistic since the prior information of categories is unknown.
UAN [37],CMU [16] and TNT[9] design sample-level uncertainty criteria to measure domain transferability.
Samples with loweruncertainty are encouraged for adversarial adaptation with higher weights.
Most UniDA methodsdetect common samples with the sample-level criteria, which requires some manually specifiedand hand-tuned threshold values.
Moreover, over-reliance on source supervision under categoryneglects discriminative representation in the target domain.
DANCE [31] proposed neighborhoodclustering as a self-supervised technique to learn features useful for discriminating “unknown”categories.
DCC [22] enumerated cluster numbers of the target domain to obtain optimal crossdomain consensus clusters as common classes, but the consensus clusters are not robust enough dueto
the hard assignment of K-Means [25].
MATHS [8] detected the existence of the target private classby Hartigan’s dip test in the first stage, then trained hybrid prototypes by learning from K-Meansassignment where a fixed hyper-parameter for K-Means was adopted.
In this paper, we use OT tohandle both common sample detection and target representation learning under a unified framework.
It is worth noting that our method is adaptive for various unbalanced compositions of common andprivate classes, which does not require any predefined threshold.
In addition, our method encouragesboth the global discrimination and the local consistency of samples for target representation learning.
Optimal Transport in Domain Adaptation Optimal transport (OT) was first proposed by Kantorovich [21] to obtain an efficient solution for moving one distribution of mass to another.
Interior point methods can handle OT problem with computational complexity at least O ( d 3 log ( d )) .
Cuturi[11] first proposed to use Sinkorn’s algorithm [33] to compute an approximate transport coupling with an entropic regularization.
This method is lightspeed and can handle large-scale problemsefficiently.
Later, DeepJDOT[12] used the method in [11] to address the domain adaptation problemsto pair samples in the source and target domain.
Accordingly, a new classifier is trained on thetransported source data representation.
Since classical OT does not allow partial displacement intransport plan, unbalanced optimal transport is proposed to relax equality constraint by replacingthe hard marginal constraints of OT with soft penalties using Kullback-Leibler (KL) divergence[15].
JUMBOT[13] proposed an unbalanced optimal transport (UOT) to yield more robust behavior, whichcan also effectively be solved by generalized Sinkhorn’s algorithm[10].
Also, [26] TS-POT addressed
PDA problem with partial OT [14] to allow partial displacement.
However, these methods consider across-domain sample-to-sample mapping under the closed-set condition and are not suitable for thecategory gap problem in UniDA.
Besides, a sample-to-sample mapping neglects the global consensusof two domains and intrinsic differences among common and target private samples.
In this paper,we propose an inter-domain partial alignment based on UOT for common class detection and intradomain representation learning based on OT for private class discovery, where a sample-to-prototypemapping is designed to encourage a more global-aware assignment.
Clustering for
Representation Learning
Deep clustering has raised increasing attention in deeplearning community due to its capability of representation learning for discriminative clusters.
DeepCluster[6] proposed an end-to-end framework that iteratively obtains clustering assignment by
K-Means[25] and updates feature representation by the assignment, while PICA[20] simultaneouslylearns both feature representation and clustering assignment.
A two-step approach was proposedby SCAN[34], which firstly learns a better initial representation, then implements an end-to-endclustering learning in the next stage.
These deep clustering methods assume the number of clusters isknown beforehand.
Self-labelling[1] and SwAV[7] formulated the clustering assignment probleminto an OT problem for better feature representation learning.
Notably, the number of clusters doesnot rely on the ground-truth categories, benefiting from the equality constraints in OT.
Therefore,
OT-based representation learning is more suitable for UniDA, where the number of private class isunknown.
However, existing OT-based representation learning [1, 7] consider global discriminationof clusters but ignores local consistency of samples, limiting its direct applications in UniDA.
In this
[Figure]
H
L
[Figure]
[Figure] [Figure]
OT
[Figure]
by argmax paper, we develop an OT-based target representation learning technique for both better global andlocal structures in UniDA.
Methodology
In Universal Domain Adaptation (UniDA), there is a labeled source domain D s =
{ x si , y si } n s i =1 andan unlabeled target domain D t
=
{ x ti } n t i =1 .
We use C s and C t to denote the label set of source andtarget domain respectively.
Denote C =
C s ∩ C t as the common label set shared by both domains.
Let C s =
C s \ C and C t =
C t \ C represent label sets of source private and target private, respectively.
UniDA aims to classify target samples into |C| + 1 classes, where target private samples are treatedas unknown class uniformly.
UniDA aims to transfer knowledge from source to target under domain gap and category gap (i.e.,different label sets among source and target domain), which makes it challenging to align commonsamples and reject private samples.
Moreover, over-reliance on common knowledge extracted fromthe source domain neglects the target intrinsic structure.
Therefore, in this paper, we propose a unifiedoptimal transport framework to address UniDA problem from two perspectives, i.e., common classdetection and private class discovery, as shown in Fig. 3.1.
3.1 Preliminary
Optimal Transport is a constrained optimization problem that seeks an efficient solution of transportingone distribution of mass to another.
We now briefly recall the well-known Sinkhorn’s algorithm foroptimal transport.
Let Σ r := { x ∈ R r + | x ⊤ 1 r = 1 } denote the probability simplex, where 1 d is a d -dimensional vectorof ones.
Given two simplex vectors α ∈ Σ r and β ∈ Σ c , we can write the transport polytope of αand β as follows:
[Equation]
The transport polytope U ( α, β ) can also be interpreted as a set of all possible joint probabilities of( X, Y ) , where X and Y are two d -dimensional random variables with marginal distribution α and β ,respectively.
Following SwAV [7], given a similarity matrix M instead of distance matrix in [11], the coupling matrix (or joint probability) Q ∗ mapping α to β can be quantified by optimizing thefollowing maximization problem:
[Equation]
where H ( Q ) = − (cid:80) ij Q ij log Q ij is the entropic regularization term.
The optimal Q ∗ has beenshown to be unique with the form Q ∗ =
Diag ( u ) exp ( M /ε )
Diag ( v ) , where u and v can be solvedby Sinkhorn’s Algorithm [11].
Since Q ∗ satisfies the mass constraint (1) U ( α, β ) strictly, the objective OT ε ( M , α, β ) is not suitablefor partial displacement problem.
In light of this, Unbalanced OT is proposed to relax the conservationof marginal constraints by allowing the system to use soft penalties [10].
The unbalanced OT isformulated as
[Equation]
where D KL is Kullback-Leibler Divergence.
Then the optimization problem (3) can be solved bygeneralized Sinkhorn’s algorithm [10].
3.2 Inter-domain Partial Alignment for Common Class Detection
To extract common knowledge, we propose an Unbalanced OT-based Common Class Detection(CCD) method, which considers a partial mapping problem between target samples and sourceprototypes from the perspective of cross domain.
Note that our method uses prototypes to encourage amore global-aware alignment, which discovers intrinsic differences among common and target privatesamples by exploiting the statistical information of the assignment matrix obtained from the optimaltransport.
Since UniDA allows private classes in both source and target domains, partial alignmentshould be considered to avoid misalignment between target private samples and source classes.
Hencewe formulate the problem mapping target features to the source prototypes C s =
[ c s 1 , · · · , c s |C s | ] ⊤with unbalanced OT objective to obtain the optimal assignment matrix Q st ,
i.e.
[Equation]
where S st = Z t C ⊤ s refers to cosine similarity between target samples and source prototypes, andthe mini-batch target ℓ 2 -normalized features Z t =
[ z t 1 , · · · , z tB ] ⊤ with batch-size B are extracted byfeature extractor f , i.e. z ti = f
( x ti ) /
∥
f ( x ti ) ∥ 2 .
Here we use the same prototype-based classifier in[31] to learn source prototypes.
Note that the private samples will be assigned with relatively lowweights in Q st .
Based on this observation, our CCD selects target samples with top confidence ascommon classes.
Firstly, the assignment matrix is normalized as ¯ Q st = Q st / (cid:80) Q st .
Then we discover statisticalproperty from normalized ¯ Q st and generate score from the statistical perspectives of both targetsamples and source prototypes.
For the i -th row in ¯ Q st , it can be seen as a vanilla predictionprobability vector and we can get pseudo-label ˆ y ti by argmax operation.
And we assign target samplesconfidence score w ti with the maximum value of the i -th row in ¯ Q st , i.e.
[Equation]
A higher score w ti means z ti is relatively closer to a source prototype than any other samples and ismore likely from a common class.
Meanwhile, to select target common samples with top confidence,we also evaluate source prototypes confidence score w si with the sum of the j -th column, i.e.
[Equation] [Equation]
Analogously, a higher score w s j means C s j is more likely a common source prototype, which isassigned to target samples more frequently.
Then the common samples are detected by statisticsmean, i.e.
[Equation]
where δ i = 1 indicates the sample x ti is detected as common sample with top confidence, which canbe assigned with pseudo label ˆ y ti by argmax operation.
We can use pseudo labels of selected targetsamples to compute inter-domain common class detection loss by standard cross-entropy loss, i.e.
[Equation]
To ensure that statistics mean can discover the intrinsic difference between common and privateclasses, we need to guarantee that the input features for partial mapping should be sampled from thetarget domain distribution with enough sampling density.
Therefore, we implement a FIFO memoryqueue[18] to save previous target features for filling mini-batch features, which avoids the limitationled by the lack of statistical property for target samples.
Adaptive filling for unbalanced proportion of positive and negative.
Unfortunately, usingstatistics mean as a threshold in Eq.
(7) may misclassify some common samples as private class insome extremely unbalanced cases, such as there are few private samples indeed.
Essentially, theassignment weight is determined by the relative distance between samples and prototypes, and privatesamples with low similarity set negative examples for the others.
To automatically detect targetsamples, we design an adaptive filling mechanism for positive and negative unbalanced proportions.
Our strategy is to fill the gap of unbalanced proportion to be balanced like Fig.
3.1 depicted.
Firstly,samples with high similarity larger than a rough boundary γ are defined as positive and the othersare negative.
Then we implement adaptive filling to be balanced when the proportion of positive ornegative samples exceeds 50% .
For negative filling, we synthesize fake private feature z ti by mixingup target feature z ti and its farthest source prototypes evenly, i.e. ˆ z ni = 12 (cid:0)
z ti + argmin c sk
( z ti c sk ⊤ ) (cid:1) .
For positive filling, we first do the CCD without adaptive filling by Eq.(7) to obtain confident featuresand then reuse these filtered confident features for positive filling.
Note that the filling samples arerandomly sampled from mix-up negatives or CCD positives and the filling size is the gap betweenunfilled positive and negative.
After adaptive filling for balanced proportion, our CCD is adaptive fordetecting confident common samples.
Adaptive update for marginal probability vector.
The marginal probability vector β in Objective(4) can be interpreted as a weight budget for source prototypes to be mapped with target samples.
However, the source domain may own source private class, which is unreasonable to set the weightbudget equally for each source prototype, i.e. β = 1 |C s | 1 |C s | .
Therefore, we use the ¯ Q st obtained inthe last iteration to update β with the moving average for adapting to the real source distribution , i.e.
[Equation]
where (cid:101) β ( t ) is the sum of column in ¯ Q st solved in the β ( t ) case.
3.3 Intra-domain Representation Learning for Private Class Discovery
To exploit target global and local structure and feature representation, we propose an OT-based Private
Class Discovery (PCD) clustering technique that considers a full mapping problem between targetsample features and target prototypes.
Especially, our PCD avoids reliance on the number of targetclasses and encourages global discrimination of clusters as well as local consistency of samples.
To solve this, we pre-define K learnable target prototypes C t and initialize them randomly, wherelarger K can be determined by the larger number of target samples.
Then we assign target features¯ Z t with the prototypes C t by solving the optimal transport optimization problem:
[Equation]
where S tt = ¯ Z t C ⊤ t refers to cosine similarity between target samples and target prototypes, and ¯ Z tis the concatenation of the mini-batch target features Z t and their nearest neighbor features.
Themarginal constraint of uniform distribution enforces that on average each prototype is selected at least2 B/K times in the batch.
The solution Q tt satisfies the constraint strictly, i.e. the sum of each row equals to 1 /K .
To obtainprototype assignment for each target sample, the soft pseudo-label matrix ˜ Q tt =
Q tt × K ensures the i -th row q tti is a probability vector.
To encourage global discrimination of clusters, the learnableprototypes C t and the feature extractor f are optimized by minimizing L global , i.e.
[Equation]
where the cross-entropy loss ℓ ( q i , z i ) is presented as [Equation] i [Equation]
k log p
i,k , where p i,k =
[Equation]
To encourage local consistency of samples, we swap prediction between anchor feature z ti andits nearest neighbor feature ˜
z
ti retrieved from the memory queue.
Note that instead of swappedprediction between the different views of anchor image in SwAV[7], our swapped L local is based onanchor-neighbor swapping, i.e.
[Equation]
Eventually, to encourage both global discrimination of clusters and local consistency of samples,
our
L PCD is presented as
[Equation] [Equation]
Similar to Sec.3.2, since OT encourages a more deliberative mapping under marginal constraint, wealso need to fill input features with the previous target features in memory queue before solving theoptimization problem (10).
3.4 A Unified Framework for UniDA by Optimal Transport
In UniDA, the partial alignment for common class detection and representation learning for privateclass discovery are based on UOT and OT models, respectively.
Therefore, we can summarize theabove two models in a unified framework, i.e. UniOT.
Considering cross-entropy loss L cls on sourcesamples, Common Class Discovery loss L CCD and Private Class Discovery loss L PCD , our overallobjective can be expressed as
[Equation]
In the testing phase, only feature extractor and prototype-based classifier are preserved.
We solve theoptimization problem (4) for all the concatenated target features and adaptive filled features.
Thencompute w tj in Eq.
5 for each sample.
For those samples who satisfy w tj ≥ 1 n are assigned with theclosest source class, where n is the sum of the total size of target samples n t and adaptive filling size.
Otherwise, the samples are marked as unknown .
Experiments
4.1 Experimental Settings
Dataset.
Our method will be validated in 4 popular datasets in Domain Adaptation.
Office [30]contains 31 categories and about 4K images in 3 domains: Amazon( A ), DSLR( D ) and Webcam( W ).
Office-Home [35] contains 65 categories and about 15K images in 4 domains: Artistic images( Ar ),
Clip-Art images( Cl ), Product images( Pr ) and Real-World images( Rw ).
VisDA [29] is a large datasetwhich constains 12 categories in source domain with 15K synthetic images and target domain with5K real-world images.
DomainNet [28] is the largest domain adaptation dataset which contains 345categories and about 0.6 million in 6 domains but we only use 3 domain: Painting ( P ), Real ( R ), and
Sketch ( S ) like [16].
Note that we use A 2 W to denote that transfer task from Amazon to DSLR.
Wefollow the dataset split in [16] to conduct experiments.
Evaluation metric.
For UniDA, the trade-off between the accuracy of common and private classesis important in evaluating performance.
Thus, we use the H-score[16] to calculate the harmonic meanof the instance accuracy on common class a C and accuracy on the single private class a ¯ C t .
However,
H-score treats all private samples as a single class.
For target private class discovery purposes,we further introduce H 3 -score, as the harmonic mean of accuracy on common class, accuracy onunknown class and Normalized Mutual Information (NMI) for target-private clusters, i.e.
[Equation]
(16) where NMI is a well-known measure for the quality of clustering, and NMI is obtained by K-Meanswith the ground truth of the number of private classes in the inference stage.
4.2 Results
Comparison with state-of-the-arts.
Tab. 1 and 2 shows the H-score results for Office, Office-Home,
VisDA and DomainNet.
Our UniOT achieves the best performance on all benchmarks.
H-score onnon-UniDA methods are reported from [16], and UniOT methods are reported from original papersaccordingly, except for DANCE[31] since they did not report H-score and we reproduce it with theirreleased code for fair comparison, marked as ‡ .
In particular, with respect to H 3 -score in Tab. 3, our
UniOT surpasses other methods by 7% in both Office and Office-Home datasets, which demonstratesthat our UniOT achieves balanced performance for both common class detection and private classdiscovery. [Table] [Table] ResNet
[Table]
[Table]
DANN[17]
[Table]
[Table]
[Table]
[Table]
[Table]
Evaluation of the effectiveness of the proposed CCD and PCD.
To evaluate the contribution of
L CSD , L global and L local , we train the model with different combination of each component.
Asshown in Tab.4, the combination of L global and L local , i.e. L PCD , brings significant contributionsince target representation is crucial for distinguishing common and private samples.
Especially,
L global contributes more than L local , which demonstrates the benefit of global discrimination of [Figure] [Figure] clusters for representation learning.
Besides, we also conduct experiments for CCD without adaptivefilling and denote the loss as L † CCD , which verifies the effectiveness of adaptive filling design.
[Figure]
[Table]
✓ ✓
[Table]
✓ ✓ ✓
[Table]
Robustness in realistic UniDA.
We compare the performance of our UniOT with DCC, DANCEand UAN under different categories split.
In this analysis, we perform on A2W of Office with a fixedcommon label set C and target private label set C t , but the different size of source private label set
C s .
We set 10 categories for C , 11 categories for C t and vary the number of C s .
Fig.
3(a) shows thatour UniOT always outperforms other methods.
Besides, UniOT is not sensitive to the variation ofsource private classes while other methods are not stable enough.
Additionally, we also analyze thebehavior of UniOT under the different size of target private label set C t .
We also perform on A2Wof Office with a fixed common label set C for 10 categories and source private label set C s for 10categories.
Fig.
3(b) shows that our UniOT still performs better, while the other methods presentsensitivity to the number of target private classes.
Therefore, UniOT is more robust to the variation oftarget private classes.
Conclusion
In this paper, we have proposed to use Optimal Transport to handle common class detection andprivate class discovery for UniDA under a unified framework, namely UniOT.
More precisely, an
OT-based cross-domain partial alignment is designed to detect common class with an adaptive fillingstrategy to handle the diverse UniDA settings.
In addition, we have proposed an OT-based targetrepresentation learning for private class discovery, which encourages both global discrimination andlocal consistency of samples.
In the future, we will modify memory model for more efficient featuresfilling in OT.
Experimental results on four benchmark datasets validate the superiority of our UniOTover a wide range of state-of-the-art methods.
Checklist
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