Learning visual similarity is of great importance for a wide range of vision tasks, such as image clustering(Bouchacourt et al., 2018) et al., 2015) and image retrieval (Wu et al., 2017)
. Measuring similarity requires learning an embedding space which captures images and reasonably reflects their similarity using a defined distance metric. One of the most adopted classes of algorithms for this task is Deep Metric Learning (DML) which leverages deep neural networks to learn such a distance preserving embedding.
Due to the growing interest in DML, a large corpus of literature has been proposed contributing to its success. However, as recent DML approaches explore more diverse research directions such as architectures (Xuan et al., 2018; Jacob et al., 2019), objectives functions (Wang et al., 2019a; Yuan et al., 2019) and additional training tasks (Roth et al., 2019; Lin et al., 2018), an unbiased comparison of their results becomes more and more difficult. Further, undisclosed technical details such as data augmentations and training regularization pose a challenge to the reproducibility of such methods, which is of great concern in the machine learning community in general (Bouthillier et al., 2019). One goal of this work is to counteract this worrying trend by providing a comprehensive comparison of the most important and widely used DML baselines under identical training conditions on standard benchmark datasets (Fig. 1). In addition, we thoroughly review common design choices of DML models which strongly influence generalization performance to allow for better comparability of current and future work.
On that basis, we extend our analysis to: (i) The process of data sampling which is well-known to impact the DML optimization (Schroff et al., 2015). While previous works only studied this process in the specific context of triplet mining strategies for ranking-based objectives (Wu et al., 2017; Harwood et al., 2017), we examine the model-agnostic case of sampling informative mini-batches. (ii) The generalization capabilities of DML models by analyzing the structure of their learned embedding spaces. While we are not able to reliably link typically targeted concepts such as large inter-class margins (Liu et al., 2017; Deng et al., 2018)
and intra-class variance(Lin et al., 2018) to generalization performance, we uncover a strong correlation to the compression of the learned representations. Lastly, based on this observation, we propose a simple, yet effective, regularization technique which effectively boosts the performance of ranking-based approaches on standard benchmark datasets as also demonstrated in Fig. 1. In summary, our most important contributions can be described as follows:
We provide an exhaustive analysis of recent DML objective functions, their training strategies, the influence of data-sampling, and model design choices to set a standard benchmark. To this end, we will publish all code.
We provide new insights into DML generalization by analyzing its correlation to the embedding space compression (as measured by its spectral decay), inter-class margins and intra-class variance.
Based on the result above, we propose a simple technique to regularize the embedding space compression which we find to boost generalization performance of ranking-based DML approaches.
2 Related Works
Deep Metric Learning:
Deep Metric Learning (DML) has become increasingly important for applications ranging from image retrieval (Movshovitz-Attias et al., 2017; Roth et al., 2019; Wu et al., 2017; Lin et al., 2018) to zero-shot classification (Schroff et al., 2015; Sanakoyeu et al., 2019) and face verification (Hu et al., 2014; Liu et al., 2017). Many approaches use ranking-based objectives based on tuples of samples such as pairs (Hadsell et al., 2006), triplets (Wu et al., 2017; Yu et al., 2018), quadruplets(Chen et al., 2017) or higher-order variants like N-Pairs(Sohn, 2016), lifted structure losses (Oh Song et al., 2016; Yu et al., 2018) or NCA-based criteria(Movshovitz-Attias et al., 2017). Further, classification-based methods adjusted to DML (Deng et al., 2018; Zhai and Wu, 2018) have proven to be effective for learning distance preserving embedding spaces. To address the computational complexity of tuple-based methods111As an example, the number of triplets scales with , where is the dataset size., different sampling strategies have been introduced (Schroff et al., 2015; Wu et al., 2017; Ge, 2018). Moreover, proxy-based approaches address this issue by approximating class distributions using only few virtual representatives (Movshovitz-Attias et al., 2017; Qian et al., 2019).
Additionally, more involved research extending above objectives has been proposed: Sanakoyeu et al. (2019) follow a divide-and-conquer strategy by splitting and subsequently merging both the data and embedding space; Opitz et al. (2018); Xuan et al. (2018) employ an ensemble of specialized learners and Roth et al. (2019)
combine DML with self-supervised learning. Moreover,Lin et al. (2018) and Zheng et al. (2019) generate artificial samples to effectively augment the training data, thus learning more complex ranking relations. The majority of these methods are trained using the essential objective functions and, further, hinge on the training parameters discussed in our study, thus directly benefiting from our findings. Moreover, we propose an effective regularization technique to improve ranking-based objectives.
Mini-batch selection: The benefits of large mini-batches for training are well studied (Smith et al., 2017; Goyal et al., 2017; Keskar et al., 2016). However, there has been limited research examining effective strategies for the creation of mini-batches. Research into mini-batch creation has been done to improve convergence in optimization methods for classification tasks(Mirzasoleiman et al., 2020; Johnson and Guestrin, 2018) or to construct informative mini-batches using core-set selection to optimize generative models (Sinha et al., 2019)
. Similarly, we analyze mining strategies maximizing data diversity and compare their impact to standard heuristics employed in DML(Wu et al., 2017; Roth et al., 2019; Sanakoyeu et al., 2019)).
Generalization in DML: Generalization capabilities of representations (Achille and Soatto, 2016; Shwartz-Ziv and Tishby, 2017) and, in particular, of discriminative models has been well studied (Jiang* et al., 2020; Belghazi et al., 2018; Goyal et al., 2017), e.g. in the light of compression (Tishby and Zaslavsky, 2015; Shwartz-Ziv and Tishby, 2017) which is covered by strong experimental support (Goyal et al., 2019; Belghazi et al., 2018; Alemi et al., 2016). Verma et al. (2018) link compression to a ’flattening’ of a representation in the context of classification. We apply this concept to analyze generalization in DML and find that strong compression actually hurts DML generalization. Existing works on generalization in metric learning focus on robustness of linear or kernel-based distance metrics (Bellet and Habrard, 2015; Bellet, 2013) and examine bounds on the generalization error (Huai et al., 2019). In contrast, we examine the correlation between generalization and structural characteristics of the learned embedding space.
3 Training a Deep Metric Learning Model
In this section, we briefly summarize key components for training a DML model and motivate the main aspects of our study. We first introduce the common categories of training objectives which we consider for comparison in Sec. 3.1. Next, in Sec. 3.2 we examine the data sampling process and present strategies for sampling informative mini-batches. Finally, in Sec. 3.3, we discuss components of a DML model which impact its performance and exhibit an increased divergence in the field, thus impairing objective comparisons.
3.1 The objective function
In Deep Metric Learning we learn an embedding function mapping datapoints into an embedding space , which allows to measure the similarity between as with being a predefined distance function.
For that, let be a deep neural network parametrised by with its output typically normalized to the real hypersphere for regularization purposes (Wu et al., 2017; Huai et al., 2019).
In order to train to reflect the semantic similarity defined by given labels , many objective functions have been proposed based on different concepts which we now briefly summarize.
The most popular family are ranking-based loss functions operating on pairs(Hadsell et al., 2006), triplets (Schroff et al., 2015; Wu et al., 2017) or larger sets of datapoints (Sohn, 2016; Oh Song et al., 2016; Chen et al., 2017; Wang et al., 2019a). Learning is defined as an ordering task, such that the distances between an anchor and positive of the same class, , is minimized and the distances of to negative samples with different class labels, , is maximized. For example, triplet-based formulations typically optimize their relative distances as long as a margin is violated, i.e. as long as . Further, ranking-based objectives are also extended to histogram matching, as proposed in (Ustinova and Lempitsky, 2016).
Classification-based: As DML is essentially solving a discriminative task, some approaches (Zhai and Wu, 2018; Deng et al., 2018; Liu et al., 2017)
can be derived from softmax-logits. For example, Deng et al. (2018) exploit the regularization to the real hypersphere and the equality to maximize the margin between classes by direct optimization over angles . Further, also standard cross-entropy optimization proves to be effective under normalization (Zhai and Wu, 2018).
Proxy-based: These methods approximate the distributions for the full class by one (Movshovitz-Attias et al., 2017) or more (Qian et al., 2019) learned representatives. By considering the class representatives for computing the training loss, individual samples are directly compared to an entire class. Additionally, proxy-based methods help to alleviate the issue of tuple mining which is encountered in ranking-based loss functions.
3.2 Data sampling
The synergy between tuple mining strategies and ranking losses has been widely studied (Wu et al., 2017; Schroff et al., 2015; Ge, 2018). To analyze the impact of data-sampling on performance in the scope of our study, we consider the process of mining informative mini-batches . This process is independent of the specific training objective and so far has been commonly neglected in DML research. Following we present batch mining strategies operating on both labels and the data itself: label samplers, which are sampling heuristics that follow selection rules based on label information only, and embedded samplers, which operate on data embeddings themselves to create batches of diverse data statistics.
Label Samplers: To control the class distribution within , we examine two different heuristics based on the number, , of ’Samples Per Class’ (SPC-) heuristic:
SPC-2/4/8: Given batch-size , we randomly select unique classes from which we select samples randomly.
SPC-R: We randomly select samples from the dataset and choose the last sample to have the same label as one of the other samples to ensure that at least one triplet can be mined from . Thus, we effectively vary the number of unique classes within mini-batches.
Embedded Samplers: Increasing the batch-size has proven to be beneficial for stabilizing optimization due to an effectively larger data diversity and richer training information (Mirzasoleiman et al., 2020; Brock et al., 2018; Sinha et al., 2019). As the DML training is commonly performed on a single GPU (limited especially due to tuple mining process on the mini-batch), the batch-size is bounded by memory. Nevertheless, in order to ‘virtually’ maximize the data diversity, we distill the information content of a large set of samples into a mini-batch by matching the statistics of and under the embedding . To avoid computational overhead, we sample from a continuously updated memory bank of embedded training samples. Similar to Misra and van der Maaten (2019), is generated by iteratively updating its elements based on the steady stream of training batches . Using , we mine mini-batches by first randomly sampling from with and subsequently find a mini-batch to match its data statistics by using one of the following criteria:
Greedy Coreset Distillation (GC): Greedy Coreset (Agarwal et al., 2005) finds a batch by iteratively adding samples which maximize the distance from the samples that have already been selected , thereby maximizing the covered space within by solving .
Matching of distance distributions (DDM): DDM aims to preserve the distance distribution of . We randomly select candidate mini-batches and choose the batch with smallest Wasserstein distance between normalized distance histograms of and (Rubner et al., 2000).
FRD-Score Matching (FRD): Similar to the recent GAN evaluation setting, we compute the frechet distance (Heusel et al., 2017)) between and to measure the similarity between their distributions using , with being the mean and covariance of the embedded set of samples. Like in DDM, we select the closest batch to among randomly sampled candidates.
3.3 Training parameters, regularization and architecture
Recall performance of commonly used network architectures after ImageNet pretraining. Final linear layer is randomly initialized and normalized.
Next to the objective function and data sampling process, successfully learning a DML model hinges on a reasonable choice of the training environment. While there is a multitude of parameters to be set, we identify several factors which both influence performance and exhibit an divergence in lately proposed works.
Architectures: In recent DML literature predominantly three basis network architectures are used: GoogLeNet (Szegedy et al., 2015) (GN, typically with embedding dimensionality 512), Inception-BN (Ioffe and Szegedy, 2015) (IBN, 512) and ResNet50 (He et al., 2016)
(R50, 128) (with optionally frozen Batch-Normalization layers for improved convergence and stability across varying batch sizes222Note that Batch-Normalization is still performed, but no parameters are learned., see e.g. Roth et al. (2019); Cakir et al. (2019)). Due to the varying number of parameters and configuration of layers, each architecture exhibits a different starting point for learning, based on its initialization by ImageNet pretraining (Deng et al., 2009). Table 1 compares their initial DML performance measured in Recall@1 (R@1). The reference to differences in architecture is one of the main arguments used by individual works not compare themselves to competing approaches. Disconcertingly, even when reporting additional results using adjusted networks is feasible, typically only results using a single architecture are reported. Consequently, a fair comparison between approaches is heavily impaired.
Weight Decay: Commonly, network optimization is regularized using weight decay/L2-regularization (Krogh and Hertz, 1992). In DML, particularly on small datasets its careful adjustment is crucial to maximize generalization performance. Nevertheless, many works do not report this.
Embedding dimensionality: Choosing a dimensionality of the embedding space influences the learned manifold and consequently generalization performance. While each architecture typically uses an individual, standardized dimensionality in DML, recent works differ without reporting proper baselines using an adjusted dimensionality. Again, comparison to existing works and the assessment of the actual contribution is impaired.
Advanced DML methodologies There are many extensions to objective functions, architectures and the training setup discussed so far. However, although extensions are highly individual, they still rely on these components and thus benefit from findings in the following experiments, evaluations and analysis.
4 Analyzing DML training strategies
The three examined benchmarking datasets are:
CUB200-2011: Contains 11,788 images of birds over 200 classes. Train/Test sets are made up of the first/last 100 classes and 5,864/5,924 images respectively (Wah et al., 2011). Samples are distributed evenly across classes.
CARS196: Contains 16,185 images of cars in 196 classes, with even sample distribution. Train/Test sets are made up of the first/last 98 classes and 8054/8131 images respectively (Krause et al., 2013).
Stanford Online Products (SOP): Contains 120,053 product images divided into 22,634 classes. Train/Test sets are provided, contain 11,318 classes/59,551 images in the Train and 11,316 classes/60,502 images in the Test set (Oh Song et al., 2016). In SOP, unlike the other benchmarks, most classes have few instances, leading to significantly different data distribution compared to CUB200-2011 and CARS196.
4.1 Experimental Protocol
Our training protocol follows parts of Wu et al. (2017), which utilize a ResNet50 architecture with frozen Batch-Normalization layers. We set the embedding dimensionality to 128 to be comparable with already proposed results with this architecture. While both GoogLeNet (Szegedy et al., 2015) and Inception-BN (Ioffe and Szegedy, 2015) are also often employed in DML literature, we choose ResNet50 with due to its success in recent state-of-the-art approaches (Roth et al., 2019; Sanakoyeu et al., 2019). In line with standard practices we randomly resize and crop images to to resolution for training and center crop to the same size for evaluation. During training, random horizontal flipping () is used. Optimization is performed using Adam (Kingma and Ba, 2015) with learning rate fixed to . For all evaluations and experiments, no learning rate scheduling is used for unbiased comparison. Weight decay, if not noted otherwise, is set to a constant value of , as motivated in section 4.2
. We implemented all models using the PyTorch framework(Paszke et al., 2017)
, and experiments are performed on individual Nvidia Titan X, V100 and T4 GPUs with memory usage limited to 12GB. Each training is run over 150 epochs for CUB200-2011/CARS196 and 100 epochs for Stanford Online Products, if not stated otherwise. For batch sampling we utilize the the SPC-2 strategy, as motivated in section4.3
. Finally, each result is averaged over multiple seeds to avoid seed-based performance fluctuations. All loss-specific hyperparameters are discussed in the supplementary material, along with their original implementation details. For our study, we examine the following evaluation metrics (described further in the supplementary): Recall at 1 and 2(Jegou et al., 2011), Normalized Mutual Information (NMI) (Manning et al., 2010), F1 score (Sohn, 2016) and mean average precision measured on recall (mAP).
4.2 Studying DML parameters and architectures
|Imagenet (Deng et al., 2009)|
|Angular (Wang et al., 2017)|
|Arcface (Deng et al., 2018)|
|Contrastive(D) (Hadsell et al., 2006)|
|GenLifted (Yu et al., 2018)|
|Hist. (Ustinova and Lempitsky, 2016)|
|Margin(D) (Wu et al., 2017)|
|MultiSim. (Wang et al., 2019b)|
|N-Pair (Sohn, 2016)|
|Pnca (Movshovitz-Attias et al., 2017)||-||-|
|Quadruplet(D) (Chen et al., 2017)|
|SNR(D) (Yuan et al., 2019)|
|SoftTriplet (Qian et al., 2019)||-||-|
|Softmax (Zhai and Wu, 2018)|
|Triplet(R) (Schroff et al., 2015)|
|Triplet(S) (Schroff et al., 2015)|
|Triplet(D) (Wu et al., 2017)|
|R-Margin, (D, )|
|R-Margin, (D, )|
Now we study the influence of the parameter and architectures discussed in Sec. 3.3 using five different objective functions. For each experiment performance is measured across all metrics noted in Sec. 4.1. For each loss, every metric is normalized by the maximum across the evaluated value range. This enables to report an aggregated summary of performance across all metrics, where differences correspond to metric-agnostic relative improvement.
Fig. 2 analyzes each component by evaluating a reasonable range of values, while fixing the other parameters to the experimental setup of Sec. 4.1. For weight decay we observe a heavily model and dataset dependent behavior, while affecting the relative performance up to . This underlines the importance of a complete declaration of the training protocol to facilitate reproducibility and comparability. Similar results are observed for the embedding dimensionality . Our analysis shows that training objectives perform differently given a certain dimensionality and seem to converge at . However, e.g. in the case of R50, is typically fixed to , thus disadvantaging some training objectives over others. Finally, comparing common DML architectures reveals a strong impact on performance. In addition, the variance in performance between loss functions for each network is varying, with R50 and IBN being more consistent than GN.
Implications: In order to warrant unbiased comparability, equal training protocols, architectures and transparent model evaluation are essential, as even small deviations can result in large deviations in performance.
4.3 Batch sampling impacts DML training
We now analyze how the data sampling process for mini-batches impacts the performance of DML models using the sampling strategies presented in Sec. 3.2. To conduct an unbiased study, we experiment with six conceptually different objective functions: Marginloss with Distance-Weighted Sampling, Triplet Loss with Random Sampling, ProxyNCA, Multi-Similarity Loss, Histogram loss and Normalized Softmax loss. To aggregate our evaluation metrics (cf. 4.1), we utilize the same normalization procedure discussed in Sec. 4.2. Fig. 3 summarizes the results for each sampling strategy by reporting the distributions of normalized scores of all pairwise combinations of training loss and evaluation metrics. Our analysis reveals that the batch sampling process indeed effects DML training with a difference in mean performance up to . While there is no clear winner across all datasets, we observe that the SPC-2 and FRD samplers perform very well and, in particular, consistently outperform the SPC-4 strategy which is commonly reported to be used in literature (Wu et al., 2017; Schroff et al., 2015).
Implications: Our study indicates that DML benefits from data diversity in mini-batches, independent of the chosen training objective. While complex mining strategies may perform better, simple heuristics like SPC-2 are sufficient.
4.4 Comparing DML models
Based on our training parameter and batch-sampling evaluations we now compare a large selection of different DML objectives under fixed training conditions noted in sections 4.1 and 4.2. For ranking-based models, we employ distance-based tuple mining (D) (Wu et al., 2017) which proved most effective, except for the tuple mining study using the classic triplet loss, for which we also include random and semihard sampling (Schroff et al., 2015). Loss-specific hyperparameters are determined via a small cross-validation gridsearch around originally proposed values to adjust for our training setup. Exact parameters and method details including the originally utilized setup are listed in the supplementary. Fig. 2 summarizes our evaluation results on all benchmarks, while Fig. 4 measures correlations between all evaluation metrics. We observe particularly on CUB200-2011 and CARS196 a higher performance saturation between methods as compared to the SOP dataset due to the strong difference in data distribution. We find that representatives of ranking based objectives in general outperform their classification based counterparts. On average, margin loss offers the best performance across datasets. Remarkably, under our carefully chosen training setting, a multitude of losses compete or even outperform more involved state-of-the-art DML approaches (including strong ensemble methods) on the SOP dataset. For a detailed comparison to the state-of-the-art, we refer to the supplementary material.
Implications: Carefully trained baseline models are able to outperform state-of-the-art approaches which use considerable stronger architectures. To evaluate the true benefit of proposed contributions, baseline models need to be competitive.
5 Generalization in Deep Metric Learning
In the previous section we showed how different model and training parameter choices result in models of vastly different performance. However, how such differences can be explained best on basis of the learned embedding space is an open question and, for instance, studied under the concept of compression (Tishby and Zaslavsky, 2015). Recent work (Verma et al., 2018)
links compression to a class-conditioned flattening of representation, indicated by an increased decay of singular values obtained by Singular Value Decomposition (SVD) on the data representations. As a result, the class representations occupy a more compact volume, thus reducing the number of directions with significant variance. The subsequent strong focus on the most discriminative directions is shown to be beneficial for classic classification scenario with i.i.d. training and test distributions. However, this effect also overly discards features which could be useful for capturing data characteristics outside the training distribution. As a result, generalization in transfer learning problems like DML is hindered due to the shift in training and testing distribution(Bellet and Habrard, 2015). Given this observation, we hypothesize that actually retaining a considerable amount of directions of significant variance is an important requirement to learn a well generalizing embedding function .
To verify this assumption, we analyze the spectral decay of an embedding space by performing SVD on the embedded training data instead of considering individual training class representations, as testing and training distribution are shifted333For comparison we also analyse the class-conditioned singular value spectra as Verma et al. (2018) in the supplementary.. Next, we normalize the sorted spectrum of singular values
by their sum and compute the KL-divergence to a D-dimensional discrete uniform distribution, i.e. 444For simplicity we use the notation instead of ., which is proportional to the entropy of . Lower values of indicate more directions of significant variance. Using this measure, we analyze DML models trained with a large selection of different objectives in Fig. 5 (rightmost) on CUB200-2011, CARS196 and SOP dataset555A detailed comparison can be found in the supplementary. Comparing their R@1 accuracy and reveals a significant inverse correlation () between generalization performance and the spectral decay of the embedding spaces . This observation strongly confirms the positive effect of more directions of variance in the presence of training-testing distribution shifts.
We now compare our finding to commonly exploited concepts for training such as (i) larger margins between classes (Deng et al., 2018; Liu et al., 2017), i.e. an increase in average inter-class distances ; (ii) explicitly introducing intra-class variance (Lin et al., 2018), which is indicated by an increase in average intra-class distance ; and (iii) their relation by using the ratio . Here, denotes the set of embedded samples of a class , their mean embedding and normalization constants. Fig. 5 compares these measures with . It is evident that neither of the distance related measures consistently exhibits significant correlation with generalization performance when taking all three datasets into account. Individually on SOP, only exhibits similarly strong correlation to generalization performance due to the strong imbalance between dataset size and amount of samples per class.
Implications: Generalization performance in DML exhibits strong inverse correlation to the decay of the singular value spectrum of a learned representation. This indicates that representation learning under considerable shifts between training and testing distribution is hurt by excessive compression.
5.1 -regularization for improved generalization
We now exploit our findings from the previous section to propose a simple -regularization for ranking-loss based approaches by counteracting the compression of the representation. We randomly perform a switch operation within tuples by exchanging negative samples with the positive in a given ranking-loss formulation (cf. Sec. 3.1
) with fixed probability. This regularization pushes samples of the same class apart, thus enabling a DML model to capture extra non-label-discriminative features. Simultaneously, this process dampens the compression induced by strong discriminative training signals.
Fig. 6 depicts a 2D toy example (details in supplementary) which illustrates the effect of our proposed regularization and further highlights the issue of overly compressed data representations. Even though the training distribution exhibits features needed to separate all test classes, these features are disregarded by the strong discriminative training signal. Regularizing the compression by attenuating the spectral decay enables the model to capture more information and as a result exhibits stronger generalization to the unseen test classes. In addition, Fig. 8 verifies that the -regularization also leads to a decreased spectral decay on DML benchmark datasets, resulting in improved recall performance (cf. Tab. 2 (bottom)). We further observe that the vast amounts of classes for datasets such as SOP naturally counteract the compression of a representation, thus already exhibiting a considerable amount of directions of significant variance. Finally, we control the degree of the regularization by varying the probability and study the influence on performance in Fig. 7. Increasing the probability boosts the generalization performance until class boundaries get too close and, thus, discriminativity is lost.
In this work, we counteract the worrying trend of diverging training protocols in Deep Metric Learning (DML). We conduct a large, comprehensive study of important training components and objectives for DML to contribute to improved comparability of recent and future approaches. On this basis, we study generalization performance in DML and uncover a strong correlation to the level of compression of learned data representation. Our findings reveal that highly compressed representations disregard helpful features for capturing data characteristics that transfer to unknown test distributions. To this end, we propose a simple technique for ranking-based methods to regularize the compression of the learned embedding space, which results in boosted performance across all benchmark datasets.
We thank Alex Lamb (Mila) for insightful discussions that helped with the design of the toy examples. We would also like to thank Sharan Vaswani and Dmitry Serdyuk for their feedback (both Mila). We would also like to thank Nvidia for donating NVIDIA DGX-1, and Compute Canada for providing resources for this research.
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Appendix A Description of Methods
In this section, we briefly describe each DML training objective and triplet mining strategy used in our study, as well as the choice of their individual hyperparameters. General training parameters and details of the training protocol are already discussed in the main paper in Sec. 4.1. For notation, we refer to the embedding of an image including output normalization as . The non-normalized version is denoted as . All methods operate on the mini-batch containing image indices. If not mentioned otherwise, all embeddings operate in dimension .
a.1 Training Criteria
Contrastive (Hadsell et al., 2006)
The contrastive training formalism is simple: Given embedding pairs (sampled from a mini-batch of size ) containing an anchor from class and either a positive with or a negative from a different class, , the network is trained to minimize
with margin , which we set to . The margin ensures that embeddings are not projected arbitrarily far apart from each other. For our distance function we utilize the standard euclidean distance . We combine the contrastive loss with the distance-weighting negative sampling mentioned below.
Triplet (Hu et al., 2014)
Triplets extend the contrastive formalism to provide a concurrent ranking surrogate for both negative and positive sample embeddings using triplets sampled from a mini-batch:
with margin , thus following recent implementations in e.g. Roth et al. (2019) or (Wu et al., 2017). Initial works (Schroff et al., 2015) using the triplet loss commonly utilized random or semihard triplet sampling and a GoogLeNet-based architecture. Recent methods typically employ the more effective distance-weighted sampling (Wu et al., 2017) and more powerful networks (Roth et al., 2019; Sanakoyeu et al., 2019). For completeness, we compare the triplet-loss performance combinde with random, semihard and distance-weighted sampling schemes introduced below.
Generalized Lifted Structure (Hermans et al., 2017)
The Generalized Lifted Structure loss extends the standard lifted structure loss (Oh Song et al., 2016) to include all available anchor-positive and anchor-negative distance pairs within a mini-batch , instead of utilizing only a single anchor-positive combination:
with mini-batch samples of a class grouped into , and sets of contained in . denotes the non-normalized version of . The margin serves the standard purpose of avoiding over-distancing already correct image pairs. To account for increasing values, regularizes the embeddings.
N-Pair (Sohn, 2016)
N-Pair or N-Tuple losses extend the triplet formalism to incorporate all negatives in the mini-batch by
with embedding regularization , as (Sohn, 2016) noted a slow convergence for normalized embeddings.
Angular (Wang et al., 2017)
By introducing an angle-based penalty, the angular loss effectively introduces scale invariance and higher-order geometric constraints that are not explicitly introduced in normal contrastive losses:
with angular margin , which, as proposed in the original paper, is set to . is the trade-off between standard ranking losses and the angular constraint.
Arcface (Deng et al., 2018)
Arcface transforms the standard softmax formulation typically used in classification problem to retrieval-based problems by enforcing an angular margin between the embeddings and an approximate center for each class, resulting in
Further, this training objective also introduces the additive angular margin penalty for increased inter-class discrepancy, while the scaling denotes the radius of the effective utilized hypersphere . The class centers are optimized with learning rate .
Histogram (Ustinova and Lempitsky, 2016)
In contrast to many sample-based ranking objective functions, Histogram Loss learns to minimize the probability of a positive sample pair having a higher similarity score than a negative pair. Given a mini-batch , the sets of positive similarities and negative similarities , one optimises
resulting in soft, differentiable histogram assignments. The final objective then penalizes strong overlap between the probability of positive pairs having higher distance (i.e. its cumulative distribution to point ) than respective negative pairs. Such a histogram loss introduces a single hyperparameter, namely the degree of histogram discretisation , which we set to for our study. In general, our implementation borrows from the original code base used in (Ustinova and Lempitsky, 2016).
Margin (Wu et al., 2017)
Margin loss extends the standard triplet loss by introducing a dynamic, learnable boundary between positive and negative pairs. This transfers the common triplet ranking problem to a relative ordering of pairs :
The learning rate of the boundary is set to , with initial value either or and triplet margin . For our implementation, we utilise the distance-weighted triplet sampling method highlighted below.
MultiSimilarity (Wang et al., 2019b)
Unlike contrastive and triplet based ranking methods, the MultiSimilarity loss concurrently evaluates similarities between anchor and negative, anchor and positive, as well as positive-positive and negative-negative pairs in relation to an anchor:
where and denote the set of positive and negative samples for a sample
, with cosine similarity
for two normalized vectors. For our hyperparameters, we use , , and .
ProxyNCA (Movshovitz-Attias et al., 2017)
The sampling complexity of tuples heavily affects the training convergence. ProxyNCA introduces a remedy by introducing class proxies, which act as approximations to entire classes. This way only an anchor is sampled and compared against the respective positive and negative class proxies. Utilizing one proxy per class , ProxyNCA is then defined as
Quadruplet (Chen et al., 2017)
The quadruplet loss is an extension to the triplet loss, which introduces higher level ordering constraint on sample embeddings. By using an anchor, a positive and two exclusive negatives, the quadruplet criterion is defined as:
with margin parameters and . We utilize distance-weighted sampling to propose the first negative sample , which we found to work better than the quadruplet sampling scheme originally proposed in the paper.
Snr (Yuan et al., 2019)
The Signal-to-Noise-Ratio loss (SNR) introduces a novel distance metric based on the ratio between anchor embedding variance and variance of noise, which is simply defined as the difference between anchor and compared embedding. This optimises the embedding space directly for informativeness. The complete loss can then be written as
with margin parameter and regularization to ensure zero-mean distributions. Note that .
SoftTriple (Qian et al., 2019)
Similar to ProxyNCA, the SoftTriple objective function utilizes learnable data proxies to tackle the sampling problem. However, instead of class-discriminative proxies, a set of normalized intra-class proxies per class are learned using the NCA-based similarity measure of a sample to all proxies of a class . Denoting the set of available classes as and the total set of proxies as , we get
The second term denotes a regularization on the learned proxies to ensure sparseness in the class set of proxies. For our tests, we utilised the following hyperparameter values (borrowing from the official implementation in (Qian et al., 2019)): , , , and the number of proxies per class . The proxy learning rate is set to .
Normalized Softmax (Zhai and Wu, 2018)
Similar to other classification-based losses in DML that are based on re-formulations of the standard softmax function (such as ), the normalized softmax loss is optimized by comparing input embeddings to class proxies per class :
for gradient boosting and class proxy learning rate set to.
a.2 Tuple Mining
Basic contrastive, triplet or higher order ranking losses commonly need to mine their training tuples from the available mini-batch. In our study, we measure the influence of tuple sampling on the standard triplet loss, while utilising Distance-Weighted Mining for all ranking-based objective functions except N-Pair based methods.
Random Tuple Mining (Hu et al., 2014)
The trivial way involves the random sampling of tuples. Simply put, per sample we select a respective positive or negative sample .
Semihard Triplet Mining (Schroff et al., 2015)
The potential number of triplets scales cubic in training set size. During learning, more and more of those triplets are correctly ordered and effectively provide no training signal (Schroff et al., 2015), thus impairing the remaining training process. To alleviate this, negative samples are carefully selected based on the anchor-positive sample distance (which are sampled at random). Given an anchor embedding and its positive , the negative is sampled randomly from the set
This way, only negatives are considered which are reasonably hard to separate from an anchor. Moreover, this mining strategy avoids the sampling of overlay hard negatives, which often correspond to data noise and potentially lead to model collapses and bad local minima (Schroff et al., 2015).
Distance-Weighted Tuple Mining (Wu et al., 2017)
In DML, the embedding spaces are typically normalized to a -dimensional (unit) hypersphere for regularisation purposes (Wu et al., 2017). The analytical distribution of pairwise distances on a hypersphere follows
for arbitrary embedding pairs . In order to sample negatives from the whole range of possible distances to an anchor, Wu et al. (2017) propose to sample negatives based on a distance distribution inverse to , i.e.
We set and limit the distances to .
a.3 Evaluation Metrics
In this section, we examine the evaluation metrics to measure the performance of the studied models on a the testset .
Recall@k (Jegou et al., 2011)
be the set of the first nearest neighbours of a sample , then we measure Recall@k as
which measures the average number of cases in which for a given query there is at least one sample among its top nearest neighbours with the same class, i.e. .
Normalized Mutual Information (NMI) (Manning et al., 2010)
To measure the clustering quality using NMI, we embed all samples to obtain and perform a clustering (e.g. -Means (Lloyd, 1982)). Following, we assign all samples a cluster label indicating the closest cluster center and define with and being the number of classes and clusters. Similarly for the true labels we define with . The normalized mutual information is then computed as
with mutual Information between cluster and labels, and entropy on the clusters and labels respectively.
F1-Score (Sohn, 2016)
Mean Average Precision measured on Recall (mAP):
The mAP-score measured on recall follows the same definition as standard mAP. In our case, the mAP is equivalent to the mean over the class-wise average precision@ with being the number of samples with label . With defined as in eq. 23, this gives
Appendix B Correlation between performance and spectral decay
Similar to Fig. 5 (rightmost) in the main paper, we now provide a more detailed illustration in Fig. 9 comparing the performance of the training objectives and their corresponding spectral decay . For ranking losses, we further include the results using -regularization while training, which further shows that in each case a gain in performance is related to a decrease of . Especially the contrastive loss (Hadsell et al., 2006) greatly profits from our proposed regularization, as also indicated by the analysis of the singular value spectra (cf. Fig. 8 of main paper). Its large gains, more then on the CARS196 dataset, is well explained by comparison of its training objective with those of triplet-based formulations. The latter optimizes over relative positive ()) and negative distances () up to a fixed margin , which counteracts a compression of the embedding space to a certain extend. On the other hand, the constrastive loss, while controlling only the negative distances by , is able to perform an unconstrained contraction of entire classes, which facilitates overly compressed embedding spaces .
Appendix C Analysis of per-class singular value spectra
In Sec. 5 of our main paper we analyze generalization in DML by considering the decay of the singular value spectrum over all embedded samples . Thus, we analyze the general compression of the entire embedding space as unseen test classes can be projected anywhere in , in contrast to Verma et al. (2018) which conduct a class-conditioned analysis for i.i.d. classification problems. In order to show that the effect of -regularization (as shown in Fig. 8 in main paper) is also reflected in the class-conditioned singular value spectrum, we perform SVD on and subsequently average over all classes . Fig. 10 compares the sorted, first singular values for both, models trained with and without -regularization. We clearly see that the regularization decreases the average decay of singular values similar to the total singular value spectra shown in the main paper.
Appendix D Comparison to state-of-the-art approaches on SOP dataset
|DVML(Lin et al., 2018)||GoogLeNet||512||70.2||85.2||93.8||90.8|
|MIC(Roth et al., 2019)||ResNet50||128||77.2||89.4||95.6||90.0|
|D&C(Sanakoyeu et al., 2019)||ResNet50||128||75.9||88.4||94.9||90.2|
|Rank(Wang et al., 2019a)||Inception-BN||1536||79.8||91.3||96.3||90.4|
|ABE(Kim et al., 2018)||GoogLeNet||512||76.3||88.4||94.8||-|
|Margin (ours)(Wu et al., 2017)||ResNet50||128||78.3||-||-||90.8|
In this section we provide a detailed comparison between current state-of-the-art DML approaches and our strongest baseline model, margin loss (D, ) (Wu et al., 2017), on the SOP dataset in Tab. 6. The results for these approaches are taken from their public manuscripts. We observe that our baseline model outperforms each of the models using varying architectures, but especially other ResNet50-based implementations. While R50 proves to be a stronger base network (cf. Fig. 2 of main paper) than GoogLeNet based model, improvements over MIC and D&C using the same backbone by at least and methods based on the similarly strong Inception-BN showcase the relevance of a well-defined baseline. Additionally, even though Rank and ABE employ considerable more powerful network ensembles, our carefully motivated baseline exhibits competitive performance.
Appendix E 2D Toy Examples
For our toy examples, we use a fully-connected network with two 30 neuron layers. Both input and embedding dimension are 2D, while the latter is normalized onto the unit circle. Each of the four training and test lines contain 15 samples taken from either the diagonal or vertical/horizontal line segments, respectively. We train the networks both with and without regularization foriterations, a batchsize of and learning rate of using a standard contrastive loss (eq. 1) with margin . For regularisation, we set . Similar to Fig.6 in the main paper, Fig. 11 shows another 2D toy example based on vertical lines which again demonstrates the effect of compression and of our proposed -regularization. The example consists of four training lines that are separable only by their -coordinate and a test set of lines which are separable by their -coordinate. As we observe, the test samples are collapsed onto a single point in the non-regularized embedding space, thus can not be distinguished. In contrast, the regularized representation allows us to separate the test classes and, further, exhibits a decreased decay in the singular value spectrum.
Appendix F Influence of Manifold Mixup on DML
Now, we examine the effect of applying the regularization proposed in ManifoldMixup (Verma et al., 2018) on the DML transfer learning setting. As ManifoldMixup has been proposed to increase the compression of a learned representation in the context of standard supervised classification, it is expected to decrease the performance of DML models. For that, we train three different DML models on the CUB200-2011 dataset: (1) Normalized Softmax, (2) Triplet with Distance Sampling and (3) Margin loss with and Distance Sampling. For (1), the implementation directly follows the standard implementation noted in Verma et al. (2018). For the ranking-based training objectives, we perform mixup in our ResNet50 and generate the mixed class labels, which consequently have either one (if image from the same class are mixed) or two entries (if images from different classes are mixed). Per (mixed) anchor embedding, this gives rise to up to two possible sets of triplets, for which we compute the loss and weigh it by the respective mixup coefficient :