# Evaluating Contrastive Models for Instance-based Image Retrieval

Tarun Krishna, Kevin McGuinness and Noel O’Connor<EMAIL_ADDRESS>SFI Insight Centre for Data Analytics, Dublin City University, Ireland

(2021)

###### Abstract.

In this work, we evaluate contrastive models for the task of image retrieval.
We hypothesise that models that are learned to encode semantic similarity
among instances via discriminative learning should perform well on the task of
image retrieval, where relevancy is defined in terms of instances of the same
object. Through our extensive evaluation, we find that representations from
models trained using contrastive methods perform on-par with (and outperforms)
a pre-trained supervised baseline trained on the ImageNet labels in retrieval
tasks under various configurations. This is remarkable given that the
contrastive models require no explicit supervision. Thus, we conclude that
these models can be used to bootstrap base models to build more robust image
retrieval engines.

Deep learning, Contrastive learning, Self-supervised learning

††journalyear: 2021††copyright: rightsretained††conference: Proceedings of the
2021 International Conference on Multimedia Retrieval; August 21–24, 2021;
Taipei, Taiwan††booktitle: Proceedings of the 2021 International Conference on
Multimedia Retrieval (ICMR ’21), August 21–24, 2021, Taipei, Taiwan††doi:
10.1145/3460426.3463585††isbn: 978-1-4503-8463-6/21/08††submissionid:
108††copyright: rightsretained††ccs: Computing methodologies Visual content-
based indexing and retrieval

##  1\. Introduction

Large scale image retrieval, where the task is to search a large image
collection for the most relevant image/content for a given query, is a
fundamental task in computer vision. Since their inception, convolutional
neural networks (ConvNets) (Krizhevsky et al., 2012; Simonyan and Zisserman,
2014) have become the prominent approach for extracting descriptors for image
retrieval. These descriptors perform very well in capturing the global
semantics of an image and this has led to state-of-the-art results on many
benchmark computer vision tasks (Ren et al., 2016; Chen et al., 2014; He et
al., 2016).

The activations in the intermediate layers in ConvNets can be used as a
descriptor for an image. These descriptors are often followed by some encoding
methods for a compact representation. These encoding techniques range from
traditional approaches of VLAD (Jégou et al., 2010), BoW (Mohedano et al.,
2016), and Fisher vectors (Perronnin and Dance, 2007), to simple pooling
methods like Maximum Activation of Convolution (MAC) (Azizpour et al., 2015a),
Sum Pooling of Convolution (SPoC) (Babenko and Lempitsky, 2015), Regional-MAC
(Tolias et al., 2015), etc. The drawback of these methods is that these (off-
the-shelf) network are trained to reduce inter-class variance through
supervision on ImageNet classes, this might affect the performance of instance
retrieval (i.e. retrieving images that represent the same object or scene as
in a query), which is a more fine-grained task.

This drawback has been addressed in the literature by fine-tuning (Gordo et
al., 2016; Radenović et al., 2016; Gordo et al., 2017; Revaud et al., 2019).
We hypothesise that a simpler approach could be to retain a traditional off-
the-shelf regime but instead use models that are trained based on instance-
wise supervision using similarity-based learning. To this end, we investigate
contrastive learning based methods, i.e. trained in an unsupervised fashion
using contrastive loss (Wu et al., 2018; Gutmann and Hyvärinen, 2010). This
learning regime relies on learning a meaningful embedding that captures
inherent similarity between instances using discriminative approaches (Wu et
al., 2018). This work investigates the effectiveness of contrastive methods
that capture this very idea of instance similarity. To summarize our
contributions:

  * •

we extensively evaluate contrastive methods as a fixed feature extractor
across different benchmark;

  * •

we provide experimental evidence showing that these models (trained without
any explicit supervision) perform on par with a pre-trained supervised
baseline (Table 1 and 2);

  * •

we further investigate the role of the dimensionality of the feature
embeddings for this task (Table 3).

##  2\. Related Work

Conventional image retrieval methods (Sivic and Zisserman, 2003; Nister and
Stewenius, 2006) relied on bag-of-words models that exploit local invariant
features such as SIFT (Lowe, 2004) and large visual vocabularies (e.g.
(Philbin et al., 2007)). To aggregate local patches and build a global
summary, encoding methods such as Fisher vectors (Perronnin and Dance, 2007)
or VLAD (Jégou et al., 2010), have also been proposed (Perronnin et al., 2010;
Gordoa et al., 2012; Radenović et al., 2015).

Since the introduction of Deep-ConvNets, (Krizhevsky et al., 2012; Simonyan
and Zisserman, 2014; Donahue et al., 2014; He et al., 2016) there has been a
paradigm shift to exploit deep features instead of hand-crafted ones.
Intermediate layers in convolutional nets can be used as global or local
descriptors. As a result, so-called off-the-shelf (Azizpour et al., 2015b;
Razavian et al., 2014) features can be used for retrieval. Based on this
authors in (Babenko and Lempitsky, 2015) used sum pooling with a centre prior
for aggregating features across spatial dimensions. Other conventional
encoding techniques like VLAD (Gong et al., 2014) or Fisher kernels (Perronnin
and Larlus, 2015) have also been used in combination with these local feature
maps. For example, in (Mohedano et al., 2016) proposed BoW-encodings of
convolutional features for instance retrieval, whilst (Tolias et al., 2015)
proposed R-MAC using max activations over a grid of windows of different
scales to obtain compact representations.

Most of the off-the-shelf features are trained on ImageNet (Russakovsky et
al., 2015) to reduce inter-class variance. However, this may degrade the
performance of an instance-based retrieval system. One way to address this is
to finetune the model as shown in (Babenko et al., 2014; Gordo et al., 2016;
Radenović et al., 2016; Gordo et al., 2017; Revaud et al., 2019). In the
context of image retrieval, most of the finetuning has been performed on
Landmark datasets (Babenko et al., 2014), which further requires cleaning of
non-related images and potentially expensive post-processing.

Another way is to exploit methods that are trained to reduce intra-class
variance, as is the case in contrastive learning. Unlike in supervised
learning, these approaches learn to discriminate among individual instances
without any concept of categories. This work (Wu et al., 2018) discusses this
notion of instance discrimination. Building on this, a simple formulation is
presented in SimCLR (Chen et al., 2020b). The intuition behind these
approaches is to maximize the agreement among augmented views of the same
instance using Noise Contrastive Estimation (NCE) (Gutmann and Hyvärinen,
2010). Minimising NCE is equivalent to maximizing mutual information (MI) as
was formally shown in CPC (Oord et al., 2018) as InfoNCE. DIM (Hjelm et al.,
2018) and AMDIM (Bachman et al., 2019) further extend the idea of InfoNCE
across multiple views and scales. One of the downsides of these approaches is
that they require large batch sizes on large GPU clusters. To address this
drawback, the authors in (He et al., 2020; Chen et al., 2020a) introduced
MoCo, which uses an online and momentum updated offline network that views
contrastive learning as a dictionary lookup task. Intuitively, the ability to
discriminate among individual instances inherently encoded through the
learning makes contrastive learning a good candidate for the task of instance
retrieval.

##  3\. Contrastive Models

Contrastive learning refers to learning by comparison. This comparison is
performed between positive pairs of “similar” and negative pairs of
“dissimilar” inputs, which is achieved via a contrastive loss (Chen et al.,
2020b; Le-Khac et al., 2020) derived from Noise Contrastive Estimation (NCE)
(Gutmann and Hyvärinen, 2010).

This work targets representative contrastive models for investigation, which
are diverse in terms of the way they fuse information. Intuitively, this
should lead to each model spanning different feature spaces from an image
understanding perspective. The following briefly describes the models
considered in this study.

AMDIM: Augmented Multiscale DIM (Bachman et al., 2019) extends the Deep Info-
Max (DIM) framework (Hjelm et al., 2018) by learning features extracted from
multiple views of a shared context. Local DIM maximizes MI across features
extracted independently from augmented views of each input along with features
across multiple scales with more powerful encoders.

MoCo: Momentum Contrast Representation Learning (He et al., 2020; Chen et al.,
2020a) alleviates the need for storing offline representations of the entire
dataset in memory (Wu et al., 2018) through the use of a dynamic memory queue.
The samples in the dictionary are progressively replaced. This approach looks
at contrastive learning as a way of building a discrete dictionary (queue) of
inputs (data samples) for a high-dimensional space. We consider MoCov1 (He et
al., 2020) and MoCov2 (Chen et al., 2020a).

SimCLR: Simple framework for Contrastive Learning (Chen et al., 2020b) is a
simplified framework for contrastive learning compared to the previous ones.
Here stochastic data augmentations are applied on an input $\mathbf{x}$ to get
two views of $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$. Sequentially these
augmented inputs are passed through a base encoder $\mathbf{f(.)}$ followed by
projection head, a small network that projects representations from
$\mathbf{f(.)}$ to space where a contrastive loss is applied. We investigate
SimClr1x,2x,4x.

Table 1. Comparing mAP (%) score across different models. Bold (red) best performing ensemble if it exists. |  | G | G | G | G | G | G | G | G
---|---|---|---|---|---|---|---|---|---
|  |  | +AQE | +DFS | +DBA | +AQE | +DBA | +DBA | +DBA
|  |  |  |  |  | +DFS | +AQE | +DFS | +AQE
Dataset | Method |  |  |  |  |  |  |  | +DFS
| Baseline | 55.12 | 67.85 | 74.29 | 62.63 | 82.64 | 70.6 | 77.95 | 80.4
| SimClr1x | 51.47 | 62.96 | 71.42 | 61.00 | 76.40 | 66.32 | 72.98 | 73.38
| SimClr2x | 58.59 | 70.49 | 79.11 | 67.34 | 82.34 | 72.62 | 79.02 | 79.96
| SimClr4x | 59.40 | 69.80 | 80.07 | 67.52 | 82.00 | 71.03 | 78.67 | 76.98
| MoCov1 | 56.76 | 66.89 | 74.78 | 64.12 | 76.44 | 69.77 | 73.39 | 73.95
Oxford 5k | MoCov2 | 58.36 | 67.49 | 75.26 | 65.09 | 78.07 | 69.86 | 72.52 | 72.86
| AmDim | 36.95 | 43.44 | 48.13 | 39.97 | 52.99 | 44.69 | 46.3 | 47.74
| SimClr2x+SimClr4x | 61.89 | 72.64 | 81.66 | 69.27 | 86.02 | 74.43 | 80.45 | 81.99
| SimClr2x+AmDim | 57.06 | 67.37 | 75.63 | 64.69 | 77.77 | 69.29 | 72.84 | 74.51
| SimClr2x+MoCov2 | 61.33 | 71.58 | 78.97 | 68.47 | 82.33 | 72.94 | 78.36 | 79.03
| SimClr4x+MoCov2 | 61.87 | 71.97 | 80.86 | 68.20 | 84.02 | 72.94 | 78.56 | 79.57
| SimClr4x+AmDim | 58.13 | 68.76 | 77.66 | 64.81 | 80.05 | 69.78 | 75.93 | 77.32
| GeM (AP) | 66.90 | 78.57 | 90.36 | 75.70 | 94.34 | 81.92 | 89.65 | 91.12
| Baseline | 41.46 | 61.76 | 83.74 | 64.71 | 90.17 | 75.56 | 90.17 | 90.62
| SimClr1x | 42.28 | 61.15 | 85.16 | 64.36 | 90.17 | 74.45 | 89.37 | 90.49
| SimClr2x | 44.48 | 59.65 | 89.39 | 63.28 | 91.61 | 71.91 | 90.82 | 91.48
| SimClr4x | 44.96 | 56.3 | 88.98 | 60.40 | 90.18 | 68.46 | 88.18 | 88.74
| MoCov1 | 43.22 | 60.34 | 83.13 | 63.00 | 86.9 | 72.11 | 87.91 | 88.73
Paris 6k | MoCov2 | 49.72 | 64.46 | 88.02 | 67.06 | 89.86 | 75.01 | 88.65 | 89.01
| AmDim | 34.87 | 49.2 | 59.55 | 47.72 | 63.7 | 56.06 | 64.18 | 65.08
| SimClr2x+SimClr4x | 45.27 | 56.84 | 89.97 | 59.88 | 91.60 | 68.07 | 89.19 | 90.28
| SimClr2x+AmDim | 44.45 | 60.06 | 86.71 | 63.02 | 89.06 | 71.12 | 88.46 | 88.76
| SimClr2x+MoCov2 | 47.93 | 61.46 | 90.92 | 64.96 | 92.47 | 72.37 | 90.42 | 91.08
| SimClr4x+MoCov2 | 47.44 | 58.44 | 90.40 | 61.88 | 91.97 | 69.48 | 89.00 | 89.93
| SimClr4x+AmDim | 44.74 | 57.05 | 87.59 | 60.66 | 88.97 | 68.60 | 87.06 | 87.45
| GeM (AP) | 49.87 | 63.46 | 93.99 | 67.99 | 95.56 | 76.78 | 90.15 | 94.07
| Baseline | 36.64 | 64.42 | 71.7 | 63.96 | 76.43 | 74.32 | 16.41 | 76.13
| SimClr1x | 27.51 | 47.85 | 55.38 | 47.34 | 60.61 | 57.73 | 21.00 | 61.15
| SimClr2x | 36.68 | 57.04 | 66.47 | 56.89 | 69.48 | 65.27 | 24.93 | 67.18
| SimClr4x | 44.76 | 63.73 | 73.51 | 63.26 | 75.02 | 70.17 | 28.79 | 70.54
| MoCov1 | 33.44 | 54.88 | 66.31 | 54.84 | 70.1 | 64.65 | 29.25 | 69.8
INSTRE | MoCov2 | 33.36 | 51.79 | 60.66 | 50.47 | 62.56 | 58.83 | 22.38 | 59.77
| AmDim | 24.34 | 37.92 | 45.47 | 36.69 | 47.14 | 42.74 | 29.47 | 47.71
| SimClr2x+SimClr4x | 45.58 | 63.94 | 73.54 | 63.54 | 74.81 | 70.54 | 28.73 | 70.75
| SimClr2x+AmDim | 39.37 | 59.02 | 68.85 | 59.09 | 71.14 | 66.68 | 26.19 | 67.93
| SimClr2x+MoCov2 | 41.99 | 60.99 | 70.42 | 60.64 | 72.24 | 68.41 | 26.09 | 68.69
| SimClr4x+MoCov2 | 47.07 | 65.23 | 74.64 | 64.93 | 76.12 | 71.42 | 27.09 | 71.26
| SimClr4x+AmDim | 45.83 | 63.88 | 74.25 | 63.84 | 75.28 | 70.83 | 30.07 | 71.60
| GeM (AP) | 20.78 | 31.37 | 34.85 | 30.29 | 38.06 | 35.93 | 18.48 | 37.54
Table 2. Comparison on global ranking across different model. Bold (red) best performing ensemble if it exists. |  | Easy | Medium | Hard
---|---|---|---|---
Dataset | Method | mAP | mp@5 | mp@10 | mAP | mp@5 | mp@10 | mAP | mp@5 | mp@10
| Baseline | 45.65 | 67.55 | 61.81 | 32.89 | 64.00 | 58.57 | 12.10 | 23.71 | 19.43
| SimClr1x | 47.17 | 69.78 | 63.90 | 31.98 | 65.24 | 57.52 | 9.52 | 20.57 | 15.86
| SimClr2x | 54.95 | 76.69 | 69.49 | 38.54 | 73.14 | 65.33 | 14.41 | 31.21 | 22.93
| SimClr4x | 54.65 | 74.80 | 68.37 | 38.57 | 73.71 | 65.05 | 14.05 | 30.07 | 22.79
| MoCov1 | 48.29 | 68.48 | 64.36 | 33.57 | 64.43 | 58.58 | 9.27 | 20.05 | 15.62
rOxford 5k | MoCov2 | 52.69 | 72.33 | 65.86 | 36.49 | 67.24 | 59.24 | 10.72 | 23.38 | 18.67
| AmDim | 21.24 | 38.31 | 31.35 | 17.54 | 36.38 | 32.57 | 4.11 | 5.92 | 6.33
| SimClr2x+SimClr4x | 55.23 | 76.15 | 68.87 | 39.60 | 74.29 | 65.64 | 15.01 | 29.43 | 23.97
| SimClr2x+AmDim | 52.08 | 72.28 | 65.51 | 36.19 | 66.86 | 58.57 | 10.20 | 21.64 | 17.64
| SimClr2x+MoCov2 | 55.11 | 76.47 | 69.35 | 38.46 | 70.52 | 62.57 | 12.51 | 27.12 | 21.83
| SimClr4x+MoCov2 | 53.98 | 72.40 | 67.90 | 38.31 | 70.10 | 64.69 | 13.38 | 28.43 | 23.13
| SimClr4x+AmDim | 49.23 | 70.76 | 64.00 | 35.72 | 67.62 | 61.33 | 11.72 | 23.79 | 18.80
| GeM (AP) | 64.07 | 84.93 | 80.56 | 51.03 | 89.43 | 83.86 | 30.30 | 54.86 | 44.00
| Baseline | 47.20 | 91.14 | 87.00 | 31.63 | 93.71 | 91.71 | 10.44 | 57.14 | 43.43
| SimClr1x | 49.35 | 93.14 | 89.52 | 31.61 | 95.71 | 92.00 | 8.93 | 46.57 | 38.43
| SimClr2x | 54.28 | 94.00 | 91.14 | 34.47 | 96.00 | 92.86 | 10.87 | 62.00 | 49.71
| SimClr4x | 54.97 | 93.71 | 91.71 | 35.39 | 96.29 | 94.86 | 12.34 | 66.57 | 56.29
| MoCov1 | 50.47 | 92.29 | 89.71 | 31.48 | 95.14 | 92.57 | 8.13 | 46.57 | 36.29
rParis 6k | MoCov2 | 55.77 | 92.86 | 89.24 | 36.32 | 94.86 | 92.14 | 11.71 | 60.00 | 50.43
| AmDim | 38.62 | 80.29 | 73.71 | 25.92 | 82.57 | 75.71 | 6.36 | 25.14 | 19.57
| SimClr2x+SimClr4x | 55.54 | 94.00 | 91.86 | 35.59 | 96.00 | 94.00 | 12.37 | 66.29 | 56.00
| SimClr2x+AmDim | 53.71 | 93.71 | 90.43 | 34.57 | 95.14 | 93.14 | 10.46 | 55.71 | 46.00
| SimClr2x+MoCov2 | 57.00 | 93.43 | 91.00 | 36.66 | 95.14 | 93.29 | 11.88 | 64.00 | 52.43
| SimClr4x+MoCov2 | 56.86 | 94.29 | 91.71 | 36.61 | 96.29 | 94.86 | 12.68 | 69.43 | 55.57
| SimClr4x+AmDim | 54.65 | 92.86 | 90.57 | 35.06 | 95.71 | 93.43 | 11.41 | 61.14 | 48.71
| GeM (AP) | 54.90 | 94.38 | 91.95 | 37.36 | 98.86 | 97.00 | 14.65 | 76.86 | 63.71
Table 3. Comparison of mAP (%) across different PCA dimension and the true
dimension.

|  | PCA-Whitening | True
---|---|---|---
Dataset | Method | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 | dim.
| Baseline | 48.31 | 54.61 | 56.68 | 57.52 | 55.12 | 49.44 | 37.23 | 58.47
| SimClr1x | 34.96 | 44.96 | 54.17 | 54.79 | 51.47 | 44.16 | 34.94 | 50.63
| SimClr2x | 36.79 | 47.96 | 60.32 | 61.68 | 58.59 | 49.82 | 24.62 | 53.72
Oxford 5k | SimClr4x | 35.32 | 45.86 | 57.77 | 61.70 | 59.40 | 53.83 | 44.71 | 41.94
| MoCov1 | 29.90 | 41.95 | 54.62 | 57.38 | 56.76 | 50.20 | 39.37 | 38.73
| MoCov2 | 39.92 | 47.88 | 57.92 | 60.43 | 58.36 | 52.14 | 40.05 | 51.76
| AmDim | 11.52 | 15.30 | 21.71 | 28.71 | 36.95 | 37.58 | 30.73 | 16.10
| Baseline | 72.22 | 71.83 | 63.65 | 53.40 | 41.46 | 28.99 | 17.89 | 68.36
| SimClr1x | 69.90 | 73.29 | 66.62 | 54.58 | 42.28 | 30.7 | 20.06 | 66.60
| SimClr2x | 75.21 | 77.90 | 69.25 | 57.83 | 44.48 | 33.05 | 23.27 | 72.20
Paris 6k | SimClr4x | 77.16 | 78.04 | 69.19 | 57.29 | 44.96 | 35.02 | 25.79 | 72.89
| MoCov1 | 56.84 | 63.63 | 61.87 | 54.07 | 43.22 | 32.57 | 21.63 | 53.66
| MoCov2 | 70.08 | 75.11 | 71.24 | 61.66 | 49.72 | 35.73 | 22.98 | 69.99
| AmDim | 21.49 | 33.96 | 41.19 | 40.85 | 34.87 | 26.82 | 17.85 | 25.50
| Baseline | 26.44 | 34.92 | 38.68 | 38.55 | 36.64 | 29.25 | 20.20 | 33.03
| SimClr1x | 16.47 | 23.27 | 29.01 | 30.35 | 27.51 | 21.85 | 15.97 | 21.85
| SimClr2x | 18.42 | 26.83 | 35.88 | 39.67 | 36.68 | 29.64 | 21.57 | 25.94
INSTRE | SimClr4x | 18.92 | 28.97 | 40.45 | 46.99 | 44.76 | 36.68 | 27.65 | 28.98
| MoCov1 | 20.08 | 27.85 | 33.77 | 36.24 | 33.44 | 26.66 | 18.46 | 23.01
| MoCov2 | 19.00 | 27.38 | 34.34 | 36.22 | 33.36 | 26.86 | 18.79 | 26.22
| AmDim | 10.65 | 15.88 | 20.55 | 24.15 | 24.34 | 20.45 | 14.45 | 10.25

##  4\. Experiments

This section describes our experimental setup for the evaluation.

###  4.1. Setup

We evaluate models on three standard benchmark datasets: Oxford5k (Philbin et
al., 2007), Paris6k (Philbin et al., 2008), and INSTRE (Wang and Jiang, 2015).
Retrieval performance is measured using mean Average Precision (mAP) following
standard procedures for Oxford 5k and Paris 6k benchmarks and for INSTRE
evaluating mAP over 1200 images as described in (Iscen et al., 2017). We
further evaluate the performance on revised rOxford 5k and rParis 6k using the
new evaluation protocol based on easy, medium, and hard ground truth labels
(Radenović et al., 2018). For the revised benchmarks we report both mAP and
mean precision@(10,5) (mp@10, mp@5).

The goal is not to fine-tune the models but instead evaluate them as a fixed
feature extractor to obtain visual descriptors. The base encoder of each of
the models is some flavour of ResNet111apart from simClr2x,4x each uses
ResNet50 as a backbone encoder with varying complexity. To this end, we
consider the output of the last convolutional layer, i.e. just before the
adaptive pooling layer, as our descriptor, which leads to feature maps of size
$\mathbb{R}^{C\times H\times W}$. To obtain compact representations we use
R-MAC ($L=3a$) (Tolias et al., 2015) over spatial dimensions to get a fixed
representation of size $\mathbb{R}^{C}$. We further post-process the vectors
by applying $L_{2}$ normalization, PCA-whitening, and $L_{2}$ normalization
again.

We resize our input images to a fixed resolution of $724\times 724$ giving a
feature spatial dimension of $23\times 23$ except in the case of AMDIM where
the dimensions are $40\times 40$. However, we downsample this to $23\times 23$
to keep uniformity across the evaluation. Also, before running the final
evaluation we first run each of the models in training mode (PyTorch
model.train() just feed-forward) to tune the batch-normalization statistics to
the current dataset and then finally test models in evaluation mode
(model.eval()).

Baseline. For comparing across all the contrastive models we use ResNet50 (He
et al., 2016) trained on ImageNet as a fixed feature extractor as our pre-
trained supervised baseline model. Note. For completeness we also evaluate a
fine-tuned model (Revaud et al., 2019), which uses Generalized Mean Pooling
(instead of R-MAC) trained with Average Precision loss (GeM (AP))
222https://github.com/naver/deep-image-retrieval. The purpose of this is to
provide an indicative upper bound to the evaluation scores.

Ranking. We consider global search (G) in this evaluation. We further
integrate Global search with Average Query Expansion (Chum et al., 2007)
(AQE), DataBase Augmentation (DBA), (Arandjelović and Zisserman, 2012) and
Diffusion (Yang et al., 2019) (DFS). For AQE we consider nearest neighbour
$N=10$, for DBA we consider $N^{{}^{\prime}}=20$ while combining both of these
we consider $N=1$ and $N^{{}^{\prime}}=20$, based on the findings in (Gordo et
al., 2017).

We use a PCA dimension of 512 and evaluate on a global search for R-MAC
representations unless otherwise stated.

###  4.2. Results

Table 1 compares different models along with different expansion
techniques333here and in Table 2 GeM (AP) serves as a upper bound indicator
rather than a benchmark.. For a naive global search on Oxford 5k, the
contrastive approach achieves an mAP (%) of 59.40 while the baseline achieves
55.12. Overall best performance is achieved with AQE and DFS for the baseline
model (82.64) but this mAP score for SimCLR2x (82.34) is in the same range as
the former. We also include results of ensembling contrastive methods444The
R-MAC representations are concatenated and dimensionally reduced via PCA,
which seems to give a further performance boost (in red in Table 1). A similar
inference could be drawn for Paris 6k where the best mAP for global is
achieved by MoCOv2 (49.72) while the overall best (91.61) is achieved with AQE
and DFS for SimClr2x. A further boost can be observed for the ensemble. In the
case of the INSTRE dataset, we see a similar pattern for global search with
mAP 44.76 corresponding to SimClr4x while the overall best is achieved with
AQE and DFS for the baseline (76.43) while the best result achieved for
contrastive models is 75.02 for SimClr4x. This clearly indicates that
contrastive methods trained to reduce intra class variance capture the notion
of instance similarity which is being reflected in this evaluation. Also
expansion techniques further boosts the performance over global search.

To further consolidate our findings, we also conducted an evaluation on the
revised rOxford 5k and rParis 6k datasets as depicted in Table 2. On rOxford
5k SimClr2x gives the best performance on all labels. mP@10 is almost 70% for
the easy category, with the drop in performance for hard label. Similarly, on
rParis 6k, SimClr4x gives the best performance with mP@10 over 90 for easy and
medium, but again this drops off for hard labels. As in the Table 1, here
ensembling further boosts performance. Again, contrastive models surpass the
baseline pre-trained supervised model.

Effect of descriptor dimension on performance. Table 3 reports our findings
for global search555Comparison is across the horizontal dimension (columns).
Interestingly, true dimensions ($L_{2}$ normalized R-MAC representations)
appear to perform worse for almost all the models. The best dimension varies
across the dataset but it is never the true dimension. This could be
attributed to dimensions with small principal components being noisy and
redundant and adversely affecting performance.

##  5\. Conclusion

This work evaluated contrastive models for the task of instance-based image
retrieval. Our evaluation found that these methods are on par with those
trained on class labels. In fact, in many settings in Table 1, 2 contrastive
approaches surpass the supervised model. The quantitative evaluation shows
that these contrastive methods can easily surpass supervised models without
any explicit supervision.

##  6\. Acknowledgment

This work has emanated from research supported by Science Foundation Ireland
(SFI) under Grant Number SFI/12/RC/2289_P2, co-funded by the European Regional
Development Fund and Xperi FotoNation.

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∎  e1e-mail<EMAIL_ADDRESS>11institutetext: Laboratoire de Physique des 2
Infinis Irène Joliot-Curie, Université Paris-Saclay, CNRS/IN2P3, France

# Deeply virtual Compton scattering on the neutron with positron beam

S. Niccolaie1,addr1  P. Chatagnonaddr1  M. Hoballahaddr1  D. Marchandaddr1  C.
Munoz Camachoaddr1  E. Voutieraddr1

(Received: date / Revised version: date)

###### Abstract

Measuring DVCS on a neutron target is a necessary step to deepen our
understanding of the structure of the nucleon in terms of Generalized Parton
Distributions (GPDs). The combination of neutron and proton targets allows to
perform a flavor decomposition of the GPDs. Moreover, neutron-DVCS plays a
complementary role to DVCS on a transversely polarized proton target in the
determination of the GPD $E$, the least known and constrained GPD that enters
Ji’s angular momentum sum rule. A measurement of the beam-charge asymmetry
(BCA) in the $e^{\pm}d\to e^{\pm}n\gamma(p)$ reaction can significantly impact
the experimental determination of the real parts of the $E$ and, to a lesser
extent, $\widetilde{H}$ GPDs.

###### pacs:

PACS-keydiscribing text of that key PACS-keydiscribing text of that key

††journal: Eur. Phys. J. A

##  1 Introduction

It is well known that the fundamental particles which form hadronic matter are
the quarks and the gluons, whose interactions are described by the QCD
Lagrangian. However, exact QCD-based calculations cannot yet be performed to
explain all the properties of hadrons in terms of their constituents.
Phenomenological functions need to be used to connect experimental observables
with the inner dynamics of the constituents of the nucleon, the partons.
Typical examples of such functions include form factors, parton densities, and
distribution amplitudes. The GPDs are nowadays the object of intense research
in the perspective of unraveling nucleon structure. They describe the
correlations between the longitudinal momentum and transverse spatial position
of the partons inside the nucleon, they give access to the contribution of the
orbital momentum of the quarks to the nucleon, and they are sensitive to the
correlated $q$-$\bar{q}$ components. The original articles and general reviews
on GPDs and details of the formalism can be found in Refs. muller -revrady .

Figure 1: The handbag diagram for the DVCS process on the nucleon $eN\to
e^{\prime}N^{\prime}\gamma^{\prime}$. Here $x+\xi$ and $x-\xi$ are the
longitudinal momentum fractions of the struck quark before and after
scattering, respectively, and $t=(N-N^{\prime})^{2}$ is the squared four-
momentum transfer between the initial and final nucleons. $\xi\simeq
x_{B}/(2-x_{B})$ is proportional to the Bjorken scaling variable
$x_{B}=Q^{2}/2M\nu$, where $M$ is the nucleon mass and $\nu$ the energy
transfer to the quark.

The GPDs of the nucleon are the structure functions which are accessed in the
measurement of the exclusive leptoproduction of a photon (DVCS) or of a meson
on the nucleon, at sufficiently large virtual-photon virtuality ($Q^{2}$) for
the reaction to happen at the quark level. Figure 1 illustrates the leading
process for DVCS, also called “handbag diagram”. At leading-order QCD and at
leading twist, considering only quark-helicity conserving quantities and the
quark sector, the process is described by four GPDs:
$H^{q},\widetilde{H^{q}},E^{q},\widetilde{E^{q}}$, one for each quark flavor
$q$, that account for the possible combinations of relative orientations of
the nucleon spin and quark helicities between the initial and final states.
$H^{q}$ and $E^{q}$ do not depend on the quark helicity and are therefore
called unpolarized GPDs while $\widetilde{H^{q}}$ and $\widetilde{E^{q}}$
depend on the quark helicity and are called polarized GPDs. $H^{q}$ and
$\widetilde{H^{q}}$ conserve the spin of the nucleon, whereas $E^{q}$ and
$\widetilde{E^{q}}$ correspond to a nucleon-spin flip.
The GPDs depend upon three variables, $x$, $\xi$ and $t$: $x+\xi$ and $x-\xi$
are the longitudinal momentum fractions of the struck quark before and after
scattering, respectively, and $t$ is the squared four-momentum transfer
between the initial and final nucleon (see caption of Fig. 1 for definitions).
The transverse component of $t$ is the Fourier-conjugate of the transverse
position of the struck parton in the nucleon. Among these three variables,
only $\xi$ and $t$ are experimentally accessible with DVCS.
The DVCS amplitude is proportional to combinations of integrals over $x$ of
the form

$\int_{-1}^{1}dx\,F(\mp
x,\xi,t)\left[\frac{1}{x-\xi+i\epsilon}\pm\frac{1}{x+\xi-i\epsilon}\right],$
(1)

where $F$ represents one of the four GPDs. The top combination of the plus and
minus signs applies to unpolarized GPDs ($H^{q},E^{q}$), and the bottom
combination of signs applies to the polarized GPDs
($\widetilde{H^{q}},\widetilde{E^{q}}$). Each of these 4 integrals, or Compton
Form Factors (CFFs), can be decomposed into its real and imaginary parts, as
following:

$\displaystyle\Re{\rm e}\left[{\cal F}(\xi,t)\right]$ $\displaystyle=$
$\displaystyle{\cal
P}\int_{-1}^{1}dx\left[\frac{F(x,\xi,t)}{x-\xi}\mp\frac{F(x,\xi,t)}{x+\xi}\right]\,\,\,\,\,\,\,\,$
(2) $\displaystyle\Im{\rm m}\left[{\cal F}(\xi,t)\right]$ $\displaystyle=$
$\displaystyle-\pi[F(\xi,\xi,t)\mp F(-\xi,\xi,t)],$ (3)

where ${\cal P}$ is Cauchy’s principal value integral and the sign convention
is the same as in Eq. 1. The information that can be extracted from the
experimental data at a given ($\xi,t$) point depends on the measured
observable. $\Re{\rm e}[{\cal F}]$ is accessed primarily measuring observables
which are sensitive to the real part of the DVCS amplitude, such as double-
spin asymmetries, beam-charge asymmetries or unpolarized cross sections.
$\Im{\rm m}[{\cal F}]$ can be obtained measuring observables which are
sensitive to the imaginary part of the DVCS amplitude, such as single-spin
asymmetries or the difference of polarized cross-sections. However, knowing
the CFFs does not define the GPDs uniquely. A model input is necessary to
deconvolute their $x$-dependence. The DVCS process is accompanied by the
Bethe-Heitler (BH) process, in which the final-state real photon is radiated
by the incoming or scattered electron and not by the nucleon itself. The BH
process, which is not sensitive to GPDs, is experimentally indistinguishable
from DVCS and interferes with it at the amplitude level. However, considering
that the nucleon form factors are well known at small $t$, the BH process is
precisely calculable.

## Neutron GPDs and flavor separation

The importance of neutron targets in the DVCS phenomenology was clearly
established in the pioneering Hall A experiments at 6 GeV, where the
polarized-beam cross section difference off a neutron, from a deuterium
target, was measured malek ; benali . Measuring neutron GPDs in complement to
proton GPDs allows for a quark-flavor separation. For instance, the ${\cal
E}$-CFF of the proton and the neutron can be expressed as

$\displaystyle{\cal E}_{p}(\xi,t)$ $\displaystyle=$
$\displaystyle\frac{4}{9}{\cal E}^{u}(\xi,t)+\frac{1}{9}{\cal E}^{d}(\xi,t)$
(4) $\displaystyle{\cal E}_{n}(\xi,t)$ $\displaystyle=$
$\displaystyle\frac{1}{9}{\cal E}^{u}(\xi,t)+\frac{4}{9}{\cal E}^{d}(\xi,t)$
(5)

(and similarly for ${\cal H}$, ${\widetilde{\cal H}}$ and ${\widetilde{\cal
E}}$). From this it follows that

$\displaystyle{\cal E}^{u}(\xi,t)=\frac{9}{15}\left[4{\cal E}_{p}(\xi,t)-{\cal
E}_{n}(\xi,t)\right]$ (6) $\displaystyle{\cal
E}^{d}(\xi,t)=\frac{9}{15}\left[4{\cal E}_{n}(\xi,t)-{\cal
E}_{p}(\xi,t)\right]\,.$ (7)

An extensive experimental program dedicated to the measurement of the DVCS
reaction on a proton target has recently started at Jefferson Lab, in
particular with CLAS12. Single-spin asymmetries with polarized beam and/or
linearly or transversely polarized proton targets, as well as unpolarized and
polarized cross sections, will be measured with high precision over a vast
kinematic coverage. If a similar program is performed on the neutron, the
flavor separation of the various GPDs will be possible. The beam-spin
asymmetry for nDVCS, particularly sensitive to the GPD $E_{n}$ is currently
being measured at CLAS12, using an experimental technique different from the
initial Hall A measurement and involving the detection of the active neutron
proposal . Additionally, the measurement of single- and double-spin
asymmetries with a longitudinally polarized deuteron target is also foreseen
for the nearby future with CLAS12. Here we focus on the extraction of one more
observable, the beam-charge asymmetry. The next section outlines the benefits
of this observable for the CFFs determination.

## Beam charge asymmetry

Considering unpolarized electron and positron beams, the sensitivity of the
cross-section to the lepton beam charge can be expressed with the beam charge
asymmetry observable

$A_{\rm
C}(\phi)=\frac{d^{4}\sigma^{+}-d^{4}\sigma^{-}}{d^{4}\sigma^{+}+d^{4}\sigma^{-}}=\frac{d^{4}\sigma_{UU}^{\rm
I}}{d^{4}\sigma_{UU}^{\rm BH}+d^{4}\sigma_{UU}^{\rm DVCS}}$ (8)

which isolates the BH-DVCS interference contribution at the numerator and the
DVCS amplitude at the denominator. Following the harmonic decomposition
proposed in Ref. belitski ,

$\displaystyle d^{4}\sigma_{UU}^{\rm I}$ $\displaystyle=$
$\displaystyle\frac{K_{1}}{\mathcal{P}_{1}(\phi)\,\mathcal{P}_{2}(\phi)}\sum_{n=0}^{3}c_{n,\rm{unp}}^{\rm
I}\cos(n\phi)$ (9) $\displaystyle d^{4}\sigma_{UU}^{\rm DVCS}$
$\displaystyle=$
$\displaystyle\frac{K_{2}}{Q^{2}}\sum_{n=0}^{2}c_{n,\rm{unp}}^{\rm
DVCS}\cos(n\phi),$ (10)

where $K_{i}$’s are kinematical factors, and $P_{i}(\phi)$’s are the BH
propagators. Because of the $1/Q^{2}$ kinematical suppression of the DVCS
amplitude, the dominant contribution to the denominator originates from the BH
amplitude. This approximation depends on the kinematics and, in the most
general case, the DVCS contribution in the denominator complicates the
extraction of CFFs. At leading twist, the dominant coefficients of the
numerator are $c_{0,{\rm{unp}}}^{\rm I}$ and $c_{1,{\rm{unp}}}^{\rm I}$

$\displaystyle c_{0,\rm{unp}}^{\rm I}$ $\displaystyle\propto$
$\displaystyle-\frac{\sqrt{-t}}{Q}\,c_{1,\rm{unp}}^{\rm I}$ (11)
$\displaystyle c_{1,{\rm{unp}}}^{\rm I}$ $\displaystyle\propto$
$\displaystyle\Re{\rm e}\left[F_{1}{\cal H}+\xi(F_{1}+F_{2})\widetilde{\cal
H}-\frac{t}{4M^{2}}F_{2}{\cal E}\right]\,.$ (12)

Given the relative strength of $F_{1}$ and $F_{2}$ at small $t$ for a neutron
target, the beam charge asymmetry becomes

$A_{\rm C}(\phi)\propto\frac{1}{F_{2}}\,\Re{\rm e}\left[\xi\widetilde{\cal
H}_{n}-\frac{t}{4M^{2}}{\cal E}_{n}\right].$ (13)

Therefore, the BCA is mainly sensitive to the real part of the GPD $E_{n}$,
and, for selected kinematics, to the real part of the GPD $\widetilde{H_{n}}$.

##  2 Projections for CLAS12 experiment

Projections for a neutron-DVCS experiment with positron and electron beams and
the JLab CLAS12 spectrometer were produced. The n-DVCS/BH final state will be
reconstructed by detecting the scattered electron/positron and the DVCS/BH
photon in the forward part of CLAS12 and the recoil neutron mostly in the
Central Neutron Detector (CND), as very few neutrons are emitted in the
forward direction with enough momentum to be detected in EC with appreciable
efficiency. The expected number of reconstructed events for n-DVCS/BH has been
calculated, as a function of the kinematics, with a GPD-based event generator
including Fermi motion for a deuterium target and the Fast-MC simulation of
CLAS12. The forward-CLAS12 fiducial cuts have been included, and an overall
10% neutron-detection efficiency for neutrons with $\theta>40^{o}$ has been
assumed. The detection efficiencies for electrons/positrons and photons
efficiencies for the Forward Detector have been assumed to be 100%, within the
fiducial cuts. Figure 2 shows the coverage in $Q^{2}$, $x_{B}$ and $t$ that is
obtained with an electron/positron beam energy of 11 GeV.

Figure 2: Distributions of kinematic variables for n-DVCS events. Forward-
CLAS12 acceptance cuts are included. Top: $Q^{2}$ as a function of $x_{B}$.
Middle: $t$ as a function of $x_{B}$. Bottom: $t$ as a function of $Q^{2}$.

The count-rate calculation has been done for a luminosity $L=10^{35}$ cm-2s-1
per nucleon (corresponding to roughly 60 nA of beam current) and for 80 days
(40 with positron beam, 40 with electron beam) of running time. A
4-dimensional ($Q^{2}$, $x_{B}$, $-t$, $\phi$) grid of bins has been adopted.

The BCA is defined, for each 4-dimensional bin, as:

$\displaystyle
A=\frac{d^{4}\sigma^{+}-d^{4}\sigma^{-}}{d^{4}\sigma^{+}+d^{4}\sigma^{-}}=\frac{\frac{N^{+}}{Q^{+}}-\frac{N^{-}}{Q^{-}}}{\frac{N^{+}}{Q^{+}}+\frac{N^{-}}{Q^{-}}},$
(14)

where $Q^{\pm}$ is the integrated charge for events with positron and electron
beam. Here we assumed $Q^{+}=Q^{-}$.

The number of events $N^{\pm}$, for each 4-dimensional bin ($Q^{2}$, $x_{B}$,
$t$ and $\phi$), has been computed as:

$\displaystyle N=\frac{d\sigma}{dQ^{2}dx_{B}dtd\phi}\cdot\Delta t\cdot\Delta
Q^{2}\cdot\Delta x_{B}\cdot\Delta\phi\cdot L\cdot T\cdot A,$ (15)

where $\frac{d\sigma}{dQ^{2}dx_{B}dtd\phi}$ is the 4-fold differential cross
section, $T$ is the running time, $L$ the luminosity, $Acc$ is the bin-by-bin
acceptance multiplied by the neutron-detection efficiency.

The statistical errors on the beam-charge asymmetries will then depend on the
values of the BCA itself ($A$), through the formula:

$\displaystyle\sigma_{A}=\frac{\sqrt{1-A^{2}}}{\sqrt{N}}.$ (16)

Figure 3 shows the expected accuracy on the n-DVCS/BH beam-charge asymmetry,
computed using the VGG model and assuming various values for $J_{u}=.3$ and
$J_{d}=.1$, for all 4-dimensional kinematic bins within the CLAS12 acceptance.

Figure 3: Beam-charge asymmetry for n-DVCS/BH as predicted by the VGG model.
Projections for different sets of values of $J_{u}$/$J_{d}$ are compared:
0.3/0.1 (black), 0.2/0.0 (red), 0.1/-0.1 (green), 0.3/-0.1 (blue)), plotted as
a function of $\phi$ for all the covered bins in $Q^{2}$, $x_{B}$, and $-t$.
The vertical axis scale ranges from -0.3 to 0.1 for the top plot and from -0.3
to 0.2 for the bottom plot. The error bars reflect the expected uncertainties
for our CLAS12 experiment, corresponding to 80 hours of beam time at a
luminosity of $10^{35}$ cm-2s-1 per n ucleon.

## Extraction of Compton form factors

In order to establish the impact of a beam-charge asymmetry measurement on the
nDVCS experimental program planned with CLAS12 at Jefferson Lab, projections
for four kinds of asymmetries (beam-spin asymmetry, BSA, longitudinal single,
TSA, and double targe-spin asymmetry, DSA, and the BCA) were produced using
the VGG model and realistic count rates and acceptances. The projected
observables were then processed using a fitting procedure fitmick ; mick_herve
to extract the neutron CFFs. This approach is based on a local-fitting method
at each given experimental $(Q^{2},x_{B},-t)$ kinematic point. In this
framework, there are eight real CFF-related quantities

$\displaystyle F_{Re}(\xi,t)$ $\displaystyle=$ $\displaystyle\Re{\rm
e}\left[{\cal F}(\xi,t)\right]$ (17) $\displaystyle F_{Im}(\xi,t)$
$\displaystyle=$ $\displaystyle-\frac{1}{\pi}\Im{\rm m}\left[{\cal
F}(\xi,t)\right]$ (18) $\displaystyle=$ $\displaystyle\left[F(\xi,\xi,t)\mp
F(-\xi,\xi,t)\right],$

where the sign convention is the same as for Eq. 1. These CFFs are the almost-
free111The values of the CFFs are allowed to vary within $\pm 5$ times the
values predicted by the VGG model vgg1 ; vgg2  parameters to be extracted from
DVCS observables using the well-established theoretical description of the
process based on the DVCS and BH mechanisms. The BH amplitude is calculated
exactly while the DVCS one is determined at the QCD leading twist vgg1 . As
there are eight CFF-related free parameters, including more observables,
measured at the same kinematic points, will result in tighter constraint on
the fit and will increase the number and accuracy of the extracted CFFs. In
the adopted version of the fitter code, $\widetilde{E_{Im}}(n)$ is set to
zero, as $\widetilde{E_{n}}$ is assumed to be purely real. Thus, seven out of
the eight real and imaginary parts of the CFFs are left as free parameters in
the fit. The results for the 7 neutron CFFs are shown in Figs. 10-7, as a
function of $-t$, and for each bin in $Q^{2}$ and $x_{B}$. The blue points are
the CFFs resulting from the fits of the four observables, while the red ones
are the CFFs obtained fitting only the projections of the currently approved
n-DVCS experiments. The error bars reflect both the statistical precision of
the fitted observables and their sensitivity to that particular CFFs. Only
results for which the error bars are non zero, and therefore the fits properly
converged, are included in the figures.
The major impact of the BCA measurement is, as expected, on $E_{Re}(n)$, for
which the already approved projections have hardly any sensitivity. Thanks to
the BCA, $E_{Re}(n)$ can be extracted over basically the whole phase. A
considerable extension in the coverage can be also obtained for
$\tilde{H}_{Re}(n)$. An overall improvement to the precision on the other
CFFs, as well as an extension in their kinematic coverage can also be induced
by the nDVCS BCA dataset.

Figure 4: $E_{Re}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and
$x_{B}$. The blue points are the results of the fits including the proposed
BCA while the red ones include only already approved experiments. Figure 5:
$\tilde{H}_{Re}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and
$x_{B}$. The blue points are the results of the fits including the proposed
BCA while the red ones include only already approved experiments. Figure 6:
$H_{Re}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and $x_{B}$. The
blue points are the results of the fits including the proposed BCA while the
red ones include only already approved experiments. Figure 7:
$\tilde{E}_{Re}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and
$x_{B}$. The blue points are the results of the fits including the proposed
BCA while the red ones include only already approved experiments. Figure 8:
$E_{Im}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and $x_{B}$. The
blue points are the results of the fits including the proposed BCA while the
red ones include only already approved experiments. Figure 9:
$\tilde{H}_{Im}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and
$x_{B}$. The blue points are the results of the fits including the proposed
BCA while the red ones include only already approved experiments. Figure 10:
$H_{Im}(n)$ as a function of $-t$, for all bins in $Q^{2}$ and $x_{B}$. The
blue points are the results of the fits including the proposed BCA while the
red ones include only already approved experiments.

## Summary

The strong sensitivity to the real part of the GPD $E^{q}$ of the beam-charge
asymmetry for DVCS on a neutron target makes the measurement of this
observable particularly important for the experimental GPD program of
Jefferson Lab. This sensitivity is maximal for values of $x_{B}$ which are
attainable only with a 11 GeV beam. Model predictions show that for possible
CLAS12 kinematics, this asymmetry can be comparable in size to the BSA
obtained for p-DVCS.
The addition of the beam-charge asymmetry to the already planned measurements
with CLAS12, permits the model-independent extraction of the real parts of the
${\cal E}_{n}$ and $\widetilde{{\cal H}_{n}}$ CFFs of the neutron over the
whole available phase space. Combining all the neutron and the proton CFFs,
obtained from the fit of n-DVCS and p-DVCS observables to be measured at
CLAS12, will ultimately allow the quark-flavor separation of all GPDs.

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