# Multi-Tower Multi-Interest Recommendation with User Representation Repel

Tianyu Xiong  Huawei ResearchWenquanzhen Jingtian Road Zhongguancun
Environmental Protection Park, BeijingBeijingChina<EMAIL_ADDRESS>and
Xiaohan Yu  Huawei ResearchWenquanzhen Jingtian Road Zhongguancun
Environmental Protection Park, BeijingBeijingChina<EMAIL_ADDRESS>
(2023; 5 October 2023; 12 October 2023)

###### Abstract.

In the era of information overload, the value of recommender systems has been
profoundly recognized in academia and industry alike. Multi-interest
sequential recommendation, in particular, is a subfield that has been
receiving increasing attention in recent years. By generating multiple user
representations, multi-interest learning models demonstrate superior
expressiveness than single user representation models, both theoretically and
empirically. Despite major advancements in the field, three major issues
continue to plague the performance and adoptability of multi-interest learning
methods, the difference between training and deployment objectives, the
inability to access item information, and the difficulty of industrial
adoption due to its single-tower architecture. We address these challenges by
proposing a novel multi-tower multi-interest framework with user
representation repel. Experimental results across multiple large-scale
industrial datasets proved the effectiveness and generalizability of our
proposed framework.

Recommender System, Candidate Matching, Multi-Interest Learning, Sequential
Recommendation, Metric Learning

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: The 47th International ACM SIGIR Conference on
Research and Development in Information Retrieval; July 14-18, 2024;
Washington, DC, USA††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Information systems Probabilistic retrieval models

##  1\. Introduction

Sequential recommendation systems play a vital role in addressing the issue of
information overload for users, particularly in domains like e-commerce,
social media, and music streaming. They are instrumental in optimizing key
business metrics such as click-through rates (CTR). These systems organize
items based on when users interact with them, focusing on mining sequential
patterns to predict the next item of interest to users. Many existing methods
combine user preferences and item characteristics to make accurate
predictions. As a result, research in sequential recommendation primarily
revolves around enhancing the quality of how users and items are represented.

Due to the practical significance of sequential recommendation systems,
various approaches have been proposed, yielding promising results. For
instance, GRU4Rec [1] was the pioneering work that applied Recurrent Neural
Networks (RNN) to model sequential information for recommendations. Kang and
McAuley [2] introduced an attention-based method to capture complex, dynamic
patterns in sequences. More recently, some approaches, like PinSage [3], have
harnessed Graph Neural Network (GNN) techniques to derive user and item
representations for downstream tasks. Nevertheless, it’s worth noting that
most prior studies have focused on creating a single, comprehensive
representation of a user’s behavior sequence, which may not effectively
capture a user’s diverse interests. Few studies in the literature have
attempted to address the challenge of representing a user’s multiple interests
adequately within a single vector.

In recent times, there has been a notable rise in the adoption of multi-
interest learning-based approaches [4, 5], demonstrating significant potential
in enhancing matching performance. These methods explicitly address the
challenge of representing users’ varied interests by deriving diverse interest
representations from their behavioral sequences, effectively overcoming the
limitations associated with a single, generic user embedding. One such method,
MIND [4], achieves this by initially capturing a user’s multiple interests
through dynamic routing, employing a Capsule Network [6]. Subsequently,
ComiRec [5] takes the concept of diversity into account and extends the
approach by utilizing multi-head attention mechanisms to encode the diverse
interests of users. Recent research efforts [7, 8] have gone even further,
incorporating considerations of periodicity, interactivity, and user profiles
into the modeling process. Additionally, REMI[9] has introduced an interest-
aware hard negative mining strategy and used routing regularization to address
the problem of routing collapse. These advancements reflect the growing
recognition of the importance of accommodating users’ multiple interests and
improving the overall performance of matching algorithms in various
applications.

Despite the various model architectures and information explored in the realm
of multi-interest learning, the current paradigm for multi-interest learning
frames the challenge of candidate matching as an extreme multiclass
classification problem. In this prevailing approach, the user’s behavioral
sequence is initially transformed into a sequence of item embeddings, which
are subsequently translated into multiple interest representations.

When these multi-interest representations are deployed, they individually
retrieve sets of items using the k-nearest neighbors (kNN) algorithm. However,
during the training phase to create these multi-interest representations, a
label-aware attention mechanism or argmax selection process is employed to
identify the representation that is closest to the positive target. Once this
representation is selected, it becomes the active user representation and is
trained in a manner similar to general candidate matching. This typically
involves applying a softmax function, either uniformly or according to a log-
uniform distribution [10, 11], to efficiently train the model on a large
dataset. Nevertheless, it’s important to acknowledge that there are two
significant drawbacks associated with this approach. Firstly, the practice of
selecting the most suitable representation using the target item during
training creates a disconnect between the objectives of the training and
deployment phases, leading to overly simplistic training. Secondly, framing
candidate matching as a multiclass classification problem reduces each
candidate item to a binary label (0 or 1), which fundamentally limits the
utilization of valuable item-side information and semantic knowledge.

To tackle these issues, we introduce a novel approach called Multi-Tower
Multi-Interest Learning (MTMI) as an alternative paradigm for multi-interest
learning. Our approach not only addresses the aforementioned issues but also
facilitates the seamless adaptation of multi-interest learning to two-tower
candidate matching models, which are widely employed in the industry today.

Figure 1. The MTMI candidate matching system using a multi-tower architecture
with embedding layers. Item features(right), are use to compute a item
representation(Right). User behaviors and user profiles are mapped to
embedding vectors, which are then used to compute multiple user interst
representations. User interest representations and item representation are
used to compute similarity scores and match candidates to users.

MTMI draws inspiration from the Deep Structured Semantic Models (DSSM)[12]
approach, where user and item information are independently processed to
produce user embeddings and item embeddings. Unlike the common practice of
employing a single-user tower and a single-item tower (the two-tower design),
MTMI allows for the incorporation of multiple user towers, each generating a
distinct user representation. We use an Inverse Distance Weighted Loss to
assess the weighted distance between the generated user representations and
the target item representation. Fig. 1 illustrates a motivating example of our
MTMI framework.

We conducted extensive experiments, comparing MTMI-based models to MIND-based
models under basic settings and applying state-of-the-art techniques. These
experiments utilized three real-world, large-scale public recommendation
datasets. The results demonstrate that MTMI-based methods significantly
outperform MIND-based methods in basic settings. While they may not surpass
the state-of-the-art in multi-interest learning, they pave the way for future
advancements and improvements in this field.

In summary, our contributions can be distilled into three key aspects:

  * •

Problem Reevaluation: We reevaluated the prevailing
multi-interest learning paradigm, identifying three significant issues
regarding the disparity between training and deployment objectives, the
constraints on accessing item information, and the difficulty for industrial
adoption.

  * •

MTMI Paradigm Introduction: We introduced a groundbreaking Multi-Tower Multi-
Interest learning paradigm
(MTMI) designed to address these identified problems effectively. MTMI also
enhances the applicability of Multi-Interest Learning, particularly for two-
tower candidate generation systems commonly used in the industry.

  * •

Empirical Validation: We conducted comprehensive experiments, revealing that
MTMI yields substantial improvements over existing multi-interest learning
methods, under the most basic settings. These findings underscore the
practical value and promise of MTMI in the field of recommendation systems.

##  2\. Related Work

###  2.1. Candidate Matching

In the context of large-scale industrial recommender systems, candidate
matching holds significant importance as it effectively narrows down a
selection from a vast pool of items for subsequent refined ranking processes
[8, 13, 14]. Given the imperative for efficiency, candidate matching models
typically employ lightweight architectures and often do not incorporate
candidate awareness during user modeling. In earlier stages, solutions based
on Collaborative Filtering (CF) [15, 16] introduced learnable mechanisms for
matching users with candidates, while Neural network-based Collaborative
Filtering (NCF) [17] enhanced traditional CF with multi-layer perceptrons.
Subsequently, the adoption of two-tower Deep Neural Network (DNN) structures
[10, 12] gained popularity due to their computational efficiency, avoiding
early interactions between user and candidate modeling. Furthermore, there
have been investigations into tree-based and graph-based structures for deep
candidate matching [18, 19, 20]. For instance, PDNP [18] devised a retrieval
architecture based on a 2-hop graph, facilitating online retrieval with
minimal latency and computational costs. However, these approaches typically
represent user preferences as a single vector, potentially limiting their
ability to capture the multi-interest nature of users.

###  2.2. Sequential Recommendation

The issue of sequential recommendation stands as a pivotal challenge within
the realm of recommender systems, and it has garnered substantial attention in
recent research endeavors. Several noteworthy contributions have focused on
addressing this challenge. For instance, FPMC [21] encapsulates both a
conventional Markov chain and the standard matrix factorization model to
handle sequential basket data. HRM [22] extends the FPMC framework by
implementing a two-layer structure that facilitates the construction of a
hybrid representation encompassing users and items based on the most recent
transaction. In a pioneering fashion, GRU4Rec [23] introduces an RNN-based
approach to comprehensively model entire user sessions, thereby enhancing the
precision of recommendations. DREAM [24], founded on Recurrent Neural Networks
(RNN), leverages dynamic user representations to unveil evolving user
interests. Fossil [25] seamlessly integrates similarity-based techniques with
Markov Chains to enable personalized sequential predictions, particularly
suited for sparse and long-tailed datasets. TransRec [26] adopts an approach
that embeds items into a vector space, representing users as vectors that
operate on item sequences, thereby enabling large-scale sequential prediction.
RUM [27], on the other hand, combines a memory-augmented neural network with
insights from collaborative filtering to facilitate recommendation. SASRec
[28] employs a self-attention-based sequential model, adept at capturing long-
term semantics, and employs an attention mechanism to generate predictions
based on relatively limited user actions. In contrast, DIN [29] introduces a
local activation unit designed to adaptively learn user interest
representations based on past behaviors pertaining to specific advertisements.
Finally, SDM [14] encodes behavior sequences through a multi-head self-
attention module, allowing for the capture of multiple types of user
interests, while a long-short-term gated fusion module is employed to
incorporate long-term preferences. These prior works collectively contribute
to the body of research addressing the challenges posed by sequential
recommendation in recommender systems.

###  2.3. Multi-interest Learning

Recent studies have highlighted the inadequacy of representing users’
interests as a single vector, leading to increased interest in multi-interest
learning for both the matching and ranking stages of recommender systems. In
the matching stage, MIND [4] introduced a dynamic routing mechanism to
aggregate users’ historical behaviors into multiple interest capsules,
followed by ComiRec [5], which explored multi-head attention-based multi-
interest routing for a more diverse user interest representation. PIMIRec[7]
and UMI [8] extended these concepts by incorporating time information,
interactivity, and user profiles, while Re4 [30] considered backward flow
regularization. Furthermore, REMI[9] has presented a strategy for mining hard
negatives that takes into account user interests and has employed routing
regularization as a means to mitigate the issue of routing collapse.

###  2.4. Deep Metric Learning

The main objective of Metric Learning is to measure the similarity of samples
using an optimally learned distance metric[31, 32]. In recent years, Deep
Metric Learning, which empowers metric learning with deep architectures has
been gaining traction due to its superior expressibility of nonlinearities. It
has been widely adopted in the state-of-the-art (SOTA) face recognition and
information retrieval models[33]. [34] first proposed a contrastive loss for
face recognition, which separates dissimilar samples by a given margin. The
triplet loss, being one of the foundations of this article, is first proposed
in [35], which minimizes the distance between the anchor and the positive
sample simultaneously maximizing the distance between the anchor and negative
samples. [36] proposed quadruplet loss to minimize intraclass variation, and
[37, 38] improves sample effectiveness by making full use of samples in a
batch, The learning objective was improved by taking the overall structure of
the dataset into account as in [39, 40]. Despite these innovations. The
triplet loss and contrastive loss remain to be relevant due to their
simplicity and computational efficiency.

##  3\. Method

###  3.1. Problem Formulation

Table 1. Notation Notation | Description
---|---
$u$ | an user
$i$ | an item
$e$ | the embedding of a feature
$t$ | a target item
$v_{u}$ | a user representation
$v_{i}$ | a item representation
$v_{t}$ | a target item representation
$E_{p}$ | the matrix of a user’s profile embeddings
$E_{t}$ | the matrix of a target item embeddings
$E_{u}$ | the matrix of a user’s behavior embeddings
$\mathcal{U}$ | the set of users
$\mathcal{I}$ | the set of items
$d$ | the dimension of an embedding
$K$ | the number of user representations
$N$ | the number of candidate items

The Candidate Matching phase of the industrial recommendation system is
designed to select a subset of items from a vast item pool (denoted as
$\mathcal{I}$) containing billions of items for each user $u$ belonging to the
set of users $\mathcal{U}$. This subset should consist of only a few thousand
items, and each item should be relevant to the user’s interests. To accomplish
this objective, historical data generated by the RS is gathered to construct a
matching model. More specifically, each instance can be represented as a tuple
$(E_{u},E_{p},E_{t})$, where $E_{u}$ represents the embedding matrix of items
that user $u$ has interacted with (referred to as user behavior), $E_{p}$ is
the embedding matrix that encompasses the user’s basic profiles (e.g., gender
and age), and $E_{t}$ is the embedding matrix that encompasses the features of
the target item (e.g., item ID, category ID, brand ID, seller ID, title,
etc.).

The main objective of single-tower Multi-Interest Learning algorithms is to
learn the mapping $f_{user}$ that maps raw feature embeddings into user
representation vectors.

(1) $V_{u}=f_{user}(E_{p},E_{u})$

which minimize

$\mathcal{L_{\mathcal{SSM}}}\left(\Upsilon(v_{t},V_{u},V_{u})\right)$

Given the user representation matrix $V_{u}$

$V_{u}=\left({v_{u}^{(1)},v_{u}^{(2)},v_{u}^{(3)},...,v_{u}^{(K)}}\right)\in
R^{d\times K}$

and the label aware attention $\Upsilon$

$\Upsilon(v_{t},V_{u},V_{u})=V_{u}Softmax(Pow(V_{u}^{T}v_{t},p))$

For a detailed description of the notation used in our formulation, please
refer to Table 1.

###  3.2. Embedding Layer & Pooling Layer

The embedding layer in the Multi-Tower Multi-Interest (MTMI) framework plays a
crucial role in capturing the essence of user behavior and item features. This
layer acts as a bridge between raw user interactions and item attributes,
transforming them into dense, continuous vectors that can be easily processed
by subsequent layers of the model. The embedding layer takes user behavior
sequences, user profiles, and item features as input and represents them as
numerical embeddings in a continuous vector space.

###  3.3. Representation/Interest Extraction

In MTMI-basic, a simplified approach is adopted for the representation
extraction module, referred to as ”the towers.” Instead of employing complex
architectural components, the representation extraction module in this variant
relies on straightforward attention fusion modules as illustrated in 3. [41,
42]

(2) $\displaystyle w$ $\displaystyle=Softmax(proj_{L}(e_{1}||...||e_{L}))$ (3)
$\displaystyle f$ $\displaystyle=\sum^{L}_{i=1}w_{i}e_{i}$

###  3.4. Multi-tower Architecture

The Multi-Tower Multi-Interest (MTMI) framework incorporates a unique multi-
tower architecture designed to enhance the modeling of user interests and item
characteristics. This architecture comprises multiple user towers and an item
tower, all of which share a common embedding layer. This is demonstrated in
Fig. 1.

Figure 2. A visual representation of the Candidate Matching process,
consisting of approximate kNN algorithm with K centroid for serving(Left), and
IDW pulling and User Presentation Repel(Right)

During the training phase, the multiple user towers independently generate
user representations, while the item tower simultaneously produces a target
item representation. These user representations are designed to exhibit
repulsion from one another, reflecting the diversity of user interests within
the system. At the same time, they are drawn towards the target item
representation.

In the deployment phase, the user representations produced by the user tower
individually gather a collection of nearby item representations, along with
the associated distances, using the K-Nearest Neighbor algorithm.
Subsequently, these sets of items are consolidated and arranged based on the
estimated distances, resulting in a ranked order. An illustrative graph of the
training phase and deployment phase can be found in Fig. 2.

###  3.5. Invert Distance Weighting Loss(IDWLoss)

Given cosine similarity $\mathcal{S}_{c}(A,B)$, cosine distance
$\mathcal{D}_{c}(A,B)$ and inverted distance $\mathcal{ID}_{c}(A,B)$

(4) $\displaystyle\mathcal{S}_{c}(A,B)$ $\displaystyle=\frac{A\cdot
B}{||A||||B||}$ (5) $\displaystyle\mathcal{D}_{c}(A,B)$
$\displaystyle=1-\mathcal{S}_{c}(A,B)$ (6)
$\displaystyle\mathcal{ID}_{c}(A,B)$
$\displaystyle=\frac{1}{\mathcal{D}_{c}(A,B)^{\alpha}}$

Our novel IDW loss $\mathcal{L_{IDW}}$ can be formulated as

(7) $\displaystyle\mathcal{L_{IDW}}$
$\displaystyle\left(V_{u},v_{+},V_{-}\right)=$
$\displaystyle\sum^{K}_{j}\mathcal{ID}_{c}\left(v_{u}^{(j)},v_{t}\right)\mathcal{L}_{triplet}\left(v_{u}^{(j)},v_{i+},v_{i-}^{(j)}\right)$
$\displaystyle+\beta\sum^{K}_{j}\mathcal{L}_{triplet}\left(v_{u}^{(j)},v_{i+},v_{u-}^{(j)}\right)$

where $\mathcal{L}_{triplet}$ is a triplet loss commonly used in metric
learning. [35, 43]

(8)
$\displaystyle\mathcal{L}_{triplet}(v_{u},v_{+},v_{-})=\max\left(0,\gamma-\mathcal{S}_{c}\left(v_{u},v_{+}\right)+\mathcal{S}_{c}\left(v_{u},v_{-}\right)\right)$

$v_{i+}$ is the representation of the positive item.

(9) $\displaystyle v_{i+}=v_{t}$

$v_{u-}$ represents user representations other than the currently selected
one.

(10) $\displaystyle
v_{u-}^{(j)}=\left[v_{u}^{(i)};i\in\\{1,...,K\\},i!=j\right]$

Negative items $i-$ are sampled from a multinomial distribution with $120$
samples and $N$ classes.

##  4\. Experiments

Table 2. Statistics of Datasets Dataset | Users | Goods | Interactions | Cutoff
---|---|---|---|---
MovieLens 100k | 943 | 1,349 | 99,287 | 20
Retail Rocket | 33,708 | 81,635 | 356,840 | 20
Amazon Books | 603,668 | 367,982 | 8,898,041 | 20
Gowalla | 976,779 | 1,708,530 | 85,384,110 | 40

We perform extensive experiments on three real-world recommendation datasets
in order to address the following research inquiries:

  * •

RQ1: Can MTMI enhance common two-tower Candidate Matching models?

  * •

RQ2: Can MTMI beat single tower MIL algorithms under the most basic settings?

  * •

RQ3: What is the effect of IDWLoss and User Representation Repel?

  * •

RQ4: What is the Optimal Hyperparameter setting for MTMI?

###  4.1. Experimental Settings

####  4.1.1. Dataset

We have opted for the utilization of three extensive public datasets to
conduct an assessment of MTMI’s efficacy:

  * •

Amazon[44]: This dataset encompasses a diverse array of product views derived
from the widely recognized Amazon platform. For our evaluation, we
specifically select the largest subset focusing on books, characterized by
various book types.

  * •

Gowalla [45]: This dataset collects check-in data from Gowalla, a location-
based social networking website.

  * •

RetailRocket[46]: This dataset mirrors real-world e-commerce interactions and
spans a four-month duration, capturing various user behaviors. For the purpose
of our study, we exclusively consider the view events within this dataset.

  * •

MovieLens[47, 48]: This dataset was acquired from the MovieLens website over a
seven-month duration. Subsequently, a data cleaning process was undertaken,
wherein users with fewer than 20 ratings or those lacking comprehensive
demographic information were excluded from the dataset. We use the 100k
version of this dataset for the hyperparameter scan.

To ensure consistency and comparability, we have implemented dataset
preprocessing methodologies in alignment with a prior research study [4, 5,
9]. This process involves the removal of items and users with occurrences
falling below a predefined threshold of 5, maximum user behavior sequence
length set to 20. Moreover, all user interactions within these datasets are
treated as implicit feedback. For a detailed summary of the statistical
attributes of the three datasets post-preprocessing, we refer to Table 2.

####  4.1.2. Training and Evaluation Setup

In adherence to the methodologies established in prior studies [4, 5, 9], we
perform the partitioning of our dataset into training, validation, and test
sets, maintaining a ratio of 8:1:1 concerning distinct users. In the training
phase, models are trained to utilize the complete user behavior sequences from
the training set. During the evaluation process, we employ the initial 80% of
the user behavior sequence to infer user embeddings, subsequently calculating
the designated metrics using the remaining 20% of items within the sequence.
For further elaboration, additional comprehensive information can be obtained
from references [4, 5, 9]. We employ widely recognized evaluation metrics,
specifically Recall, Hit Rate, and NDCG (Normalized Discounted Cumulative
Gain), to assess the effectiveness of our proposed solution. These metrics are
computed based on the top 20/50 matched candidates.

####  4.1.3. Baseline Models.

To demonstrate the effectiveness of our MTMI framework, we compare it with
classic single-tower MIL models such as MIND and ComiRec, some of the general
sequential recommendation algorithms, as well as the SOTA single-tower MIL
model REMI.

  * •

Most Popular An algorithm recommending the most popular item to its users.

  * •

YouTube DNN An influential industrial Deep Neural Network (DNN) model that
aggregates behavior. embeddings, subsequently employing Multi-Layer Perceptron
(MLP) layers to derive user representation.

  * •

GRU4Rec The first sequential recommendation model based on Recurrent Neural
Networks (RNN) that captures sequential patterns.

  * •

Que2Search An influential industrial DNN model that aggregates behavior
embeddings, subsequently employing Simple Attention Fusion layers to derive
user representations.

  * •

MIND Pioneering the realm of multi-interest learning frameworks, it leverages
capsule networks to capture diverse user interests.

  * •

ComiRec An advanced multi-interest framework that permits control over
diversity and introduces a multi-head attention mechanism for modeling various
user interests. The multi-head attention version, ”ComiRec-SA”, became the
standard backbone for Multi-Interest Learning frameworks for its simplicity
and stability.

  * •

REMI The SOTA single-tower multi-interest learning framework that effectively
tackled issues related to the challenge of ”easy negative” instances and the
problem of ”routing collapse.” This successful intervention has led to
significant enhancement in performance.

####  4.1.4. Implimentation Details

Our model was constructed using Torch 2.0.1, and for the purpose of k-nearest
neighbors (kNN) search, Faiss 1.7.3 was employed. To visualize the outcomes,
Matplotlib and Wandb were utilized. The optimization of hyperparameters
involved an exhaustive grid search encompassing alpha, beta, and the number of
towers or interests (referred to as K). This meticulous exploration led to the
identification of the most effective parameter set
$\\{\alpha=5.5,\beta=6,K=8\\}$, which subsequently became the default
configuration for our experiments unless otherwise specified. Similarly, the
embedding dimension was firmly set at 64, while the batch size was established
at 128 to ensure efficient model training. The learning rate was defined as
$1\times 10^{-3}$ to strike a balance between convergence speed and model
accuracy. For addressing negative training requirements and maintaining
consistency, the batch-wise shared negative training sample size was set at
128 * 10, and the number of negative samples was consistently maintained at
120 for all applicable experiments. These implementation details are pivotal
for the reproducibility and robustness of our experimental procedures.

###  4.2. Enhancement to two tower models (RQ1)

We employed the MTMI scheme on three popular two-tower vector retrieval
models, YouTube DNN, GRU4Rec, Que2Search to test if MTMI can effectively and
consistently enhance existing two-tower models.

Table 3. Comparing Performance of existing dual tower models and their MTMI enhanced version.  Dataset | Metric | Pop | GRU4Rec | MTMI-GRU | Youtube DNN | MTMI-DNN | Que2Search | MTMI-Q2S
---|---|---|---|---|---|---|---|---
Books | R@20 | 0.0158 | 0.0441 | 0.0478 | 0.0467 | 0.049 | 0.0453 | 0.0485
HR@20 | 0.0345 | 0.1004 | 0.114 | 0.1043 | 0.1155 | 0.0969 | 0.0939
ND@20 | 0.0143 | 0.0378 | 0.0363 | 0.0391 | 0.0394 | 0.038 | 0.0419
R@50 | 0.0281 | 0.0706 | 0.0877 | 0.0722 | 0.0686 | 0.0702 | 0.0794
HR@50 | 0.0602 | 0.1553 | 0.1607 | 0.1607 | 0.1594 | 0.154 | 0.1801
ND@50 | 0.0193 | 0.0443 | 0.0542 | 0.0457 | 0.0404 | 0.045 | 0.0549
Gowalla | R@20 | 0.0231 | 0.09 | 0.0884 | 0.0864 | 0.0959 | 0.0809 | 0.0893
HR@20 | 0.1121 | 0.3359 | 0.3804 | 0.3211 | 0.3642 | 0.3132 | 0.377
ND@20 | 0.0483 | 0.1433 | 0.1582 | 0.1384 | 0.1632 | 0.1346 | 0.1629
R@50 | 0.0365 | 0.1458 | 0.1617 | 0.1388 | 0.1603 | 0.1546 | 0.17
HR@50 | 0.1582 | 0.4577 | 0.5496 | 0.439 | 0.4086 | 0.4791 | 0.6471
ND@50 | 0.0569 | 0.1494 | 0.1717 | 0.1434 | 0.1544 | 0.1558 | 0.1683
Retail Rocket | R@20 | 0.0129 | 0.0827 | 0.0918 | 0.105 | 0.1175 | 0.0982 | 0.1168
HR@20 | 0.0252 | 0.1376 | 0.1431 | 0.1711 | 0.2297 | 0.1216 | 0.1562
ND@20 | 0.0098 | 0.0517 | 0.0669 | 0.0641 | 0.0825 | 0.0703 | 0.0678
R@50 | 0.0244 | 0.1371 | 0.1628 | 0.1608 | 0.1956 | 0.1321 | 0.1641
HR@50 | 0.0462 | 0.2132 | 0.2088 | 0.2518 | 0.3075 | 0.12 | 0.1408
ND@50 | 0.0139 | 0.0593 | 0.0623 | 0.0701 | 0.0785 | 0.0647 | 0.0656

As illustrated in Table 3, the MTMI-enhanced models significantly outperform
their vanilla counterparts.

###  4.3. Comparison with Basic MIL models (RQ2)

We also compared the performance of the MTMI-GRU model with basic MIL models.

Table 4. Comparing the performance of basic MIL models with MTMI-GRU Dataset | Metric | MIND | ComiRec | MTMI-DNN
---|---|---|---|---
Books | R@20 | 0.042 | 0.0557 | 0.049
HR@20 | 0.0986 | 0.1142 | 0.1155
ND@20 | 0.0357 | 0.0446 | 0.0394
R@50 | 0.0687 | 0.0863 | 0.0686
HR@50 | 0.1533 | 0.1796 | 0.1594
ND@50 | 0.0433 | 0.0511 | 0.0404
Gowalla | R@20 | 0.0901 | 0.0805 | 0.0959
HR@20 | 0.3129 | 0.2901 | 0.3642
ND@20 | 0.1331 | 0.121 | 0.1632
R@50 | 0.1456 | 0.132 | 0.1603
HR@50 | 0.4442 | 0.4086 | 0.4086
ND@50 | 0.1424 | 0.131 | 0.1544
Retail Rocket | R@20 | 0.1171 | 0.1304 | 0.1175
HR@20 | 0.1883 | 0.1904 | 0.2297
ND@20 | 0.0698 | 0.0689 | 0.0825
R@50 | 0.1899 | 0.1922 | 0.1956
HR@50 | 0.2927 | 0.2895 | 0.3075
ND@50 | 0.0795 | 0.0786 | 0.0785

As illustrated in the able, the MTMI models also outperform basic MIL models.

###  4.4. Comparison with SOTA MIL models (RQ3)

Having comprehensively assessed MTMI in contrast to conventional single-tower
models, we have additionally conducted a comparative analysis with the state-
of-the-art (SOTA) single-tower multi-interest learning framework, REMI.
Acknowledging that this comparison may not be entirely equitable, as REMI
incorporates numerous advanced techniques and strategies specialized for
single-tower MIL.

###  4.5. Abalation Study(RQ4)

To properly address the effectiveness of IDWLoss and User Representation
Repel(URR). We conduct an ablation study for each of these methods.

The result of quadruple-user-tower IDWLoss(without representation repel) is
compared with the single-user-tower triplet loss and quadruple-user-tower
minimum loss. As we can see in Fig. 3, the IDWLoss empowered model outperforms
the triplet loss model and the performance of the model that uses Label Aware
Attention and Sampled Softmax quickly regresses due to information leakage and
overfitting.

Figure 3. Study of the effect of IDWLoss. We use ML-100k dataset for these
comparisons.

We also compared turning the URR on and off. As we can see in Fig. 4, the URR-
enhanced model demonstrated significantly better performance.

Figure 4. Study on the effect of URR. We use ML-100k dataset for these
comparisons.

###  4.6. Hyper-parameter Studies(RQ5)

Figure 5. Hyperparameter Scan

Within this segment, an exhaustive examination of three pivotal
hyperparameters in the context of the MTMI model is conducted, specifically
addressing the number of interests denoted by $K$, the coefficient of distance
scaling $\alpha$, and the divergence parameter $\beta$. The variable $K$ is
selected from the set ${2,4,6,8}$, $\alpha$ is ascertained from the interval
$[0.5,6]$, and $\beta$ is chosen from the discrete series ${1,10,100,1000}$. A
comprehensive grid search methodology is employed to meticulously explore
every conceivable permutation of these parameters. The entirety of this
hyperparameter exploration is carried out utilizing the Movielens-100k
dataset, which is of a modest magnitude, thereby facilitating rapid iteration
across successive experimental runs. The results of this exhaustive search can
be found at 5, where the performance peaks at $K=4,\alpha=2,\beta=1000$.

##  5\. Conclusions and Futurework

In conclusion, this paper has addressed three critical issues in the field of
Multi-interest learning for Candidate Matching: the discrepancy between
training and deployment objectives, the challenge of accessing item
information, and the complexities of industrial deployment. To resolve these
challenges, we introduced the innovative Multi-Tower Multi-Interest Learning
framework, known as MTMI. MTMI aligns training and deployment objectives
through a novel IDWLoss, offers the capability to access item information via
a dedicated item tower, and facilitates the adaptation of industry-standard
two-tower models for multi-interest learning. Comparative analyses between
traditional Multi-Interest Learning methods and MTMI have revealed substantial
improvements in performance.

Although MTMI has significantly enhanced the applicability of Multi-Interest
Learning techniques and outperformed single-tower methods under basic
settings, it has yet to surpass the state-of-the-art methods in Multi-Interest
Learning. Future advancements and refinements hold the potential to elevate
MTMI’s performance to surpass state-of-the-art methods. There exist several
potential avenues for enhancing MTMI, including the refinement of hard
negative mining strategies, the development of more intricate and balanced
loss functions, and the incorporation of effective inter-tower communication
mechanisms. We eagerly anticipate witnessing the progress and innovations in
this evolving field.

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# Performance Assessment of the KASI-Deep Rolling Imaging Fast-optics
Telescope pathfinder

Woowon Byun Korea Astronomy and Space Science Institute, Daejeon 34055,
Republic of Korea  Jongwan Ko Korea Astronomy and Space Science Institute,
Daejeon 34055, Republic of Korea  University of Science and Technology, Korea,
Daejeon 34113, Republic of Korea  Yunjong Kim Korea Astronomy and Space
Science Institute, Daejeon 34055, Republic of Korea  Kwang-Il Seon Korea
Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
University of Science and Technology, Korea, Daejeon 34113, Republic of Korea
Seunghyuk Chang  Center for Integrated Smart Sensors, Daejeon 34141, Republic
of Korea  Dohoon Kim  Green Optics Co., Ltd., Cheongju 28126, Republic of
Korea  Changsu Choi  Korea Astronomy and Space Science Institute, Daejeon
34055, Republic of Korea  Sang-Hyun Chun Korea Astronomy and Space Science
Institute, Daejeon 34055, Republic of Korea  Young-Beom Jeon  Korea Astronomy
and Space Science Institute, Daejeon 34055, Republic of Korea  Jae-Woo Kim
Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
Chung-Uk Lee Korea Astronomy and Space Science Institute, Daejeon 34055,
Republic of Korea  Yongseok Lee Korea Astronomy and Space Science Institute,
Daejeon 34055, Republic of Korea  School of Space Research, Kyung Hee
University, Yongin, Kyeonggi 17104, Republic of Korea  Hong Soo Park Korea
Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
University of Science and Technology, Korea, Daejeon 34113, Republic of Korea
Eon-Chang Sung  Korea Astronomy and Space Science Institute, Daejeon 34055,
Republic of Korea  Jaewon Yoo Korea Astronomy and Space Science Institute,
Daejeon 34055, Republic of Korea  University of Science and Technology, Korea,
Daejeon 34113, Republic of Korea  Gayoung Lee  Kyungpook National University,
Daegu 41566, Republic of Korea  Hyoungkwon Lee  LeO SPACE Inc., Daejeon 34014,
Republic of Korea

(Accepted July 28, 2022)

###### Abstract

In a $\Lambda$CDM universe, most galaxies evolve by mergers and accretions,
leaving faint and/or diffuse structures, such as tidal streams and stellar
halos. Although these structures are a good indicator of galaxies’ recent mass
assembly history, they have the disadvantage of being difficult to observe due
to their low surface brightness (LSB). To recover these LSB features by
minimizing the photometric uncertainties introduced by the optical system, we
developed a new optimized telescope named K-DRIFT pathfinder, adopting a
linear astigmatism free-three mirror system. Thanks to the off-axis design, it
is expected to avoid the loss and scattering of light on the optical path
within the telescope. To assess the performance of this prototype telescope,
we investigate the photometric depth and capability to identify LSB features.
We find that the surface brightness limit reaches down to $\mu_{r,1\sigma}\sim
28.5$ mag arcsec-2 in $10^{\prime\prime}\times 10^{\prime\prime}$ boxes,
enabling us to identify a single stellar stream to the east of NGC 5907. We
also examine the characteristics of the point spread function (PSF) and find
that the PSF wing reaches a very low level. Still, however, some internal
reflections appear within a radius of $\sim$6 arcmin from the center of
sources. Despite a relatively small aperture (0.3 m) and short integration
time (2 hr), this result demonstrates that our telescope is highly efficient
in LSB detection.

Multiple mirror telescopes (1080), Astronomy data reduction (1861), Galaxies
(573), Stellar streams (2166)

††journal: PASP††facilities: The K-DRIFT pathfinder, SDSS, IRSA,
NED.††software: Astropy (Astropy Collaboration et al., 2013, 2018), Scipy
(Virtanen et al., 2020), Source Extractor (Bertin & Arnouts, 1996), SWarp
(Bertin et al., 2002), Photutils (Bradley et al., 2021), Matplotlib (Hunter,
2007), PSFEx (Bertin, 2011)

##  1 Introduction

In the $\Lambda$CDM model, galaxies undergo numerous mergers and accretions
(White & Rees, 1978). Small galaxies orbiting large ones are occasionally
disrupted by tidal forces, leaving them as stellar streams (see Bullock et
al., 2001; Bullock & Johnston, 2005). These structures gradually disperse for
a few Gyr and eventually become a part of stellar halos of host galaxies (cf.
De Lucia & Helmi, 2008; Johnston et al., 2008). That is, most large galaxies
in the universe are expected to have extended, faint tidal remnants (e.g.,
Moore et al., 1999; Cooper et al., 2010, 2013). In other words, we can
directly infer the recent assembly history of a host galaxy via the existence
of stellar streams and stellar halos and the nature of progenitors via their
shapes, colors, and total luminosities. Such a description can be applied to
not only the structures in the outskirts of individual galaxies but also the
intracluster light in cluster environments (e.g., Conroy et al., 2007; Contini
et al., 2014).

To verify this paradigm, decisive observational evidence is necessary.
Unfortunately, however, there is a critical limitation that it is difficult to
observe such structures because of their low stellar surface density, which
generally corresponds to a surface brightness of at least $\sim$27–28 mag
arcsec-2 (cf. Courteau et al., 2011; Bakos & Trujillo, 2012).

Thanks to its relative closeness, many observational studies have been
dedicated to the Local Group, including the Milky Way and Andromeda, using the
star count method (e.g., Ibata et al., 1997; Belokurov et al., 2007; Radburn-
Smith et al., 2011; Martin et al., 2014; McConnachie et al., 2018). The
photometric depth achieved by this method corresponds to at least $\sim$30 mag
arcsec-2. To acquire more samples beyond the Local Group, the measurement of
smooth integrated light of galaxies has been carried out as an alternative
approach (e.g., Malin & Hadley, 1997; Shang et al., 1998; Mihos et al., 2005;
van Dokkum, 2005; Tal et al., 2009; Martínez-Delgado et al., 2010; Atkinson et
al., 2013; Ko & Jee, 2018; Yoo et al., 2021). Still, it appeared to barely
achieve surface brightness levels of $\sim$28–29 mag arcsec-2. Thus,
overcoming the existing observational limitation is essential to investigating
the nature of extremely low surface brightness (LSB) galaxies.

The reason why LSB imaging is difficult is not just because of insufficient
exposure time. Systematic uncertainties are more likely to be the primary
cause of this difficulty. For example, reaching deeper surface brightness
levels will be hindered by local background fluctuations, introduced by a
combination of zodiacal light, moonlight, stray light from stars, point spread
function (PSF), and the Galactic cirrus, etc. (see Knapen et al., 2017;
Holwerda, 2021). Those factors can cause inaccuracy in the calibration
process, such as sky determination, resulting in significant photometric
uncertainty in the LSB regime. In addition, they can directly interfere with
LSB detection. For example, spurious light derived from local background
fluctuations may be mistaken for LSB features.

Specific observation strategies, e.g., observing objects with high galactic
altitudes in a dark time, can potentially reduce fluctuations caused by
external factors. The remaining problem is that ordinary telescopes may hold a
risk of the loss and/or scattering of light on the optical path. Large
diameters are indeed advantageous for collecting lots of photons. However,
larger telescopes are likely to have more complex optical systems, which can
cause various problems. For example, a significant PSF wing may affect the
background, thereby obscuring LSB features or leading to incorrect estimates
of the LSB galaxy’s properties such as shape and brightness (e.g., Trujillo &
Fliri, 2016). In addition, internal reflection can create extended asymmetric
artifacts around bright stars, commonly referred to as ghosts, which
significantly contaminate images. Indeed, we may fail to detect the LSB
objects if they are overlapped with the artifacts brighter than 27 mag
arcsec-2.

While software processing through ray tracing and machine learning can be
applied to remove these artifacts (e.g., Slater et al., 2009; Tanoglidis et
al., 2022), we still need to develop telescopes optimized for LSB studies that
suppress loss and scattering of light fundamentally, through the careful
baffling, anti-reflective coating, as well as simple optics design. The
Dragonfly Telephoto Array made with multiple commercial CCD cameras is a good
example of a novel attempt to overcome surface brightness limitation (see
Abraham & van Dokkum, 2014). In brief, it was designed to be able to reach a
deep surface brightness level and obtain a wide field of view (FoV) at the
expense of a high spatial resolution.

As another independent attempt to do so, we launched a project at the Korea
Astronomy and Space Science Institute (KASI) in 2019. It is called “KASI-Deep
Rolling Imaging Fast-optics Telescope (K-DRIFT)”. This project is planning to
develop new telescopes optimized for LSB studies and conduct deep imaging
surveys. We adopted an off-axis design with a small focal ratio to improve the
response to faint light, even with a small aperture. As a prototype telescope,
the K-DRIFT pathfinder is significant because it is the first ground-based
telescope manufactured with this concept. Over the next decade, we plan to
build several 0.3–0.5 m ground-based telescopes and even launch a small space
telescope (see details in Ko et al. 2022 in preparation).

The manufacture of the K-DRIFT pathfinder was completed in early 2021, and we
installed it at the Bohyunsan Optical Astronomy Observatory (BOAO) in Korea
for test observation. In this paper, we assess the performance of the K-DRIFT
pathfinder and discuss the potential of the project by understanding the
systems that must be improved in the next production of telescopes. The paper
is organized as follows: Section 2 describes the specification of the
telescope, and Section 3 presents an outline of the observation and data
reduction. Section 4 provides the observational results and describes the
telescope’s performance. Section 5 discusses some shortcomings of the
telescope, and Section 6 summarizes all the assessment.

##  2 Telescope

Figure 1: Exterior and interior designs of the K-DRIFT pathfinder. The
diameter of the aperture is 300 mm.

Figure 1 shows the configuration of the K-DRIFT pathfinder. It was designed as
a linear astigmatism free-three mirror system (LAF-TMS) that is an all-
reflective imaging system consisting of three free-form mirrors (cf. Chang,
2013; Park et al., 2020). The off-axis design can minimize the loss and
scattering of light due to not requiring the internal structures located on
the optical path. In addition, it is analytically optimized to remove linear
astigmatism and reduce aberrations. Thus, it can produce a small and nearly-
uniform spot size across the entire FoV, thereby avoiding quality degradation
for the wide-field image. Specifically, our system was designed to have an
average spot size of 3.6 $\mu$m, corresponding to 0.3 pixels. The mirrors are
composed of a concave primary mirror (M1), a convex secondary mirror (M2), and
a concave tertiary mirror (M3). All the mirrors were coated with silver to
enhance the reflectivity. Triple baffling can also robustly remove stray light
from stars outside the line of sight.

The diameter of the aperture is 300 mm, and the focal length is 1200 mm,
resulting in the focal ratio of $f/4$. A 2k$\times$2k sCMOS camera (KL400 BI)
is installed and the FoV corresponds to $\sim$$1\times 1$ deg2, yielding a
pixel scale of $\sim$1.89 arcsec. Only a clear luminance ($L$) filter, which
covers a range of 3500–8500 $\mathrm{\AA}$, is equipped to collect as many
photons as possible. The CMOS camera’s quantum efficiency has been improved
significantly compared to the past, resulting in more than 90% at the
effective wavelength.

Note that CCD cameras, widely used in astronomy, have characteristics that
readout time inevitably increases as the FoV and angular resolution increase.
Dividing a CCD chip into several amplifiers is often used to resolve this
issue. However, this can introduce artificial patterns and uncertainty when
flattening images, thus hampering the detection of LSB signals. In this
regard, CMOS has the advantage of being able to operate relatively fast, even
for a single large chip.111With the telescope’s specifications mentioned
above, readout time may not be a significant problem even if a CCD camera was
used. However, we had to consider that the next K-DRIFT telescopes will adopt
imaging sensors with a resolution of at least 6k $\times$ 6k. This advantage
allows the development of an optimized observation strategy, including
frequent and rolling dithering, to process imaging data more precisely. Also,
the small pixels of CMOS sensors make it possible to reduce the optical
system’s size, power consumption, and cost. However, there are still
shortcomings, such as high dark current and poor electronic stability, but we
expect it will improve rapidly soon. Despite some issues, the combination of
novel telescopes and CMOS is expected to secure originality from previous
studies.

It is worth mentioning that this is the first time the whole development
process of a LAF-TMS was completed and did not end up as a laboratory
experiment but led to actual observations. By doing so, we could clarify the
advantages and disadvantages of the telescope. These assessments will be
utilized in the upcoming development of the next K-DRIFT telescopes. The
details about the fabrication, assembly, and alignment process can be found in
Y. Kim et al. (2022 in preparation).

##  3 Data

###  3.1 Observation

The observation to verify the telescope’s performance was conducted on 2021
June 5 and 6 at BOAO in Korea. As the first target to observe LSB structures,
we selected an edge-on spiral galaxy, NGC 5907. This galaxy, which has a
stellar mass of $\sim$$8\times 10^{10}\ M_{\odot}$ (Laine et al., 2016), is
one of the well-known galaxies with giant stellar stream(s) in the Local Group
($D\sim 17$ Mpc; Radburn-Smith et al. 2011). The stream was first reported by
Shang et al. (1998), and many observational verifications and analyses have
been carried out until recently (e.g., Zheng et al., 1999; Martínez-Delgado et
al., 2008; Laine et al., 2016; Müller et al., 2019; van Dokkum et al., 2019).
This allows us to compare our results with other well-studied results and
effectively assess the performance of the K-DRIFT pathfinder. In addition, NGC
5907 and its vicinity are likely to be less contaminated by the Galactic
cirrus since the far-infrared flux appears to be low ($F_{\mathrm{100\mu
m}}\lesssim 1.5$ MJy sr-1) from the IRAS map.222https://irsa.ipac.caltech.edu
Thus, the following results are likely to mainly reflect the telescope’s
performance without being significantly affected by external factors.

Figure 2: Schematic dithering pattern with red crosses plotted over a raw
image of NGC 5907 taken by the K-DRIFT pathfinder. The FoV of the image
corresponds to $\sim$$1\times 1$ deg2. The gap between adjacent dithering
positions is $\sim$15 arcmin. Amplifier glow is distinct at the image
boundary.

Figure 2 shows a raw image of NGC 5907 together with the schematic dithering
pattern we used. We observed NGC 5907 by positioning it at nine points during
one night. Since the diameter of NGC 5907 corresponds to $\sim$ 13 arcmin
(Nilson, 1973), we chose the gap of nine points to be at least $\sim$15 arcmin
so that the structure of NGC 5907 is not overlapped between every dithers. We
performed a non-constant shift so that each position has a slight offset in
practice. A 30-sec exposure was taken ten times continuously at each place.
Due to the low readout noise of the CMOS camera (1.6 electrons), each frame
would be sky-noise limited in short exposure times. This ensures a large
number of dithered object frames in a relatively short period, which can be
advantageous for minimizing the background noise in the final co-added image.
As a result of two-day iterative observations, the total exposure time is 2
hr.

As described in the next section, we performed flat fielding using a dark-sky
flat generated by combining object-masked science images. To do so,
considerable dithering was necessary. Such dithering provides enough pixels of
the blank sky to generate a dark-sky flat robustly. It can also remove
artifacts, such as cosmic rays and satellite tracks, when co-adding images in
the final data processing.

Figure 3: Variation in the levels of bias (blue) and bias-subtracted dark
(orange). The error bars indicate the standard deviation of each frame. Frames
“1-10” and “11-20” were continuously obtained at once after target
observation. The bias level appears nearly constant, but the dark level shows
a decreasing trend over time.

###  3.2 data reduction

The following procedures were performed using a custom-developed pipeline
based on PYTHON, including astropy and photutils libraries. This pipeline is
expected to be flexible in dealing with data of the next K-DRIFT telescopes,
which will be developed for a decade. The overall concept of data reduction
was adopted from Byun et al. (2018).

Before data processing, we checked the variation of bias and dark levels to
ensure the reliability of master calibration frames. Figure 3 shows the median
values of bias and bias-subtracted dark frames. Here, frames “1–10” and
“11–20” were independently obtained after different target observations. In
fact, we carried out object and calibration observations alternately to
examine the detector’s response. The bias level appears almost constant with a
slight deviation, while the dark level seems pretty complicated.

The dark level, $\sim$420–430 ADU, steadily decreased over 300 sec and reached
down to $\sim$370–380 ADU. Since the detector has been cooled down to
approximately $-20^{\circ}\mathrm{C}$, we ruled out the possibility of
contamination from external heat sources. At the same time, we found that the
dark frames had false signals where the bright stars were placed on the last
object frame, and these signals diminished over time. It is very similar to
the phenomenon known as “residual bulk image (RBI)”. Thus, we suspect that the
dark (object) frames seem to be affected by the object image taken just
before, which has a background level of $\sim$13000 ADU on average. This may
be a critical issue in that the detector cannot completely flush the charge of
each exposure. We will discuss it again in Section 5.

Meanwhile, the peak-to-peak deviation of the dark level was revealed as
$\sim$0.5% of the background level of object images. Therefore, it should be
considered that using a master dark can lead to a non-negligible uncertainty
in dark subtraction. Fortunately, however, the deviation is not too large, and
each dark frame seems not to have any distinct pattern over the images. Thus,
this may not be a critical flaw in identifying the existence of any LSB
structures. However, there is still a potential error in measuring surface
brightness.

Figure 4: Comparison of background variations of images divided into 16 blocks
using a $4\times 4$ grid. The dashed black line represents the median value of
randomly-selected 90 images before flat fielding. The solid blue and red lines
represent the median values of the same frames but after flat fielding with
twilight flat and dark-sky flat, respectively. The error bars indicate the
standard deviation between 90 images. The block numbers on the x-axis follow
the order row-by-row, bottom-to-top, and each block is color-coded to clarify
the order.

After bias and dark subtraction, flat fielding was carried out using a dark-
sky flat. To create a dark-sky flat, we first masked all bright objects in the
science images with a 2$\sigma$ threshold of the background noise. The mask
size was then slightly expanded by convolution using a Gaussian kernel. In
addition, we manually added circular masks for bright stars and galaxies with
a radius twice the major axis of each mask to cover unmasked diffuse light.
Finally, the normalized object-masked science images were median-combined. The
data may have different photometric conditions between two other nights, so we
created two dark-sky flats using only the images obtained on the same night
and separately used them for flat fielding.

Figure 4 shows the results of flat fielding for a subsample of randomly-
selected 90 images. To visualize the background variation effectively, we
divided each image into 16 blocks using a $4\times 4$ grid. We found a
gradient from left to right in the images before the flat fielding, and the
peak-to-peak deviation among 16 blocks was revealed as $\sim$2% of the
original background level (dashed black line). On the other hand, the sky
gradient became negligible after the flat fielding with the dark-sky flat, and
the deviation decreased to as low as $\sim$0.02% (solid red line).

It is almost impossible to secure a flat, wide light source surface.
Therefore, obtaining the actual pixel-to-pixel sensitivity variation in a
wide-field image is very difficult. Using a dark-sky flat can be a good
approach, but there is a risk of eliminating the real sky gradients, mainly
introduced by the variation of airglow depending on the zenith angle. Instead,
we expect that the ‘median-combining’ of a sufficiently large number of frames
provides a master flat almost unrelated to each frame’s real gradient. Indeed,
the gradients of all the object frames used to create the dark-sky flat varied
even from frame to frame. The deviation was less than 0.3%, but the trend
appeared nearly consistent.

On the other hand, twilight flat frames may significantly differ in trend, as
they are literally obtained at different times and/or from different skies. To
verify whether the twilight flat is suitable for flattening images, we carried
out flat fielding with the twilight flat. A total of 45 twilight flat frames
were carefully taken at zenith with a slight offset. As shown in Figure 4
(solid blue line), there was still a significant fluctuation of 1% level. This
fluctuation might be removed by subtracting the background using complex sky
models in the following procedure. However, this approach can cause another
photometric uncertainty. In addition, unlike the dark-sky flat, the twilight
flat could not effectively remove the amplifier glow because of different
exposure times. Thus, using a dark-sky flat is likely to be more robust in
flattening wide-field images than using a twilight flat.

We performed sky subtraction in the individual images. Although the flat
fielding almost mitigated the sky fluctuations, two-dimensional background
models were used to eliminate the remaining trivial sky gradients. Briefly, we
divided an object-masked image into a $16\times 16$ grid and measured the
median values of each. Then, we generated a pixel-scaled background model with
a second-order two-dimensional polynomial fitting. Using a smaller or larger
grid may change the background model, but it did not significantly affect the
result of this study.

The world coordinate system of the resulting images was updated using
astrometry.net,333https://astrometry.net and then all images were co-added
using SWarp (Bertin et al., 2002) with median-combine. Note that the dark
subtraction removed most of the amplifier glow but left a small spot on the
left side of the images. For that reason, we masked that area in all images
before co-addition so that it does not affect the result at all.

Figure 5: SDSS $r$ magnitudes of the field stars in the final co-added image
as a function of their total counts estimated using aperture photometry. The
contaminated or saturated ones (gray) were excluded by iterative sigma
clipping, and the rest (red) were used for a linear fit (solid line).

Standardization was performed using the Sloan Digital Sky Survey (SDSS)
$r$-filter data since no archive data is available in the $L$-band. We
utilized the magnitudes of $\sim$870 field stars within a radius of 30 arcmin
from the center of NGC 5907 and measured the fluxes of the stars in units of
counts using FLUX_BEST in SExtractor (Bertin & Arnouts, 1996). As shown in
Figure 5, we estimated the zero point by using $\sim$90% of the matched stars,
excluding defects and saturated ones. It appears that stellar magnitudes
versus counts are nicely fitted by linear regression, resulting in the zero
point of $m_{0,r}\simeq 29.08\pm 0.12$. This nominal value can go down one mag
deeper if the $g$ filter is applied instead. Here, one can see that the
luminance filter has advantageous for LSB detection, but there is a limitation
in quantitative comparison with data obtained using more common filters.
Therefore, we are considering the application of various filter sets for
detecting and analyzing LSB features in the future.

##  4 Results

###  4.1 Identification of LSB features

We remind the readers that one of the main goals of this project is to detect
LSB features such as tidal streams in the outskirts of galaxies, (diffuse)
dwarf galaxies in the nearby universe, and intracluster light of galaxy
clusters. The typical size of these sources is expected to be considerably
larger than the pixel size of our data. Therefore, correction for their actual
angular sizes must be applied to estimate the practical detection limit.

Figure 6: Left: the final co-added image of NGC 5907 field. Right: the zoomed-
in image of NGC 5907 in ours (top) and the $g$-filter image provided from van
Dokkum et al. (2019) (bottom). The FoV of the two is the same, and all bright
objects except NGC 5907 were masked. Both images were smoothed with a Gaussian
filter to emphasize LSB structures. A stellar stream to the east of the galaxy
is evident in both, but our result does not clearly identify a long tail to
the northwest.

Referring to the definition of Román et al. (2020), we estimated the 1$\sigma$
surface brightness limit from the background noise, yielding
$\mu_{r,1\sigma}\sim 28.5$ mag arcsec-2 in $10^{\prime\prime}\times
10^{\prime\prime}$ boxes. Note that the surface brightness limit can vary
depending on ways of measuring surface brightness. For instance, a one-
dimensional radial surface brightness profile of galaxies can be used to
detect much fainter structures. Therefore, the above nominal value may be
considered an upper limit to our sensitivity.

Figure 6 shows the final co-added and zoomed-in images of NGC 5907, which
directly demonstrate the telescope’s performance in detecting LSB features. To
emphasize the LSB features in the zoomed-in image, all sources but those
related to NGC 5907 were masked, and the image was smoothed with a Gaussian
function. In short, we were able to clearly identify a single curved stellar
stream to the east of NGC 5907, as already reported in the literature (e.g.,
Shang et al., 1998; Laine et al., 2016; Müller et al., 2019; van Dokkum et
al., 2019). Our result can be another observational evidence that disagrees
with the result of a double loop structure reported in Martínez-Delgado et al.
(2008).

Meanwhile, van Dokkum et al. (2019) reported that the stellar stream on the
east side appears to pass through NGC 5907 and extends long to the northwest.
For direct comparison, we present the zoomed-in images of both ours and van
Dokkum et al. (2019)444This $g$-filter image is provided from the website
https://www.pietervandokkum.com/ngc5907 with the same FoV (the right panels in
Figure 6). We could find no definite evidence of any structure extending to
the northwest in our image. This result seems to be reasonable since van
Dokkum et al. (2019) reported that the structure on the west side of the
galaxy has a surface brightness fainter than 28 mag arcsec-2 in both $g$\- and
$r$-filters, which is marginally below our surface brightness limit.

One might argue that there seem to be sparse, faint clumps along the reported
long tail. However, it cannot be evidence of the tail because it is difficult
to distinguish whether it is a part of a tail structure or a fluctuation
caused by unmasked diffuse light from bright sources. Back to the point, the
existence of the stellar stream on the east side seems quite certain due to
the continuity. On the other hand, in the case of the west side, we would not
have been able to claim the existence of the tail at all without comparing it
with the result of van Dokkum et al. (2019).

###  4.2 Comparison with other studies

Describing our telescope’s performance by comparing the imaging result with
those of other surveys may be inappropriate because all imaging surveys have
been conducted under different photometric conditions and using various
observation strategies such as different exposure times and dithering
patterns. In addition, multiple factors in the observation system, such as
aperture size, detector type, and filter sets, are also involved with the
telescope’s performance. Despite this complexity, comparing several
characteristics of imaging surveys that have achieved similar results can help
us understand our telescope’s efficiency in LSB detection.

Table 1: Key characteristics of the studies that observed the stellar stream
of NGC 5907

Reference | Diameter | Total Exposure | filters | Telescope
---|---|---|---|---
| (m) | (hr) |  |
Shang et al. (1998) | 0.6/0.9 | 26.17 | m6660, m8020 | Schmidt telescope
Martínez-Delgado et al. (2008) | 0.5 | 11.4 | $L$ | Ritchey–Chrétien telescope
Laine et al. (2016) | 8.2 | 0.2 |  $g$, $r$, $i$ | Subaru telescope
Müller et al. (2019) | 1.4 | 7.2 | $L$ | Milanković telescope
van Dokkum et al. (2019) | 0.143 (1) | 115 (4.8) |  $g$, $r$ | Dragonfly Telephoto Array
This study | 0.3 | 2.0 | $L$ | K-DRIFT pathfinder

Note. — The values presented here are only for a single aperture with a single
filter. Therefore, the Dragonfly Telephoto Array can practically be regarded
as a 1 m telescope with a total exposure time of 4.8 hr.

Table 1 presents the key characteristics of previous observations that
identified the stellar stream of NGC 5907. Note that not all the studies have
been aimed at LSB research, so their data reduction may not have been
optimized. Our telescope has the smallest aperture size555The Dragonfly
Telephoto Array consists of $48\times 0.143$ m lenses. Its single aperture is
smaller than ours, but the array can practically be regarded as a single 1 m
telescope. and moderately short total exposure time but could achieve
comparable performance in LSB detection.

It is worth noting that van Dokkum et al. (2019) achieved a deeper surface
brightness limit than ours, owing to the time-efficient array system. Their
observation time was about 2.5 times longer than ours, but the practical total
integration time was nearly 60 times longer. According to the principle that
background noise is reduced by the square root of the number of co-added
images, using our telescope would not require such a long integration time to
achieve a result similar to that of van Dokkum et al. (2019). It is expected
to achieve a sufficient surface brightness limit if a total integration time
of only 2–3 times longer than the present.

Instead, as a telescope with a single aperture, the description comparing
characteristics with the Subaru telescope can be more concise. Since Laine et
al. (2016) did not specify the surface brightness limit from the background
noise, let us assume the photometric depth is similar to ours. The aperture
size of our telescope is about 30 times smaller than that of the Subaru
telescope, while our total exposure time is ten times longer. Due to adopting
the luminance filter, our telescope can collect about twice as much light over
the same time. This suggests that our telescope is $\sim$$30^{2}/10/2=50$
times more efficient than the Subaru telescope, at least in some sense, in
detecting LSB features.

Figure 7: Example of a 30-sec exposure of Arcturus. Not only scattered light
but also reflection features are clearly shown. The dashed and dotted arcs
indicate the reflection edges, which might be created from the dewar window
and filter. The radii of annuli are $\sim$4 and $\sim$5.6 arcmin,
respectively.  Figure 8: Comparison of the PSF of our telescope (red) with
those of the Burrell Schmidt Telescope (dotted; Slater et al., 2009) and the
Dragonfly Telephoto Array (dashed; Abraham & van Dokkum, 2014). The fluxes of
the three PSF profiles were adjusted so that the fluxes at a radius of 0.1
arcmin were identical. The left and right panels show the PSF profiles as a
function of radius on linear and logarithmic scales, respectively.

###  4.3 Scattering and reflection

Apart from the deeper surface brightness limit of van Dokkum et al. (2019), a
significant unmasked light can be seen in the Dragonfly Telephoto Array image,
especially in the lower right corner (Figure 6). Even considering the
different object-mask sizes, our data seems less affected by scattered light.
Thus, in addition to identifying the faint stellar stream of NGC 5907, it
would be worthwhile to examine the scattered and reflected light. Indeed, a
large PSF can have a significant impact on estimating the brightness and shape
of LSB features (see de Jong, 2008; Tal & van Dokkum, 2011; Sandin, 2014;
Trujillo & Fliri, 2016).

The detailed analysis of scattering and reflection is out of the scope of this
paper. Instead, we examined the PSF profile of our telescope using a bright
star. Before that, it would be helpful to check the appearance of the
scattering and reflection visually. Figure 7 presents an example of a 30-sec
exposure of Arcturus. The most conspicuous artifact is a vertical jet-like
feature on a scale of tens of arcmin. It appears to have a slightly misaligned
and diffuse shape, unlike the typical saturation bleed. The right panel
emphasizes an asterisk-like scattered light and reflected light from the dewar
window and filter. The outer annulus was revealed to be out to a radius of 5.6
arcmin. We will discuss these features in the next section.

The PSF core was parameterized using PSFEx (Bertin, 2011) that extracts a
precise model of unsaturated field stars. The extended PSF wing was modeled
from Arcturus using a simple circular isophote fitting to avoid the effect of
the complex structures. Here, we could construct the profile in the saturated
area of the star by connecting the fragmented wing profiles modeled from the
images with different exposure times (e.g., 1, 10, 30, and 60 sec).

Figure 8 shows the radial profile of the PSF model of our telescope. The PSF’s
core size is pretty large (FWHM $\sim$ 6–8 arcsec), but the wing appears very
compact and faint. To compare its shape, we also presented the PSF models of
the Burrell Schmidt telescope (Slater et al., 2009) and the Dragonfly
Telephoto Array (Abraham & van Dokkum, 2014). The PSF wing reaches a very low
level, as low as about 100 times fainter than the former on average. The
overall trend of the PSF profile is comparable to the latter but seems
slightly different. While the outermost wing of our PSF model is fainter than
that of the Dragonfly Telephoto Array, an excess exists between the radii of
$\sim$0.5–8 arcmin in our PSF. This seems to be caused by the scattering and
reflection of light shown in Figure 7. Nevertheless, we conclude that this is
a very encouraging result compared to the PSF level of conventional
telescopes. We remind that this result is preliminary because we used only a
single bright star centered on the image. Indeed, the characterization of the
wide-angle PSF may be sensitive to sky subtraction, particularly in crowded
regions (cf. Liu et al., 2022). We will improve the telescope’s PSF model
analysis in future work.

##  5 Discussion

Although this study achieved rather encouraging results in LSB detection,
several unexpected issues were revealed in the detector and the mirror. This
section introduces and discusses the issues that must be addressed in future
work to improve data quality and photometric depth.

First, our data seems to be affected by false signals, namely the RBI.666In
general, the RBI effect is known to mainly occur in front-illuminated CCD
chips. The cause is unknown, but the same phenomenon occurred in this study
with a back-illuminated sCMOS chip. In particular, the RBI effect can be
clearly seen in the dark frame taken after object exposure, specifically as
follows: (1) the dark level was slightly raised by the high background level,
and (2) faint spots were left where bright stars were placed on the last
object frame. These traces gradually faded and disappeared after at least
$\sim$10 min. This means that the RBI effect may affect almost all science
images taken at 30-sec intervals. However, the impact on the background
fluctuation in the final co-added image is not expected to be very significant
since the RBI level is only $\sim$0.5% of the original signal and disappears
during dithering observation.

Nevertheless, the RBI effect must be eliminated. If the detector is the cause,
it can be solved by replacing it with another detector. Otherwise, we must
find other ways to minimize this effect. The RBI effect is known to be
temperature-dependent. The lower the chip’s temperature, the longer trapped
electrons can remain. In addition, the brighter the object, the stronger the
residual in a shorter time. In future work, we will characterize the RBI
effect of the CMOS chip depending on temperature and exposure time so that we
can determine an optimized observation strategy.

Second, the PSF’s core size is twice as large as expected from the optics
design. Even considering the average seeing of the observatory ($\sim$2–3
arcsec), it must be smaller than the present. Some test observations not
introduced in this paper suggest that the large PSF is unlikely due to
focusing or tracking problems. In addition to this, the (vertically) scattered
light was clearly seen around bright stars. Ideally, this problem should not
occur because our telescope was designed not to scatter any light on the
optical path by removing internal structures. We suspect that the secondary
mirror with insufficient surface precision mainly causes these problems based
on optics simulation using mirror models provided through surface measurement.
The secondary mirror will be replaced by a new one manufactured more precisely
in mid-2022, and then the problem is expected to be solved or at least
reduced. Another possibility is that it might be caused by diffraction from
the microlenses on the CMOS array. We will also check this issue after
replacing the secondary mirror.

Lastly, internal reflections by the dewer window and filter mimic an extended
halo and contaminate the vicinity of bright stars. This problem can be reduced
by using anti-reflective coatings on the filter and dewer window. However,
faint reflections may still appear in practice. Eventually, a software
solution must be invented to eliminate these features. The reflection features
may have diverse brightness and central position depending on the position and
brightness of the object on the image. Therefore, we will characterize the
reflection through photon simulation with the exact specifications of our
telescope in future work.

##  6 Summary

A simple optics design is advantageous for studying LSB nature because it can
reduce photometric uncertainty arising from the instrument, thereby achieving
extremely deep surface brightness limits. For this reason, we have been
developing new telescopes adopting an optics design with a clean pupil to
observe LSB features effectively. In early 2021, we successfully manufactured
a prototype telescope called the K-DRIFT pathfinder. This telescope has an
off-axis design consisting of a linear astigmatism free-three mirror system.
Since there is no structure on the optical path, it was expected to minimize
the loss and scattering of light.

To assess the telescope’s performance, we observed NGC 5907, which is well
known to have a large stellar stream. The data taken for two days were
processed through the optimized data reduction, including flat fielding using
a dark-sky flat and dedicated sky subtraction. We obtained a wide and deep
image with a total integration time of 2 hr by co-adding reduced images. As a
result, we were able to achieve a 1$\sigma$ surface brightness limit of
$\sim$28.5 mag arcsec-2 in $10^{\prime\prime}\times 10^{\prime\prime}$ boxes
and identify a single curved stellar stream to the east of NGC 5907. Compared
to other observational studies that have shown similar detection ability, one
can say that our telescope is more efficient in detecting LSB features at face
value. In addition, we found that the PSF wing reached a very low level
despite being affected by some reflection features.

We highlight that this is an encouraging result, considering the relatively
small aperture and short integration time. Data with similar conditions to
this study would help detect tidal features and faint dwarf galaxies. However,
it seems insufficient to identify fainter structures such as stellar halos in
galaxies because of much lower surface brightness levels and their
diffuseness. We expect this limitation to be overcome by increasing exposure
time and applying more effective observation strategies that allow the
background to be as flat as possible, such as adding rotation to the dithering
pattern.

We note that the K-DRIFT pathfinder is a prototype of four to five telescopes
to be built over the next decade. Through this study, we understood what to
upgrade to improve the performance of the next K-DRIFT telescopes. For
example, more precise mirror manufacturing is required, and the detector’s
characteristics, including the RBI effect, must be clarified. At the same
time, data must be expanded through intensive surveys for scientific
achievements. By adopting a larger FoV and various filter sets, we expect to
detect LSB features more effectively and analyze the LSB nature in detail.

We are grateful to an anonymous referee for constructive comments and
suggestions. This research was supported by the Korea Astronomy and Space
Science Institute under the R&D program (Project No. 2022-1-830-05),
supervised by the Ministry of Science and ICT. This research was supported by
‘National Research Council of Science & Technology (NST)’ - ‘Korea Astronomy
and Space Science (KASI)’ Postdoctoral Fellowship Program for Young Scientists
at KASI in South Korea.

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