# Galactic-Seismology Substructures and Streams Hunter with LAMOST and Gaia. I. Methodology and Local Halo Results

Guan-Yu Wang, Hai-Feng Wang, Yang-Ping Luo, Yuan-Sen Ting, Thor Tepper-García, Joss Bland-Hawthorn, Jeffrey Carlin

► To cite this version:

Guan-Yu Wang, Hai-Feng Wang, Yang-Ping Luo, Yuan-Sen Ting, Thor Tepper-García, et al. Galactic-Seismology Substructures and Streams Hunter with LAMOST and Gaia. I. Methodology and Local Halo Results. The Astrophysical Journal, 2024, 974, 10.3847/1538-4357/ad6d59. insu-04821654

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# Galactic-Seismology Substructures and Streams Hunter with LAMOST and Gaia. I. Methodology and Local Halo Results

Guan-Yu Wang¹, Hai-Feng Wang¹,²,³,⁴,⁵ ID, Yang-Ping Luo¹ ID, Yuan-Sen Ting⁶,⁷,⁸,⁹ ID, Thor Tepper-García¹⁰,¹¹ ID, Joss Bland-Hawthorn¹⁰,¹¹ ID, and Jeffrey Carlin¹² ID

¹Department of Astronomy, China West Normal University, Nanchong, 637002, People's Republic of China; haifeng.wang.astro@gmail.com

²Centro Ricerche Enrico Fermi, Via Panisperna 89a, I-00184 Rome, Italy

³Department of Physics, Sapienza, Università di Roma, P.le A. Moro 5, I-00185 Rome, Italy

⁴GEPI, Observatoire de Paris, Université PSL, CNRS, Place Jules Janssen F-92195, Meudon, France

⁵Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universita di Padova, Vicolo Osservatorio 3, I-35122, Padova, Italy

⁶Research School of Astronomy & Astrophysics, Australian National University, Canberra, ACT 2611, Australia

⁷School of Computing, Australian National University, Acton, ACT 2601, Australia

⁸Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA

⁹Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA

¹⁰Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia

¹¹Centre of Excellence for All-Sky Astrophysics in Three Dimensions (ASTRO-3D), Australia

¹²Rubin Observatory/Legacy Survey of Space and Time (LSST), 950 N. Cherry Ave., Tucson, AZ 85719, USA

Received 2024 January 10; revised 2024 August 1; accepted 2024 August 2; published 2024 October 14

## Abstract

We present a novel, deep-learning-based method—dubbed Galactic-Seismology Substructures and Streams Hunter, or GS³ Hunter for short—to search for substructures and streams in stellar kinematics data. GS³ Hunter relies on a combined application of Siamese neural networks to transform the phase space information and the K-means algorithm for the clustering. As a validation test, we apply GS³ Hunter to a subset of the Feedback in Realistic Environments (FIRE) cosmological simulations. The stellar streams and substructures thus identified are in good agreement with corresponding results reported earlier by the FIRE team. In the same vein, we apply our method to a subset of local halo stars from the Gaia Early Data Release 3 and GALAH DR3 data sets and recover several previously known dynamical groups, such as Thamnos 1+2, the hot thick disk, ED-1, L-RL3, Helmi 1+2, Gaia-Sausage-Enceladus, Sequoia, Virgo Radial Merger, Cronus, and Nereus. Finally, we apply our method without fine-tuning to a subset of K giant stars located in the inner halo region, obtained from the LAMOST Data Release 5 data set. We recover three previously known structures (Sagittarius, Hercules-Aquila Cloud, and the Virgo Overdensity), but we also discover a number of new substructures. We anticipate that GS³ Hunter will become a useful tool for the community dedicated to the search for stellar streams and structures in the Milky Way (MW) and the Local Group, thus helping advance our understanding of the stellar inner and outer halos and the assembly and tidal stripping history in and around the MW.

*Unified Astronomy Thesaurus concepts: Milky Way (1054); Milky Way stellar halo (1060); Milky Way disk (1050); Local Group (929)*

*Materials only available in the online version of record: figure sets*

## 1. Introduction

The assembly history of the Milky Way (MW), which has formed over billions of years as the result of many accretion and merger events, is imprinted in its stellar substructures and streams (e.g., M. Chiba & T. C. Beers 2000; K. Freeman & J. Bland-Hawthorn 2002; S. R. Majewski 2004; E. J. Bernard et al. 2016; H. J. Newberg & J. L. Carlin 2016; V. Belokurov et al. 2018; A. Helmi et al. 2018; R. A. Ibata et al. 2019; H. H. Koppelman et al. 2019; R. P. Naidu et al. 2020; Z. Yuan et al. 2020; K. Malhan et al. 2022).

In recent years, many research groups have discovered new Galactic stellar streams across different chemical and dynamical spaces as well as over a range of stellar orbital parameters (e.g., T. S. Li et al. 2019, 2022; A. Bonaca et al. 2021a, 2021b). C. J. Grillmair & J. L. Carlin (2016) have pinpointed the location and estimated the kinematic properties as well as the

chemical abundances for some of the major Galactic streams, such as the Sagittarius stream (Sgr), the Virgo Stellar stream, and the Orphan stream or Orphan-Chenab large stream. Recently, C. Mateu (2023) published the *Galstreams* library, a uniform compilation of the orbital parameters (position and velocity) for nearly 100 known Galactic stellar streams. In addition, that study presents an analysis of the issues and uncertainties in the parameters of individual streams across the library, providing clear pointers of needed improvement. The library provides a convenient platform to catalog known stellar streams and promotes the addition and follow-up of new discoveries, greatly assisting the study and our understanding of the stellar streams and structures in and around the MW.

Theoretical models suggest that, under the hierarchical cosmological assembly process, accreting objects will retain their original orbital properties and kinematic states after structurally phase mixing (e.g., A. Helmi & P. T. de Zeeuw 2000). A long-standing problem is to determine which substructures (or streams) were originally formed in situ (i.e., in the Galactic stellar halo) and which ones formed in external systems that were then accreted onto the MW as part of later

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events. One possible solution to this is to estimate as accurately
as possible the kinematic and orbital parameters of stellar
substructures and streams around galaxies, since their orbital
properties are effective tracers of the galaxy formation history
and gravitational potential (D. R. Law & S. R. Majewski 2010).
Thus, the availability of such tracers is essential for our
understanding of the background and history of galaxy
formation, as well as the origin and birth environment of
substructures and streams.

Streams can be formed when globular clusters or dwarf
galaxies are tidally stripped (or entirely destroyed) as a result of
the gravitational interaction with the MW. Some disk
substructures or disk streams can also form by internal and
external perturbations (H. F. Wang et al. 2018a, 2019,
2020b, 2020c, 2020a, 2023). A prime example of the former
process is the Sagittarius stellar (e.g., D. R. Law & S. R. Maj-
ewski 2010; C. W. Purcell et al. 2011) and gaseous (T. Tepp-
er-García & J. Bland-Hawthorn 2018) streams, which formed
during the infall of the Sagittarius dwarf (R. A. Ibata et al.
1994) onto the MW. One stream formed during the dwarf’s first
orbital wrap and is located at roughly $\tilde{\Lambda}_{\odot} \in [-180^{\circ}, 180^{\circ}]$, the
second is at $\tilde{\Lambda}_{\odot} \in [-540^{\circ}, -180^{\circ}] \cup [180^{\circ}, 540^{\circ}]$, and the third
is at $\tilde{\Lambda}_{\odot} \ge 540^{\circ}$ (P. Ramos et al. 2022). Many models for stellar
streams and their structure have also been continuously
proposed, and some colleagues have suggested that the streams
provide an effective way to study the distribution of dark matter
and potential in galaxies (e.g., D. R. Law & S. R. Majew-
ski 2010; J. Peñarrubia et al. 2010; E. Vasiliev et al. 2021;
H.-F. Wang et al. 2022).

A promising approach to identifying signatures of (Galactic) accretion events is based on the use of machine learning (clustering) methods coupled to (ideally) comprehensive data sets. In this respect, large photometric and spectroscopic surveys such as the Sloan Digital Sky Survey (D. G. York et al. 2000), the Two Micron All-Sky Survey (T. H. Jarrett et al. 2000), or the Large Sky Area Multi-Objective Fiber Optic Spectroscopic Telescope (LAMOST; G. Zhao et al. 2012; L.-C. Deng et al. 2012), in addition to the full six-dimensional (6D) phase-space information of millions of stars in the solar neighborhood provided by the Gaia satellite, in particular its Early (E) Data Release 3 (D. Katz et al. 2023), provide an exquisite data set to which these clustering techniques can be seamlessly applied.

The central idea of any these methods is to search for
significant clusters in the distribution of stars in a suitable
projection space. For instance, some earlier studies have looked
at the distribution of stars in 6D phase space or on a plane
defined by the corresponding integrals of motions (IOMs) such
as total energy (E), the component of angular momentum
parallel to the z-axis ($L_z$), and the in-plane component of
angular momentum $L_{\perp} (=\sqrt{L_x^2 + L_y^2})$ (S. S. Lövdal et al.
2022), while other studies have made use of action-angle
coordinates to detect the clustering signal of streams and
substructures based on the ENLINK algorithm (S. Sharma &
K. V. Johnston 2009; Y. Wu et al. 2021).

Previous detection methods for stellar streams include the
matched filter technique (C. M. Rockosi et al. 2002; E. Balbinot
et al. 2011), the detection of comoving groups of stars (C. Martin
et al. 2013), and the pole count technique (K. V. Johnston et al.
1996). Some comparisons for these three methods may be found
in K. Malhan & R. A. Ibata (2018).

Other clustering algorithms, developed and improved over
many years, have been used to find and identify streams and
substructures (M. Davis et al. 1985; K. Malhan &
R. A. Ibata 2018; Z. Yuan et al. 2018; S. S. Lövdal et al.
2022). These include the following.

(i) *Friends-of-friends*. The friends-of-friends (FOF) algorithm (M. Davis et al. 1985) is a classical algorithm for detecting (sub)structures. It is also a purely geometric method in which particles that are closer than a given scale (link length) calculated by Euclidean distance are connected. Different particles will be distributed in different regions. This method needs identifying substructures within the halo at a fixed linkage setting. Using the LAMOST Data Release 5 (DR5) K giant sample with distances $d \approx 5-120$ kpc and FOF methods, C. Yang et al. (2019) have identified streams and substructures in the Galactic halo, such as the Sgr, the Monoceros Ring, the Virgo Overdensity (VOD), the Hercules-Aquila Cloud (HAC), the Orphan streams, and other previously unknown substructures.

(ii) *Star's Galactic Origin*. Developed by Z. Yuan et al. (2018), this is a substructure identification method based on the self-organizing map (SOM) technique (T. Kohonen 2001). The advantage of the algorithm is that the SOM grid automatically finds the nearest node and updates the position based on iteration and Euclidean distance calculation and has successfully identified structures such as the Cetus stream (H. J. Newberg et al. 2009), among others. SOM is used to generate a low-dimensional space for training samples, which can transform complex nonlinear statistical relationships among high-dimensional data into simple geometric relationships and display them in a low-dimensional way to provide a convenient method for further calculations. And the aim of SOM is to map n-D input data to a 2D neural map while retaining the topological structures within the data at the same time. The main point we should note about SOM is that this algorithm needs to provide weight information to a large quantity of neurons in order to cluster the input data. To be successful, the input to the weight layer must be carefully chosen. Using Star's Galactic Origin (StarGO), the metal-poor outer part of the stellar halo of the MW ($d > 15$ kpc) and the northern counterpart of the Cetus structure were unraveled in Z. Yuan et al. (2019).

(iii) *Single linkage cluster*. The single linkage algorithm (S. S. Lövdal et al. 2022; T. Ruiz-Lara et al. 2022) is a hierarchical clustering method for detecting (sub)structures. The distance between the nearest two nodes is calculated using the Mahalanobis distance. Applying this technique, S. S. Lövdal et al. (2022) found 67 local halo clusters in the IOM space within $d = 2.5$ kpc from the Sun. This algorithm is computationally intensive because it has to calculate the two-by-two distance of all data points in multiple clusters each time. In addition, since hierarchical clustering uses the greedy algorithms, one of the characteristics of such algorithms is that the results obtained may be locally optimal and not necessarily globally optimal. But this is not necessarily a disadvantage, because the locally optimal solution is also sufficient to solve the problem in the search for agglomerative structures.

E. Dodd et al. (2023) have adopted the single linkage
clustering method proposed by S. S. Lövdal et al. (2022) and
found seven main dynamic substructures in the local Galactic
stellar halo, some of which are believed to have formed in situ,
and others which likely formed during earlier accretion events.
Most of the identified groups belong to larger structures such as Gaia-Sausage-Enceladus (GSE; C. B. Brook et al. 2003; A. Helmi et al. 2018; V. Belokurov et al. 2018), L-RL3 (E. Dodd et al. (2023),<fnref n="13" /> the hot thick disk (A. Helmi et al. 2018; P. Di Matteo et al. 2019), Sequoia (H. Koppelman et al. 2018), the Helmi stream (T. Ruiz-Lara et al. 2022), and Thamnos (H. Koppelman et al. 2018). In addition, E. Dodd et al. (2023) discovered a previously unknown substructure, dubbed ED-1, and identified 11 independent clumps, five of which were previously unknown (ED-2, 3, 4, 5, and 6).

T. Antoja et al. (2012) have applied the clustering method of wavelet transform to identify the overdensity distribution in velocity space around the Sun. They found that the main kinematic groups are consistent with belonging to large-scale features, and they also identified a new group in the local disk region.

With the help of the hierarchical clustering method and the Gaia Early DR3 (EDR3) data set, as well as catalogs from ground-based spectroscopic surveys, using a single linkage algorithm defined by the usual integration of the motion energy $E$ and the two components of the angular momentum $L_z$ and $L_\perp$, S. S. Lövdal et al. (2022) actually unveiled six main groups or substructures; more population properties of these substructures were also discussed in a later work (T. Ruiz-Lara et al. 2022). In contrast, T. Donlon & H. J. Newberg (2023) have also analyzed the components in the local halo using the Bayesian Gaussian mixed model regression algorithm to characterize the local stellar halo, and they found that the data fit the model best with four components: Virgo Radial Merger (VRM), Cronus, Nereus, and Thamnos. They suggest that the GSE merger event is probably a combination of these four components.

The Stream Finder method (K. Malhan & R. A. Ibata 2018) has been developed to explicitly identify dynamically cold and thin stream structures based on their orbit parameters. Using this method, K. Malhan & R. A. Ibata (2018) have succeeded in detecting the GD-1 stellar stream based on the Pan-STARRS1 data set (K. Malhan et al. 2018). Applying this method to the Gaia DR2 data set, R. Ibata et al. (2020) have presented a complete 6D panorama of the Sgr. Moreover, Z. Yuan et al. (2022) combined the StarGO method together with Stream Finder and found two structures in the southern extension of the northern Cetus stream: the Palca stream and a new southern stream that wraps around the sky extending over 100°. Moreover, by combining data sets of 70 globular clusters, 41 streams, and 46 satellites together, K. Malhan et al. (2022) detected a total of six groups, including the known Sagittarius, Cetus, GSE, LMS-1/Wukong, and Arjuna/Sequoia/I'toi and a new merger event they refer to as Pontus, also with the help of the ENLINK software (S. Sharma & K. V. Johnston 2009).

In this paper, we introduce a new concept for the identification of stellar streams or substructures based on a novel clustering technique. Our method relies mainly on the application of the Siamese neural networks (U. Shaham & R. R. Lederman 2018) to transform the 6D phase-space information upon which the groups of structures (substructures) are identified by a K-means clustering algorithm.

This paper is organized as follows. In Section 2, we describe the data set selection and processing required before it is subject to our analysis. In Section 3, we describe the technical

details of our novel algorithm. In Section 4, we present some properties of the stellar structures we find by applying our algorithm to the data set, mainly including velocity space. We also present the structures identified in the inner halo as a by-product of applying our method to the LAMOST data set and show their properties and spatial distributions. In Section 5, we summarize our results and compare them with the results from other groups obtained by means of other methods. In Section 6, we conclude our discussion.

## 2. Data

### 2.1. Data Set for the Local Halo Structures (Also Belonging to the Disk Region)

For the local halo search, we utilized the data set compiled by S. S. Lövdal et al. (2022), which incorporates the Gaia EDR3 Radial Velocity Spectrometer (RVS; Gaia Collaboration et al. 2021) sample of stars with the line-of-sight velocities, supplemented by radial velocities observed by ground-based spectroscopy such as the sixth data release (DR6) of the LAMOST low-resolution (J. Wang et al. 2020) and medium-resolution (N. Liu et al. 2019) survey, the DR6 from the RA dial Velocity Experiment (M. Steinmetz et al. 2020), the third data release from Galactic Archaeology and HERMES (GALAH DR3; S. Buder et al. 2021), and the 16th data release from the Apache Point Observatory Galactic Evolution Experiment (R. Ahumada et al. 2020).

Stars with distances greater than 2.5 kpc have been removed from this catalog; the distance was computed by inverting the parallaxes after correcting for a zero-point offset of 0.017 mas (L. Lindegren et al. 2021). To improve the quality of the data, stars for which the normalized unit weight error is larger than 1.4 are removed. For this work, the local standard of rest $v_{LSR} = 232 \text{ km s}^{-1}$, the distance of the Sun from the Galactic center is 8.178 kpc (GRAVITY Collaboration et al. 2019), and $(v_\odot, V_\odot, W_\odot) = (11.1, 12.24, 7.25) \text{ km s}^{-1}$ is the peculiar motion of the Sun (R. Schönrich et al. 2010; M. J. Reid & A. Brunthaler 2020), all of which are consistent with the Galaxy parameter set recommendations of J. Bland-Hawthorn & O. Gerhard (2016).

Here the total energy is $E = V^2/2 + \Phi(x)$, where $V$ is the total velocity, $\Phi(x)$ is the gravitational potential of the galaxy at the location of the star, and the potential is calculated from H. H. Koppelman et al. (2019) and S. S. Lövdal et al. (2022), that is, the potential includes a Miyamoto–Nagai disk, a Hernquist bulge, and a Navarro–Frenk–White halo. The MW parameters are $(a_d, b_d) = (6.5, 0.26)$ kpc, $M_d = 9.3 \times 10^{10} M_\odot$ for the disk; $c_b = 0.7$ kpc, $M_p = 3.0 \times 10^{10} M_\odot$ for the bulge; and $r_s = 21.5$ kpc, $c_h = 12$, $M_{\text{halo}} = 10^{12} M_\odot$ for the halo (after tests, we find that different masses will not change our final conclusions). More details can be found in S. S. Lövdal et al. (2022). As mentioned in S. S. Lövdal et al. (2022), the halo stars are selected by requiring $|v - v_{\text{LSR}}| > 210 \text{ km s}^{-1}$.

We make full use of this publicly available data set with $N = 51,671$ stars in our deep learning methods.<fnref n="14" /> Figure 1 shows the recovered sample distribution for the local halo stars in IOM space, colored according to the density. As seen, the energy peak is around $-1.5 \times 10^5 (\text{km}^2 \text{s}^{-2})$, and the vertical angular momentum peak is around $0 \times 10^3 \text{ kpc km s}^{-1}$. The spatial and Galactic velocity distribution are shown in Figure 2.

<footnote n="13">¹³ We adopt the nomenclature introduced by E. Dodd et al. (2023).</footnote>

<footnote n="14">¹⁴ We adopt EDR3 for this project as a proof of concept because this work was initiated during the time when only EDR3 was available.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Local halo sample distribution in IOM space (angular momentum and energy), colored by the normalized star counts. The sample includes 51,671 stars in the solar neighborhood with 6D phase-space information.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Local halo sample distribution in the 6D phase space as a complement of Figure 1 in Galactic coordinates.</figcaption></figure>

## 2.2. Data Set for Inner Halo Region

For the inner halo streams search, the data set is based on the LAMOST DR5 catalog of about 9 million spectra, of which about 5 million have information on stellar parameters and radial velocities. Using the criteria mentioned in C. Liu et al. (2014) and H. F. Wang et al. (2018a), we select a sample of 1 million K giant stars. After cross-matching with Gaia DR3 using TOPCAT (M. B. Taylor 2005) with a matching radius of 1", we obtain detailed information about the sky position, distance, line-of-sight velocity, and proper motion of 248,040 K giant sources after cross-matching with the Gaia DR3 data set. Figure 3 (upper panel) shows the distribution of the line-of-sight velocity with longitude for the 248,040 K giants we have selected, most of which are located in the disk.

Since we wish to focus on the Galactic inner halo, we exclude K giants within 5 kpc of the Galactic disk ($|z| < 5$ kpc) according to the selection criteria in C. Yang et al. (2019).

After removing stars with $|z| < 5$ kpc, most of the disk stars are excluded, as shown in the lower panel of Figure 3. Finally, we build a sample of 8099 halo K giants with 3D position, 3D velocity, and metallicity. The spatial distribution of our final sample in the x-z plane is shown in Figure 4, and the sky distribution of halo K giants is also displayed in Figure 5. Most

of the K giants have Galactocentric distances between 5 and 60 kpc. Next, we will detect and identify substructures from the selected sample of halo K giants using the Galactic-Seismology Substructures and Streams Hunter (GS³ Hunter) method, which will be demonstrated in Section 4.

## 3. Methodology and Validations

In our study, we tackle the challenge of analyzing data within a complex 6D kinematic space. The inherent sparsity and irregularity of these data demand a sophisticated approach for effective clustering. Our methodology integrates neural network architectures and clustering techniques to simplify the data and enhance their interpretability.

At the heart of our method is the Siamese network (U. Shaham & R. R. Lederman 2018), an architecture adept at comparing and identifying relationships between pairs of input data. This network is ideal for our purpose as it processes two parallel input streams, learning to discern the similarities or dissimilarities between them. Specifically, the Siamese network is trained to map the input data into some embedding space z, and dissimilar pairs are farther apart. Next, by calculating the distance of data pairs in the embedding space, their similarity can be determined. Two data points, $x_0$ and $x_1$, are considered
<figure><img src="image_6.png" /><figcaption>Figure 3. Galactic longitude vs. line-of-sight velocity of LAMOST K giants. The top panel shows the whole sample of 248,040 stars cross-matched with Gaia DR3 within 1". The bottom panel is the sample $|z| > 5$ kpc, and the edge-on projection is shown in Figure 4.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4. Spatial distribution on the x-z plane for 8099 LAMOST inner halo K giant stars, which will be used to for a quick investigation of the inner halo substructures, as a by-product of this work. The K giant catalog and selection criteria are based on C. Liu et al. (2014) and C. Yang et al. (2019), respectively. More detailed inner and, in particular, outer halo streams will be shown in future work.</figcaption></figure>

positively related (similar) if their Euclidean distance is below a specific threshold ($\text{siam}_D$) in the embedding space. Our selection process for the $\text{siam}_D$ value involved the following. (a) Referencing the Mahalanobis distance results from the previous step. (b) Conducting multiple tests to verify that the final number of candidate groups met our filtering criteria. (c) Empirically choosing the most appropriate $\text{siam}_D$ value based on these tests. While we aim to make this parameter more flexible in future iterations of our method, we believe that a fixed value is reasonable for the current study. The sensitivity of our results to this parameter choice is an important consideration, and we acknowledge that a more detailed analysis of this relationship could provide valuable insights into the robustness of our method.

To quantify the similarity of the data points in the input parameter space and get the value of $\text{siam}_D$, we used the Mahalanobis distance.<fnref n="15" /> Compared with the Euclidean distance, the Mahalanobis distance takes the measurement covariances between different features more into account. The Mahalanobis distance is calculated as

$$D_M(x_0, x_1) = \sqrt{(x_0 - x_1)^T \Sigma^{-1} (x_0 - x_1)}. \quad (1)$$

In our case, the 6D covariance $\Sigma$ is derived from the Gaia phase-space data.

Training the Siamese network focuses on minimizing a contrastive loss objective that draws similar data points closer in the embedding space while distancing the dissimilar ones. The loss function (which we refer to U. Shaham et al. 2018) is

$$L(x_0, x_1) = \begin{cases} ||z_0 - z_1||^2, & (x_0, x_1) \text{ is positive,} \\ \max(c - ||z_0 - z_1||, 0)^2, & (x_0, x_1) \text{ is negative,} \end{cases} \quad (2)$$

where $c$ is a margin (typically set to 1) and the Siamese net maps the data point $x_0$ into the embedding space $z_0$ and clusters closely related samples in $z$-space within a specified threshold ($\text{siam}_D$). The distance in the $z$-space is assumed to be Euclidean. The input $x$ is in $\mathbb{R}^6$, which consists of the phase-space information. $\text{siam}_D$ here serves to generate pseudolabels for the contrastive learning, since the data are unlabeled.

The architecture of the Siamese network follows closely from U. Shaham et al. (2018). Briefly, in this work, it consists of a multilayer perceptron (MLP) of four layers, with the activation of a rectified linear unit (RELU) function, which is a commonly used activation function in neural networks. It introduces nonlinearity into the network, enhancing its capacity to learn complex patterns from data. A batch normalization layer has been added to normalize the data and accelerate network convergence. Additionally, a custom lambda layer has been included to compute the Euclidean distance in the embedding space $z$.

To further enhance the contrast in the clustering, especially in the high-density region, we further feed the output of the Siamese network into an MLP, mapping into the final transformed space of $y \in \mathbb{R}^s$. The MLP has a structure consisting of four layers with a RELU activation function, followed by a layer with a tanh activation function. We also assume the L2 regularization, which improves generalization and reduces overfitting during training. The MLP trained with a self-supervision loss is

$$L(\theta) = \mathbb{E}_{(x_i, x_j)} [\omega(z_0, z_1) ||y_0 - y_1||^2], \quad (3)$$

where $\omega(z_0, z_1)$ is the affinity matrix calculated by k-nearest neighbors (KNN)<fnref n="16" /> in the embedding space and $||y_0 - y_1||$ is the similarity between $y_0$ and $y_1$ calculated by the Euclidean distance ($y_0, y_1 \in \mathbb{R}^s$). The expectation is done over all pairs, although due to the affinity matrix, only a few pairs contributed. The affinity matrix of the embedding is used as the weights for the loss function in order to put more weights on “nodes” with higher density and therefore require the network to put more focus on this challenging regime. In practice, we found that setting $s$ to be the number of clusters is a good rule of thumb.

Finally, the output embedding ($y \in \mathbb{R}^s$) undergoes unsupervised clustering through the K-means algorithm,<fnref n="17" /> an empirically determined number of clusters sufficient to capture structures in the $\mathbb{R}^s$ embedding space.

<footnote n="15">¹⁵ The relevant documentation can be found at https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.mahalanobis.html.</footnote>

<footnote n="16">¹⁶ The relevant documentation can be found at https://scikit-learn.org/stable/modules/generated/sklearn_neighbors.KNeighborsClassifier.html.</footnote>

<footnote n="17">¹⁷ The relevant documentation can be found at https://scikit-learn.org/stable/modules/generated/sklearn.cluster.KMeans.html.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Galactic sky coverage of the LAMOST final halo K giant sample we adopt in this work. Stars are colored according to Galactocentric distance $r_{gc}$, line-of-sight velocity $V_{los}$, and tangential velocities ($v_l, v_b$).</figcaption></figure>

We note that *K*-means clustering can produce minor, nonrobust clusters. To mitigate this, we introduce a density-based validation step using the density peaks clustering algorithm (DPCA; A. Rodriguez & A. Laio 2014). For each cluster identified by *K*-means, we determine the peak via DPCA. From this peak, we define a radius $r = \sigma \times \sqrt{\lambda}$, where $\sigma$ represents a confidence interval based on the $\chi^2$ distribution ($\chi^2 = 12.95$). The choice of the $\chi^2$ distribution is motivated by the consistency of the algorithm compared to a $\chi^2$ distribution with 6 degrees of freedom. And $\lambda$ is the eigenvalue from principal component analysis (PCA) performed on the cluster, considering the cluster's orientation in the 6D space. A cluster is deemed robust and significant if it contains 95% of its data points within this radius.

This methodology presents a rigorous yet accessible strategy for interpreting data from a high-dimensional space. It offers a distinct advantage over classical methods like FOF, which primarily rely on Euclidean distance in feature space. Our use of the Siamese network enables effective initial data transformation. We employ a machine-learning-driven method to automatically determine appropriate similarity through Siamese network training and map the originally loose and irregular structure in the initial data space into more isotropic groups in the $\mathbb{R}^s$ space by neural network transformation. Crucially, we also account for the inherent covariance among the 6D input parameters by initially assessing the similarity between data points using the Mahalanobis distance. Additionally, $\text{GS}^3$ Hunter is proficient at detecting both dynamically cold and moderately hot stellar streams. When combined with

sophisticated clustering and validation techniques, our method is well equipped to uncover meaningful patterns and structures within the kinematic–dynamic space. This multifaceted approach ensures the robustness and relevance of the identified clusters over traditional clustering methodologies.

We have tested the algorithm using the Feedback in Realistic Environments (FIRE) simulation,<fnref n="18" /> mainly from the FIRE-2 “Latte” and “ELVIS” suites. These simulations, detailed in previous literature (A. R. Wetzel et al. 2016; P. F. Hopkins et al. 2018; S. Garrison-Kimmel et al. 2019; J. Samuel et al. 2020; N. Panithanpaisal et al. 2021; A. Arora et al. 2022; A. Wetzel et al. 2023; N. Shipp et al. 2023; T. Nguyen et al. 2024), offer a rich data set for our purpose. The “Latte” suite includes simulations of galaxies such as m12i, m12f, m12m, m12b, m12c, and m12w, while the “ELVIS on FIRE” suite encompasses the m12 Thelma & Louise and m12 Remus & Romulus simulations. These simulations implement stellar evolution and feedback models, including processes such as stellar winds, core collapse, Type Ia supernovae, photoelectric heating, etc. The FIRE-2 simulations achieve parsec-scale resolution. In the work, they have released three different suites as well as the accompanying (sub)halo catalogs of structures, and the FIRE data are currently publicly available.<fnref n="19" /> Other analysis and details can be found in A. Wetzel et al. (2023).<fnref n="20" /> Our choice of the FIRE sample is influenced by its dominance

<footnote n="18">¹⁸ https://fire.northwestern.edu/milky-way/</footnote>

<footnote n="19">¹⁹ http://flathub.flatironinstitute.org/fire</footnote>

<footnote n="20">²⁰ https://wetzel.ucdavis.edu/fire-simulations/</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 6.** The spatial distribution of the positions and velocities of the FIRE simulation data, where the different colors represent different labeled structures (N. Panithanpaisal et al. 2021); more details can also be found in A. R. Wetzel et al. (2016) and A. Wetzel et al. (2023).</figcaption></figure>

in satellite and phase-mixed substructures, as classified by N. Panithanpaisal et al. (2021). “Satellite” is defined as a candidate with $120–10^5$ star particles and maximum pairwise distances less than 120 kpc. “Phase-mixed” is defined as a candidate with $120–10^5$ star particles, maximum pairwise distances exceeding 120 kpc, and a median local velocity dispersion above the value specified in their Equation (1). “Stream” is defined as a candidate with $120–10^5$ star particles, maximum pairwise distances over 120 kpc, and a median local velocity dispersion below the value specified in their Equation (1). The differences in criteria for defining structures could explain some differences in classification. Specifically, we apply the GS³ Hunter algorithm to the simulation data, which resulted in the identification of 33 distinct groups. Notably, among these groups, eight true progenitors were recognized. The distribution of the original FIRE structures in Figure 6, Then we apply the GS³ Hunter algorithm to the FIRE data, the effectiveness of the GS³ Hunter algorithm in recovering these progenitors is detailed in Figure 7, the fraction shown in Figure 8.It should be noticed that multiple clusters being linked to a single structure is normal results of the current work.

We also note that there are some differences between the original mock data and the reconstructed results, which can be attributed to the following factor: our method utilizes 6D kinematic parameters (positions and velocities) as input. According to N. Panithanpaisal et al. (2021), phase-mixed structures may appear more dispersed in this space, affecting our results. Additionally, the choice of input space significantly impacts our results. The cutoff distance ($siam_D$), which

determines if two sample particles belong to the same group/cluster, is fixed. Consequently, simulations with more phase-mixed structures and stellar streams will result in relatively poorer reconstructions using our method.

Additionally, in FIRE simulations, higher-mass progenitor structures tend to become phase-mixed with the host galaxy more rapidly. Structures with larger masses generally exhibit greater velocity dispersion (see Figures 10, 11, and 12 in N. Panithanpaisal et al. 2021), which can challenge our method’s ability to detect diffuse structures. Considering these factors, the current results are reasonable for these differences.

There are two key observations from this exercise. First, the GS³ Hunter algorithm successfully recovers the majority of the substructures, underscoring the robustness of our method. This achievement demonstrates the algorithm’s capability in accurately identifying complex structures within the simulated data. Second, it is important to note that the eight identified progenitors were found across different groups. This outcome suggests that, while the downstream MLP network aids in grouping data closer to their true progenitors, the process is not flawless. The groups recovered by the GS³ Hunter algorithm can, and in some cases do, originate from different progenitors. This observation highlights the nuanced nature of the data and the complexity involved in perfectly aligning the algorithmically identified groups with their true progenitors.

We also show the results for the detailed fraction in Table 1 in Appendix C. Next, we will compute and analyze the data sets for group finding from GS³ Hunter, and we also summarize
<figure><img src="image_10.png" /><figcaption>**Figure 7.** The results after applying GS³ Hunter in FIRE are shown in 6D space, where the different colors correspond to the different structures shown by the labels below.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** The fraction analysis for the results of Figures 6 and 7. The horizontal coordinate is the name of the real structure in FIRE, and the vertical coordinate is the number. The dashed part of the bar graph is the total number of each real structure, while the colored bars indicate the corresponding part of FIRE found by GS³ Hunter. We label each bar with the percentage of the corresponding structure in the results. The low fraction for a few mock galaxies (e.g., m12b) might be caused by the relatively higher stellar mass, velocity dispersion, and phase mixing time in FIRE simulations and the grouping classification of FIRE-2; in addition, the drawback of the method is another factor, which is not good at finding the relatively diffuse or very hot streams.</figcaption></figure>

the results in Tables 2 and 3, presented separately in the Appendix.

# 4. Results

## 4.1. Local Halo Results

We apply the above algorithm to the two data sets primarily by inputting 6D phase-space parameters to the network and then obtaining the groups or substructure candidates. When we have candidates, we first check whether they are reported in other literature. Those that are left over we assign as new candidates to be confirmed in the future. We have described the criteria by which we assign a group to a substructure (which might correspond to one or more groups) above.

We run GS³ Hunter using the 6D parameters (x, y, z, $v_x$, $v_y$, $v_z$) from the local halo sample of 51,671 Gaia RVS stars, setting the initial parameter *siamD* as 3. GS³ Hunter detects 38 groups, 13 groups are associated with known structure and among them 12 are corresponding to the six well-known structures from previous work (one structure might correspond to a few groups), that is, GSE (C. B. Brook et al. 2003; A. Helmi et al. 2018; V. Belokurov et al. 2018), the hot thick disk (A. Helmi et al. 2018; P. Di Matteo et al. 2019), L-RL3 (E. Dodd et al. 2023), Thamnos (H. Koppelman et al. 2018), the Helmi streams (T. Ruiz-Lara et al. 2022), and ED-1 (E. Dodd et al. 2023). More comparisons are shown in the discussion. Our results are shown in Figure 9, which illustrates the clustering features in the *E–L* and velocity space. Different
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Substructures of the solar neighborhood halo identified by GS³ Hunter, where the different colors indicate different groups or clusters. The background in gray shows the stars that are not clustered in the method. The top panels show the IOM energy–angular momentum space, and the bottom panels show the velocity distributions, using the same colors for each component. As seen, six well-known substructures are revealed here.</figcaption></figure>

colors represent different previously known structures, and the gray dots are background.

GSE is colored midnight blue in Figure 9. It is the most important substructure in the local halo because it contributes much to the inner halo, and some estimate that it accounts for ~40% of the halo in the solar neighborhood (V. Belokurov et al. 2018; J. T. Mackereth & J. Bovy 2020). GSE debris is a major contributor to the inner halo and was slightly more massive than the Magellanic Clouds prior to infall also mentioned in A. Helmi et al. (2018). Recently, T. Donlon & H. J. Newberg (2023) also pointed out that the inner halo cannot be primarily composed of debris from a single massive, ancient merger event like GSE; there are at least four components contributing to the inner halo. In this work, we have found one group that is clearly related to this structure, which contains 2572 stars. Here, the references we have used to identify this structure are A. Helmi et al. (2018) and S. S. Lövdal et al. (2022; see their Extended Data Figure Captions and Section 5.2).

The hot thick disk is shown in orange in Figure 9. This structure is also dominant in halolike stars close to the Sun, with a metal-rich [Fe/H] > −0.7 dex population. Also known as the “Splash,” it is linked to the thick disk. This structure may have been born in the thick disk of the MW before the massive accretion event that also changed their orbits (V. Belokurov et al. 2020). In our result, we find that five groups are related to this structure; to identify this structure, we refer to E. Dodd et al. (2023; see their Section 4.1).

The Thamnos structure is colored maroon in Figure 9. It was originally discovered by H. H. Koppelman et al. (2019) using a sample of RVS data from Gaia DR2; stars were identified with $v_{\phi} \sim -200 \text{ km s}^{-1}$ and $v_{\phi} \sim -150 \text{ km s}^{-1}$, respectively, as Thamnos 1 and 2. Stars in Thamnos are in a low-inclination,

mildly eccentric retrograde orbit. D. Horta et al. (2023) summarize the chemical characterization of Thamnos. The distribution of Thamnos in the chemical plane is observed to be similar to that of the accretion population of low-mass satellite galaxies and the MW. It is also noted that the abundance pattern of Thamnos is relatively unique and not aligned with any of the other substructures they have studied. In this work, we find two groups that are related to this feature, as defined by H. H. Koppelman et al. (2019; see their Section 4).

The Helmi stream is shown in lime green in Figure 9. We found two groups related to this structure, as defined by E. Dodd et al. (2022). Through the T. Matsuno et al. (2022) analysis, Helmi stream stars were found to have lower elemental abundances of Na, Mg, and Ca compared to other halo stars, including stars accreting from Gaia-Enceladus. At [Fe/H] $\lesssim -1.5$ dex, the difference in chemical abundances between the Helmi stream and Gaia-Enceladus stars is significant, while in the region of high metallicity, some Helmi stream stars have the same elemental abundance ratios as Gaia-Enceladus stars.

L-RL3 is shown in aquamarine in Figure 9. E. Dodd et al. (2023) discovered that this group is very similar to Cluster 3 identified in S. S. Lövdal et al. (2022) and T. Ruiz-Lara et al. (2022), and they therefore refer to it as L-RL3. L-RL3 contains a mixed population of stars. While most of the stars fit the profile of thick disk stars, a metal-poor fraction is also present. At the same time, T. Ruiz-Lara et al. (2022) found no correlation between it and Kraken/Heracles by comparing their energy and angular momentum distributions. We continue to use this term to find one group that is related to this structure.

We also find the structure ED-1, shown in Dodger blue in Figure 9, which was first detected in E. Dodd et al. (2023), using our method we have one group related to this structure.
In addition, we find that GSH-11 (cluster #11) is very similar to cluster #38 shown in T. Ruiz-Lara et al. (2022). GSH-11 is loosely distributed in the energy versus angular momentum space in Appendix Figure 17, about 1.5 kpc from the Sun. This structure has a significant velocity distribution in both the x- and z-directions in 3D space, while most of the velocity values in the y-direction are around 0 km s⁻¹.

Note that we compare our results with some previous literature mentioned above, but different groups might have different structure names and locations using different methods and data set; one possibility is that they might have detected different parts of the same structure. For example, in the discussion, we compare the results with the work using the GALAH DR3 data set (Figure 16).

## 4.2. Inner Halo Results

Although we do not have enough sampling, careful fine-tuning, and detailed data processes for the inner halo larger region, as a by-product and quick test, here we input the 6D information (l, b, d, $v_r$, $v_l$, $v_b$) from the 8099 selected LAMOST DR5 K giants in the halo to the deep learning network, with the initial parameter $\text{siam}_D$ set to 4.

Here we show the 29 candidate groups detected in the x-z plane (see Figure 10). The sky coverage of the three well-known structures identified in the work are also shown in Figure 11; as seen, these three structures have some overlap. Meanwhile, Figure 12 shows the position and velocity distributions of the 13 unknown groups discovered in the inner halo, as well as their metallicity distributions for the inner halo region.

To be more specific, in the inner halo region, we obtained a total of 45 groups using 29 are candidates, 16 of these are related to known structures for the moment: Sgr (R. Ibata et al. 2001; S. R. Majewski et al. 2003), we find four groups related to this structure; VOD (H. J. Newberg et al. 2002; S. Duffau et al. 2006; J. L. Carlin et al. 2012), we find four groups related to this structure; and HAC (V. Belokurov et al. 2007), we find two groups related to this structure; and others in the previous literature.

We have also analyzed Sgr, VOD, and HAC for chemodynamical patterns obtained from GS³ Hunter shown in the Appendix (see Figures 18, 19, and 20), which are the properties of these two structures, but we are not planning to discuss the origins in this work.

Meanwhile, we also find some structures that seem to have been identified in the previous work, and here we observe the characteristics of these structures. Cluster #14 shown in Figure 21 might be C-22 mentioned in N. F. Martin et al. (2022); cluster #15 shown in Figure 21 might be the Slidr mentioned in R. Ibata et al. (2021); cluster #22 shown in Figure 21 might be the combination of Slidr and Sylgr (see R. Ibata et al. 2021); cluster #26 shown in Figure 21 must be a part of Gaia-1 (see N. F. Martin et al. 2022); cluster #33 shown in Figure 21 contains C-1 (see N. F. Martin et al. 2022); and cluster #35 shown in Figure 21 might be the combination of the C-1 and Fjörm structures (see R. Ibata et al. 2021), six in total in the Appendix, for the inner halo part analysis.

In addition, 13 groups (1891 K giants) cannot be linked to any known substructure (see Figure 12) as far as we know, so they may be related to unknown new substructures that will be investigated in more detail in our second work about the tomography of the MW halo streams and substructure in the

<figure><img src="image_13.png" /><figcaption>**Figure 10.** x-z distribution of the 29 candidate groups identified from the LAMOST halo K giants. The dashed line is the $r_{gc}$ per 20 kpc. We use different colors and symbols to represent the different groups. One structure can have a few clustering groups.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Sky coverage of the three structures identified from the LAMOST halo K giant: Sgr-leading part (red), Sgr-trailing part (blue), HAC (dark orange), and VOD (dark green). The background (gray dots) is the sky coverage of the K giant sample.</figcaption></figure>

much broader region. The inner halo is not the key target for this work, and we aim to refine our method for the inner halo in future studies.

# 5. Discussion

## 5.1. Local Halo Substructures

In this part, we show the six substructures corresponding to the 12 groups identified with GS³ Hunter and compare them with the results in R. P. Naidu et al. (2020) and S. S. Lövdal et al. (2022) for Gaia-Enceladus, Sequoia, Thamnos, etc. The phase-space properties of these candidates may represent either a single structure (e.g., GSE) or several components of a structure. It is important to note that in our current work, it is normal for one structure or stream to correspond to one or more groups.
<figure><img src="image_15.png" /><figcaption>**Figure 12.** The figure shows the position and velocity distributions of the 13 unknown groups in the inner halo, as well as their metallicity distributions for the inner halo region.</figcaption></figure>

As shown in Figure 13, we recover clustering features associated with these substructure in similar regions, in particular S. S. Lövdal et al. (2022). We have approximately marked the locations of these substructures with different color labels in the figure. It seems that Thamnos, Helmi, and GSE structures in the energy–angular momentum space are more extended in our work, which does not come as a surprise, since it might mainly be due to the differences in the clustering methods different groups adopt. The distribution of these substructures in chemodynamical space was revealed and analyzed by E. Dodd et al. (2023). The substructures or streams found here will be worthwhile for further studying their internal properties, such as their specific populations, metallicity, and chemical abundances, in future works.

The main substructures we found are GSE, the Helmi stream, the hot thick disk, Thamnos, ED-1, and L-RL3, but we did not find the Sequoia structure in the purely kinematics space (G. C. Myeong et al. 2019) with our method, compared to the results of R. P. Naidu et al. (2020) and S. S. Lövdal et al. (2022). This may be due to the fact that the structure is

relatively dispersed in the 6D kinematic space and is not well identified by our method, or it might be mixed with other structures. We speculate that after adding the energy or other IOM parameter, we might recover this main structure. As a test, after adding the *E* in our algorithm to search, this structure is successfully found, as shown in Figure 14. One may notice that our method, at this stage, can only be applied well to small batches, such as a few-kiloparsec region. When it is applied to a large area, such as the inner halo region, it needs more patches or regions to search. This drawback will be solved step-by-step, but it will not much affect our conclusions focusing on the 6D parameters for the current work. To improve our methodology, we propose the following steps. (a) Conduct a more detailed analysis of the specific phase-space distribution of different structures in future work. (b) Enhance the flexibility of our method by removing the fixed threshold parameter, allowing the network to learn and train different thresholds for various cluster distributions. However, for the current study, using a fixed value is reasonable.
<figure><img src="image_16.png" /><figcaption>**Figure 13.** The left panel shows the positions of the six main groups in the *E*-*L*<sub>z</sub> plane obtained by S. S. Lövdal et al. (2022). Here we have identified the positions by circles, while different labels correspond to different structures. The middle panel shows the results of our work, where the structures represented by the different colors are shown by the name labels. Also, we have circled the main part of the result here with a solid line using a corresponding color by hand, while the dashed part is based on the left figure region S. S. Lövdal et al. (2022) but here we use 3σ region to mark our substructure region. Note that our results are more extended. The right panel shows the results from R. P. Naidu et al. (2020) obtained using H3 data; different colors correspond to different groups identified by ellipses. As seen, due to the different sample sky coverage and method, the left two have some differences from the right panel but still have some of the same structures. Some of the substructures in this work are more extended than the left panel, such as the hot thick disk. The possible drawback in the purely kinematic space is not detecting the Sequoia clearly (see Figure 14).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 14.** Substructures of the solar neighborhood halo identified by GS³ Hunter, which is similar to Figure 9, but here Sequoia is detected.</figcaption></figure>

We also present the density distribution of the results integrated by GS³ Hunter across different parameter spaces. Our findings indicate that structures are clearly mixed in the *xyz* spatial position space, whereas individual structures are more distinctly shown in the velocity space and the IOM space, as shown in Figure 15. It shows that the choice of input parameter space significantly impacts on our results.

Note that here we mainly use the sample in the local halo to compare with S. S. Lövdal et al. (2022), so the structure Sgr, alpha, high-α disk + in situ halo in the right panel of Figure 13 are not shown well, our sample is not distributed in this range. But it seems that the right two panels show that the GSE can be

more extended than in the left panel. At the same time, some of the structures we have found, such as the hot thick disk and Helmi stream, have a few stars that are relatively looser and not compact, which might imply that these structures may have other extensions. In this work, we mainly present our new method; further analysis of the structure will be further explored in the subsequent work.

In the meantime, we also analyze the local halo component using the GALAH DR3 data. Our data selection steps are equivalent to those employed by T. Donlon & H. J. Newberg (2023). We input the GALAH DR3 sample into GS³ Hunter and identified a total of 30 clusters/groups. Comparing with
<figure><img src="image_18.png" /><figcaption>**Figure 15.** The first and second row present the density distribution of the result from GS³ Hunter in spatial and velocity space, respectively, and the third row represents the result in IOM space.</figcaption></figure>

T. Donlon & H. J. Newberg (2023) for each structure in Figure 16, it is noticed that our results are in good agreement. Because the local stellar halo contains a variety of substructures as in Figure 16, different GSE stellar selection methods may choose different mixtures of these substructures; more details can also be found in T. Donlon et al. (2022, 2024) and T. Donlon & H. Newberg (2023).

## 5.2. Unknown Structure Properties of the Local Halo

In this work, we discover many unknown new structures (as far as we know), substructures, or streams in the local halo, and here we briefly analyze and introduce the characteristics of some of the unknown structures with a high star fraction (at least 95%) in the definition. The unknown structure properties can also be found in the Appendix of this paper.

### 5.2.1. GSH-8

GSH-8 has a similar distribution in energy versus angular momentum space as the hot thick disk structure shown in Figure 17 and is more diffuse in the 3D position space. The distribution of GSH-8 is more concentrated in the 3D velocity space. We can note that the velocity values of the GSH-8 structure in the *y*-direction are mainly concentrated around $-125 \text{ km s}^{-1}$, with the color–magnitude clustering features in the last panel. Based on the isochrone in the last H-R subplot, we can see that most of the stars in this structure have a log (age) of about 9.8.

### 5.2.2. GSH-9

GSH-9 is a group of 1924 stars shown in Figure 18, which has a relatively large number of stars and may contain some contaminations. This group is mainly in the low-energy region
<figure><img src="image_19.png" /></figure>

**Figure 16.** The comparison of our results of this method to T. Donlon & H. J. Newberg (2023). The left panel is from T. Donlon & H. J. Newberg (2023) directly in the GALAH DR3 data part of Figure 9. The different components are shown in green (VRM), red (Cronus), blue (Nereus), and yellow (Thamnos), respectively, and here we label them with different colored labels. In the right panel, we show the results obtained by GS³ Hunter using a similar data set, with the different components represented in green (VRM), purple (Cronus), blue (Nereus), and yellow (Thamnos). It can be seen that there is a good agreement between the two results, although it is not exactly the same. We notice there might be some controversial analysis in different groups, but here we only focus on the substructure or overdensity hunting; moreover, we currently speculate that different groups have detected different parts of the same structure or stream using different methods and data sets, and then they have different names for the structure in the halo.

$(E \sim [-1.8, -1.2] (10^5 \text{ km}^2 \text{ s}^{-2}))$ and has a large scatter in the $v_z-v_\phi$ region. In the last panel, the isochrone shows that most of the stars in this structure have a log(age) of about 9.5. Note Figure 17 has been discussed in the previous Section 4.1.

### 5.2.3. GSH-14

GSH-14 is a more relaxed structure in the energy-angular momentum space as shown in Figure 20. The continuous part in the velocity space is mostly larger in the direction of $v_z$, which ranges from 200 km s⁻¹ to 400 km s⁻¹. In the last panel, the isochrone shows that most of the stars in this structure have a log(age) of about 9.2.

### 5.2.4. GSH-16

The group GSH-16, as shown in Figure 21, shows a certain overlap in energy-angular momentum space with the position of the new structure ED-1 found by E. Dodd et al. (2023), but it is less consistent in velocity space, and GSH-16 covers a slightly larger region. As seen in the isochrone of the last H-R diagram, we can see that most of the stars in this structure have a log(age) of about 9.6.

### 5.2.5. GSH-20

GSH-20 seems to give a good linear structure, shown in Figure 17; based on the narrow structure, the progenitor of the GSH-20 structure may be a globular cluster, which needs further investigation. The component in the z-direction of the galactic plane and the y-direction of velocity space is about 0 km s⁻¹, while the distribution of velocity direction in x and z is about 200–500 km s⁻¹. In the last panel, when the isochrone is taken into account, it can be observed that this structure has a log(age) of about 8.05.

### 5.2.6. GSH-27

The structure GSH-27 in Figure 17 is in the low-energy space and is relatively dispersed. At the same time, the distributions of GSH-27 and Thamnos in the energy-angular momentum space overlap each other. When the isochrone is taken into account in the last panel, it can be observed that this structure has a log(age) of about 9.7.

### 5.2.7. GSH-30

In Figure 17, this structure is in the low-energy space region, and it can be seen that GSH-30 also overlaps with the Thamnos structure. However, the distribution of the Thamnos structure in the velocity space is more widely distributed, mainly in $v_R-v_\phi$ and $v_R-v_z$. The distribution of this structure, GSH-30, in the velocity space is more concentrated. When the isochrone is taken into account in the last H-R diagram, it can be observed that this structure has a log(age) of about 9.6.

### 5.2.8. GSH-36

In Figure 17, we show the properties of the cluster GSH-36. This cluster has a more diffuse, stream-like distribution in the energy-angular momentum space, which needs more analysis. It is also observed that the structure is 500 km s⁻¹ in $v_\phi$ and has a larger distribution in the $v_z$-direction from -400 km s⁻¹ to 200 km s⁻¹. The isochrone shows that most of the stars in this structure have a log(age) of about 7.5.

## 5.3. Inner Halo Structures

Our results for the LAMOST DR5 K giants do not include the Monoceros Ring substructure because we exclude the stars with $z = 5$ kpc. B. Yanny et al. (2003) also have pointed out that the structure is mainly located in the −20° to 30° disk
latitude range, so we think the main reason is that we do not
cover the Monoceros Ring range in our data selection, and the
origin of it might be from the disk flaring (H. F. Wang et al.
2018b). By analysis of the dynamical and chemical abundance
distributions of the VOD and HAC member stars, H. H. Yan
et al. (2023) find that among the orbital parameters, the
distributions of the eccentricity, the apocenter, and the
maximum height from the Galactic disk show consistency.
Both structures are located in low angular momentum and
high-energy orbits with similar spatial distribution of chemical
abundances. Ultimately, they conclude that the two structures
may have originated from the merger of the same dwarf galaxy.
I. T. Simion et al. (2019) utilized RR Lyrae stars to study the
orbital properties of HAC and VOD, and they revealed a very
similar spatial distribution of kinematics for HAC and VOD
stars. From this they concluded that the two structures may be
part of the same accretion event. We also have some
observational analyses in the Appendix but defer the origin
discussion to future work.

6. Conclusions

In this work, we have developed a new method (in Section 3) that we call GS³ Hunter to find the structures, substructures, or streams of the Galactic solar neighborhood halo (local halo) and the Galactic inner halo. We first verify the feasibility of our algorithm with the FIRE suites of mock MW-like simulated galaxies, which contain a total of 594,409 stars in the FIRE simulation data, and each star is labeled with a real structure. After applying GS³ Hunter, we successfully find the corresponding eight real structures with different numbers of stars (in Section 3).

The data used in this work are Gaia EDR3 publicly available RVS sample and LAMOST DR5 K giants, we obtain the sample by analyzing the data (in Section 2), finally 51,671 stars from Gaia RVS sample (local halo) and 8099 stars from LAMOST DR5 K giants (Inner halo as by-product). After applying our method to the two data sets, we have detected a total of 38 clusters in the stellar local halo near the Sun, of which 13 are identified as associated with previously identified halo substructures. We find one group associated with the GSE structure, five groups associated with the hot thick disk structure, one group associated with the L-RL3 structure, two groups associated with the Thamnos structure, two groups associated with the Helmi stream structure,<fnref n="21" /> and one group linked to the ED-1 structure. One cluster in this work is named cluster #11, which corresponds to cluster #38 shown in T. Ruiz-Lara et al. (2022).

Moreover, we also detect 45 clusters in the inner halo ($|z| > 5$ kpc). Sixteen groups within these cluster features are identified as being related to known substructures. We found four groups related to the Sagittarius structure,<fnref n="22" /> four groups related to the VOD structure, and two groups related to HAC. Six groups are mentioned in the recent literature. Furthermore, we have also analyzed some of these structures in more detail

with chemical and dynamical information, and we present the
results in Section 4.

More specifically, for the local halo, we find a total of 38 clusters, 21 of which we used as candidates (data fraction >95%), and 13 of these clusters were selected to be associated with known structures. Eight clusters are currently considered as unknown structures and are presented above as our new discoveries. For the inner halo, we find a total of 45 clusters, 29 of which are considered as candidates (data fraction >95%); 16 of these clusters are selected to be related to known structures, and 13 of these are unknown.

In the discussion part of Section 5.1, we also have recovered the four substructures mentioned in T. Donlon & H. J. Newb-erg (2023), which implies that different groups might have found different parts of the same stream with different methods and names. The inner halo or local halo will need more investigation in the future; for example, as mentioned in A. Arora et al. (2022), the fixed potential might give an incorrect stream conclusion in action space.

As a prospect, we will investigate more of the inner and outer halo region with action angles and some other dynamical–chemical parameters as a second step; in the future, we will apply our method to more MW and Local Group stream research.

Acknowledgments

We would like to thank the anonymous referee for the very
helpful and insightful comments. We are grateful for some
comments or interactions from Heidi Newberg, Changjiang,
etc. We acknowledge the National Key R & D Program of
China (Nos. 2021YFA1600401 and 2021YFA1600400). T.T.
G. acknowledges financial support from the Australian
Research Council (ARC) through an Australian Laureate
Fellowship awarded to J.B.-H. Y.S.T. acknowledges financial
support from the Australian Research Council through DECRA
Fellowship DE220101520. L.Y.P is also supported by the
National Natural Science Foundation of China (NSFC) under
grant 12173028, the Sichuan Science and Technology Program
(grant No. 2020YFSY0034), and the Sichuan Youth Science
and Technology Innovation Research Team (grant No.
21CXTD0038). The Guo Shou Jing Telescope (the Large
Sky Area Multi-Object Fiber Spectroscopic Telescope,
LAMOST) is a National Major Scientific Project built by the
Chinese Academy of Sciences. Funding for the project has
been provided by the National Development and Reform
Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of
Sciences. This work has also made use of data from the
European Space Agency (ESA) mission Gaia (https://www.
cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC; https://www.cosmos.esa.
int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the
institutions participating in the Gaia Multilateral Agreement.

Appendix A
Local Halo Unknown Clusters

Figure 17 can be found in Appendix A, more details can be
found in the caption.

<footnote n="21">²¹ In addition, we would like to further analyze cluster 24. Due to the small size of this cluster and the loose distribution of a few stars, our calculated fraction of cluster 24 is only 80%. But based on the location of most of the points of cluster 24, we have also included it as part of the Helmi stream.</footnote>

<footnote n="22">²² It is worth mentioning that we have also grouped GSH-5 into part of the leading arm of Sgr. Since some of the stars in this group we identified have a looser distribution, they did not pass our filters in data fraction value. However, by carefully analyzing the distribution of most of the stars in this cluster, we decide to classify it as part of the leading arm of Sgr.</footnote>
<figure><img src="image_20.png" /><figcaption>**Figure 17.** The three panels in the first row show the spatial distribution of the IOM for our unknown cluster #8, while the second and third rows present the 6D distribution of cluster #8 in velocity and spatial space, respectively. For reference, the gray background shows all our halo samples. The last panel on the bottom left presents the color–magnitude diagram of cluster #8, while we show its PARSEC isochrones (A. Bressan et al. 2012) using a solid gray line, and the corresponding age information is also labeled in this panel.</figcaption></figure>

(The complete figure set (9 images) is available in the online article.)
## Appendix B
### Inner Halo Known Clusters

Figures 18–21 can be found in Appendix B, more analysis can be found in the caption.

<figure><img src="image_21.png" /><figcaption>**Figure 18.** Detailed analysis of Sgr obtained in our GS³ Hunter cross-matched with the result of P. Ramos et al. (2022). The blue color indicates the leading arm, and the red color indicates the trailing arm. Top left panel: the velocity distribution relative to the Galactic center. Top right panel: distribution of the angular momentum and the energy (IOM) in Sgr. Bottom left panel: the Kiel diagram of Sgr. Bottom right panel: we present the metallicity distribution of Sgr.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 19.** Detailed analysis of the VOD obtained in our GS³ Hunter. Top left panel: the velocity distribution relative to the Galactic center. Top right panel: distribution of the angular momentum and the energy (IOM) in the VOD. Bottom left panel: the Kiel diagram of VOD. Bottom right panel: we present the metallicity distribution of VOD.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Figure 20.** Detailed analysis of the HAC obtained in our GS³ Hunter. Top left panel: the velocity distribution relative to the Galactic center. Top right panel: distribution of the angular momentum and the energy in the HAC, which is more compact as shown. Bottom left panel: the Kiel diagram of HAC. Bottom right panel: we present the metallicity distribution of HAC.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Figure 21.** The first row shows the sky position distribution of cluster #14 in our inner halo search, the distance distribution with longitude, and the proper motion of this structure. The second row shows the cluster #14 distribution in 3D phase space. The first panel of the third row shows the Kiel distribution, the second panel shows its line-of-sight velocity distribution with longitude, and the third panel shows the [Fe/H] distribution. For reference, the gray background shows all our inner halo samples. Cluster #14 might be C-22, mentioned in N. F. Martin et al. (2022). Note that if we want to investigate the inner halo more carefully, we need to redesign the search strategy; we defer a more detailed analysis of more inner halo structures to a later contribution.</figcaption></figure>

(The complete figure set (6 images) is available in the online article.)
## Appendix C
Tables for Substructures

Tables 1–3 can be found in Appendix C, the 6D information and fraction can be found in these tables.

<table><caption>**Table 1**
FIRE Simulation Validations for Structures</caption><thead><tr><th>N (Cluster)</th><th>Corresponding Structure</th><th>Fraction (of Corresponding Structure)</th></tr></thead><tbody><tr><td>0</td><td>3</td><td>35.29%</td></tr><tr><td>1</td><td>9</td><td>52.03%</td></tr><tr><td>2</td><td>2</td><td>53.65%</td></tr><tr><td>3</td><td>0</td><td>49.83%</td></tr><tr><td>4</td><td>1</td><td>67.81%</td></tr><tr><td>5</td><td>6</td><td>43.12%</td></tr><tr><td>6</td><td>3</td><td>15.02%</td></tr><tr><td>7</td><td>3</td><td>15.65%</td></tr><tr><td>8</td><td>6</td><td>23.30%</td></tr><tr><td>9</td><td>2</td><td>3.97%</td></tr><tr><td>10</td><td>8</td><td>5.36%</td></tr><tr><td>11</td><td>4</td><td>34.24%</td></tr><tr><td>12</td><td>9</td><td>17.12%</td></tr><tr><td>13</td><td>0</td><td>9.78%</td></tr><tr><td>14</td><td>3</td><td>7.59%</td></tr><tr><td>15</td><td>2</td><td>13.50%</td></tr><tr><td>16</td><td>0</td><td>2.87%</td></tr><tr><td>17</td><td>0</td><td>3.93%</td></tr><tr><td>18</td><td>3</td><td>3.18%</td></tr><tr><td>19</td><td>8</td><td>19.24%</td></tr><tr><td>20</td><td>0</td><td>3.70%</td></tr><tr><td>21</td><td>8</td><td>21.89%</td></tr><tr><td>22</td><td>0</td><td>1.16%</td></tr><tr><td>23</td><td>8</td><td>3.57%</td></tr><tr><td>24</td><td>8</td><td>7.21%</td></tr><tr><td>25</td><td>3</td><td>16.66%</td></tr><tr><td>26</td><td>1</td><td>14.42%</td></tr><tr><td>27</td><td>2</td><td>4.74%</td></tr><tr><td>28</td><td>2</td><td>1.52%</td></tr><tr><td>29</td><td>9</td><td>1.84%</td></tr><tr><td>30</td><td>8</td><td>0.63%</td></tr><tr><td>31</td><td>8</td><td>0.94%</td></tr><tr><td>32</td><td>6</td><td>32.71%</td></tr></tbody></table>

<table><caption>**Table 2**
The Definition of Each Cluster/Group (Local Halo)</caption><thead><tr><th>N</th><th>x<br>(kpc)</th><th>y<br>(kpc)</th><th>z<br>(kpc)</th><th>v<sub>x</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>y</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>z</sub><br>(km s<sup>-1</sup>)</th><th>Star Fraction</th><th>Yes/No</th></tr></thead><tbody><tr><td>0</td><td>(-2.5, 2.4)</td><td>(-2.4, 2.3)</td><td>(-2.4, 2.5)</td><td>(-466.3, -112.4)</td><td>(-547.7, -125.4)</td><td>(-40.1, 146.7)</td><td>75.4%</td><td>No</td></tr><tr><td>1</td><td>(-2.4, 2.4)</td><td>(-2.3, 2.4)</td><td>(-2.4, 2.5)</td><td>(78.1, 442.2)</td><td>(-658.1, -96.8)</td><td>(-284.5, 10.5)</td><td>83.7%</td><td>No</td></tr><tr><td>2</td><td>(-2.5, 2.5)</td><td>(-2.5, 2.4)</td><td>(-2.4, 2.5)</td><td>(-369.8, -67.0)</td><td>(-609.2, -94.4)</td><td>(-22.2, 337.7)</td><td>56.1%</td><td>No</td></tr><tr><td>3</td><td>(-2.5, 2.5)</td><td>(-2.5, 2.4)</td><td>(-2.5, 2.5)</td><td>(35.0, 286.1)</td><td>(-583.1, -156.6)</td><td>(-26.3, 173.7)</td><td>92.6%</td><td>No</td></tr><tr><td>4</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.5)</td><td>(-235.9, -24.8)</td><td>(-737.1, -135.9)</td><td>(-83.2, 417.7)</td><td>63.9%</td><td>No</td></tr><tr><td>5</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.2)</td><td>(-2.3, 2.5)</td><td>(-415.1, -67.8)</td><td>(-605.9, -121.9)</td><td>(-183.4, 24.8)</td><td>65.1%</td><td>No</td></tr><tr><td>6</td><td>(-2.3, 2.2)</td><td>(-2.4, 2.3)</td><td>(-2.4, 2.4)</td><td>(-140.1, 6.5)</td><td>(-721.3, -206.9)</td><td>(-141.8, -2.7)</td><td>97.0%</td><td>Yes</td></tr><tr><td>7</td><td>(-2.2, 2.4)</td><td>(-2.2, 2.3)</td><td>(-2.4, 2.4)</td><td>(2.8, 444.9)</td><td>(-548.1, -0.6)</td><td>(48.3, 424.6)</td><td>75.0%</td><td>No</td></tr><tr><td>8</td><td>(-2.0, 2.2)</td><td>(-2.2, 1.9)</td><td>(-1.9, 2.3)</td><td>(113.6, 350.8)</td><td>(-267.9, -49.8)</td><td>(-270.6, -79.9)</td><td>96.5%</td><td>Yes</td></tr><tr><td>9</td><td>(-2.3, 2.3)</td><td>(-2.4, 2.3)</td><td>(-2.4, 2.5)</td><td>(-114.7, 145.5)</td><td>(-726.2, -177.4)</td><td>(-0.5, 300.5)</td><td>98.2%</td><td>Yes</td></tr><tr><td>10</td><td>(-2.3, 2.3)</td><td>(-2.3, 2.3)</td><td>(-2.4, 2.5)</td><td>(96.2, 463.8)</td><td>(-538.3, -13.9)</td><td>(-22.9, 134.4)</td><td>80.6%</td><td>No</td></tr><tr><td>11</td><td>(-1.6, 2.3)</td><td>(-1.8, 1.5)</td><td>(-1.3, 2.3)</td><td>(28.3, 450.9)</td><td>(-108.5, 222.4)</td><td>(-232.9, -4.7)</td><td>96.3%</td><td>Yes</td></tr><tr><td>12</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.5)</td><td>(-2.4, 2.4)</td><td>(-95.7, 124.7)</td><td>(-763.7, -152.5)</td><td>(-195.2, 408.9)</td><td>82.5%</td><td>No</td></tr><tr><td>13</td><td>(-2.4, 2.3)</td><td>(-2.2, 2.4)</td><td>(-1.9, 2.2)</td><td>(-531.1, -151.9)</td><td>(-249.0, 126.9)</td><td>(-234.6, -0.9)</td><td>95.2%</td><td>Yes</td></tr><tr><td>14</td><td>(-1.8, 2.3)</td><td>(-2.1, 1.9)</td><td>(-1.8, 2.4)</td><td>(-384.1, 148.9)</td><td>(-342.5, 33.7)</td><td>(97.1, 515.8)</td><td>96.1%</td><td>Yes</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>N</th><th>x<br>(kpc)</th><th>y<br>(kpc)</th><th>z<br>(kpc)</th><th>v<sub>x</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>y</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>z</sub><br>(km s<sup>-1</sup>)</th><th>Star Fraction</th><th>Yes/No</th></tr></thead><tbody><tr><td>15</td><td>(-2.5, 2.4)</td><td>(-1.7, 2.4)</td><td>(-1.7, 2.4)</td><td>(-526.1, -14.9)</td><td>(-133.9, 197.6)</td><td>(-36.4, 317.2)</td><td>96.3%</td><td>Yes</td></tr><tr><td>16</td><td>(-1.8, 2.4)</td><td>(-2.5, 2.3)</td><td>(-2.0, 2.5)</td><td>(39.0, 243.4)</td><td>(-439.6, -119.9)</td><td>(-292.4, -70.4)</td><td>98.9%</td><td>Yes</td></tr><tr><td>17</td><td>(-1.8, 2.4)</td><td>(-2.4, 2.3)</td><td>(-2.1, 2.5)</td><td>(-376.9, 96.3)</td><td>(-382.2, 92.9)</td><td>(-524.9, -93.9)</td><td>95.6%</td><td>Yes</td></tr><tr><td>18</td><td>(-1.2, 2.4)</td><td>(-1.6, 1.6)</td><td>(-1.5, 2.2)</td><td>(138.8, 392.9)</td><td>(-95.7, 6.6)</td><td>(-222.8, -194.)</td><td>98.9%</td><td>Yes</td></tr><tr><td>19</td><td>(-2.3, 2.5)</td><td>(-2.2, 2.3)</td><td>(-2.3, 2.4)</td><td>(119.1, 387.2)</td><td>(-377.8, -82.3)</td><td>(-145.1, 0.4)</td><td>76.2%</td><td>No</td></tr><tr><td>20</td><td>(-2.5, 2.4)</td><td>(-2.4, 1.0)</td><td>(-0.5, 2.1)</td><td>(160.3, 495.7)</td><td>(-12.8, 79.0)</td><td>(-44.6, 33.6)</td><td>97.2%</td><td>Yes</td></tr><tr><td>21</td><td>(-2.0, 2.3)</td><td>(-2.4, 2.3)</td><td>(-2.1, 2.2)</td><td>(-171.9, 57.1)</td><td>(-600.1, -62.4)</td><td>(-443.7, -98.9)</td><td>85.8%</td><td>No</td></tr><tr><td>22</td><td>(-2.4, 2.5)</td><td>(-2.0, 2.4)</td><td>(-2.3, 2.4)</td><td>(-570.2, -154.0)</td><td>(-43.3, 10.4)</td><td>(-70.2, 217.5)</td><td>83.4%</td><td>No</td></tr><tr><td>23</td><td>(-2.4, 2.3)</td><td>(-2.3, 2.2)</td><td>(-2.2, 2.4)</td><td>(-54.7, 299.9)</td><td>(-616.3, -98.5)</td><td>(-442.3, -73.5)</td><td>98.4%</td><td>Yes</td></tr><tr><td>24</td><td>(-1.7, 2.0)</td><td>(-1.6, 2.2)</td><td>(-1.9, 2.5)</td><td>(-249.6, 296.1)</td><td>(-154.5, 271.2)</td><td>(-460.9, 476.9)</td><td>80.8%</td><td>No</td></tr><tr><td>25</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.2)</td><td>(-2.0, 2.3)</td><td>(132.1, 493.5)</td><td>(-235.1, -9.7)</td><td>(-170.1, 18.7)</td><td>98.4%</td><td>Yes</td></tr><tr><td>26</td><td>(-2.3, 2.4)</td><td>(-2.3, 2.3)</td><td>(-2.2, 2.5)</td><td>(48.7, 368.5)</td><td>(-583.7, -110.8)</td><td>(-242.6, -52.3)</td><td>92.1%</td><td>No</td></tr><tr><td>27</td><td>(-2.3, 2.5)</td><td>(-2.2, 2.2)</td><td>(-2.3, 2.4)</td><td>(-212.8, 63.5)</td><td>(-702.4, -189.6)</td><td>(-219.4, -44.5)</td><td>97.8%</td><td>Yes</td></tr><tr><td>28</td><td>(-2.2, 2.4)</td><td>(-2.4, 2.3)</td><td>(-2.4, 2.4)</td><td>(-440.8, -9.7)</td><td>(-407.9, -41.9)</td><td>(-385.4, -33.3)</td><td>67.8%</td><td>No</td></tr><tr><td>29</td><td>(-2.2, 2.5)</td><td>(-2.1, 2.3)</td><td>(-2.2, 2.4)</td><td>(-282.7, 386.6)</td><td>(-411.2, 82.8)</td><td>(-394.4, 108.9)</td><td>87.8%</td><td>No</td></tr><tr><td>30</td><td>(-2.4, 2.5)</td><td>(-2.5, 2.4)</td><td>(-2.4, 2.4)</td><td>(-11.8, 254.3)</td><td>(-718.9, -119.2)</td><td>(-225.9, 209.3)</td><td>97.5%</td><td>Yes</td></tr><tr><td>31</td><td>(-2.2, 2.3)</td><td>(-2.0, 2.3)</td><td>(-1.9, 2.4)</td><td>(-15.5, 497.8)</td><td>(-400.4, 4.6)</td><td>(-6.7, 402.9)</td><td>97.5%</td><td>Yes</td></tr><tr><td>32</td><td>(-2.2, 2.4)</td><td>(-2.3, 2.3)</td><td>(-2.3, 2.4)</td><td>(-320.1, 96.1)</td><td>(-533.4, -91.9)</td><td>(-505.8, -55.9)</td><td>90.5%</td><td>No</td></tr><tr><td>33</td><td>(-2.1, 2.4)</td><td>(-2.1, 2.3)</td><td>(-2.2, 2.4)</td><td>(111.0, 396.8)</td><td>(-345.0, -75.7)</td><td>(-10.9, 127.6)</td><td>91.3%</td><td>No</td></tr><tr><td>34</td><td>(-2.5, 2.2)</td><td>(-2.4, 2.3)</td><td>(-2.5, 2.4)</td><td>(-69.1, 19.9)</td><td>(-778.3, -211.2)</td><td>(-81.7, 71.2)</td><td>98.4%</td><td>Yes</td></tr><tr><td>35</td><td>(-2.3, 1.8)</td><td>(-1.6, 1.8)</td><td>(-1.9, 2.3)</td><td>(14.2, 388.8)</td><td>(-128.2, 148.9)</td><td>(-432.7, 41.4)</td><td>97.8%</td><td>Yes</td></tr><tr><td>36</td><td>(-1.0, 1.8)</td><td>(-1.1, 2.0)</td><td>(-2.2, 1.4)</td><td>(-118.1, 341.3)</td><td>(56.3, 259.0)</td><td>(-371.2, 247.1)</td><td>97.4%</td><td>Yes</td></tr><tr><td>37</td><td>(-2.4, 2.4)</td><td>(-2.4, 2.3)</td><td>(-2.3, 2.4)</td><td>(226.9, 483.9)</td><td>(-444.4, -59.1)</td><td>(-117.5, 81.7)</td><td>99.8%</td><td>Yes</td></tr></tbody></table>

Note. The first column, *N*, shows the sequence number of the group/cluster in our local halo results. The second to seventh columns show the range of *x*, *y*, *z*, *v*<sub>x</sub>, *v*<sub>y</sub>, and *v*<sub>z</sub> for each group/cluster separately. The eighth column shows the star fraction of each group/cluster to define a group candidate according to the 95% criterion. The last column is whether or not the cluster is a candidate.

<table><caption>**Table 3**
The Definition of Each Cluster/Group (Inner Halo)</caption><thead><tr><th>N</th><th>l<br>(deg)</th><th>b<br>(deg)</th><th>d<br>(kpc)</th><th>x<sub>v</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>l</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>b</sub><br>(km s<sup>-1</sup>)</th><th>Star Fraction</th><th>Yes/No</th></tr></thead><tbody><tr><td>0</td><td>(20.2, 337.7)</td><td>(-65.2, 89.4)</td><td>(5.0, 51.8)</td><td>(-249.0, 97.3)</td><td>(-12.2, 39.3)</td><td>(-9.9, 11.7)</td><td>73.9%</td><td>No</td></tr><tr><td>1</td><td>(68.2, 240.7)</td><td>(-65.6, 85.4)</td><td>(5.3, 44.5)</td><td>(-153.9, 19.0)</td><td>(-4.5, 32.0)</td><td>(-8.0, 14.4)</td><td>92.9%</td><td>No</td></tr><tr><td>2</td><td>(35.8, 359.4)</td><td>(-48.1, 89.2)</td><td>(5.0, 69.8)</td><td>(-296.8, 77.1)</td><td>(-10.1, 17.1)</td><td>(-10.1, 15.1)</td><td>86.9%</td><td>No</td></tr><tr><td>3</td><td>(171.3, 359.9)</td><td>(26.7, 88.6)</td><td>(5.1, 60.5)</td><td>(-124.6, 157.8)</td><td>(-58.6, 58.8)</td><td>(-60.7, 12.2)</td><td>98.2%</td><td>Yes</td></tr><tr><td>4</td><td>(1.2, 133.8)</td><td>(-58.8, 86.4)</td><td>(5.5, 49.8)</td><td>(-335.9, -141.8)</td><td>(-11.5, 5.4)</td><td>(-13.1, 18.5)</td><td>95.2%</td><td>Yes</td></tr><tr><td>5</td><td>(21.8, 358.8)</td><td>(-16.7, 88.7)</td><td>(5.3, 58.0)</td><td>(-217.1, 177.5)</td><td>(-8.4, 17.2)</td><td>(-10.1, 14.2)</td><td>81.5%</td><td>No</td></tr><tr><td>6</td><td>(38.2, 355.2)</td><td>(20.7, 89.1)</td><td>(5.1, 34.9)</td><td>(21.4, 209.3)</td><td>(-8.6, 9.2)</td><td>(-10.1, 12.4)</td><td>90.4%</td><td>No</td></tr><tr><td>7</td><td>(5.5, 162.9)</td><td>(-53.1, 82.3)</td><td>(6.3, 58.7)</td><td>(-479.4, -276.0)</td><td>(-12.4, 10.8)</td><td>(-22.1, 14.0)</td><td>99.2%</td><td>Yes</td></tr><tr><td>8</td><td>(0.1, 169.5)</td><td>(-65.3, 88.4)</td><td>(5.4, 53.7)</td><td>(-310.9, -76.8)</td><td>(-12.4, 15.6)</td><td>(-11.3, 8.5)</td><td>92.2%</td><td>No</td></tr><tr><td>9</td><td>(3.6, 187.5)</td><td>(-61.4, 82.6)</td><td>(5.5, 36.8)</td><td>(-278.9, -20.7)</td><td>(-8.3, 5.9)</td><td>(-6.9, 15.9)</td><td>81.8%</td><td>No</td></tr><tr><td>10</td><td>(0.2, 182.9)</td><td>(-46.7, 87.5)</td><td>(5.1, 36.7)</td><td>(38.6, 333.1)</td><td>(-16.0, 5.9)</td><td>(-30.7, 9.2)</td><td>95.1%</td><td>Yes</td></tr><tr><td>11</td><td>(50.3, 256.8)</td><td>(-60.2, 63.7)</td><td>(6.7, 118.1)</td><td>(17.3, 251.7)</td><td>(-2.8, 12.2)</td><td>(-9.4, 9.1)</td><td>89.3%</td><td>No</td></tr><tr><td>12</td><td>(67.1, 359.1)</td><td>(-58.3, 55.3)</td><td>(6.6, 63.8)</td><td>(-43.4, 89.6)</td><td>(-2.0, 25.9)</td><td>(-54.3, 8.4)</td><td>99.4%</td><td>Yes</td></tr><tr><td>13</td><td>(50.3, 173.8)</td><td>(-61.7, 23.3)</td><td>(5.8, 45.6)</td><td>(-281.8, -84.6)</td><td>(-6.9, 7.4)</td><td>(-10.4, 1.8)</td><td>98.9%</td><td>Yes</td></tr><tr><td>14</td><td>(299.2, 358.1)</td><td>(47.0, 69.9)</td><td>(5.3, 10.5)</td><td>(177.3, 243.0)</td><td>(-18.9, -2.2)</td><td>(-2.5, 12.2)</td><td>100.0%</td><td>Yes</td></tr><tr><td>15</td><td>(211.9, 291.0)</td><td>(29.4, 60.5)</td><td>(7.1, 20.1)</td><td>(298.9, 373.2)</td><td>(-3.7, 2.8)</td><td>(-6.3, 0.6)</td><td>100.0%</td><td>Yes</td></tr><tr><td>16</td><td>(0.1, 174.8)</td><td>(15.9, 85.2)</td><td>(5.2, 44.7)</td><td>(-241.2, 10.9)</td><td>(-17.8, 3.8)</td><td>(-11.5, 8.4)</td><td>70.1%</td><td>No</td></tr><tr><td>17</td><td>(157.7, 358.5)</td><td>(6.3, 63.2)</td><td>(6.1, 57.8)</td><td>(28.2, 120.4)</td><td>(-4.3, 22.5)</td><td>(-14.3, 0.5)</td><td>93.1%</td><td>No</td></tr><tr><td>18</td><td>(0.4, 187.4)</td><td>(-63.4, 86.3)</td><td>(5.4, 40.3)</td><td>(-272.6, -1.5)</td><td>(-16.1, 10.1)</td><td>(-11.8, 8.5)</td><td>70.6%</td><td>No</td></tr><tr><td>19</td><td>(214.1, 301.3)</td><td>(39.6, 76.8)</td><td>(5.5, 18.2)</td><td>(231.1, 343.6)</td><td>(-9.5, 2.1)</td><td>(-7.9, -1.4)</td><td>97.2%</td><td>Yes</td></tr><tr><td>20</td><td>(0.7, 160.4)</td><td>(33.3, 85.9)</td><td>(5.4, 37.9)</td><td>(-292.4, -54.2)</td><td>(-9.5, 1.6)</td><td>(-3.7, 18.6)</td><td>98.1%</td><td>Yes</td></tr><tr><td>21</td><td>(155.9, 331.1)</td><td>(40.2, 87.1)</td><td>(6.1, 25.2)</td><td>(132.3, 249.7)</td><td>(-7.0, -2.5)</td><td>(-8.7, -0.6)</td><td>97.1%</td><td>Yes</td></tr><tr><td>22</td><td>(181.9, 344.6)</td><td>(34.7, 83.7)</td><td>(6.0, 31.5)</td><td>(194.1, 328.8)</td><td>(-7.7, 3.8)</td><td>(-7.4, -0.1)</td><td>97.8%</td><td>Yes</td></tr><tr><td>23</td><td>(161.1, 338.6)</td><td>(11.7, 83.8)</td><td>(5.3, 55.8)</td><td>(91.8, 206.6)</td><td>(-7.1, 8.3)</td><td>(-18.2, 0.3)</td><td>98.7%</td><td>Yes</td></tr><tr><td>24</td><td>(1.1, 223.2)</td><td>(24.4, 88.8)</td><td>(5.0, 51.9)</td><td>(-244.3, 66.6)</td><td>(-14.7, 10.9)</td><td>(-7.7, 18.2)</td><td>79.6%</td><td>No</td></tr><tr><td>25</td><td>(170.6, 278.9)</td><td>(21.8, 84.3)</td><td>(6.3, 23.1)</td><td>(181.3, 304.7)</td><td>(-1.5, 8.6)</td><td>(-7.6, -0.1)</td><td>97.9%</td><td>Yes</td></tr><tr><td>26</td><td>(318.4, 353.7)</td><td>(52.7, 60.2)</td><td>(6.1, 12.6)</td><td>(236.6, 284.7)</td><td>(-8.0, -3.7)</td><td>(0.1, 3.9)</td><td>100%</td><td>Yes</td></tr></tbody></table>
<table><caption>**Table 3**
(Continued)</caption><thead><tr><th>N</th><th>l<br>(deg)</th><th>b<br>(deg)</th><th>d<br>(kpc)</th><th>x<sub>v</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>l</sub><br>(km s<sup>-1</sup>)</th><th>v<sub>b</sub><br>(km s<sup>-1</sup>)</th><th>Star Fraction</th><th>Yes/No</th></tr></thead><tbody><tr><td>27</td><td>(188.8, 329.9)</td><td>(26.7, 82.8)</td><td>(5.5, 18.6)</td><td>(175.0, 317.5)</td><td>(-15.7, 3.2)</td><td>(-4.9, 0.2)</td><td>98.3%</td><td>Yes</td></tr><tr><td>28</td><td>(0.0, 160.6)</td><td>(23.3, 88.4)</td><td>(5.1, 58.4)</td><td>(-265.2, -43.5)</td><td>(-13.1, 6.4)</td><td>(-4.2, 14.2)</td><td>95.1%</td><td>Yes</td></tr><tr><td>29</td><td>(1.4, 137.3)</td><td>(-57.5, 87.1)</td><td>(5.1, 34.3)</td><td>(-302.6, -204.1)</td><td>(-16.1, 8.4)</td><td>(-8.2, 11.1)</td><td>96.0%</td><td>Yes</td></tr><tr><td>30</td><td>(0.1, 166.1)</td><td>(-61.6, 83.1)</td><td>(5.5, 66.9)</td><td>(-259.9, -21.7)</td><td>(-14.6, 12.4)</td><td>(-10.0, 18.2)</td><td>82.3%</td><td>No</td></tr><tr><td>31</td><td>(0.1, 172.5)</td><td>(16.9, 88.6)</td><td>(5.1, 48.4)</td><td>(-227.0, 36.5)</td><td>(-13.9, 11.2)</td><td>(-8.5, 12.4)</td><td>86.9%</td><td>No</td></tr><tr><td>32</td><td>(131.6, 355.8)</td><td>(21.9, 84.7)</td><td>(5.7, 40.9)</td><td>(124.2, 312.2)</td><td>(-5.2, 7.5)</td><td>(-9.5, 3.9)</td><td>98.2%</td><td>Yes</td></tr><tr><td>33</td><td>(296.9, 358.8)</td><td>(44.1, 87.7)</td><td>(5.1, 17.7)</td><td>(30.6, 144.3)</td><td>(-23.5, -0.1)</td><td>(-7.3, 11.6)</td><td>98.9%</td><td>Yes</td></tr><tr><td>34</td><td>(-277.0, 349.6)</td><td>(54.1, 74.6)</td><td>(6.8, 20.1)</td><td>(221.3, 358.1)</td><td>(-5.1, -0.8)</td><td>(-7.8, 2.7)</td><td>100.0%</td><td>Yes</td></tr><tr><td>35</td><td>(298.1, 359.1)</td><td>(58.2, 79.6)</td><td>(5.3, 12.5)</td><td>(155.7, 215.9)</td><td>(-9.7, -1.2)</td><td>(-7.2, 5.9)</td><td>97.1%</td><td>Yes</td></tr><tr><td>36</td><td>(160.5, 359.8)</td><td>(27.9, 78.2)</td><td>(5.3, 73.9)</td><td>(-41.8, 263.1)</td><td>(-14.1, 3.8)</td><td>(-8.4, 7.7)</td><td>96.6%</td><td>Yes</td></tr><tr><td>37</td><td>(199.8, 358.5)</td><td>(20.5, 56.3)</td><td>(-7.4, 20.5)</td><td>(101.3, 210.3)</td><td>(-8.3, 3.3)</td><td>(-2.7, 5.6)</td><td>72.4%</td><td>No</td></tr><tr><td>38</td><td>(47.8, 173.4)</td><td>(-66.0, -14.4)</td><td>(6.8, 40.1)</td><td>(-213.9, -37.1)</td><td>(-5.1, 12.3)</td><td>(-15.8, 0.2)</td><td>96.2%</td><td>Yes</td></tr><tr><td>39</td><td>(224.6, 332.9)</td><td>(41.6, 76.4)</td><td>(6.1, 24.4)</td><td>(170.2, 265.7)</td><td>(-7.3, 1.5)</td><td>(-7.2, 1.6)</td><td>97.8%</td><td>Yes</td></tr><tr><td>40</td><td>(50.3, 199.4)</td><td>(-66.1, -17.6)</td><td>(5.9, 42.1)</td><td>(-236.1, -17.5)</td><td>(-2.9, 8.0)</td><td>(-10.7, -0.1)</td><td>93.8%</td><td>No</td></tr><tr><td>41</td><td>(139.1, 299.3)</td><td>(23.9, 86.4)</td><td>(6.0, 24.9)</td><td>(173.8, 344.6)</td><td>(-3.7, 12.8)</td><td>(-11.2, 1.5)</td><td>98.3%</td><td>Yes</td></tr><tr><td>42</td><td>(182.4, 359.5)</td><td>(26.9, 85.5)</td><td>(5.1, 24.5)</td><td>(128.3, 390.9)</td><td>(-7.8, 4.6)</td><td>(-6.2, 5.2)</td><td>91.7%</td><td>No</td></tr><tr><td>43</td><td>(259.3, 260.9)</td><td>(49.2, 58.5)</td><td>(7.8, 17.8)</td><td>(390.9, 433.4)</td><td>(-1.2, 2.4)</td><td>(-8.9, -6.5)</td><td>100.0%</td><td>Yes</td></tr><tr><td>44</td><td>(55.1, 185.9)</td><td>(-64.2, -15.8)</td><td>(5.9, 58.7)</td><td>(-176.9, 0.46)</td><td>(-9.1, 21.8)</td><td>(-11.7, 0.7)</td><td>96.9%</td><td>Yes</td></tr></tbody></table>

Note. The first column, *N*, shows the sequence number of the group/cluster in our inner halo results. The second to seventh columns show the range of 1, *b*, *d*, *x*, *v*, and *v*<sub>b</sub> for each group/cluster separately. The eighth column shows the star fraction of each group/cluster to define the candidate of structures according to the 95% criterion. The last column shows whether or not the group is a candidate.

## ORCID iDs

Hai-Feng Wang https://orcid.org/0000-0001-8459-1036

Yang-Ping Luo https://orcid.org/0000-0003-3736-6076

Yuan-Sen Ting https://orcid.org/0000-0001-5082-9536

Thor Tepper-García https://orcid.org/0000-0002-1081-883X

Joss Bland-Hawthorn https://orcid.org/0000-0001-7516-4016

Jeffrey Carlin https://orcid.org/0000-0002-3936-9628

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