# Searching for white dwarf variables in TESS data

Rhorom Priyatikanto  Space Science Center, National Research and Innovation
Agency, Indonesia<EMAIL_ADDRESS>
###### Abstract

A sample of 4 thousands Gaia sources with apparent $G$-band magnitudes of
$<17$ and trigonometric parallaxes of $>3.33$ milli-arcseconds were cross-
matched to the Transiting Exoplanet Survey high level science products and the
Full-Frame Images in order to extract the light curves of possible white dwarf
variables. Most of the targets in the sample were observed in at least 27-day
observing cycle with 30-minutes cadence. Based on Lomb-Scargle periodogram
constructed from de-trended light curves, more than 600 sources have
indication of periodic variability. This paper presents the early results from
the identification.

##  1 Introduction

Most of transiting exoplanets were discovered using Kepler space telescope
which observes a small patch of starry sky in the constellation of Cygnus
[Borucki and _et al_ , 2010]. Following the great success of this mission,
NASA is progressing with the second light of Kepler mission known as K2
[Howell et al., 2014] and also by launching and operating all-sky space-borne
survey named Transiting Exoplanet Survey Satellite or TESS in short [Ricker
and itet al, 2015]. TESS was launched in 2018 and started producing valuable
data since July that year.

Orbiting in an elliptical orbit with period of $13.7$ days which is
approximately half of lunar month, TESS operates 4 cameras with telescopic
lenses. Every camera consists of 4 CCDs with modest sensitivity from 600 to
1000 nm. In total, TESS has $24\times 90$ square degrees field of view with
small portion of blind areas between CCDs. For 27.4 days, TESS observes a
particular sector and records data with various cadence. To date, TESS has
entered its fourth productive year and acquiring abundant data from 42 sectors
covering $\sim 90\%$ of the sky. Some sectors overlap such that the sky
coverage at some areas may reach a year. Until August 2021, TESS identified
4471 objects of interest. There are at least 154 confirmed exoplanets among
them.

The normal operating cycle of TESS starts with capturing the sky with 2-second
exposure time which can be stacked into 20-sec, 120-sec, and 1800-sec cadence
data. After considering the limited storage posed by the satellite, high
cadence data only cover a small portion of the sky around targeted objects.
There are approximately $1,000$ small postage stamps (or TESS pixel files,
TPF) for astroseismology target stars recorded with 20-sec cadence, $100,000$
postage stamps for priority exoplanet search. The full frame images (FFIs)
covering the whole field of view are recorded for every 30 minutes. Based on
the acquired FFIs, independent processes can be applied to extract quality
data from much more targets of interest. Several groups developed and
implemented pipelines to perform such extraction tasks for their specific
inquiries. Among others, there are STELLA for stellar flare detection
[Feinstein et al., 2020], DIAMANTE for detecting exoplanets orbiting FGKM
dwarfs and subgiants [Montalto et al., 2020], TASOC for astroseismology [Lund
et al., 2017], TESS-SPOC as the extension of the TESS main products [Caldwell
et al., 2020], and QLP which produced more than 24-million light curve
segments from southern and northern ecliptic hemisphere [Huang et al., 2020].

Apart from its major mission for detecting exoplanet and producing target for
follow-up observation, data from TESS also triggers discoveries and advanced
research in other areas of astronomy. Among others, there are studies about
subdwarf blue variables [Sahoo et al., 2020, Baran et al., 2021], up to the
dwarf novae in the accreting binary system [Court et al., 2019]. Inspired by
those studies, this current work’s objectives are to obtain light curves from
TESS data especially for the target stars which are located below main
sequence and close to white dwarf. Although TESS observes millions of bright
objects, this satellite has its own limitation, mainly related to spatial and
spectral resolution. Consequently, this work positions the TESS in the
important part of target selection for further follow-up observations. If
white dwarf is discussed, then short period binaries and accreting system
become interesting issues.

##  2 Data and Methods

To achieve the objectives, a list of target sources was created based on the
Gaia Early Data Release 3, just published at the end of 2020 [Gaia
Collaboration and _et al_ , 2016, Gaia Collaboration and _et al_ , 2021].
Among $2.9$ million sources closer than $300$ pc and brighter than $17$ mag in
Gaia $G$ band, objects situated below the main-sequence and bluer than
$B_{P}-R_{P}=1.5$ were selected. This criteria produced a list containing more
than $40,000$ sources while stricter criteria related to the photometric
quality was implemented to cut the list leaving approximately 10% from its
initial size. There are 4593 sources were selected as the targets.

Light curves for some of those targets were acquired from the Mikulski Archive
for Space Telescopes (MAST) since some high level science products were
archived there. TESS-SPOC and QLP produced a large amount of light curves
beyond the main TESS products. Both TESS-SPOC and QLP analyzed full frame
images and extracted light curves mostly from stars brighter than $13.5$ in
$T_{mag}$. The outputs of those two pipelines can be accessed in the MAST
programmatically using lightkurve python package [Lightkurve Collaboration et
al., 2018]. As the complement, eleanor package [Feinstein et al., 2019] was
used to perform independent aperture photometry from target pixel files or
TEES cut. This independent analysis enabled the extraction of light curves for
dimmer targets. In principle, the output of aperture photometry performed
either by TESS-SPOC, QLP, or independently is time series data containing
corrected flux and associated error. The flux can be transformed into TESS-
magnitude ($T_{mag}$) with baseline from the TESS Input Catalog [Stassun and
_et al_ , 2019].

After acquisition, the next process was light curve median filtering (1-day
kernel) to correct the systematic trends or transient variation. Any object
with brightening of more than $0.5$ magnitude after filtering is flagged as
possible transient. The residual of the filtering was processed to get Lomb-
Scargle periodogram as the key for finding periodic signal [VanderPlas, 2018,
Lomb, 1976, Scargle, 1982]. The period it self ranges from approximately 35
minutes (close to the data cadence) up to 10 days. Based on the established
periodogram, the target is considered as periodic if the maximum power exceeds
3 times the false alarm level.

##  3 Result

In general, TESS data for two-third of the targets are processed by TESS-SPOC
and/or QLP. Light curves from $2,985$ targets were downloaded and analyzed
using procedure described before. Independent light curve extraction using
eleanor was performed to $3,928$ targets. Interesting objects with clear sign
of periodic variability or brightening were flagged. Among 634 of those
objects, 382 are periodic with various amplitude (some milli-magnitude to more
than one magnitude). There are 46 objects identified having variability period
of less than 2 hours. Some well-studied cataclysmic variables were on the list
and such object can be used as the benchmark. Figure 1 presents the folded
light curves that come from objects observed by TESS a couple of times. From
these objects, TESS-SPOC and/or QLP processed the data really well, but flux
variation across sectors are obvious for some objects (e.g. GD 1068 in figure
1). Variable background levels among frames acquired within 27-day orbital
period or beyond become the main source of problem to be corrected more
properly. Different imaging scheme [Montalto et al., 2020] or point spread
function photometry [Nardiello, 2020] may provide better solution for this
problem.

Figure 1: Folded light curves for some frequently observed objects. The name
printed on the upper right corner for each panel is the main identification
listed in Simbad. Figure 2: Same as figure 1, but for some faintest targets.

Additionally, light curves from some faintest targets can be seen in figure 2.
For the QLP products, Huang _et al_. [Huang et al., 2020] demonstrated the
precision of $10^{3}$ part per million (ppm) for objects with $T_{mag}\approx
13.5$. If the trend is extrapolated, then the expected precision at
$T_{mag}\approx 17.0$ is at the order of $10^{2}$ ppt. Crowding field and
imperfect data stitching across sectors may decrease the precision of some
objects.

The distribution of the targets and interesting/flagged objects is displayed
in figure 3. Based on the number distribution as a function of $G$, it is
obvious that the flagged objects tend to be brighter. On the other hand, a
portion of nearby targets are overlooked by TESS-SPOC and QLP as indicated by
significantly lower number density at distance of $\sim 0.1$ kpc.

Figure 3: The distribution of $4,593$ targets (black), 2,985 targets with
publicly available light curves (orange), and flagged objects (red) in color-
magnitude diagram (left), distance space (upper right), and apparent magnitude
(lower right).

The complete table of white dwarf variables needs to be curated and compared
to relevant existing catalog, such as the white dwarf catalog [Gentile Fusillo
et al., 2019, Gentile Fusillo et al., 2021], confirmed cataclysmic variables
[Abril et al., 2020], or all-sky catalog of UV sources [Bianchi et al., 2017].
Additionally, Baran _et al_ [Baran et al., 2021]. identified 506 blue subdwarf
variables from TESS dataset which may coincide with the objects identified in
this study.

However, manual screening of the produced light curve was indeed producing
list of fascinating objects for further exploration. On top of the list, there
is TIC 60040774 which was identified by Gentile Fusillo _et al_. [Gentile
Fusillo et al., 2019] as probable white dwarf with mass of $0.277$ M⊙. It is
noteworthy that white dwarf which such low mass is expected to stay in the
main-sequence for a period longer than the current age of the universe. Thus,
it is likely that this object actually consist of unresolved white dwarf
binary with larger mass and observed as a single brighter object. More
interestingly, TIC 60040774 shows excess in infrared implying the existence of
companion with substellar mass. It can either be planet or brown dwarf,
depends on its mass. The system has orbital period of $9.715378$ hours with
flux depression of $\sim 60\%$ though the actual depression need to be
confirmed as the flux from the object is blended with the fluxes from
neighboring brighter sources. However, pixel-by-pixel analysis supports the
notion that the 9.7-hrs variability comes from a dim blue source previously
identified as high proper motion white dwarf. Further photometric and
spectroscopic observations are required to solve the mystery about this
object. If it is indeed a white dwarf-brown dwarf pair, then this object will
be the fourth of its kind [Casewell et al., 2020].

Some objects with outburst are indeed observed by TESS. However, most of them
are well-studied by astronomers, even monitored by amateur astronomer
community. Finding a new understudied object is a challenging process. The
other fascinating object is TIC 48448653 with possible outburst observed by
TESS around Julian Date $JD=2458695$. At another time, possible outburst with
more than 1 mag brightening was observed by ground-based ASAS-SN telescopes.
Apart from that, the periodic variability is unclear.

##  4 Closing Remarks

Even though the analysis of 3,900 objects was performed and the variability of
$634$ objects were identified, this is not a finished work. There are more
works to do in the future such as cross-matching with other catalog, data
mining from databases of multi-wavelength data and machine learning for
automatic classification. In near future, this study is expected to generate a
list of fascinating objects for follow-up observations to be conducted at
astronomical observatories in Indonesia.

## Acknowledgment

This work is a part of research visit at the University of Southampton hosted
by Prof. Christian Knigge and supported by Dr. Simone Scaringi from the Durham
University. This work has made use of data from the European Space Agency
(ESA) mission Gaia, processed by the Gaia Data Processing and Analysis
Consortium (DPAC). Funding for the DPAC has been provided by national
institutions, in particular the institutions participating in the Gaia
Multilateral Agreement.

## References

## References

  * [Abril et al., 2020] Abril, J., Schmidtobreick, L., Ederoclite, A., and López-Sanjuan, C. (2020).  Disentangling cataclysmic variables in Gaia’s HR diagram.  Mon. Not. Roy. Astron. Soc., 492(1):L40–L44.
  * [Baran et al., 2021] Baran, A. S., Sahoo, S. K., Sanjayan, S., and Ostrowski, J. (2021).  A search for variable subdwarf B stars in TESS Full Frame Images - II. Variable objects in the northern ecliptic hemisphere.  Mon. Not. Roy. Astron. Soc., 503(3):3828–3847.
  * [Bianchi et al., 2017] Bianchi, L., Shiao, B., and Thilker, D. (2017).  Revised Catalog of GALEX Ultraviolet Sources. I. The All-Sky Survey: GUVcat_AIS.  Astrop. J. Supp., 230(2):24.
  * [Borucki and _et al_ , 2010] Borucki, W. J. and _et al_ (2010).  Kepler Planet-Detection Mission: Introduction and First Results.  Sci., 327(5968):977.
  * [Caldwell et al., 2020] Caldwell, D. A., Tenenbaum, P., Twicken, J. D., Jenkins, J. M., Ting, E., Smith, J. C., Hedges, C., Fausnaugh, M. M., Rose, M., and Burke, C. (2020).  TESS Science Processing Operations Center FFI Target List Products.  Res. Notes Amer. Astron. Soc., 4(11):201.
  * [Casewell et al., 2020] Casewell, S. L., Belardi, C., Parsons, S. G., Littlefair, S. P., Braker, I. P., Hermes, J. J., Debes, J., Vanderbosch, Z., Burleigh, M. R., Gänsicke, B. T., Dhillon, V. S., Marsh, T. R., Winget, D. E., and Winget, K. I. (2020).  WD1032 + 011, an inflated brown dwarf in an old eclipsing binary with a white dwarf.  Mon. Not. Roy. Astron. Soc., 497(3):3571–3580.
  * [Court et al., 2019] Court, J. M. C., Scaringi, S., Rappaport, S., Zhan, Z., Littlefield, C., Castro Segura, N., Knigge, C., Maccarone, T., Kennedy, M., Szkody, P., and Garnavich, P. (2019).  The eclipsing accreting white dwarf Z chameleontis as seen with TESS.  Mon. Not. Roy. Astron. Soc., 488(3):4149–4160.
  * [Feinstein et al., 2020] Feinstein, A. D., Montet, B. T., Ansdell, M., Nord, B., Bean, J. L., Günther, M. N., Gully-Santiago, M. A., and Schlieder, J. E. (2020).  Flare Statistics for Young Stars from a Convolutional Neural Network Analysis of TESS Data.  Astron. J., 160(5):219.
  * [Feinstein et al., 2019] Feinstein, A. D., Montet, B. T., Foreman-Mackey, D., Bedell, M. E., Saunders, N., Bean, J. L., Christiansen, J. L., Hedges, C., Luger, R., Scolnic, D., and Cardoso, J. V. d. M. (2019).  eleanor: An Open-source Tool for Extracting Light Curves from the TESS Full-frame Images.  Pub. Astron. Soc. Pacific, 131(1003):94502.
  * [Gaia Collaboration and _et al_ , 2016] Gaia Collaboration and _et al_ (2016).  The Gaia mission.  Astron. Astroph., 595:A1.
  * [Gaia Collaboration and _et al_ , 2021] Gaia Collaboration and _et al_ (2021).  Gaia Early Data Release 3. Summary of the contents and survey properties.  Astron. Astroph., 649:A1.
  * [Gentile Fusillo et al., 2021] Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., Vorontseva, A., Lallement, R., Hollands, M., Gänsicke, B. T., Burdge, K. B., McCleery, J., and Jordan, S. (2021).  A catalogue of white dwarfs in Gaia EDR3.  Mon. Not. Roy. Astron. Soc., 508(3):3877–3896.
  * [Gentile Fusillo et al., 2019] Gentile Fusillo, N. P., Tremblay, P.-E., Gänsicke, B. T., Manser, C. J., Cunningham, T., Cukanovaite, E., Hollands, M., Marsh, T., Raddi, R., Jordan, S., Toonen, S., Geier, S., Barstow, M., and Cummings, J. D. (2019).  A Gaia Data Release 2 catalogue of white dwarfs and a comparison with SDSS.  Mon. Not. Roy. Astron. Soc., 482(4):4570–4591.
  * [Howell et al., 2014] Howell, S. B., Sobeck, C., Haas, M., Still, M., Barclay, T., Mullally, F., Troeltzsch, J., Aigrain, S., Bryson, S. T., Caldwell, D., Chaplin, W. J., Cochran, W. D., Huber, D., Marcy, G. W., Miglio, A., Najita, J. R., Smith, M., Twicken, J., and Fortney, J. J. (2014).  The K2 Mission: Characterization and Early Results.  Pub. Astron. Soc. Pacific, 126(938):398.
  * [Huang et al., 2020] Huang, C. X., Vanderburg, A., Pál, A., Sha, L., Yu, L., Fong, W., Fausnaugh, M., Shporer, A., Guerrero, N., Vanderspek, R., and Ricker, G. (2020).  Photometry of 10 Million Stars from the First Two Years of TESS Full Frame Images: Part I.  Res. Notes Amer. Astron. Soc., 4(11):204.
  * [Lightkurve Collaboration et al., 2018] Lightkurve Collaboration, Cardoso, J., Hedges, C., Gully-Santiago, M., Saunders, N., Cody, A., Barclay, T., Hall, O., Sagear, S., Turtelboom, E., Zhang, J., Tzanidakis, A., Mighell, K., Coughlin, J., Bell, K., Berta-Thompson, Z., Williams, P., Dotson, J., and Barentsen, G. (2018).  Lightkurve: Kepler and TESS time series analysis in Python.  Astrophysics Source Code Library.
  * [Lomb, 1976] Lomb, N. (1976).  Least-Squares Frequency Analysis of Unequally Spaced Data.  Astroph. Sp. Sci., 39(2):447–462.
  * [Lund et al., 2017] Lund, M. N., Handberg, R., Kjeldsen, H., Chaplin, W. J., and Christensen-Dalsgaard, J. (2017).  Data preparation for asteroseismology with TESS.  In European Physical Journal Web of Conferences, volume 160 of European Physical Journal Web of Conferences, page 1005.
  * [Montalto et al., 2020] Montalto, M., Borsato, L., Granata, V., Lacedelli, G., Malavolta, L., Manthopoulou, E., Nardiello, D., Nascimbeni, V., and Piotto, G. (2020).  A search for transiting planets around FGKM dwarfs and subgiants in the TESS full frame images of the Southern ecliptic hemisphere.  Mon. Not. Roy. Astron. Soc., 498(2):1726–1749.
  * [Nardiello, 2020] Nardiello, D. (2020).  A PSF-based Approach to TESS High quality data Of Stellar clusters (PATHOS) - III. Exploring the properties of young associations through their variables, dippers, and candidate exoplanets.  Mon. Not. Roy. Astron. Soc., 498(4):5972–5989.
  * [Ricker and itet al, 2015] Ricker, G. R. and itet al (2015).  Transiting Exoplanet Survey Satellite (TESS).  J. of Astron. Tel. Ins. and Sys., 1:14003.
  * [Sahoo et al., 2020] Sahoo, S. K., Baran, A. S., Sanjayan, S., and Ostrowski, J. (2020).  A search for variable subdwarf B stars in TESS full frame images - I. Variable objects in the southern ecliptic hemisphere.  Mon. Not. Roy. Astron. Soc., 499(4):5508–5526.
  * [Scargle, 1982] Scargle, J. (1982).  Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data.  Astroph. J., 263:835–853.
  * [Stassun and _et al_ , 2019] Stassun, K. G. and _et al_ (2019).  The revised TESS input catalog and candidate target list.  Astron. J., 158(4):138.
  * [VanderPlas, 2018] VanderPlas, J. T. (2018).  Understanding the Lomb-Scargle Periodogram.  Astroph. J. Supp., 236(1):16.


# Sharpness-Aware Graph Collaborative Filtering

Huiyuan Chen<EMAIL_ADDRESS>Visa Research ,  Chin-Chia Michael Yeh
<EMAIL_ADDRESS>Visa Research ,  Yujie Fan<EMAIL_ADDRESS>Visa Research ,  Yan
Zheng<EMAIL_ADDRESS>Visa Research ,  Junpeng Wang<EMAIL_ADDRESS>Visa
Research ,  Vivian Lai<EMAIL_ADDRESS>Visa Research ,  Mahashweta Das
<EMAIL_ADDRESS>Visa Research and  Hao Yang<EMAIL_ADDRESS>Visa Research

(2023)

###### Abstract.

Graph Neural Networks (GNNs) have achieved impressive performance in
collaborative filtering. However, GNNs tend to yield inferior performance when
the distributions of training and test data are not aligned well. Also,
training GNNs requires optimizing non-convex neural networks with an abundance
of local and global minima, which may differ widely in their performance at
test time. Thus, it is essential to choose the minima carefully. Here we
propose an effective training schema, called gSAM, under the principle that
the flatter minima has a better generalization ability than the sharper ones.
To achieve this goal, gSAM regularizes the flatness of the weight loss
landscape by forming a bi-level optimization: the outer problem conducts the
standard model training while the inner problem helps the model jump out of
the sharp minima. Experimental results show the superiority of our gSAM.

Collaborative Filtering, Sharpness-aware Minimization

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
46th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 23–27, 2023; Taipei, Taiwan††booktitle:
Proceedings of the 46th International ACM SIGIR Conference on Research and
Development in Information Retrieval (SIGIR ’23), July 23–27, 2023, Taipei,
Taiwan††price: 15.00††doi: 10.1145/3539618.3592059††isbn:
978-1-4503-9408-6/23/07††ccs: Information systems Collaborative filtering

##  1\. Introduction

Collaborative Filtering (CF) has been widely used in recommender systems due
to its efficiency (Schafer et al., 2007; Koren et al., 2009; Rendle et al.,
2009; Covington et al., 2016). As the user-item interaction data can be
naturally represented as a bipartite graph, Graph Neural Networks (GNNs) (Ying
et al., 2018; Wang et al., 2019; Yeh et al., 2022; He et al., 2020; Yu et al.,
2022; Wang et al., 2022; Chen et al., 2021, 2022a) have gained considerable
attention to fully leverage graph structural information. As such, higher-
order dependencies between users and items can be captured to refine the node
embeddings.

However, real-world graphs often exhibit a power-law distribution (Hu et al.,
2020), where the long tail contains low-degree items that lack collaborative
signals. Hence, the neighborhood aggregation scheme in GNNs is inevitably
biased towards high-degree items (Wu et al., 2021), neglecting the impact of
low-degree items. Also, the training and the test data distribution may differ
significantly due to distribution shifts (Wang et al., 2020). This notoriously
makes the training of GNNs suffer from overfitting issues with poor
generalization and stability.

On the other hand, GNNs, inheriting the proprieties of neural networks, are
often overparameterized and have the capacity to fit even a random labeling of
the training data (Zhang et al., 2021). Thus, a small training loss does not
necessarily guarantee good generalization. Additionally, training GNNs often
requires optimizing complex and non-convex neural networks, with an abundance
of local and global minima. Indeed, not all minima are created equal, and
several studies (Neyshabur et al., 2017; Foret et al., 2020; Liu et al., 2020;
Kawaguchi, 2016) show that the standard Stochastic Gradient Descent (SGD) can
be easily made to converge to bad minima (e.g., saddle points or sharp minima)
that could not generalize well.

(a) NGCF.

(b) NGCF+gSAM.

(c) LightGCN.

(d) LightGCN+gSAM.

Figure 1. Compared to vanilla baselines (e.g., NGCF and LightGCN) with sharp
surfaces, our training schema gSAM produces a much smoother surface on the
Amazon-Book dataset.

To explain the above concerns, we explore the loss landscape of NGCF and
LightGCN (Li et al., 2018; Foret et al., 2020). Given a well-trained GNN model
$f_{\bm{\theta}}$, we compute the loss values when moving the model parameters
$\bm{\theta}$ along two random directions to generate a 3D loss landscape (Li
et al., 2018). Figure 1 (a) and (c) display the loss landscapes111Noted that
the figure only shows partial loss surfaces since we only sample a 2D grid to
plot the curves. The real loss landscapes should be much more complex. of NGCF
and LightGCN for Amazon dataset. First, despite using the same loss function,
the loss surfaces of NGCF are relatively complicated compared to LightGCN, due
to the use of non-linear transformation in NGCF. Second, we notice that both
NGCF and LightGCN may converge at sharp minima, whose curvatures are unstable
since the loss values could change quickly around their neighborhood, leading
to poor generalization (Keskar et al., 2017; Li et al., 2018; Foret et al.,
2020; Andriushchenko and Flammarion, 2022). Therefore, we argue that the
first-order optimizers (e.g., Adam) only seek the model parameters that
minimizes the training error, but they dismiss the high-order information like
the sharpness of the loss landscape.

In this work, we propose an effective training schema, called Graph-based
Sharpness-Aware Minimization (gSAM), under the principle that the flatter
minima has a better generalization ability than the sharper ones (Keskar et
al., 2017; Foret et al., 2020; Andriushchenko and Flammarion, 2022). gSAM
explicitly penalizes the sharp minima and biases the convergence to a flat
region by forming a bi-level optimization: the outer problem conducts the
standard model training while the inner problem helps the model jump out of
the sharp minima. By doing so, our gSAM is able to produce smoother loss
surfaces as shown in Figure 1 (b) and (d). Finally, we extensively evaluate
our proposed gSAM on several benchmark datasets, obtaining favorable results
compared with exiting GNNs.

##  2\. Related Work

###  2.1. Graph Neural Network

GNNs learn node representations by aggregating structural messages from their
neighbor, which have been successfully applied to collaborative filtering
(Ying et al., 2018; Wang et al., 2019; He et al., 2020; Chen et al., 2022b,
c). Notably, GNN-based models, such as PinSage (Ying et al., 2018), NGCF (Wang
et al., 2019), LightGCN (He et al., 2020), and MixGCF (Huang et al., 2021)
have achieved superior performance in many applications. However, GNNs tend to
yield inferior performance when the distributions of training and test data
are not aligned well (Wu et al., 2021; Yu et al., 2022). One prominent
direction to improve generalization is contrastive learning (Wu et al., 2021;
Yu et al., 2022; Lin et al., 2022) that applies different data augmentation
methods to extract features from unlabeled data. Nevertheless, contrastive
frameworks (e.g., SGL (Wu et al., 2021)) often require a large batch size of
comparing pairs. It has been observed that a large batch size easily makes the
models converge to bad or sharp minima, resulting in unexpected performance
(Keskar et al., 2017).

###  2.2. Sharpness-Aware Minimization

Recent studies (Jiang* et al., 2020; Keskar et al., 2017; Kwon et al., 2021;
Li et al., 2018; Andriushchenko and Flammarion, 2022; Kim et al., 2022) have
shown a strong correlation between the sharpness of loss landscape and the
generalization error on a large set of neural networks. In particular,
Sharpness-Aware Minimization (SAM) (Foret et al., 2020) aims to minimize both
the loss value and loss sharpness within a maximization region around each
parameter during training. SAM has inspired several follow-up works. For
example, ASAM (Kwon et al., 2021) introduces adaptive sharpness with a scale-
invariant property that adjusts the maximization region of weight space.
FisherSAM (Kim et al., 2022) replaces SAM’s Euclidean balls with ellipsoids
induced by the Fisher information, which obtains more accurate manifold
structures. However, the use of a one-step gradient within SAMs is unstable
with large variance. Recent efforts (Liu et al., 2022; de Jorge et al., 2022)
surprisingly find that adding random noise perturbations does not hurt the
inner gradient ascent, implying the one-step gradient provides little gradient
information. To overcome this limitation, we make a rigorous connection
between SAM and bi-level optimization. We further put forward an implicit
differentiation algorithm to consider the Hessian to achieve better
generalization for GNNs.

##  3\. PRELIMINARIES

###  3.1. Problem Setup

In this work, we focus on implicit recommendation, in which the behavior data
involves a set of users $\mathcal{U}=\\{u\\}$, a set of items
$\mathcal{I}=\\{i\\}$, and the observed user-item interactions
$\mathcal{O}^{+}=\\{y_{ui}|u\in\mathcal{U},i\in\mathcal{I}\\}$, where $y_{ui}$
denotes that user $u$ has interacted with item $i$ before.

One can view user-item interactions as a bipartite graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ (Wang et al., 2019; He et al., 2020),
where the nodes set $\mathcal{V}=\mathcal{U}\cup\mathcal{I}$ contains all
users and items, and the edge set $\mathcal{E}=\mathcal{O}^{+}$ denotes the
observed user-item links. The goal of collaborative filtering is to recommend
a ranked list of items that are of interest to the user $u\in\mathcal{U}$, in
the same sense that performing link prediction on the bipartite graph
$\mathcal{G}$.

###  3.2. Graph Neural Network

####  3.2.1. Message-passing Schema

The core idea of GNNs is to update the representation of each node by
aggregating messages from its neighbors, which can be expressed as:

(1) $\small\mathbf{E}^{(l+1)}=f_{\text{agg}}(\mathbf{E}^{(l)},\mathcal{G}),$

where $\mathbf{E}^{(l)}$ is the nodes’ embeddings at the $l$-th layer, and
$\mathbf{E}^{(0)}$ can be initialized via lookup tables;
$f_{\text{agg}}(\cdot)$ can be any differentiable aggregation functions (Ying
et al., 2018; Wang et al., 2019; He et al., 2020). After $L$ layers, one may
adopt a readout function to generate the final embeddings as:

(2) $\small\mathbf{e}_{u}=f_{\text{readout}}(\\{\mathbf{e}^{(l)}_{u},0\leq
l\leq L\\}),\mathbf{e}_{i}=f_{\text{readout}}(\\{\mathbf{e}^{(l)}_{i},0\leq
l\leq L\\}),$

where the readout function $f_{\text{readout}}(\cdot)$ is usually permutation
invariant like concatenation (Wang et al., 2019) and weighted sum (He et al.,
2020). Based on the final representations $(\mathbf{e}_{u},\mathbf{e}_{i})$,
we can use inner product to predict how likely user $u$ would interact with
item $i$ as: $\hat{y}_{ui}=\mathbf{e}_{u}^{\top}\mathbf{e}_{i}$.

####  3.2.2. Bayesian Personalized Ranking Loss

One common objective function is the pairwise Bayesian Personalized Ranking
(BPR) loss (Rendle et al., 2009), which enforces the prediction of an observed
interaction to be scored higher than its unobserved counterparts:

(3)
$\small\mathcal{L}_{\text{bpr}}(\bm{\theta})=\sum_{(u,i,j)\in\mathcal{O}}-\ln\sigma(\hat{y}_{ui}-\hat{y}_{uj}),$

where
$\mathcal{O}=\\{(u,i,j)|(u,i)\in\mathcal{O}^{+},(u,j)\in\mathcal{O}^{-}\\}$ is
the training data, and $\mathcal{O}^{-}$ contains the unobserved user-item
interactions; $\bm{\theta}$ denotes the model parameters in GNNs. However,
obtaining good generalizations for GNNs is challenging. We next revisit
Sharpness-Aware Minimization (Keskar et al., 2017; Kwon et al., 2021; Li et
al., 2018; Andriushchenko and Flammarion, 2022), which exploits the
relationship between sharpness/flatness of local minima and their
generalization.

###  3.3. Sharpness-Aware Minimization

####  3.3.1. PAC-Bayesian Bound

We begin by introducing the PAC-Bayesian theory (Neyshabur et al., 2017,
2018), which provides a foundation for deriving an upper bound of the
generalization gap between the training and test error. The PAC-Bayesian
theory is as follows:

###### Theorem 1 (PAC-Bayesian Bound (Neyshabur et al., 2017, 2018)).

Given a prior distribution $\mathcal{P}$ on the weight $\bm{\theta}$ of a
neural network, any $\tau\in(0,1]$, an expected error loss
$\hat{\mathcal{L}}(\bm{\theta},\hat{\mathcal{D}})$ for a data distribution
$\hat{\mathcal{D}}$, its empirical loss
${\mathcal{L}}(\bm{\theta},{\mathcal{D}})$, for any posterior distribution
$\mathcal{Q}$ of the weight $\bm{\theta}$, and let ${\mathcal{D}}$ drawn $m$
i.i.d. samples from $\hat{\mathcal{D}}$, the following inequality holds with a
probability at least $1-\tau$,

(4)
$\small\mathbb{E}_{\bm{\theta}\in\mathcal{Q}}\hat{\mathcal{L}}(\bm{\theta},\hat{\mathcal{D}})\leq\mathbb{E}_{\bm{\theta}\in\mathcal{Q}}{\mathcal{L}}(\bm{\theta},{\mathcal{D}})+4\sqrt{\frac{1}{m}\text{KL}(\mathcal{Q}||\mathcal{P})+\log\frac{6m}{\tau}}.$

The goal of PAC-Bayesian learning is to optimize $\mathcal{Q}$ on the RHS of
Eq. (4) in order to obtain a tight upper bound on the test error
$\mathbb{E}_{\bm{\theta}\in\mathcal{Q}}\hat{\mathcal{L}}(\bm{\theta},\hat{\mathcal{D}})$
(LHS). However, directly optimizing RHS of Eq. (4) over $\mathcal{Q}$ is
difficult because of the square root term.

Instead of optimizing the RHS, Sharpness-Aware Minimization (SAM) (Foret et
al., 2020) replaces it with
$\max_{\|\bm{\delta}\|_{2}\leq\rho}{\mathcal{L}}(\bm{\theta}+\bm{\delta},{\mathcal{D}})$
and uses $l_{2}$ norm on $\|\bm{\theta}\|_{2}$.

As such, SAM aims to minimize two losses (Foret et al., 2020): 1) the vanilla
loss ${\mathcal{L}}(\bm{\theta},{\mathcal{D}})$ that optimizes the model
parameters, and 2) the loss associated to the sharpness term
${\mathcal{R}}(\bm{\theta},{\mathcal{D}})$ that maximizes change of the
training loss within the local neighborhood:

(5)
$\displaystyle\bm{\theta}^{*}=\arg\min_{\bm{\theta}}{\mathcal{L}}(\bm{\theta},{\mathcal{D}})+{\mathcal{R}}(\bm{\theta},{\mathcal{D}}),\text{where}$
$\displaystyle{\mathcal{R}}(\bm{\theta},{\mathcal{D}})=\max_{\|\bm{\delta}\|_{2}\leq\rho}{\mathcal{L}}(\bm{\theta}+\bm{\delta},{\mathcal{D}})-{\mathcal{L}}(\bm{\theta},{\mathcal{D}}),$

where $\rho$ is a constant radius. The Eq. (5) is originally developed for
i.i.d. data, recent efforts (Liao et al., 2021; Ma et al., 2021) show that the
PAC-Bayesian generalization bound also holds for non-i.i.d. graph data.
Inspired by these findings, we can extend the Eq. (5) to our GNN-based
collaborative filtering Eq. (3), and rewrite the sharpness-aware minimization
problem as the following minimax optimization:

(6)
$\small\min_{\bm{\theta}}\max_{\|\bm{\delta}\|_{2}\leq\rho}{\mathcal{L}_{\text{bpr}}}(\bm{\theta}+\bm{\delta}).$

Intuitively, Eq. (6) minimizes the maximum loss around the neighborhood of
$\bm{\theta}$. In this way, the maximum loss within the $\bm{\theta}$’s
neighborhood area could be close to the loss of model parameters
$\bm{\theta}$. Therefore, it expects to converge to a flatter minimum compared
to minimizing the loss ${\mathcal{L}_{\text{bpr}}}(\bm{\theta})$ only.

####  3.3.2. One-step Gradient

However, finding the exact solution for Eq. (6) is NP-hard. SAM uses a one-
step gradient ascent to approximate the optimal solution. According to the
first-order Taylor expansion, we can find the $\bm{\delta}^{*}$ as:

(7)
$\small\bm{\delta}^{*}\approx\underset{\|\bm{\delta}\|_{2}\leq\rho}{\arg\max}~{}\mathcal{L}_{\text{bpr}}(\bm{\theta})+\bm{\delta}\cdot\nabla_{\bm{\theta}}\mathcal{L}_{\text{bpr}}(\bm{\theta})=\rho\cdot\frac{\nabla_{\bm{\theta}}\mathcal{L}_{\text{bpr}}(\bm{\theta})}{\left\|\nabla_{\bm{\theta}}\mathcal{L}_{\text{bpr}}(\bm{\theta})\right\|_{2}}.$

With the determined $\bm{\delta}^{*}$, SAM approximates
$\nabla_{\bm{\theta}}\mathcal{L}_{\text{bpr}}(\bm{\theta}+\bm{\delta}^{*})$ to
$\nabla_{\bm{\theta}}\mathcal{L}_{\text{bpr}}(\bm{\theta})$ at
$\bm{\theta}=\bm{\theta}+\bm{\delta}^{*}$, which could be addressed by the
gradient descent framework to obtain the $\bm{\theta}^{*}$.

Figure 2. Results of gSAM, SAM, and LightGCN for 30 runs. Compared to
baselines, our proposed gSAM is more stable.

####  3.3.3. Limitation

The one-step gradient algorithm works well for i.i.d. image data. However, we
empirically find it lacks stability for non-i.i.d. graph data. Figure 2 shows
the results of LightGCN, SAM, and our gSAM for $50$ runs on three benchmark
datasets. Clearly, it can be seen that SAM (one-step gradient) has a large
variance in performance. In some cases, e.g., Yelp dataset, the SAM performs
worse than LightGCN. This implies that the one-step gradient algorithm is not
sufficient to solve the minimax optimization. In contrast, our gSAM
consistently performs better than SAM and LightGCN. We next introduce our
effective training schema in detail.

##  4\. The proposed gSAM

Here we present our gSAM by leveraging the implicit function theorem to
understand the relationship between $\bm{\theta}$ and $\bm{\delta}$. In
particular, we reformulate Eq. (6) into a Bi-level Optimization problem (Liu
et al., 2021):

(8)
$\small\min_{\bm{\theta}}\quad\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta})),\quad\text{s.t.}\quad\bm{\delta}^{*}(\bm{\theta})=\operatorname*{arg\,min}_{\|\bm{\delta}\|_{2}\leq\rho}\mathcal{L}_{\text{in}}(\bm{\theta},\bm{\delta}),$

where
$\mathcal{L}_{\text{out}}=-\mathcal{L}_{\text{in}}=\mathcal{L}_{\text{bpr}}$,
and the implicit function $\bm{\delta}^{*}(\bm{\theta})$ is the best-response
of the model weights $\bm{\delta}$ to $\bm{\theta}$. As such, we use the outer
problem to conduct the standard model training, while the inner problem helps
the model jump out of the sharp minima.

For the inner objective $\mathcal{L}_{\text{in}}$, we can use Projected
Gradient Descent to update the $\bm{\delta}$, while for the outer objective
$\mathcal{L}_{\text{out}}$, we can decompose the hypergradient
$\nabla_{\bm{\theta}}\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta}))$
into:

(9)
$\small\nabla_{\bm{\theta}}\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta}))=\frac{\partial\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta}))}{\partial\bm{\theta}}+\frac{\partial\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta}))}{\partial\bm{\delta}^{*}(\bm{\theta})}\times\frac{\partial\bm{\delta}^{*}(\bm{\theta})}{\partial\bm{\theta}},$

where the first term denotes the direct gradient that is easy to compute,
while the second term is the indirect gradient where we must compute the
Jacobian $\frac{\partial\bm{\delta}^{*}(\bm{\theta})}{\partial\bm{\theta}}$.
Inspired by Cauchy Implicit Function theorem (Lorraine et al., 2020), we can
estimate the Jacobian as:

(10)
$\small\left.\frac{\partial\bm{\delta}^{*}(\bm{\theta})}{\partial\bm{\theta}}\right|_{\bm{\theta^{\prime}}}=\left.-\left[\frac{\partial^{2}\mathcal{L}_{\text{in}}(\bm{\theta},\bm{\delta})}{\partial\bm{\delta}\partial\bm{\delta}^{T}}\right]^{-1}\times\frac{\partial^{2}\mathcal{L}_{\text{in}}(\bm{\theta},\bm{\delta})}{\partial\bm{\delta}\partial\bm{\theta}^{T}}\right|_{\bm{\delta}^{*}(\bm{\theta^{\prime}}),\bm{\theta^{\prime}}}$

Moreover, we can efficiently compute the inverse Hessian using Neumann series
(Liao et al., 2018):

(11)
$\small\left[\frac{\partial^{2}\mathcal{L}_{\text{in}}(\bm{\theta},\bm{\delta})}{\partial\bm{\delta}\partial\bm{\delta}^{T}}\right]^{-1}=\lim_{i\to\infty}\sum_{j=0}^{i}\left[\mathbf{I}-\frac{\partial^{2}\mathcal{L}_{\text{in}}(\bm{\theta},\bm{\delta})}{\partial\bm{\delta}\partial\bm{\delta}^{T}}\right]^{j},$

where $\mathbf{I}$ is an identity matrix. Usually, only the first $J$ terms of
the Neumann series are enough for approximation. As such, the final
hypergradient
$\nabla_{\bm{\theta}}\mathcal{L}_{\text{out}}(\bm{\theta},\bm{\delta}^{*}(\bm{\theta}))$
can be computed as:

(12)
$\small\nabla_{\bm{\theta}}\mathcal{L}_{\text{out}}\approx\frac{\partial\mathcal{L}_{\text{out}}}{\partial\bm{\theta}}-\frac{\partial\mathcal{L}_{\text{out}}}{\partial\bm{\delta}}\times\sum_{j=0}^{J}\left[\mathbf{I}-\frac{\partial^{2}\mathcal{L}_{\text{in}}}{\partial\bm{\delta}\partial\bm{\delta}^{T}}\right]^{j}\times\frac{\partial^{2}\mathcal{L}_{\text{in}}}{\partial\bm{\delta}\partial\bm{\theta}^{T}}.$

To this end, we can summarize the proposed gSAM as:

  1. (1)

Update
$\bm{\delta}\leftarrow\mathcal{P}_{\rho}[\bm{\delta}-\eta_{1}\nabla_{\bm{\delta}}\mathcal{L}_{\text{in}}]$,
where
$\mathcal{P}_{\rho}[\mathbf{x}]=\rho\frac{\mathbf{x}}{\max\\{\rho,\|\mathbf{x}\|_{2}\\}}$
denotes the projection onto the $l_{2}$ ball of radius $\rho$.

  2. (2)

Update model parameters
$\bm{\theta}\leftarrow\bm{\theta}-\eta_{2}\nabla_{\bm{\theta}}\mathcal{L}_{\text{out}}$
via Eq. (12).

Unlike the one-step gradient algorithm, our hypergradient algorithm can adjust
each dimension of the gradient by Hessian, which is more tolerant to the
change of the loss curvature. More importantly, our implicit hypergradient can
be implemented in a memory-efficient manner compared to the explicit way of
solving bi-level optimization (Liu et al., 2021), which is scalable to
millions of parameters (Lorraine et al., 2020).

Table 1. Statistics of three benchmark datasets. Dataset | #user | #item | #inter | inter/user | density
---|---|---|---|---|---
Book | 52.6k | 91.6k | 2984.1k | 56.7 | 0.06%
Gowalla | 29.9k | 41.0k | 1027.4k | 56.7 | 0.06%
Yelp | 31.7k | 38.0k | 1561.4k | 49.3 | 0.13%

##  5\. Experiments

###  5.1. Experimental Settings

####  Dataset.

We conduct experiments on three
datasets222https://github.com/kuandeng/LightGCN/tree/master/Data: Amazon-Book,
Gowalla, and Yelp-2018. The statistics of the datasets are summarized in Table
1. For each dataset, we randomly split each user’s historical interactions
into training/validation/test sets with the ratio 8:1:1. Also we adopt two
common used Top-$k$ metrics: Recall@$k$ and NDCG@$k$ ( $k=20$ by default) with
the all-ranking protocol (He et al., 2020).

####  Baselines.

Our gSAM is fully compatible with existing GNN-based recommenders to obtain
better generalization. We choose the following baselines: 1) NGCF (Wang et
al., 2019), which applies the message-passing scheme to exploit the high-order
neighbors’ information; 2) LightGCN (He et al., 2020), which omits the non-
linear transformation to obtain node representations; 3) MixGCF (Huang et al.,
2021), which uses the mix-up strategy to generate hard negative samples; 4)
SimGCL (Yu et al., 2022) is a contrastive framework that adds uniform noise to
the representations.

####  Parameter Settings.

For all baselines, the size of user/item representation is searched among
$\\{32,64,128,256\\}$. For NGCF, LightGCN, MixGCF, and SimGCL, their
hyperparameters are initialized the same as their original settings, and are
then carefully tuned to achieve optimal performance. For gSAMs, we choose the
same hyperparameters as their backbones, such as batch size, stopping
criteria, learning rate, etc. For the radius of the neighbor ball $\rho$ in
Eq. (8), we vary $\rho$ within $\\{0.01,0.05,0.1,0.5,1.0\\}$.

###  5.2. Experimental Results

####  5.2.1. Overall Performance

The results of different models in terms of Recall$@20$ and NDCG$@20$ are
summarized in Table 2. From the experimental results, we mainly have the
following observations. First, MixGCF and SimGCL perform better than NGCF and
LightGCN. MixGCF synthesizes hard negative samples using mix-up data
augmentation, while SimGCL adopts random noises data augmentation. This
indicates that the GNN models generally get benefits from data augmentation
during the training. Second, NGCF+gSAM and LightGCN+gSAM yield better
performance than their backbones for all datasets. For example, by comparing
the LightGCN and LightGCN+gSAM, LightGCN+gSAM has on average $9.34\%$
improvement with respect to Recall$@20$ and over $8.85\%$ improvements in
terms of NDCG$@20$. This verifies the necessity of explicitly smoothing the
loss geometry during model training. Third, gSAM can further improve the
accuracy of MixGCF and SimGCL with a large margin, which shows the potential
benefit of integrating sharpness-aware minimization and data augmentation.

Table 2. The performance of gSAM with different backbones.

| Book | Gowalla | Yelp
---|---|---|---
Method | recall | ndcg | recall | ndcg | recall | ndcg
NGCF | 0.0759 | 0.0466 | 0.1373 | 0.0810 | 0.0716 | 0.0440
+gSAM |  | 0.0822
---
(+8.30%)
| 0.0504
---
(+8.15%)
| 0.1551
---
(+12.96%)
| 0.0954
---
(+17.77%)
| 0.0805
---
(+12.43%)
| 0.0488
---
(+10.91%)
LightGCN | 0.0875 | 0.0576 | 0.1865 | 0.1086 | 0.0833 | 0.0514
+gSAM |  | 0.0993
---
(+13.48%)
| 0.0641
---
(+11.28%)
| 0.1977
---
(+6.01%)
| 0.1163
---
(+7.09%)
| 0.0904
---
(+8.52%)
| 0.0556
---
(+8.17%)
MixGCF | 0.0922 | 0.0601 | 0.2011 | 0.1221 | 0.0889 | 0.0546
+gSAM |  | 0.0998
---
(+8.36%)
| 0.0655
---
(+9.03%)
| 0.2162
---
(+7.54%)
| 0.1335
---
(+9.32%)
| 0.0962
---
(+8.26%)
| 0.0597
---
(+9.33%)
SimGCL | 0.0941 | 0.0642 | 0.1986 | 0.1189 | 0.0937 | 0.0571
+gSAM |  | 0.1016
---
(+7.97%)
| 0.0721
---
(+12.31%)
| 0.2152
---
(+8.36%)
| 0.1287
---
(+8.24%)
| 0.1024
---
(+9.28%)
| 0.0629
---
(+10.16%)

####  5.2.2. Further Probe

GNNs are known to be biased towards high-degree items, neglecting the impact
of low-degree items. Here we investigate the generalization ability of our
gSAM on Book dataset (the other two have similar results and are omitted
here). Following (Yu et al., 2022), we divide the test set into three subsets
in proportion to the popularity of items: ’Unpopular’, ’Normal’, and
’Popular’. From Figure 3(a), we can find that the performance of gSAM is
consistently better than LightGCN and SimGCL. This implies that gSAM generally
has better generalization as it can still perform high-quality recommendation
with sparse data.

Our gSAM has an additional hyperparameter: the radius $\rho$ in Eq. (8). To
analyze the influence of $\rho$, we vary $\rho$ in the range of $0.01$ to
$1.0$ and report the experimental results in Figure 3(b). We observe that our
gSAM is stable with respect to $\rho$. Specifically, when $\rho$ is set to
around $0.5$, we obtain the best performance on Amazon dataset.

Figure 3. (a) Performance comparison for different item groups. (b) Influence
of the radius $\rho$.

##  6\. conclusion

Training GNN-based recommenders can easily fall into sharp minima, which may
lead to poor generalization. To address this issue, we present a novel
training framework, call gSAM, to smooth out the loss landscapes during the
training of GNNs. The core idea behind gSAM is to explicitly penalize the
sharp minima and guide the convergence to a flatter region by solving a bi-
level optimization. Extensive experimental results demonstrate the positive
impact of our gSAM for personalized ranking with better generalization.

## References

  * (1)
  * Andriushchenko and Flammarion (2022) Maksym Andriushchenko and Nicolas Flammarion. 2022.  Towards understanding sharpness-aware minimization. In _International Conference on Machine Learning_. 639–668.
  * Chen et al. (2022a) Huiyuan Chen, Xiaoting Li, Kaixiong Zhou, Xia Hu, Chin-Chia Michael Yeh, Yan Zheng, and Hao Yang. 2022a.  TinyKG: Memory-Efficient Training Framework for Knowledge Graph Neural Recommender Systems. In _Proceedings of the 16th ACM Conference on Recommender Systems_. 257–267.
  * Chen et al. (2021) Huiyuan Chen, Lan Wang, Yusan Lin, Chin-Chia Michael Yeh, Fei Wang, and Hao Yang. 2021\.  Structured graph convolutional networks with stochastic masks for recommender systems. In _Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 614–623.
  * Chen et al. (2022b) Huiyuan Chen, Chin-Chia Michael Yeh, Fei Wang, and Hao Yang. 2022b.  Graph Neural Transport Networks with Non-local Attentions for Recommender Systems. In _Proceedings of the ACM Web Conference 2022_. 1955–1964.
  * Chen et al. (2022c) Huiyuan Chen, Kaixiong Zhou, Kwei-Herng Lai, Xia Hu, Fei Wang, and Hao Yang. 2022c.  Adversarial Graph Perturbations for Recommendations at Scale. In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 1854–1858.
  * Covington et al. (2016) Paul Covington, Jay Adams, and Emre Sargin. 2016\.  Deep neural networks for youtube recommendations. In _Proceedings of the 10th ACM conference on recommender systems_. 191–198.
  * de Jorge et al. (2022) Pau de Jorge, Adel Bibi, Riccardo Volpi, Amartya Sanyal, Philip Torr, Grégory Rogez, and Puneet K. Dokania. 2022.  Make Some Noise: Reliable and Efficient Single-Step Adversarial Training. In _Advances in Neural Information Processing Systems_.
  * Foret et al. (2020) Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. 2020.  Sharpness-aware Minimization for Efficiently Improving Generalization. In _International Conference on Learning Representations_.
  * He et al. (2020) Xiangnan He, Kuan Deng, Xiang Wang, Yan Li, Yongdong Zhang, and Meng Wang. 2020\.  Lightgcn: Simplifying and powering graph convolution network for recommendation. In _Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval_. 639–648.
  * Hu et al. (2020) Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. 2020\.  Open graph benchmark: Datasets for machine learning on graphs. In _Advances in neural information processing systems_. 22118–22133.
  * Huang et al. (2021) Tinglin Huang, Yuxiao Dong, Ming Ding, Zhen Yang, Wenzheng Feng, Xinyu Wang, and Jie Tang. 2021.  Mixgcf: An improved training method for graph neural network-based recommender systems. In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_. 665–674.
  * Jiang* et al. (2020) Yiding Jiang*, Behnam Neyshabur*, Hossein Mobahi, Dilip Krishnan, and Samy Bengio. 2020\.  Fantastic Generalization Measures and Where to Find Them. In _International Conference on Learning Representations_.
  * Kawaguchi (2016) Kenji Kawaguchi. 2016\.  Deep learning without poor local minima. In _Advances in neural information processing systems_.
  * Keskar et al. (2017) Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. 2017.  On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima. In _International Conference on Learning Representations_.
  * Kim et al. (2022) Minyoung Kim, Da Li, Shell X Hu, and Timothy Hospedales. 2022\.  Fisher sam: Information geometry and sharpness aware minimisation. In _International Conference on Machine Learning_. 11148–11161.
  * Koren et al. (2009) Yehuda Koren, Robert Bell, and Chris Volinsky. 2009\.  Matrix factorization techniques for recommender systems.  _Computer_ 42, 8 (2009), 30–37.
  * Kwon et al. (2021) Jungmin Kwon, Jeongseop Kim, Hyunseo Park, and In Kwon Choi. 2021\.  Asam: Adaptive sharpness-aware minimization for scale-invariant learning of deep neural networks. In _International Conference on Machine Learning_. 5905–5914.
  * Li et al. (2018) Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. 2018.  Visualizing the loss landscape of neural nets. In _Proceedings of the 32nd International Conference on Neural Information Processing Systems_. 6391–6401.
  * Liao et al. (2021) Renjie Liao, Raquel Urtasun, and Richard Zemel. 2021\.  A PAC-Bayesian Approach to Generalization Bounds for Graph Neural Networks. In _International Conference on Learning Representations_.
  * Liao et al. (2018) Renjie Liao, Yuwen Xiong, Ethan Fetaya, Lisa Zhang, KiJung Yoon, Xaq Pitkow, Raquel Urtasun, and Richard Zemel. 2018\.  Reviving and improving recurrent back-propagation. In _International Conference on Machine Learning_. 3082–3091.
  * Lin et al. (2022) Zihan Lin, Changxin Tian, Yupeng Hou, and Wayne Xin Zhao. 2022\.  Improving Graph Collaborative Filtering with Neighborhood-enriched Contrastive Learning. In _Proceedings of the ACM Web Conference 2022_. 2320–2329.
  * Liu et al. (2021) Risheng Liu, Yaohua Liu, Shangzhi Zeng, and Jin Zhang. 2021\.  Towards gradient-based bilevel optimization with non-convex followers and beyond. In _Advances in Neural Information Processing Systems_.
  * Liu et al. (2020) Shengchao Liu, Dimitris Papailiopoulos, and Dimitris Achlioptas. 2020\.  Bad global minima exist and sgd can reach them. In _Advances in Neural Information Processing Systems_. 8543–8552.
  * Liu et al. (2022) Yong Liu, Siqi Mai, Minhao Cheng, Xiangning Chen, Cho-Jui Hsieh, and Yang You. 2022\.  Random Sharpness-Aware Minimization. In _Advances in Neural Information Processing Systems_.
  * Lorraine et al. (2020) Jonathan Lorraine, Paul Vicol, and David Duvenaud. 2020\.  Optimizing millions of hyperparameters by implicit differentiation. In _International Conference on Artificial Intelligence and Statistics_. 1540–1552.
  * Ma et al. (2021) Jiaqi Ma, Junwei Deng, and Qiaozhu Mei. 2021.  Subgroup generalization and fairness of graph neural networks.  _Advances in Neural Information Processing Systems_ 34 (2021), 1048–1061.
  * Neyshabur et al. (2017) Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nati Srebro. 2017.  Exploring generalization in deep learning. In _Advances in neural information processing systems_.
  * Neyshabur et al. (2018) Behnam Neyshabur, Srinadh Bhojanapalli, and Nathan Srebro. 2018\.  A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks. In _International Conference on Learning Representations_.
  * Rendle et al. (2009) Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Lars Schmidt-Thieme. 2009.  BPR: Bayesian personalized ranking from implicit feedback. In _Proceedings of the 25th conference on uncertainty in artificial intelligence_. AUAI Press, 452–461.
  * Schafer et al. (2007) J Ben Schafer, Dan Frankowski, Jon Herlocker, and Shilad Sen. 2007.  Collaborative filtering recommender systems.  In _The adaptive web_. Springer, 291–324.
  * Wang et al. (2020) Junshan Wang, Guojie Song, Yi Wu, and Liang Wang. 2020\.  Streaming graph neural networks via continual learning. In _Proceedings of the 29th ACM International Conference on Information & Knowledge Management_. 1515–1524.
  * Wang et al. (2019) Xiang Wang, Xiangnan He, Meng Wang, Fuli Feng, and Tat-Seng Chua. 2019.  Neural graph collaborative filtering. In _Proceedings of the 42nd international ACM SIGIR conference on Research and development in Information Retrieval_. 165–174.
  * Wang et al. (2022) Yu Wang, Yuying Zhao, Yushun Dong, Huiyuan Chen, Jundong Li, and Tyler Derr. 2022\.  Improving fairness in graph neural networks via mitigating sensitive attribute leakage. In _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. 1938–1948.
  * Wu et al. (2021) Jiancan Wu, Xiang Wang, Fuli Feng, Xiangnan He, Liang Chen, Jianxun Lian, and Xing Xie. 2021.  Self-supervised graph learning for recommendation. In _Proceedings of the 44th international ACM SIGIR conference on research and development in information retrieval_. 726–735.
  * Yeh et al. (2022) Chin-Chia Michael Yeh, Mengting Gu, Yan Zheng, Huiyuan Chen, Javid Ebrahimi, Zhongfang Zhuang, Junpeng Wang, Liang Wang, and Wei Zhang. 2022\.  Embedding Compression with Hashing for Efficient Representation Learning in Large-Scale Graph. In _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. 4391–4401.
  * Ying et al. (2018) Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. 2018\.  Graph convolutional neural networks for web-scale recommender systems. In _Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining_. 974–983.
  * Yu et al. (2022) Junliang Yu, Hongzhi Yin, Xin Xia, Tong Chen, Lizhen Cui, and Quoc Viet Hung Nguyen. 2022\.  Are graph augmentations necessary? simple graph contrastive learning for recommendation. In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 1294–1303.
  * Zhang et al. (2021) Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. 2021.  Understanding deep learning (still) requires rethinking generalization.  _Commun. ACM_ (2021), 107–115.


#  A family of conformally flat Hamiltonian-minimal Lagrangian tori in
$\mathbb{CP}^{3}$

Andrey E. Mironov and Dafeng Zuo  Mironov, Sobolev Mathematical institute,
Siberian Branch of the Russian Academy of Sciences and Novosibirsk state
university, Novosibirsk  Zuo, School of Mathematics, Korea Institute for
Advanced Study 207-43 Cheongnyangni 2-dong, Dongdaemun-gu Seoul, 130-722 Korea
and Department of Mathematics, University of Science and Technology, Hefei
230026, P.R.China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper by reduction we construct a family of conformally flat
Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^{3}$ as the image of the
composition of the Hopf map $\mathcal{H}:\mathbb{S}^{7}\to\mathbb{CP}^{3}$ and
a map $\psi:\mathbb{R}^{3}\to\mathbb{S}^{7}$ with certain conditions.

MSC(2000): 05C42, 53D12, 35Q51;
Key words: conformally flat, Hamiltonian-minimal Lagrangian tori

##  1\. Introduction

A Lagrangian submanifold of a Kähler manifold is said to be Hamiltonian-
minimal (briefly, H-minimal) if it is a critical point of the volume under
Hamiltonian deformations. This notion was introduced by Oh in [17], who also
gave an example, that is, Clifford tori in $\mathbb{C}^{n}$ with standard
Hermitian metric

$\mathbb{S}^{1}(r_{1})\times\dots\times\mathbb{S}^{1}(r_{n})\subset{\mathbb{C}}^{n},$

where $\mathbb{S}^{1}(r_{j})$ is a circle of radius $r_{j}$ in ${\mathbb{C}}$.

In [7], Hélein and Romon studied a general construction of H-minimal tori in
$\mathbb{C}^{2}$ from the point of view of completely integrable systems. They
provided new explicit nontrivial examples of H-minimal Lagrangian tori which
include the examples previously constructed by Castro and Urbano in
$\mathbb{C}^{2}$ [3]. A similar construction has been generalized to the cases
of Hermitian symmetric spaces, e.g., in $\mathbb{CP}^{2}$, see [8, 11, 10] for
details. However, in the non-flat cases, the underlying equations are no
longer linear, which makes the problem much harder. Although in [10], Ma
introduced a spectral parameter $\lambda\in\mathbb{S}^{1}$, as she pointed
out, a description of H-minimal Lagrangian tori in $\mathbb{CP}^{2}$ in terms
of theta functions seems to be possible. But owing to this spectral parameter
$\lambda\not\in\mathbb{C}$, thus it is still open about the integrability of
this problem in classical sense. In [13, 14] it is shown that if a Lagrangian
conformal map from ${\mathbb{R}}^{2}$ to ${\mathbb{C}}P^{2}$ is given as
composition of maps
$\varphi:={\mathcal{H}}\circ\psi:{\mathbb{R}}^{2}\rightarrow
S^{5}\rightarrow{\mathbb{C}}P^{2},$ where ${\mathcal{H}}$ is the Hopf map,
then components of $\varphi$ satisfy Shrödinger equation

$\Delta\varphi_{j}+i(\theta_{x}\partial_{x}\varphi_{j}+\theta_{y}\partial_{y}\varphi_{j})+4e^{u}\varphi_{j}=0,$

where $ds^{2}=2e^{u}(dx^{2}+dy^{2})$ is an induced metric and $\theta(x,y)$ is
Lagrangian angle. In the case of H-minimal Lagrangian tori $\theta$ is a
linear function. So in order to construct finite gap tori it is necessary to
use spectral data of finite gap Shrödiger operators, see [6], [18] for
details.

In [2], Castro and Urbano constructed a family of minimal Lagrangian tori in
$\mathbb{CP}^{2}$, which are characterized by their invariance under a one-
parameter group of holomorphic isometries of $\mathbb{CP}^{2}$. By using this
idea, in [13, 9], they independently reduced this problem to a one dimensional
system and obtained an equivariant solution in terms of elliptic functions,
and then constructed H-minimal Lagrangian tori in $\mathbb{CP}^{2}$.

In high dimensional case one of the present authors constructed some examples
of H-minimal and minimal Lagrangian immersions and embeddings in
${\mathbb{C}}^{n}$ and $\mathbb{CP}^{n}$. Recently we can find some of works
about this topic, see [1, 4, 5] and references therein. But it is far from the
complete characterization.

In this paper we address to construct a family of conformally flat H-minimal
Lagrangian tori in $\mathbb{CP}^{3}$ by reduction methods, which generalizes
the results in [13, 15, 9], with the metric

$ds^{2}=e^{u}(dx^{2}+dy^{2}+dz^{2}).$

In $\mathbb{CP}^{3}$, it seems to be a little harder. We thus restrict to
discuss a very special case, that is, $u=u(z)$ and the Lagrangian angle
$\theta=ax+by$, where $a$ and $b$ are arbitrary real constants.

##  2\. Preliminaries

In this section, we review some well-known facts without proofs and sketch our
strategy about the construction of H-minimal Lagrangian cone or tori in
$\mathbb{C}^{4}$ and $\mathbb{CP}^{3}$.

###  2.1. Notations

Let $\mathbb{C}^{4}$ be the canonical complex space of dimension $4$ endowed
with an Hermitian product $\langle
u,v\rangle=\displaystyle\sum_{k=1}^{4}u_{k}\bar{v}_{k}.$ Let us denote
$\omega=\mbox{Im}\langle~{},~{}\rangle$ and
$(~{},~{})=\mbox{Re}\langle~{},~{}\rangle$. Let
$\psi:\mathbb{R}^{3}\to\mathbb{S}^{7}$ be an oriented Lagrangian immersion
$L$, i.e. $\psi^{*}\omega=0$, where $\mathbb{S}^{7}$ is the unit sphere in
${\mathbb{C}}^{4}$. Wolfson in [19] introduced a Lagrangian angle $\theta$ and
obtained a criterion of H-minimality of $L$ in terms of $\theta$, that is, the
Lagrangian immersion $L$ is H-minimal if and only if the Lagrangian angle
$\theta$ is a harmonic function on $L$. The Lagrangian angle $\theta$ of $L$
in $\mathbb{C}^{4}$ is defined by the formula

$e^{i\,\,\theta(p)}=dz_{1}\wedge\dots\wedge dz_{4}(\Psi),\quad p\in L,$

where $z_{j},j=1,\cdots,4$ are coordinates on ${\mathbb{C}}^{4}$ and $\Psi$ is
an orthonormal tangent frame at $p\in L$ with the same orientation of $L$. For
the general case, see [19] for details.

Let $\mathcal{H}:S^{7}\rightarrow{\mathbb{C}}P^{3}$ be the Hopf map. An
induced Hermitian product on ${\mathbb{C}P}^{3}$ is called to be the Fubini-
Study metric defined by
$<\zeta_{1},\zeta_{2}>:=\langle\tilde{\zeta}_{1},\tilde{\zeta}_{2}\rangle,$
where $\zeta_{i},~{}i=1,2$ are tangent to ${\mathbb{C}P}^{3}$ and
$\tilde{\zeta}_{i}$ are the corresponding horizontal lifting by $\mathcal{H}$.
Let $\mathcal{C}$ be a Lagrangian cone in ${\mathbb{C}}^{4}$ with the vertex
at the origin. It follows from the definition of $<~{},~{}>$ that
$\mathcal{H}(\widetilde{\mathcal{C}})$ is a Lagrangian submanifold in
${\mathbb{C}P}^{3}$, where $\widetilde{\mathcal{C}}=\mathcal{C}\cap S^{7}$.
Moreover, if the cone $\mathcal{C}$ is H-minimal in $\mathbb{C}^{4}$, then
$\mathcal{H}(\widetilde{\mathcal{C}})$ is also H-minimal in
${\mathbb{C}P}^{3}$, see [12, 1] for details.

###  2.2. On conformally flat Lagrangian immersions

In the following we only consider conformally flat immersions in
$\mathbb{C}^{4}$ and $\mathbb{CP}^{3}$.

Let
$\psi=(\psi^{1},\psi^{2},\psi^{3},\psi^{4}):\mathbb{R}^{3}\to\mathbb{S}^{7}\subset\mathbb{C}^{4}$
be an oriented immersion with a conformally flat metric

$ds^{2}=e^{u(x,y,z)}(dx^{2}+dy^{2}+dz^{2})$

satisfying the following properties

$\langle\psi,\psi_{x}\rangle=\langle\psi,\psi_{y}\rangle=\langle\psi,\psi_{z}\rangle=0,$
$\langle\psi_{x},\psi_{y}\rangle=\langle\psi_{y},\psi_{z}\rangle=\langle\psi_{z},\psi_{x}\rangle=0,$

by the above arguments, thus $\mathcal{H}\circ\psi$ is a Lagrangian immersion
in $\mathbb{CP}^{3}$ and

(2.1)
$\Phi=(\psi,e^{-u}\psi_{x},e^{-u}\psi_{y},e^{-u}\psi_{z})^{t}\in\mbox{U}(4)$

and the Lagrangian angle $\theta(x,y,z)$ is given by

(2.2) $e^{i\,\,\theta}={\rm det}(\Phi).$

By using (2.1) and (2.2), we have

(2.3)
$\Psi=(e^{i\,\,\theta}\psi,e^{-u}\psi_{x},e^{-u}\psi_{y},e^{-u}\psi_{z})^{t}\in\mbox{SU}(4)$

and denote

(2.4)
$\mathcal{U}:=\Psi_{x}\Psi^{-1},\quad\mathcal{V}:=\Psi_{y}\Psi^{-1},\quad\mathcal{W}:=\Psi_{z}\Psi^{-1}\in\mbox{SU}(4).$

The compatibility condition of (2.4) is

(2.5)
$\mathcal{U}_{y}-\mathcal{V}_{x}+[\mathcal{U},\mathcal{V}]=0,\,\mathcal{V}_{z}-\mathcal{W}_{y}+[\mathcal{V},\mathcal{W}]=0,\,\mathcal{W}_{x}-\mathcal{U}_{z}+[\mathcal{W},\mathcal{U}]=0.$

Moreover, if $\triangle\theta(x,y,z)=0$, then the immersion
$\mathcal{H}\circ\psi$ is a conformally flat H-minimal Lagrangian immersion in
$\mathbb{CP}^{3}$, where $\triangle$ is the corresponding Lapalacian operator
given by the formula

$\triangle:=-e^{-u(z)}\,[\,\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial
z^{2}}+\frac{u_{x}}{2}\frac{\partial}{\partial
x}+\frac{u_{y}}{2}\frac{\partial}{\partial
y}+\frac{u_{z}}{2}\frac{\partial}{\partial z}\,].$

Conversely, given $\mbox{SU}(4)$-valued $\mathcal{U},\ \mathcal{V},\
\mathcal{W}$ satisfying (2.5), we solve the system (2.4) with (2.3) for
$\Psi\in\mbox{SU}(4)$ and then obtain the immersion
$\psi:\mathbb{R}^{3}\to\mathbb{S}^{7}$ and the Lagrangian angle $\theta$. If
$\triangle\theta=0$ and $\psi$ satisfies certain periodic conditions, then the
immersion $\mathcal{H}\circ\psi$ gives a conformally flat H-minimal Lagrangian
torus in $\mathbb{CP}^{3}$. Notice that for general case, this method does not
work.

##  3\. Conformally flat H-minimal Lagrangian tori in $\mathbb{CP}^{3}$

In this section, by using the above method, we will construct a special class
of conformally flat H-minimal Lagrangian tori in $\mathbb{CP}^{3}$ with
$u=u(z)$ and the Lagrangian angle $\theta=ax+by$, where $a$ and $b$ are
arbitrary real constants.

### 3.1.

The first step is to choose $\mathcal{U}$, $\mathcal{V}$ and $\mathcal{W}$. By
a direct calculation, we have the following lemma.

######  Lemma 3.1.

Suppose $\mathcal{U}=(u_{kl})$, $\mathcal{V}=(v_{kl})$, $\mathcal{W}=(w_{kl})$
are $\hbox{SU}(4)$-valued matrices satisfying (2.4) and $u_{kl},v_{kl},w_{kl}$
depend only one variable $z$ for $2\leq k,l\leq 4$, then they must be the
following form

$\small{\mathcal{U}=\left(\begin{array}[]{cccc}ia&e^{i\,\theta+u}&0&0\\\
-e^{-i\,\theta+u}&-i(a+c_{1})&ic_{2}&ic_{3}e^{-3u}-u^{\prime}\\\
0&ic_{2}&ic_{1}&0\\\ 0&ic_{3}e^{-3u}+u^{\prime}&0&0\\\ \end{array}\right),}$

and

$\small{\mathcal{V}=\left(\begin{array}[]{cccc}ib&0&e^{i\,\theta+u}&0\\\
0&ic_{2}&ic_{1}&0\\\
-e^{-i\,\theta+u}&ic_{1}&-i(b+c_{2})&ic_{3}e^{-3u}-u^{\prime}\\\
0&0&ic_{3}e^{-3u}+u^{\prime}&0\\\ \end{array}\right),}$

and

$\small{\mathcal{W}=\left(\begin{array}[]{cccc}0&0&0&e^{i\,\theta+u}\\\
0&ic_{3}e^{-3u}&0&0\\\ 0&0&ic_{3}e^{-3u}&0\\\
-e^{-i\,\theta+u}&0&0&-2ic_{3}e^{-3u}\\\ \end{array}\right)},$

where $u=u(z)$ satisfies the equation

(3.1)
$u^{\prime}\,{}^{2}+e^{2u}+c_{3}^{2}e^{-6u}-\mathfrak{C}=0,~{}~{}u^{\prime}=\frac{du}{dz}.$

and $c_{1}$, $c_{2}$, $c_{3}$ are arbitrary real constants and
$\mathfrak{C}=ac_{1}+bc_{2}+2c_{1}^{2}+2c_{2}^{2}$.

### 3.2.

The second step is to solve the following system

(3.2)
$\Psi_{x}=\mathcal{U}\Psi,\quad\Psi_{y}=\mathcal{V}\Psi,\quad\Psi_{z}=\mathcal{W}\Psi.$

Write

(3.3)
$\Psi=(e^{i\,\,\theta}\psi,e^{-u}\psi_{x},e^{-u}\psi_{y},e^{-u}\psi_{z})^{t},$

where $\psi=\psi(x,y,z)$ is a smooth function. By using (3.3), the system
(3.2) can be rewritten as

(3.4) $\displaystyle\psi_{xz}-(u^{\prime}+ic_{3}e^{-3u})\psi_{x}=0,$ (3.5)
$\displaystyle\psi_{yz}-(u^{\prime}+ic_{3}e^{-3u})\psi_{y}=0,$ (3.6)
$\displaystyle\psi_{xy}-i(c_{2}\psi_{x}+ic_{1}\psi_{y})=0,$

and

(3.7)
$\displaystyle\psi_{xx}+e^{2u}\psi+i(a+c_{1})\psi_{x}-ic_{2}\psi_{y}+((u^{\prime}-ic_{3}e^{-3u})\psi_{z}=0,$
(3.8) $\displaystyle\psi_{yy}+e^{2u}\psi-
ic_{1}\psi_{x}+i(c_{2}+b)\psi_{y}+((u^{\prime}-ic_{3}e^{-3u})\psi_{z}=0,$
(3.9)
$\displaystyle\psi_{zz}+e^{2u}\psi+(2ic_{3}e^{-3u}-u^{\prime})\psi_{z}=0.$

From (3.4) and (3.5), we know that $\psi$ must be of the form

(3.10) $\psi=P(z)\varphi(x,y)+Q(z),\quad\varphi(x,y)\neq\mbox{constant},$

and then (3.4) and (3.5) reduce to

(3.11) $P(z)-(u^{\prime}+ic_{3}e^{-3u})P(z)=0.$

The solution of (3.11) is

(3.12) $P(z)=a_{1}e^{u+ic_{3}\int e^{-3u}dz},\quad a_{1}\in\mathbb{R}.$

Substituting (3.10) and (3.12) into (3.6), we get

(3.13) $\varphi_{xy}-i(c_{2}\varphi_{x}+c_{1}\varphi_{y})=0.$

Substituting (3.10) into (3.7) and (3.8), and then using (3.1) and (3.13), we
get the following

(3.14) $\displaystyle
Q^{\prime}(z)=\frac{e^{5u}Q(z)}{ic_{3}-u^{\prime}e^{3u}},$ (3.15)
$\displaystyle\varphi_{xx}+i(a+c_{1})\varphi_{x}-ic_{2}\varphi_{y}+\mathfrak{C}\,\varphi=0,$
(3.16)
$\displaystyle\varphi_{yy}-ic_{1}\varphi_{x}+i(b+c_{2})\varphi_{y}+\mathfrak{C}\,\varphi=0.$

Write

(3.17) $Q(z)=H(z)e^{iG(z)},$

where $G(z)$ and $H(z)$ are real smooth functions. By differentiating (3.1),
we obtain

(3.18) $u^{\prime\prime}+e^{2u}-3c_{3}^{2}e^{-6u}=0.$

By using (3.18), and separating the real part and the imaginary part of
(3.14), we obtain

$H^{\prime}(z)(\mathfrak{C}-e^{2u})+u^{\prime}e^{2u}H(z)=0,\quad
G^{\prime}(z)(e^{2u}-\mathfrak{C})-c_{3}e^{-u}=0,$

thus

$H(z)=a_{2}\sqrt{\mathfrak{C}-e^{2u}},~{}~{}a_{2}\in\mathbb{R},\quad
G(z)=\int\frac{c_{3}e^{-u}}{e^{2u}-\mathfrak{C}}dz+a_{3},~{}~{}a_{3}\in\mathbb{R}.$

From (3.13), without loss of generality, we could assume that $\varphi(x,y)$
has the form

$\varphi(x,y)=\sum
a_{\alpha_{j}\beta_{j}}e^{i(x\alpha_{j}+y\beta_{j})},~{}~{}\beta_{j}=\frac{c_{2}\alpha_{j}}{\alpha_{j}-c_{1}},~{}~{}a_{\alpha_{j}\beta_{j}}\in\mathbb{C},~{}~{}a_{00}=0.$

It follows from (3.17),(3.15) and (3.16) that $\alpha=\alpha_{j}$ is a root of
the equation

${\alpha}^{3}+a{\alpha}^{2}-\mathfrak{B}\alpha+{c_{{1}}}\mathfrak{C}=0.$

where $\mathfrak{B}=2\,c_{{1}}a+3\,{c_{{1}}}^{2}+c_{{2}}b+3\,{c_{{2}}}^{2}$.

Notice that up to now we only use (3.4)—(3.8) to obtain an explicit form of
$\psi$ as follows

(3.19) $\displaystyle\psi(x,y,z)$ $\displaystyle=\sum
a_{1}a_{\alpha_{j}\beta_{j}}e^{u(z)+ic_{3}\int{e^{-3u(z)}dz}}e^{i(x\alpha_{j}+y\beta_{j})}$
$\displaystyle\quad+~{}a_{2}e^{ia_{3}}\sqrt{\mathfrak{C}-e^{2u(z)}}e^{ic_{3}\int{\dfrac{e^{-u(z)}}{e^{2u(z)}-\mathfrak{C}}}dz}.$

Furthermore, it is easy to check that this function $\psi$ also satisfies
(3.9). Thus we solve the system (3.2). Summarizing the above discussions, we
have the following proposition.

######  Proposition 3.2.

If we suppose that $u=u(z)$ is a smooth solution of
$u(z)^{\prime}\,{}^{2}+e^{2u(z)}+c_{3}^{2}e^{-6u(z)}-\mathfrak{C}=0$; and
$\alpha$ is a root of the equation

(3.20)
${\alpha}^{3}+a\,{\alpha}^{2}-\mathfrak{B}\,\alpha+{c_{{1}}}\mathfrak{C}=0$

Then
$\Psi=(e^{i\,\,\theta}\psi,e^{-u}\psi_{x},e^{-u}\psi_{y},e^{-u}\psi_{z})^{t}$
is a solution of the system (3.2) with $\theta=a\,x+b\,y$ and

$\psi(x,y,z)=\kappa_{1}e^{i(\alpha\,x+\beta\,y)}P(z)+\kappa_{2}Q(z),\quad\beta=\dfrac{c_{2}\alpha}{\alpha-
c_{1}},$

where $\kappa_{1}$ and $\kappa_{2}$ are arbitrary complex constants and

$P(z)=e^{u(z)+ic_{3}\int{e^{-3u(z)}dz}},\quad
Q(z)=\sqrt{\mathfrak{C}-e^{2u(z)}}e^{ic_{3}\int{\dfrac{e^{-u(z)}}{e^{2u(z)}-\mathfrak{C}}}dz}.$

###  3.3. Main results

We now state our main theorem.

######  Theorem 3.3.

Suppose that the equation (3.20) has three distinct roots, denoted by
$\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$. Write
$\beta_{j}=\frac{c_{2}\alpha_{j}}{\alpha_{j}-c_{1}},~{}j=1,2,3$. Then the map
$\mathcal{H}\circ\psi:\mathcal{R}^{3}\to\mathbb{CP}^{3}$ defines a conformally
flat H-minimal Lagrangian immersion in $\mathbb{CP}^{3}$, where
$\mathcal{H}:\mathbb{S}^{7}\to\mathbb{CP}^{3}$ is the Hopf map and the map
$\psi:\mathcal{R}^{3}\to\mathbb{S}^{7}\subset\mathbb{C}^{4}$ is given by the
formula

$\psi=(\gamma_{1}P(z)e^{i(x\alpha_{1}+y\beta_{1})},\
\gamma_{2}P(z)e^{i(x\alpha_{2}+y\beta_{2})},\
\gamma_{3}P(z)e^{i(x\alpha_{3}+y\beta_{3})},\ \gamma_{4}Q(z)).$

Here $\gamma_{4}=\sqrt{\frac{1}{\mathfrak{C}}}$ and

$\tiny{\gamma_{1}=\sqrt{\frac{\mathfrak{C}+\alpha_{2}\alpha_{3}}{\mathfrak{C}(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})}}\,,\,\gamma_{2}=\sqrt{\frac{\mathfrak{C}+\alpha_{1}\alpha_{3}}{\mathfrak{C}(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})}}\,,\,\gamma_{3}=\sqrt{\frac{\mathfrak{C}+\alpha_{1}\alpha_{2}}{\mathfrak{C}(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})}}}\,.$

###### Proof.

It suffices to check that

$\Psi=(e^{i\,\,\theta}\psi,e^{-u}\psi_{x},e^{-u}\psi_{y},e^{-u}\psi_{z})^{t}\in\mbox{SU}(4).$

By using (3.20), we have

(3.21)
$\alpha_{1}+\alpha_{2}+\alpha_{3}=-a,~{}~{}\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{3}=-\mathfrak{B},~{}~{}\alpha_{1}\alpha_{2}\alpha_{3}=-\mathfrak{C}.$

It follows from (3.21) and the explicit forms of $\gamma_{j}$ that

$\displaystyle\displaystyle\sum_{j=1}^{3}\gamma_{j}^{2}=\gamma_{4}^{2},$
$\displaystyle\quad\sum_{j=1}^{3}\gamma_{j}^{2}\alpha_{j}=0,\quad\displaystyle\sum_{j=1}^{3}\gamma_{j}^{2}\alpha_{j}^{2}=1,$
$\displaystyle\displaystyle\sum_{j=1}^{3}\gamma_{j}^{2}\alpha_{j}\beta_{j}=0,$
$\displaystyle\quad\displaystyle\sum_{j=1}^{3}\gamma_{j}^{2}\beta_{j}=0,\quad\sum_{j=1}^{3}\gamma_{j}^{2}\beta_{j}^{2}=1.$

These identities yield that

$\displaystyle\langle\psi,\psi\rangle=1,\langle\psi_{x},\psi_{x}\rangle=\langle\psi_{y},\psi_{y}\rangle=e^{2u},$
$\displaystyle\langle\psi,\psi_{x}\rangle=\langle\psi,\psi_{y}\rangle=\langle\psi,\psi_{z}\rangle=0,$
$\displaystyle\langle\psi_{x},\psi_{y}\rangle=\langle\psi_{y},\psi_{z}\rangle=\langle\psi_{z},\psi_{x}\rangle=0,$
$\displaystyle\langle\psi_{z},\psi_{z}\rangle=P^{\prime}(z)\overline{P^{\prime}(z)}\sum_{j=1}^{3}\gamma_{j}^{2}+\gamma_{4}^{2}Q^{\prime}(z)\overline{Q^{\prime}(z)}$
$\displaystyle\qquad=\gamma_{4}^{2}[e^{2u}(u^{\prime}2+c_{3}^{2}e^{-6u})+\frac{e^{4u}(u^{\prime}2+c_{3}^{2}e^{-6u})}{\mathfrak{C}-e^{2u}}]$
$\displaystyle\qquad=e^{2u}.\qquad\mbox{by using \eqref{MZ3.3}}$

That is to say, $\Psi\in\mbox{SU}(4)$. Thus we complete the proof of the
theorem.∎

We finish this section to discuss how to obtain conformally flat H-minimal
Lagrangian tori in $\mathbb{CP}^{3}$.

Notice that in (3.1) if we make the following change

(3.22) $u=u(z):=-\log(2\sqrt{-q(z)}\,),$

then we have

(3.23) $q^{\prime}(z)^{2}=256c_{3}^{2}q(z)^{5}+4\mathfrak{C}q(z)^{2}+q(z).$

Thus if we choose three real constants $c_{1}$, $c_{2}$ and $c_{3}$ such that
the equation

$256c_{3}^{2}t^{5}+4\mathfrak{C}t^{2}+t=0$

has two negative roots and does not have multiple roots, then this assures
that (3.23) has a smooth periodic solution of the period $\tau$, see [16] for
details. It follows from (3.22) that so is (3.1). We here remark that in this
case $\mathfrak{C}=ac_{1}+bc_{2}+2c_{1}^{2}+2c_{2}^{2}>0$.

We next discuss the condition such that the function $\psi$ in (3.19) is a
periodic function of $x$, $y$,$z$ repesctively. According to the form of
$\psi$ in (3.19), if we assume that $c_{1}\in\mathbb{Q}$ and $\alpha_{2}$,
$\alpha_{2}$ and $\alpha_{3}$ are three distinct rational roots of (3.20),
then $\psi$ is periodic w.r.t. $x$ and $y$. Notice that $u(z+\tau)=u(z)$ and
there exists a periodic function $h(z)$ of the periodic $\tau$ such that

$\int\dfrac{e^{-3u(z)}\mathfrak{C}}{e^{2u(z)}-\mathfrak{C}}dz=h(z)+z\int_{0}^{\tau}\dfrac{e^{-3u(z)}\mathfrak{C}}{e^{2u(z)}-\mathfrak{C}}dz.$

This implies that if we assume

$\frac{c_{3}\mathfrak{C}\tau}{2\pi}\int_{0}^{\tau}\dfrac{e^{-3u(z)}}{e^{2u(z)}-\mathfrak{C}}dz\in\mathbb{Q}\,,$

then $\psi$ is periodic in $z$ with the period $n\tau$ for some
$n\in\mathbb{N}$.

Thus, combining with Theorem 3.3, we have

######  Theorem 3.4.

If we suppose that

$1$. $u=u(z)$ is a periodic solution of (3.1) with the period $\tau$;

$2$.
$c_{1}\in\mathbb{Q},\quad\dfrac{c_{3}\mathfrak{C}\tau}{2\pi}\int_{0}^{\tau}\dfrac{e^{-3u(z)}}{e^{2u(z)}-\mathfrak{C}}dz\in\mathbb{Q}\,$;

$3$. $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are distinct rational roots
of (3.20).

Then the map $\mathcal{H}\circ\psi:\mathbb{R}^{3}\to\mathbb{CP}^{3}$ defines a
conformally flat H-minimal Lagrangian torus in $\mathbb{CP}^{3}$.

## Acknowledgment

A.E.Mironov is grateful to generous supports by KIAS, where part of the work
was done in KIAS. The work of A.E.Mironov was partially supported by Russian
Federation foundation for basic research (grant no.06-01-00094a) and grant
MK-9651.2006.1 of the president of Russian Federation.

Zuo would like to thank Qing Chen and Bumsig Kim and Youjin Zhang for their
constant guidance and supports. Zuo also thanks Chengmin Bai for hospitality
during stay at Chern Institute of mathematics in Nankai university, where part
of the work was done. The work of Zuo was partially supported by a post-doc
fellowship from KIAS and the Natural Science Foundation of China (grant no.
10501043).

Both of the authors thank organizers for the hospitality during the ISLAND-3
on Islay, Scotland.

## References

  * [1] Castro I., Li H. and Urbano F., Hamiltonian-minimal Lagrangian submanifolds in Complex space forms, Pacific J. Math., 227 (2006) 43–63.
  * [2] Castro I. and Urbano F., New examples of minimal Lagrangian tori in the complex projective plane, Manuscripta Math., 85(1994) 265–281.
  * [3] Castro I. and Urbano F., Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbb{C}^{2}$., Compositio Math., 111 (1998)1–14.
  * [4] Chen B.-Y. and Garay O.J., Classification of Hamiltonian-stationary Lagrangian sumbanifolds of constant in ${\mathbb{C}}P^{3}$ with positive nullity, Nonlinear Analysis (2007) doi:10.1016/j.na.2007.06.005.
  * [5] Dong Y.-X., Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries, Nonlinear Anal., 67 (2007) 865–882.
  * [6] Dubrovin B.A., Krichever I.M. and Novikov S.P., The Schrödinger equation in a periodic field and Riemann surfaces, Sov. Math. Dokl., 17 (1976) 947–951.
  * [7] Hélen F. and Romon P.; Hamiltonian stationary Lagrangian surfaces in $\mathbb{C}^{2}$., Comm. Anal. Geom. 10 (2002)79–126.
  * [8] Hélen F. and Romon P. Hamiltonian stationary tori in ${\mathbb{C}P}^{2}$, Comm. Anal. Geom., 10 (2002) 79–126.
  * [9] Ma Hui and Schmies M., Examples of hamiltonian Stationary Lagrangian tori in ${\mathbb{C}}P^{2}$, Geometriae Dedicata, 118(2006) 173–183.
  * [10] Ma Hui, Hamiltonian stationary Lagrangian surfaces $\mathbb{CP}^{2}$ , Ann. Global. Anal Geom., 27(2005) 1–16.
  * [11] Ma Hui and Ma Y.-J., Totally real minimal minimal tori in $\mathbb{CP}^{2}$, Math. Z. 249(2005) 241–267.
  * [12] Mironov A.E., New examples of Hamiltionian-minimal and minimal Lagrangian manifolds in $\mathbb{C}^{n}$ and $\mathbb{CP}^{n}$, (Russian), Mat. Sb. 195 (2004) 89–102.
  * [13] Mironov A.E., On the Hamiltonian-minimal Lagrangian tori in the ${\mathbb{C}}P^{2}$, Siberian Math. J., 44(2003) 1039–1042.
  * [14] Mironov A.E., The Veselov–Novikov hierarchy of equations, and integrable deformations of minimal Lagrangian tori in ${\mathbb{C}}P^{2}$ (in Russian), Sib. Elektron. Mat. Izv., 1(2004) 38–46.
  * [15] Mironov A.E., On family of conformal flat minimal Lagrangian tori in ${\mathbb{C}P}^{3}$, Math. Notes., 81 (2007) 329–337.
  * [16] Novikov S.P.(Ed.), Theory of Solitons. The Inverse Sacttering Method, Plenum, New York, 1984.
  * [17] Oh,Y-G., Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math. Z.,212(1993) 175–192.
  * [18] Veselov A.P. and Novikov, S.P., Finite-zone two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations, Sov. Math. Dokl., 30(1984) 588–591.
  * [19] Wolfson J., Minimal Lagrangian diffeomophisms and the Monge-Ampére equation, J. Differential Geom. 45 (1997) 335–373.