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+Draft version February 1, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+A Pilot Study of Nulling in 22 Pulsars Using Mixture Modeling
+Akash Anumarlapudi
+,1 Joseph K. Swiggum
+,1, 2 David L. Kaplan
+,1 and Travis D. J. Fichtenbauer1
+1Center for Gravitation, Cosmology, and Astrophysics, Department of Physics, University of Wisconsin-Milwaukee, PO Box 413,
+Milwaukee, WI, 53201, USA
+2Dept. of Physics, 730 High St., Lafayette College, Easton, PA 18042, USA
+ABSTRACT
+The phenomenon of pulsar nulling, observed as the temporary inactivity of a pulsar, remains poorly
+understood both observationally and theoretically. Most observational studies that quantify nulling
+employ a variant of Ritchings (1976)’s algorithm which can suffer significant biases for pulsars where
+the emission is weak. Using a more robust mixture model method, we study pulsar nulling in a sample
+of 22 recently discovered pulsars, for which we publish the nulling fractions for the first time. These
+data clearly demonstrate biases of the former approach and show how an otherwise non-nulling pulsar
+can be classified as having significant nulls. We show that the population-wide studies that find a
+positive correlation of nulling with pulsar period/characteristic age can similarly be biased because
+of the bias in estimating the nulling fraction. We use our probabilistic approach to find the evidence
+for periodicity in the nulls in a subset of three pulsars in our sample. In addition, we also provide
+improved timing parameters for 17 of the 22 pulsars that had no prior follow-up.
+Keywords: Pulsar Nulling — Neutron Stars — Radio Astronomy
+1. INTRODUCTION
+Pulsar nulling, initially observed by Backer (1970a),
+is the absence of observed emission from a pulsar for
+one or more pulse periods.
+Observationally, the phe-
+nomenon of pulsar nulling remains poorly understood.
+It is clear that nulling is a broadband phenomenon, ob-
+served from 102 MHz (Davies et al. 1984) to 8.35 GHz
+(Honnappa et al. 2012).
+However, it is not firmly
+established whether nulling is simultaneous over this
+frequency range using a large sample of nulling pul-
+sars.
+Prior studies found contradictory conclusions.
+For example, observing over two frequency ranges, 50-
+140 MHz and 275-430 MHz, Taylor et al. (1975) found
+that nulls are simultaneous in two different pulsars (PSR
+B0031−07, PSR B0809+74), while Davies et al. (1984)
+found the evidence for excessive nulls in single pulses at
+102 MHz compared to 406 MHz in PSR B0809+74. A
+more recent study by Gajjar et al. (2014a) found that the
+nulls are highly coherent in three pulsars at four different
+frequencies — 313, 607, 1380, and 4850 MHz. In addi-
+Corresponding author: Akash Anumarlapudi
+[email protected]
+tion, it is also not clear whether pulsars null randomly.
+Redman & Rankin (2009) and Gajjar et al. (2012) found
+that nulls might not occur randomly but might be clus-
+tered, where nulls and bursts tend to occur in groups,
+but the latter found that the null durations can be ran-
+dom. However, for many of these results the dependency
+of the nulling inferences on signal-to-noise ratio makes
+it hard to robustly interpret their findings.
+Although the formation of a pair cascade and the radi-
+ation from these accelerated pairs in the pulsar magne-
+tosphere is often invoked to explain the observed emis-
+sion from a pulsar (Ruderman & Sutherland 1975), a
+full theory of pulsar magnetospheres and its emission to
+explain the diverse morphology in pulse profiles and phe-
+nomenology is yet to be developed. As such, the theory
+of pulsar nulling remains largely speculative, though it is
+often attributed to one of two classes: i) inherent to the
+magnetosphere itself such as loss of coherence condition
+required for radio emission, e.g., Filippenko & Radhakr-
+ishnan (1982), or the depletion of pairs in the magneto-
+sphere themselves, e.g., Kramer et al. (2006) or ii) geo-
+metrical factors external to the magnetosphere such as
+the line of sight traversing through the ‘empty’ region
+between rotating emission carousels, e.g., Herfindal &
+Rankin (2007, 2009). Further progress may require ad-
+arXiv:2301.13258v1  [astro-ph.HE]  30 Jan 2023
+
+ID2
+Anumarlapudi et al.
+ditional observational data to understand how the prop-
+erties of nulling relate to the properties of the pulsars
+themselves.
+Nulling as a phenomenon may be related to other more
+extreme forms of intensity modulation, where the pulses
+can disappear for hours to months in the cases of rotat-
+ing radio transients (RRATs; McLaughlin et al. 2006)
+or intermittent pulsars (Kramer et al. 2006; Lyne 2009).
+However, the connection between these populations is
+not clear. Furthermore, pulsar nulling is often discussed
+in tandem with two other forms of single pulse vari-
+ations: mode changing – a phenomenon in which an
+otherwise stable pulse profile switches between multiple
+shapes (or modes) (Backer 1970b) and sub-pulse drift-
+ing – a phenomenon in which the single pulse phase
+shows a uniform periodic drift (Drake & Craft 1968).
+Regardless, in all of these cases the appearance of these
+phenomena can be limited by instrumental sensitivity:
+without enough sensitivity to probe single pulses at high
+significance, one cannot be certain whether the pulsar
+emission is truly missing during the nulls or the pulsar
+switches to an alternate mode with lower intensity. To-
+gether all three are often thought of as different represen-
+tatives of a larger underlying phenomenon of sub-pulse
+intensity variations (Lorimer & Kramer 2004).
+Nulling is usually quantified by the fraction of pulses
+where there is no discernible emission, called the Nulling
+Fraction (NF). NF can vary from 0 – in the case of stan-
+dard emission picture that shows no nulls – to 1, in the
+extreme case where the pulsar emission is visible only
+between long nulls (intermittent pulsars and RRATs).
+NF has been measured in roughly 8% of pulsars, but
+this has more to do with the lack of single pulse studies
+as opposed to nulling being restricted to a small subset
+of pulsars. This smaller data set of nulling pulsars is en-
+tirely restricted to normal (not recycled) pulsars, owing
+to the high sensitivity demands that would be needed
+to observe single pulses of millisecond pulsars (MSPs),
+although some recent studies (Rajwade et al. 2014) have
+been conducted in a sample of bright MSPs which did
+not find a signature of nulling with high confidence. In
+addition, there can be a bias against discovering normal
+pulsars which tend to have a high NF, or are intermit-
+tent. Hence the fraction (8%), can only be considered
+as a conservative lower limit.
+Such a small data set restricts our ability to infer
+population-wide properties, which might give clues to
+the origin of the phenomenon, and hence studies done
+thus far have not reached a consensus. An initial study
+done by Ritchings (1976) claimed a correlation between
+NF and pulsar period (with longer period pulsars expe-
+riencing higher NF) and also a stronger correlation with
+the characteristic age. Wang et al. (2007) also suggested
+a correlation with spin-down age, albeit qualitatively,
+with older pulsars experiencing higher NF, before even-
+tually crossing the death line.
+Konar & Deka (2019)
+found that there may be two different populations of
+pulsars separated by a NF of ∼ 40% but did not find
+correlations with any intrinsic pulsar properties, while
+Sheikh & MacDonald (2021) claimed that there is no
+strong evidence for the existence of two sub-populations.
+All of these studies may be significantly biased since the
+samples used are restricted to the pulsars that explicitly
+showed nulling.
+In general, most studies (Wang et al. 2007; Gajjar
+et al. 2012, 2014b,a; Herfindal & Rankin 2009) estimate
+NF using the methodology (or a variant) proposed by
+Ritchings (1976). But as Kaplan et al. (2018) demon-
+strated, this method can suffer strong biases in the case
+of weaker pulsars which can lead to overestimating the
+NF and classifying an otherwise standard weak pulsar
+as a nulling pulsar. This can also lead to systematic bi-
+ases in population inferences. In addition, Kaplan et al.
+(2018) proposed an alternate method in which they use
+Gaussian Mixtures to model the single pulse intensities
+and estimate the NF , and demonstrate the reliability of
+this method in accurately measuring the NF in weaker
+pulsars. In this study, we expand on the Gaussian Mix-
+ture Model (GMM) of Kaplan et al. (2018)1 to general-
+ize their method and apply it to a larger sample of 22
+pulsars2.
+Pulsars selected for this study were discovered as a
+part of the Green Bank North Celestial Cap (GBNCC)
+pulsar survey (Stovall et al. 2014) in 2-min drift scans
+at 350 MHz with a 100 MHz bandwidth and with data
+sampled every 81.92 µs. At 350 MHz the beam size is
+36′ (Full Width at Half Maximum; FWHM) and hence
+the astrometric precision prior to a coherent timing so-
+lution is limited by the beam size depending on the
+Signal-to-noise ratio (SNR) of the discovery candidate.
+These were later followed up at the Green Bank Tele-
+scope (GBT) and Arecibo Observatory (AO) to improve
+their timing solutions and establish their nulling char-
+acteristics.
+The structure of this paper is as follows: In Section 2,
+we detail our data acquisition and reduction methods,
+and provide updated timing solutions for the pulsars in
+this study. We then describe the mixture model and pro-
+vide our basic results in Section 3. Finally, we present
+1 As noted in Kaplan et al. (2018), a similar method may have
+been used in Arjunwadkar et al. (2014).
+2 All of our code is available at https://github.com/AkashA98/
+pulsar nulling
+
+Pulsar Nulling with Mixture Models
+3
+the implications of the results in Section 4 and conclude
+in Section 5.
+2. DATA ANALYSIS
+2.1. Observations and Data Reduction
+A sample of 22 recently discovered pulsars was selected
+for this pilot study if they showed any signs of intermit-
+tency in their discovery plots3. Data for 15 out of 22
+pulsars were collected using the 100-m Robert C. Byrd
+Green Bank Telescope (GBT) (hereafter referred to as
+the GBT sample), operating at 820 MHz with a band-
+width of 200 MHz, in 2 hr contiguous scans, with the
+primary aim of determining the pulsars’ nulling charac-
+teristics (project code 18A−436; PI: J. Swiggum). Data
+for another nine pulsars were collected at the 300-m
+William E. Gordon Arecibo Observatory (AO) operat-
+ing at 430 MHz over a bandwidth of 24 MHz, with the
+goals to both establish coherent timing solutions and de-
+termine nulling characteristics (project code P3436; PI:
+J. Swiggum) (hereafter referred to as the AO sample).
+Two pulsars in our sample, PSR J0414+31, and PSR
+J1829+25, were observed at both observatories.
+Six of the 15 pulsars in the GBT sample already had
+coherent timing solutions (Lynch et al. 2018) and the
+data for these were collected in coherent search mode us-
+ing the Green Bank Ultimate Pulsar Processing Instru-
+ment (GUPPI; Ransom et al. 2009) with 128 frequency
+channels sampled at 10.24 µs and retaining full polar-
+ization information. The remaining nine pulsars had no
+prior follow-up campaigns and so we first improved their
+positions using gridding observations and then observed
+them in incoherent search mode with 2048 frequency
+channels sampled at 40.96 µs. Data for the AO sample
+were collected in coherent search mode using the Puerto
+Rico Ultimate Pulsar Processing Instrument4 (PUPPI),
+with 64 channels sampled at 40.96 µs, over a span of ∼
+six months to establish coherent timing solutions in ad-
+dition to studying the nulling properties. A summary of
+observations for each pulsar is provided in Tables 1 and
+2.
+Starting with the raw search mode data, we used
+dspsr (van Straten & Bailes 2011) to fold the data.
+We then used pazi, the interactive zapping routine in
+psrchive (van Straten et al. 2011) to remove radio fre-
+quency interference (RFI)-affected frequency channels
+and single pulses. For GBT data, we also made use of
+RFI scans taken at the observatory5, when available, to
+3 See the GBNCC discovery page:
+http://astro.phys.wvu.edu/
+GBNCC.
+4 http://www.naic.edu/puppi-observing/
+5 https://greenbankobservatory.org/rfi-gui-user-guide/
+Table 1. Times and durations of GBT ob-
+servations
+Pulsar
+Observations
+Total Time
+MJD (hr)
+(hr)
+J0054+6946
+58163 (2.00)
+2.00
+J0111+6624
+58163 (2.24)
+2.24
+J0325+6744
+58163 (1.52)
+2.00
+58164 (0.48)
+· · ·
+J0414+31a
+58164 (1.50)
+1.50
+J0614+83
+58164 (1.90)
+1.90
+J0738+6904
+58209 (2.00)
+2.00
+J1529−26
+58209 (1.50)
+1.50
+J1536−30
+58209 (1.50)
+1.50
+J1629+33
+58209 (1.50)
+1.50
+J1821+4147
+58209 (1.69)
+1.69
+J1829+25a
+58246 (1.50)
+1.50
+J1901−04
+58246 (1.50)
+1.50
+J2040−21
+58246 (1.50)
+1.50
+J2131−31
+58246 (0.33)
+0.33
+J2310+6706
+58246 (1.75)
+1.75
+Note—For each pulsar we give the individual
+Modified Julian Date (MJD) and duration of
+each session, as well as the total observing
+time.
+aThis pulsar was observed at both AO and
+GBT
+identify the frequency bands that are affected by RFI,
+which are otherwise not obvious visually. In some cases,
+we found that one of the polarization channels was per-
+sistently affected by RFI, and in such cases we excluded
+data from that polarization channel at the cost of SNR.
+Fortunately, this did not have a significant impact on
+the determination of the nulling fractions. Some of the
+AO data had periodic “drop-outs” in the data with sub-
+millisecond periodicity at zero dispersion measure (DM),
+caused by data rate overflow during the observations.
+We cleaned these “drop-outs” by replacing the data with
+NaN values and being careful to exclude those when fold-
+ing/averaging.
+After cleaning the RFI, both for tim-
+ing and estimating nulling, we averaged polarizations to
+measure the total intensity.
+2.2. Timing
+For the 16 pulsars in our sample that had no prior
+follow-up, we first tried to improve the timing parame-
+ters. We used paas from psrchive (van Straten et al.
+2011) to make a standard template and then used pat
+
+4
+Anumarlapudi et al.
+Table 2. Times and durations of Arecibo observations
+Pulsar
+Observations
+Total Time
+MJD (hr)
+(hr)
+J0355+28
+58890 (0.25), 58922 (0.33)
+2.95
+58924 (0.42), 58928 (0.39)
+· · ·
+58936 (0.39), 58951 (0.39)
+· · ·
+58982 (0.39), 59013 (0.39)
+· · ·
+J0414+31a
+58890 (0.50), 58922 (0.38)
+3.46
+58924 (0.50), 58928 (0.35)
+· · ·
+58936 (0.30), 58951 (0.40)
+· · ·
+58982 (0.63), 59013 (0.40)
+· · ·
+J1822+02
+58941 (0.22), 58968 (0.17)
+1.55
+58970 (0.17), 58974 (0.17)
+· · ·
+58981 (0.17), 59000 (0.33)
+· · ·
+59029 (0.17), 59063 (0.17)
+· · ·
+J1829+25a
+58852 (0.17), 58941 (0.17)
+1.03
+58968 (0.14), 58970 (0.11)
+· · ·
+58974 (0.11), 58981 (0.11)
+· · ·
+59029 (0.11), 59063 (0.11)
+· · ·
+J1904+33
+58852 (0.17), 58882 (0.17)
+1.34
+58941 (0.17), 58968 (0.14)
+· · ·
+58970 (0.14), 58974 (0.14)
+· · ·
+58981 (0.14), 59029 (0.14)
+· · ·
+59063 (0.14)
+· · ·
+J1928+28
+58852 (0.17), 58882 (0.17)
+1.98
+58941 (0.17), 58968 (0.14)
+· · ·
+58970 (0.17), 58974 (0.17)
+· · ·
+58981 (0.17), 59000 (0.50)
+· · ·
+59029 (0.17), 59063 (0.17)
+· · ·
+J1941+02
+58852 (0.17), 58882 (0.17)
+1.5
+58912 (0.14), 58941 (0.17)
+· · ·
+58968 (0.14), 58970 (0.14)
+· · ·
+58974 (0.14), 58981 (0.10)
+· · ·
+59029 (0.17), 59063 (0.17)
+· · ·
+J2000+29
+58852 (0.39), 58882 (0.17)
+1.83
+58941 (0.10), 58968 (0.14)
+· · ·
+58970 (0.14), 58974 (0.14)
+· · ·
+58981 (0.14), 59000 (0.33)
+· · ·
+59029 (0.14), 59063 (0.14)
+· · ·
+J2044+28
+58852 (0.17), 58882 (0.17)
+1.18
+58968 (0.07), 58970 (0.14)
+· · ·
+58974 (0.14), 58981 (0.14)
+· · ·
+59000 (0.07), 59029 (0.14)
+· · ·
+59063 (0.14)
+· · ·
+aThis pulsar was observed at both AO and GBT
+to extract the Times of Arrival (TOAs) from the data.
+For the GBT data, our goal was to improve the spin fre-
+quency (F0) and DM measurements since we had only
+2 hour scan at a single epoch for each source. For the
+AO data, the data spanned ∼3–6 months depending on
+the pulsar and hence we can generate a phase-connected
+solution. However, the relatively narrow bandwidth of
+the observations (24 MHz) restricted our ability to fit
+for DM using sub-banded TOAs and hence we used the
+DM of the discovery candidate found on the GBNCC
+discovery page.
+The timing solutions for all the pulsars in this study
+are given in Table 3. For pulsars observed at GBT we
+improved the positions through gridding, and F0 and
+DM estimates through timing.
+For pulsars observed
+at AO, we improved the gridded positions, F0 and the
+frequency derivative F1 = ˙F0 through coherent timing.
+For the two overlapping pulsars observed at both GBT
+and AO, a timing solution was obtained by combining
+the TOAs from both observatories. In the case of pul-
+sars observed at AO for only ∼3 months (J0355+28,
+J0414+31, J1822+02), and pulsars where a combina-
+tion of low SNR and nulling resulted in few TOAs with
+SNR > 8 (J1928+28), it is difficult to estimate both po-
+sition and F1 precisely (they are highly covariant). In
+such cases, we rely on the
+F-statistic, given by
+F = (χ2
+0 − χ2)/(p − p0)
+χ2/p
+where χ2
+0 and χ2 are the chi-squared values of the timing
+residuals, and p0 and the p are the degrees of freedom
+before and after the addition of F1 (or any additional
+parameter(s), in general). This F-statistic follows an F-
+distribution (Lomax 2007) and hence we include F1 in
+the fit if the improvement in the goodness of fit (χ2) due
+to F1 is <1% by chance. The resulting timing residuals
+are shown in Figure 1.
+2.3. ON/OFF histograms
+Once we had improved the timing solution, we used
+dspsr in single pulse mode to generate single pulses for
+all scans and used psradd, from psrchive, to phase
+align pulses from different scans after cleaning the data
+for RFI. We then averaged the data along the polariza-
+tion and frequency axes to obtain the pulse intensity of
+the single pulses as a function of the rotational phase
+and generated single pulse stacks such as that shown in
+Figure 2.
+The most important aspect in estimating the nulling
+fraction is determining the “ON”-pulse and “OFF”-
+
+Pulsar Nulling with Mixture Models
+5
+Table 3. Timing Parameters for the GBNCC pulsars used to study nulling
+Pulsar
+Position (J2000)
+Period
+Period derivative
+DM
+RA
+RA error
+DEC
+DEC error
+(′′)
+(′′)
+(s)
+(10−15 s/s)
+(pc/cm3)
+GBT sample
+J0054+6946a
+00h 54m 59.s1
+00.1
++69◦ 46′ 16.′′8
+00.0(3)
+0.832911328744(4)
+−0.7194(8)
+116.52(5)
+J0111+6624a
+01h 11m 21.s9
+01.7
++66◦ 24′ 10.′′9
+00.6
+4.3018721007(3)
+−8.4(2)
+111.20(3)
+J0325+6744a
+03h 25m 05.s1
+00.3
++67◦ 44′ 59.′′4
+00.1
+1.36467876728(1)
+−1.553(9)
+65.28(5)
+J0414+31b
+04h 14m 35.s6
+02.6
++31◦ 38′ 35.′′4
+25.3
+1.0805116(1)
+−3.6(5)
+64.64(3)
+J0614+83c
+06h 14m 03.s4
+34.6
++83◦ 13′ 46.′′2
+34.6
+1.03918794(5)
+· · ·
+44.2(1)
+J0738+6904a
+07h 38m 22.s6
+00.5
++69◦ 04′ 20.′′0
+00.3
+6.8276928023(5)
+−26.97(4)
+17.22(2)
+J1529−26c
+15h 29m 07.s2
+38.9
+−26◦ 26′ 35.′′5
+38.9
+0.79857094(5)
+· · ·
+44.7(1)
+J1536−30c
+15h 36m 33.s4
+17.3
+−30◦ 06′ 14.′′4
+17.3
+0.190084143(9)
+· · ·
+63.40(7)
+J1629+33c
+16h 29m 22.s6
+99.2
++33◦ 23′ 35.′′9
+99.2
+1.5247311(3)
+· · ·
+34.8(5)
+J1821+4147a
+18h 21m 52.s3
+00.1
++41◦ 47′ 02.′′6
+00.0(4)
+1.26185719(3)
+−1.7292(9)
+40.63(5)
+J1829+25b
+18h 30m 31.s8
+01.8
++25◦ 08′ 00.′′4
+01.4
+2.85769207(9)
+−1.9(4)
+73.64(9)
+J1901−04c
+19h 01m 37.s1
+62.0
+−04◦ 54′ 44.′′9
+62.0
+1.8255459(8)
+· · ·
+105.4(9)
+J2040−21c
+20h 40m 40.s6
+09.7
++21◦ 52′ 51.′′6
+09.7
+0.562564125(4)
+· · ·
+23.77(1)
+J2131−31c
+21h 31m 30.s9
+65.9
+−31◦ 32′ 53.′′4
+65.9
+3.32537(3)
+· · ·
+31.753
+J2310+6706a
+23h 10m 42.s1
+02.9
++67◦ 06′ 52.′′1
+00.9
+1.944788973(1)
+−0.06(5)
+97.7(2)
+AO sample
+J0355+28
+03h 55m 22.s8
+00.4
++28◦ 38′ 50.′′1
+00.8
+0.36492919909(3)
+· · ·
+48.788
+J0414+31b
+04h 14m 35.s6
+02.6
++31◦ 38′ 35.′′4
+25.3
+1.0805116(1)
+−3.6(5)
+64.64(3)
+J1822+02
+18h 22m 43.s6
+01.4
++02◦ 28′ 53.′′8
+01.2
+1.5081132778(9)
+· · ·
+103.22
+J1829+25b
+18h 30m 31.s8
+01.8
++25◦ 08′ 00.′′4
+01.4
+2.85769207(9)
+−1.9(4)
+73.64(9)
+J1904+33
+19h 04m 40.s2
+00.2
++33◦ 58′ 25.′′9
+00.1
+0.417032327(1)
+−0.247(5)
+81.139
+J1928+28
+19h 27m 58.s4
+01.1
++28◦ 59′ 12.′′4
+01.0
+1.0630373062(5)
+· · ·
+79.34
+J1941+02
+19h 40m 34.s1
+00.8
++02◦ 39′ 21.′′7
+01.0
+1.23229077(1)
+−0.18(9)
+87.478
+J2000+29
+20h 00m 16.s5
+00.4
++29◦ 20′ 07.′′6
+00.1
+3.07377646(2)
+−37.37(8)
+132.62
+J2044+28
+20h 43m 36.s9
+00.4
++28◦ 28′ 37.′′3
+00.2
+1.61816650(1)
+−3.99(4)
+90.169
+Note—Quantities in parentheses are 1σ uncertainties on the last digit.
+aCoherent timing solutions are given in Lynch et al. (2018)
+b Timing solution is obtained by combining AO and GBT data.
+c Astrometric positions are estimated from gridding and the positional uncertainties are estimated from the beam size (15′)
+and the Signal to Noise Ratio (SNR)
+
+6
+Anumarlapudi et al.
+-1.0
+0
+1.0
+PSR J0355+28
+-3.0
+0
+3.0
+PSR J0414+31
+-11.0
+0
+11.0
+PSR J1822+02
+-3.0
+0
+3.0
+PSR J1829+25
+-1.0
+0
+1.0
+Residuals (ms)
+PSR J1904+33
+-2.0
+0
+2.0
+PSR J1928+28
+-2.0
+0
+2.0
+PSR J1941+02
+-2.0
+0
+2.0
+PSR J2000+29
+58800
+58850
+58900
+58950
+59000
+59050
+59100
+Modified Julian Date (MJD)
+-2.0
+0
+2.0
+PSR J2044+28
+-0.002
+0
+0.002
+-0.002
+0
+0.002
+-0.007
+0
+0.007
+-0.001
+0
+0.001
+-0.002
+0
+0.002
+Residuals (cycles)
+-0.001
+0
+0.001
+-0.001
+0
+0.001
+-0.0006
+0
+0.0006
+-0.001
+0
+0.001
+Figure 1. Timing residuals for the pulsars observed in the timing/nulling campaign at the AO. The red dots are the residuals
+(in milliseconds) from the timing model with the error bars representing the 1-σ error on the TOAs. The timing model solutions
+are presented in Table 3.
+
+Pulsar Nulling with Mixture Models
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+200
+400
+Single pulses
+ON
+OFF
+−0.1
+0.0
+0.1
+0
+2
+0
+0.5
+1
+NP
+NP= 0.5
+Intensity
+(a) Single pulse stack of PSR J0325+6744
+−1
+0
+1
+2
+3
+4
+5
+Normalized Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Density
+OFF window
+ON window
+(b) Pulse intensity histogram for PSR J0325+6744
+Figure 2. (a)The bottom left panel shows the single pulse
+stack with the ON and OFF windows marked with black
+dashed lines. Null probabilities (NP) for every single pulse
+are calculated using the method described in §3.2 and are
+shown in the bottom right plot. The distribution of NP is
+shown in the top right panel where we can clearly see the
+evidence for two classes of pulses. The summed profile of
+all the single pulses with null probability < 0.5 is shown in
+the top left panel, while the summed profile for pulses with
+null probability > 0.5 is shown in the middle panel. (b) The
+pulse intensities in the OFF and ON windows are shown
+in blue and orange histograms.
+The presence of excessive
+counts in the ON histogram (the null component) at the
+background noise level separated from a second component
+at higher intensities (the emission component) is evidence for
+the nulling behavior.
+pulse phase windows. The single pulse intensities in the
+“OFF”-pulse window should be entirely due to radiome-
+ter noise, while the intensities in the “ON”-pulse window
+should be the sum of the radiometer noise component
+(same as the “OFF”-pulse window) and the pulsar emis-
+sion component. We first generated the average pulse
+profile to visually select on and off windows of the same
+widths. We then fit a sixth-order polynomial as a func-
+tion of pulse phase to each single pulse (similar to Rosen
+et al. 2013; Lynch et al. 2013; Kaplan et al. 2018) after
+masking the ON/OFF windows to remove any trends
+and construct a flat baseline. We recorded the ON/OFF
+intensities as the sum of the baseline-subtracted inten-
+sities across the windows. Finally, we constructed his-
+tograms of the ON/OFF intensities which we used to
+determine the nulling properties.
+Figure 2 shows the
+single pulse intensity distribution in the ON/OFF win-
+dow. The OFF histogram can be accurately described
+by a single component (Gaussian noise), but the ON
+histogram can have multiple components — “null” and
+“emission” components. The presence of nulling man-
+ifests in the ON histogram as an excess of samples at
+levels consistent with the OFF component, which we
+refer to as the null component. The residual distribu-
+tion, after removing the null component, is supposed to
+be a realization of pulsar’s emission distribution (here-
+after referred to as ‘emission’ component). The emission
+component can be a single distribution or a combination
+of multiple distributions. The ON distribution can be
+thought of as the sum of the null and the emission com-
+ponents.
+3. METHODS & RESULTS
+3.1. Determining Nulling Frations
+As demonstrated by Kaplan et al. (2018), Ritchings’
+method can give biased estimates for NF (hereafter re-
+ferred as NFr) in pulsars where the emission compo-
+nent is close to the noise level. Therefore, following Ka-
+plan et al. (2018) we adopt a method which models the
+ON/OFF histograms using a mixture model (MM). This
+means that the intensities x can be considered as ran-
+dom draws from the probability density function (PDF)
+p(x|¯θ) =
+m
+�
+n=1
+cn Fn(x|{θn}),
+(1)
+where the Fn functions are the individual probability
+density functions parameterized by the set {θn}, cn are
+the weights. In the case where all the Fn functions are
+the same and are normal distributions
+Fn(x; µn, σn) = N(x; µn, σn) =
+1
+√
+2πσn
+e− 1
+2(
+x−µn
+σn )
+2
+,
+where {µn} and {σn} are the means and standard devia-
+tions of component n, this reduces to a Gaussian mixture
+model (GMM), but more general models are considered.
+
+8
+Anumarlapudi et al.
+There is an additional constraint that the weights cn add
+to one:
+m
+�
+n=1
+cn = 1,
+which comes from the normalization of the PDF, which
+leaves the total number of free parameters to be deter-
+mined as �m
+n=1 dim({θn}) model parameters, and m−1
+latent parameters.
+In general, the OFF histogram can be well-described
+by a Gaussian as expected of radiometer noise (assum-
+ing that RFI has been sufficiently removed), and this is
+what we observe in our data. The emission component
+usually can be described by a single Gaussian as well.
+However, there are cases when it deviates from a single
+Gaussian component. More than one component is a
+possibility considered in Kaplan et al. (2018), which can
+be tested against the single-component model through a
+model comparison test. However, we also consider non-
+Gaussian models here. Specifically, multi-path propaga-
+tion of the pulses through the interstellar medium (ISM)
+(Smith 1973; Bhat et al. 2003; Lorimer & Kramer 2004),
+can result in the emission distribution having long tails
+towards higher intensities. This effect can be reasonably
+well described by the intensity distribution
+F(x; µ, σ, τ) = 1
+2τ exp
+� σ2
+2τ 2
+�
+exp
+�
+−x − µ
+τ
+�
+erfc
+�
+−x − (µ + σ2/τ)
+√
+2σ
+�
+(2)
+which is a convolution of a Gaussian N(x; µ, σ) and a
+one-sided exponential 1
+τ exp(−x/τ)U(x), where U(x) is
+the Heaviside or step function, erfc(x) is the complemen-
+tary error function, and τ is the decay time of the ex-
+ponential (McKinnon 2014). Hence we try to model the
+emission component using multi-component Gaussians
+and Gaussians with exponential tails and rank them us-
+ing their Bayesian Information Criterion (BIC) values
+to choose the best-fit model.
+We employ the scikit-learn Gaussian mixture
+model (Pedregosa et al. 2011) to derive an initial fit for
+the ON and OFF histograms. This is based on the ex-
+pectation–maximization (EM) algorithm, in which pa-
+rameters are estimated by maximizing the likelihood
+function L(data | ¯θ) (see Ivezi´c et al. 2020, for details).
+This produces a very good fit for the OFF histogram.
+However, in the case of weaker pulsars where the emis-
+sion can be confused with the background, Kaplan et al.
+(2018) showed that this method can still fail in produc-
+ing a reliable fit for the null and emission components
+of the ON histogram simultaneously, although this bias
+can be small compared to the Ritchings’ algorithm. As
+such, a refined fit for the null and emission components
+can be obtained by performing a Markov-Chain Monte
+Carlo (MCMC) analysis.
+For MCMC analysis, the likelihood function is given
+by
+L(¯x|¯θ) =
+�
+i
+p(xi|¯θ)
+(3)
+following p(xi|¯θ) from Equation 1.
+The priors chosen are:
+• Initial Gaussian fit from the EM algorithm for the
+off-pulse mean and standard deviation
+• Uniform between the bounds dictated by the on-
+pulse intensities for the parameters governing the
+pulsar emission component
+• Dirichlet distribution for the m coefficients cm
+(Wilks 2008)
+We use the emcee (Foreman-Mackey et al. 2013) en-
+semble sampler to sample the posterior. We initialize
+32 walkers within a ±5σ range of the initial fit values
+of the parameters. To account for the finite correlation
+length of the chains and produce independent samples,
+we first let the walkers “burn-in” to erase their start-
+ing conditions, and we then let the walkers explore the
+parameter space until we have at least 100 independent
+samples for each walker.
+Figure (3, left column) shows the pulse intensity his-
+tograms for PSR J0325+6744: a pulsar in which the
+emission component is easily discernible from the noise;
+and PSR J1529−26: a pulsar where these two start to
+blend into each other. Looking at the null component in
+the ON histogram for the two pulsars, the evidence for
+nulling is clear in J0325+6744 while J1529−26 behaves
+like a non-nulling pulsar whose emission is weak. The
+blue, green and orange-filled regions show the fit for the
+OFF, null, and emission components respectively, and
+the black dotted line shows the overall fit for the ON
+component.
+The posteriors for the model parameters
+are presented in Figure (3, right column) with the point
+estimates (median6) of the NF from MM given in Ta-
+ble 4.
+For PSR J0325+6744, where the null and emission
+components are well separated (bright pulsars), our
+method yields a NF = 53.92 ± 0.81% while Ritchings’
+6 In the case of non-nulling pulsars where the distribution of NF
+is one-sided, the median will be over-estimated compared to the
+true value. Even so, the uncertainty on NF is larger than the
+difference between the median and mode and hence NF is still
+consistent with 0.
+
+Pulsar Nulling with Mixture Models
+9
+Table 4. Nulling properties of the GBNCC pulsars
+Pulsar
+Model
+NF
+NFr
+Null period
+Lengths
+Null
+Em.
+(%)
+(%)
+(pulse periods)
+GBT sample
+J0054+6946
+G3
+27.5±5.1
+36.8
+· · ·
+2
+3
+J0111+6624
+G2
+10.2±1.7
+17.9
+· · ·
+2
+7
+J0325+6744
+G2
+53.9±0.8
+55.1
+· · ·
+3
+4
+J0414+31
+G2
+27.5±1.9
+40.7
+28.4c
+2
+4
+J0614+83
+G2
+06.7±3.1
+52.3
+· · ·
+1-2a
+· · ·
+J0738+6904
+Eg2
+66.6±1.5
+64.9
+42.7c
+9
+4
+J1529−26
+G2
+05.4±4.3
+48.5
+· · ·
+1-2a
+· · ·
+J1536−30
+G2
+43.1±2.2
+57.5
+· · ·
+4
+J1629+33
+G2
+83.8±1.9
+83.9
+· · ·
+12
+1-2a
+J1821+4147
+G2
+00.0±0.6
+20.9
+· · ·
+1-2a
+· · ·
+J1829+25
+G2
+00.0±0.6
+07.8
+· · ·
+0b
+· · ·
+J1901−04
+G2
+13.9±4.1
+50.4
+1a
+· · ·
+J2040−21
+G2
+25.4±1.8
+42.4
+23.3c
+2
+5
+J2131−31
+G2
+49.8±8.6
+54.2
+· · ·
+3
+3
+J2310+6706
+Eg2
+54.1±2.7
+52.7
+3
+3
+AO sample
+J0355+28
+G2
+01.6±1.1
+30.3
+· · ·
+1-2a
+· · ·
+J0414+31
+G2
+33.0±0.7
+37.1
+28.4c
+2
+4
+J1822+02
+G2
+00.1±0.7
+09.3
+· · ·
+1a
+· · ·
+J1829+25
+G2
+00.0±0.6
+05.5
+· · ·
+0b
+· · ·
+J1904+33
+G2
+00.0±0.1
+09.4
+· · ·
+1a
+· · ·
+J1928+28
+G2
+47.6±2.4
+71.9
+· · ·
+3
+3
+J1941+02
+G2
+00.2±1.7
+31.1
+· · ·
+1-3a
+· · ·
+J2000+29
+G2
+19.3±1.1
+23.4
+· · ·
+1-2a
+3
+J2044+28
+G2
+15.2±0.9
+17.4
+· · ·
+1-2a
+6
+Note—Naming convention for the model represents the model used
+to describe the emission histogram (G=Gaussian, Eg=Exponentially
+modified Gaussian) followed by the number of components in the ON
+histogram.
+aWe find that in extreme cases (non-nulling/highly-nulling), one of the
+distributions is confined to very few bins and so we quote this range
+rather than fitting for it.
+b We find that there are no single pulses with NP>0.5.
+c We observe quasi-periodicity in these cases.
+method (see Ritchings 1976; Wang et al. 2007; Kaplan
+et al. 2018, for implementation) gives a comparable esti-
+mate of 55.01%. However in the case of a weaker pulsar,
+PSR J1529−26, where the emission component is closer
+to the background noise, our method gives a best-fit
+value of NF = 5.55 ± 4.4% compared to 48.1% given
+by the Ritchings’ method.
+The latter is significantly
+overestimated and can easily lead to (mis)classifying the
+source as a nulling pulsar, further illuminating the bias
+of Ritchings’ method in weaker pulsars.
+Full results for all the 23 pulsars, including the single
+pulse stacks, posteriors from the MCMC run and the
+resultant ON/OFF histogram model fits are shown in
+Appendix A.
+3.2. Nulling Correlations
+After determining the nulling properties we wish to
+know whether the locations and durations of nulls are
+completely random, or if there is any correlation be-
+tween different nulling and emission episodes in a pulsar.
+Specifically, given a single pulse that shows emission (or
+that nulls), how likely are we to see emission for the next
+pulse, and are there any patterns of longer duration?
+We test this using the probability of a null (the nulling
+“responsibility”) evaluated for each individual pulse,
+given by
+NPI =
+c0F0(I|{θ0})
+�m
+n=1 cn Fn(I|{θn}).
+(4)
+We divided the data into stacks of 256 pulses (similar to
+Ritchings 1976; Herfindal & Rankin 2009) to calculate
+more robust estimates and to be less sensitive to long-
+term variations like scintillation and system temperature
+changes, and use equation 4 to calculate the probabil-
+ity of a given single pulse being a null. We then looked
+for periodic signature by taking the Fourier transform
+(FT) within each stack and co-adding the power from all
+stacks incoherently. Figure 4 shows the resultant spec-
+trum for PSR J0414+31, in which a certain pattern of
+combination of emission and nulls seems to be periodic
+over ∼28 pulse periods. We estimate the significance of
+peaks in the stacked power spectra assuming that the
+null distribution from n stacks follows a χ2 distribution
+with 2n degrees of freedom (this assumes white noise).
+We see significant periodic or quasi-periodic (a signifi-
+cant broad peak in the power spectrum) signatures in a
+few other pulsars, and tabulate their periods in Table 4.
+In the case of precise period measurements, we estimate
+the uncertainty as described in Ransom et al. (2002).
+However, this only points to the periodic nature of
+a certain pattern of emission and nulls.
+To find how
+emissions and nulls are ‘bunched’, we look for the dis-
+tribution of continuous emissions and nulls, where we
+use NPI=0.5 to be the boundary between an emission
+and a null. Figure 5 shows the emission and null length
+
+10
+Anumarlapudi et al.
+−1
+0
+1
+2
+3
+4
+5
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+(a1) PSR J0325+6744 – NF = 53.92 ± 0.81% vs NFr =
+55.01%
+µ0
+µ0=-0.001
+2.10
+2.16
+2.22
+µ1
+µ1=2.183
+0.345
+0.360
+σ0
+σ0=0.355
+0.80
+0.85
+0.90
+σ1
+σ1=0.853
+−0.015
+0.000
+0.015
+µ0
+0.52
+0.54
+0.56
+NF
+2.10
+2.16
+2.22
+µ1
+0.345
+0.360
+σ0
+0.80
+0.85
+0.90
+σ1
+0.52
+0.54
+0.56
+NF
+NF=0.54
+(a2) Model parameter posteriors for PSR J0325+6744
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Raw Intensity
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+(b1) PSR J1529−26 – NF = 5.4 ± 4.4% vs NFr =48.5%
+µ0
+µ0=0.011
+1.05
+1.20
+µ1
+µ1=1.082
+1.3
+1.4
+σ0
+σ0=1.35
+1.30
+1.35
+1.40
+σ1
+σ1=1.389
+−0.06
+0.00
+0.06
+µ0
+0.08
+0.16
+NF
+1.05
+1.20
+µ1
+1.3
+1.4
+σ0
+1.30
+1.35
+1.40
+σ1
+0.08
+0.16
+NF
+NF=0.054
+(b2) Model parameter posteriors for PSR J1529−26
+Figure 3. Left (a1, b1) Two-component Gaussian model fits for the ON and OFF histograms. Individual ON/OFF histograms
+are shown in solid black lines. The blue, green and orange-filled regions shows the OFF, the null (NF × OFF) and the emission
+(ON − NF × OFF) components respectively, where this estimate of NF is obtained using the mixture model. The black dotted
+line shows the overall fit for the ON pulse distribution. Right (a2, b2) Corner plots for 2-component Gaussian fit to the ON/OFF
+histograms parameterized by the means {µ1, µ2}, standard deviations {σ1, σ2} and the nulling fraction NF. The dashed vertical
+lines are the quoted median point estimates of the parameters
+
+Pulsar Nulling with Mixture Models
+11
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Fourier frequency (in 1/P)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Power (arbitrary units)
+NP FFT (ON)
+NP FFT (OFF)
+analytical limit
+(a) PSR J0414+31 (GBT)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Fourier frequency (in 1/P)
+0
+100
+200
+300
+400
+500
+Power (arbitrary units)
+NP FFT (ON)
+NP FFT (OFF)
+analytical limit
+(b) PSR J0414+31 (AO)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Fourier frequency (in 1/P)
+0
+50
+100
+150
+200
+250
+Power (arbitrary units)
+NP FFT (ON)
+NP FFT (OFF)
+analytical limit
+(e) PSR J2040−21
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Fourier frequency (in 1/P)
+0
+20
+40
+60
+80
+Power (arbitrary units)
+NP FFT (ON)
+NP FFT (OFF)
+analytical limit
+(f) PSR J0738+6904
+Figure 4. Fourier transform of the null probability for the pulsars in our sample that show periodicity.
+Power combined
+incoherently from multiple stacks of 256 pulses is shown at 129 discrete frequencies (in the units of 1/pulse period) in the
+blue line. The orange curve shows the same for the OFF component (background noise) which can be used to eliminate any
+instrumental variations/artifacts and/or RFI. The black dotted line shows the upper limit that allows for 1 false positive in 1000
+trails, corresponding to a 99.9% confidence limit. The gray curves are the normalized power from the individual stacks (not to
+scale) that are used to look for quasi-periodicity. The value of the periodicities are given in Table 4
+
+12
+Anumarlapudi et al.
+0
+5
+10
+15
+20
+Pulse periods
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Normalized counts
+null lengths
+em. lengths
+null fit τem=0.49
+null fit τem=0.3
+Figure 5. Distribution of emission lengths and null lengths
+for J0414+31. The gray-filled and the black-open histograms
+show the distribution of null and emission episodes respec-
+tively.
+The orange curve shows an exponential fit for the
+emission length distribution with decay constant τem=0.3,
+whereas the blue curve shoes the same for the null length
+distribution with τnull=0.49.
+distributions for the single pulses of PSR J0414+31. We
+find that these distributions can be well described by an
+exponential distribution (p(x) = τ −1 exp(−x/λ)), where
+x is the null or emission length and the mean duration
+of the episode is λ. We find that for PSR J0414+31, the
+emission episodes have a characteristic period of four
+periods, whereas the nulls are two periods long, which
+is consistent with the observed nulling fraction of ∼ 33%
+(see Table 4). We repeat this for all the pulsars and the
+results are tabulated in Table 4.
+3.3. Sub-pulse Drifting
+Beyond nulling, we also look for any correlations be-
+tween nulling and sub-pulse drifting. Drifting is usually
+characterized by two periods: the drifting period P3,
+defined as the period for which the pulse is seen at the
+same longitude (phase), and P2, the spacing between
+two sub-pluses within the same single pulse (see Figure
+6). To estimate both, we prepared the data by selecting
+only the on-pulse window of data (np phase bins) for all
+the single pulses (ns single pulses). We then calculated
+Longitude Resolved Fluctuation Spectra (LRFS, Backer
+1970c), where we take a 1-D Fourier transform of the
+(ns × np) data along the ns axis. Figure 6 shows one
+of the two pulsars in our sample, J1822+02, that shows
+clear signs of drifting. A period P3 of ∼ 28 pulse periods
+and P2 of ∼ 35/1024 pulse periods can be clearly seen.
+We also find the evidence for drifting in PSR J1829+25
+(see figure 7), with a P3 of ∼ three pulse periods and a
+P2 of 1/128 pulse periods, with similar inferences in the
+data from both AO and GBT.
+4. DISCUSSION
+4.1. Biases in Nulling Models
+Kaplan et al. (2018) demonstrated the bias of Ritch-
+ings’ method for weaker pulsars through simulated data,
+where the mixture model was able to recover the true in-
+jected nulling fraction. They also showed that for Gaus-
+sian mixtures, an analytical correction can correct the
+biased estimate of Ritchings’ method to find the true
+value. We extend the same technique using our sam-
+ple of 22 pulsars. Figure 8 shows the comparison of the
+NF estimates derived using both methods.
+The blue
+points show the NF estimate derived using Ritchings’
+algorithm (NFr), the orange points show NFr estimate
+corrected for the bias (as in Kaplan et al. 2018), and the
+green points show the NF derived using mixture model-
+ing. In the case of highly nulling pulsars, the contamina-
+tion of the null component from the emission component
+can be small, and both methods perform comparably.
+However, in the case of pulsars with small NF a system-
+atic bias can be seen as the pulsar emission component
+becomes blended with the background noise, and the
+fact that the green and orange points agree quite well
+demonstrates our confidence in estimating the bias in
+the Ritchings method and the utility of mixture models.
+4.2. Is the Nulling Fraction Correlated with Pulsar
+Properties?
+Comparing the nulling estimates from the mixture
+modelling and Ritchings’ method in Table 4, it can be
+seen that there can be significant differences between
+these estimates. Such a scenario can lead to significant
+biases in population-wide studies that look for corre-
+lation between nulling fraction and pulsar properties.
+Figure 9 shows the most complete list of nulling pulsars,
+extended from Konar & Deka (2019), on the P − ˙P dia-
+gram. We do not find any clear visual trends of NF with
+respect to period (P), spin-down rate ( ˙P), characteris-
+tic age (τc), or surface magnetic field (Bsurf), although
+we emphasize that most of the pulsars here (142/164)
+have their NF estimates derived using some variant of
+the Ritchings method.
+Our sample size of 22 pulsars is too small to derive
+reliable correlations. However, we can test the similar-
+ity/disparity in the correlations obtained using nulling
+estimates derived with mixture models versus the Ritch-
+ings algorihtm. We use the Spearman correlation test, a
+non-parametric correlation test to quantify any correla-
+tions between the relevant parameters (P/ ˙P/Bsurf/τc)
+and NF. Table 5 shows the correlation coefficients of
+nulling fraction with parameters of interest (P, ˙P, Bsurf,
+τc). In no case do we see an evidence for strong cor-
+relations but we can see large differences between these
+coefficients obtained using the NF derived using the two
+
+Pulsar Nulling with Mixture Models
+13
+0.1
+0.16
+0.21
+0.27
+0.33
+0.39
+Phase
+0
+50
+100
+150
+200
+250
+Singlepulse number
+P2
+P3
+0.21
+0.23
+0.25
+0.27
+0.29
+Pulse phase
+0.6
+0.8
+1.0
+Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Frequency (in units of 1/Period)
+0.25
+0.50
+0.75
+1.00
+Power
+Figure 6. Left: A stack of 300 single pulses of PSR J1822+02 clearly showing the sub-pulse drifting phenomenon. The drifting
+periods P2 and P3 are shown. Right: LFRS of the single pulse stack of J1822+02. The 2D spectrogram shows the Fourier
+transform of data along the axis of single pulses. The evidence of a single drifting frequency across the phase bins is evident
+from the spectrogram. The bottom panel shows the 2D spectrogram scrunched along the phase axis and the right-hand plot
+shows the same scrunched along the frequency axis.
+methods. We emphasize that the values of these have to
+be taken with a high degree of caution, given the relative
+sample size under study and the presence of outliers. In
+particular we find that PSR J2310+6706 turns out to be
+a strong outlier, especially in the τc and Bsurf space and
+this significantly affects the results (see Table 5), further
+illustrating the limitations of a small sample size.
+Previously, using a sample size (23) comparable to
+ours, Wang et al. (2007) qualitatively found that NF is
+related to age with older population experiencing larger
+nulling fractions. Ritchings (1976) found a positive cor-
+relation both with the pulsar period and age in a sample
+(32) slightly larger than the one in this study. However,
+as mentioned above those and most other nulling esti-
+mates in the literature are derived using some variant
+of Ritchings’ algorithm. Computing the Spearman coef-
+ficient for all of the archival sources we cannot confirm
+either correlation and suggest caution in interpreting re-
+sults using Ritchings’ algorithm.
+However, we also note that the source of this dispar-
+ity does not seem to be straightforward: For a sam-
+ple of pulsars with a given SNR, the energy per sin-
+gle pulse will be lower for pulsars with shorter peri-
+ods, which means that the NF estimates for the short-
+period pulsars should experience larger biases and have
+higher nulling fractions measured with the Richtings’
+method.
+Under the (overly simplistic) assumption of
+a uniform distribution of luminosity with period (cf.
+Faucher-Gigu`ere & Kaspi 2006; Bates et al. 2014), the
+correlation of inferred nulling fraction with period will
+then be negative which is contrary to the previous stud-
+ies.
+This suggests that the source of this bias is not
+simple and needs careful understanding of the under-
+Table 5. Spearman rank correlation coefficients for our sam-
+ple data set and archival data set.
+Parameter
+MM
+Ritchings
+Catalog
+P
+0.356
+0.008
+0.311
+0.314
+−0.064
+· · ·
+| ˙P|
+0.274
+0.035
+−0.013
+0.457
+0.057
+· · ·
+τc
+−0.353
+−0.088
+0.149
+−0.557
+−0.207
+· · ·
+Bsurf
+0.291
+−0.006
+0.110
+0.450
+0.071
+· · ·
+Note—Not all the pulsars in the sample have
+˙P measurements. Hence the sample size used
+for period is larger. The two rows for each pa-
+rameter correspond to the rank coefficients
+including and excluding PSR J2310+6706
+(see Figure 10).
+lying distribution of NF with pulsar properties and a
+larger sample of pulsars with more robust and unbiased
+NF estimates.
+4.3. Is Nulling Periodic?
+As shown in Section 3.2, we find that nulling appears
+periodic/quasi-periodic in a subset of pulsars, with their
+periods noted in Table 4. Herfindal & Rankin (2007,
+2009) also find evidence for such signatures and at-
+tributd this to the line of sight passing through a struc-
+tured rotating carousel. In addition we also find that
+
+14
+Anumarlapudi et al.
+0.36
+0.41
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+Pulse phase
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Single pulses
+(a) AO data single pulse stack
+0.556
+0.56
+0.565
+0.57
+0.574
+Pulse phase
+0.25
+0.50
+0.75
+1.00
+Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Frequency (in units of (1/P))
+0.0
+0.5
+1.0
+Power
+(b) LRFS (AO data)
+0.36
+0.41
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+Pulse phase
+0
+25
+50
+75
+100
+125
+150
+175
+Single pulses
+(a) GBT data single pulse stack
+0.487
+0.492
+0.496
+0.5
+0.505
+Pulse phase
+0.4
+0.6
+0.8
+1.0
+Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Frequency (in units of (1/P))
+0.25
+0.50
+0.75
+1.00
+Power
+(b) LRFS (GBT data)
+Figure 7. Sub-pulse drifting in PSR J1829+25: The left panels shows the stack of single pulses, in the data taken at AO and
+GBT, which shows the signature of drifting phenomenon. The right panels shows the LRFS (see §3.3) of the single pulse stacks.
+Data from AO (top right) shows a strong feature with a periodicity ∼ 3 pulse periods. Data from GBT (bottom right) shows a
+quasi-periodic (broad) peak consistent with the period from AO data.
+in PSR J0414+31, which was observed at two differ-
+ent frequencies with different instruments, this period is
+the same. It should be noted that the frequency reso-
+lution here is ∼ 0.004 pulse period−1 (from the stacks of
+256 pulses) and so we will be insensitive to any changes
+that are finer than this. Although significant correla-
+tions can not be drawn from these periodicities given
+our sample size and the number of pulsars that show
+periodic nulling, the occurrence of such a phenomenon
+in modest set of pulsars in our sample suggests that this
+might not be uncommon and should be searched for in
+future data.
+5. CONCLUSIONS
+In this study, we have extended the Gaussian mixture
+model of Kaplan et al. (2018) to study nulling behav-
+ior in 22 pulsars, spanning a wider range of properties
+than in the initial paper but still not selected indepen-
+dent of nulling behavior. We find that all pulsars can
+be well-represented by mixture model, but we find that
+a single Gaussian is not sufficient to describe the emis-
+
+Pulsar Nulling with Mixture Models
+15
+0.2
+1.0
+1.3
+2.6
+6.5
+Emission component SNR
+0.0
+0.2
+0.4
+0.6
+0.8
+NF
+Uncorrected NFr
+Corrected NFr
+NF
+Figure 8. Comparison of NF estimates from Ritchings’ al-
+gorithm and mixture model as a function of pulsar emission
+component (significance; in units of σOFF). The blue error
+bars show the estimates from Ritchings’ algorithm while the
+orange error bars are from mixture model.
+The green er-
+ror bars are derived by estimating the systematic bias from
+the Ritchings’ method and clearly depict the bias in the cases
+where the emission component is weak compared to the back-
+ground.
+10−1
+100
+Period (s)
+10−18
+10−17
+10−16
+10−15
+10−14
+10−13
+10−12
+10−11
+Period derivative (s/s)
+109 yr
+107 yr
+105 yr
+1013 G
+1012 G
+1011 G
+ATNF catalog
+Archival NF
+This work
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Nulling Fraction
+Figure 9. Period-period derivative (P − ˙P) diagram high-
+lighting nulling pulsars.
+Shown in grey circles are all the
+pulsar from the ATNF catalog (Manchester et al. 2005), in
+colored circles are the archival nulling pulsars from Konar
+& Deka (2019) and in diamonds are the pulsars from this
+study. The contours represent lines of constant character-
+istic age τc and dipolar surface magnetic field (Bsurf). The
+color bar shows the nulling fraction which ranges from 0 to 1.
+No clear discernible trend of NF with any of P/ ˙P/Bsurf/τc
+is visible.
+sion component in some pulsars7.
+Similar to Kaplan
+et al. (2018), we find that previous methods used to
+estimate NF can suffer significant biases when the pul-
+sar emission is weak compared to the background noise.
+Such biases may lead to misinterpreting weak pulsars
+as nulling pulsars. We also show that these biases may
+lead to spurious correlations between the NF and pulsar
+properties in population-wide studies.
+Drawing on the more robust statistics that we calcu-
+late, we find that nulling can appear periodic, with three
+pulsars in our sample showing this behavior. Two pul-
+sars in our sample, PSR J1822+02 and PSR J1829+25,
+shows clear signs of sub-pulse drifting, and they have an
+inferred nulling fraction consistent with 0. In contrast,
+studies like Gajjar et al. (2014a); Davies et al. (1984)
+find sub-pulse drifting in pulsars that exhibit moderate
+nulling, indicating that sub-pulse drifting and nulling
+might be two independent manifestations of sub-pulse
+intensity variations. In all cases we look forward to us-
+ing larger, less-biased samples to more robustly explore
+the nulling population and seeing if it is related to other
+phenomenology.
+Two pulsars in our sample, PSR J0414+31 and PSR
+J1829+25, were observed at two different frequencies
+(430 MHz and 820 MHz), albeit not simultaneously.
+PSR J1829+25 has nulling estimates that agree at both
+frequencies, consistent with 0, but we find that PSR
+J0414+31, has NF estimates in tension at the ∼ 2σ
+level, with the NF higher at lower frequencies.
+Al-
+though it is hard to draw definite conclusions from these
+two pulsars since the observations are not simultane-
+ous, it emphasizes the need for simultaneous observa-
+tions at multiple frequencies (or across a larger band-
+width). Observing at 4 different frequencies (325, 610,
+1400, 4850 MHz), Gajjar et al. (2014a) find coherent
+nulling in three different pulsars whereas Bhat et al.
+(2007) find the evidence for null excess at lower frequen-
+cies in PSR B1133+16 further emphasizing the need for
+multi-frequency observations in a larger sample to find
+whether nulling is universally broadband.
+One of the pulsars in our sample (PSR J2310+6706)
+has a two-component profile with a faint leading peak in
+addition to the primary peak. The very low SNR of the
+leading component limits our ability to find a stringent
+estimate of the NF independent of the primary com-
+ponent, but we find that the NF values obtained from
+each component is consistent. Analyzing nulling charac-
+teristics in pulsars with multi-component pulse profiles
+7 PSR J0054+6946 is better described by 2 different emission com-
+ponents, one at lower amplitude and the other at higher ampli-
+tude, as seen in Figure 11.
+
+16
+Anumarlapudi et al.
+0
+2
+4
+6
+Period (s)
+0.0
+0.2
+0.4
+0.6
+0.8
+NF
+10−16
+10−15
+10−14
+Period derivative (s/s)
+107
+108
+109
+Characteristic Age (yr)
+J2310+6706
+1012
+1013
+Surface Magnetic field (G)
+J2310+6706
+Figure 10. Scatter plot showing the NF of the pulsars in this study vs their properties. It can be seen that the pulsars appear
+scattered in the P/ ˙P space. However, with the exclusion of PSR J2310+6706 which appears as an outlier in the τc/Bsurf space,
+a rough trend can be seen that of NF decreasing with the age τc and increasing with the surface magnetic field Bsurf. The
+correlation coefficients are given in Table 5.
+with a robust method like mixture modeling can provide
+insights into the simultaneous nulling in the different re-
+gions of the pulsar’s magnetosphere.
+So far we have only analyzed normal, non-recycled
+pulsars. Current sensitivity limitations restrict the sam-
+ple of nulling pulsars to normal pulsars (as is evident
+from Figure 9), while MSPs are largely unexplored. Ini-
+tial single pulse studies done by Rajwade et al. (2014)
+do not find any compelling evidence for nulling in MSPs.
+Using the mixture model technique, which does not suf-
+fer from the same biases at low signal-to-noise, for MSPs,
+together with newer higher-sensitivity facilities may help
+explore whether the nulling phenomenon affects all pul-
+sars, or is limited to a sub-population.
+We thank an anonymous referee for helpful suggestions
+that clarified this work. AA, JS, and DK receive sup-
+port from National Science Foundation (NSF) Physics
+Frontiers Center award numbers 1430284 and 2020265.
+AA thanks Alex McEwen for helpful discussions dur-
+ing the data reduction stage.
+The Arecibo Observa-
+tory is a facility of the NSF operated under cooperative
+agreement (#AST-1744119) by the University of Cen-
+tral Florida (UCF) in alliance with Universidad Ana G.
+M´endez (UAGM) and Yang Enterprises (YEI), Inc. The
+Green Bank Observatory is a facility of the NSF oper-
+ated under cooperative agreement by Associated Uni-
+versities, Inc.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+Facilities: GBT (GUPPI), Arecibo (PUPPI)
+Software:
+PINT (Luo et al. 2019), PSRCHIVE (van
+Straten et al. 2011), dspsr (van Straten & Bailes 2011),
+NumPy (Harris et al. 2020), Matplotlib (Hunter 2007),
+AstroPy (Astropy Collaboration et al. 2013, 2018),
+emcee (Foreman-Mackey et al. 2013)
+APPENDIX
+A. NULLING RESULTS FOR ALL PULSARS
+We show pulse profiles, MCMC corner plot results,
+and nulling histograms for all of the pulsars in our sam-
+ple.
+
+Pulsar Nulling with Mixture Models
+17
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+Pulsar Nulling with Mixture Models
+19
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.001
+0.25
+0.50
+0.75
+µ1
+µ1=0.498
+1.6
+1.8
+2.0
+µ2
+µ2=1.821
+0.425
+0.450
+0.475
+σ0
+σ0=0.442
+0.45
+0.60
+σ1
+σ1=0.509
+0.80
+0.88
+0.96
+σ2
+σ2=0.91
+0.15
+0.30
+c0 (NF)
+NF=0.273
+−0.02
+0.00
+0.02
+µ0
+0.15
+0.30
+0.45
+c1
+0.25
+0.50
+0.75
+µ1
+1.6
+1.8
+2.0
+µ2
+0.425
+0.450
+0.475
+σ0
+0.45
+0.60
+σ1
+0.80
+0.88
+0.96
+σ2
+0.15
+0.30
+c0 (NF)
+0.15
+0.30
+0.45
+c1
+c1=0.257
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission comps.
+MM null fit
+Figure 11. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0054+6946. In this case the best fit model is a 3-component Gaussian mixture
+
+20
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.016
+1.08
+1.14
+1.20
+µ1
+µ1=1.141
+0.24
+0.28
+σ0
+σ0=0.268
+0.64
+0.68
+0.72
+σ1
+σ1=0.667
+0.000
+0.025
+µ0
+0.08
+0.12
+0.16
+NF
+1.08
+1.14
+1.20
+µ1
+0.24
+0.28
+σ0
+0.64
+0.68
+0.72
+σ1
+0.08
+0.12
+0.16
+NF
+NF=0.108
+−2
+−1
+0
+1
+2
+3
+4
+Raw Intensity
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 12. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0111+6624. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+21
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.001
+2.10
+2.16
+2.22
+µ1
+µ1=2.183
+0.345
+0.360
+σ0
+σ0=0.355
+0.80
+0.85
+0.90
+σ1
+σ1=0.853
+−0.015
+0.000
+0.015
+µ0
+0.52
+0.54
+0.56
+NF
+2.10
+2.16
+2.22
+µ1
+0.345
+0.360
+σ0
+0.80
+0.85
+0.90
+σ1
+0.52
+0.54
+0.56
+NF
+NF=0.54
+−1
+0
+1
+2
+3
+4
+5
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 13. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0325+6744. In this case the best fit model is a 2-component Gaussian mixture
+
+22
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.0
+1.000
+1.025
+1.050
+µ1
+µ1=1.014
+0.66
+0.68
+0.70
+σ0
+σ0=0.672
+0.960
+0.975
+σ1
+σ1=0.972
+−0.015
+0.000
+0.015
+µ0
+0.02
+0.04
+NF
+1.000
+1.025
+1.050
+µ1
+0.66
+0.68
+0.70
+σ0
+0.960
+0.975
+σ1
+0.02
+0.04
+NF
+NF=0.018
+−4
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 14. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0355+28. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+23
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.042
+1.35
+1.50
+µ1
+µ1=1.351
+0.60
+0.65
+σ0
+σ0=0.618
+1.10
+1.15
+1.20
+σ1
+σ1=1.136
+0.00
+0.04
+0.08
+µ0
+0.24
+0.32
+NF
+1.35
+1.50
+µ1
+0.60
+0.65
+σ0
+1.10
+1.15
+1.20
+σ1
+0.24
+0.32
+NF
+NF=0.275
+−4
+−2
+0
+2
+4
+6
+8
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 15. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0414+31 (GBT). In this case the best fit model is a 2-component Gaussian mixture
+
+24
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.024
+1.44
+1.50
+µ1
+µ1=1.476
+0.285
+0.300
+σ0
+σ0=0.296
+0.99
+1.02
+σ1
+σ1=1.011
+0.015
+0.030
+µ0
+0.300
+0.325
+0.350
+NF
+1.44
+1.50
+µ1
+0.285
+0.300
+σ0
+0.99
+1.02
+σ1
+0.300
+0.325
+0.350
+NF
+NF=0.329
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 16. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0414+31 (arecibo). In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+25
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.015
+1.05
+1.20
+µ1
+µ1=1.09
+1.7
+1.8
+1.9
+σ0
+σ0=1.795
+1.50
+1.56
+1.62
+σ1
+σ1=1.568
+−0.08
+0.00
+µ0
+0.08
+0.16
+NF
+1.05
+1.20
+µ1
+1.7
+1.8
+1.9
+σ0
+1.50
+1.56
+1.62
+σ1
+0.08
+0.16
+NF
+NF=0.074
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Raw Intensity
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 17. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0614+83. In this case the best fit model is a 2-component Gaussian mixture
+
+26
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.003
+0.50
+0.75
+1.00
+µ1
+µ1=0.857
+0.020
+0.022
+σ0
+σ0=0.021
+0.4
+0.6
+σ1
+σ1=0.462
+0.4
+0.5
+0.6
+λ
+λ=0.47
+0.002
+0.004
+µ0
+0.60
+0.65
+0.70
+NF
+0.50
+0.75
+1.00
+µ1
+0.020
+0.022
+σ0
+0.4
+0.6
+σ1
+0.4
+0.5
+0.6
+λ
+0.60
+0.65
+0.70
+NF
+NF=0.666
+0
+100
+101
+Raw Intensity
+0
+100
+101
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 18. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J0738+6904. In this case the best fit model is a 2-component Exponential convolved Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+27
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.011
+1.05
+1.20
+µ1
+µ1=1.082
+1.3
+1.4
+σ0
+σ0=1.35
+1.30
+1.35
+1.40
+σ1
+σ1=1.389
+−0.06
+0.00
+0.06
+µ0
+0.08
+0.16
+NF
+1.05
+1.20
+µ1
+1.3
+1.4
+σ0
+1.30
+1.35
+1.40
+σ1
+0.08
+0.16
+NF
+NF=0.054
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Raw Intensity
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 19. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1529-26. In this case the best fit model is a 2-component Gaussian mixture
+
+28
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.018
+1.50
+1.75
+2.00
+µ1
+µ1=1.786
+0.55
+0.60
+0.65
+σ0
+σ0=0.607
+1.20
+1.35
+σ1
+σ1=1.33
+−0.05
+0.00
+µ0
+0.40
+0.48
+NF
+1.50
+1.75
+2.00
+µ1
+0.55
+0.60
+0.65
+σ0
+1.20
+1.35
+σ1
+0.40
+0.48
+NF
+NF=0.429
+−4
+−2
+0
+2
+4
+6
+8
+10
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 20. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1536-30. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+29
+0
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.153
+4.5
+6.0
+7.5
+µ1
+µ1=5.61
+2.55
+2.70
+σ0
+σ0=2.655
+4.8
+5.6
+σ1
+σ1=4.952
+0.00
+0.15
+0.30
+µ0
+0.78
+0.84
+0.90
+NF
+4.5
+6.0
+7.5
+µ1
+2.55
+2.70
+σ0
+4.8
+5.6
+σ1
+0.78
+0.84
+0.90
+NF
+NF=0.84
+−10
+0
+10
+20
+30
+Raw Intensity
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 21. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1629+33. In this case the best fit model is a 2-component Gaussian mixture
+
+30
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.0
+1.00
+1.04
+1.08
+µ1
+µ1=1.028
+0.60
+0.65
+0.70
+σ0
+σ0=0.646
+0.850
+0.875
+σ1
+σ1=0.861
+−0.04
+0.00
+0.04
+µ0
+0.015
+0.030
+NF
+1.00
+1.04
+1.08
+µ1
+0.60
+0.65
+0.70
+σ0
+0.850
+0.875
+σ1
+0.015
+0.030
+NF
+NF=0.004
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 22. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1821+4147. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+31
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.002
+0.99
+1.02
+1.05
+µ1
+µ1=1.003
+0.30
+0.33
+0.36
+σ0
+σ0=0.334
+0.58
+0.60
+0.62
+σ1
+σ1=0.596
+−0.02
+0.00
+0.02
+µ0
+0.02
+0.04
+NF
+0.99
+1.02
+1.05
+µ1
+0.30
+0.33
+0.36
+σ0
+0.58
+0.60
+0.62
+σ1
+0.02
+0.04
+NF
+NF=0.007
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 23. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1822+02. In this case the best fit model is a 2-component Gaussian mixture
+
+32
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.014
+1.00
+1.05
+1.10
+µ1
+µ1=1.04
+0.36
+0.42
+σ0
+σ0=0.389
+0.57
+0.60
+0.63
+σ1
+σ1=0.584
+−0.04
+0.00
+0.04
+µ0
+0.02
+0.04
+NF
+1.00
+1.05
+1.10
+µ1
+0.36
+0.42
+σ0
+0.57
+0.60
+0.63
+σ1
+0.02
+0.04
+NF
+NF=0.004
+−1
+0
+1
+2
+3
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 24. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1829+25 (GBT). In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+33
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+50
+100
+150
+200
+250
+300
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.014
+1.00
+1.05
+1.10
+µ1
+µ1=1.04
+0.36
+0.42
+σ0
+σ0=0.389
+0.57
+0.60
+0.63
+σ1
+σ1=0.584
+−0.04
+0.00
+0.04
+µ0
+0.02
+0.04
+NF
+1.00
+1.05
+1.10
+µ1
+0.36
+0.42
+σ0
+0.57
+0.60
+0.63
+σ1
+0.02
+0.04
+NF
+NF=0.004
+−1
+0
+1
+2
+3
+Raw Intensity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 25. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1829+25 (AO). In this case the best fit model is a 2-component Gaussian mixture
+
+34
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.015
+1.0
+1.2
+1.4
+µ1
+µ1=1.17
+1.4
+1.6
+1.8
+σ0
+σ0=1.646
+1.3
+1.4
+1.5
+σ1
+σ1=1.384
+−0.15
+0.00
+0.15
+µ0
+0.15
+0.30
+NF
+1.0
+1.2
+1.4
+µ1
+1.4
+1.6
+1.8
+σ0
+1.3
+1.4
+1.5
+σ1
+0.15
+0.30
+NF
+NF=0.147
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Raw Intensity
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 26. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1901-04. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+35
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.002
+0.990
+1.005
+µ1
+µ1=0.997
+0.475
+0.500
+0.525
+σ0
+σ0=0.503
+0.59
+0.60
+σ1
+σ1=0.593
+−0.015
+0.000
+0.015
+µ0
+0.003
+0.006
+NF
+0.990
+1.005
+µ1
+0.475
+0.500
+0.525
+σ0
+0.59
+0.60
+σ1
+0.003
+0.006
+NF
+NF=0.001
+−2
+−1
+0
+1
+2
+3
+4
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 27. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1904+33. In this case the best fit model is a 2-component Gaussian mixture
+
+36
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0=-0.015
+1.8
+2.1
+µ1=1.836
+1.4
+1.5
+1.6
+σ0=1.467
+2.40
+2.55
+σ1=2.517
+−0.08
+0.00
+0.08
+0.40
+0.48
+0.56
+1.8
+2.1
+1.4
+1.5
+1.6
+2.40
+2.55
+0.40
+0.48
+0.56
+NF=0.476
+−10
+−5
+0
+5
+10
+15
+Raw Intensity
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 28. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1928+28. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+37
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.006
+1.04
+1.12
+µ1
+µ1=1.027
+0.78
+0.84
+0.90
+σ0
+σ0=0.831
+0.92
+0.96
+1.00
+σ1
+σ1=0.979
+−0.05
+0.00
+0.05
+µ0
+0.04
+0.08
+NF
+1.04
+1.12
+µ1
+0.78
+0.84
+0.90
+σ0
+0.92
+0.96
+1.00
+σ1
+0.04
+0.08
+NF
+NF=0.013
+−4
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 29. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J1941+02. In this case the best fit model is a 2-component Gaussian mixture
+
+38
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.001
+1.20
+1.25
+1.30
+µ1
+µ1=1.246
+0.120
+0.135
+0.150
+σ0
+σ0=0.137
+0.64
+0.68
+0.72
+σ1
+σ1=0.686
+0.000
+0.015
+µ0
+0.18
+0.21
+NF
+1.20
+1.25
+1.30
+µ1
+0.120
+0.135
+0.150
+σ0
+0.64
+0.68
+0.72
+σ1
+0.18
+0.21
+NF
+NF=0.197
+−1
+0
+1
+2
+3
+4
+5
+Raw Intensity
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 30. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J2000+29. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+39
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.016
+1.2
+1.3
+1.4
+µ1
+µ1=1.328
+0.68
+0.72
+σ0
+σ0=0.697
+1.05
+1.10
+1.15
+σ1
+σ1=1.119
+0.00
+0.04
+µ0
+0.20
+0.25
+0.30
+NF
+1.2
+1.3
+1.4
+µ1
+0.68
+0.72
+σ0
+1.05
+1.10
+1.15
+σ1
+0.20
+0.25
+0.30
+NF
+NF=0.254
+−4
+−2
+0
+2
+4
+6
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 31. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J2040-21. In this case the best fit model is a 2-component Gaussian mixture
+
+40
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=-0.001
+1.16
+1.20
+µ1
+µ1=1.188
+0.180
+0.195
+0.210
+σ0
+σ0=0.198
+0.475
+0.500
+0.525
+σ1
+σ1=0.499
+−0.015
+0.000
+0.015
+µ0
+0.14
+0.16
+NF
+1.16
+1.20
+µ1
+0.180
+0.195
+0.210
+σ0
+0.475
+0.500
+0.525
+σ1
+0.14
+0.16
+NF
+NF=0.152
+0
+1
+2
+3
+4
+Raw Intensity
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 32. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J2044+28. In this case the best fit model is a 2-component Gaussian mixture
+
+Pulsar Nulling with Mixture Models
+41
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+50
+100
+150
+200
+250
+300
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.02
+1.6
+2.4
+µ1
+µ1=2.032
+0.75
+1.00
+1.25
+σ0
+σ0=0.998
+0.8
+1.2
+1.6
+σ1
+σ1=1.026
+−0.25
+0.00
+0.25
+µ0
+0.25
+0.50
+NF
+1.6
+2.4
+µ1
+0.75
+1.00
+1.25
+σ0
+0.8
+1.2
+1.6
+σ1
+0.25
+0.50
+NF
+NF=0.485
+−2
+0
+2
+4
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 33. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J2131-31. In this case the best fit model is a 2-component Gaussian mixture
+
+42
+Anumarlapudi et al.
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pulse phase
+0
+100
+200
+300
+400
+Single pulses
+ON
+OFF
+Intensity
+µ0
+µ0=0.03
+0.0
+0.3
+0.6
+µ1
+µ1=0.371
+0.70
+0.75
+0.80
+σ0
+σ0=0.727
+0.2
+0.4
+0.6
+σ1
+σ1=0.349
+0.54
+0.60
+0.66
+λ
+λ=0.589
+0.00
+0.05
+0.10
+µ0
+0.30
+0.45
+0.60
+NF
+0.0
+0.3
+0.6
+µ1
+0.70
+0.75
+0.80
+σ0
+0.2
+0.4
+0.6
+σ1
+0.54
+0.60
+0.66
+λ
+0.30
+0.45
+0.60
+NF
+NF=0.533
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Raw Intensity
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Probability Density
+MM On fit
+ON/OFF histograms
+MM Off fit
+MM emission fit
+MM null fit
+Figure 34. Single pulse stack (upper left), MCMC corner plot (bottom), and pulse intensity histogram (upper right) for PSR
+J2310+6706. In this case the best fit model is a 2-component Exponential convolved Gaussian mixture
+

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+Fairness-aware Vision Transformer via Debiased Self-Attention
+Yao Qiang
+Chengyin Li
+Prashant Khanduri
+Dongxiao Zhu
+Department of Computer Science, Wayne State University
+{yao,cyli,khanduri.prashant,dzhu}@wayne.edu
+Abstract
+Vision Transformer (ViT) has recently gained significant
+interest in solving computer vision (CV) problems due to
+its capability of extracting informative features and mod-
+eling long-range dependencies through the self-attention
+mechanism. To fully realize the advantages of ViT in real-
+world applications, recent works have explored the trust-
+worthiness of ViT, including its robustness and explainabil-
+ity. However, another desiderata, fairness has not yet been
+adequately addressed in the literature. We establish that
+the existing fairness-aware algorithms (primarily designed
+for CNNs) do not perform well on ViT. This necessitates
+the need for developing our novel framework via Debiased
+Self-Attention (DSA). DSA is a fairness-through-blindness
+approach that enforces ViT to eliminate spurious features
+correlated with the sensitive attributes for bias mitigation.
+Notably, adversarial examples are leveraged to locate and
+mask the spurious features in the input image patches. In
+addition, DSA utilizes an attention weights alignment reg-
+ularizer in the training objective to encourage learning in-
+formative features for target prediction. Importantly, our
+DSA framework leads to improved fairness guarantees over
+prior works on multiple prediction tasks without compro-
+mising target prediction performance.
+1. Introduction
+Recently, Visual Transformer (ViT) [11, 30] has emerged
+as an architectural paradigm and a viable alternative to the
+standard Convolutional Neural Network (CNN) [19,27,42]
+for computer vision (CV) tasks. Unlike CNN, ViT is ca-
+pable of extracting global relationships via self-attention
+mechanism as well as informative features from the input
+images, leading to impressive feature representation capa-
+bilities. Consequently, ViT has demonstrated improved per-
+formance in a variety of CV tasks, including image classi-
+fication [11, 30], object detection [3, 9], semantic segmen-
+tation [55, 67], and image generation [21], to name a few.
+Due to its promising performance, it is anticipated that ViT
+will form the architectural backbone of CV algorithms in
+the near-future for real-world applications. This has led the
+(a) Original Image
+(b) Vanilla 
+(c) DSA 
+Figure 1. An illustration example. The prediction target is hair
+color and the sensitive attribute is gender. The heatmap of atten-
+tion weights show that Vanilla ViT (b) uses gender-sensitive fea-
+tures, e.g., ‘red lip’ and ‘eye shadow’, whereas our fairness-aware
+ViT DSA (c) uses informative features, e.g., ‘hair’, for predictions.
+researchers to analyze the trustworthiness of ViT for solving
+CV tasks.
+Studying the robustness of ViT has recently attracted a
+growing interest [2, 13, 38, 50, 68]. It is critical to improve
+ViT’s robustness in order to deploy them safely in the real-
+world. On the other hand, investigating ViT’s vulnerability
+to attacks can give us a deeper understanding of its underly-
+ing working mechanism. In the past, researchers have dis-
+sected the self-attention mechanism [1,47] and the gradient-
+based attribution [4] to offer a faithful explanation of the
+inner workings of ViT or Transformer at large.
+Besides robustness and explainability, fairness also
+stands as a core trustworthy desiderata for both industry
+[20] and academia [7].
+Several studies show that many
+deep-learning-based CV models simply make predictions
+by exploiting spurious correlations with the input features
+[23, 58]. These spurious features are statistically informa-
+tive features that work for a majority of training examples
+but do not capture the underlying relationship between the
+input features and the target labels.
+For illustration, let
+us consider the example in Figure 1 (taken from CelebA
+dataset). Since the target label, hair color, is spuriously cor-
+related with the gender-related sensitive attributes, e.g., ‘eye
+shadow’ or ‘red lips’ in Figure 1(b), a vanilla ViT model
+would simply learn these spurious features as a shortcut to
+predict the hair color whereas our fairness-aware ViT model
+learns the informative features, e.g., ‘hair’ in Figure 1(c), to
+make prediction.
+arXiv:2301.13803v1  [cs.CV]  31 Jan 2023
+
+Such spurious correlations can cause ViT to behave in
+a biased manner, e.g., a lower performance on some pop-
+ulation subgroups [54, 61]. Although an array of debias-
+ing algorithms have been proposed for image classification
+tasks [23, 36, 46, 60, 65, 66], most are designed for learn-
+ing with the CNN models. Whether these algorithms are
+compatible or even transferable to the ViT architecture is
+unclear. Regardless of the neural network architecture, lim-
+iting the spurious correlation between the input features and
+the target labels for bias mitigation is still a challenging
+problem. The difficulty arises from the fact that automat-
+ically locating the spurious features in the input images is
+computationally intractable. For example, one simple solu-
+tion is to have domain experts and/or crowd workers curate
+the entire training set, which neither works well with un-
+known bias [29] nor is scalable to large-scale datasets [39].
+Moreover, even if one can identify the spurious features,
+the major challenge is how to make the classifier blind to
+such features? Image in-painting [35, 59] appears to be a
+promising approach to remove the undesired features; nev-
+ertheless, significant challenges remain regarding what old
+features to cut out for debiasing and what new features to
+fill up to repair the corrupted images.
+To address the above challenges, we propose a novel
+framework for ensuring bias mitigation training of ViT via
+Debiasing Self-Attention (DSA) to decouple the target pre-
+diction from the spurious features.
+DSA takes a hierar-
+chical approach, where, in the first stage, we first localize
+the spurious features from the input imaging patches. This
+is achieved by training a bias-only model which exploits
+the spurious features to explicitly predict the sensitive at-
+tributes (e.g., gender and race). We then use adversarial
+attacks against the bias-only model to identify and perturb
+(or mask) the top patches that are responsible for the de-
+creased accuracy in predicting the sensitive attributes. No-
+tably, our approach for the fair ViTs is a novel addition to
+the growing body of work on “adversarial examples for fair-
+ness” [62,66].
+DSA relies on the intuitive hypothesis that the adversar-
+ial attacks initially designed to evaluate and understand the
+robustness of ViT can also be a viable approach for identi-
+fying and removing the spurious features towards training
+the fair ViT models. Meanwhile, whether the approaches
+that generate adversarial examples for CNN are transferable
+to ViT remains a matter of contention [17, 41, 44, 45, 53],
+the work in [13] propose Patch-Fool as one of the first ap-
+proaches to fool the self-attention mechanism by attack-
+ing image patches (as opposed to pixels) during ViT’s self-
+attention computations. In this work, we apply Patch-Fool
+to attack the bias-only model with the goal of capturing the
+most important patches for learning the sensitive attributes.
+As a result, the effect of sensitive features can be miti-
+gated with adversarial examples, which are constructed by
+directly perturbing (attacking) the sensitive patches.
+In the second stage, in addition to augmenting the
+original training set with these adversarial examples as the
+debiased training set, we also align the biased examples
+and their corresponding (unbiased) adversarial examples
+via an attention weights aligning regularizer tailor-made
+for self-attention mechanism in ViT. This leads to a novel
+training objective that encourages learning informative
+features while ensuring fairness of the trained ViT models.
+Major contributions: We summarize our major contribu-
+tions as follows: (1) We tackle the under-addressed fair-
+ness problem in ViT from a novel perspective of leveraging
+adversarial examples to eliminate spurious features while
+utilizing attention weights alignment to retain informative
+features. (2) We design a novel DSA framework for ViT
+to mitigate bias in both training set and learning algorithm
+via identifying and decorrelating the sensitive features from
+the target label. (3) DSA presents a flexible and modular
+debiasing approach that can be used either standalone or
+with other fairness-aware training algorithms to ensure ViT
+fairness. (4) Experimental results show that DSA improves
+group fairness with respect to demographic parity (DP) and
+equality of odds (EO) metrics while achieving a competitive
+or even better prediction accuracy compared to the base-
+lines. The qualitative analysis further indicates that DSA
+has reduced attention on sensitive features.
+2. Related Work
+ViT based Classification.
+The earlier exploration of ViT
+either used a hybrid architecture combining convolution and
+self-attention [3] or a pure self-attention architecture with-
+out convolution [48]. The work in [11] proposed a ViT
+that achieves impressive results on image classification us-
+ing ImageNet data set. This success has motivated a se-
+ries of subsequent works to further exploit ViT’s expressive
+power from various perspectives, such as incorporating lo-
+cality into ViT [28,30,63], and finding well-performing ViT
+using neural architecture search (NAS) [6].
+Fairness and Debiased Learning.
+The field of fairness in
+deep learning has grown significantly over the past several
+years as a result of bias in training data and algorithms [36,
+46]. The existing techniques for debiased learning can be
+roughly categorized into pre-, in-, and post-processing.
+– Pre-processing methods attempt to debias and increase
+the quality of the training set with the assumption that fair
+training sets would result in fair models [8, 25, 66]. The
+work in [66] proposed to balance the data distribution over
+different protected attributes by generating adversarial ex-
+amples to supplement the training dataset. Similarly, [25]
+generated the bias-swapped image augmentations to bal-
+ance protected attributes, which would remove spurious
+
+correlation between the target label and protected attributes.
+In [8], the authors presented fair mixup as a new data aug-
+mentation method to generate interpolated samples to find
+middle-ground representation for different sensitive groups.
+The work [46] described a novel generative data augmen-
+tation approach to create counterfactual samples that d-
+separates the sensitive attributes and the targets ensuring
+fairness and attribution-based explainability.
+– In-processing approaches aim to mitigate bias during the
+training process by directly modifying the learning algo-
+rithm and model weights with specifically designed fair-
+ness penalties/constraints or adversarial mechanism [24,36,
+40, 49, 65]. To enforce the fairness constraints, one line
+of works either disentangles the association between model
+predictions and the sensitive attributes via an auxiliary reg-
+ularization term [40] or minimize the performance differ-
+ence between protected groups with a novel objective func-
+tion [49]. However, the issue is that the trained models may
+behave differently at the inference stage even though such
+fairness constraints are satisfied during the training. An-
+other line of works [24, 36, 60, 65] enforce the model to
+generate fair outputs with adversarial training techniques
+through the min-max objective: maximizing accuracy while
+minimizing the ability of a discriminator to predict the pro-
+tected (sensitive) attribute. Nevertheless, this process can
+compromise the model performance on the main prediction
+task. Additional line of works impose either orthogonal-
+ity [51], disentanglement [32], or feature alignment [23]
+constraints on the feature representation and force the repre-
+sentation to be agnostic to the sensitive attributes. We note
+that most of these approaches are exclusively designed for
+CNN architectures, and whether these approaches are trans-
+ferable to the ViT has not yet been demonstrated.
+– Post-processing techniques directly calibrate or modify
+the classifier’s decisions to certain fairness criteria at infer-
+ence time [26, 33]. These methods require access to the
+sensitive attribute for fair inference, which may not be fea-
+sible in real-world applications due to the salient security
+and privacy concerns.
+Fairness in ViT.
+Recently, [16] explored how the spuri-
+ous correlations are manifested in ViT and performed exten-
+sive experiments to understand the role of the self-attention
+mechanism in debiased learning of ViT. Despite the new
+insights, the authors did not provide any debiasing tech-
+niques for ViT. The authors in [56] proposed a new method,
+named TADeT, for debiasing ViT that aims to discover and
+remove bias primarily from query matrix features. To our
+knowledge, this is the only published work along the line
+of fairness ViT. Nevertheless, this pioneering work TADeT
+has two weaknesses: first, it requires parameter sharing
+across the key and value weights in self-attention mecha-
+nism, which may conflict with most ViT architectures; sec-
+ond, the complex alignment strategy on the query matrix
+is not straightforward and well investigated. Thus, TADeT
+does not even outperform the compared baselines that pri-
+marily designed for CNNs.
+In contrast to the above works, this work tackles the de-
+biasing problem through a novel perspective of fairness-
+through-adversarial-attack. The proposed DSA framework
+combines the strengths of both pre- and in-processing ap-
+proaches via leveraging data augmentation (for ensuring
+fairness in the training set) and feature alignment for bias
+mitigation. The adversarial examples are used to both dis-
+entangle spurious features from informative features and to
+align attention weights, specifically, tailor-made for the self-
+attention mechanism in ViT.
+3. Preliminaries
+3.1. Overview of Vision Transformer
+Similar to the Transformer architecture [57], the ViT model
+expects the input to be a linear sequence of token/patch
+embeddings. An input image is first partitioned into non-
+overlapping fixed-size square patches with resolution p×p,
+resulting in a sequence of flattened 2D patches. For ex-
+ample, given an image of size 384 × 384 and patch size
+p = 16, the image is divided into patches of resolution
+16 × 16, resulting in 576 image patches. These patches are
+then mapped to constant-size embeddings using a trainable
+linear projection. In this example, the projection layer will
+produce 576 embedding vectors of fixed dimensions. Next,
+position embeddings are added to the patch embeddings to
+imbibe relative positional information of the patches. Fi-
+nally. the ViT model prepends a learnable embedding (class
+token) to the sequence of embedded patches following [10],
+which is used as image representation at the model’s output.
+The core architecture of ViT mainly consists of mul-
+tiple stacked encoder blocks, where each block primarily
+consists of a Multi-head Self Attention (MSA) layer and
+a Feed-Forward Network (FFN) layer.
+Within the MSA
+layer, multiple self-attention heads learn relationships be-
+tween each pair of input patches. Using three different lin-
+ear transformations, the input patch xi is first projected to
+a query qi, a key ki, and a value vi in each self-attention
+head, i here is the index of the patches. The query qi then
+computes the dot products with all the keys k, which are
+further scaled and normalized by the softmax layer to obtain
+the attention weights. After this, it outputs hi by weighted
+sum up all the values v with the obtained attention weights.
+Finally, the outputs from all heads are concatenated and
+re-projected by a linear layer into an output patch. FFN
+consists of two linear layers, which are connected by the
+GeLU activation function and process each hi ∈ Rd from
+the precedent MSA layer individually. Both MSA and FFN
+layers function as the residual connection.
+
+3.2. Fairness Metrics
+Many different notions of fairness have been proposed in
+the literature [12, 18]. In this work, we mainly focus on
+the two most widely used definitions: demographic par-
+ity [12] and equalized odds [18] as the metrics to assess
+group fairness of the model. Demographic Parity (DP) mea-
+sures whether the true positive rates across all groups (de-
+fined by a sensitive attribute s, e.g., gender) are equal, par-
+ticularly between the vulnerable minority group (s = 0)
+and others (s = 1), formally: DP = TPRs=1 − TPRs=0.
+Equalized Odds (EO) is used to understand the dispari-
+ties in both the true positive rates and the false positive
+rates in the vulnerable group compared to others: EO =
+1
+2[TPRs=1 − TPRs=0] + 1
+2[FPRs=1 − FPRs=0]. In ad-
+dition, we also use Accuracy (ACC) and Balanced Accu-
+racy (BA) [43], where BA =
+1
+4[TPRs=0 + TNRs=0 +
+TPRs=1 + TNRs=1], to evaluate the utility of the model.
+However, when a dataset is class imbalanced, BA will have
+an implicit bias against the minority class. Therefore, we
+introduce Difference of Balanced Accuracy (DBA) as a
+way to measure the difference in a model’s performance
+across groups defined by a sensitive attribute while account-
+ing for class imbalance, formally: DBA = 1
+2[TPRs=1 +
+TNRs=1] − 1
+2[TPRs=0 + TNRs=0].
+4. The Proposed Framework
+4.1. Problem Formulation
+We consider a supervised classification task with training
+samples {x, y, s} ∼ pdata, where x ∈ X is the input fea-
+ture, y ∈ Y is the target label, and s ∈ S is an annotated
+sensitive categorical attribute that we wish to protect. Some
+examples of s include gender, race, age or other attributes
+that can identify a certain protected group. We assume that
+the sensitive attributes S can only be used during training
+phase, and are not accessible during the inference (post-
+training phase). Moreover, we suppose that each input fea-
+ture x can be split into two parts, one with sensitive features
+xs that are highly relevant to the sensitive attribute s, and
+the rest xt that are relevant to the prediction of the target
+label y, i.e., we have x = (xs, xt) ∈ X.
+We develop a two-step hierarchical approach for bias
+mitigation, wherein, in the first stage, we localize and mask
+the sensitive attributes xs from the input x in order to
+disentangle xs from xt.
+This is accomplished by trans-
+forming the model prediction from p(x) = p(y|xs, xt) to
+p(x) ∝ p(x′) = p(y|x′
+t), where x′ is the sample constructed
+after masking the sensitive attributes xs from x via adver-
+sarial attacks. In the second stage, we utilize the original
+x and the augmented data x′ to train a ViT model f(·) for
+generating the prediction, as ˆy = f(x), while at the same
+time satisfying certain fairness requirements (i.e., DP, EO,
+and DBA) with respect to the sensitive attributes s.
+4.2. Bias in Training Set and ViT Model
+The tendency of neural networks (including ViT) to learn
+spurious correlations makes them particularly vulnerable
+to utilizing sensitive features to make predictions, thereby,
+propagating biases towards a particular group [15]. This
+issue is particularly salient with the current deep learning
+models that follow the data-driven learning paradigm and
+are trained with imbalanced data set where some sensitive
+features could have a high correlation with certain class la-
+bels. Our work is motivated by the empirical observation
+that the bias in learning is mainly caused by the model’s
+reliance on sensitive features for prediction. Note that the
+sensitive features xs are parts of the input features x, that
+are highly predictive of the sensitive attribute s. In Figure
+1, we visualize the attention weights from the ViT model to
+analyze the importance of different features. In this exam-
+ple, gender is the sensitive attribute that is highly correlated
+with the prediction task of hair color. The Vanilla model
+may pay more attention on the gender related features, in-
+dicating that it has associated gender with the hair color.
+This association might lead the ViT model to discriminate
+against the female group. We have thus established that, for
+the image classification task using CelebA dataset, the ViT
+model is heavily biased as it relies on the sensitive features
+for prediction. This observation naturally leads to our DSA
+framework for bias mitigation discussed next.
+4.3. Debiased Self-Attention (DSA) Framework
+The discussion in Section 4.2 demonstrates that the reliance
+of ViT on the sensitive features for prediction can lead to
+bias. Therefore, to mitigate the bias originating from the
+sensitive features, we propose to achieve fairness by miti-
+gating the influence of sensitive features on the prediction
+task. However, note that it is a challenging task to locate the
+sensitive features in the input. To address this challenge, we
+propose a hierarchical framework as discussed in Section
+4.1. Specifically, our DSA framework follows a two-step
+procedure (Figure 2):
+Step 1: Firstly, we train a bias-only model that deliberately
+maximizes the usage of sensitive features for prediction,
+followed by adversarial attack on the bias-only model to lo-
+calize and mask the sensitive attributes.
+Step 2: Second, we train a debiased model with augmented
+adversarial examples and attention weights alignment.
+4.3.1
+Training the Bias-only Model
+Recall that the input feature x = (xs, xt) ∈ X where xs are
+the sensitive features while xt are the target related features.
+The goal of Step 1 (see Section 4.2) is to learn only the sen-
+sitive features xs, during training the bias-only model. To
+achieve this, we first build a bias-only ViT model which
+maximally utilizes the sensitive features for prediction. We
+
+Debiased Self-Attention 
+Sensitive Label (s) Prediction 
+11
+15
+16
+6
+7
+Bias-Only 
+0
+2
+1
+11
+Target Label (y) Prediction 
+0
+2
+1
+15
+16
+6
+7
+Adversarial Attack 
+16
+15
+0
+2
+1
+6
+7
+11
+16
+15
+0
+2
+1
+6
+7
+11
+cls
+low
+attention 
+pos
+Attention Weights
+Alignment
+adversarial attack
+train bias-only model
+train debiased model
+attention weights
+alignment
+Figure 2. The DSA framework. The bias-only model is first trained to learn the spurious features (the green patches) for predicting sensitive
+attribute (s ∈ S) (see Section 4.3.1). Adversarial attack is then applied against the bias-only model to generate the adversarial examples,
+(x′), by perturbing the sensitive patches (the grid shadow patches) of the original inputs (x ∈ X) (see Section 4.3.2). Finally, both x
+and x′ are used to train a fairness-aware ViT with an attention weights alignment objective (see Eq. (10)) and learn the target (y)-related
+informative features (the red patches) (see Sections 4.3.3 and 4.3.4). Best viewed in color.
+denote the bias-only model by fB(x, s) = c(h(x), s),
+where h(x) is the intermediate representation of the input
+x, and c(·) maps the intermediate representation to the final
+prediction. Note that h(x) contains only m elements from
+the categories in S, e.g., m = 2 in most of our experimental
+settings. The key motivation of using the m elements for
+input representation h(x) is to force the bias-only model
+to only utilize sensitive attributes to obtain the prediction
+fB(x, s).
+Given N samples of the input, xi, and the sensitive at-
+tribute, si, pairs {xi, si}N
+i=1, the bias-only model minimizes
+the following loss.
+LB(x) = − 1
+N
+N
+�
+i=1
+si log(fB(xi, si))
++ (1 − si) log(1 − fB(xi, si)).
+(1)
+We illustrate the idea using the example in Figure 2. We
+consider the hair color classification tasks with gender bias.
+Input representation h(x) is denoted using two elements,
+indicating the sensitive attributes male and female, respec-
+tively. The bias-only model fB(x, s) mainly relies on the
+sensitive features, like ‘eye shadow’ and/or ‘red lips’, to
+predict the label as female, while at the same time pay-
+ing nearly no attention to the hair color related features like
+‘hair’ themselves.
+4.3.2
+Adversarial Attack Against the Bias-only Model
+After obtaining the bias-only model, the following proce-
+dure in Step 2 of the DSA framework localizes and masks
+the spurious (sensitive) features via adversarial attacks that
+are generated using the Patch-Fool construction proposed
+in [13].
+Specifically, Patch-Fool is designed to fool the
+self-attention mechanism in ViTs by attacking their basic
+component (i.e., a single patch) with a series of attention-
+aware optimization techniques, demonstrating that the ViTs
+are more vulnerable to adversarial attacks than the CNNs.
+However, in contrast to [13], instead of applying Patch-Fool
+as an adversarial attack method to evaluate the robustness of
+ViT, we utilize it to efficiently localize and mask the sensi-
+tive features in the inputs. To this end, we adapt the objec-
+tive function of Patch-Fool in order to attack the bias-only
+model on the sensitive labels instead of the target labels.
+Specifically, given the objective function LB(x) and a se-
+ries of input image patches X = [x1, · · · , xp, · · · , xn]T ∈
+Rn×d with its associated sensitive label s, the objective of
+the adversarial algorithm is
+arg max
+1≤p≤n,E∈Rn×dLB(X + 1 ⊙ E, s),
+(2)
+where E denotes the adversarial perturbation; 1 ∈ Rn is the
+identifying one-hot vector demonstrating whether current p-
+th patch is selected or not; ⊙ represents the penetrating face
+product [13]. Thus, the Patch-Fool needs to (1) select the
+adversarial patch p, and (2) optimize the corresponding ad-
+versarial attack, E.
+Selection of p: For encoder blocks in the ViT, we define:
+t(l)
+j
+= �
+h,i a(l,h,i)
+j
+to measure the importance of the j-th
+patch in the l-th block based on its contributions to other
+patches in the self-attention calculation, where a(l,h,i) =
+[a(l,h,i)
+1
+, · · · , a(l,h,i)
+n
+] denotes the attention distribution for
+
+the ith patch of the hth head in the lth block. The moti-
+vation behind applying Patch-Fool is to localize the most
+influence patch p according to the predicted sensitive at-
+tribute s. Here, we derive the top k (which is a tunable
+hyper-parameter) important patches from arg max t(l)
+j .
+Optimize E: Given the selected adversarial patch index p
+from the previous step, an attention-aware loss is applied for
+the lth block as: LAttn = �
+h,i a(l,h,i)
+p
+. This loss is expected
+to be maximized so that the adversarial patch p, serving as
+the target adversarial patch, can attract more attention from
+other patches for effectively fooling ViTs. The perturbation
+E is then updated based on both the final sensitive classifi-
+cation loss and a layer-wise attention-aware loss:
+L(X′, s, p) = LB(X′, s) + α
+�
+l
+LAttn(X′, p),
+(3)
+where X′ ≜ X + 1 ⊙ E and α is a weight hyper-parameter
+set to 0.5 in the experiments. Moreover, PCGrad [64] is
+adopted to avoid the gradient conflict of the two losses and
+E is updated using:
+δE = ∇EL(X′, s, p) − α
+�
+l
+βl∇ELB(X′, s),
+(4)
+where
+βl =
+�
+�
+�
+0,
+⟨∇ELB(X′, s), ∇ELAttn(X′, p)⟩ > 0
+⟨∇ELB(X′, s), ∇ELAttn(X′, p)⟩
+∥∇ELB(X′, s)∥2
+,
+otherwise.
+(5)
+Following PGD [37], we iteratively update E using an
+Adam optimizer: Et+1 = Et + η · Adam(δEt), where η
+is the step-size for each update.
+4.3.3
+Attention Weights Alignment
+After Step 1, the DSA framework generates the adversarial
+example x′, whose top k patches containing sensitive at-
+tributes are perturbed through the adversarial attack. Here,
+besides using these adversarial examples as augmentation
+during training of the debiased ViT models, we also lever-
+age them via attention weights alignment to further guide
+the model to pay more attention to the target-related fea-
+tures. This also allows more sensitive features to be dis-
+covered and ignored by self-attention mechanism in the
+ViT models as shown in Figure 2. In particular, we ap-
+ply three different feature discrepancy metrics D(·, ·), i.e.,
+Mean Squared Error (MSE), Kullback-Leibler Divergence
+(KL-Div), and Attention Transfer (AT), to evaluate the dis-
+crepancy between the attention weights Ax and Ax′ from
+the original sample x and the adversarial example x′, re-
+spectively. We define the three metrics as:
+LA = D⋆(Ax, Ax′),
+(6)
+where D⋆ is either
+DMSE(Ax, Ax′) = 1
+2
+�
+j∈I
+∥Ax
+j − Ax′
+j ∥2
+(7)
+DKL−Div(Ax∥Ax′) =
+�
+j∈I
+Ax
+j log Ax
+j
+Ax′
+j
+(8)
+DAT(Ax, Ax′) = 1
+2
+�
+j∈I
+����
+Ax
+j
+∥Ax
+j ∥2
+−
+Ax′
+j
+∥Ax′∥2
+����
+2
+,
+(9)
+where I denotes the indices of all the adversarial examples
+and the original example attention weights pairs for which
+we perform alignment. Finally, to incorporate the attention
+distributions of Ax and Ax′ in the objective, we add LA as
+a regularizer in the overall objective.
+4.3.4
+Overall Loss Function
+Putting the above Steps 1 and 2 together, the overall objec-
+tive for training the proposed debiased model is:
+L = λ1LCE(x, y) + λ2LCE(x′, y) + λ3LA,
+(10)
+where LCE denotes the standard cross entropy (CE) loss;
+λ1, λ2, and λ3 are three weighted coefficients for control-
+ling the three losses. These parameters are designed for
+controlling the fairness-utility trade-off. We provide further
+ablation study on these terms in the experiments.
+5. Experimental Settings
+5.1. Datasets
+We evaluate the DSA framework on two popular CV
+datasets, namely, Waterbirds [49] and CelebA [31]. Wa-
+terbirds dataset contains spurious correlation between the
+background features S = {water, land} and target label Y =
+{waterbird, landbird}. The spurious correlation is injected
+by pairing waterbirds with the water background and land-
+birds with the land background more frequently, as com-
+pared to other combinations. CelebA dataset, which has
+been widely used in the fairness literature, contains 200k
+celebrity face images with annotations for 40 binary at-
+tributes. We present the results on three settings follow-
+ing [16,56], each with a corresponding binary task (Y) that
+the model is trained to predict, and a binary sensitive at-
+tribute (S) over which we wish the model to be unbiased.
+The three settings described as a tuple (Y, S) are as follows:
+(Gray Hair, Gender), (Wavy Hair, Gender), and (Smiling,
+High Cheekbones). We provide more details of these set-
+tings in the Supplementary Materials.
+5.2. Implementation Details
+We train the ViT-S/16 models from scratch for each pre-
+diction task. The ViT-S/16 model consists of 196 patches
+
+ACC
+DP
+BA
+DBA
+EO
+82.4 84.8 87.2 89.6 92.0
+Vanila
+TADeT
+MMD
+MFD
+DANN
+LAFTRE
+AM
+DSA (Ours)
+0.25
+0.28
+0.3
+0.33
+0.35
+79.0
+80.0
+81.0
+82.0
+83.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.26
+0.28
+0.31
+0.33
+0.35
+(a) Y: Gray hair S: Gender
+ACC
+DP
+BA
+DBA
+EO
+68.0 71.0 74.0 77.0 80.0
+Vanila
+TADeT
+MMD
+MFD
+DANN
+LAFTRE
+AM
+DSA (Ours)
+0.23
+0.29
+0.34
+0.4
+0.45
+59.0
+63.0
+67.0
+71.0
+75.0
+1.4
+2.8
+4.2
+5.6
+7.0
+0.2
+0.25
+0.3
+0.35
+0.4
+(b) Y: Wavy hair S: Gender
+ACC
+DP
+BA
+DBA
+EO
+85.2 86.4 87.6 88.8 90.0
+Vanila
+TADeT
+MMD
+MFD
+DANN
+LAFTRE
+AM
+DSA (Ours)
+0.31
+0.34
+0.36
+0.39
+0.42
+72.4
+74.8
+77.2
+79.6
+82.0
+0.0
+0.01
+0.01
+0.02
+0.02
+0.32
+0.35
+0.37
+0.4
+0.42
+(c) Y: Smiling S: High Cheeckbones
+Figure 3. Fairness and accuracy evaluation for all methods over different combinations of target (y) and sensitive (s) on CelebA dataset. For
+DSA, we use LA = DAT . The test accuracies of the bias-only model used in AM and DSA for predicting gender and high cheekbones are
+82.62% and 80.71%, respectively. The success rates of adversarial attacks are reported in Supplementary Material. The ↙ signs indicate
+the lower value of the corresponding metric is better, while ↗ denotes the higher value is better. Best viewed in color.
+(each representing a 16x16 sub-image), 1 class token patch,
+12 transformer encoder layers, and 8 attention heads. We
+flatten and project each patch into a 64-dimensional vec-
+tor and add positional embeddings. The embedded patches
+are fed into the ViT encoder. After the ViT encoder pro-
+cesses the patch embeddings, the class token patch is fed
+into 2 fully-connected layers (with hidden state size as 256)
+and a sigmoid layer to produce a single normalized output
+score (since we deal with binary classification). We train the
+ViT models using momentum Stochastic Gradient Descent
+(SGD) with a momentum parameter of 0.9 and an initial
+learning rate of 3e-2 for 20 epochs. We use a fixed batch
+size of 32, gradient clipping at global norm 1, and a cosine
+decay learning rate schedule with a linear warmup follow-
+ing [16]. We select the model with the best accuracy on the
+validation sets.
+5.3. Baselines
+We select the following debiasing algorithms as baselines
+for performance evaluation, for which we select the best
+model that yields the highest validation performance. To
+our knowledge, besides the proposed DSA and AM as a
+home run method, TADeT is the only third-party fairness-
+aware algorithm tailor-made for ViT while all the others are
+designed for CNN. We consider the following baselines:
+Vanilla [11]: The ViT models are only trained with CE
+loss for target prediction. Attention Masking (AM): The
+self-attention mechanism is critical in ViT as it provides
+important weights for extracting visual features. We pro-
+pose the AM method as a home run that directly masks
+the top-k patches with highest attention scores for the bias-
+only model. Mitigating Bias in ViT via Target Alignment
+(TADeT) [56] uses a targeted alignment strategy for debi-
+asing ViT that aims to identify and remove bias primarily
+from query matrix features. Maximum Mean Discrepancy
+(MMD) [34] calculates the mean of penultimate layer fea-
+ture activation values for each sensitive attribute setting and
+then minimizes their L2 distance. MMD-based Fair Dis-
+tillation (MFD) [23] adds a MMD-based regularizer that
+utilizes the group-indistinguishable predictive features from
+the teacher model while discouraging the student model
+from discriminating against any protected group. Domain
+Adversarial Neural Network (DANN) [14] employs a sen-
+sitive attribute adversary learned on top of the penultimate
+layer activation. The adversarial head consists of two linear
+layers in the same dimension as the class token, followed by
+a sigmoid function. Learning Adversarially Fair and Trans-
+ferable Representation (LAFTR) [36] trains a model with
+a modified adversarial objective that attempts to meet the
+EO fairness criterion. This objective is implemented by
+minimizing the average absolute difference on each task.
+6. Main Results and Discussion
+In this Section, we report the results of fairness and accu-
+racy evaluations, the ablation study, and the effects of model
+size and patch size.
+In Supplementary Materials, many
+more experimental results are reported, including the im-
+pact of several tunable hyper-parameters, results with dif-
+ferent D⋆ in the regularizer LA, and some qualitative eval-
+uations.
+6.1. Fairness and Accuracy Evaluations
+We report the fairness and accuracy performance on the
+three aforementioned settings (see Section 5.1) on CelebA
+dataset in Figure 3. We make the following observations.
+First, DSA outperforms all the baselines, demonstrated with
+the largest area (enclosed by the red lines) in the radar
+charts, significantly improving the ViT fairness with lower
+EO, DP, and DBA while maintaining higher accuracy in
+terms of BA and ACC. Second, several baseline methods
+
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+EO
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+DP
+Vanilla(62.36)
+TADeT(69.05)
+MMD(67.81)
+MFD(67.36)
+DANN(60.04)
+LAFTRE(64.80)
+AM(61.70)
+DSA(69.58)
+Figure 4. Fairness and accuracy evaluation on Waterbirds dataset.
+The ACCs are shown in the legends. All tunable hyper- parameters
+and other settings are same as in Figure 3.
+(e.g., MMD, MFD, and DANN) that have shown strong per-
+formance with CNN models, do not even outperform the
+vanilla model on some fairness metrics (e.g., EO), partic-
+ularly under the (Wavy Hair, Gender) setting. This may
+happen because ViT primarily learns global image fea-
+tures by modeling long-range dependencies using the self-
+attention mechanism, which is fundamentally different form
+convolution-based local feature leaning with CNN. As such,
+these baseline methods (designed for the CNNs) are not
+transferable for bias mitigation with the ViT models. Third,
+we note the home run method AM is also designed by blind-
+ing the sensitive attributes in the input based on only the
+attention weights of the bias-only model. However, sev-
+eral works [1, 22, 52] have questioned whether highly at-
+tentive inputs would significantly impact the model outputs.
+Since self-attention mechanism involves the computation of
+queries, keys, and values, reducing it only to the derived
+attention weights (inner products of queries and keys) can
+be insufficient to capture the importance of the features.
+Hence, the home run AM method fails to achieve a com-
+parable performance with the proposed DSA method.
+Similarly, we observe the same patterns on the results of
+Waterbirds dataset as shown in Figure 4. DSA outperforms
+all other baselines in terms of fairness evaluations, i.e., DP
+and EO, as well as accuracy performance.
+6.2. Ablating DSA
+The training objective of DSA contains three essential com-
+ponents for bias mitigation. We conduct ablation study us-
+ing the (Gray Hair, Gender) setting to analyze their indi-
+vidual contributions and report the results in Table 1 (the
+other two settings are reported in the Supplementary Ma-
+terials). We summarize our major findings. First, all of
+the components contribute towards the improved fairness
+performance across all three fairness metrics (i.e., EO, DP
+and DBA). Second, both the target (task) related CE losses
+in Eq.(10) are critical in preventing DSA from compro-
+mising the prediction performance (otherwise, the accu-
+racy drops from 90.95 to 88.32 and 88.54, respectively).
+Third, the training objective LA in Eq.(10) contributes the
+most to the higher fairness measures, as is clearly shown
+by: EO (0.2934→0.2558), DP (0.2865→0.2337), and DBA
+(0.0206→0.0031).
+Table 1. Ablation study of the three training objectives. Best re-
+sults are bold faced. ‘w/o’ represents without.
+Models
+Y : Gray Hair S: Gender
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+L(all)
+0.2558
+0.2337
+0.0031
+82.92
+90.95
+w/o LCE(x, y)
+0.2754
+0.2541
+0.0175
+81.21
+88.32
+w/o LCE(x′, y)
+0.2641
+0.2503
+0.0129
+80.65
+88.54
+w/o LA
+0.2934
+0.2865
+0.0206
+81.54
+89.91
+6.3. Effect of ViT Model Size and Patch Size
+We examine the effect of ViT model size and patch size
+on DSA in Table 2. The ViT-B model is much larger than
+the ViT-S model, which has 12 self-attention heads in each
+block and 256 hidden state size in the two fully-connected
+layers. Each patch is flattened and projected into a vector
+of 768 dimensions. We draw two conclusions from Table 2.
+First, the larger ViT-B models outperform the smaller ViT-S
+on most of the fairness and accuracy metrics, demonstrat-
+ing better feature learning capabilities with higher feature
+dimensions and more self-attention heads. Second, smaller
+patch size (16) achieves a better performance on both fair-
+ness and accuracy measurements because small patches en-
+ables extracting more fine-grained features [5].
+Table 2. Evaluations with different ViT models (i.e., ViT-B (B)
+and ViT-S (S)) and patch sizes (i.e., 16 and 32). All tunable hyper-
+parameters are set same as Figure 3. VA is short for Vanilla.
+Model
+Y : Gray Hair S: Gender
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+B/16
+VA
+0.2984
+0.2841
+0.0142
+81.95
+91.05
+DSA
+0.2424
+0.2205
+0.0081
+83.42
+91.24
+S/16
+VA
+0.2763
+0.3185
+0.0422
+81.84
+90.25
+DSA
+0.2558
+0.2337
+0.0031
+82.92
+90.95
+B/32
+VA
+0.2982
+0.2976
+0.0205
+81.11
+90.16
+DSA
+0.2629
+0.2520
+0.0109
+82.73
+91.03
+S/32
+VA
+0.3014
+0.3213
+0.0198
+80.64
+89.18
+DSA
+0.2935
+0.3165
+0.0086
+80.86
+89.45
+7. Conclusion
+In this work, we proposed a novel hierarchical fairness-
+aware ViT training framework named DSA for bias mitiga-
+tion in both the training set and the learning algorithm. The
+DSA framework eliminates the spurious features through
+adversarial attacks on the bias-only model while retaining
+the informative features through an attention weights align-
+ment regularizer. The experimental results demonstrate the
+effectiveness of DSA for bias mitigation without compro-
+mising prediction performance.
+
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+8. Supplementary Materials
+In this Section, we provide additional experiments for per-
+formance evaluation of the proposed DSA framework on
+CelebA and Waterbird datasets.
+8.1. Dataset Statistics
+Recall from Section 5.1 that we choose three settings from
+the CelebA dataset and one setting from the Waterbird
+dataset to evaluate the baselines against the proposed DSA
+framework. We describe these four settings using the tuples
+(Y, S) as follows: a) (Smiling, High Cheekbones), b) (Wavy
+Hair, Gender), c) (Gray Hair, Gender), and d) (Waterbird,
+Place). Note that the first three settings are considered for
+the CelebA dataset while the last setting is considered for
+the Waterbird dataset. We first provide the data statistics
+for all these settings in Figure 5. We note that significant
+biases exist in all these settings. For example, a majority
+of “Smiling” faces are correlated with “High Cheekbones”
+whereas the majority of “Not Smiling” faces are correlated
+with “Not High Cheekbones”. Similar spurious correlations
+are also observed in other settings as well, which can lead to
+biased models. We establish this by further analyzing and
+reporting the True Positive Rate (TPR) of the vanilla ViT
+models trained on these biased datasets in Table 3. Clearly,
+the biased ViT models perform significantly worse on the
+minority groups, e.g., predicting “Smiling” when the indi-
+vidual does not have “High Cheekbones” (S = 0: 63.93%)
+compared to the ones that have “High Cheekbones” (S = 1:
+96.50%). Next, we analyze the effect of the tunable hyper-
+parameters on the performance of DSA.
+Table 3. Disparities of true positive rate (TPR) among different
+task and sensitive attribute tuples.
+Y
+S
+TPR%↑
+∆TPR%
+Smiling
+Not High Cheekbones (0)
+63.93
+32.57
+High Cheekbones (1)
+96.50
+Wavy Hair
+Female (0)
+77.62
+20.04
+Male (1)
+57.58
+Gray Hair
+Female (0)
+63.31
+31.85
+Male (1)
+95.16
+Waterbird
+Land (0)
+55.75
+14.14
+Water (1)
+69.89
+8.2. Effect of Discrepancy Metrics
+Since We apply three different feature discrepancy metrics
+D(·, ·), i.e., MSE, KL-Div, and AT, to evaluate the discrep-
+ancy between the attention weights Ax and Ax′ in (6), we
+report the effect of these discrepancy metrics in Table 4.
+Although the differences between these discrepancy met-
+rics are relatively small, AT clearly achieves the best per-
+formance, especially on the fairness metrics. Since AT can
+capture the most significant differences between Ax and
+Ax′ as shown in (9), the regularizer LA is more efficient
+to minimize their differences.
+Table 4. Evaluations with different discrepancy metrics in the reg-
+ularizer (6).
+D⋆
+Y : Gray Hair S: Gender
+EO↓
+DP↓
+DBA↓
+BA↑
+Acc%↑
+MSE
+0.2706
+0.2488
+0.0136
+82.07
+90.13
+KL-Div
+0.2608
+0.2467
+0.0106
+83.26
+89.48
+AT
+0.2558
+0.2337
+0.0031
+82.92
+90.95
+8.3. Effect of Tunable Hyper-parameters
+There are several tunable hyper-parameters in the proposed
+DSA framework, including the various coefficient weights
+in the objective function and the number of masked patches
+learned during the adversarial attack.
+We tune the three coefficient weights in the objective
+function (10) to identify the best-performing model as
+shown in Table 5. To improve model performance, we be-
+lieve that these coefficient weights should be carefully tuned
+and selected under different settings and datasets.
+The effect of the number of masked patches learned dur-
+ing the adversarial attack is shown in Table 6. In our ex-
+periments, the ViT model with k = 3 patches achieves the
+best performance among all compared metrics in most set-
+tings. Looking into more details of the adversarial examples
+shown in Figure 6, if we perturb only one patch out of all
+the input patches, some sensitive attributes may not be lo-
+calized and masked. On the contrary, perturbing excessive
+patches (e.g., 5 patches) would increase the risk of masking
+the related attributes to the target task, resulting in a worse
+prediction performance. For example, the ACC drops from
+90.95 to 88.55 in the setting of (Gray Hair, Gender) with 5
+perturbed patches, as shown in Table 6.
+Table 5. Evaluations with different tunable coefficient weights in
+the objective function (10).
+λ1, λ2, λ3
+Y : Gray Hair S: Gender
+EO↓
+DP↓
+DBA↓
+BA↑
+Acc%↑
+1.0, 0.5, 0.5
+0.2843
+0.2675
+0.0125
+81.45
+91.12
+0.5, 1.0, 0.5
+0.2633
+0.2578
+0.0106
+81.32
+89.26
+1.0, 1.0, 0.5
+0.2558
+0.2337
+0.0031
+82.92
+90.95
+8.4. Ablation Studies and Effect of Patch Size
+We report the adversarial success rates of DSA on the sen-
+sitive attributes as target with different number of masked
+patches in Table 7. Note that we only generate adversarial
+examples for the training sets.
+In Table 8, we report additional ablation study results for
+the DSA framework on the other two settings from CelebA
+dataset. It is straightforward to make a similar conclusion
+
+Smiling
+Not Smiling
+0
+20000
+40000
+60000
+80000
+78899
+13290
+18770
+91640
+High CheekBones
+Not High CheekBones
+(a) Y: Smiling S: High Cheeckbones
+Wavy Hair
+Not Wavy Hair
+0
+10000
+20000
+30000
+40000
+50000
+60000
+70000
+11892
+72542
+52852
+65313
+Male
+Female
+(b) Y: Wavy hair S: Gender
+Gray Hair
+Not Gray Hair
+0
+1000
+2000
+3000
+4000
+5000
+6000
+6136
+1262
+1262
+6136
+Male
+Female
+(c) Y: Gray hair S: Gender
+Waterbird
+Landbird
+0
+1000
+2000
+3000
+4000
+5000
+6000
+6220
+831
+2905
+1832
+Water
+Land
+(d) Y: Waterbird S: Place
+Figure 5. Spurious correlation between tasks (Y ) and sensitive attributes (S) tuples (Y, S). Note that Figures 5a, 5b and 5c represent the
+data statistics for the CelebA dataset while Figure 5d represents the data statistics of the Waterbird dataset.
+Table 6. Performance of DSA with different number of masked or perturbed patches.
+k
+Y : Smiling
+S: High Cheekbones
+Y : Wavy Hair
+S: Gender
+Y : Gray Hair
+S: Gender
+EO↓
+DP↓
+DBA↓
+BA(%)↑
+Acc(%)↑
+EO↓
+DP↓
+DBA↓
+BA(%)↑
+Acc(%)↑
+EO↓
+DP↓
+DBA↓
+BA(%)↑
+Acc(%)↑
+1
+0.3502
+0.3341
+0.0046
+77.84
+87.18
+0.1822
+0.2036
+0.0098
+72.79
+77.26
+0.2946
+0.3075
+0.0110
+81.77
+90.61
+3
+0.3012
+0.2864
+0.0034
+80.10
+89.23
+0.1618
+0.1844
+0.0056
+73.34
+79.34
+0.2558
+0.2337
+0.0031
+82.92
+90.95
+5
+0.3218
+0.3179
+0.0040
+79.88
+88.12
+0.1604
+0.1776
+0.0087
+72.13
+78.16
+0.2776
+0.2560
+0.0216
+81.91
+88.55
+Table 7. Adversarial attack success rates of DSA on the sensitive
+attributes target with different number of masked patches, k.
+S
+k
+Success Rate%↑
+Gender
+1
+88.52
+3
+91.47
+5
+93.69
+High Cheekbones
+1
+85.41
+3
+88.64
+5
+91.58
+as in Section 6.2. We note that all the terms in the objec-
+tive function in (10) contribute towards better fairness and
+accuracy performance.
+Additional evaluations capturing the effect of different
+patch sizes on the performance of DSA are reported in Ta-
+ble 9. Similar to our conclusion in Section 6.3, the ViT
+models with smaller patch sizes, i.e., 16, achieve the best
+performance on two other settings from the CelebA dataset.
+Figure 6. Adversarial examples with different number of masked
+patches in the (Gray Hair, Gender) setting.
+8.5. Qualitative Evaluations
+In Figures 7, 8, and 9, we demonstrate some more qualita-
+tive evaluations to further demonstrate the effectiveness of
+the DSA approach. We note that the distribution of the at-
+tention weights for the ViT models trained with the vanilla
+method simply focuses on the sensitive attributes, e.g., “eye
+shadow”. This demonstrates that the vanilla ViT models are
+biased and simply leverage the sensitive features to predict
+the target labels. On the contrary, DSA reduces the atten-
+tion on these sensitive features but pays more attention on
+the target-related features, e.g., the hair, which actually de-
+termines the target label Gray and Wavy hair.
+8.6. Summary
+We summarize the major findings of our experimental study
+here. First, DSA reduces the attention on the sensitive fea-
+tures while focusing on the target-related features as an ef-
+fective approach to bias mitigation. Second, the additional
+ablation studies demonstrate that each term in the objec-
+tive function (10) contributes towards the improved fairness
+and accuracy performance of DSA. Third, we noted that the
+smaller patch size results in better performance of DSA due
+to their capability of efficiently extracting fine-grained fea-
+tures.
+(a) Original Image
+(b) Vanilla 
+(c) DSA 
+Figure 7.
+Qualitative evaluation of DSA. Y: Smiling S: High
+Cheeckbones.
+
+VANN
+.(a) Original Image
+(b) Vanilla 
+(c) DSA 
+Figure 8. Qualitative evaluation. Y: Gray hair S: Gender.
+(a) Original Image
+(b) Vanilla 
+(c) DSA 
+Figure 9. Qualitative evaluation. Y: Wavy hair S: Gender.
+Table 8. Ablation study of DSA for the three training objectives on two other settings from the CelebA data set. Best results are bold faced.
+‘w/o’ represents without.
+Models
+Y : Wavy Hair S: Gender
+Y : Smiling S: High Cheekbones
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+L(all)
+0.1618
+0.1844
+0.0056
+73.34
+79.34
+0.3012
+0.2864
+0.0034
+80.10
+89.23
+w/o LCE(x, y)
+0.2114
+0.22.03
+0.0288
+72.42
+77.56
+0.3105
+0.3014
+0.0231
+79.54
+88.06
+w/o LCE(x′, y)
+0.2004
+0.2154
+0.0275
+72.45
+77.98
+0.3198
+0.2987
+0.0129
+78.39
+88.54
+w/o LA
+0.1942
+0.2012
+0.0312
+72.33
+77.45
+0.3125
+0.2955
+0.0198
+79.21
+87.95
+Table 9. Performance evaluation of DSA with different patch sizes (i.e., 16 and 32). All tunable hyper-parameters are set same as Figure
+3. VA is short for Vanilla.
+Model
+Y : Wavy Hair S: Gender
+Y : Smiling S: High Cheekbones
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+EO↓
+DP↓
+DBA↓
+BA↑
+ACC↑
+S/16
+VA
+0.2193
+0.2204
+0.0310
+72.69
+78.78
+0.3382
+0.3256
+0.0125
+77.80
+88.04
+DSA
+0.1618
+0.1844
+0.0056
+73.34
+79.34
+0.3012
+0.2864
+0.0034
+80.10
+89.23
+S/32
+VA
+0.2236
+0.2319
+0.0450
+72.05
+78.68
+0.3398
+0.3315
+0.0213
+78.12
+87.54
+DSA
+0.1805
+0.2196
+0.0254
+72.58
+79.02
+0.3209
+0.3177
+0.0156
+79.27
+88.15
+

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@@ -0,0 +1,1123 @@
+arXiv:2301.03279v1  [cs.GT]  9 Jan 2023
+Revisiting the Distortion of Distributed Voting
+Aris Filos-Ratsikas1 and Alexandros A. Voudouris2
+1School of Informatics, University of Edinburgh, UK
+2School of Computer Science and Electronic Engineering, University of Essex, UK
+Abstract
+We consider a seting with agents that have preferences over alternatives and are partitioned
+into disjoint districts. Te goal is to choose one alternative as the winner using a mechanism
+which first decides a representative alternative for each district based on a local election with the
+agents therein as participants, and then chooses one of the district representatives as the winner.
+Previous work showed bounds on the distortion of a specific class of deterministic plurality-based
+mechanisms depending on the available information about the preferences of the agents in the
+districts. In this paper, we first consider the whole class of deterministic mechanisms and show
+asymptotically tight bounds on their distortion. We then initiate the study of the distortion of
+randomized mechanisms in distributed voting and show bounds based on several informational
+assumptions, which in many cases turn out to be tight. Finally, we also experimentally compare
+the distortion of many different mechanisms of interest using synthetic and real-world data.
+1
+Introduction
+Voting is a ubiquitous method for making decisions with a large number of applications, such as electing
+political representatives, deciding how to split a public budget between projects, or choosing which
+services (restaurants, hotels, etc) to recommend to new users based on past user experiences. As such, it
+has been at the epicenter of research within multiple disciplines including political sciences, economics
+and computer science [Brandt et al., 2016]. Te most prominent question in this research agenda is to
+identify the best voting rule to use to collectively aggregate the preferences of agents over alternative
+options into a single winning alternative, with most of the earlier literature focusing on axiomatic
+properties that good voting rules should have. An alternative way to tackle this question that has been
+proposed in computer science is through the distortion framework [Anshelevich et al., 2021] which
+allows to compare different voting rules based on how well they approximate the optimal choice as
+measured in terms of a social objective function like the utilitarian social welfare.
+Since its inception in 2006 by Procaccia and Rosenschein [2006], the distortion framework has been
+applied to several utilitarian social choice setings (e.g., [Boutilier et al., 2015, Anshelevich et al., 2018,
+Gkatzelis et al., 2020]). Te lion’s share of previous work has focused on centralized models with a
+single pool of agents whose preferences are directly given as input to a voting rule, which thus can
+utilize all the given information at once to make a decision. However, there are many applications
+with multiple pools of agents which make independent local decisions that can be thought of as rec-
+ommendations for the final decision. To give a concrete example, in most political election systems,
+the citizens are partitioned into districts based on geographic or other criteria, and vote within their
+districts to propose the candidate (party) they would like to be selected as the winner.
+Inspired by situations like the one described above, Filos-Ratsikas et al. [2020] initiated the study
+of the distortion of mechanisms in a distributed single-winner seting where a set of n agents with
+1
+
+Deterministic
+Randomized-of-Deterministic
+Randomized-of-Randomized
+Ordinal
+Θ(km2)
+Θ(km2)
+Ω(√m), O(√m log m)
+Cardinal
+Θ(km)
+Θ(k)
+Θ(k)
+Strategyproof
+Θ(nm)
+Θ(nm)
+Ω(√m), O(√m log m)
+Table 1: An overview of our results. Specific details can be found in the appropriate sections.
+cardinal preferences over a set of m alternatives are partitioned into k disjoint districts. Te authors
+focused on deterministic mechanisms of the form Plurality-of-f, which first choose a representative
+alternative for each district according to some rule f, by holding a local election with the agents of
+the district as the voters, and then picking the winner to be the alternative that is representative of
+the most districts (i.e., using the Plurality rule). Filos-Ratsikas et al. considered mechanisms for
+which the rule f can be cardinal or ordinal, i.e., it may use the actual numerical information about
+the preferences of the agents within the districts or just consistent rankings. Te authors showed
+that, when the districts are symmetric (that is, each of them contains the same number of agents), the
+distortion of a cardinal mechanism, namely Plurality-of-Range-Voting is O(km), and provided an
+asymptotically matching lower bound of Ω(km) on the distortion of any Plurality-of-f mechanism.
+For ordinal mechanisms, they showed that Plurality-of-Plurality achieves a distortion of O(km2),
+and that this is asymptotically best among all ordinal Plurality-of-f mechanisms.
+1.1
+Revisiting the distortion of distributed voting
+A first observation about the results of Filos-Ratsikas et al. [2020] is that there is a-priori no reason
+to restrict our atention to only mechanisms in the class Plurality-of-f, as using other over-districts
+rules could potentially lead to beter distortion. Indeed, follow-up work considered distributed social
+choice setings with metric preferences [Anshelevich et al., 2022, Filos-Ratsikas and Voudouris, 2021]
+without such restrictions on the over-districts rule. In addition, all of the previous work on these
+setings only considered deterministic mechanisms that use deterministic in-district and over-districts
+rules. Randomization has proven out to be a very useful tool in achieving beter (expected) distortion
+bounds in the centralized seting (see Boutilier et al. [2015], Ebadian et al. [2022]), so it is only natural
+to consider randomized mechanisms in the distributed seting as well. Finally, an important question
+is how the distortion bounds are affected in case the participants act selfishly, and whether there are
+strategyproof mechanisms with good distortion bounds. Tis question has been considered in the
+centralized seting [Filos-Ratsikas and Miltersen, 2014, Bhaskar and Ghosh, 2018, Bhaskar et al., 2018,
+Ebadian et al., 2022] and also in the distributed metric seting [Filos-Ratsikas and Voudouris, 2021]; we
+consider it in the context of the normalized seting of Filos-Ratsikas et al. [2020] as well.
+1.2
+Our Contributions
+We consider the class of all mechanisms for distributed voting in the seting of [Filos-Ratsikas et al.,
+2020]. In particular, we consider the fover-of-fin class of mechanisms, where fin is an in-district rule that
+takes as input the preferences of the agents within each district and outputs a representative alternative
+for the district, while fover is a rule that takes as input the representative alternatives of all districts and
+chooses one of them as the overall winner. We consider several different cases depending on the nature
+of fover and fin (deterministic or randomized), and the type of information they can utilize (cardinal or
+ordinal). We show the following results; see Table 1 for an overview.
+Deterministic Mechanisms. When fover and fin are both deterministic and the districts are symmet-
+ric, we show that the best possible distortion is Θ(km) when the valuation functions of the agents are
+2
+
+accessible (cardinal mechanisms), and is Θ(km2) when only ordinal information about the preferences
+of the agents is available (ordinal mechanisms). Te upper bounds were shown by Filos-Ratsikas et al.
+[2020] and here we provide asymptotically tight lower bounds. Tese results show that for general,
+unstructured (normalized) valuations, employing different over-district rules in fact does not result in
+improvements on the distortion. We present these results in Section 3.
+Randomized Mechanisms. In Section 4, we consider for the first time the distortion of randomized
+mechanisms in distributed voting. We first prove a simple composition theorem, which shows that
+using an in-district rule with known distortion δ in the centralizedseting and then selecting the winner
+uniformly at random from the set of representatives, defines a distributed mechanism with distortion
+O(kδ). Using this, complemented with new lower bounds, we show that the best possible distortion
+for cardinal unanimous mechanisms is Θ(k); in fact, this is true even when the districts are asymmetric
+and when fover is randomized but fin is deterministic.
+For ordinal mechanisms, we consider two cases: (a) mechanisms that use deterministic in-district
+rules fin, and (b) fully-randomized mechanisms, where both fover and fin are randomized rules. For
+(a), we show that the best possible distortion is Θ(km2). Te upper bound follows from the bound on
+Plurality-of-Plurality proven in [Filos-Ratsikas et al., 2020]; here, we provide an asymptotically
+matching lower bound assuming a natural universal tie-breaking rule. For (b), we prove a simple but
+very interesting result: For a well-studied class of randomized centralized voting rules called point-
+voting schemes (e.g., see Gibbard [1977], Barbera [1978]), there exists a distributed implementation so
+that there is no effect on the induced probability distribution, even for asymmetric districts. Simply put,
+using such rules it is possible to escape the ill effects of districts in terms of the distortion, even when
+the districts are asymmetric. From this result, it follows that there exists a distributed implementation
+of a well-known mechanism of Boutilier et al. [2015] that achieves distortion O(√m log m), almost
+matching the best possible lower bound of Ω(√m).
+Strategyproof Mechanisms. For strategyproof mechanisms, which are resilient to strategic manip-
+ulation, we show that a best-possible distortion of Θ(nm) for deterministic mechanisms (and more
+generally mechanisms with a deterministic in-district rule) is easy to achieve by a variation of a dic-
+tatorship rule. For randomized mechanisms, since point-voting schemes are strategyproof, the bound
+O(√m log m) carries over to this class as well. Results about deterministic strategyproof mechanisms
+are presented in Section 3, and about randomized strategyproof mechanisms in Section 4.
+Experiments. Finally, in Section 5, we perform experiments using real-world data and synthetic data
+to evaluate the effect of distributed decision making to the distortion in setings closer to practice. Te
+main conclusions of our experimental results mirror that of our theoretical results in Sections 3 and 4.
+1.3
+Further Related Work
+Te distortion literature is by now rather extensive, including topics such as single-winner voting
+[Boutilier et al., 2015, Anshelevich et al., 2018, Gkatzelis et al., 2020, Kizilkaya and Kempe, 2022],
+multi-winner voting [Caragiannis et al., 2017, 2022], matching problems [Filos-Ratsikas et al., 2014,
+Amanatidis et al., 2022a], and participatory budgeting [Benad`e et al., 2017]. Generally speaking, most
+works can be categorized as either studying a normalized utilitarian seting (e.g., [Procaccia and Rosen-
+schein, 2006, Boutilier et al., 2015, Filos-Ratsikas et al., 2014, Benad`e et al., 2017, Ebadian et al., 2022]) or
+a metric preference seting (e.g., [Anshelevich and Sekar, 2016, Anshelevich et al., 2018, Gkatzelis et al.,
+2020, Caragiannis et al., 2022, Kizilkaya and Kempe, 2022]). Some more recent works have also studied
+the interplay between information and distortion [Amanatidis et al., 2021, 2022a,b, Mandal et al., 2019,
+2020, Abramowitz et al., 2019], and there have also been several works on strategyproofness in the con-
+text of distortion [Filos-Ratsikas and Miltersen, 2014, Filos-Ratsikas et al., 2014, Bhaskar and Ghosh,
+3
+
+2018, Bhaskar et al., 2018, Ebadian et al., 2022]. We refer the reader to the survey of Anshelevich et al.
+[2021] for a detailed overview of the related literature.
+Besides the aforementioned works on distributed voting, Borodin et al. [2019] studied a related
+two-stage seting in which the voters participate in a central election, but the candidates themselves
+come from local elections within the political parties’ electorates. Beyond distortion, in the context of
+district-based elections, there have also been other works that have considered the degree of deviation
+from proportional representation (e.g., see [Bachrach et al., 2016] and references therein), and some
+works that have studied the complexity of manipulation (e.g., see [Elkind et al., 2021, Lewenberg et al.,
+2017, Lev and Lewenberg, 2019, Borodin et al., 2018]).
+2
+Preliminaries
+An instance I of our problem is given by a tuple I = (N, A, v, D). Tere is a set N of n agents (or
+voters) that have preferences over a set A of m alternatives (or candidates). Te preferences of each
+agent i ∈ N are captured by a valuation function vi : A → R≥0 that maps every alternative a ∈ A to a
+real non-negative value vi(a) = via. Following previous work, we assume that the valuation functions
+are normalized such that �
+a∈A via = 1 for every i ∈ N (unit-sum assumption). Let v = (vi)i∈N be
+the valuation profile consisting of the valuation functions of all agents. Te agents are also partitioned
+into a set D of k disjoint districts.
+For every district d ∈ D, let Nd be the set of agents it contains, such that �
+d∈D Nd = N. In the
+symmetric case, each district d contains exactly λ = n/k agents. In the asymmetric case, each district
+d contains a number nd of agents. All our lower bounds follow by instances consisting of symmetric
+districts, whereas our upper bounds in Section 4 hold for asymmetric districts.
+2.1
+Mechanisms
+Our goal is to choose an alternative to satisfy several criteria of interest. Tis choice must be done
+using a distributed mechanism that uses an in-district voting rule fin and an over-districts voting rule
+fover to implement the following two independent steps:
+• Step 1: For each district d, choose a representative alternative ad ∈ A by holding a local election
+based on fin.
+• Step 2: Choose a district representative as the winner based on fover by considering the districts
+as voters and their representatives as the candidates they approve.
+For simplicity we refer to such mechanisms as fover-of-fin. Different choices of fin and fover lead to
+different distributed mechanisms. Note that the in-district rule can in general use various types of
+information about the preferences of the agents. For instance, it may be able to use exact cardinal
+information about the valuation functions, or only ordinal information that is induced by the values
+(i.e., rankings of alternatives that are consistent to the values of the agents for them). In the later case,
+we will use σi to denote the preference ranking of agent i ∈ N so that σi(a) is the rank of alternative
+a ∈ A in the ranking of i, and σi(a) < σi(b) if vi(a) ≥ vi(b); let σ = (σi)i∈N be the ordinal
+profile consisting of the preference rankings of all agents. To be concise in the definitions below, let
+δ(I) be the information about the preferences of the agents in instance I = (N, A, v, D) that is used
+by a mechanism; that is, δ(I) = v in case of cardinal information, or δ(I) = σ in case of ordinal
+information.
+We will focus on different classes of distributed mechanisms depending on the available informa-
+tion about the preferences of the agents at the district level (cardinal or ordinal), and also on whether
+4
+
+their decision is deterministic or randomized (that is, they choose the district representatives or final
+winner based on probability distributions).
+2.2
+Social Welfare and Distortion
+Given an instance I, the social welfare of an alternative a ∈ A is the total value that the agents have for
+a, that is, SW(a|I) = �
+i∈N via. So, the expected social welfare achieved by a randomized distributed
+mechanism M that chooses alternative a ∈ A as the winner w with probability PrM[w = a] is
+E[SW(M(I))] =
+�
+a∈A
+Pr
+M [w = a] · SW(a|I).
+Te efficiency of a distributed mechanism is measured by the notion of distortion. Te distortion of a
+distributed mechanism M is the worst-case ratio (over all possible instances with n agents, m alterna-
+tives, and k districts) of the maximum social welfare achieved by any alternative over the (expected)
+social welfare of the alternative chosen by the mechanism as the winner w, that is,
+dist(M) = sup
+I
+maxa∈A SW(a|I)
+E[SW(M(δ(I))] .
+Clearly, dist(M) ≥ 1. When the denominator in the definition of the distortion tends to 0, we will
+say that the distortion is infinite or unbounded. Our goal is to identify the best possible distributed
+mechanisms in terms of distortion.
+2.3
+Strategyproofness
+Another important property that we would like our mechanisms to satisfy is that of strategyproof-
+ness. A strategyproof mechanism makes decisions such that providing false information never leads to
+the selection of an alternative that an agent prefers over the alternative chosen when the agent pro-
+vides truthful information. In particular, for any instance I, it must be the case that vi(M(δ(I))) ≥
+vi(M(δ(I′))) for any agent i ∈ N, where I′ is the instance obtained when only agent i reports infor-
+mation different than that in I.
+2.4
+Some useful observations and properties
+Before we present our technical results, let us briefly discuss some useful properties.
+Locality of distributed mechanisms: First, observe that any distributed mechanism fover-of-fin
+satisfies a locality property in the following sense. A district d (that is, the preferences of a number
+of agents) appears in different instances if in each of these instances there is a district with the same
+number of agents and the same information about theirs preferences as in d (depending on what is
+required by the mechanism). Since the information is the same, the in-district rule fin must decide the
+same alternative as the representative of the district in all these instances. Similarly, in all instances
+where the mechanism has decided the same set of district representatives, the over-districts rule fover
+must decide the same final winner.
+Distortion of distributed vs centralized: Another useful observation is that the distortion of a
+distributed mechanism fover-of-fin is at least as much as the distortion of the in-district centralized
+voting rule fin. Indeed, when k = 1, there is only one representative alternative chosen by fin, and
+thus this alternative must be chosen as the winner by fover; this is also true for instances with k ≥ 2
+districts which are all copies of one district. Consequently, the distortion of fin is a lower bound on
+the distortion of fover-of-fin.
+5
+
+Strategyproofness: Observe that for a distributed mechanism fover-of-fin to be strategyproof it is
+necessary that the in-district rule fin is strategyproof. Tis again follows by how the mechanism would
+work in instances with a single district, in which case the over-districts rule fover does not play any
+role in the selection of the final winner.
+Unanimity: A few of our results will require the in-district rules fin to be unanimous. Unanimity
+stipulates that if all of the agents have the same alternative as the top preference, that alternative
+must be selected (with probability 1). Unanimity is a very natural property of “reasonable” voting
+rules, especially deterministic ones. For randomized rules, there might be reasons to consider non-
+unanimous choices, e.g., see Gibbard [1977], Filos-Ratsikas and Miltersen [2014].
+3
+Deterministic mechanisms
+We start with deterministic distributed mechanisms and focus explicitly on the case of symmetric
+districts in this section (that is, the size of each district is λ). When full information about the valuations
+of the agents is known at the district level, Filos-Ratsikas et al. [2020] showed that the mechanism
+Plurality-of-Range-Voting, which chooses the representative of each district to be the alternative
+with maximum social welfare for the agents in the district, has distortion O(km). We show that this
+mechanism is asymptotically best possible over all possible deterministic distributed mechanisms that
+use unanimous in-district rules (but may not use Plurality as the over-districts rule).
+Teorem 3.1. Te distortion of any deterministic distributed mechanism with a unanimous in-district
+rule is Ω(km).
+Proof. Let M be some deterministic distributed mechanism with a unanimous in-district rule. Without
+loss of generality, whenever there are k distinct district representatives {a1, . . . , ak}, we assume that
+M chooses a1 as the overall winner. Let ε > 0 be some positive infinitesimal and consider the following
+instance with k districts {d1, . . . , dk} and m > k alternatives:
+• In district d1, all agents have value 1/m + ε for alternative a1, and value 1/m − ε/(m − 1) for
+any other alternative.
+• For any ℓ ∈ {2, . . . , k}, in district dℓ, all agents have value 1/2 + ε for alternative aℓ, value
+1/2 − ε for alternative x, and value 0 for any other alternative.
+Since the in-district rule is unanimous, the district representatives are alternatives {a1, . . . , ak}, and
+the overall winner is thus a1. Te social welfare of alternative a1 is approximately λ/m, whereas the
+social welfare of alternative x is approximately k · λ/2, leading to distortion Ω(km).
+When only ordinal information about the preferences of the agents is available, Filos-Ratsikas
+et al. [2020] showed that Plurality-of-Plurality, which chooses the favorite alternative of most of
+the agents in a district as its representative and then the alternative that represents the most districts
+as the winner, has distortion O(km2). We show that this mechanism is asymptotically best possible
+among all ordinal distributed mechanisms (without any restrictions), thus improving upon the result
+of Filos-Ratsikas et al. [2020] who showed that Plurality-of-Plurality is best possible only within
+the class of mechanisms they studied.
+We first prove an easy but important lemma showing that when only ordinal information is avail-
+able, to achieve finite distortion, it is necessary the representative of each district to be some alternative
+that is the favorite of at least one agent in the district.
+6
+
+Lemma 3.2. Te representative of any district must be some top-ranked alternative, otherwise the distor-
+tion is infinite.
+Proof. Let d be a district and let T be the set of top-ranked alternatives. Suppose that the representative
+of d is chosen to be some alternative x ̸∈ T. Ten, in any instance consisting of copies of d, the winner
+must be x. However, the valuation profile might be such that all agents have value 1 for their favorite
+alternative and 0 for any other alternative. Consequently, the social welfare of x might be 0, whereas
+the social welfare of any top-ranked alternative is positive, leading to infinite distortion.
+We say that a district is divided if its λ agents are partitioned into m/2 equal-sized sets such that all
+the 2λ/m agents in each set rank the same alternative first and different sets of agents have different
+top-ranked alternatives. By Lemma 3.2, the representative of such a district must be one of the top-
+ranked alternatives. Te following lemma shows that choosing the representative of a divided district
+as the winner is, under some circumstances, a bad choice.
+Lemma 3.3. Suppose that some alternative y1 is chosen as the winner by a deterministic ordinal dis-
+tributed mechanism when the set of representatives is {y1, . . . , yk}. If there exists a divided district that
+is represented by y1, then there are k − 1 districts with representatives y2, . . . , yk, and altogether these k
+districts define an instance such that the distortion of the mechanism is Ω(km2).
+Proof. Let M be a deterministic ordinal distributed mechanism that selects y1 as the winner when
+the set of representatives is {y1, . . . , yk}, and let d be the divided district that is represented by y1.
+Consider the following k districts:
+• Te first district is a copy of d.
+• For every ℓ ∈ {2, . . . , k}, the ℓ-th district is such that all agents therein rank yℓ first, x ̸∈
+{y1, . . . , yk} second, and then all other alternatives. By Lemma 3.2, M must choose yℓ as the
+representative of the ℓ-th district, as this is the only top-ranked alternative.
+So, indeed the set of representatives is {y1, . . . , yk} and M chooses y1 as the winner by assumption.
+One possible valuation profile is the following:
+• In the first, divided district, the 2λ/m agents that rank y1 first have value 1/m for all alternatives,
+and the remaining agents all have value 1 for their favorite alternative.
+• For every ℓ ∈ {2, . . . , k}, all agents in the ℓ-th district have value 1/2 for their two favorite
+alternatives (yℓ and x).
+Consequently, the social welfare of y1 is λ/m2 whereas the social welfare of x is approximately k·λ/2,
+and thus the distortion is Ω(km2).
+Lemma 3.3 shows that deterministic ordinal distributed mechanisms with distortion o(km2) must
+not output the representative of a divided district as the winner when it is given a set of districts with
+different representatives. However, as we show in the proof of the next theorem, there are instances
+where such a choice is inevitable, and thus the distortion is Ω(km2).
+Teorem 3.4. Te distortion of any deterministic ordinal distributed mechanism is Ω(km2).
+Proof. Let M be a deterministic ordinal distributed mechanism. We focus on instances with k districts
+and sets of alternatives A ∪ B ∪ C ∪ {x}, where A = {a1, . . . , ak}, B = {b1, . . . , bm/2+k−1}, and
+7
+
+C = {c1, . . . , cm−2k}. Without loss of generality, suppose that when the district representatives are
+{a1, . . . , ak}, M chooses a1 as the overall winner.
+Let d1 be a divided district with set of top-ranked alternatives {a1, b1, . . . , bm/2−1}. By Lemma 3.3,
+if a1 is the representative of d1, then there exists an instance such that the distortion of M is Ω(km2).
+So, suppose that the representative of d1 is some other top-ranked alternative, say b1. Again by
+Lemma 3.3, if b1 is chosen as the winner whenever she is part of a representative set consisting of
+k distinct alternatives, then the distortion of M would be Ω(km2). So, let us assume that when the
+district representatives are {b1, a2, . . . , ak}, the winner is an alternative different than b1, say a2.
+We can now repeat this argument step by step for each alternative aℓ, ℓ ∈ {2, . . . , k}. In particular,
+let dℓ be a divided district with top-ranked alternatives {aℓ, bℓ, . . . , bm/2+ℓ−2} (note that alternatives
+b1, . . . , bℓ−1 do not appear as top-ranked alternatives in dℓ). By Lemma 3.3, if aℓ is the representative
+of dℓ then the distortion of M is Ω(km2), so the representative is some other alternative from the set
+{bℓ, . . . , bm/2+ℓ−2}, say bℓ. Again by Lemma 3.3, if bℓ is chosen as the winner whenever she is part of
+a representative set consisting of k distinct alternatives, then the distortion of M would be Ω(km2).
+So, when the district representatives are {b1, . . . , bℓ, aℓ+1, . . . , ak}, the winner is an alternative not in
+{b1, . . . , bℓ}, say aℓ.
+Te last step of this repeated argument leads to the lower bound of Ω(km2): We have reached an
+instance with set of representatives {b1, . . . , bk} all of whom are representative of some divided district,
+and thus no mater who of them is chosen as the winner, by Lemma 3.3 there exists an instance that
+includes the corresponding divided district and k − 1 unanimous districts (like in the proof of the
+lemma) such that the distortion is Ω(km2).
+Finally, let us discuss the case of deterministic strategyproof distributed mechanisms. Bhaskar and
+Ghosh [2018] showed that the distortion of any deterministic centralized strategyproof voting rule
+(including those that have access to the valuation functions) is Θ(nm). From the discussion Section 2.4,
+we directly obtain a lower bound of Ω(nm) for the distributed seting as well. A tight upper bound is
+also not hard to derive by considering the straightforward First-of-First mechanism which works as
+follows:
+• For each district d, choose the favorite alternative of the first agent therein as the representative.
+• Choose the representative of the first district as the winner.
+Teorem 3.5. First-of-First is strategyproof and achieves an asymptotically best possible distortion of
+Θ(nm) within the class of deterministic strategyproof distributed mechanisms.
+Proof. Te mechanism is clearly strategyproof since the winner is the favorite alternative of the first
+agent of the first district who acts as a dictator. Since the winner is ranked first by an agent, the social
+welfare of the mechanism is at least 1/m. Te maximum possible social welfare is n, and thus the
+distortion is O(nm).
+4
+Randomized mechanisms
+We start our discussion on randomized distributed mechanisms by analyzing a general class of mech-
+anisms that we call Uniform-of-δ-Approximate. A mechanism M in this class works as follows:
+• For each district d, M chooses the representative ad according to some centralized voting rule
+fin that has distortion at most δ.
+• M chooses the winner uniformly at random from the set of representatives.
+8
+
+Picking the winner uniformly at random from the representatives that have been selected seems to be
+the most natural choice as there is not much information about the preferences of the agents in the
+districts, and essentially all we can do is assign higher proportional probability to an alternative that
+is representative of more districts. We have the following result.
+Teorem 4.1. Te distortion of any Uniform-of-δ-Approximate mechanism is O(kδ).
+Proof. Consider an arbitrary instance. Let o be the optimal alternative, ad the representative of district
+d, and w the final winner. Denote by SWd(x) the social welfare of alternative x only from the agents
+in d; clearly, SW(x) = �
+d∈D SWd(x). Te expected social welfare of the mechanism is
+E[SW(M)] =
+�
+a∈A
+Pr[w = a] · SW(a)
+= 1
+k
+�
+a∈A
+��
+d∈D
+Pr[ad = a]
+�
+SW(a)
+= 1
+k
+�
+d∈D
+�
+a∈A
+Pr[ad = a] · SW(a)
+= 1
+k
+�
+d∈D
+E[SW(ad)]
+≥ 1
+k
+�
+d∈D
+E[SWd(ad)]
+Since ad is chosen based on a voting rule with distortion at most δ, we have that E[SW(ad)] ≥ 1
+δ ·
+SWd(o). Combining this together with the fact that SW(o) = �
+d∈D SWd(o), and using the linearity
+of expectation, we obtain
+E[SW(M)] ≥ 1
+k
+�
+d∈D
+E[SWd(ad)]
+≥ 1
+k
+�
+d∈D
+1
+δ · SWd(o)
+= 1
+kδ · SW(o).
+Hence, the distortion of the mechanism is at most kδ.
+Teorem 4.1 is a simple composition theorem, analogous to the one presented by Anshelevich
+et al. [2022] for the metric seting. Based on it, we can define randomized distributed mechanisms
+with proven distortion guarantees by appropriately choosing the in-district rule. Before we continue,
+observe that the sizes of the districts do not appear in the proof of Teorem 4.1, and thus the distortion
+of any Uniform-of-δ-Approximate mechanism is O(kδ) even if the districts are asymmetric. So, the
+distortion of the mechanism depends on the number of agents only if the distortion δ of the in-district
+rule depends on the number of agents.
+If cardinal information is available at the district level, by using Range-Voting with δ = 1 as the
+in-district rule, we obtain the following.
+Corollary 4.2. Te distortion of Uniform-of-Range-Voting is O(k).
+If only ordinal information about the preferences of the agents is given at the district level, then we
+can use Plurality with δ = O(m2) and the randomized rule Stable-Lottery mechanism of Ebadian
+et al. [2022] with δ = O(√m) as the in-district rule to obtain the following results.
+9
+
+Corollary 4.3. Te distortion of Uniform-of-Plurality is O(km2).
+Corollary 4.4. Te distortion of Uniform-of-Stable-Lottery is O(k√m).
+An important question to ask next is under what circumstances the aforementioned upper bounds
+of Corollaries 4.2, 4.3 and 4.4 are tight. First, we show that Uniform-of-Range-Voting is the best
+among mechanisms with unanimous in-district rules which may even use cardinal information.
+Teorem 4.5. Te distortion of any randomized distributed mechanism with a unanimous in-district rule
+is Ω(k).
+Proof. Let ε > 0 be a positive infinitesimal. Consider an instance with the following k symmetric
+districts: For any ℓ ∈ [k], in district dℓ, all λ agents therein have value 1/2 + ε for alternative aℓ,
+1/2 − ε for alternative x, and 0 for any other alternative. Since, the in-district rule is unanimous, the
+representative of district dℓ must be aℓ with probability 1. Hence, no mater what the probability of
+choosing a district representative as the winner is, the expected social welfare of the mechanism is
+λ · (1/2 + ε). However, the social welfare of alternative x is k · λ · (1/2 − ε), and thus the distortion
+is Ω(k).
+If we consider non-unanimous in-district rules, but require the in-district rule to be deterministic,
+then we can show a weaker lower bound of Ω(
+√
+k); notice that the theorem also implies the same
+bound for fully deterministic distributed mechanisms without unanimous in-district rules.
+Teorem 4.6. Te distortion of any randomized distributed mechanism with a deterministic in-district
+rule is Ω(
+√
+k).
+Proof. Consider a district dℓ in which all agents have value 1/2 for alternative aℓ, value 1/(2
+√
+k) for
+each alternative in {b1, . . . , b√
+k}, and 0 for any other alternative. If the representative of this district is
+not aℓ, then in instances consisting of copies of this district, the distortion is at least
+√
+k; in particular, it
+is at least that much if some alternative in {b1, . . . , b√
+k} is chosen and infinite if any other alternative
+is chosen. So, suppose that the representative of dℓ is aℓ.
+Next, consider an instance with k symmetric districts d1, . . . , dk. By the above discussion, for any
+ℓ ∈ [k], the representative of dℓ is alternative aℓ with social welfare λ/2 (note that only the agents
+of dℓ have positive value, equal to 1/2, for aℓ). Hence, no mater which district representative is
+chosen as the winner (or the probability distribution over the representatives), the (expected) social
+welfare of the mechanism is λ/2. In contrast, the social welfare of any alternative in {b1, . . . , b√
+k} is
+k · λ/(2
+√
+k) =
+√
+k · λ/2, and thus the distortion is
+√
+k.
+Next, we show that Uniform-of-Plurality is the best possible among ordinal randomized dis-
+tributed mechanisms with deterministic in-district rules, assuming an arbitrary but fixed ordering of
+the alternatives. Tis is quite surprising, as it shows that randomization over the districts is not beter
+than just choosing an arbitrary alternative that is representative of the most districts (i.e., not beter
+than Plurality-of-Plurality).
+Teorem 4.7. Te distortion of any ordinal distributed mechanism with a deterministic in-district rule is
+Ω(km2), when there exists an arbitrary but fixed tie-breaking ordering of the alternatives.
+Proof. Without loss of generality, suppose that the tie-breaking ordering of the alternatives is a1 ≻
+. . . ≻ ak ≻ b1 ≻ . . . ≻ bm/2−1 ≻ x ≻ c1 ≻ . . . ≻ cm/2−k; the naming of the alternatives is arbitrary
+but is assumed to be known and can be exploited. For simplicity, for any set of alternatives X, denote
+by [X] an arbitrary ordering of the alternatives in X.
+10
+
+Consider an instance with k symmetric districts such that in district dℓ there is a set of 2λ/m
+agents with preference ordering aℓ ≻ x ≻ [A\{aℓ, x}], a set of 2λ/m agents with preference ordering
+b1 ≻ x ≻ [A \ {b1, x}], . . ., and a set of 2λ/m agents with preference ordering bm/2−1 ≻ x ≻
+[A \ {bm/2−1, x}]. By Lemma 3.2, the representative of dℓ must be one of the top-ranked alternatives
+(otherwise the distortion of the mechanism would be infinite). Since aℓ is ranked above the other
+alternatives in the tie-breaking ordering, she chosen as the representative of dℓ. Hence, the set of
+representatives is {a1, . . . , ak}, and the winner is chosen according to some probability distribution
+over this set.
+Te valuation profile may be such that the 2λ/m agents in district dℓ that rank aℓ first have value
+1/m for all alternatives, while all other agents in dℓ have value 1/2 for their two favorite alterna-
+tives. Consequently, the social welfare of alternative aℓ is 2λ/m2, and thus the social welfare of the
+mechanism is also this much, no mater the probability distribution over the district representatives.
+In contrast, the social welfare of x is approximately kλ/2, leading to a distortion of Ω(km2).
+When randomization at the district level can be leveraged by ordinal distributed mechanisms, then
+we achieve distortion much beter than what is implied by Corollary 4.4, while also achieving strat-
+egyproofness. In particular, there are several centralized voting rules that can be implemented as
+distributed mechanisms, in the sense that they define the same probability distribution over the alter-
+natives. One such important class of voting rules is that of point-voting schemes, which is part of a
+larger class of strategyproof mechanisms [Barbera, 1978, Hylland, 1980, Gibbard, 1977] and includes
+rules with almost best possible distortion guarantees [Boutilier et al., 2015, Ebadian et al., 2022].
+4.1
+Point-voting schemes
+A point-voting scheme chooses an agent uniformly at random and then outputs her t-th favorite al-
+ternative with probability pt, where p1 ≥ . . . ≥ pm ≥ 0 and �m
+t=1 pt = 1. Hence, the probability
+according to which the point-voting scheme using the probability vector p = (p1, . . . , pm) chooses
+alternative a ∈ A as the winner w is Pr[w = a] = 1
+n
+�
+i∈N pσi(a), where σi(a) is the position that i
+ranks a in her preference ranking σ.
+Tere are many point-voting schemes of interest. For every positional scoring rule using the scor-
+ing vector s = (s1, . . . , sm), we can define a point-voting scheme f(s) by normalizing the scoring
+vector, that is, define pt = st/
+��
+j∈[m] sj
+�
+for every t ∈ [m] so that the winning probability of
+alternative a is
+Pr[w = a] = 1
+n
+�
+i∈N
+sσi(a)
+�
+j∈[m] sj
+=
+�
+i∈N sσi(a)
+n · �
+j∈[m] sj
+.
+Another important point-voting scheme is the rule that chooses each alternative uniformly at random;
+in this case, we have pt = 1/m for every t ∈ [m] so that Pr[w = a] = 1
+n
+�
+i∈N
+1
+m = 1
+m.
+For any point-voting scheme f that uses a probability vector p, we consider the distributed mech-
+anism Proportional-of-f-Point-Voting, which works as follows:
+• For every district d, choose the representative ad to be alternative a ∈ A with probability
+1
+λ
+�
+i∈Nd pσi(a).
+• Choose the winner to be the representative of district d with probability nd/n.
+11
+
+Teorem 4.8. Proportional-of-f-Point-Voting defines the same probability distribution as the point-
+voting scheme f.
+Proof. Te probabilitythat alternativea is chosen as the winner by Proportional-of-f-Point-Voting
+is
+Pr[w = a] =
+�
+d∈D
+Pr[w = ad] · Pr[ad = a]
+=
+�
+d∈D
+nd
+n · 1
+nd
+�
+i∈Nd
+pσi(a)
+= 1
+n
+�
+i∈N
+pσi(a),
+that is, Proportional-of-f-Point-Voting chooses a with the same probability as f.
+Teorem 4.8 shows that Proportional-of-f-Point-Voting achieves the same distortion bound
+as the point-voting scheme f it uses as the in-district rule, and also that it inherits its strategyproofness
+property. Tis is extremely useful, as there are centralized voting rules that are based on point-voting
+schemes and achieve almost the best possible distortion.
+Boutilier et al. [2015] considered a voting rule that is a convex combination of two point-voting
+schemes: With probability 1/2 choose an alternative uniformly at random, and with probability 1/2
+run the point-voting scheme defined by normalizingthe harmonic scoring rule H = (1, 1/2, . . . , 1/m).
+We will refer to this mechanism as BCHLPS. Boutilier et al. [2015] showed that this voting rule has
+distortion O(√m log m). An important property of point-voting schemes is that any rule that is a
+convex combination of point-voting schemes is also a point-voting scheme. Te following lemma is
+similar to lemmas proved before in the literature (e.g., see Filos-Ratsikas and Miltersen [2014], Barbera
+[1978]); we provide a proof for completeness.
+Lemma 4.9. Let f1, . . . , fκ be point-voting schemes defined by the probability vectors p1, . . . , pκ. For
+any non-negative numbers q1, . . . , qκ such that �
+j∈[κ] qj = 1, the voting rule f that chooses the outcome
+of fj with probability qj is a point-voting scheme.
+Proof. Let σ be an arbitrary preference profile. For any j ∈ [κ], denote the t-th coordinate of pj as pj,t,
+and let Pj(a) = Pr[a = fj(σ)] be the probability of choosing a as the winner according to point-voting
+scheme fj. Ten, the voting rule f chooses alternative a as the winner w with probability
+Pr[w = a] =
+�
+j∈[κ]
+qj · Pj(a)
+=
+�
+j∈[κ]
+qj ·
+�
+1
+n
+�
+i∈N
+pj,σi(a)
+�
+= 1
+n
+�
+i∈N
+�
+j∈[κ]
+qj · pj,σi(a).
+Hence, f is a point-voting scheme defined by the probability vector p with pt = �
+j∈[κ] qj · pj,t.
+Consequently, by Teorem 4.8 and Lemma 4.9, we can construct a randomized ordinal distributed
+mechanism based on the point-voting scheme of Boutilier et al. [2015] that achieves the same distortion
+bound and is strategyproof.
+12
+
+Corollary 4.10. Tere exists a randomized ordinal strategyproof distributed mechanism with distortion
+O(√m log m).
+Tis distortion bound is almost best possible as the lower bound of Ω(√m) for randomized cen-
+tralized rules holds trivially for distributed mechanisms by considering single-district instances.
+5
+Experiments
+In this section, we perform experiments with real and synthetic datasets, aiming to identify paterns in
+the distortion of several well-known voting rules and examine whether these support our theoretical
+findings. It is well-documented in the literature (e.g., see [Boutilier et al., 2015, Filos-Ratsikas et al.,
+2020]) that when working with real or realistic preferences, it ofen is the case that the distortions
+bounds are small numbers quite close to 1. In this sense, our goal is not primarily to demonstrate the
+distortion bounds themselves, but rather the dependence of these bounds on the distributed decision-
+making process, in particular the number of districts, as well as the use of randomization. We perform
+two main experiments, one with real-world preferences and valuation data, and one with synthetic
+data. All our experiments are with symmetric districts.
+5.1
+Experiments with the Jester Dataset
+For our first experiment, we use the Jester Joke Dataset [Goldberg et al., 2001]. Te dataset contains
+ratings for 100 different jokes in the range [−10, 10], provided by 70000 users. We chose to work
+with this dataset as it has also been employed by Boutilier et al. [2015] in the context of centralized
+distortion bounds, and also by Filos-Ratsikas et al. [2020] for the distortion of deterministic distributed
+mechanisms that use plurality as the over-district rule.
+Following the methodology developed in these works, we construct inputs consisting of ratings
+for the 8 most-rated jokes. In particular, we perform 1000 random runs in which we sample 100 users
+from the set of all users that have provided rankings for all eight jokes, and then partition them into
+k equal-sized districts uniformly at random, for k ∈ {1, 2, 5, 10, 20, 25}. Clearly, the case of k = 1
+corresponds to the centralized seting and will be used as a reference point. We interpret the ratings
+of the jokes as cardinal valuations: to be consistent with our seting (and with the experiments of
+[Boutilier et al., 2015, Filos-Ratsikas et al., 2020]), we add 10 to each user’s rating vector, to ensure that
+the values are positive and then apply the unit-sum normalization. For these inputs, we compute the
+average distortion of a set of 20 voting rules over the 1000 runs of the experiment. In particular, we
+consider distributed mechanisms fover-of-fin, where for fover we use Plurality or Uniform, whereas
+for fin we have:
+Deterministic Rules: We use simple voting scoring rules, namely Plurality (PL), Veto, Borda and
+Harmonic, as well as Range-Voting (RV), which in the case of k = 1 finds the optimal alternative.
+Randomized Rules: Here we use several natural point-voting schemes with probability vectors that
+are proportional to the aforementioned scoring rules (recall the definition from Section 4), namely
+• Proportional to Plurality Score (PropPL);
+• Proportional to Borda Score (PropBorda);
+• Proportional to Veto Score (PropVeto);
+• Proportional to Harmonic Score (PropHarmonic).
+13
+
+k
+RV
+PL
+Veto
+Borda
+Harmonic
+PropPL
+PropVeto
+PropBorda
+PropHarmonic
+BCHLPS
+1
+1
+1.049
+1.035
+1.007
+1.017
+1.135
+1.166
+1.155
+1.156
+1.166
+2
+1.017
+1.070
+1.059
+1.018
+1.020
+1.137
+1.166
+1.155
+1.156
+1.165
+5
+1.018
+1.064
+1.070
+1.020
+1.036
+1.133
+1.162
+1.155
+1.156
+1.165
+10
+1.019
+1.066
+1.082
+1.021
+1.044
+1.133
+1.162
+1.153
+1.154
+1.163
+20
+1.024
+1.066
+1.107
+1.030
+1.050
+1.134
+1.165
+1.154
+1.155
+1.164
+25
+1.022
+1.067
+1.142
+1.031
+1.107
+1.133
+1.165
+1.153
+1.154
+1.164
+Table 2: Distortion bounds of various voting rules based on valuations defined by the provided scores of the Jester dataset and random district partitions.
+RV
+PL
+Veto
+Borda
+Harmonic
+PropPL
+PropVeto
+PropBorda
+PropHarmonic
+BCHLPS
+k = 1
+Uniform
+1
+1.038
+1.045
+1.006
+1.019
+1.079
+1.087
+1.085
+1.085
+1.087
+Beta
+1
+1.086
+1.105
+1.029
+1.050
+1.140
+1.152
+1.147
+1.147
+1.150
+Exponential
+1
+1.032
+1.096
+1.018
+1.013
+1.118
+1.137
+1.132
+1.131
+1.134
+k = 2
+Uniform
+1.026
+1.052
+1.056
+1.030
+1.039
+1.079
+1.087
+1.084
+1.084
+1.086
+Beta
+1.044
+1.111
+1.118
+1.064
+1.080
+1.140
+1.152
+1.147
+1.147
+1.150
+Exponential
+1.039
+1.062
+1.115
+1.055
+1.051
+1.118
+1.136
+1.132
+1.130
+1.135
+k = 5
+Uniform
+1.031
+1.050
+1.057
+1.029
+1.038
+1.076
+1.084
+1.081
+1.081
+1.084
+Beta
+1.052
+1.113
+1.125
+1.074
+1.094
+1.143
+1.155
+1.151
+1.150
+1.154
+Exponential
+1.039
+1.069
+1.110
+1.055
+1.056
+1.119
+1.137
+1.133
+1.131
+1.134
+k = 20
+Uniform
+1.031
+1.055
+1.077
+1.039
+1.042
+1.077
+1.085
+1.082
+1.082
+1.084
+Beta
+1.055
+1.105
+1.145
+1.073
+1.084
+1.141
+1.154
+1.149
+1.149
+1.152
+Exponential
+1.047
+1.069
+1.123
+1.060
+1.058
+1.115
+1.133
+1.128
+1.127
+1.129
+k = 25
+Uniform
+1.031
+1.056
+1.071
+1.036
+1.044
+1.077
+1.085
+1.082
+1.0824
+1.084
+Beta
+1.054
+1.124
+1.149
+1.084
+1.094
+1.148
+1.155
+1.150
+1.150
+1.151
+Exponential
+1.042
+1.069
+1.129
+1.060
+1.054
+1.116
+1.134
+1.129
+1.128
+1.131
+Table 3: Distortion bounds of various voting rules based on valuations defined according to several probability distributions and random district
+partitions. Results for deterministic mechanisms are presented at the lef of the bold vertical line, and results for randomized mechanisms are at the
+right of the bold vertical line.
+
+We also use the rule of Boutilier et al. [2015] (we refer to it as BCHLPS in the following); recall that this
+is a point-voting scheme that with probability 1/2 selects an alternative at random and with probability
+1/2 runs the PropHarmonic rule defined above. As established in Corollary 4.10 (and the discussion
+before the statement of the corollary), this is best possible in terms of the worst-case distortion.
+Te results of our experiments can be seen in Table 2. In the table we only present the results
+where as fover, we used Plurality for deterministic rules and Uniform for randomized rules. Tis
+is in accordance to our approach in the theoretical results in previous sections. Te bounds for the
+cases not shown are quite similar, and slightly larger in general. For each of the randomized rules,
+we perform 300 runs and calculate their expected social welfare, which we then use to calculate the
+distortion.
+From the results of Table 2 we observe that, as expected, the existence of multiple districts has an
+adverse effect on the distortion of deterministic mechanisms, which becomes worse compared to the
+centralized case k = 1. For these rules, we can also observe that the distortion generally increases as k
+increases. In contrast, the distortion of randomized rules remains virtually unchanged for any value of
+k. Tis is in complete accordance with our theoretical findings, where we established that these rules
+induce the same probability distribution. Te experiments showcase that this does not only hold in
+expectation, but also in practice (given sufficiently many runs).
+Another crucial observation is that, in terms of the absolute distortion numbers, randomization
+does not seem to help; if anything, it makesthe distortion bounds worse! Tis can be justified by the fact
+that real-world instances like those from the Jester dataset display a large degree of homogeneity, which
+results in the simple deterministic rules performing quite well. On the other hand, randomization ofen
+leads to suboptimal choices even on such “well-behaved” instances, demeaning the distortion bounds
+on average. Surprisingly, among ordinal voting rules, Borda seems to perform best across the board
+even though the theoretical distortion of Borda is in fact unbounded.
+5.2
+Experiments with Synthetic Datasets
+We also perform experiments with datasets that are generated from probability distributions. In par-
+ticular, and to be consistent with the Jester experiment presented above, we create instances with
+100 agents and 8 alternatives, by first drawing the values of the agents from a certain distribution,
+and then constructing the induced ordinal preference profile from those values. We use the following
+distributions:
+• Uniform distribution in [1, 100]. Tis is the simplest case, where all possible values are equally
+likely.
+• Beta distribution with α = 1/10 and β = 1/10. Tis distribution has a symmetric convex pdf
+function centered around a mean of 1/2, assigning higher probabilities to values very close to 1
+or 0.
+• Exponential distribution with exponent 4, i.e., the pdf is f(x) = 4e4 for x ≥ 0 and f(x) = 0
+otherwise. Tis distribution generates values close to 0 with high probability, and as the values
+increase, the probability of them being generated decreases exponentially.
+For the rest of the experiment, we perform similar steps as in the case of the Jester dataset: We nor-
+malize the values to sum up to 1, and run the set of mechanisms described above. For each ran-
+domized mechanism we now perform 150 individual runs and calculate its expected welfare. We
+calculate the average distortions over 500 runs of the experiment for k symmetric districts, where
+k ∈ {1, 2, 5, 20, 25}. Note that the number of runs and the number of district sizes is slightly smaller
+15
+
+in this experiment, because it is more computationally intensive (as we need to calculate bounds for 3
+different distributions). Again, we use Plurality as fover for deterministic and Uniform for random-
+ized mechanisms; the results for the other cases were similar and are not reported.
+Te results can be found in Table 3. Similarly to the Jester experiment, it is evident that the distor-
+tion of the deterministic mechanisms becomes worse for k ≥ 2, whereas it remains prety much the
+same for randomized mechanisms. Again, we observe that randomization results in worse distortion
+bounds overall, and that Borda performs best among deterministic mechanisms. Interestingly, con-
+trary to the Jester dataset, here we do not see a clear patern of the distortion increasing as k increases
+for deterministic mechanisms (other than the jump from k = 1 to k = 2). Tis is probably due to the
+fact that the synthetic instances are highly homogeneous, and with uniform random district partitions,
+the districts end up being quite uniform, regardless of their number and size.
+Te role of unit-sum. We remark here that normalizing the values to sum up to 1 effectively makes the
+Uniform and Exponential distributions prety similar, and this is reflected in the results. To get a sense
+of the effect of normalization, we also ran the experiments without it. We observe that the distortions
+for the exponential distribution are now larger than those of the uniform distribution. In general, the
+distortion bounds still lie in the range [1.03, 1.15] for all distributions, but their average values (over
+all documented distortion bounds) are larger for all distributions except Uniform. It is also the case
+that for the Beta distribution, the bounds of deterministic mechanisms are much closer to those of
+randomized ones. Te distortion of randomized mechanisms is still almost the same for any number
+of districts.
+6
+Open Problems
+From our results, an interesting technical challenge is to remove the requirement for a consistent tie-
+breaking ordering from the statement of Teorem 4.7. Similarly, we could atempt to remove unanimity
+from the lower bound of Teorem 3.1; although unanimity is usually prety natural, removing it would
+make the theorem stronger. More interestingly, our result about point-voting schemes in Teorem 4.8
+crucially does not depend on the normalization of the valuations, and hence also could be applied
+verbatim to the metric distributed social choice seting studied by Anshelevich et al. [2022], where
+randomized mechanisms have never been considered; this seems like a natural starting point for such
+an investigation.
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+18
+

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+arXiv:2301.00978v1  [math.NT]  3 Jan 2023
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Abstract. Let K be either a locally compact non-discrete field
+of characteristic p > 2 or K = Qp, and Q be a non-degenerate
+isotropic quadratic form with coefficients in K. We obtain asymp-
+totic estimates for the number of solutions in the two fold product
+of certain discrete set inside K, of the inequalities of the form
+|Q(x, y)| < δ for some δ > 0, where | · | is an ultrametric abso-
+lute value on K. The estimates are obtained in terms of continued
+fraction expansions of the coefficients of the quadratic form Q.
+Mathematics Subject Classification: 11E16, 11E08, 11D88, 11A55,
+11J70, 11K50, 37A44.
+Keywords: Quadratic forms, locally compact fields, asymptotic esti-
+mates, continued fractions.
+Contents
+1.
+Introduction
+1
+2.
+K has positive characteristic (> 2)
+3
+3.
+K is the field of p-adic numbers
+10
+References
+14
+1. Introduction
+The Oppenheim conjecture, solved by Margulis in 1987 (see [13]
+for more details), states that if Q is a real non-degenerate indefinite
+quadratic form which is not proportional to a form with rational coeffi-
+cients, then Q(Zn) is dense in R if n ≥ 3. After Oppenheim conjecture
+was settled, people got interested in studying finer questions related to
+the distribution of the values of Q on integral points. Given a quadratic
+form as above, and a, b, ρ ∈ R with ρ > 0, let
+NQ(a, b, ρ) := # {v ∈ Zn : a < Q(v) < b, v ∈ B(ρ)},
+B(ρ) being the ball of radius ρ around the origin in Rn. Also let
+VQ(a, b, ρ) := Vol ({v ∈ Rn : a < Q(v) < b, v ∈ B(ρ)}).
+1
+
+2
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Then it was shown by Dani and Margulis in [7] that
+lim inf
+ρ→∞
+NQ(a, b, ρ)
+VQ(a, b, ρ) = 1.
+Asymptotic upper bound for the quantity NQ(a,b,ρ)
+VQ(a,b,ρ) was found by Eskin,
+Margulis and Mozes (see [9] for instance), and combining the result of
+[7], they showed that if Q is a quadratic form as above such that the
+signature of Q is neither (2, 1) nor (2, 2), then
+lim
+ρ→∞
+NQ(a, b, ρ)
+VQ(a, b, ρ) = 1.
+The Oppenheim conjecture fails for binary quadratic forms due to
+the existence of badly approximable numbers. A real number α is called
+badly approximable if there exists c > 0 such that
+���α − p
+q
+��� > c
+q2 for any
+rational number p
+q. Now, let Q be the binary quadratic form defined
+by
+Q(x, y) = (x + αy)y,
+α being a badly approximable number. Then Q(Z2) avoids the neigh-
+bourhood (−c, c) of zero. Nevertheless, one can study the distribution
+of the values taken by such forms at integral points. This was done
+in [6] with the interval (a, b) being a neighbourhood of 0.
+In case
+of binary quadratic forms, the asymptotic estimates depend on the
+quadratic form under consideration, and they are given in terms of
+the partial quotients of the continued fraction expansions of the coeffi-
+cients of the quadratic form. There is a natural connection between the
+values of non-degenerate indefinite binary quadratic forms at integral
+points, and certain geometric and dynamical aspects of the orbits of
+geodesic flow associated with the modular surface. In [6], the authors
+explored this connection, and used a method of coding of geodesics on
+the modular surface via nearest integer continued fraction which was
+introduced by S. Katok and I. Ugarcovicci (see [10] for instance), to
+obtain the estimates (see [18] for a different proof which does not uses
+the mechinary of geodesic flow etc.). The method of [6] can be adopted
+to obtain similar type of estimates in terms of a more general class of
+continued farctions as well, see Remark 3.4 of [5] for more details.
+In the present article, we do a similar study for non-degenerate
+isotropic binary quadratic forms whose coefficients are coming from
+a non-discrete locally compact field K such that either K has char-
+acteristic p > 2, or K is the field of p-adic numbers. In the following
+sections, we first deal with the positive characteristic case and then con-
+sider quadratic forms with coefficients in Qp. Note that an analogue
+of Oppenheim conjecture holds in S-arithmetic setting for isotropic
+quadratic forms in n ≥ 3 variables (see [2] for more details) as well.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+3
+2. K has positive characteristic (> 2)
+By the classification of non-discrete locally compact fields, if K is
+of positive characteristic, then K is the Laurent series fields in one
+indeterminate over a finite field. Let p be an odd prime, q be a power of
+p, and Fq be the finite field of characteristic p consisting of q elements.
+We denote by Z the polynomial ring Fq[X] in one variable over Fq.
+Let Fq(X) be the field of rational functions with coefficients in Fq and
+K := Fq((X−1)) be the field of formal Laurent series in X−1 over Fq.
+More precisely, if α ∈ Fq((X−1)), then
+α =
+�
+j≥n0
+ajX−j,
+aj ∈ Fq, n0 ∈ Z.
+Whenever α ∈ Fq((X−1))\Fq(X), we call α an irrational element. We
+define a valuation ν on K as follows: if α =
+�
+n≥n0
+anX−n, then
+ν(α) := inf {j ∈ Z : aj ̸= 0}.
+This valuation gives rise to an absolute value on K as follows: if α(̸=
+0) ∈ K and ν(α) = dα, then
+|α| := qdα,
+and the absolute value of the zero element in K is 0.
+Then K is
+the completion of Fq(X) with respect to this absolute value. As ν is
+a non-Archimedean valuation, the absolute value defined above is an
+ultrametric absolute value. Being a locally compact field, K admits a
+Haar measure (see [14] for details) which we denote by µ. For a ∈ K
+and r ∈ Z, let
+B(a, qr) := {α ∈ K : |α − a| < qr}
+be the open disc around a of radius qr, then µ(B(a, qr)) = qr. Let µ⊗µ
+be the corresponding product measure on K2 which is denoted by η.
+As in the case of real numbers, any α in K has a unique continued
+fraction expansion
+α = b0 +
+1
+b1 +
+1
+b2 +
+1
+b3 + ....
+,
+also written as
+α = [b0, b1, b2, ....]
+with bj ∈ Z for j ≥ 0 and bj has positive degree for j ≥ 1. Given any
+α =
+�
+j≥n0
+ajX−j in K, let
+⌊α⌋ =
+
+
+
+
+
+
+
+0
+�
+j=n0
+ajX−j
+if
+n0 ≤ 0
+0
+if
+n0 ≥ 1.
+
+4
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Then the continued fraction algorithm is defined as follows:
+α0 := α, αn+1 := (αn − bn)−1 and bn = ⌊αn⌋.
+Here bn’s are called partial quotients and αn’s are called complete quo-
+tients of the continued fraction expansion of α (see [16] for more de-
+tails).
+Now let sn
+tn be the nth convergent of the continued fraction
+expansion of α, i.e.,
+sn
+tn
+= [b0, b1, b2, ..., bn].
+Then the sequences (sn)n≥0 and (tn)n≥0 in Z satisfy the following re-
+currence relations:
+(1)
+sn = bnsn−1 + sn−2,
+tn = bntn−1 + tn−2.
+They also satisfy the following equation:
+(2)
+sn+1tn − sntn+1 = (−1)n
+which tells us that sn and tn are coprime, i.e., they do not have any
+common factor other than the constant polynomials in Fq[X]. The fol-
+lowing equalities which are special features of continued fraction theory,
+will be quite useful for this article. If α, bn, sn, tn are as above, then
+(3)
+|tn| = |bn · · · b1| ; ∀n ≥ 1,
+(4)
+����α − sn
+tn
+���� =
+1
+|bn+1||tn|2,
+and
+(5)
+����α − sn
+tn
+���� =
+1
+|tn+1||tn|.
+Note that in the case of continued fraction for real numbers, inequal-
+ities hold instead of equalities in (4) and (5). This is because of the
+ultrametric nature of the absolute value on K. The following lemma is
+a simple characterization of the convergents of the continued fraction
+expansion of any element in K, the proof of which can be found in [16].
+Lemma 1. Let s, t ∈ Z with t ̸= 0. Then s
+t is a convergent to α if and
+only if
+(6)
+����α − s
+t
+���� < 1
+|t|2.
+Now, let us consider binary quadratic forms with coefficients in K.
+It is well-known that if Q is a non-degenerate isotropic quadratic form
+with coefficients in a field F of characteristic not equal to 2, then there
+exists a basis {v1, v2} of F 2 such that if a1, a2 ∈ F, then
+Q(a1v1 + a2v2) = a1a2.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+5
+This says in particular that if Q0 is the quadratic from on K2 defined
+by
+Q0(x, y) = xy for x, y ∈ K,
+then for any isotropic quadratic form Q on K2, there is a matrix AQ
+in SL(2, K) and γ in K, such that
+(7)
+Q(x, y) = γ Q0(AQ(x, y)).
+So, to study the asymptotic behaviour of the set of values of an isotropic
+quadratic form with coefficients in K, it is enough to consider quadratic
+form Q given as follows:
+Q(x, y) = (ax + by)(cx + dy)
+with a, b, c, d ∈ K, bc − ad = 1.
+Now let Q be a quadratic form of the type Q(x, y) = (ax+by)(cx+dy)
+with a, b, c, d ∈ K, bc − ad = 1 (there is no loss of generality because
+one may replace γ by −γ in (7)) such that ba is an irrational element of
+K. Also let p be the set of primitive elements of Z2, i.e., p is the set of
+those (s, t) in Z2 such that s and t do not have a common factor except
+constant polynomials. For fixed real numbers k and δ with k > 1 and
+0 < δ < 1, let
+G(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)| < δ, ||(s, t)|| ≤ ρ, |cs + dt| > k},
+where ||(s, t)|| = max{|s|, |t|}.
+Let α = −ba and β = ac, and the
+continued fraction expansion of α be given by
+α = [b0, b1, b2, ...]
+with sn
+tn being the nth convergent. Also let
+H(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)| < δ, ||(x, y)|| ≤ ρ, |cx + dy| > k}.
+In this article, we find asymptotic lower and upper bound of the quo-
+tient # G(ρ)
+η (H(ρ)) as ρ → ∞. Now let
+α− := lim inf
+n→∞
+1
+n
+n
+�
+j=1
+log |bj|
+and
+α+ := lim sup
+n→∞
+1
+n
+n
+�
+j=1
+log |bj|.
+Also for 0 < δ < 1, let
+e(δ) := lim inf
+n→∞
+1
+n#
+�
+j, 1 ≤ j ≤ n : |bj+1| ≥ 1
+δ
+�
+and
+f(δ) := lim sup
+n→∞
+1
+n#
+�
+j, 1 ≤ j ≤ n : |bj+1| ≥ 1
+δ
+�
+.
+The main result of this article is contained in the following theorem.
+
+6
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Theorem 2. Let Q be a quadratic form defined by
+Q(x, y) = (ax + by)(cx + dy) with a, b, c, d ∈ K, bc − ad = 1,
+and ba an irrational element of K. Also let G(ρ), H(ρ), α+, α−, e(δ),
+f(δ) be as defined above. If α− < ∞, then we have the followings:
+lim inf
+ρ→∞
+# G(ρ)
+η (H(ρ)) ≥ c e(δ)
+α+
+and
+lim sup
+ρ→∞
+# G(ρ)
+η (H(ρ)) ≤ c f(δ)
+α− ,
+where c is a constant depending on δ and q.
+Remark 3. Let
+I(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)| < δ, ||(s, t)|| ≤ ρ, |as + bt| > k}
+and
+J(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)| < δ, ||(x, y)|| ≤ ρ, |ax+by| > k}.
+Then one can obtain a similar estimates for # I(ρ)
+η (J(ρ)) in terms of the
+continued fraction expansion of −dc provided dc is an irrational element
+of K.
+Proof of Theorem 2:
+Let
+G′(ρ) := {(s, t) ∈ p : |t(tα − s)| < δ, |t| ≤ ρ}.
+It is easy to see that
+(8)
+Q(s, t) = (tα − s)(t + β(tα − s)).
+If |Q(s, t)| < δ with |cs+dt| > k then |as+bt| < δ
+k, which implies that
+|tα − s| < δ|a|
+k , i.e., |tα − s| is bounded. Now by (8),
+|Q(s, t)|
+|q(tα − s)| =
+�����1 + β
+t (tα − s)
+����� .
+Since |tα − s| is bounded, it follows that
+|Q(s, t)|
+|t(tα − s)| = 1 if |t| is suffi-
+ciently large. Note that when |tα − s| is bounded, ||(s, t)|| → ∞ if and
+only if |t| → ∞. Also, if |q(tα−s)| < δ, then clearly |tα−s| is bounded
+and
+|Q(s, t)|
+|t(tα − s)| = 1 for sufficiently large |t|. Combining all these facts,
+we can say that there exists a constant C > 0 such that
+#G
+′(ρ) − C ≤ #G(ρ) ≤ #G
+′(ρ) + C
+for sufficiently large ρ. Since 0 < δ < 1, it follows from Lemma 1, that if
+(s, t) ∈ G
+′(ρ), then s = sj and t = tj, where sj
+tj is a convergent of α in its
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+7
+continued fraction expansion. Also G
+′(ρ) = G
+′(|tn|) if |tn| ≤ ρ < |tn+1|.
+Note that if (sj, tj) ∈ G
+′(|tn|), then (asj, atj) ∈ G
+′(|tn|) as well for any
+a ∈ F∗
+q.
+Now let us calculate the measure of H(ρ). Let A be the set given by
+A := {(x, y) ∈ K2 : 0 < |xy| < δ, ||(x, y)|| ≤ ρ, |y| > k},
+then
+η(H(ρ)) = |det(M)| η(A)
+where M =
+�
+a
+b
+c
+d
+�
+. Since bc−ad = 1, we have that η(H(ρ)) = η(A).
+Note that for 0 < δ < 1, k > 1 and ρ ≥ k, there exist unique
+m0, m
+′
+0, t and i ∈ Z such that qm0 ≤ δ < qm0+1, qm
+′
+0 ≤
+√
+δ <
+qm
+′
+0+1, qm
+′
+0+t ≤ k < qm
+′
+0+t+1 and qm
+′
+0+t+i ≤ ρ < qm
+′
+0+t+i+1. Also for
+1 ≤ n ≤ i, let
+An := {(x, y) ∈ K2 : |x| ≤ qm0−m
+′
+0−t−n and |y| = qm
+′
+0+t+n}.
+Clearly An’s are disjoint, and it is easy to see that A = ∪i
+n=1An. Hence,
+η(A) =
+i�
+n=1 η(An). Now
+{y ∈ K : |y| ≤ qm
+′
+0+t+n}
+= {y ∈ K : |y| < qm
+′
+0+t+n} ∪ {y ∈ K : |y| = qm
+′
+0+t+n}.
+Therefore,
+η(An) = µ({x ∈ K : |x| ≤ qm0−m
+′
+0−t−n}) · µ({y ∈ K : |y| = qm
+′
+0+t+n})
+= µ({x ∈ K : |x| ≤ qm0−m
+′
+0−t−n})
+· (µ({y ∈ K : |y| ≤ qm
+′
+0+t+n}) − µ({y ∈ K : |y| < qm
+′
+0+t+n}))
+= (qm0−m
+′
+0−t−n+1) · (qm
+′
+0+t+n+1 − qm
+′
+0+t+n)
+= (qm0−m
+′
+0−t−n+1)(qm
+′
+0+t+n)(q − 1)
+= qm0+1(q − 1),
+and consequently,
+η(H(ρ)) = η(A) =
+i
+�
+n=1
+η(An) = iqm0+1(q − 1).
+Since qm′
+0+t+i ≤ ρ < qm′
+0+t+i+1, it follows that
+(m′
+0 + t + i) log q ≤ log ρ < (m′
+0 + t + i + 1) log q
+which implies that
+log ρ
+log q − m′
+0 − t − 1 < i ≤ log ρ
+log q − m′
+0 − t.
+
+8
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Hence,
+�log ρ
+log q − m
+′
+0 − t − 1
+�
+(q − 1)qm0+1 < η(H(ρ)) ≤
+�log ρ
+log q − m
+′
+0 − t
+�
+(q − 1)qm0+1.
+(9)
+Now,
+lim inf
+ρ→∞
+#G(ρ)
+η(H(ρ)) ≥ lim inf
+ρ→∞
+#G′(ρ) − C
+η(H(ρ))
+= lim inf
+n→∞
+#G′(|tn|) − C
+η(H(|tn|))
+(for |tn| ≤ ρ < |tn+1|)
+= lim inf
+n→∞
+1
+n(#G′(|tn|) − C)
+1
+n(η(H(|tn|)))
+≥
+lim inf
+n→∞
+1n(#G′(|tn|))
+lim sup
+n→∞
+1n(η(H(|tn|)))
+≥
+lim inf
+n→∞
+1
+n(q − 1) #
+�
+j : 1 ≤ j ≤ n, |bj| ≥ 1
+δ
+�
+lim sup
+n→∞
+1
+n
+�log |tn|
+log q
+− m′
+0 − t
+�
+qm0+1(q − 1)
+(by (4) and (9))
+≥
+lim inf
+n→∞
+1
+n #
+�
+j : 1 ≤ j ≤ n, |bj| ≥ 1
+δ
+�
+lim sup
+n→∞
+1
+n
+�log |b1b2 · · · bn|
+log q
+− m′
+0 − t
+�
+qm0+1
+(by (3))
+≥
+lim inf
+n→∞
+1
+n #
+�
+j : 1 ≤ j ≤ n, |bj| ≥ 1
+δ
+�
+lim sup
+n→∞
+1
+n
+
+
+
+
+
+n�
+j=1 log |bj|
+log q
+− m′
+0 − t
+
+
+
+
+ qm0+1
+= e(δ)
+α+
+log q
+qm0+1.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+9
+A similar calculation yields
+lim sup
+ρ→∞
+#G(ρ)
+η(H(ρ)) ≤ f(δ)
+α−
+log q
+qm0+1.
+Corollary 4. Let Q be a quadratic form as in Theorem 2, and 0 < δ <
+1 be fixed. Then there exist a subset K′ of K with µ(K′) = µ(K) such
+that if α = −ba ∈ K′, then
+lim
+ρ→∞
+#G(ρ)
+η(H(ρ)) =
+q − 1
+q⌈δ−1⌉+m0+1,
+where ⌈δ−1⌉ denotes the smallest integer greater or equal to δ−1.
+Proof. Let [b0, b1, b2, . . .] be the continued fraction expansion of α =
+−ba as above.
+It follows from Theorem 6 of [1] that there is a full
+measure subset K′ of K such that if α = −ba ∈ K′, then
+(10)
+lim
+n→∞ |b1b2 · · · bn| 1n = q
+q
+q − 1.
+This implies that
+lim
+n→∞
+1
+n
+n
+�
+j=1
+log |bj| =
+q
+q − 1 log q,
+and, therefore, α− = α+ =
+q
+q−1 log q. Also for any 0 < δ < 1, there
+exists a unique l ∈ N such that l = ⌈δ−1⌉. Then by Theorem 14 of
+[12], for α in a full measure set which without loss of generality we may
+assume to be K′,
+lim
+n→∞
+1
+n #{1 ⩽ j ⩽ n : |bj| ⩾ ql} =
+1
+ql−1
+which implies that e(δ) = f(δ) =
+1
+ql−1 =
+1
+q⌈δ−1⌉−1. Then it follows from
+Theorem 2 above that, if α = −ba ∈ K
+′, then
+lim
+ρ→∞
+#G(ρ)
+η(H(ρ)) =
+1
+q⌈δ−1⌉ − 1
+�
+q
+q − 1 log q
+� log q
+qm0+1
+=
+q − 1
+q⌈δ−1⌉ + m0 + 1.
+□
+Remark 5. Let Q, α be as in Theorem 2. Now, if the absolute values
+of the partial quotients in the continued fraction expansion of α are
+
+10
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+bounded by some real numbers, then it is easy to see that e(δ) = f(δ) =
+0 if δ is sufficiently small. In this case,
+lim
+ρ→∞
+#G(ρ)
+η(H(ρ)) = 0.
+3. K is the field of p-adic numbers
+In this section, we consider isotropic quadratic forms with coefficients
+in the field of p-adic numbers for a prime p. Recall that the field of
+p-adic numbers, denoted by Qp, is the collection of all formal series of
+the form
+�
+j≥n0
+ajpj, with n0 ∈ Z and aj ∈ {0, 1, . . ., p − 1}.
+The ultrametric absolute value on Qp is defined as follows: if
+α (̸= 0) =
+�
+j≥n0
+ajpj,
+then
+|α|p := p−νp(α), and |0|p = 0,
+where νp(α) := inf {j ∈ Z : aj ̸= 0}. The integer νp(α) is also known
+as the valuation of α. For α ∈ Qp and r ∈ Z, let
+B(a, pr) := {α ∈ K : |α − a|p < pr}
+be the open disc of radius pr around the point α. The Haar measure µ
+(say) on Qp is defined in such a way that µ(B(a, pr)) = pr. We denote
+by η again the product measure µ ⊗ µ on Qp × Qp.
+As in the case of real numbers and elements of Laurent series fields
+over finite fields, continued fraction expansion exists for p-adic num-
+bers as well. There are mainly two types of continued fractions for
+p-adic numbers, one of them was introduced by Schneider (see [17] for
+instance), and the other one was introduced by Ruban (see [15] for
+instance) and modified later by Brokwin (see [3], [4]). In this article,
+we are going to consider the continued fraction introduced by Ruban
+which has some similarity with the simple continued fraction for real
+numbers. From now on, unless otherwise stated, we will be considering
+Ruban’s continued fraction only. Let Z be the subset of Qp given by
+Z := {a0 + a1
+1
+p + . . . an
+1
+pn : ai ∈ {0, 1, . . ., p − 1} for 0 ≤ i ≤ n}.
+It is easy to see that Z is a discrete set in the topology coming from
+the p-adic abosolute value. For α (̸= 0) = �
+j≥n0
+ajpj, let
+⌊α⌋ =
+
+
+
+
+
+
+
+0
+�
+j=n0
+ajpj
+if
+n0 ≤ 0
+0
+if
+n0 ≥ 1.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+11
+Given α ∈ Qp, we define two sequences (αn) and (bn) as follows: α0 =
+α, b0 = ⌊α0⌋; for n ≥ 0, if bn = αn, then αn+1 and bn+1 are not defined,
+otherwise, αn+1 = (αn − bn)−1 and bn+1 = ⌊αn+1⌋. Any p-adic number
+α has a unique continued fraction expansion as α = [b0, b1, . . . , bn, . . . ]
+which can be obtained by using the algorithm discussed above. Note
+that the partial quotients bn’s are elements of Z. The nth convergent
+is given by sn
+tn = [b0, b1, . . . , bn] where sn and tn satisfy the recurrence
+relation as in (1), and equation (2) as well.
+The p-adic versions of
+equation (3), (4) and (5) are valid as well with the absolute value in
+the Laurent series field replaced by the p-adic absolute value. As we
+could not find a proper reference for a p-adic version of Lemma 1, we
+include a proof here following the proof of Lemma 1 given in [16].
+Lemma 6. Let s, t ∈ Z with t ̸= 0. Then s
+t is a convergent to α if and
+only if
+(11)
+����α − s
+t
+����
+p < 1
+|t|2p
+Proof. By the p-adic version of equation (4),
+����α − sn
+tn
+����
+p
+=
+1
+|bn+1|p |tn|2
+p
+<
+1
+|tn|2
+p
+for any convergent sn
+tn corresponding to the continued fraction expan-
+sion of α.
+Conversely, assume that s, t ∈ Z with t ̸= 0 such that
+����α − s
+t
+����
+p < 1
+|t|2
+p
+.
+There is a unique n such that |tn|p ≤ |t|p < |tn+1|p. Then
+����α − s
+t
+����
+p <
+1
+|t|p|tn|p
+,
+and
+����α − sn
+tn
+����
+p
+=
+1
+|tn|p|tn+1|p
+(by p-adic version of (5))
+<
+1
+|t|p|tn|p
+,
+so that
+����
+s
+t − sn
+tn
+����
+p
+=
+����
+s
+t − α + α − sn
+tn
+����
+p
+≤ max
+�����α − s
+t
+����
+p ,
+����α − sn
+tn
+����
+p
+�
+<
+1
+|t|p|tn|p
+.
+
+12
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+Thus, s
+t = sn
+tn .
+□
+Now, let Q be a non-degenerate isotropic binary quadratic form with
+coefficients in Qp. Since Qp has characteristic zero, as explained in the
+previous section, it is enough to consider Q defined by
+Q(x, y) = (ax + by)(cx + dy)
+with a, b, c, d in Qp and bc−ad = 1. We also assume that ba is not of the
+form s
+t for some s, t ∈ Z with t ̸= 0. Let p denote the set of all those
+(s, t) ∈ Z such that s and t does not have a common factor except the
+constant polynomials in 1p inside Z. For k > 1 and 0 < δ < 1, we
+define G(ρ) and H(ρ) as in the previous section as follows:
+G(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)|p < δ, ||(s, t)|| ≤ ρ, |cs + dt|p > k},
+H(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)|p < δ, ||(x, y)|| ≤ ρ, |cx+dy|p > k},
+here ||(s, t)|| = max { |s|p, |t|p }. Also let α = −ba and β = ac, and the
+continued fraction expansion of α be given by
+α = [b0, b1, b2, ...].
+The quantities e(δ), f(δ), α− and α+ are defined similarly as in the
+previous section with the absolute value replaced by the p-adic absolute
+value wherever applicable. Then an analogue of Theorem 2 holds in
+this set up as well.
+Theorem 7. With all the notations as above, if α− < ∞, then
+lim inf
+ρ→∞
+#G(ρ)
+η(H(ρ)) ≥ c e(δ)
+α+ ,
+and
+lim sup
+ρ→∞
+#G(ρ)
+η(H(ρ)) ≤ c f(δ)
+α− ,
+where c =
+log p
+pm0 + 1.
+Let X = B(0, 1) and T : X → X be the continued fraction map
+defined by
+T(α) = 1
+α −
+� 1
+α
+�
+.
+It is known that the map T is ergodic (see [15] for details) with respect
+to the Haar measure µ. As an application of the ergodicity, we obtain
+a result similar to Theorem 14 of [12].
+Lemma 8. Let α ∈ X and [0, b1, b2, . . . ] be the continued fraction
+expansion of α. Then for any natural number l,
+lim
+n→∞ #{1 ≤ j ≤ n : −νp(bj) ≥ l} =
+1
+pl−1
+almost everywhere with respect to the Haar measure µ.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+13
+Proof. Note that b1 = b1(α) can be thought of as a function on B(0, 1).
+Then it is easy to check that the function
+f(α) = χ[pl,∞)(|b1(α)|p), α ∈ B(0, 1)
+is integrable on B(0, 1). Now, by the pointwise ergodic theorem
+(see Theorem 2.30 of [8] for instance),
+lim
+n→∞
+1
+n#{1 ≤ j ≤ n : −νp(bj) ≥ l} = lim
+n→∞
+1
+n#{1 ≤ j ≤ n : |bj|p ≥ pl}
+= lim
+n→∞
+1
+n
+n
+�
+j=1
+χ[pl,∞)(|b1(T j(α))|p)
+=
+�
+B(0,1)
+χ[pl,∞)(|b1(α)|p)dµ
+= µ{α ∈ B(0, 1) : |b1(α)|p ≥ pl}
+= µ{α ∈ B(0, 1) : |α|p ≤ p−l}
+= p−l+1
+=
+1
+pl−1
+□
+Now, using Theorem 8 of [15] and Lemma 8 above, we obtain a p-adic
+version of Corollary 4.
+Corollary 9. Let Q be a quadratic form as in Theorem 7, and 0 < δ <
+1 be fixed. Then there exist a subset K
+′ of K with µ(K
+′) = µ(K) such
+that if α = −ba ∈ K
+′, then
+lim
+ρ→∞
+#G(ρ)
+η(H(ρ)) =
+p − 1
+p⌈δ−1⌉+m0+1.
+It is easy to see that a version of Remark 5 is true in the p-adic set
+up as well. As the statements are similar, we do not write it separately
+here. Rather, we give an example of a p-adic number whose continued
+fraction expansion consists of partial quotients with bounded absolute
+values. One may look at [11] and references cited there in for similar
+examples in Laurent series field over finite fields. Let α be the p-adic
+number given by α =
+�
+j≥−1 ajpj, with aj = 1 for all j ≥ −1. Let the
+continued fraction expansion of α be [b0, b1, b2, ...]. Then b0 = p0 + p−1
+
+14
+MANOJ CHOUDHURI AND PRASHANT J. MAKADIYA
+and |b0|p = p. Now
+α1 = (α0 − b0)−1
+=
+
+�
+j≥1
+pj
+
+
+−1
+= p−1 +
+�
+j≥0
+(p − 1)pj.
+Then b1 = (p − 1)p0 + p−1 and |b1|p = p. Again
+α2 = (α1 − b1)−1
+=
+
+�
+j≥1
+(p − 1)pj
+
+
+−1
+=
+�
+j≥−1
+(p − 1)pj.
+Then b2 = (p − 1)p0 + (p − 1)p−1 and |b2|p = p. Observe that α3 =
+(α2 − b2)−1 = α2, and hence |b3|p = p. In a similar manner we get
+αn+1 = αn, bn+1 = bn, |bn+1|p = p for n ≥ 3 as well. Therefore, the
+absolute values of all the partial quotients of the continued fraction
+expansion of α are bounded by p.
+Remark 10. As in the case of binary real quadratic forms, the Op-
+penheim conjecture fails to hold for non-degenerate isotropic quadratic
+form with coefficients in a non-discrete locally compact non-Archimedean
+field as well. To see this, let us consider the quadratic form Q given by
+Q(x, y) = (x + αy)y
+with α ∈ Fq((X−1)) (or Qp). Now if the partial quotients in the con-
+tinued fraction expansion of α have bounded absolute values, then us-
+ing Lemma 1 (or Lemma 6), it is easy to see that the set of values
+{|Q(s, t)| : s, t ∈ Z} (Z is either as in Section 1 or as in Section 2)
+avoids certain neighbourhood of zero.
+Acknowledgement . Prashant J. Makadiya acknowledges the support
+of Government of Gujarat thorugh the SHODH (ScHeme Of Developing
+High Quality Research) fellowship. Manoj Choudhuri thanks L. Singhal
+for helpful discussions.
+References
+[1] Val´erie Berth´e and Hitoshi Nakada. On continued fraction expansions in pos-
+itive characteristic: equivalence relations and some metric properties. Expo.
+Math., 18(4):257–284, 2000.
+[2] Armand Borel and Gopal Prasad. Values of isotropic quadratic forms at S-
+integral points. Compositio Math., 83(3):347–372, 1992.
+
+ON VALUES OF ISOTROPIC QUADRATIC FORMS
+15
+[3] Jerzy Browkin. Continued fractions in local fields. i. Demonstratio Math.,
+11(1):67–82, 1978.
+[4] Jerzy
+Browkin.
+Continued
+fractions
+in
+local
+fields.
+ii.
+Math.
+Comp.,
+70(235):1281–1292, 2001.
+[5] Manoj Choudhuri. On certain orbits of geodesic flow and (a, b)-continued frac-
+tions. Proc. Indian Acad. Sci. Math. Sci., 131(1):Paper No. 2, 19, 2021.
+[6] Manoj Choudhuri and S. G. Dani. On values of binary quadratic forms at
+integer points. Math. Res. Lett., 22(4):1023–1045, 2015.
+[7] S. G. Dani and G. A. Margulis. Limit distributions of orbits of unipotent flows
+and values of quadratic forms. In I. M. Gelfand Seminar, volume 16 of Adv.
+Soviet Math., pages 91–137. Amer. Math. Soc., Providence, RI, 1993.
+[8] Manfred Einsiedler and Thomas Ward. Ergodic theory with a view towards
+number theory, volume 259 of Graduate Texts in Mathematics. Springer-Verlag
+London, Ltd., London, 2011.
+[9] Alex Eskin, Gregory Margulis, and Shahar Mozes. On a quantitative ver-
+sion of the Oppenheim conjecture. Electron. Res. Announc. Amer. Math. Soc.,
+1(3):124–130, 1995.
+[10] Svetlana Katok and Ilie Ugarcovici. Arithmetic coding of geodesics on the
+modular surface via continued fractions. In European women in mathematics—
+Marseille 2003, volume 135 of CWI Tract, pages 59–77. Centrum Wisk. In-
+form., Amsterdam, 2005.
+[11] Alain Lasjaunias and Jean-Jacques Ruch. Algebraic and badly approximable
+power series over a finite field. Finite Fields Appl., 8(1):91–107, 2002.
+[12] Poj Lertchoosakul and Radhakrishnan Nair. On the metric theory of continued
+fractions in positive characteristic. Mathematika, 60(2):307–320, 2014.
+[13] G. A. Margulis. Oppenheim conjecture. In Fields Medallists’ lectures, volume 5
+of World Sci. Ser. 20th Century Math., pages 272–327. World Sci. Publ., River
+Edge, NJ, 1997.
+[14] Dinakar Ramakrishnan and Robert J. Valenza. Fourier analysis on number
+fields, volume 186 of Graduate Texts in Mathematics. Springer-Verlag, New
+York, 1999.
+[15] A. A. Ruban. Certain metric properties of the p-adic numbers. Sibirsk. Mat.
+ˇZ., 11:222–227, 1970.
+[16] Wolfgang M. Schmidt. On continued fractions and Diophantine approximation
+in power series fields. Acta Arith., 95(2):139–166, 2000.
+[17] Th. Schneider. ¨Uber p-adische Kettenbr¨uche. In Symposia Mathematica, Vol.
+IV (INDAM, Rome, 1968/69), pages 181–189. Academic Press, London, 1970.
+[18] David Simmons. The Hurwitz continued fraction expansion as applied to real
+numbers. Enseign. Math., 62(3-4):475–485, 2016.
+Institute of Infrastructure, Technology, Research and Manage-
+ment, Near Khokhara Circle, maninagar (East), Ahmedabad 380026,
+Gujarat, India.
+Email address: [email protected]
+Email address: [email protected]
+

19AzT4oBgHgl3EQfDfqo/content/tmp_files/load_file.txt

@@ -0,0 +1,382 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf,len=381
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='00978v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='NT]  3 Jan 2023  ON VALUES OF ISOTROPIC QUADRATIC FORMS  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let K be either a locally compact non-discrete field  of characteristic p > 2 or K = Qp, and Q be a non-degenerate  isotropic quadratic form with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' We obtain asymp-  totic estimates for the number of solutions in the two fold product  of certain discrete set inside K, of the inequalities of the form  |Q(x, y)| < δ for some δ > 0, where | · | is an ultrametric abso-  lute value on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The estimates are obtained in terms of continued  fraction expansions of the coefficients of the quadratic form Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Mathematics Subject Classification: 11E16, 11E08, 11D88, 11A55,  11J70, 11K50, 37A44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Keywords: Quadratic forms, locally compact fields, asymptotic esti-  mates, continued fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  K has positive characteristic (> 2)  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  K is the field of p-adic numbers  10  References  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Introduction  The Oppenheim conjecture, solved by Margulis in 1987 (see [13]  for more details), states that if Q is a real non-degenerate indefinite  quadratic form which is not proportional to a form with rational coeffi-  cients, then Q(Zn) is dense in R if n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' After Oppenheim conjecture  was settled, people got interested in studying finer questions related to  the distribution of the values of Q on integral points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Given a quadratic  form as above, and a, b, ρ ∈ R with ρ > 0, let  NQ(a, b, ρ) := # {v ∈ Zn : a < Q(v) < b, v ∈ B(ρ)},  B(ρ) being the ball of radius ρ around the origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also let  VQ(a, b, ρ) := Vol ({v ∈ Rn : a < Q(v) < b, v ∈ B(ρ)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  1  2  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Then it was shown by Dani and Margulis in [7] that  lim inf  ρ→∞  NQ(a, b, ρ)  VQ(a, b, ρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Asymptotic upper bound for the quantity NQ(a,b,ρ)  VQ(a,b,ρ) was found by Eskin,  Margulis and Mozes (see [9] for instance), and combining the result of  [7], they showed that if Q is a quadratic form as above such that the  signature of Q is neither (2, 1) nor (2, 2), then  lim  ρ→∞  NQ(a, b, ρ)  VQ(a, b, ρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  The Oppenheim conjecture fails for binary quadratic forms due to  the existence of badly approximable numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' A real number α is called  badly approximable if there exists c > 0 such that  ���α − p  q  ��� > c  q2 for any  rational number p  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now, let Q be the binary quadratic form defined  by  Q(x, y) = (x + αy)y,  α being a badly approximable number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then Q(Z2) avoids the neigh-  bourhood (−c, c) of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Nevertheless, one can study the distribution  of the values taken by such forms at integral points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' This was done  in [6] with the interval (a, b) being a neighbourhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  In case  of binary quadratic forms, the asymptotic estimates depend on the  quadratic form under consideration, and they are given in terms of  the partial quotients of the continued fraction expansions of the coeffi-  cients of the quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' There is a natural connection between the  values of non-degenerate indefinite binary quadratic forms at integral  points, and certain geometric and dynamical aspects of the orbits of  geodesic flow associated with the modular surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In [6], the authors  explored this connection, and used a method of coding of geodesics on  the modular surface via nearest integer continued fraction which was  introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Katok and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Ugarcovicci (see [10] for instance), to  obtain the estimates (see [18] for a different proof which does not uses  the mechinary of geodesic flow etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The method of [6] can be adopted  to obtain similar type of estimates in terms of a more general class of  continued farctions as well, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='4 of [5] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  In the present article, we do a similar study for non-degenerate  isotropic binary quadratic forms whose coefficients are coming from  a non-discrete locally compact field K such that either K has char-  acteristic p > 2, or K is the field of p-adic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In the following  sections, we first deal with the positive characteristic case and then con-  sider quadratic forms with coefficients in Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Note that an analogue  of Oppenheim conjecture holds in S-arithmetic setting for isotropic  quadratic forms in n ≥ 3 variables (see [2] for more details) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' K has positive characteristic (> 2)  By the classification of non-discrete locally compact fields, if K is  of positive characteristic, then K is the Laurent series fields in one  indeterminate over a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let p be an odd prime, q be a power of  p, and Fq be the finite field of characteristic p consisting of q elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  We denote by Z the polynomial ring Fq[X] in one variable over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Let Fq(X) be the field of rational functions with coefficients in Fq and  K := Fq((X−1)) be the field of formal Laurent series in X−1 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  More precisely, if α ∈ Fq((X−1)), then  α =  �  j≥n0  ajX−j,  aj ∈ Fq, n0 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Whenever α ∈ Fq((X−1))\\Fq(X), we call α an irrational element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' We  define a valuation ν on K as follows: if α =  �  n≥n0  anX−n, then  ν(α) := inf {j ∈ Z : aj ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  This valuation gives rise to an absolute value on K as follows: if α(̸=  0) ∈ K and ν(α) = dα, then  |α| := qdα,  and the absolute value of the zero element in K is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then K is  the completion of Fq(X) with respect to this absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' As ν is  a non-Archimedean valuation, the absolute value defined above is an  ultrametric absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Being a locally compact field, K admits a  Haar measure (see [14] for details) which we denote by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' For a ∈ K  and r ∈ Z, let  B(a, qr) := {α ∈ K : |α − a| < qr}  be the open disc around a of radius qr, then µ(B(a, qr)) = qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let µ⊗µ  be the corresponding product measure on K2 which is denoted by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  As in the case of real numbers, any α in K has a unique continued  fraction expansion  α = b0 +  1  b1 +  1  b2 +  1  b3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='.  ,  also written as  α = [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='.]  with bj ∈ Z for j ≥ 0 and bj has positive degree for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Given any  α =  �  j≥n0  ajX−j in K, let  ⌊α⌋ =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  0  �  j=n0  ajX−j  if  n0 ≤ 0  0  if  n0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  4  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Then the continued fraction algorithm is defined as follows:  α0 := α, αn+1 := (αn − bn)−1 and bn = ⌊αn⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Here bn’s are called partial quotients and αn’s are called complete quo-  tients of the continued fraction expansion of α (see [16] for more de-  tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Now let sn  tn be the nth convergent of the continued fraction  expansion of α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=',  sn  tn  = [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', bn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then the sequences (sn)n≥0 and (tn)n≥0 in Z satisfy the following re-  currence relations:  (1)  sn = bnsn−1 + sn−2,  tn = bntn−1 + tn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  They also satisfy the following equation:  (2)  sn+1tn − sntn+1 = (−1)n  which tells us that sn and tn are coprime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', they do not have any  common factor other than the constant polynomials in Fq[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The fol-  lowing equalities which are special features of continued fraction theory,  will be quite useful for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' If α, bn, sn, tn are as above, then  (3)  |tn| = |bn · · · b1| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' ∀n ≥ 1,  (4)  ����α − sn  tn  ���� =  1  |bn+1||tn|2,  and  (5)  ����α − sn  tn  ���� =  1  |tn+1||tn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Note that in the case of continued fraction for real numbers, inequal-  ities hold instead of equalities in (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' This is because of the  ultrametric nature of the absolute value on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The following lemma is  a simple characterization of the convergents of the continued fraction  expansion of any element in K, the proof of which can be found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let s, t ∈ Z with t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then s  t is a convergent to α if and  only if  (6)  ����α − s  t  ���� < 1  |t|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Now, let us consider binary quadratic forms with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It is well-known that if Q is a non-degenerate isotropic quadratic form  with coefficients in a field F of characteristic not equal to 2, then there  exists a basis {v1, v2} of F 2 such that if a1, a2 ∈ F, then  Q(a1v1 + a2v2) = a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  5  This says in particular that if Q0 is the quadratic from on K2 defined  by  Q0(x, y) = xy for x, y ∈ K,  then for any isotropic quadratic form Q on K2, there is a matrix AQ  in SL(2, K) and γ in K, such that  (7)  Q(x, y) = γ Q0(AQ(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  So, to study the asymptotic behaviour of the set of values of an isotropic  quadratic form with coefficients in K, it is enough to consider quadratic  form Q given as follows:  Q(x, y) = (ax + by)(cx + dy)  with a, b, c, d ∈ K, bc − ad = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Now let Q be a quadratic form of the type Q(x, y) = (ax+by)(cx+dy)  with a, b, c, d ∈ K, bc − ad = 1 (there is no loss of generality because  one may replace γ by −γ in (7)) such that ba is an irrational element of  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also let p be the set of primitive elements of Z2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', p is the set of  those (s, t) in Z2 such that s and t do not have a common factor except  constant polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' For fixed real numbers k and δ with k > 1 and  0 < δ < 1, let  G(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)| < δ, ||(s, t)|| ≤ ρ, |cs + dt| > k},  where ||(s, t)|| = max{|s|, |t|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Let α = −ba and β = ac, and the  continued fraction expansion of α be given by  α = [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=']  with sn  tn being the nth convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also let  H(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)| < δ, ||(x, y)|| ≤ ρ, |cx + dy| > k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  In this article, we find asymptotic lower and upper bound of the quo-  tient # G(ρ)  η (H(ρ)) as ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now let  α− := lim inf  n→∞  1  n  n  �  j=1  log |bj|  and  α+ := lim sup  n→∞  1  n  n  �  j=1  log |bj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Also for 0 < δ < 1, let  e(δ) := lim inf  n→∞  1  n#  �  j, 1 ≤ j ≤ n : |bj+1| ≥ 1  δ  �  and  f(δ) := lim sup  n→∞  1  n#  �  j, 1 ≤ j ≤ n : |bj+1| ≥ 1  δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  The main result of this article is contained in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  6  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let Q be a quadratic form defined by  Q(x, y) = (ax + by)(cx + dy) with a, b, c, d ∈ K, bc − ad = 1,  and ba an irrational element of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also let G(ρ), H(ρ), α+, α−, e(δ),  f(δ) be as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' If α− < ∞, then we have the followings:  lim inf  ρ→∞  # G(ρ)  η (H(ρ)) ≥ c e(δ)  α+  and  lim sup  ρ→∞  # G(ρ)  η (H(ρ)) ≤ c f(δ)  α− ,  where c is a constant depending on δ and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let  I(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)| < δ, ||(s, t)|| ≤ ρ, |as + bt| > k}  and  J(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)| < δ, ||(x, y)|| ≤ ρ, |ax+by| > k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then one can obtain a similar estimates for # I(ρ)  η (J(ρ)) in terms of the  continued fraction expansion of −dc provided dc is an irrational element  of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Proof of Theorem 2:  Let  G′(ρ) := {(s, t) ∈ p : |t(tα − s)| < δ, |t| ≤ ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It is easy to see that  (8)  Q(s, t) = (tα − s)(t + β(tα − s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  If |Q(s, t)| < δ with |cs+dt| > k then |as+bt| < δ  k, which implies that  |tα − s| < δ|a|  k , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', |tα − s| is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now by (8),  |Q(s, t)|  |q(tα − s)| =  �����1 + β  t (tα − s)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Since |tα − s| is bounded, it follows that  |Q(s, t)|  |t(tα − s)| = 1 if |t| is suffi-  ciently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Note that when |tα − s| is bounded, ||(s, t)|| → ∞ if and  only if |t| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also, if |q(tα−s)| < δ, then clearly |tα−s| is bounded  and  |Q(s, t)|  |t(tα − s)| = 1 for sufficiently large |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Combining all these facts,  we can say that there exists a constant C > 0 such that  #G  ′(ρ) − C ≤ #G(ρ) ≤ #G  ′(ρ) + C  for sufficiently large ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Since 0 < δ < 1, it follows from Lemma 1, that if  (s, t) ∈ G  ′(ρ), then s = sj and t = tj, where sj  tj is a convergent of α in its  ON VALUES OF ISOTROPIC QUADRATIC FORMS  7  continued fraction expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also G  ′(ρ) = G  ′(|tn|) if |tn| ≤ ρ < |tn+1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Note that if (sj, tj) ∈ G  ′(|tn|), then (asj, atj) ∈ G  ′(|tn|) as well for any  a ∈ F∗  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Now let us calculate the measure of H(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let A be the set given by  A := {(x, y) ∈ K2 : 0 < |xy| < δ, ||(x, y)|| ≤ ρ, |y| > k},  then  η(H(ρ)) = |det(M)| η(A)  where M =  �  a  b  c  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Since bc−ad = 1, we have that η(H(ρ)) = η(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Note that for 0 < δ < 1, k > 1 and ρ ≥ k, there exist unique  m0, m  ′  0, t and i ∈ Z such that qm0 ≤ δ < qm0+1, qm  ′  0 ≤  √  δ <  qm  ′  0+1, qm  ′  0+t ≤ k < qm  ′  0+t+1 and qm  ′  0+t+i ≤ ρ < qm  ′  0+t+i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also for  1 ≤ n ≤ i, let  An := {(x, y) ∈ K2 : |x| ≤ qm0−m  ′  0−t−n and |y| = qm  ′  0+t+n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Clearly An’s are disjoint, and it is easy to see that A = ∪i  n=1An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Hence,  η(A) =  i�  n=1 η(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now  {y ∈ K : |y| ≤ qm  ′  0+t+n}  = {y ∈ K : |y| < qm  ′  0+t+n} ∪ {y ∈ K : |y| = qm  ′  0+t+n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Therefore,  η(An) = µ({x ∈ K : |x| ≤ qm0−m  ′  0−t−n}) · µ({y ∈ K : |y| = qm  ′  0+t+n})  = µ({x ∈ K : |x| ≤ qm0−m  ′  0−t−n})  (µ({y ∈ K : |y| ≤ qm  ′  0+t+n}) − µ({y ∈ K : |y| < qm  ′  0+t+n}))  = (qm0−m  ′  0−t−n+1) · (qm  ′  0+t+n+1 − qm  ′  0+t+n)  = (qm0−m  ′  0−t−n+1)(qm  ′  0+t+n)(q − 1)  = qm0+1(q − 1),  and consequently,  η(H(ρ)) = η(A) =  i  �  n=1  η(An) = iqm0+1(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Since qm′  0+t+i ≤ ρ < qm′  0+t+i+1, it follows that  (m′  0 + t + i) log q ≤ log ρ < (m′  0 + t + i + 1) log q  which implies that  log ρ  log q − m′  0 − t − 1 < i ≤ log ρ  log q − m′  0 − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  8  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Hence,  �log ρ  log q − m  ′  0 − t − 1  �  (q − 1)qm0+1 < η(H(ρ)) ≤  �log ρ  log q − m  ′  0 − t  �  (q − 1)qm0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  (9)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  lim inf  ρ→∞  #G(ρ)  η(H(ρ)) ≥ lim inf  ρ→∞  #G′(ρ) − C  η(H(ρ))  = lim inf  n→∞  #G′(|tn|) − C  η(H(|tn|))  (for |tn| ≤ ρ < |tn+1|)  = lim inf  n→∞  1  n(#G′(|tn|) − C)  1  n(η(H(|tn|)))  ≥  lim inf  n→∞  1n(#G′(|tn|))  lim sup  n→∞  1n(η(H(|tn|)))  ≥  lim inf  n→∞  1  n(q − 1) #  �  j : 1 ≤ j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' |bj| ≥ 1  δ  �  lim sup  n→∞  1  n  �log |tn|  log q  − m′  0 − t  �  qm0+1(q − 1)  (by (4) and (9))  ≥  lim inf  n→∞  1  n #  �  j : 1 ≤ j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' |bj| ≥ 1  δ  �  lim sup  n→∞  1  n  �log |b1b2 · · · bn|  log q  − m′  0 − t  �  qm0+1  (by (3))  ≥  lim inf  n→∞  1  n #  �  j : 1 ≤ j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' |bj| ≥ 1  δ  �  lim sup  n→∞  1  n  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  n�  j=1 log |bj|  log q  − m′  0 − t  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8 qm0+1  = e(δ)  α+  log q  qm0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  9  A similar calculation yields  lim sup  ρ→∞  #G(ρ)  η(H(ρ)) ≤ f(δ)  α−  log q  qm0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let Q be a quadratic form as in Theorem 2, and 0 < δ <  1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then there exist a subset K′ of K with µ(K′) = µ(K) such  that if α = −ba ∈ K′, then  lim  ρ→∞  #G(ρ)  η(H(ρ)) =  q − 1  q⌈δ−1⌉+m0+1,  where ⌈δ−1⌉ denotes the smallest integer greater or equal to δ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='] be the continued fraction expansion of α =  −ba as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It follows from Theorem 6 of [1] that there is a full  measure subset K′ of K such that if α = −ba ∈ K′, then  (10)  lim  n→∞ |b1b2 · · · bn| 1n = q  q  q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  This implies that  lim  n→∞  1  n  n  �  j=1  log |bj| =  q  q − 1 log q,  and, therefore, α− = α+ =  q  q−1 log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also for any 0 < δ < 1, there  exists a unique l ∈ N such that l = ⌈δ−1⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then by Theorem 14 of  [12], for α in a full measure set which without loss of generality we may  assume to be K′,  lim  n→∞  1  n #{1 ⩽ j ⩽ n : |bj| ⩾ ql} =  1  ql−1  which implies that e(δ) = f(δ) =  1  ql−1 =  1  q⌈δ−1⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then it follows from  Theorem 2 above that, if α = −ba ∈ K  ′, then  lim  ρ→∞  #G(ρ)  η(H(ρ)) =  1  q⌈δ−1⌉ − 1  �  q  q − 1 log q  � log q  qm0+1  =  q − 1  q⌈δ−1⌉ + m0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let Q, α be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now, if the absolute values  of the partial quotients in the continued fraction expansion of α are  10  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  bounded by some real numbers, then it is easy to see that e(δ) = f(δ) =  0 if δ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In this case,  lim  ρ→∞  #G(ρ)  η(H(ρ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' K is the field of p-adic numbers  In this section, we consider isotropic quadratic forms with coefficients  in the field of p-adic numbers for a prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Recall that the field of  p-adic numbers, denoted by Qp, is the collection of all formal series of  the form  �  j≥n0  ajpj, with n0 ∈ Z and aj ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', p − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  The ultrametric absolute value on Qp is defined as follows: if  α (̸= 0) =  �  j≥n0  ajpj,  then  |α|p := p−νp(α), and |0|p = 0,  where νp(α) := inf {j ∈ Z : aj ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The integer νp(α) is also known  as the valuation of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' For α ∈ Qp and r ∈ Z, let  B(a, pr) := {α ∈ K : |α − a|p < pr}  be the open disc of radius pr around the point α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The Haar measure µ  (say) on Qp is defined in such a way that µ(B(a, pr)) = pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' We denote  by η again the product measure µ ⊗ µ on Qp × Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  As in the case of real numbers and elements of Laurent series fields  over finite fields, continued fraction expansion exists for p-adic num-  bers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' There are mainly two types of continued fractions for  p-adic numbers, one of them was introduced by Schneider (see [17] for  instance), and the other one was introduced by Ruban (see [15] for  instance) and modified later by Brokwin (see [3], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In this article,  we are going to consider the continued fraction introduced by Ruban  which has some similarity with the simple continued fraction for real  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' From now on, unless otherwise stated, we will be considering  Ruban’s continued fraction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let Z be the subset of Qp given by  Z := {a0 + a1  1  p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' an  1  pn : ai ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', p − 1} for 0 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It is easy to see that Z is a discrete set in the topology coming from  the p-adic abosolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' For α (̸= 0) = �  j≥n0  ajpj, let  ⌊α⌋ =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  0  �  j=n0  ajpj  if  n0 ≤ 0  0  if  n0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  11  Given α ∈ Qp, we define two sequences (αn) and (bn) as follows: α0 =  α, b0 = ⌊α0⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' for n ≥ 0, if bn = αn, then αn+1 and bn+1 are not defined,  otherwise, αn+1 = (αn − bn)−1 and bn+1 = ⌊αn+1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Any p-adic number  α has a unique continued fraction expansion as α = [b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' , bn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' ]  which can be obtained by using the algorithm discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Note  that the partial quotients bn’s are elements of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' The nth convergent  is given by sn  tn = [b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' , bn] where sn and tn satisfy the recurrence  relation as in (1), and equation (2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  The p-adic versions of  equation (3), (4) and (5) are valid as well with the absolute value in  the Laurent series field replaced by the p-adic absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' As we  could not find a proper reference for a p-adic version of Lemma 1, we  include a proof here following the proof of Lemma 1 given in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let s, t ∈ Z with t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then s  t is a convergent to α if and  only if  (11)  ����α − s  t  ����  p < 1  |t|2p  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' By the p-adic version of equation (4),  ����α − sn  tn  ����  p  =  1  |bn+1|p |tn|2  p  <  1  |tn|2  p  for any convergent sn  tn corresponding to the continued fraction expan-  sion of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Conversely, assume that s, t ∈ Z with t ̸= 0 such that  ����α − s  t  ����  p < 1  |t|2  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  There is a unique n such that |tn|p ≤ |t|p < |tn+1|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then  ����α − s  t  ����  p <  1  |t|p|tn|p  ,  and  ����α − sn  tn  ����  p  =  1  |tn|p|tn+1|p  (by p-adic version of (5))  <  1  |t|p|tn|p  ,  so that  ����  s  t − sn  tn  ����  p  =  ����  s  t − α + α − sn  tn  ����  p  ≤ max  �����α − s  t  ����  p ,  ����α − sn  tn  ����  p  �  <  1  |t|p|tn|p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  12  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  Thus, s  t = sn  tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  □  Now, let Q be a non-degenerate isotropic binary quadratic form with  coefficients in Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Since Qp has characteristic zero, as explained in the  previous section, it is enough to consider Q defined by  Q(x, y) = (ax + by)(cx + dy)  with a, b, c, d in Qp and bc−ad = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' We also assume that ba is not of the  form s  t for some s, t ∈ Z with t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let p denote the set of all those  (s, t) ∈ Z such that s and t does not have a common factor except the  constant polynomials in 1p inside Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' For k > 1 and 0 < δ < 1, we  define G(ρ) and H(ρ) as in the previous section as follows:  G(ρ) := {(s, t) ∈ p : 0 < |Q(s, t)|p < δ, ||(s, t)|| ≤ ρ, |cs + dt|p > k},  H(ρ) := {(x, y) ∈ K2 : 0 < |Q(x, y)|p < δ, ||(x, y)|| ≤ ρ, |cx+dy|p > k},  here ||(s, t)|| = max { |s|p, |t|p }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Also let α = −ba and β = ac, and the  continued fraction expansion of α be given by  α = [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  The quantities e(δ), f(δ), α− and α+ are defined similarly as in the  previous section with the absolute value replaced by the p-adic absolute  value wherever applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then an analogue of Theorem 2 holds in  this set up as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' With all the notations as above, if α− < ∞, then  lim inf  ρ→∞  #G(ρ)  η(H(ρ)) ≥ c e(δ)  α+ ,  and  lim sup  ρ→∞  #G(ρ)  η(H(ρ)) ≤ c f(δ)  α− ,  where c =  log p  pm0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Let X = B(0, 1) and T : X → X be the continued fraction map  defined by  T(α) = 1  α −  � 1  α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It is known that the map T is ergodic (see [15] for details) with respect  to the Haar measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' As an application of the ergodicity, we obtain  a result similar to Theorem 14 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let α ∈ X and [0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' ] be the continued fraction  expansion of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then for any natural number l,  lim  n→∞ #{1 ≤ j ≤ n : −νp(bj) ≥ l} =  1  pl−1  almost everywhere with respect to the Haar measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Note that b1 = b1(α) can be thought of as a function on B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then it is easy to check that the function  f(α) = χ[pl,∞)(|b1(α)|p), α ∈ B(0, 1)  is integrable on B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now, by the pointwise ergodic theorem  (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='30 of [8] for instance),  lim  n→∞  1  n#{1 ≤ j ≤ n : −νp(bj) ≥ l} = lim  n→∞  1  n#{1 ≤ j ≤ n : |bj|p ≥ pl}  = lim  n→∞  1  n  n  �  j=1  χ[pl,∞)(|b1(T j(α))|p)  =  �  B(0,1)  χ[pl,∞)(|b1(α)|p)dµ  = µ{α ∈ B(0, 1) : |b1(α)|p ≥ pl}  = µ{α ∈ B(0, 1) : |α|p ≤ p−l}  = p−l+1  =  1  pl−1  □  Now, using Theorem 8 of [15] and Lemma 8 above, we obtain a p-adic  version of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let Q be a quadratic form as in Theorem 7, and 0 < δ <  1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then there exist a subset K  ′ of K with µ(K  ′) = µ(K) such  that if α = −ba ∈ K  ′, then  lim  ρ→∞  #G(ρ)  η(H(ρ)) =  p − 1  p⌈δ−1⌉+m0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  It is easy to see that a version of Remark 5 is true in the p-adic set  up as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' As the statements are similar, we do not write it separately  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Rather, we give an example of a p-adic number whose continued  fraction expansion consists of partial quotients with bounded absolute  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' One may look at [11] and references cited there in for similar  examples in Laurent series field over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let α be the p-adic  number given by α =  �  j≥−1 ajpj, with aj = 1 for all j ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Let the  continued fraction expansion of α be [b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Then b0 = p0 + p−1  14  MANOJ CHOUDHURI AND PRASHANT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' MAKADIYA  and |b0|p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now  α1 = (α0 − b0)−1  =  \uf8eb  \uf8ed�  j≥1  pj  \uf8f6  \uf8f8  −1  = p−1 +  �  j≥0  (p − 1)pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then b1 = (p − 1)p0 + p−1 and |b1|p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Again  α2 = (α1 − b1)−1  =  \uf8eb  \uf8ed�  j≥1  (p − 1)pj  \uf8f6  \uf8f8  −1  =  �  j≥−1  (p − 1)pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Then b2 = (p − 1)p0 + (p − 1)p−1 and |b2|p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Observe that α3 =  (α2 − b2)−1 = α2, and hence |b3|p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In a similar manner we get  αn+1 = αn, bn+1 = bn, |bn+1|p = p for n ≥ 3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Therefore, the  absolute values of all the partial quotients of the continued fraction  expansion of α are bounded by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' As in the case of binary real quadratic forms, the Op-  penheim conjecture fails to hold for non-degenerate isotropic quadratic  form with coefficients in a non-discrete locally compact non-Archimedean  field as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' To see this, let us consider the quadratic form Q given by  Q(x, y) = (x + αy)y  with α ∈ Fq((X−1)) (or Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Now if the partial quotients in the con-  tinued fraction expansion of α have bounded absolute values, then us-  ing Lemma 1 (or Lemma 6), it is easy to see that the set of values  {|Q(s, t)| : s, t ∈ Z} (Z is either as in Section 1 or as in Section 2)  avoids certain neighbourhood of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Acknowledgement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Prashant J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Makadiya acknowledges the support  of Government of Gujarat thorugh the SHODH (ScHeme Of Developing  High Quality Research) fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Manoj Choudhuri thanks L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Singhal  for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  References  [1] Val´erie Berth´e and Hitoshi Nakada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' On continued fraction expansions in pos-  itive characteristic: equivalence relations and some metric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Expo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
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+page_content='  [2] Armand Borel and Gopal Prasad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Values of isotropic quadratic forms at S-  integral points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', 83(3):347–372, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ON VALUES OF ISOTROPIC QUADRATIC FORMS  15  [3] Jerzy Browkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Continued fractions in local fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Demonstratio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=',  11(1):67–82, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [4] Jerzy  Browkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Continued  fractions  in  local  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=',  70(235):1281–1292, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [5] Manoj Choudhuri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' On certain orbits of geodesic flow and (a, b)-continued frac-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Indian Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', 131(1):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' 2, 19, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [6] Manoj Choudhuri and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Dani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' On values of binary quadratic forms at  integer points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', 22(4):1023–1045, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Dani and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Margulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Limit distributions of orbits of unipotent flows  and values of quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Gelfand Seminar, volume 16 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Soviet Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', pages 91–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', Providence, RI, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [8] Manfred Einsiedler and Thomas Ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Ergodic theory with a view towards  number theory, volume 259 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Springer-Verlag  London, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=', London, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  [9] Alex Eskin, Gregory Margulis, and Shahar Mozes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' On a quantitative ver-  sion of the Oppenheim conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Announc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
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+page_content='  [10] Svetlana Katok and Ilie Ugarcovici.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
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+page_content='  Institute of Infrastructure, Technology, Research and Manage-  ment, Near Khokhara Circle, maninagar (East), Ahmedabad 380026,  Gujarat, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='  Email address: manojchoudhuri@iitram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='in  Email address: prashant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='makadiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='20pm@iitram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}
+page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf'}

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+arXiv:2301.02831v1  [cs.IT]  7 Jan 2023
+1
+Joint Beamforming and Phase Shift Design for
+Hybrid-IRS-aided Directional Modulation Network
+Rongen Dong, Hangjia He, Feng Shu, Riqing Chen, and Jiangzhou Wang, Fellow, IEEE
+Abstract—To make a good balance between performance,
+cost, and power consumption, a hybrid intelligent reflecting
+surface (IRS)-aided directional modulation (DM) network is
+investigated in this paper, where the hybrid IRS consists of
+passive and active reflecting elements. To maximize the achievable
+rate, two optimization algorithms, called maximum signal-to-
+noise ratio (SNR)-fractional programming (FP) (Max-SNR-FP)
+and maximum SNR-equal amplitude reflecting (EAR) (Max-
+SNR-EAR), are proposed to jointly design the beamforming
+vector and IRS phase shift matrix by alternately optimizing one
+and fixing another. The former employs the successive convex
+approximation and FP methods to solve the beamforming vector
+and hybrid IRS phase shift matrix, while the latter uses the
+maximum signal-to-leakage-noise ratio method and the criteria
+of phase alignment and EAR to design them. Simulation results
+show that the rates harvested by the proposed two methods
+are slightly lower than that of active IRS with higher power
+consumption, which are 35 percent higher than those of no IRS
+and random phase IRS, while passive IRS achieves only about
+17 percent rate gain over the latter. Moreover, compared to Max-
+SNR-FP, the proposed Max-SNR-EAR method makes an obvious
+complexity reduction at the cost of a slight rate performance loss.
+Index Terms—Intelligent reflecting surface, directional modu-
+lation, fractional programming, beamforming, phase shift
+I. INTRODUCTION
+Directional modulation (DM) is a promising solution to sig-
+nificantly improve the performance of physical layer security
+in wireless networks [1]. The design of DM synthesis is mainly
+implemented in the radio frequency (RF) frontend or baseband.
+For example, in [2], the signal was produced in a given
+direction by shifting the phase of each antenna element at the
+RF frontend. In [3], a multi-beam DM scenario was considered
+to maximize the secure rate (SR), where the precoder and
+the artificial noise (AN) were designed by maximizing signal-
+to-leakage-noise ratio and maximizing the signal-to-AN ratio
+methods, respectively.
+Intelligent reflecting surface (IRS), as a cost and energy-
+efficient solution to enhance the performance of the wire-
+less communication system, has been adopted to aid various
+This work was supported in part by the National Natural Science Foundation
+of China (Nos.U22A2002, and 62071234), the Major Science and Technology
+plan of Hainan Province under Grant ZDKJ2021022, and the Scientific
+Research Fund Project of Hainan University under Grant KYQD(ZR)-21008.
+Rongen Dong and Feng Shu are with the School of Information and Com-
+munication Engineering, Hainan University, Haikou, 570228, China (Email:
+[email protected]).
+Hangjia He is with the School of Electronic and Optical Engineering,
+Nanjing University of Science and Technology, Nanjing, 210094, China.
+Riqing Chen is with the Digital Fujian Institute of Big Data for Agriculture,
+Fujian Agriculture and Forestry University, Fuzhou 350002, China (Email:
+[email protected]).
+Jiangzhou Wang is with the School of Engineering, University of Kent,
+Canterbury CT2 7NT, U.K. (Email: [email protected]).
+wireless communication directions: unmanned aerial vehicle
+communication [4], single-cell wireless communication [5],
+multi-cell communication [6], etc. Recently, IRS-aided DM
+system have also been investigated. To maximize the SR
+of IRS-aided DM system, the general alternating iterative
+and null-space projection algorithms were proposed to jointly
+obtain the transmit beamforming vectors and IRS phase shift
+matrix in [7]. To maximize the receive power sum, the authors
+in [8] proposed the general alternating optimization and zero-
+forcing algorithms to jointly design the receive beamforming
+vectors and IRS phase shift matrix.
+However, all the above work was considered in the scenarios
+of passive IRS, and the system may not be able to guarantee
+a satisfactory achievable rate due to the presence of double
+path loss in the cascaded channels. To overcome the “double
+fading” effect and enhance the performance of the passive
+IRS-aided wireless network, the fully active IRS has been
+investigated [9], [10]. Due to the high power consumption and
+hardware design of active IRS, a hybrid active-passive IRS
+was proposed to overcome the limitation of passive and active
+IRSs [11], [12]. The main idea of the hybrid IRS is to employ
+some active elements to replace the one of the passive IRS,
+these active elements of hybrid IRS with signal amplification
+can efficiently compensate for the path loss and increase the
+achievable rate. To the best of the authors’ knowledge, the
+hybrid IRS-aided DM system have not been investigated yet.
+In this paper, we employ the hybrid IRS to further enhance
+the performance of passive IRS-aided DM network. The main
+contributions of this paper are summarized as follows:
+1) To make a good balance between performance, cost,
+and power consumption, a hybrid IRS-aided DM system
+model is proposed. To maximize the achievable rate,
+the optimization problem of maximizing the signal-to-
+noise ratio (SNR) is established, and the maximum SNR-
+fractional programming (FP) (Max-SNR-FP) scheme is
+proposed to jointly obtain the beamforming vector and
+hybrid IRS phase shift matrix by optimizing one and
+fixing another. In this scheme, the beamforming vector
+and passive IRS phase shift matrix are solved by the
+successive convex approximation algorithm, and the
+active IRS phase shift matrix is computed by the FP
+method.
+2) To reduce the high computational complexity of the
+above scheme, a low-complexity maximum SNR-equal
+amplitude reflecting (EAR) (Max-SNR-EAR) method is
+proposed. By utilizing the maximum signal-to-leakage-
+noise ratio (SLNR) method, the beamforming vector is
+
+2
+obtained. Moreover, the hybrid IRS phase shift matrix is
+computed based on the criteria of phase alignment and
+EAR. Simulation results show that the achievable rates
+harvested by both the proposed methods are higher than
+those of no IRS, random phase IRS, and passive IRS.
+In addition, the difference in achievable rates between
+these two methods is trivial when the number of hybrid
+IRS elements tends to large scale.
+The remainder of this paper is organized as follows. Section
+II describes the system model of hybrid IRS-aided DM net-
+work. The Max-SNR-FP scheme is presented in Section III.
+Section IV describes the Max-SNR-EAR scheme. Numerical
+simulation results are presented in Section V. Finally, we draw
+conclusions in Section VI.
+Notations: throughout this paper, boldface lower case and
+upper case letters represent vectors and matrices, respectively.
+Signs (·)T , (·)∗, (·)H, Tr(·), ℜ{·}, and diag{·} denote the
+transpose, conjugate, conjugate transpose, trace, real part,
+and diagonal operations, respectively. The sign | · | is the
+determinant of a matrix or the absolute value of a scalar. The
+symbol CN×N denotes the space of N × N complex-valued
+matrix. The notation IN is the N × N identity matrix.
+II. SYSTEM MODEL
+As shown in Fig. 1, a hybrid IRS-aided DM system is
+considered, where the base station (BS) is equipped with
+N antennas, and the user (Bob) is equipped with single
+antenna. The hybrid IRS is equipped with M elements, which
+consists of Ma active and Mp passive IRS reflecting elements
+(M = Ma + Mp, 1 ≤ Ma ≤ Mp). It is assumed that the
+active elements can tune both the phase and amplitude while
+the passive ones can only shift the phase of the incident
+signal. The signals reflected more than once on the hybrid IRS
+are negligible due to the severe path loss [6]. All channels
+are assumed to be line-of-sight channels since DM is only
+applicable to line-of-sight channels. It is assumed that all the
+channel state information is perfectly known through channel
+estimation [13].
+Fig. 1. System model of Hybrid-IRS-aided directional modulation network.
+Similar to the conventional passive IRS, it is assumed that
+each elements of hybrid IRS can independently reflect the inci-
+dent signals. Let us denote the set of the Ma active elements by
+Ω. Θ = diag{θ∗} = diag{θ1, · · · , θm, · · · , θM} ∈ CM×M,
+Ψ = diag{ψ∗} ∈ CM×M, and Φ = diag{φ∗} ∈ CM×M are
+the reflection coefficients of total elements, active elements,
+and passive elements of hybrid IRS, respectively, where
+θm =
+�
+|βm|ejµm,
+if m ∈ Ω,
+ejµm,
+otherwise,
+(1)
+µm
+∈ [0, 2π) is the phase, and |βm| is the amplifying
+coefficient and determined by the total power of the active
+elements. Let us define
+Ψ = EMaΘ, Φ = EMpΘ,
+(2)
+where
+EMa + EMp = IM, EMaEMp = 0M,
+(3)
+EMa is an M × M diagonal matrix whose non-zero elements
+are all unity and have positions determined by Ω.
+The transmitted signal at BS is
+s =
+√
+Pvx,
+(4)
+where P denotes the transmit power, v ∈ CN×1 and x are the
+beamforming vector and the information symbol, satisfying
+vHv = 1 and E[∥x∥2] = 1, respectively.
+Taking the path loss into consideration, the received signal
+at Bob is
+yb = (√ρsrbhH
+rbΘHsr + √ρsbhH
+sb)s + √ρrbhH
+rbΨnr + nb
+=
+√
+P(√ρsrbhH
+rbΨHsr + √ρsrbhH
+rbΦHsr + √ρsbhH
+sb)vx
++ √ρrbhH
+rbΨnr + nb,
+(5)
+where ρsrb = ρsrρrb is the equivalent path loss coefficient
+of BS-to-IRS channel and IRS-to-Bob channel, ρsb and ρrb
+are the path loss coefficient of BS-to-Bob channel and IRS-
+to-Bob channel, respectively. nr ∼ CN(0, σ2
+rIMa) and nb ∼
+CN(0, σ2
+b) denote the complex additive white Gaussian noise
+(AWGN) at the Ma active elements of the hybrid IRS and
+at Bob, respectively. hsb ∈ CN×1, hrb ∈ CM×1, and Hsr =
+hsrhH
+sr ∈ CM×N are the BS-to-Bob, IRS-to-Bob, and BS-to-
+IRS channels, respectively. Let us define the channel htr =
+h(θtr), the normalized steering vector h(θ) is
+h(θ) =
+1
+√
+N
+[ej2πΨθ(1), . . . , ej2πΨθ(n), . . . , ej2πΨθ(N)]T , (6)
+and the phase function Ψθ(n) is given by
+Ψθ(n)
+∆= −(n − (N + 1)/2)d cosθ
+λ
+, n = 1, . . . , N,
+(7)
+where θ represents the direction angle of arrival or departure,
+n denotes the index of antenna, d is the spacing of adjacent
+transmitting antennas, and λ represents the wavelength.
+In accordance with (5), the achievable rate at Bob can be
+written as
+Rb = log2 (1 + SNR) ,
+(8)
+where
+SNR = P|(√ρsrbhH
+rbΨHsr + √ρsrbhH
+rbΦHsr + √ρsbhH
+sb)v|2
+σ2r|√ρrbhH
+rbΨ|2 + σ2
+b
+.
+(9)
+
+Hybrid IRS
+Active
+Passive
+H
+H
+rb
+((())
+H
+5
+sb
+User
+(Bob)
+Base station3
+The transmit power of the active elements at the hybrid IRS
+is given by
+Pr = Tr
+�
+Ψ
+�
+ρsrPHsrvvHHH
+sr + σ2
+rIM
+�
+ΨH�
+,
+(10)
+which satisfies Pr ≤ P max
+r
+, where P max
+r
+represents the maxi-
+mum transmit power of Ma active elements.
+In this paper, we maximize the SNR by jointly optimizing
+beamforming vector v, passive IRS phase shift matrix Φ, and
+active IRS phase shift matrix Ψ. The optimization problem
+can be formulated as
+max
+v,Φ,Ψ
+SNR
+(11a)
+s.t.
+vHv = 1, Pr ≤ P max
+r
+,
+(11b)
+|Φ(m, m)| = 1, if m ̸∈ Ω,
+(11c)
+|Φ(m, m)| = 0, otherwise,
+(11d)
+|Ψ(m, m)| ≤ βmax, if m ∈ Ω,
+(11e)
+|Ψ(m, m)| = 0, otherwise,
+(11f)
+where βmax is the amplification budget. It is notes that this
+optimization problem is a non-convex problem with a constant
+modulus constraint, and it is challenging to solve it directly in
+general. In what follows, we propose the alternating optimiza-
+tion algorithm to design the beamforming vector and hybrid
+IRS phase shift matrix, respectively.
+III. PROPOSED MAX-SNR-FP SCHEME
+In this section, we construct a Max-SNR-FP method to
+jointly optimize the beamforming vector v, passive IRS phase
+shift matrix Φ, and active IRS phase shift matrix Ψ. In what
+follows, we will alternately solve for v, Φ, and Ψ.
+A. Optimize v given Φ and Ψ
+Firstly, we transform the power constraint in (11b) into a
+convex constraint with respect to v as follows
+Pr = vH �
+ρsrPHH
+srΨHΨHsr
+�
+v + Tr
+�
+σ2
+rΨΨH�
+≤ P max
+r
+.
+(12)
+Then, given Φ and Ψ, the optimal beamforming vector v can
+be found by solving the following problem
+max
+v
+vHA¯v
+s.t. vHv = 1, (12),
+(13)
+where
+A =(√ρsrbhH
+rbΦHsr + √ρsrbhH
+rbΨHsr + √ρsbhH
+sb)H
+(√ρsrbhH
+rbΦHsr + √ρsrbhH
+rbΨHsr + √ρsbhH
+sb).
+(14)
+It is clear that this problem is not convex, and in accordance
+with the Taylor series expansion, we have
+vHAv ≥ 2ℜ{¯vHAv} − ¯vHA¯v,
+(15)
+where ¯v is a given vector. Then (13) can be recasted as
+max
+v
+2ℜ{¯vHAv} − ¯vHA¯v
+s.t. vHv = 1, (12).
+(16)
+It is a convex optimization problem and can be solved by
+employing CVX tool.
+B. Optimize Φ given v and Ψ
+To simplify the SNR expression related to the phase shift
+matrix Φ, we regard v and Ψ as two constants, and define
+B = (√ρsrbhH
+rbΨHsr + √ρsbhH
+sb)v.
+(17)
+Then, the subproblem to optimize Φ can be expressed as
+max
+Φ
+|√ρsrbhH
+rbΦHsrv + B|2
+(18a)
+s.t. |Φ(m, m)| = 1, if m ̸∈ Ω,
+(18b)
+|Φ(m, m)| = 0, otherwise.
+(18c)
+By defining
+C = ρsrbdiag{hH
+rb}HsrvvHHH
+srdiag{hH
+rb}H,
+(19)
+and based on the fact that diag{a}b = diag{b}a for a, b ∈
+CM×1, the objective function in (18) can be recasted as
+φHCφ + 2ℜ{√ρsrbφHdiag{hH
+rb}HsrvB∗} + |B|2.
+(20)
+Based on the Taylor series expansion, we have
+φHCφ ≥ 2ℜ{ ¯φHCφ} − ¯φHC ¯φ,
+(21)
+where ¯φ is a given vector. For the unit modulus constraint
+(18b), it can be relaxed as
+|Φ(m, m)| ≤ 1, if m ̸∈ Ω.
+(22)
+At this point, the problem (18) can be rewritten as
+max
+Φ
+2ℜ{ ¯φHCφ} − ¯φHC ¯φ + |B|2 + 2ℜ{√ρsrbφH•
+diag{hH
+rb}HsrvB∗}
+s.t.
+(22), (18c).
+(23)
+We can find that it is a convex optimization problem and can
+be solved by employing CVX tool.
+C. Optimize Ψ given v and Φ
+To optimize Ψ, we regard v and Φ as two given constants,
+and transform the power constraint in (11b) into a convex
+constraint on ψ as follows
+Pr = Tr
+�
+Ψ
+�
+ρsrPHsrvvHHH
+sr + σ2IM
+�
+ΨH�
+= ψT (ρsrPdiag{vHHH
+sr}diag{Hsrv} + σ2
+rIM)ψ∗
+≤ P max
+r
+.
+(24)
+By neglecting the constant terms, the subproblem with respect
+to Ψ is given by
+max
+Ψ
+|(√ρsrbhH
+rbΨHsr + √ρsrbhH
+rbΦHsr + √ρsbhH
+sb)v|2
+σ2r|√ρrbhH
+rbΨ|2 + σ2
+b
+(25a)
+s.t.
+(11e), (11f), (24).
+(25b)
+Let us define
+D = (√ρsrbhH
+rbΦHsr + √ρsbhH
+sb)v.
+(26)
+Then, the objective function in (25) can be converted to
+ψHCψ + 2ℜ{ψH√ρsrbdiag{hH
+rb}HsrvD∗} + |D|2
+σ2rρrb|ψHdiag{hH
+rb}|2 + σ2
+b
+.
+(27)
+
+4
+At this point, the optimization problem (25) becomes a nonlin-
+ear fractional optimization problem. Based on the FP strategy
+in [14], we introduce a parameter τ and transform the objective
+function (27) as
+ψHCψ + 2ℜ{ψH√ρsrbdiag{hH
+rb}HsrvD∗} + |D|2
+− τ(σ2
+rρrb|ψHdiag{hH
+rb}|2 + σ2
+b).
+(28)
+The
+optimal
+solution
+can
+be
+achieved
+if
+and
+only
+if ψHCψ + 2ℜ{ψH√ρsrbdiag{hH
+rb}HsrvD∗} + |D|2 −
+τ(σ2
+rρrb|ψHdiag{hH
+rb}|2 + σ2
+b) = 0. We linearize the ψHCψ
+by employing Taylor series expansion at a given vector ¯ψ, the
+subproblem with respect to Ψ can be rewritten as
+max
+Ψ,τ
+2ℜ{ ¯ψHCψ} − ¯ψHC ¯ψ + 2ℜ{ψH√ρsrbdiag{hH
+rb}•
+HsrvD∗} + |D|2 − τ(σ2
+rρrb|ψHdiag{hH
+rb}|2 + σ2
+b)
+s.t.
+(11e), (11f), (24).
+(29)
+It should be noted that this problem is convex, which can be
+effectively solved by the CVX tool. The whole procedure of
+the Max-SNR-FP algorithm is described in Algorithm 1.
+Algorithm 1 Proposed Max-SNR-FP algorithm
+1: Initialize v(0), Φ(0), and Ψ(0), compute R(0)
+b
+based on (8).
+2: Set p = 0, threshold value ǫ.
+3: repeat
+4:
+Given Φ(p) and Ψ(p), solve (16) to determine v(p+1).
+5:
+Given v(p+1) and Ψ(p), solve (23) to determine Φ(p+1).
+6:
+Given v(p+1) and Φ(p+1), solve (29) to determine
+Ψ(p+1).
+7:
+Compute R(p+1)
+b
+using v(p+1), Φ(p+1), and Ψ(p+1).
+8:
+p = p + 1.
+9: until |R(p)
+b
+− R(p−1)
+b
+| ≤ ǫ.
+The computational complexity of the proposed Max-SNR-
+FP algorithm is O(L((M + 1)3 + 2MN 2 + 2M 2)In(1/ǫ) +
+M 3+N 3+5M 2+2MN+2M+2MN 2) float-point operations
+(FLOPs), where L is the numbers of alternating iterations, ǫ
+denotes the accuracy.
+IV. PROPOSED MAX-SNR-EAR SCHEME
+In the previous section, we proposed the Max-SNR-FP
+method to design the beamforming v, IRS phase shift matrices
+Φ and Ψ. However, it has a high computational complexity.
+To reduce the computational complexity, a low-complexity
+method named Max-SNR-EAR is proposed in what follows.
+A. Optimize v given Φ and Ψ
+Given IRS phase shift matrices Φ and Ψ, in accordance with
+the principle of maximizing SLNR in [15], the beamforming
+vector v can be optimized by solving the following problem
+max
+v
+SLNR =
+vHEv
+vH(σ2
+bIN)v
+s.t. vHv = 1, (12),
+(30)
+where
+E =ρsrbHH
+srΦHhrbhH
+rbΦHsr + ρsrbHH
+srΨHhrbhH
+rbΨHsr
++ hsbhH
+sb.
+(31)
+According to the Taylor series expansion and neglecting the
+constant terms, the problem (30) can be recasted as
+max
+v
+2ℜ{¯vHEv} − ¯vHE¯v
+s.t.
+vHv = 1, (12).
+(32)
+Note that it is a convex optimization problem and can be
+solved with CVX tool.
+B. Optimize Φ and Ψ given v
+Given beamforming vector v, we consider to design the
+phase of hybrid IRS firstly. The confidential message received
+by Bob through the cascade path is expressed as
+PρsrbhH
+rbΘHsrvvHHH
+srΘHhrb.
+(33)
+To maximize the confidential message of the cascade path, the
+phase alignment method is employed to design the hybrid IRS
+phase �θ, �θ is given by
+�θ = [e(−iarg(s1)), · · · , e(−iarg(sM))]T ,
+(34)
+where s = diag{hH
+rb}Hsrv, and si is the i-th element of s.
+Next, inspired by the amplitude design of fully active IRS
+in [9], we assume that all active IRS elements have the same
+amplitude. Based on the IRS power constraint in (11b), we
+have
+|β| =
+�
+P max
+r
+/Q,
+(35)
+where
+Q =Tr(�θH(ρsrPdiag{vHHH
+srEMa}diag{vHHH
+srEMa}H
++ σ2EMaEMa)�θ).
+(36)
+Based on (34) and (35), we can obtain the passive IRS phase
+shift matrix and active IRS phase shift matrix as follows
+Φ = EMpdiag{�θ}, Ψ = |β|EMadiag{�θ}.
+(37)
+Similar to Algorithm 1, we calculate v, Φ, and Ψ alternately
+until convergence, i.e., |R(p)
+b −R(p−1)
+b
+| ≤ ǫ. The computational
+complexity of Max-SNR-EAR algorithm is O(K(2M 2+N 3+
+2M 2 + 8N 2M + 2MN) FLOPs, where K is the numbers of
+alternating iterations.
+V. SIMULATION RESULTS AND DISCUSSIONS
+In this section, simulation results are presented to evaluate
+the performance of two proposed algorithms. Simulation de-
+fault parameters are chosen as follows: N = 8, M = 128,
+Ma = 32, d = λ/2, θsr = π/4, θsb = π/3, dsr = 200m,
+dsb = 220m, σ2
+b = −70dBm, σ2
+r = 2σ2
+b, P = 25dBm,
+P max
+r
+= 30dBm. The path loss at the distance d is modeled
+as g(d) = PL0 − 10γlog10
+d
+d0 , where PL0 = −30dB is the
+path loss reference distance d0 = 1m, and γ is the path loss
+exponent. The path loss exponents of all channels are chosen
+as 2. The positions of the IRS active elements are fixed to
+Ω = {1, · · · , Ma}.
+First, we make an investigation of the convergence be-
+haviour of the proposed Max-SNR-FP and Max-SNR-EAR
+algorithms. Fig. 2 shows the achievable rate versus the differ-
+ent BS power, i.e., P = 20dBm, 25dBm. It can be seen from
+the figure that both of the proposed algorithms converge within
+limited iterations. The proposed Max-SNR-EAR algorithm has
+a faster convergence rate than the Max-SNR-FP algorithm,
+regardless of P = 20dBm or 25dBm.
+
+5
+0
+5
+10
+15
+20
+25
+30
+13.5
+14
+14.5
+15
+15.5
+16
+16.5
+17
+Fig. 2. Convergence of the proposed algorithms at different BS power.
+Fig. 3 depicts the curves of the achievable rate versus the
+number of IRS phase shift elements, where Ma = M/2. We
+compare two proposed algorithms to the benchmark schemes:
+active IRS, passive IRS, no IRS, random phase IRS, and exist-
+ing method in [11]. The achievable rates of the proposed Max-
+SNR-FP and Max-SNR-EAR algorithms gradually increase
+as the number of IRS elements increases, and the former
+is better than the latter and existing method in [11]. The
+achievable rates of both the proposed algorithms are much
+better than that of the passive IRS, no IRS and random phase
+IRS. Moreover, the difference in achievable rates between both
+the proposed algorithms and active IRS gradually decreases
+when the number of IRS elements tends to large scale.
+3
+4
+5
+6
+7
+8
+11
+12
+13
+14
+15
+16
+17
+18
+19
+Fig. 3. Achievable rate versus the numbers of IRS phase shift elements.
+Fig. 4 plots the curves of the computational complexity
+versus the number of IRS elements. It can be found that the
+complexities of the proposed Max-SNR-FP method, proposed
+Max-SNR-EAR method, and existing method in [11] are
+similar at small-scale IRS. However, the complexities of the
+existing method in [11] and proposed Max-SNR-FP method
+are far higher than that of the proposed Max-SNR-EAR
+method when the number of IRS elements tends to large scale.
+VI. CONCLUSION
+In this paper, we have made an investigation of the hybrid
+IRS-aided DM network. To fully explore the advantages of
+hybrid IRS and maximize the achievable rate, the Max-SNR-
+FP and Max-SNR-EAR algorithms were proposed to jointly
+design the beamforming vector, passive IRS phase shift matrix,
+and active IRS phase shift matrix by alternately optimizing one
+and fixing rest. Simulation results showed that the achievable
+2
+3
+4
+5
+6
+7
+0
+2
+4
+6
+8
+10
+12
+14
+107
+Fig. 4. Computational complexity versus the numbers of IRS elements.
+rate of both proposed algorithms increases as the number of
+IRS elements increases, and is much better than those of
+the cases of random phase IRS, no IRS, and passive IRS.
+Moreover, the proposed Max-SNR-FP method outperforms the
+existing method in terms of the achievable rate and has lower
+complexity.
+REFERENCES
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+

1tE0T4oBgHgl3EQf_wKi/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf,len=379
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='02831v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='IT]  7 Jan 2023  1  Joint Beamforming and Phase Shift Design for  Hybrid-IRS-aided Directional Modulation Network  Rongen Dong, Hangjia He, Feng Shu, Riqing Chen, and Jiangzhou Wang, Fellow, IEEE  Abstract—To make a good balance between performance,  cost, and power consumption, a hybrid intelligent reflecting  surface (IRS)-aided directional modulation (DM) network is  investigated in this paper, where the hybrid IRS consists of  passive and active reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To maximize the achievable  rate, two optimization algorithms, called maximum signal-to-  noise ratio (SNR)-fractional programming (FP) (Max-SNR-FP)  and maximum SNR-equal amplitude reflecting (EAR) (Max-  SNR-EAR), are proposed to jointly design the beamforming  vector and IRS phase shift matrix by alternately optimizing one  and fixing another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The former employs the successive convex  approximation and FP methods to solve the beamforming vector  and hybrid IRS phase shift matrix, while the latter uses the  maximum signal-to-leakage-noise ratio method and the criteria  of phase alignment and EAR to design them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Simulation results  show that the rates harvested by the proposed two methods  are slightly lower than that of active IRS with higher power  consumption, which are 35 percent higher than those of no IRS  and random phase IRS, while passive IRS achieves only about  17 percent rate gain over the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Moreover, compared to Max-  SNR-FP, the proposed Max-SNR-EAR method makes an obvious  complexity reduction at the cost of a slight rate performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Index Terms—Intelligent reflecting surface, directional modu-  lation, fractional programming, beamforming, phase shift  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' INTRODUCTION  Directional modulation (DM) is a promising solution to sig-  nificantly improve the performance of physical layer security  in wireless networks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The design of DM synthesis is mainly  implemented in the radio frequency (RF) frontend or baseband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  For example, in [2], the signal was produced in a given  direction by shifting the phase of each antenna element at the  RF frontend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' In [3], a multi-beam DM scenario was considered  to maximize the secure rate (SR), where the precoder and  the artificial noise (AN) were designed by maximizing signal-  to-leakage-noise ratio and maximizing the signal-to-AN ratio  methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Intelligent reflecting surface (IRS), as a cost and energy-  efficient solution to enhance the performance of the wire-  less communication system, has been adopted to aid various  This work was supported in part by the National Natural Science Foundation  of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='U22A2002, and 62071234), the Major Science and Technology  plan of Hainan Province under Grant ZDKJ2021022, and the Scientific  Research Fund Project of Hainan University under Grant KYQD(ZR)-21008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Rongen Dong and Feng Shu are with the School of Information and Com-  munication Engineering, Hainan University, Haikou, 570228, China (Email:  shufeng0101@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Hangjia He is with the School of Electronic and Optical Engineering,  Nanjing University of Science and Technology, Nanjing, 210094, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Riqing Chen is with the Digital Fujian Institute of Big Data for Agriculture,  Fujian Agriculture and Forestry University, Fuzhou 350002, China (Email:  riqing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='chen@fafu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Jiangzhou Wang is with the School of Engineering, University of Kent,  Canterbury CT2 7NT, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' (Email: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='wang@kent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  wireless communication directions: unmanned aerial vehicle  communication [4], single-cell wireless communication [5],  multi-cell communication [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Recently, IRS-aided DM  system have also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To maximize the SR  of IRS-aided DM system, the general alternating iterative  and null-space projection algorithms were proposed to jointly  obtain the transmit beamforming vectors and IRS phase shift  matrix in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To maximize the receive power sum, the authors  in [8] proposed the general alternating optimization and zero-  forcing algorithms to jointly design the receive beamforming  vectors and IRS phase shift matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  However, all the above work was considered in the scenarios  of passive IRS, and the system may not be able to guarantee  a satisfactory achievable rate due to the presence of double  path loss in the cascaded channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To overcome the “double  fading” effect and enhance the performance of the passive  IRS-aided wireless network, the fully active IRS has been  investigated [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Due to the high power consumption and  hardware design of active IRS, a hybrid active-passive IRS  was proposed to overcome the limitation of passive and active  IRSs [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The main idea of the hybrid IRS is to employ  some active elements to replace the one of the passive IRS,  these active elements of hybrid IRS with signal amplification  can efficiently compensate for the path loss and increase the  achievable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To the best of the authors’ knowledge, the  hybrid IRS-aided DM system have not been investigated yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  In this paper, we employ the hybrid IRS to further enhance  the performance of passive IRS-aided DM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The main  contributions of this paper are summarized as follows:  1) To make a good balance between performance, cost,  and power consumption, a hybrid IRS-aided DM system  model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To maximize the achievable rate,  the optimization problem of maximizing the signal-to-  noise ratio (SNR) is established, and the maximum SNR-  fractional programming (FP) (Max-SNR-FP) scheme is  proposed to jointly obtain the beamforming vector and  hybrid IRS phase shift matrix by optimizing one and  fixing another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' In this scheme, the beamforming vector  and passive IRS phase shift matrix are solved by the  successive convex approximation algorithm, and the  active IRS phase shift matrix is computed by the FP  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  2) To reduce the high computational complexity of the  above scheme, a low-complexity maximum SNR-equal  amplitude reflecting (EAR) (Max-SNR-EAR) method is  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' By utilizing the maximum signal-to-leakage-  noise ratio (SLNR) method, the beamforming vector is  2  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Moreover, the hybrid IRS phase shift matrix is  computed based on the criteria of phase alignment and  EAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Simulation results show that the achievable rates  harvested by both the proposed methods are higher than  those of no IRS, random phase IRS, and passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  In addition, the difference in achievable rates between  these two methods is trivial when the number of hybrid  IRS elements tends to large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Section  II describes the system model of hybrid IRS-aided DM net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The Max-SNR-FP scheme is presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Section IV describes the Max-SNR-EAR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Numerical  simulation results are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Finally, we draw  conclusions in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Notations: throughout this paper, boldface lower case and  upper case letters represent vectors and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Signs (·)T , (·)∗, (·)H, Tr(·), ℜ{·}, and diag{·} denote the  transpose, conjugate, conjugate transpose, trace, real part,  and diagonal operations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The sign | · | is the  determinant of a matrix or the absolute value of a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The  symbol CN×N denotes the space of N × N complex-valued  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The notation IN is the N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' SYSTEM MODEL  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 1, a hybrid IRS-aided DM system is  considered, where the base station (BS) is equipped with  N antennas, and the user (Bob) is equipped with single  antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The hybrid IRS is equipped with M elements, which  consists of Ma active and Mp passive IRS reflecting elements  (M = Ma + Mp, 1 ≤ Ma ≤ Mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' It is assumed that the  active elements can tune both the phase and amplitude while  the passive ones can only shift the phase of the incident  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The signals reflected more than once on the hybrid IRS  are negligible due to the severe path loss [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' All channels  are assumed to be line-of-sight channels since DM is only  applicable to line-of-sight channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' It is assumed that all the  channel state information is perfectly known through channel  estimation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' System model of Hybrid-IRS-aided directional modulation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Similar to the conventional passive IRS, it is assumed that  each elements of hybrid IRS can independently reflect the inci-  dent signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Let us denote the set of the Ma active elements by  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Θ = diag{θ∗} = diag{θ1, · · · , θm, · · · , θM} ∈ CM×M,  Ψ = diag{ψ∗} ∈ CM×M, and Φ = diag{φ∗} ∈ CM×M are  the reflection coefficients of total elements, active elements,  and passive elements of hybrid IRS, respectively, where  θm =  �  |βm|ejµm,  if m ∈ Ω,  ejµm,  otherwise,  (1)  µm  ∈ [0, 2π) is the phase, and |βm| is the amplifying  coefficient and determined by the total power of the active  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Let us define  Ψ = EMaΘ, Φ = EMpΘ,  (2)  where  EMa + EMp = IM, EMaEMp = 0M,  (3)  EMa is an M × M diagonal matrix whose non-zero elements  are all unity and have positions determined by Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  The transmitted signal at BS is  s =  √  Pvx,  (4)  where P denotes the transmit power, v ∈ CN×1 and x are the  beamforming vector and the information symbol, satisfying  vHv = 1 and E[∥x∥2] = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Taking the path loss into consideration, the received signal  at Bob is  yb = (√ρsrbhH  rbΘHsr + √ρsbhH  sb)s + √ρrbhH  rbΨnr + nb  =  √  P(√ρsrbhH  rbΨHsr + √ρsrbhH  rbΦHsr + √ρsbhH  sb)vx  + √ρrbhH  rbΨnr + nb,  (5)  where ρsrb = ρsrρrb is the equivalent path loss coefficient  of BS-to-IRS channel and IRS-to-Bob channel, ρsb and ρrb  are the path loss coefficient of BS-to-Bob channel and IRS-  to-Bob channel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' nr ∼ CN(0, σ2  rIMa) and nb ∼  CN(0, σ2  b) denote the complex additive white Gaussian noise  (AWGN) at the Ma active elements of the hybrid IRS and  at Bob, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' hsb ∈ CN×1, hrb ∈ CM×1, and Hsr =  hsrhH  sr ∈ CM×N are the BS-to-Bob, IRS-to-Bob, and BS-to-  IRS channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Let us define the channel htr =  h(θtr), the normalized steering vector h(θ) is  h(θ) =  1  √  N  [ej2πΨθ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' , ej2πΨθ(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' , ej2πΨθ(N)]T , (6)  and the phase function Ψθ(n) is given by  Ψθ(n)  ∆= −(n − (N + 1)/2)d cosθ  λ  , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' , N,  (7)  where θ represents the direction angle of arrival or departure,  n denotes the index of antenna, d is the spacing of adjacent  transmitting antennas, and λ represents the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  In accordance with (5), the achievable rate at Bob can be  written as  Rb = log2 (1 + SNR) ,  (8)  where  SNR = P|(√ρsrbhH  rbΨHsr + √ρsrbhH  rbΦHsr + √ρsbhH  sb)v|2  σ2r|√ρrbhH  rbΨ|2 + σ2  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (9)  Hybrid IRS  Active  Passive  H  H  rb  ((())  H  5  sb  User  (Bob)  Base station3  The transmit power of the active elements at the hybrid IRS  is given by  Pr = Tr  �  Ψ  �  ρsrPHsrvvHHH  sr + σ2  rIM  �  ΨH�  ,  (10)  which satisfies Pr ≤ P max  r  , where P max  r  represents the maxi-  mum transmit power of Ma active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  In this paper, we maximize the SNR by jointly optimizing  beamforming vector v, passive IRS phase shift matrix Φ, and  active IRS phase shift matrix Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The optimization problem  can be formulated as  max  v,Φ,Ψ  SNR  (11a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  vHv = 1, Pr ≤ P max  r  ,  (11b)  |Φ(m, m)| = 1, if m ̸∈ Ω,  (11c)  |Φ(m, m)| = 0, otherwise,  (11d)  |Ψ(m, m)| ≤ βmax, if m ∈ Ω,  (11e)  |Ψ(m, m)| = 0, otherwise,  (11f)  where βmax is the amplification budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' It is notes that this  optimization problem is a non-convex problem with a constant  modulus constraint, and it is challenging to solve it directly in  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' In what follows, we propose the alternating optimiza-  tion algorithm to design the beamforming vector and hybrid  IRS phase shift matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' PROPOSED MAX-SNR-FP SCHEME  In this section, we construct a Max-SNR-FP method to  jointly optimize the beamforming vector v, passive IRS phase  shift matrix Φ, and active IRS phase shift matrix Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' In what  follows, we will alternately solve for v, Φ, and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Optimize v given Φ and Ψ  Firstly, we transform the power constraint in (11b) into a  convex constraint with respect to v as follows  Pr = vH �  ρsrPHH  srΨHΨHsr  �  v + Tr  �  σ2  rΨΨH�  ≤ P max  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (12)  Then, given Φ and Ψ, the optimal beamforming vector v can  be found by solving the following problem  max  v  vHA¯v  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' vHv = 1, (12),  (13)  where  A =(√ρsrbhH  rbΦHsr + √ρsrbhH  rbΨHsr + √ρsbhH  sb)H  (√ρsrbhH  rbΦHsr + √ρsrbhH  rbΨHsr + √ρsbhH  sb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (14)  It is clear that this problem is not convex, and in accordance  with the Taylor series expansion, we have  vHAv ≥ 2ℜ{¯vHAv} − ¯vHA¯v,  (15)  where ¯v is a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Then (13) can be recasted as  max  v  2ℜ{¯vHAv} − ¯vHA¯v  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' vHv = 1, (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (16)  It is a convex optimization problem and can be solved by  employing CVX tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Optimize Φ given v and Ψ  To simplify the SNR expression related to the phase shift  matrix Φ, we regard v and Ψ as two constants, and define  B = (√ρsrbhH  rbΨHsr + √ρsbhH  sb)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (17)  Then, the subproblem to optimize Φ can be expressed as  max  Φ  |√ρsrbhH  rbΦHsrv + B|2  (18a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' |Φ(m, m)| = 1, if m ̸∈ Ω,  (18b)  |Φ(m, m)| = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (18c)  By defining  C = ρsrbdiag{hH  rb}HsrvvHHH  srdiag{hH  rb}H,  (19)  and based on the fact that diag{a}b = diag{b}a for a, b ∈  CM×1, the objective function in (18) can be recasted as  φHCφ + 2ℜ{√ρsrbφHdiag{hH  rb}HsrvB∗} + |B|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (20)  Based on the Taylor series expansion, we have  φHCφ ≥ 2ℜ{ ¯φHCφ} − ¯φHC ¯φ,  (21)  where ¯φ is a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' For the unit modulus constraint  (18b), it can be relaxed as  |Φ(m, m)| ≤ 1, if m ̸∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (22)  At this point, the problem (18) can be rewritten as  max  Φ  2ℜ{ ¯φHCφ} − ¯φHC ¯φ + |B|2 + 2ℜ{√ρsrbφH•  diag{hH  rb}HsrvB∗}  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (22), (18c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (23)  We can find that it is a convex optimization problem and can  be solved by employing CVX tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Optimize Ψ given v and Φ  To optimize Ψ, we regard v and Φ as two given constants,  and transform the power constraint in (11b) into a convex  constraint on ψ as follows  Pr = Tr  �  Ψ  �  ρsrPHsrvvHHH  sr + σ2IM  �  ΨH�  = ψT (ρsrPdiag{vHHH  sr}diag{Hsrv} + σ2  rIM)ψ∗  ≤ P max  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (24)  By neglecting the constant terms, the subproblem with respect  to Ψ is given by  max  Ψ  |(√ρsrbhH  rbΨHsr + √ρsrbhH  rbΦHsr + √ρsbhH  sb)v|2  σ2r|√ρrbhH  rbΨ|2 + σ2  b  (25a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (11e), (11f), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (25b)  Let us define  D = (√ρsrbhH  rbΦHsr + √ρsbhH  sb)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (26)  Then, the objective function in (25) can be converted to  ψHCψ + 2ℜ{ψH√ρsrbdiag{hH  rb}HsrvD∗} + |D|2  σ2rρrb|ψHdiag{hH  rb}|2 + σ2  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (27)  4  At this point, the optimization problem (25) becomes a nonlin-  ear fractional optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Based on the FP strategy  in [14], we introduce a parameter τ and transform the objective  function (27) as  ψHCψ + 2ℜ{ψH√ρsrbdiag{hH  rb}HsrvD∗} + |D|2  − τ(σ2  rρrb|ψHdiag{hH  rb}|2 + σ2  b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (28)  The  optimal  solution  can  be  achieved  if  and  only  if ψHCψ + 2ℜ{ψH√ρsrbdiag{hH  rb}HsrvD∗} + |D|2 −  τ(σ2  rρrb|ψHdiag{hH  rb}|2 + σ2  b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' We linearize the ψHCψ  by employing Taylor series expansion at a given vector ¯ψ, the  subproblem with respect to Ψ can be rewritten as  max  Ψ,τ  2ℜ{ ¯ψHCψ} − ¯ψHC ¯ψ + 2ℜ{ψH√ρsrbdiag{hH  rb}•  HsrvD∗} + |D|2 − τ(σ2  rρrb|ψHdiag{hH  rb}|2 + σ2  b)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (11e), (11f), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (29)  It should be noted that this problem is convex, which can be  effectively solved by the CVX tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The whole procedure of  the Max-SNR-FP algorithm is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Algorithm 1 Proposed Max-SNR-FP algorithm  1: Initialize v(0), Φ(0), and Ψ(0), compute R(0)  b  based on (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  2: Set p = 0, threshold value ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  3: repeat  4:  Given Φ(p) and Ψ(p), solve (16) to determine v(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  5:  Given v(p+1) and Ψ(p), solve (23) to determine Φ(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  6:  Given v(p+1) and Φ(p+1), solve (29) to determine  Ψ(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  7:  Compute R(p+1)  b  using v(p+1), Φ(p+1), and Ψ(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  8:  p = p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  9: until |R(p)  b  − R(p−1)  b  | ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  The computational complexity of the proposed Max-SNR-  FP algorithm is O(L((M + 1)3 + 2MN 2 + 2M 2)In(1/ǫ) +  M 3+N 3+5M 2+2MN+2M+2MN 2) float-point operations  (FLOPs), where L is the numbers of alternating iterations, ǫ  denotes the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' PROPOSED MAX-SNR-EAR SCHEME  In the previous section, we proposed the Max-SNR-FP  method to design the beamforming v, IRS phase shift matrices  Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' However, it has a high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  To reduce the computational complexity, a low-complexity  method named Max-SNR-EAR is proposed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Optimize v given Φ and Ψ  Given IRS phase shift matrices Φ and Ψ, in accordance with  the principle of maximizing SLNR in [15], the beamforming  vector v can be optimized by solving the following problem  max  v  SLNR =  vHEv  vH(σ2  bIN)v  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' vHv = 1, (12),  (30)  where  E =ρsrbHH  srΦHhrbhH  rbΦHsr + ρsrbHH  srΨHhrbhH  rbΨHsr  + hsbhH  sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (31)  According to the Taylor series expansion and neglecting the  constant terms, the problem (30) can be recasted as  max  v  2ℜ{¯vHEv} − ¯vHE¯v  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  vHv = 1, (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (32)  Note that it is a convex optimization problem and can be  solved with CVX tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Optimize Φ and Ψ given v  Given beamforming vector v, we consider to design the  phase of hybrid IRS firstly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The confidential message received  by Bob through the cascade path is expressed as  PρsrbhH  rbΘHsrvvHHH  srΘHhrb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (33)  To maximize the confidential message of the cascade path, the  phase alignment method is employed to design the hybrid IRS  phase �θ, �θ is given by  �θ = [e(−iarg(s1)), · · · , e(−iarg(sM))]T ,  (34)  where s = diag{hH  rb}Hsrv, and si is the i-th element of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Next, inspired by the amplitude design of fully active IRS  in [9], we assume that all active IRS elements have the same  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Based on the IRS power constraint in (11b), we  have  |β| =  �  P max  r  /Q,  (35)  where  Q =Tr(�θH(ρsrPdiag{vHHH  srEMa}diag{vHHH  srEMa}H  + σ2EMaEMa)�θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (36)  Based on (34) and (35), we can obtain the passive IRS phase  shift matrix and active IRS phase shift matrix as follows  Φ = EMpdiag{�θ}, Ψ = |β|EMadiag{�θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  (37)  Similar to Algorithm 1, we calculate v, Φ, and Ψ alternately  until convergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=', |R(p)  b −R(p−1)  b  | ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The computational  complexity of Max-SNR-EAR algorithm is O(K(2M 2+N 3+  2M 2 + 8N 2M + 2MN) FLOPs, where K is the numbers of  alternating iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' SIMULATION RESULTS AND DISCUSSIONS  In this section, simulation results are presented to evaluate  the performance of two proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Simulation de-  fault parameters are chosen as follows: N = 8, M = 128,  Ma = 32, d = λ/2, θsr = π/4, θsb = π/3, dsr = 200m,  dsb = 220m, σ2  b = −70dBm, σ2  r = 2σ2  b, P = 25dBm,  P max  r  = 30dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The path loss at the distance d is modeled  as g(d) = PL0 − 10γlog10  d  d0 , where PL0 = −30dB is the  path loss reference distance d0 = 1m, and γ is the path loss  exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The path loss exponents of all channels are chosen  as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The positions of the IRS active elements are fixed to  Ω = {1, · · · , Ma}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  First, we make an investigation of the convergence be-  haviour of the proposed Max-SNR-FP and Max-SNR-EAR  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 2 shows the achievable rate versus the differ-  ent BS power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=', P = 20dBm, 25dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' It can be seen from  the figure that both of the proposed algorithms converge within  limited iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The proposed Max-SNR-EAR algorithm has  a faster convergence rate than the Max-SNR-FP algorithm,  regardless of P = 20dBm or 25dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  5  0  5  10  15  20  25  30  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='5  14  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='5  15  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='5  16  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='5  17  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Convergence of the proposed algorithms at different BS power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 3 depicts the curves of the achievable rate versus the  number of IRS phase shift elements, where Ma = M/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' We  compare two proposed algorithms to the benchmark schemes:  active IRS, passive IRS, no IRS, random phase IRS, and exist-  ing method in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The achievable rates of the proposed Max-  SNR-FP and Max-SNR-EAR algorithms gradually increase  as the number of IRS elements increases, and the former  is better than the latter and existing method in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' The  achievable rates of both the proposed algorithms are much  better than that of the passive IRS, no IRS and random phase  IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Moreover, the difference in achievable rates between both  the proposed algorithms and active IRS gradually decreases  when the number of IRS elements tends to large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  3  4  5  6  7  8  11  12  13  14  15  16  17  18  19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Achievable rate versus the numbers of IRS phase shift elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 4 plots the curves of the computational complexity  versus the number of IRS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' It can be found that the  complexities of the proposed Max-SNR-FP method, proposed  Max-SNR-EAR method, and existing method in [11] are  similar at small-scale IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' However, the complexities of the  existing method in [11] and proposed Max-SNR-FP method  are far higher than that of the proposed Max-SNR-EAR  method when the number of IRS elements tends to large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have made an investigation of the hybrid  IRS-aided DM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' To fully explore the advantages of  hybrid IRS and maximize the achievable rate, the Max-SNR-  FP and Max-SNR-EAR algorithms were proposed to jointly  design the beamforming vector, passive IRS phase shift matrix,  and active IRS phase shift matrix by alternately optimizing one  and fixing rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Simulation results showed that the achievable  2  3  4  5  6  7  0  2  4  6  8  10  12  14  107  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content=' Computational complexity versus the numbers of IRS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  rate of both proposed algorithms increases as the number of  IRS elements increases, and is much better than those of  the cases of random phase IRS, no IRS, and passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
+page_content='  Moreover, the proposed Max-SNR-FP method outperforms the  existing method in terms of the achievable rate and has lower  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQf_wKi/content/2301.02831v1.pdf'}
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+Prepared for submission to JINST
+Computational Models for High-Power Cyclotrons and
+FFAs
+Andreas Adelmann
+,𝑎 Chris T. Rogers,𝑏
+𝑎Paul Scherrer Institut,
+Forschungsstrasse 111 CH-5232 Villigen, Switzerland
+𝑏STFC Rutherford Appleton Laboratory,
+Harwell Science and Innovation Campus, Didcot, OX11 0QX, United Kingdom
+E-mail: [email protected], [email protected]
+Abstract: A summary of numerical modeling capabilities regarding high power cyclotrons and
+fixed field alternating gradient machines is presented. This paper focuses on techniques made
+available by the OPAL simulation code.
+Keywords: High Power Cyclotrons, High Power FFAs, Computational Models, OPAL
+1Corresponding author.
+arXiv:2301.01460v1  [physics.acc-ph]  4 Jan 2023
+
+Contents
+1
+Overview on Computational Models
+1
+1.1
+Single particle modeling
+1
+1.2
+Large Scale Multiparticle Modeling
+2
+1.3
+Surrogate Model Construction
+2
+2
+Physics Modeling
+2
+2.1
+Modeling H- Injection and Painting in Vertical and Horizontal FFAs
+2
+2.2
+Beam stripping interactions
+5
+2.3
+Spiral inflector modeling
+5
+2.4
+Neighboring Turn Modeling
+5
+3
+Path Forward
+6
+1
+Overview on Computational Models
+In all high-power particle accelerators "one of the major limitations is particle losses. Losses may
+be controlled, resulting in beam particles impinging on dedicated equipment such as collimators, or
+uncontrolled, resulting in beam particles striking other equipment around the accelerator. Uncon-
+trolled losses can damage and activate any equipment in the accelerator and so must be minimized.
+Controlled losses need to be carefully considered and also minimized. The amount and cause
+of loss are investigated by modeling accelerators using simulation codes that model numerically
+the behaviour of beams. A review of available numerical codes can be found in the article of
+Smirnov [1]. In this paper modeling capabilities available in OPAL are discussed in more detail
+[2].
+1.1
+Single particle modeling
+For conventional cyclotrons (and FFAs) the single particle tool box is established and many different
+codes variants exists [1]. For cyclotrons and (horizontal FFAs) the existing tools seem to be
+comfortable and accurate. New machines like vertical FFAs, currently studied for example at the
+Rutherford Appleton Laboratory (RAL) [3], require non–trivial modifications to the existing codes.
+These modifications are on the way for example in the code OPAL [2] and expected to be available
+in second quarter of 2022.
+Recently, in the context of very high field and ultra compact H− cyclotrons beam stripping
+losses of ion beams by interactions with residual gas and electromagnetic fields are evaluated [4].
+The beam stripping algorithm, implemented in OPAL, evaluates the interaction of hydrogen ions
+with residual gas and electromagnetic fields. In the first case, the cross sections of the processes
+are estimated according to the energy by means of analytical functions (see Sec. II-A c[4]). The
+– 1 –
+
+implementation allows the user to set the pressure, temperature, and composition of the residual
+gas, which could be selected for the calculations as either molecular hydrogen (H+
+2) or dry air in the
+usual proportion. For precise simulations, a two-dimensional pressure field map from an external
+file can be imported into OPAL, providing more realistic vacuum conditions.
+Concerning electromagnetic stripping, the electric dissociation lifetime is evaluated through
+the theoretical formalism (see Sec. II-B [4]). In both instances, the individual probability at each
+integration step for every particle is assessed.
+A stochastic process is used to evaluate if an interaction occurs. In this case the particle will
+be stripped and removed from the beam, or optionally transformed to a secondary heavy particle,
+dependent on the interaction. In this case, the secondary particle will continue its movement but
+with the new particle properties.
+1.2
+Large Scale Multiparticle Modeling
+In general, modeling losses in high intensity accelerators require 3D space-charge and sufficient
+simulation particles. Recent investigations [5] propose a sparse grid-based adaptive noise reduction
+strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto
+sparse grids, high-frequency particle noise is reduced and hence an optimal number of grid points
+and simulation particles can be obtained. For a 3D Penning trap simulation, a maximum speedup
+of 2.8 and 15 times memory reduction has been obtained. This method is already integrated into
+OPAL.
+1.3
+Surrogate Model Construction
+Cheap to evaluate surrogate models have gained a lot of interest lately. Statistical [6] or machine
+learning techniques are used [7]. These models can for example replace a computationally heavy
+model in a multi-objective optimization [8] or in the future be part of an on-line model. Some
+surrogate modeling algorithms may include an intrinsic estimator for the model uncertainty [9].
+2
+Physics Modeling
+In this section we show latest additions to the open source code OPAL [2] regarding cyclotron and
+FFA modeling capabilities.
+2.1
+Modeling H- Injection and Painting in Vertical and Horizontal FFAs
+Fixed Field Accelerators (FFAs) have fixed magnetic fields, like cyclotrons, but increase bending
+field with momentum and hence more compact designs can be realized. FFAs offer the power
+efficiency of cyclotrons combined with the energy reach of synchrotrons.
+FFAs have never been used for high power proton acceleration, however in OPAL the necessary
+models are available for design. Single particle tracking has been benchmarked against the KURNS
+FFA [10]. A design for a 3-12 MeV H- FFA prototype ring is being pursued at RAL as a prototype for
+a MW-class neutron spallation source [3]. Scaling horizontal orbit excursion (hFFA) and a vertical
+orbit excursion (vFFA) FFA are both under consideration. Both are non–isochronous machines
+using RF cavities with variable resonant frequency. Injection is planned using charge exchange of
+H− to H+ and phase space painting.
+– 2 –
+
+In hFFAs, magnetic rigidity varies with radius. The dipole field varies as [11]
+𝐵𝑧(𝑧 = 0) = 𝐵0(𝜓)
+� 𝑟
+𝑟0
+� 𝑘
+.
+(2.1)
+𝐵0(𝜓) is the dipole field as a function of a normalised azimuthal coordinate 𝜓, 𝑟 is the radial
+coordinate, 𝑟0 is a nominal (user-defined) radius, and 𝑘 is the field index. The field away from
+the midplane, at 𝑧 ≠ 0, may be calculated using a recursion relation arising from consideration
+of Maxwell’s equations in free space. OPAL has capability to calculate the expansion to arbitrary
+order, within machine precision. The normalised azimuthal coordinate
+𝜓 = 𝜙 − tan(𝛿) ln
+� 𝑟
+𝑟0
+�
+(2.2)
+is a measure of distance around the ring. Here 𝜙 is the geometrical azimuthal angle and 𝛿 is the
+spiral angle; for a sector FFA magnet 𝛿 = 0 and 𝜓 = 𝜙. The arrangement of fields in this way
+guarantees that single particle trajectories and optical parameters at all orders scale exactly with
+momentum.
+In vFFAs, magnetic rigidity varies with height.
+As particles are accelerated, the closed
+orbit changes height. Successive acceleration kicks add incoherently, so overall the beam follows
+the closed orbit with no appreciable emittance growth.
+Rectangular vFFA magnets have been
+implemented in OPAL, with a dipole field that varies as [12]
+𝐵0(𝑥𝑣 = 0) = 𝐵0(𝑠𝑣)𝑒𝑚𝑧𝑣 .
+(2.3)
+𝑧𝑣 is the height, 𝑠𝑣 is a nominal longitudinal coordinate and 𝑥𝑣 is a nominal horizontal coordinate
+in the rectangular coordinate system of the magnet. 𝐵0 describes the dipole field variation with
+longitudinal distance.
+A tanh model is available for vFFA fields.
+𝑚 is the vFFA field index,
+roughly equivalent to the field index 𝑘 in hFFAs. Fields away from the plane having 𝑥𝑣 = 0 are
+calculated using a field expansion derived from consideration of Maxwell’s laws. It is noted that
+the focusing in the magnet body is, to linear order, skew quadrupole. The fringe field has solenoid
+components parallel to 𝑠𝑣 that may be significant for short magnets. This arrangement of fields
+guarantees that trajectories and optical functions are identical as momentum increases, barring a
+vertical displacement. In particular, the path length of the beam is independent of momentum, the
+momentum compaction factor is exactly 0 and ultra-relativistic particles are isochronous.
+In order to model injection into the FFA, OPAL was extended with models for:
+• horizontal & vertical FFA magnets as described above;
+• variable frequency RF cavities;
+• arbitrary order multipoles with maxwellian fringe fields;
+• foil model (scattering and energy loss);
+• pulsed injected beam; and
+• pulsed multipoles.
+– 3 –
+
+ 
+ 
+H Bump 4
+catch H+ (x’)
+H Bump 5
+catch H+ (x)
+ 
+H Bump 2 moves H+ 
+bump orbit (r’)
+H Bump 1
+moves H+ 
+bump orbit
+(r)
+Merge H- and H+ in D 
+magnet
+Foil
+H- injection 
+Septum
+Vertical painting in
+injection line to select
+z, z’
+H = horizontal 
+bump
+H- 
+D F
+D = Defocusing
+F = Focusing
+Figure 1: Injection system for the hFFA (Left) field map of the hFFA, calculated using OPAL, with
+labels indicating the position of injection equipment (top right) closed orbits for different bump
+magnets (bottom right) required bump magnet fields.
+All but the latter two features are available in the latest version of OPAL. This enabled a fully
+four-dimensional simulation of the injection system, including consideration of effects such as
+appropriate phasing of the pulsed dipoles and transverse breathing of the beam arising due to initial
+longitudinal mismatch at injection.
+As an example, a schematic of an injection system and associated parameters for the 3-12 MeV
+test ring is shown for a horizontal FFA in Fig. 1. Owing to the compact nature of the ring, the
+injection system is spread across a number of cells. H− are brought into the ring and onto a foil.
+Bump magnets in the ring distort the proton closed orbit so that particles passing through the foil are
+returned to a nominal closed orbit. The foil is placed inside the defocusing (D) dipole magnet so that
+the distorted H+ closed orbit and H− beam, initially separated, are brought onto the same trajectory.
+Electrons are stripped from the H− leaving H+ (protons). The bump magnets are slowly varied, so
+that the proton closed orbit is moved away from the injection point for the H− and newly injected
+particles are at higher horizontal amplitude. In the H− injection line, pulsed magnets move the H−
+upwards so that newly injected particles are at higher vertical amplitude. Overall, a correlation is
+introduced between horizontal and vertical amplitude. Sample trajectories and bump magnet field
+strengths for the magnets in the ring are shown in Fig. 1. In this example vertical bumpers are not
+considered - they are all kept at 0 T field. The beam following injection is shown in fig. 2.
+– 4 –
+
+roΦ [m] for ro = 4.0 m
+0
+N
+4
+6
+8
+10
+12
+4.075
+4.050
+4.025
+4.000
+[m]
+3.975
+3.950
+3.925
+3.900
+3.875
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+[o] Φ0.00
+h bump 1
+hbump2
+hbump 3
+-0.02
+h bump 4
+hbump5
+0.04
+vbump 1
+[1] 
+v bump 2
+field
+0.06
+vbump3
+Bump
+v bump 4
+vbump 5
+-0.08
+-0.10
+0.12
+3900
+3920
+3940
+3960
+3980
+4000
+4020
+4040
+4060
+Radial position[mm]Orbit
+E
+4
+0.4
+B
+2
+0.2
+w
+0
+0.0
+-2
+0.2
+-0.4
+4
+-2
+0
+2
+4
+x [m]Figure 2: Beam (left) after injection is completed, but still on a distorted orbit (right) following
+collapse of the bump. 𝑥 is the position of the beam relative to the ring centre and 𝑦 is the height of
+the particle above the midplane. Particles are coloured according to the injection turn.
+2.2
+Beam stripping interactions
+Beam transmission optimization and loss characterization, where beam stripping interactions are
+a key issue, play an important role in the design and operation of compact cyclotrons. A beam
+stripping model has been implemented in the three-dimensional object-oriented parallel code OPAL-
+cycl, a flavor of the OPAL framework. The model includes Monte Carlo methods for interaction
+with residual gas and dissociation by electromagnetic stripping. The model has been verified with
+theoretical models and it has been applied to the AMIT cyclotron according to design conditions
+[4].
+2.3
+Spiral inflector modeling
+In [13] a spiral inflector model implemented in OPAL is presented, that enables us to run highly
+realistic simulations of the spiral inflector system of a compact cyclotron (c.f. Fig. 3). A new
+geometry class and field solver can handle the complicated boundary conditions posed by the
+electrode system in the central region of the cyclotron both in terms of particle termination, and
+calculation of self-fields. Results are benchmarked against the analytical solution of a coasting
+beam. As a practical example, the spiral inflector and the first revolution in a 1 MeV/amu test
+cyclotron, located at Best Cyclotron Systems, Inc., are modeled and compared to the simulation
+results [14, 15]. In conclusion, OPAL can handle realistic and arbitrary boundary geometries.
+Simulated injection efficiencies and beam shape compare well with measured efficiencies and a
+preliminary measurement of the beam distribution after injection.
+2.4
+Neighboring Turn Modeling
+This article presents a hardware architecture independent implementation of an adaptive mesh
+refinement Poisson solver that is integrated into the electrostatic Particle-In-Cell beam dynamics
+code OPAL. The Poisson solver is solely based on second generation Trilinos packages to ensure the
+desired hardware portability. Based on the massively parallel framework AMREX, formerly known
+– 5 –
+
+20
+[ww]
+0
+y
+-20
+3900
+3925
+3950
+3975
+4000
+4025
+4050
+4075
+4100
+x[mm]
+0.02
+0.24
+0.01
+0.22
+0.00
+0.20
+0.01
+0.18
+-0.02
+3900
+3950
+4000
+4050
+4100
+-20
+0
+20
+x[mm]
+y[mm]20
+[ww]
+0
+y
+-20
+3900
+3925
+3950
+3975
+4000
+4025
+4050
+4075
+4100
+x[mm]
+0.02
+0.24
+0.01
+0.22
+0.00
+0.20
+0.01
+0.18
+-0.02
+3900
+3950
+4000
+4050
+4100
+-20
+0
+20
+x[mm]
+y[mm]Figure 3: Spiral inflector with selected particle trajectories from an OPAL simulation.
+The
+beam enters axially (from the top) through an aperture (grey) and is bent into the mid-plane by a
+combination of the electrostatic field generated by the spiral electrodes (green and blue) and the
+cyclotron’s main magnetic field. Then it is accelerated by the two Dees (copper, Dummy-Dees not
+shown) [13].
+Figure 4: Integrated projection of the electric field component 𝐸𝑥 onto the xy-plane showing 7
+adjacent particle bunches [16].
+as BoxLib, the new adaptive mesh refinement interface provides several refinement policies in order
+to enable precise large-scale neighbouring bunch simulations in high intensity cyclotrons. The
+solver is validated with a built-in multigrid solver of AMREX and a test problem with analytical
+solution. The parallel scalability is presented as well as an example of a neighbouring bunch
+simulation that covers the scale of the later anticipated physics simulation [16].
+3
+Path Forward
+While statistical and machine learning techniques have a lot of potential, high fidelity physics
+simulations will always be used to, for example, produce the training set. In case of high-intensity
+machines we will need large numbers of particles and the associated fine mesh to solve the PDE in
+question. It is imperative that we make use of existing and future high performance infrastructure.
+– 6 –
+
+7.5
+102
+5.0
+(N)
+2.5
+cm
+101
+0
+2.5
+101
+5.0
+7.5
+1cm
+-102
+-10
+-5
+0
+5
+10
+x (cm)A performance portable implementation [16] is of utmost importance. The OPAL collaboration [2]
+is in the progress to completely rewrite the code according to the sketch in Fig. 5. With this new
+architecture we will be able to make efficient use of Exascale-Architecture that will come online
+soon. The core algorithms of OPAL are already performance portable as demonstrated in [17].
+Figure 5: Outlook of the future OPAL architecture, targeting in a performance portable way future
+exascale architectures.
+Acknowledgments
+The authors acknowledge the OPAL developer team for their continued support of this open source,
+community-driven code.
+References
+[1] V. Smirnov. Computer codes for beam dynamics analysis of cyclotronlike accelerators. Phys. Rev.
+Accel. Beams, 20:124801, 12 2017. doi: 10.1103/PhysRevAccelBeams.20.124801. URL
+https://link.aps.org/doi/10.1103/PhysRevAccelBeams.20.124801.
+[2] The OPAL Framework: Version 2.4, 2021.
+http://amas.web.psi.ch/opal/Documentation/2.4/index.html.
+[3] S. Machida, D. J. Kelliher, J-B. Lagrange, and C. T. Rogers. Optics design of vertical excursion
+fixed-field alternating gradient accelerators. Phys. Rev. Accel. Beams, 24:021601, 2 2021. doi:
+10.1103/PhysRevAccelBeams.24.021601. URL
+https://link.aps.org/doi/10.1103/PhysRevAccelBeams.24.021601.
+[4] P. Calvo, I. Podadera, D. Gavela, C. Oliver, A. Adelmann, J. Snuverink, and A. Gsell. Beam stripping
+interactions in compact cyclotrons. Phys. Rev. Accel. Beams, 24:090101, 11 2021. doi:
+10.1103/PhysRevAccelBeams.24.090101. URL
+https://link.aps.org/doi/10.1103/PhysRevAccelBeams.24.090101.
+[5] Sriramkrishnan Muralikrishnan, Antoine J. Cerfon, Matthias Frey, Lee F. Ricketson, and Andreas
+Adelmann. Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes. Journal
+of Computational Physics: X, 11:100094, 2021. ISSN 2590-0552. doi:
+– 7 –
+
+OPAL
+Kokkos aware Profiling and
+Trilinos
+IPPL
+Kokkos-Tools
+Debugging Tools)
+(Linear Solvers, Load Balancing,
+(Particles & Fields)
+Discretization,DistributedLinearAlgebra)
+Training)
+Kokkos-Kernels
+heFFTe
+(Sparse/DenseBLAS,GraphKernelsTensorKernels)
+Algorithms
+Containers
+(Random,Sort)
+(Map,CrsGraph, Mem Pool)
+Kokkos Core
+(Parallel Execution, Data Allocation, Data Transfer)
+std:thread
+OpenMP
+CUDA
+ROCmhttps://doi.org/10.1016/j.jcpx.2021.100094. URL
+https://www.sciencedirect.com/science/article/pii/S2590055221000111.
+[6] Andreas Adelmann. On nonintrusive uncertainty quantification and surrogate model construction in
+particle accelerator modeling. SIAM/ASA Journal on Uncertainty Quantification, 7(2):383–416, 2019.
+[7] Renato Bellotti, Romana Boiger, and Andreas Adelmann. Fast, efficient and flexible particle
+accelerator optimisation using densely connected and invertible neural networks. Information, 12(9),
+2021. doi: 10.3390/info12090351. URL https://www.mdpi.com/2078-2489/12/9/351.
+[8] Auralee Edelen, Nicole Neveu, Yannick Huber, Matthias Frey, and Andreas Adelmannn. Machine
+learning to enable orders of magnitude speedup in multi-objective optimization of particle accelerator
+systems’. Phys. Rev. AB, 23:044601, 2020. doi: 10.1103/PhysRevAccelBeams.23.044601. URL
+https://link.aps.org/doi/10.1103/PhysRevAccelBeams.23.044601.
+[9] Matthias Frey and Andreas Adelmann. Global sensitivity analysis on numerical solver parameters of
+particle-in-cell models in particle accelerator systems. Computer Physics Communications, 258:
+107577, 2021. ISSN 0010-4655. doi: https://doi.org/10.1016/j.cpc.2020.107577. URL
+http://www.sciencedirect.com/science/article/pii/S0010465520302770.
+[10] Suzanne Sheehy et al. Progress on Simulation of Fixed Field Alternating Gradient Accelerators. In
+6th International Particle Accelerator Conference, page MOPJE077, 2015. doi:
+10.18429/JACoW-IPAC2015-MOPJE077.
+[11] K. R. Symon, D. W. Kerst, L. W. Jones, L. J. Laslett, and K. M. Terwilliger. Fixed-field
+alternating-gradient particle accelerators. Phys. Rev., 103:1837–1859, Sep 1956. doi:
+10.1103/PhysRev.103.1837. URL https://link.aps.org/doi/10.1103/PhysRev.103.1837.
+[12] Stephen Brooks. Vertical orbit excursion fixed field alternating gradient accelerators. Phys. Rev. ST
+Accel. Beams, 16:084001, Aug 2013. doi: 10.1103/PhysRevSTAB.16.084001. URL
+https://link.aps.org/doi/10.1103/PhysRevSTAB.16.084001.
+[13] Daniel Winklehner, Andreas Adelmann, Achim Gsell, Tulin Kaman, and Daniela Campo. Realistic
+simulations of a cyclotron spiral inflector within a particle-in-cell framework. Physical Review
+Accelerators and Beams, 20(12):124201, 12 2017. doi: 10.1103/PhysRevAccelBeams.20.124201.
+URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.20.124201.
+[14] Daniel Winklehner, Andreas Adelmann, Achim Gsell, Tulin Kaman, and Daniela Campo. Realistic
+simulations of a cyclotron spiral inflector within a particle-in-cell framework. Phys. Rev. Accel.
+Beams, 20:124201, Dec 2017. doi: 10.1103/PhysRevAccelBeams.20.124201. URL
+https://link.aps.org/doi/10.1103/PhysRevAccelBeams.20.124201.
+[15] J. Alonso, S. Axani, L. Calabretta, D. Campo, L. Celona, J.M. Conrad, A. Day, G. Castro,
+F. Labrecque, and D. Winklehner. The isodar high intensity h2+ transport and injection tests. Journal
+of Instrumentation, 10(10):T10003, oct 2015. doi: 10.1088/1748-0221/10/10/T10003. URL
+https://dx.doi.org/10.1088/1748-0221/10/10/T10003.
+[16] Matthias Frey, Andreas Adelmann, and Uldis Locans. On architecture and performance of adaptive
+mesh refinement in an electrostatics particle-in-cell code (vol 247, 106912, 2020). COMPUTER
+PHYSICS COMMUNICATIONS, 265, 2021.
+[17] Sriramkrishnan Muralikrishnan, Matthias Frey, Alessandro Vinciguerra, Michael Ligotino, Antoine J.
+Cerfon, Miroslav Stoyanov, Rahulkumar Gayatri, and Andreas Adelmann. Alpine: A set of
+performance portable plasma physics particle-in-cell mini-apps for exascale computing, 2022. URL
+arXiv:2205.11052.
+– 8 –
+

5tAzT4oBgHgl3EQff_wz/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf,len=429
+page_content='Prepared for submission to JINST  Computational Models for High-Power Cyclotrons and  FFAs  Andreas Adelmann  ,𝑎 Chris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Rogers,𝑏  𝑎Paul Scherrer Institut,  Forschungsstrasse 111 CH-5232 Villigen, Switzerland  𝑏STFC Rutherford Appleton Laboratory,  Harwell Science and Innovation Campus, Didcot, OX11 0QX, United Kingdom  E-mail: andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='adelmann@psi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='ch, chris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='rogers@stfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='uk  Abstract: A summary of numerical modeling capabilities regarding high power cyclotrons and  fixed field alternating gradient machines is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' This paper focuses on techniques made  available by the OPAL simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Keywords: High Power Cyclotrons, High Power FFAs, Computational Models, OPAL  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='01460v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='acc-ph]  4 Jan 2023  Contents  1  Overview on Computational Models  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='1  Single particle modeling  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='2  Large Scale Multiparticle Modeling  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='3  Surrogate Model Construction  2  2  Physics Modeling  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='1  Modeling H- Injection and Painting in Vertical and Horizontal FFAs  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='2  Beam stripping interactions  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='3  Spiral inflector modeling  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='4  Neighboring Turn Modeling  5  3  Path Forward  6  1  Overview on Computational Models  In all high-power particle accelerators "one of the major limitations is particle losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Losses may  be controlled, resulting in beam particles impinging on dedicated equipment such as collimators, or  uncontrolled, resulting in beam particles striking other equipment around the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Uncon-  trolled losses can damage and activate any equipment in the accelerator and so must be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Controlled losses need to be carefully considered and also minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The amount and cause  of loss are investigated by modeling accelerators using simulation codes that model numerically  the behaviour of beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' A review of available numerical codes can be found in the article of  Smirnov [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In this paper modeling capabilities available in OPAL are discussed in more detail  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='1  Single particle modeling  For conventional cyclotrons (and FFAs) the single particle tool box is established and many different  codes variants exists [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' For cyclotrons and (horizontal FFAs) the existing tools seem to be  comfortable and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' New machines like vertical FFAs, currently studied for example at the  Rutherford Appleton Laboratory (RAL) [3], require non–trivial modifications to the existing codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  These modifications are on the way for example in the code OPAL [2] and expected to be available  in second quarter of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Recently, in the context of very high field and ultra compact H− cyclotrons beam stripping  losses of ion beams by interactions with residual gas and electromagnetic fields are evaluated [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  The beam stripping algorithm, implemented in OPAL, evaluates the interaction of hydrogen ions  with residual gas and electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In the first case, the cross sections of the processes  are estimated according to the energy by means of analytical functions (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' II-A c[4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The  – 1 –  implementation allows the user to set the pressure, temperature, and composition of the residual  gas, which could be selected for the calculations as either molecular hydrogen (H+  2) or dry air in the  usual proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' For precise simulations, a two-dimensional pressure field map from an external  file can be imported into OPAL, providing more realistic vacuum conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Concerning electromagnetic stripping, the electric dissociation lifetime is evaluated through  the theoretical formalism (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' II-B [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In both instances, the individual probability at each  integration step for every particle is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  A stochastic process is used to evaluate if an interaction occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In this case the particle will  be stripped and removed from the beam, or optionally transformed to a secondary heavy particle,  dependent on the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In this case, the secondary particle will continue its movement but  with the new particle properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='2  Large Scale Multiparticle Modeling  In general, modeling losses in high intensity accelerators require 3D space-charge and sufficient  simulation particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Recent investigations [5] propose a sparse grid-based adaptive noise reduction  strategy for electrostatic particle-in-cell (PIC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' By projecting the charge density onto  sparse grids, high-frequency particle noise is reduced and hence an optimal number of grid points  and simulation particles can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' For a 3D Penning trap simulation, a maximum speedup  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='8 and 15 times memory reduction has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' This method is already integrated into  OPAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='3  Surrogate Model Construction  Cheap to evaluate surrogate models have gained a lot of interest lately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Statistical [6] or machine  learning techniques are used [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' These models can for example replace a computationally heavy  model in a multi-objective optimization [8] or in the future be part of an on-line model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Some  surrogate modeling algorithms may include an intrinsic estimator for the model uncertainty [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  2  Physics Modeling  In this section we show latest additions to the open source code OPAL [2] regarding cyclotron and  FFA modeling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='1  Modeling H- Injection and Painting in Vertical and Horizontal FFAs  Fixed Field Accelerators (FFAs) have fixed magnetic fields, like cyclotrons, but increase bending  field with momentum and hence more compact designs can be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' FFAs offer the power  efficiency of cyclotrons combined with the energy reach of synchrotrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  FFAs have never been used for high power proton acceleration, however in OPAL the necessary  models are available for design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Single particle tracking has been benchmarked against the KURNS  FFA [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' A design for a 3-12 MeV H- FFA prototype ring is being pursued at RAL as a prototype for  a MW-class neutron spallation source [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Scaling horizontal orbit excursion (hFFA) and a vertical  orbit excursion (vFFA) FFA are both under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Both are non–isochronous machines  using RF cavities with variable resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Injection is planned using charge exchange of  H− to H+ and phase space painting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  – 2 –  In hFFAs, magnetic rigidity varies with radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The dipole field varies as [11]  𝐵𝑧(𝑧 = 0) = 𝐵0(𝜓)  � 𝑟  𝑟0  � 𝑘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='1)  𝐵0(𝜓) is the dipole field as a function of a normalised azimuthal coordinate 𝜓, 𝑟 is the radial  coordinate, 𝑟0 is a nominal (user-defined) radius, and 𝑘 is the field index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The field away from  the midplane, at 𝑧 ≠ 0, may be calculated using a recursion relation arising from consideration  of Maxwell’s equations in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' OPAL has capability to calculate the expansion to arbitrary  order, within machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The normalised azimuthal coordinate  𝜓 = 𝜙 − tan(𝛿) ln  � 𝑟  𝑟0  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='2)  is a measure of distance around the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Here 𝜙 is the geometrical azimuthal angle and 𝛿 is the  spiral angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' for a sector FFA magnet 𝛿 = 0 and 𝜓 = 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The arrangement of fields in this way  guarantees that single particle trajectories and optical parameters at all orders scale exactly with  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  In vFFAs, magnetic rigidity varies with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  As particles are accelerated, the closed  orbit changes height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Successive acceleration kicks add incoherently, so overall the beam follows  the closed orbit with no appreciable emittance growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Rectangular vFFA magnets have been  implemented in OPAL, with a dipole field that varies as [12]  𝐵0(𝑥𝑣 = 0) = 𝐵0(𝑠𝑣)𝑒𝑚𝑧𝑣 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='3)  𝑧𝑣 is the height, 𝑠𝑣 is a nominal longitudinal coordinate and 𝑥𝑣 is a nominal horizontal coordinate  in the rectangular coordinate system of the magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 𝐵0 describes the dipole field variation with  longitudinal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  A tanh model is available for vFFA fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  𝑚 is the vFFA field index,  roughly equivalent to the field index 𝑘 in hFFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Fields away from the plane having 𝑥𝑣 = 0 are  calculated using a field expansion derived from consideration of Maxwell’s laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' It is noted that  the focusing in the magnet body is, to linear order, skew quadrupole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The fringe field has solenoid  components parallel to 𝑠𝑣 that may be significant for short magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' This arrangement of fields  guarantees that trajectories and optical functions are identical as momentum increases, barring a  vertical displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In particular, the path length of the beam is independent of momentum, the  momentum compaction factor is exactly 0 and ultra-relativistic particles are isochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  In order to model injection into the FFA, OPAL was extended with models for:  horizontal & vertical FFA magnets as described above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  variable frequency RF cavities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  arbitrary order multipoles with maxwellian fringe fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  foil model (scattering and energy loss);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  pulsed injected beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' and  pulsed multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  – 3 –  H Bump 4  catch H+ (x’)  H Bump 5  catch H+ (x)  H Bump 2 moves H+  bump orbit (r’)  H Bump 1  moves H+  bump orbit  (r)  Merge H- and H+ in D  magnet  Foil  H- injection  Septum  Vertical painting in  injection line to select  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' z’  H = horizontal  bump  H-  D F  D = Defocusing  F = Focusing  Figure 1: Injection system for the hFFA (Left) field map of the hFFA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' calculated using OPAL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' with  labels indicating the position of injection equipment (top right) closed orbits for different bump  magnets (bottom right) required bump magnet fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  All but the latter two features are available in the latest version of OPAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' This enabled a fully  four-dimensional simulation of the injection system, including consideration of effects such as  appropriate phasing of the pulsed dipoles and transverse breathing of the beam arising due to initial  longitudinal mismatch at injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  As an example, a schematic of an injection system and associated parameters for the 3-12 MeV  test ring is shown for a horizontal FFA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Owing to the compact nature of the ring, the  injection system is spread across a number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' H− are brought into the ring and onto a foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Bump magnets in the ring distort the proton closed orbit so that particles passing through the foil are  returned to a nominal closed orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The foil is placed inside the defocusing (D) dipole magnet so that  the distorted H+ closed orbit and H− beam, initially separated, are brought onto the same trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Electrons are stripped from the H− leaving H+ (protons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The bump magnets are slowly varied, so  that the proton closed orbit is moved away from the injection point for the H− and newly injected  particles are at higher horizontal amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In the H− injection line, pulsed magnets move the H−  upwards so that newly injected particles are at higher vertical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Overall, a correlation is  introduced between horizontal and vertical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Sample trajectories and bump magnet field  strengths for the magnets in the ring are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In this example vertical bumpers are not  considered - they are all kept at 0 T field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The beam following injection is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  – 4 –  roΦ [m] for ro = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='0 m  0  N  4  6  8  10  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='075  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='050  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='025  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='000  [m]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='975  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='950  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='925  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='900  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='875  0  20  40  60  80  100  120  140  160  180  [o] Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='00  h bump 1  hbump2  hbump 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='02  h bump 4  hbump5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='04  vbump 1  [1]  v bump 2  field  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='06  vbump3  Bump  v bump 4  vbump 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='12  3900  3920  3940  3960  3980  4000  4020  4040  4060  Radial position[mm]Orbit  E  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='4  4  2  0  2  4  x [m]Figure 2: Beam (left) after injection is completed, but still on a distorted orbit (right) following  collapse of the bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 𝑥 is the position of the beam relative to the ring centre and 𝑦 is the height of  the particle above the midplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Particles are coloured according to the injection turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='2  Beam stripping interactions  Beam transmission optimization and loss characterization, where beam stripping interactions are  a key issue, play an important role in the design and operation of compact cyclotrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' A beam  stripping model has been implemented in the three-dimensional object-oriented parallel code OPAL-  cycl, a flavor of the OPAL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The model includes Monte Carlo methods for interaction  with residual gas and dissociation by electromagnetic stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The model has been verified with  theoretical models and it has been applied to the AMIT cyclotron according to design conditions  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='3  Spiral inflector modeling  In [13] a spiral inflector model implemented in OPAL is presented, that enables us to run highly  realistic simulations of the spiral inflector system of a compact cyclotron (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' A new  geometry class and field solver can handle the complicated boundary conditions posed by the  electrode system in the central region of the cyclotron both in terms of particle termination, and  calculation of self-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Results are benchmarked against the analytical solution of a coasting  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' As a practical example, the spiral inflector and the first revolution in a 1 MeV/amu test  cyclotron, located at Best Cyclotron Systems, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=', are modeled and compared to the simulation  results [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In conclusion, OPAL can handle realistic and arbitrary boundary geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Simulated injection efficiencies and beam shape compare well with measured efficiencies and a  preliminary measurement of the beam distribution after injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='4  Neighboring Turn Modeling  This article presents a hardware architecture independent implementation of an adaptive mesh  refinement Poisson solver that is integrated into the electrostatic Particle-In-Cell beam dynamics  code OPAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The Poisson solver is solely based on second generation Trilinos packages to ensure the  desired hardware portability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Based on the massively parallel framework AMREX, formerly known  – 5 –  20  [ww]  0  y  20  3900  3925  3950  3975  4000  4025  4050  4075  4100  x[mm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='02  3900  3950  4000  4050  4100  20  0  20  x[mm]  y[mm]Figure 3: Spiral inflector with selected particle trajectories from an OPAL simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  The  beam enters axially (from the top) through an aperture (grey) and is bent into the mid-plane by a  combination of the electrostatic field generated by the spiral electrodes (green and blue) and the  cyclotron’s main magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Then it is accelerated by the two Dees (copper, Dummy-Dees not  shown) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Figure 4: Integrated projection of the electric field component 𝐸𝑥 onto the xy-plane showing 7  adjacent particle bunches [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  as BoxLib, the new adaptive mesh refinement interface provides several refinement policies in order  to enable precise large-scale neighbouring bunch simulations in high intensity cyclotrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The  solver is validated with a built-in multigrid solver of AMREX and a test problem with analytical  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The parallel scalability is presented as well as an example of a neighbouring bunch  simulation that covers the scale of the later anticipated physics simulation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  3  Path Forward  While statistical and machine learning techniques have a lot of potential, high fidelity physics  simulations will always be used to, for example, produce the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In case of high-intensity  machines we will need large numbers of particles and the associated fine mesh to solve the PDE in  question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' It is imperative that we make use of existing and future high performance infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  – 6 –  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='5  102  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='0  (N)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='5  cm  101  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='5  101  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='5  1cm  102  10  5  0  5  10  x (cm)A performance portable implementation [16] is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The OPAL collaboration [2]  is in the progress to completely rewrite the code according to the sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' With this new  architecture we will be able to make efficient use of Exascale-Architecture that will come online  soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' The core algorithms of OPAL are already performance portable as demonstrated in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Figure 5: Outlook of the future OPAL architecture, targeting in a performance portable way future  exascale architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Acknowledgments  The authors acknowledge the OPAL developer team for their continued support of this open source,  community-driven code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  References  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Computer codes for beam dynamics analysis of cyclotronlike accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' Machida, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='  [9] Matthias Frey and Andreas Adelmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Global sensitivity analysis on numerical solver parameters of  particle-in-cell models in particle accelerator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='  [10] Suzanne Sheehy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Progress on Simulation of Fixed Field Alternating Gradient Accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' In  6th International Particle Accelerator Conference, page MOPJE077, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='  [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' Symon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Kerst, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Laslett, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Terwilliger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Fixed-field  alternating-gradient particle accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='  [12] Stephen Brooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Vertical orbit excursion fixed field alternating gradient accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' ST  Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' Realistic  simulations of a cyclotron spiral inflector within a particle-in-cell framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Physical Review  Accelerators and Beams, 20(12):124201, 12 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content='124201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  [14] Daniel Winklehner, Andreas Adelmann, Achim Gsell, Tulin Kaman, and Daniela Campo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content=' Realistic  simulations of a cyclotron spiral inflector within a particle-in-cell framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
+page_content='  Beams, 20:124201, Dec 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQff_wz/content/2301.01460v1.pdf'}
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+This is a post-peer-review, pre-copyedit version of an article published in Autonomous Robots.
+The final authenticated version is available online at: http://dx.doi.org/10.1007/s10514-017-9682-5
+Long-Term Online Multi-Session Graph-Based SPLAM with
+Memory Management
+Mathieu Labb´e · Fran¸cois Michaud
+Abstract For long-term simultaneous planning, local-
+ization and mapping (SPLAM), a robot should be able
+to continuously update its map according to the dy-
+namic changes of the environment and the new areas
+explored. With limited onboard computation capabili-
+ties, a robot should also be able to limit the size of the
+map used for online localization and mapping. This pa-
+per addresses these challenges using a memory manage-
+ment mechanism, which identifies locations that should
+remain in a Working Memory (WM) for online pro-
+cessing from locations that should be transferred to
+a Long-Term Memory (LTM). When revisiting previ-
+ously mapped areas that are in LTM, the mechanism
+can retrieve these locations and place them back in WM
+for online SPLAM. The approach is tested on a robot
+equipped with a short-range laser rangefinder and a
+RGB-D camera, patrolling autonomously 10.5 km in
+an indoor environment over 11 sessions while having
+encountered 139 people.
+Keywords SLAM · path planning · pose graph ·
+multi-session · loop closure detection
+1 Introduction
+The ability to simultaneously map an environment, lo-
+calize itself in it, and plan paths using this information
+This work was supported by the Natural Sciences and Engi-
+neering Research Council of Canada (NSERC), the Canada
+Research Chair program and the Canadian Foundation for
+Innovation.
+M. Labb´e
+E-mail: [email protected]
+F. Michaud
+E-mail: [email protected]
+Interdisciplinary Institute for Technological Innovation (3IT),
+Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada
+is known as Simultaneous Planning, Localization And
+Mapping, or SPLAM (Stachniss, 2009). This task can
+be particularly complex when done online on a robot
+with limited computing resources in large, unstructured
+and dynamic environments. Since SPLAM can be seen
+as an extension of Simultaneous Localization And Map-
+ping (SLAM), many approaches exist (Thrun et al.,
+2005). Our interest lies with graph-based SLAM ap-
+proaches (Grisetti et al., 2010), for which combining
+a lightweight topological map over a detailed metrical
+map reveals to be more suitable for large-scale mapping
+and navigation (Konolige et al., 2011).
+Two important challenges in graph-based SPLAM
+are :
+– Multi-session mapping, also known as the kidnapped
+robot problem or the initial state problem: when
+turned on, a robot does not know its relative po-
+sition to a map previously created, making it im-
+possible to plan a path to a previously visited loca-
+tion. A solution is to have the robot localize itself
+in a previously-built map before initiating mapping.
+This solution has the advantage of always using the
+same referential, resulting in only one map is created
+across the sessions. However, the robot must start
+in a portion already mapped of the environment.
+Another approach is to initialize a new map with
+its own referential on startup, and when a previ-
+ously visited location is encountered, a transforma-
+tion between the two maps can be computed. The
+transformations between the maps can be saved ex-
+plicitly with special nodes called anchor nodes (Mc-
+Donald et al., 2012; Kim et al., 2010), or implicitly
+with links added between each map (Konolige and
+Bowman, 2009; Latif et al., 2013). This process is
+referred to as loop closure detection. Loop closure
+detection approaches that are independent of the
+arXiv:2301.00050v1  [cs.RO]  30 Dec 2022
+
+2
+Mathieu Labb´e, Fran¸cois Michaud
+robot’s estimated position (Ho and Newman, 2006)
+can intrinsically detect if the current location is a
+new location or a previously visited one among all
+the mapping sessions conducted in the past. Popular
+loop closure detection approaches are appearance-
+based (Garcia-Fidalgo and Ortiz, 2015), exploiting
+the distinctiveness of images of the environment.
+The underlying idea is that loop closure detection
+is done by comparing all previous images with the
+new one. When loop closures are found between the
+maps, a global map can be created by combining
+the maps from each session. In graph-based SLAM,
+graph pose optimization approaches (Folkesson and
+Christensen, 2007; Grisetti et al., 2007; Kummerle
+et al., 2011; Johannsson et al., 2013) use these loop
+closures to reduce odometry errors inside each map
+and in between the maps.
+– Long-term mapping in dynamic environments. Per-
+sistent (Milford and Wyeth, 2010), lifelong (Kono-
+lige and Bowman, 2009) or continuous (Pirker et al.,
+2011) are terms generally used to describe SLAM
+approaches working in such conditions. Continu-
+ously updating and adding new data to the map in
+unbounded or dynamic environments will inevitably
+increase the map size over time. Online simulta-
+neous planning, localization and mapping requires
+that new incoming data be processed faster than
+the time to acquire them. For example, if data are
+acquired at 1 Hz, updating the map should be done
+in less than 1 sec. As the map grows, the time re-
+quired for loop closure detection and graph opti-
+mization increases, and eventually limits the size of
+the environment that can be mapped and used on-
+line.
+To address these challenges, we introduce SPLAM-
+MM, a graph-based SPLAM with a memory manage-
+ment (MM) mechanism. As demonstrated in (Labbe
+and Michaud, 2013), memory management can be used
+to limit the size of the map so that loop closure detec-
+tions are always processed under a fixed time limit, thus
+satisfying online requirements for long-term and large-
+scale environment mapping. The idea behind SPLAM-
+MM is to limit the number of nodes available for
+loop closure detection and graph optimization, keeping
+enough observations in the map for successful online
+localization and planning while still having the ability
+to generate a global representation of the environment
+that can adapt to changes over time.
+The paper is organized as follows. Section 2 reviews
+graph-based SLAM approaches that reduce the size of
+the map when revisiting the same environment while
+continuously adapting to dynamic changes. Section 3
+describes the implementation and the operating prin-
+ciples associated with the use of memory management
+with a graph-based SPLAM approach, which extends
+our previous metric-based SLAM approach (Labbe and
+Michaud, 2014) with a new planning capability. The
+implementation integrates four algorithms: loop clo-
+sure detection (Labbe and Michaud, 2013), graph opti-
+mization (Grisetti et al., 2007), metrical path planner
+(Marder-Eppstein et al., 2010) and a custom topological
+path planner. Section 4 presents experimental results of
+11 SPLAM sessions using the AZIMUT-3 robot in an
+indoor environment over 10.5 km. Section 5 discusses
+strengths and limitations of SPLAM-MM, and Section
+6 concludes the paper.
+2 Related Work
+Lifelong appearance-based SLAM requires dealing with
+dynamic environments. Glover et al. (2010) present an
+appearance-based SLAM approach that had to oper-
+ate in different lighting conditions over three weeks.
+An interesting observation from their experiments is
+that even when revisiting the same locations, the map
+still grows: in dynamic environments, the loop closure
+detector is sometimes unable to detect loop closures,
+duplicating locations in the map. A map management
+approach is therefore required to limit map size. In
+highly dynamic environments, multiple views of the
+same location may also be required for proper local-
+ization. Churchill and Newman (2012) present a graph-
+based SLAM approach where visual experiences of the
+same locations are kept in the map, to increase localiza-
+tion robustness to dynamic changes caused for instance
+by outdoor illumination conditions. If localization fails
+when revisiting an area, new experiences are added to
+the map. Even if adding new visual experiences to the
+map happens less often over time (as the robot explores
+the same location), there is no mechanism to limit this.
+Pirker et al. (2011) present a continuous monocular
+SLAM approach where new key frames are added to
+the map only when the environment has changed, to
+keep its size proportional to the explored space. But if
+the environment changes very often, there is no mech-
+anism to limit the number of key frames over the same
+physical location.
+Some
+SLAM
+approaches
+can
+handle
+dynamic
+changes of the environment while limiting the size of
+the map for long-term operation. Biber et al. (2005)
+present a sample-based representation for maps, to han-
+dle changes at different timescales, tracking both sta-
+tionary and non-stationary elements of the environ-
+ment. The idea is to refresh samples stored for each
+timescale with new sensor measurements. Map growth
+is then indirectly limited as older memories fade at
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+3
+different rates depending on the timescale. Walcott-
+Bryant et al. (2012) describe Dynamic Pose-Graph
+SLAM (DPG-SLAM), a long-term mapping approach
+that detects static and dynamic changes of the environ-
+ment through time. To keep consistency of the graph
+while reducing its size, nodes that are not observable
+anymore are removed. Johannsson et al. (2013) also re-
+move unobservable nodes to limit the size of the map
+over time when revisiting the same area. Similar nodes
+of the graph are merged together while keeping only the
+new loop closure detection. However, the graph size is
+not bounded when exploring new areas. Krajn´ık et al.
+(2016) present an occupancy grid approach where each
+cell in the map estimates its occupancy value depend-
+ing on periodical and cyclic changes occurring in the
+environment. This increases localization and navigation
+accuracy in dynamic environments compared to static
+maps, as the predicted map represents the correct state
+of the environment at that time of the day (e.g., doors
+can change to be opened or closed). The maximum
+data kept for each cell is bounded by some parameters
+(depending on the smallest and longest cyclic periods
+that should be detected), thus keeping memory usage
+fixed. However, the approach assumes that the navi-
+gation phase always occur in the same environment as
+the first mapping cycle, without possibility to extend it
+afterward.
+These problems of lifelong SLAM are also addressed
+in some SPLAM approaches. Milford and Wyeth (2010)
+present a solution to limit the size of the map (called
+experience map) while revisiting the same area: close
+nodes are merged together up to a maximum density
+threshold. This approach has the advantage of mak-
+ing the map size independent of the operating time,
+but the diversity of the observations on each location is
+somewhat lost. Konolige et al. (2011) use a view-based
+graph SLAM approach (Konolige and Bowman, 2009)
+in a SPLAM context. The approach preserves diversity
+of the images referring to the same location so that the
+map can handle dynamic changes over time, and forget-
+ting images limits the size of the graph over time when
+revisiting the same area. However, the graph still grows
+when visiting new areas.
+Overall, these approaches reduce map size when re-
+visiting the same area, while continuously adapting to
+dynamic changes. This makes them independent or al-
+most independent of the operation time of the robot in
+these conditions, but they are all limited to a maximum
+size of the environment that can be mapped online. The
+SPLAM-MM approach deals specifically with this lim-
+itation.
+Fig. 1 The AZIMUT-3 robot equipped with a URG-04LX
+laser range finder and a Xtion PRO LIVE sensor.
+3 Memory Management for SPLAM
+The underlying representation of SPLAM-MM is a
+graph with nodes and links. The nodes contain the fol-
+lowing information:
+– ID: unique time index of the node.
+– Weight: an indication of the importance of the node,
+used for memory management.
+– Bag-of-words (BOW): visual words used for loop
+closure detections. They are SURF features (Bay
+et al., 2008) quantized to an incremental vocabu-
+lary based on KD-Trees.
+– Sensor data: used to find similarities between nodes
+and to construct maps. For this paper, our imple-
+mentation of SPLAM-MM is using the AZIMUT-3
+robot (Ferland et al., 2010), equipped with an URG-
+04LX laser rangefinder and a Xtion Pro Live RGB-D
+camera, as shown by Fig. 1. The sensory data used
+are:
+– Pose: the position of the robot computed by its
+odometry system (e.g., the value given by wheel
+odometry), expressed in (x, y, θ) coordinates.
+– RGB image: used to extract visual words.
+– Depth image: used to find 3D position of the vi-
+sual words. The depth image is registered with
+the RGB image, i.e., each depth pixel corre-
+sponds exactly to the same RGB pixel.
+– Laser scan: used for loop closure transformations
+and odometry refinements, and by the Proximity
+Detection module.
+The links store rigid transformations (i.e., Eucledian
+transformation derived from odometry or loop closures)
+between nodes. There are four types of links:
+
+URG-04LX
+Xtion PRO LIVE
+AZIMUT-34
+Mathieu Labb´e, Fran¸cois Michaud
+Motion Controller 
+Waypoints 
+ Graph-based SLAM-MM 
+WM 
+STM 
+ SPLAM-MM 
+Graph-based 
+ SLAM-MM 
+Wheel Odometry 
+Laser Rangefinder 
+RGB-D Camera 
+Motion Controller 
+Topological Path 
+Planner (TPP) 
+Twist 
+Pose 
+Scan 
+RGB-D  
+Image 
+Local Map 
+Upcoming Node IDs 
+Metrical Path 
+Planner (MPP) 
+Pose 
+User 
+Goal 
+Appearance-based Loop 
+Closure Detection 
+Graph 
+Optimization 
+New  
+Link(s) 
+New 
+Node 
+Local Map 
+Proximity Detection 
+Sensor Data 
+Sensors 
+Global Map 
+LTM 
+Transferred 
+Nodes 
+Retrieved  
+Nodes 
+Global Map 
+Upcoming Node IDs  
+Patrol 
+Goal 
+Status 
+Waypoints 
+Topological Path 
+Planner (TPP) 
+Twist 
+Metrical Path 
+Planner (MPP) 
+Pose 
+User 
+Goal 
+Patrol 
+Goal 
+Status 
+Fig. 2 Memory management and control architecture of SPLAM-MM.
+– Neighbor link: created between a new node and the
+previous one.
+– Loop closure link: added when a loop closure is de-
+tected between the new node and one in the map.
+– Proximity link: added when two close nodes are
+aligned together.
+– Temporary link: used for path planning purposes. It
+is used to keep the planned path connected to the
+current map.
+Figure 2 presents a high-level representation of
+SPLAM-MM. Basically, it consists of a graph-based
+SLAM module with memory management, to which
+path planners are added. Memory management involves
+the use of a Working Memory (WM) and a Long-Term
+Memory (LTM). WM is where maps, which are graphs
+of nodes and links, are processed. To satisfy online con-
+straints, nodes can be transferred and retrieved from
+LTM. More specifically, the WM size indirectly depends
+on a fixed time limit T: when the time required to up-
+date the map (i.e., the time required to execute the pro-
+cesses in the Graph-based SLAM-MM block) reaches
+T, some nodes of the map are transferred from WM to
+LTM, thus keeping WM size nearly constant and pro-
+cessing time around T. However, when a loop closure is
+detected, neighbors in LTM with the loop closure node
+can be retrieved from LTM to WM for further loop clo-
+sure detections. In other words, when a robot revisits
+an area which was previously transferred to LTM, it
+can incrementally retrieve the area if a least one node
+of this area is still in WM. When some LTM nodes are
+retrieved, nodes in WM from other areas in the map
+can be transferred to LTM, to limit map size in WM
+and therefore keeping processing time around T.
+Therefore, the choice of which nodes to keep in
+WM is key in SPLAM-MM. The objective is to have
+enough nodes in WM from each mapping session for
+loop closure detections and to keep a maximum num-
+ber of nodes in WM for generating a map usable to
+follow correctly a planned path, while still satisfying
+online processing. Two heuristics are used to establish
+the compromise between selection of which nodes to
+keep in WM and online processing:
+– Heuristic 1 is inspired from observations made by
+psychologists (Atkinson and Shiffrin, 1968; Badde-
+ley, 1997) that people remember more the areas
+where they spent most of their time, compared to
+those where they spent less time. In terms of mem-
+ory management, this means that the longer the
+robot is at a particular location, the larger the
+weight of the corresponding node should be. Old-
+est and less weighted nodes in WM are transferred
+to LTM before the others, thus keeping in WM only
+the nodes seen for longer periods of time. As demon-
+strated in (Labbe and Michaud, 2013), this heuristic
+reveals to be quite efficient in establishing the com-
+promise between search time and space, as driven by
+the environment and the experiences of the robot.
+– Heuristic 2 is used to identifies nodes that should
+stay in WM for autonomous navigation. Nodes on a
+planned path could have small weights and may be
+identified for transfer to LTM by Heuristic 1, thus
+eliminating the possibility of finding a loop closure
+link or a proximity link with these nodes and cor-
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+5
+Map 1!
+Map 3!
+Map 4!
+Last node!
+Map 2!
+Local map!
+Global map!
+Fig. 3 Illustration of the local map (inner dashed area) and
+the global map (outer dotter area) in multi-session mapping.
+Red nodes are in LTM, while all other nodes are in WM. Loop
+closure links are shown using bidirectional green arrows.
+rectly follow the path. Therefore, Heuristic 2 must
+supersede Heuristic 1 and allow upcoming nodes to
+remain in WM, even if they are old and have a small
+weight.
+The Graph-based SLAM-MM block provides two
+types of maps derived from nodes in WM and LTM:
+– Local map, i.e., the largest connected graph that can
+be created from the last node in WM with nodes
+available in WM only. The local map is used for
+online path planning.
+– Global map, i.e., the largest connected graph that
+can be created from the last node in WM with nodes
+in WM and LTM. It is used for offline path planning.
+Figure 3 uses diamonds to represent initial and end
+nodes for each mapping session. The nodes in LTM are
+shown in red and the others are those in WM. The lo-
+cal map is created using only the nodes in WM that
+are linked to the last node. The graph linking the last
+node with other nodes in WM and LTM represents the
+global map (outer dotted area). If loop closure detec-
+tions are found between nodes of different maps, loop
+closure links can be generated, and the local map can
+span over multiple mapping sessions. Other nodes in
+WM but not included in the local map are unreachable
+from the last node, but they are still used for loop clo-
+sure detections since all nodes in WM (including those
+in Map 2 for instance) are examined.
+The modules presented in Fig. 2 are described as
+follows.
+3.1 Short-Term Memory Module
+Short-Term Memory (STM) is the entry point where
+sensor data are assembled into a node to be added to
+the map. Similarly to (Labbe and Michaud, 2013), the
+role of the STM module is to update node weight based
+on visual similarity. When a node is created, a unique
+time index ID is assigned and its weight is initialized to
+0. The current pose, RBG image, depth image and laser
+scan readings are also memorized in the node. If two
+consecutive nodes have similar images, i.e., the ratio of
+corresponding visual words between the nodes is over a
+specified threshold Y , the weight of the previous node is
+increased by one. If the robot is not moving (i.e., odom-
+etry poses are the same), the new node is deleted. To re-
+duce odometry errors on successive STM nodes, trans-
+formation refinement is done using 2D iterative-closest-
+point (ICP) optimization (Besl and McKay, 1992) on
+the rigid transformation of the neighbor link with the
+previous node and the corresponding laser scans. If the
+ratio of ICP point correspondences between the laser
+scans over the total laser scan size is greater or equal to
+C, the neighbor link’s transformation is updated with
+the correction.
+When the STM size reaches a fixed size limit of S
+nodes, the oldest node in STM is moved to WM. STM
+size is determined based on the velocity of the robot
+and at which rate the nodes are added to the map.
+Images are generally very similar to the newly added
+node, keeping S nodes in STM avoids using them for
+appearance-based loop closure detection once in WM.
+For example, at the same velocity, STM size should
+be larger if the rate at which the nodes are added to
+map increases, in order to keep nodes with consecutive
+similar images in STM. Transferring nodes with images
+very similar with the current node from STM to WM
+too early limits the ability to detect loop closures with
+older nodes in WM.
+3.2 Appearance-based Loop Closure Detection Module
+Appearance-based loop closure detection is based on
+the bag-of-words approach described in (Labbe and
+Michaud, 2013). Briefly, this approach uses a bayesian
+filter to evaluate appearance-based loop closure hy-
+potheses over all previous images in WM. When a loop
+closure hypothesis reaches a pre-defined threshold H, a
+loop closure is detected. Visual words of the nodes are
+used to compute the likelihood required by the filter. In
+this work, the Term Frequency-Inverse Document Fre-
+quency (TF-IDF) approach (Sivic and Zisserman, 2003)
+is used for fast likelihood estimation, and FLANN (Fast
+
+6
+Mathieu Labb´e, Fran¸cois Michaud
+Library for Approximate Nearest Neighbors) incremen-
+tal KD-Trees (Muja and Lowe, 2009) are used to avoid
+rebuilding the vocabulary at each iteration. To keep it
+balanced, the vocabulary is rebuilt only when it doubles
+in size.
+The RGB image, from which the visual words are
+extracted, is registered with a depth image. Using (1),
+for each 2D point (x, y) in the rectified RGB image, a
+3D position Pxyz can be computed using the calibration
+matrix (focal lengths fx and fy, optical centres cx and
+cy) and the depth information d for the corresponding
+pixel in the depth image. The 3D positions of the visual
+words are then known. When a loop closure is detected,
+the rigid transformation between the matching images
+is computed using a RANSAC (RANdom SAmple Con-
+sensus) approach which exploits the 3D visual word cor-
+respondences (Rusu and Cousins, 2011). If a minimum
+of I inliers are found, the transformation is refined us-
+ing the laser scans in the same way as the odometry
+correction in STM using 2D ICP transformation refine-
+ment. If transformation refinement is accepted, then a
+loop closure link is added with the computed transfor-
+mation between the corresponding nodes. The weight
+of the current node is updated by adding the weight
+of the loop closure hypothesis node and the latter is
+reset to 0, so that only one node with a large weight
+represents the same location.
+Pxyz =
+�(x − cx) · d
+fx
+, (y − cy) · d
+fy
+, d
+�T
+(1)
+By doing appearance-based loop closure detection
+this way, setting H high means that there is less chance
+of detecting false positives, but at the cost of detect-
+ing less loop closures (Labbe and Michaud, 2013). For
+SPLAM-MM, H can be set relatively low to detect more
+loop closures because false positives that are geometri-
+cally different will be rejected by the rigid transforma-
+tion computation step (i.e., the 3D visual word corre-
+spondences and 2D ICP transformation refinement).
+3.3 Proximity Detection Module
+Appearance-based loop closure detection is limited by
+the perceptual range of the sensory data used. For in-
+stance, when the robot is revisiting areas in opposite di-
+rection, the RGB-D camera on AZIMUT-3 is not point-
+ing in the same direction compared to when the nodes
+were created, and thus no appearance-based loop clo-
+sures can be detected. This also happens when there
+are not enough visual features under the depth range
+of the RGB-D camera (e.g., white walls or long halls).
+Simply relying on appearance-based loop closure detec-
+tions for map corrections would then limit path plan-
+ning capabilities, and make navigation difficult in such
+conditions. Figure 4a illustrates a situation where the
+robot is in a hall coming back to its starting position
+in reverse direction. Setting a goal at the starting posi-
+tion would make the planner fail because no loop clo-
+sures could be found to correct the odometry, resulting
+in having a wall directly placed on the starting posi-
+tion. One solution would be to have the robot visit the
+nodes of the graph backward so loop closures could be
+detected to correct the map, and ultimately be able
+to reach the starting position. However, it is inefficient
+and unsafe if the robot does not have sensors pointing
+backward. To deal with such situations, the Proximity
+Detection module uses laser rangefinder data to correct
+odometry drift in areas where the camera cannot de-
+tect loop closures. With a field of view of more than
+180◦, the laser scans can be aligned in reverse direc-
+tion, generating proximity links. As laser scans are not
+as discriminative as images, proximity detection is re-
+stricted to nodes of the local map located around the
+estimated position of the robot. Figure 4b illustrates
+the result.
+Figure 5 illustrates how nodes located close to the
+robot are selected by the Proximity Detection module.
+Only nodes in the local map with their pose inside ra-
+dius R centered on the robot are used. Nodes in STM
+are not considered in order to avoid adding useless links
+with nodes close by: this would increase graph optimiza-
+tion time without adding significative improvements of
+the map. The nodes are then segmented into groups
+with nodes connected only by neighbor links. A group
+must have its nearest node from the robot inside a fixed
+radius L defining close-by nodes (with L < R) to be
+considered for proximity detection, to keep the length
+of the resulting proximity links small for path planning.
+Note that Appearance-based Loop Closure Detection
+is done before Proximity Detection, thus if the near-
+est node has already a loop closure with the new node,
+the group is ignored. Proximity detection is then ap-
+plied separately on each group of nodes by doing the
+following steps:
+1. A rigid transformation between the nearest node
+of each group and the new node added to map is
+computed as in Section 3.2, and if it is accepted, a
+proximity link is added between the corresponding
+nodes, and the group of nodes is ignored for step
+2. These links are referred as visual proximity links
+because visual words are used in the transformation
+estimation.
+2. To avoid having to compare multiple nodes with
+very similar laser scans (and thus to save computa-
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+7
+a)!
+b)!
+Fig. 4 Illustration of the role of the Proximity Detection module. On the left are the raw laser scans, the blue dot is the
+starting position, and on the right the corresponding occupancy grid map at 0.05 m resolution (black, light gray and dark
+gray areas are occupied, empty and unknown spaces, respectively). In a), the yellow circle on the right locates the problematic
+situation: after the second traversal, the first nodes of the graph are located exactly over the wall, making it impossible to
+plan a path (red arrow on the right) to return to the starting position. In b), proximity links are detected using only the laser
+scans, and the local map can then be correctly optimized.
+tion), only the more recent node among those in
+the same fixed small radius L (centered on each
+node) is kept along the nodes in a remaining group.
+Then for each group, the laser scans of the nodes
+are merged together using their respective pose. 2D
+ICP transformation refinement is done between the
+merged laser scans and the one of the new node.
+If the transformation is accepted, a new proximity
+link with this transformation is added to the graph
+between the new node and the nearest one in the
+group.
+3.4 Graph Optimization Module
+TORO (Tree-based netwORk Optimizer) (Grisetti
+et al., 2007) is used for graph optimization. When loop
+closure and proximity links are added, the errors de-
+rived from odometry can be propagated to all links,
+thus correcting the local map. This also guarantees that
+nodes belonging to different maps are transformed into
+the same referential when loop closures are found.
+When only one map exists, it is relatively straight-
+forward to use TORO to create a tree because it only
+has one root. However, for multi-session mapping, each
+map has it own root with its own reference frame. When
+loop closures occur between the maps, TORO cannot
+optimize the graph if there are multiple roots. It may
+also be difficult to find a unique root when some of
+the nodes have been transferred in LTM. As a solution,
+our approach takes the root of the tree to be the latest
+a)	
+b)	
+c)	
+d)	
+R
+L
+Fig. 5 Illustration of how proximity detection works. In a),
+the larger dashed circle represents the radius R used to deter-
+mine close-by nodes, and the smaller dashed circle defined by
+L is used to limit the length of the links to be created. The
+empty dots are nodes for which the laser scans are not used,
+either because they are outside the radius R, they are too
+close from each other or they are in STM. In b) and c), nodes
+in the radius R from the two segmented groups of nodes are
+processed for proximity detection. In d), proximity links are
+added (yellow), and after graph optimization, the groups of
+nodes are connected together and the respective laser scans
+are now aligned.
+
+8
+Mathieu Labb´e, Fran¸cois Michaud
+node added to the local map, which is always uniquely
+defined across intra-session and inter-session mapping.
+All other poses in the graph are then optimized using
+the last odometry pose as the referential.
+3.5 Path Planning Modules
+Memory management has a significant effect on how to
+do path planning online using graph-based SLAM, for
+which the map changes almost at each iteration and
+with only the local map accessible while executing the
+plan. This differs from approaches that assume that the
+map is static and/or that all the previously visited loca-
+tions always remain in the map. In this paper, SPLAM-
+MM uses two path planners: a Metrical Path Planner
+(MPP) and a Topological Path Planner (TPP).
+3.5.1 Metrical Path Planning Module
+MPP receives a pose expressed in (x, y, θ) coordinates,
+and uses the local map to plan a trajectory and to make
+the robot move toward the targeted pose while avoid-
+ing obstacles. Our MPP implementation exploits the
+ROS navigation stack (Marder-Eppstein et al., 2010) to
+compute trajectories expressed as a sequence of veloc-
+ity commands (expressed as twists) sent to the robot’s
+Motion Controller module. A global Costmap is used
+to plan a trajectory to a targeted pose. MPP creates
+the global Costmap from an occupancy grid created us-
+ing the assembled laser scans from the latest local map.
+Each time the local map is updated, the occupancy grid
+is re-assembled and the trajectory is re-planned. MPP
+also uses a local Costmap for its Dynamic Window Ap-
+proach (DWA) (Fox et al., 1997) to handle dynamic
+obstacles for collision avoidance. The local Costmap is
+created directly from sensor readings. To create the lo-
+cal Costmap, only using the laser rangefinder for obsta-
+cle detection revealed to be insufficient: while the laser
+range finder can detect most of the obstacles (e.g., walls,
+people, table legs), it is located 40 cm above the floor
+and all obstacles under this height cannot be detected.
+Therefore, the depth image from the RGB-D camera
+is also used to detect these small obstacles and to add
+them to the local Costmap. Figure 6 shows an example
+where combining laser scans and RGB-D data creates a
+more robust and a safer local Costmap for navigation.
+Note that segmentation of the point cloud generated
+from the depth image is required to be able to add or
+clear small dynamic obstacles below the RGB-D cam-
+era. To segment the ground, all points with normal par-
+allel to z-axis (up to an angle Z) are labeled as ground.
+Then, all other points under a maximum height U are
+labeled as obstacles. This method would also make the
+robot capable of operating on uneven terrain.
+3.5.2 Topological Path Planning Module
+When TPP receives a goal identified by a node ID from
+a user (or a high-level module like a task planner, or
+in this paper the Patrol module), the global map is
+provided by the graph-based SLAM-MM module, and
+a topological path is computed to reach this goal. The
+topological path is a sequence of poses, expressed by
+their respective node IDs, to reach the goal. This step
+must be done offline or when the robot is not moving
+because all nodes linked to the current local map should
+be retrieved from LTM to build the global map.
+To choose which nodes to use for navigation, TPP
+computes a path from the current node to the goal node
+using Djikstra algorithm (Dijkstra, 1959). The choice
+of using Dijkstra over A* is to avoid global graph op-
+timization, which is time consuming, to know the dis-
+tance to goal required by A*. Dijkstra can also be com-
+puted directly when fetching the global map from LTM.
+Similar to (Valencia et al., 2013), to avoid losing track
+of the planned path, TPP prefers paths traversed in
+the same direction (e.g., where the camera is facing the
+same direction than on the nodes on the path) over
+shortest paths. This increases localization confidence:
+loop closure detection and visual proximity detection
+are more reliable than proximity detection using only
+laser scans because of their double verification (3D vi-
+sual word correspondences and 2D ICP transformation
+refinement). To embed this preference in Djikstra, the
+search cost is angular-based instead of distance-based,
+i.e., it finds the path with less orientation changes when
+traversing it in the forward direction.
+Then, TPP selects the farthest node on the path
+in the local map and sends its pose to MPP. While
+MPP makes the robot navigate to its targeted pose,
+TPP indicates to the graph-based SLAM-MM mod-
+ule which upcoming nodes on the topological path is
+needed, expressed as a list of node IDs from the lat-
+est node reached on the path to the farthest node in-
+side the radius R (to limit the size of the list). The re-
+quired nodes are identified by the graph-based SLAM-
+MM module with Heuristic 2 either to remain in WM or
+to be retrieved from LTM to extend the local map. The
+maximum number of retrieved nodes per map update is
+limited to M because this operation is time consuming
+as it needs to load nodes from LTM. M is set based on
+the hardware on which LTM is saved and according to
+the maximum velocity of the robot: for instance, if the
+robot is moving at the same speed or less as when it
+traversed the same area the first time, M = 1 would
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+9
+(a)
+(b)
+(c)
+Fig. 6 Example of obstacle detection using the laser rangefinder and the RGB-D camera. The red dots on the chair show
+what is detected using the laser rangefinder data. The cyan area is derived from the obstacle projection on the ground plane
+up to robot’s footprint radius, delimiting where the center of the robot should not enter to avoid collisions. In a), only the
+laser rangefinder data are used and the chair’s wheels are not detected, making unsafe for the robot to plan a path around the
+chair. In b), the point cloud generated from the camera’s depth image is used and the chair’s wheels are detected (shown by
+the orange dots), increasing the cyan area (and consequently the area to avoid colliding with the chair). Illustration c) presents
+a view from the RGB-D camera where the segmented ground is shown in green and the obstacles in orange.
+suffice to retrieve nodes on the path without having to
+slow down to wait for nodes not yet retrieved.
+Extending the local map with nodes of the topo-
+logical path is important for the robot to localize it-
+self using the Appearance-based Loop Closure Detec-
+tion module or using the Proximity Detection module,
+making it able to follow the topological path appro-
+priately. As the robot moves and new local maps are
+created, TPP always looks for the farthest node of the
+topological path that can be reached in the local map
+to update the current pose sent to MPP module. If new
+nodes are retrieved from LTM on the topological path,
+then the farthest pose is sent to MPP. TPP also de-
+tects changes in the local map after graph optimization
+(e.g., when new loop closures are detected): if so, the
+updated position of the current pose is sent to MPP.
+Up to a ratio O of the WM size, nodes identified by
+the planner and located in the radius R from the robot’s
+current position are immunized to be transferred, with
+R being the sensor range.
+Figure 7 presents an example of the interaction be-
+tween MPP and TPP to reach a goal G. While the robot
+is moving, TPP always sends the farthest pose P of the
+node on the topological path (purple links) in the lo-
+cal map. An occupancy grid is assembled with the laser
+scans contained in the nodes of the local map. MPP
+uses this occupancy grid to plan a trajectory (yellow
+arrow) to P. To keep the WM size constant, as nodes
+are retrieved from LTM on the path, older nodes are
+transferred to LTM. To follow the path appropriately,
+proximity links are detected to correct the map as the
+robot moves, otherwise the situation explained by Fig.
+4a would happen.
+TPP iterates by sending poses until the node of the
+goal (under a goal radius D expressed in m) is reached.
+Finally, handling situations where the environment has
+changed too much for proper localization must be taken
+into consideration. If no loop closures and proximity de-
+tections occur when following a path, a temporary link
+is added between the current node and the closest one
+in the path so that the topological path is always linked
+to the current node in the local map. Without this link,
+if previous nodes between the current node and those of
+the topological path are transferred to LTM, the local
+map would be divided and the nodes of the path would
+not be in the local map anymore. This temporary link
+is removed when a new link is added between the cur-
+rent node and the closest one in the path or when the
+goal is reached. If the robot has not reached the cur-
+rent pose set to MPP after F iterations of SPLAM-MM
+(e.g., MPP cannot plan to the requested pose because
+of the presence of a new obstacle or because the robot
+cannot localize itself on the path), TPP chooses another
+pose on the upcoming nodes and sends it to MPP. If all
+the upcoming nodes cannot be reached, TPP fails and
+sends a status message to its connected modules so that
+they can be notified that the goal cannot be reached.
+
+10
+Mathieu Labb´e, Fran¸cois Michaud
+P"
+G"
+(a)
+P"
+G"
+(b)
+P"
+G"
+(c)
+P"
+G"
+(d)
+P"
+G"
+(e)
+P"
+(f)
+Fig. 7 Interaction between TPP and MPP for path planning. The goal is identified by the purple G. The topological path is
+shown with purple links. The dashed yellow arrow is the trajectory computed by MPP to the targeted poses designated by the
+yellow P. Light gray, dark gray and black areas of the occupancy grid represent free, unknown and occupied cells, respectively.
+Blue nodes are in WM, and red nodes are in LTM. Yellow links are proximity links.
+3.6 Patrol Module
+We implemented the Patrol module to generate naviga-
+tion goals, referred to as waypoints so that the robot is
+programmed to continuously patrol an area. The Patrol
+module receives waypoints as inputs and sends them
+successively to TPP. By examining TPP’s status mes-
+sages, Patrol can know when a goal is reached or if TPP
+has failed. Whenever the status indicates that the goal
+is reached or not, the Patrol module sends the next
+waypoint, and restart to the first one once the whole
+list has been processed.
+4 Results
+Table 1 shows the parameters used for the trials1. The
+acquisition time A used is 1 sec (i.e., the map update
+rate is 1 Hz), which set the maximum online time al-
+lowed to process each node added to the map. For
+the trials, T is set to 200 ms to limit CPU usage for
+SPLAM-MM to around 20%, to make sure that higher
+1 In comparison with (Labbe and Michaud, 2013), T =
+Ttime, S = TST M and Y = Tsimilarity.
+frequency modules (acquisition of Sensor Data acquisi-
+tion and MPP) can run at their fixed frequency of 10
+Hz. The robot is relatively moving at the same velocity
+during the trials, and therefore M is fixed to 2 to make
+sure that nodes on a planned path are retrieved fast
+enough to avoid having the robot wait for nodes still in
+LTM. All computations are done onboard on the robot,
+which is equipped with a 2.66 GHz Intel Core i7-620M
+and a 128 GB SSD hard drive (on which the LTM is
+saved).
+To define the area over which the robot had to pa-
+trol, during session 1 we first teleoperated the robot
+and defined four waypoints (WP1 to WP4). There were
+no people in the environment during the teleoperation
+phase. After reaching WP4, the autonomous navigation
+phase is initiated by sending the waypoints to the Pa-
+trol module. Figure 8 illustrates the four waypoints on
+the global map and the first planned trajectory by TPP
+(purple path) from the current position of the robot
+(WP4) to WP1. To come back to WP1, the robot had
+to follow the path in the opposite direction from when
+these nodes were created. Proximity detection made
+it able to follow the path appropriately. To see more
+clearly the effect of proximity links, Fig. 9 shows the
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+11
+Table 1 Parameters used for the trials
+Acquisition time
+A
+1 sec
+ICP correspondence ratio
+C
+0.3
+Radius of the goal area
+D
+0.5 m
+TPP iterations before failure
+F
+10
+Loop closure hypothesis threshold
+H
+0.11
+Minimum RANSAC visual word inliers
+I
+5
+Close nodes radius
+L
+0.5 m
+Maximum retrieved close nodes
+M
+2
+Heuristics 2 close-by nodes ratio
+O
+0.25
+Laser scan range
+R
+4 m
+STM size
+S
+20
+Time limit
+T
+200 ms
+Maximum obstacle height
+U
+0.4 m
+Similarity threshold
+Y
+0.3
+Ground segmentation maximum angle
+Z
+0.1 rad
+WP4	
+WP3	
+WP2	
+WP1	
+Battery Charger	
+Fig. 8 Waypoints WP1 to WP4 identified on the global map.
+The purple path is the first path planned by TPP from the
+WP4 to WP1.
+maps after reaching WP1 with and without graph op-
+timization. Navigation would not have been possible
+without proximity links: the local map would have look
+like the map in (b) without the yellow links because no
+appearance-based similarities would have been found
+with nodes from the map on the planned path. When
+reaching WP1, the Patrol module sends the next way-
+point (WP2), making the robot continue patrolling.
+Every 45 minutes or so of operation, the robot was
+manually shutdown and moved to the battery charger
+near WP1. Once recharged, a new session of SPLAM-
+MM was initiated, creating a new node in STM with
+odometry reset, while preserving the nodes in WM
+and LTM. As the robot was initialized in the area of
+WP1 for each session, loop closures were found, con-
+WP1!
+WP1!
+WP2!
+WP2!
+WP3!
+WP3!
+WP4!
+WP4!
+(a)
+WP1!
+WP1!
+WP2!
+WP2!
+WP3!
+WP3!
+WP4!
+WP4!
+(b)
+Fig. 9 Global maps, optimized and not optimized, after
+reaching WP1. Yellow and red links are proximity and loop
+closure links, respectively.
+necting and optimizing the new map with nodes cre-
+ated from previous sessions, and allowing the Patrol
+module to provide waypoints as navigation goals to pa-
+trol the area. Overall, 11 indoor mapping sessions were
+conducted, for a total distance of 10.5 km lasting 7.5
+hours of operation spent over two weeks. The robot did
+111 patrolling cycles (i.e., traversing from WP1 through
+WP2, WP3, WP4 and coming back to WP1). The ses-
+sions were conducted during office hours, with people
+walking by. A total of 139 people were encountered by
+the robot while patrolling. Figure 10 illustrates the dy-
+namic conditions and some of the obstacles that the
+robot had to deal with during the trials.
+The main goal of the trials is to see how SPLAM is
+influenced by memory management over long-term op-
+eration, only having the local map for online process-
+ing. This can be illustrated by looking at the influences
+of memory management on SPLAM, interactions be-
+tween TPP and MPP, and the influences of LTM on
+TPP. As the robot is continuously adding new nodes,
+the trials also demonstrate how SPLAM-MM works in
+an unbounded environment.
+4.1 Influences of MM on SPLAM
+Figure 11 shows a typical navigation result when reach-
+ing the time limit T, thus limiting the size of the local
+map used for online navigation. This example shows
+the path planned between WP4 and WP1 after 4.7
+hours of operation. The local maps used for online plan-
+ning, localization and mapping are shown for different
+time steps along the trajectory. At t = 17031 sec, the
+planned path had 67 nodes and was 33 m long. It took
+1.3 sec to be generated by TPP and to have the first
+pose on the path sent to MPP. The laser scan range R
+is delimiting the upcoming nodes on the path provided
+by TPP. As the robot navigates in the environment,
+the farthest available pose in the local map on the path
+(end of the cyan line) is sent from TPP to MPP. Up-
+
+12
+Mathieu Labb´e, Fran¸cois Michaud
+a)!
+b)!
+c)!
+d)!
+e)!
+Fig. 10 Events that occurred during the trials: a) open and closed doors between traversals; b) camera exposure that led to
+the extraction of different visual features, making it difficult to find loop closures; c) someone opening a door while the robot
+is navigating; d) people walking around or blocking the robot; e) featureless images on which loop closure detection cannot
+work.
+t = 17060 sec!
+t = 17053 sec!
+t = 17031 sec!
+t = 17068 sec!
+t = 17075 sec!
+t = 17081 sec!
+t = 17108 sec!
+t = 17095 sec!
+WP4!
+WP4!
+WP4!
+WP4!
+WP4!
+WP4!
+WP4!
+WP1!
+WP1!
+WP1!
+WP1!
+WP1!
+WP1!
+WP1!
+WP1!
+Fig. 11 Example of the effect of memory management when travelling from WP4 to WP1 after 4.7 hours of operation. The
+path planned is shown in purple. The small colored icon represents the robot position at each time step. The dotted circle
+around the robot position illustrates the laser scan range R. The cyan lines represent the upcoming nodes on the planned path.
+coming nodes, if they are not in WM, are retrieved to
+make the robot able to localize itself (though loop clo-
+sures and proximity detections) on the path. Looking
+at how the local map changes in these snapshots, notice
+how starting from t = 17075 sec, the initial portion of
+the path is transferred in LTM to keep the size of the
+WM relatively constant. At t = 17108 sec, the robot
+reached WP1.
+Figure 12 compares the images between each way-
+point and the final position of the robot at the way-
+points. The robot successfully reached the waypoints
+(within D as the goal radius) 445 out of 446 times. For
+WP2, WP3 and WP4, the robot always came from be-
+hind the waypoint, and as soon the robot reached the
+waypoint within a D radius, TPP detected that the goal
+was reached. This explains why all the poses are behind
+the waypoints but inside the goal radius D. Similarly,
+for WP1, the robot came from behind from a slightly
+different direction. Spurious poses on the right part of
+the circle are those where there was an obstacle that
+caused the robot to avoid it, making it reach the way-
+point from a different direction. The one time the robot
+failed to reach a waypoint is because someone blocked
+the robot for a long time, making TPP failed after F at-
+
+SOHTESTHTELong-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+13
+tempts of reaching the upcoming nodes: a failure status
+message was then sent to the Patrol module to provide
+the next waypoint. The person left soon after the next
+waypoint was sent, and the robot reached the new way-
+point provided.
+Figure 13 illustrates the evolution of the number
+of nodes in WM and online processing time over the 11
+mapping sessions. Processing time includes all SPLAM-
+MM modules except MPP which was running concur-
+rently on a separate process (its processing time is only
+dependent of the local map size). As explained in Sec-
+tion 3.5.2, TPP occurs offline and only when a new
+goal is received from the Patrol module, and is exam-
+ined in Section 4.3. Fig. 13a illustrates that the number
+of nodes in WM and the local map was identical until T
+sec was reached. After that, nodes were transferred to
+LTM to limit the WM size for online processing, which
+is satisfied as shown by Fig. 13b. Processing time also
+remained well under the acquisition time A.
+4.2 TPP-MPP Interactions
+To illustrate with a concrete example of the situation
+described in Fig. 7, Fig. 14 presents an example of con-
+secutive poses sent by TPP to MPP while nodes from
+LTM are retrieved for the planned path. The red ar-
+row shows the pose of the farthest node on the path
+(the direction of the arrow shows the orientation of
+the pose). The red line represents the trajectory com-
+puted by MPP from the current position of the robot
+to its targeted pose, combined with obstacle avoidance.
+The blue lines represent the local map. In Fig. 14a,
+the targeted pose is on a node traversed backward (as
+shown by the arrow pointing backward). Between a)
+and b), the local map was updated with nodes loaded
+from LTM of the topological path. The targeted pose
+was updated farther on the path and at the same time,
+the occupancy grid was extended to previously mapped
+areas and MPP recomputed its trajectory. The robot
+could then move farther toward its goal and the nodes
+retrieved were used for proximity detection to correctly
+follow the planned path.
+To also illustrate the importance of obstacle detec-
+tion described in Fig. 6, Fig. 15 presents an example
+where an unexpected obstacle was encountered: as the
+laser rangefinder is 0.4 m above the ground, the forklift
+could only be detected using the RGB-D camera. MPP
+planned a slightly different path (orange) that the one
+planned by TPP (pink) to avoid the obstacle.
+4.3 Influences of LTM on TPP
+Although Fig. 13 demonstrates that SPLAM-MM is
+able to satisfy online constraints on a map increasing
+linearly in size (i.e., not bounded to a maximum size of
+environment), memory used by LTM and consequently
+TPP planning time increase linearly. For example, at
+the end of experiment, LTM contains 24002 nodes and
+113368 links. All raw sensor data in the nodes were
+also saved in the LTM’s database (for debugging and
+visualization purposes), including RGB image (JPEG
+format) and depth image (PNG format) of each node.
+The final database took 6.7 GB of hard drive space.
+With as many links at the end of the experiment, TPP
+required 2.4 sec to compute a plan to the next waypoint.
+In term of memory usage and planning time, LTM must
+be somewhat limited over time when revisiting the same
+areas.
+As a solution to limit LTM memory growth, nodes
+from STM can be merged when moved to WM if they
+have loop closure and/or visual proximity links. We
+studied this possibility by adding a graph reduction al-
+gorithm to STM, to remove the node from the graph
+and to add its neighbor links to the corresponding old
+node(s). Algorithm 1 summarizes the approach used to
+maintain the graph at the same size (same number of
+removed links and nodes than added) if there are many
+successive nodes with loop closure or visual proxim-
+ity links. If two nodes of a same location do not have
+similar images (i.e., they don’t have loop closure or vi-
+sual proximity links), they will not be merged, thus still
+keeping a variety of different images representing the
+same location. To make sure nodes to be merged are
+still in WM (to avoid to modify the LTM), nodes hav-
+ing a link to a node in STM are identified as nodes that
+must stay in WM (similarly to Heuristic 2). Figure 16
+shows how links are merged between the node moved to
+WM and its corresponding node(s) linked by loop clo-
+sure link. In a), the purple node has two loop closure
+links. On graph reduction, its two neighbor links (blue)
+are merged with the loop closure links (red) by multi-
+plying the corresponding transformations together, cre-
+ating merged neighbor links (orange). In this case, the
+same number of links are added than those removed but
+one node is removed. In b), the green node has only one
+neighbor link (with the cyan node), then the loop clo-
+sure link is only merged with it, creating only one link
+and four are removed. Merged neighbor links are ig-
+nored to be merged again to limit the number of links.
+In c), the cyan node does not have any loop closure and
+no graph reduction is done.
+To test this idea, data from the 11 sessions were
+processed again to test the influences of the graph re-
+
+14
+Mathieu Labb´e, Fran¸cois Michaud
+ID=167	
+ID = 266	
+ID = 417	
+a)	
+b)	
+c)	
+WP2	
+WP3	
+d)	
+WP4	
+ID = 26514	
+ID = 6414	
+ID = 22016	
+ID = 9896	
+ID = 19	
+−2
+−1.8
+−1.6
+−1.4
+−1.2
+−1
+−0.8
+−0.6
+−0.4
+−0.2
+−2.2
+−2
+−1.8
+−1.6
+−1.4
+−1.2
+−1
+−0.8
+−0.6
+−0.4
+wp1
+2.2
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+3.8
+4
+−8.4
+−8.2
+−8
+−7.8
+−7.6
+−7.4
+−7.2
+−7
+−6.8
+−6.6
+wp2
+15.4
+15.6
+15.8
+16
+16.2
+16.4
+16.6
+16.8
+17
+17.2
+12.2
+12.4
+12.6
+12.8
+13
+13.2
+13.4
+13.6
+13.8
+14
+wp3
+15.4
+15.6
+15.8
+16
+16.2
+16.4
+16.6
+16.8
+17
+17.2
+−4.2
+−4
+−3.8
+−3.6
+−3.4
+−3.2
+−3
+−2.8
+−2.6
+−2.4
+wp4
+WP1	
+Images	
+Laser scans	
+Fig. 12 Comparison of the corresponding images between the waypoint (left image) and at the last pose reached on one of
+the planned path (right image) for the waypoints. The top view grid shows the laser scan readings and referentials of the
+waypoint’s nodes (at the origin of the grid) and the final node. The zoomed portions represent the final poses of the robot
+(represented by blue dots), for all paths planned for each waypoint. The circle represents the goal radius D, and the grid’s
+cells used for visualization have a width of 1 m.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+x 10
+4
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+500
+Node indexes
+Nodes
+ 
+ 
+WM
+Local map
+(a) Number of nodes in WM and in the local map.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+x 10
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Time (s)
+Node indexes
+(b) Processing time (the horizontal line represents T = 0.2
+sec).
+Fig. 13 Memory size and total processing time over the 11 mapping sessions.
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+15
+Goal
+(a)
+Goal
+(b)
+Fig. 14 Example of poses sent by TPP to MPP while nodes
+from LTM are retrieved for the planned path. The goal of the
+path is somewhere outside these images in the direction shown
+by Goal. The bottom left images shows the actual RGB image
+from the RGB-D camera. The blue lines are nodes and links
+of the local map. The red line is the computed trajectory from
+MPP using the local map’s occupancy grid from its current
+pose (red arrow). The RGB point cloud and the occupancy
+grid are created using RGB-D images and laser scans stored
+in nodes from the local map, respectively. In a), the robot is
+following the red trajectory. In b), some nodes are retrieved
+from LTM and a new trajectory is computed to move further
+on the path toward the goal.
+Algorithm 1 Graph Reduction
+1: o ← node moved to WM
+2: m ← loop closure and visual proximity links of o
+3: if m is not empty then
+4:
+n ← neighbor links of o
+5:
+for all m in m do
+6:
+om ← node pointed by m
+7:
+for all n in n do
+8:
+on ← node pointed by n
+9:
+t ← m−1·n
+10:
+Add t to om
+11:
+Add t−1 to on
+12:
+end for
+13:
+end for
+14:
+Remove o from the graph
+15: end if
+duction approach using real data acquired by the robot.
+Note that even though graph reduction was validated
+offline, we carefully monitored the experiment manually
+to make sure that the robot could still localize itself cor-
+rectly on the planned paths.
+Figure 17 shows a comparison of the final global
+map without and with graph reduction. The zones with
+Fig. 15 Example where MPP plans a slightly different path
+(orange) than the one provided by TPP (pink). The yellow
+dot is the current position of the robot and the lower right
+image is the corresponding RGB image.
+STM	
+WM	
+Graph Reduction	
+STM to WM	
+WM	
+STM	
+a)	
+b)	
+c)	
+Fig. 16 Three examples illustrating how the graph reduc-
+tion algorithm works. Blue, red and orange links represent
+neighbor, loop closure and merged neighbor links, respec-
+tively. Black links and white nodes are those removed using
+graph reduction. The left column shows the rightmost node
+(the oldest) of STM moved to WM. Then on the right column,
+this node is removed if it has a loop closure link.
+less blue links indicate that there were many nodes
+merged. The zones with more blue links are where nodes
+were not merged, because of a lack of features or be-
+cause of obstacles: the robot was not able to localize
+itself perfectly on the paths every time, thus adding
+new nodes to the map.
+Figure 18 illustrates TPP planning time correspond-
+ing to LTM size with and without graph reduction. As
+the LTM became larger, TPP planning time increased:
+with graph reduction, TPP planning time was reduced
+by 89% for the last path planned (272 ms instead of 2.4
+
+16
+Mathieu Labb´e, Fran¸cois Michaud
+a)!
+b)!
+Fig. 17 Comparison between the global maps a) without
+graph reduction (24002 nodes and 113368 links); b) with
+graph reduction (6059 nodes and 18255 links).
+sec). Figure 19 illustrates hard drive usage with and
+without graph reduction. Extrapolating linearly mem-
+ory usage with a 100 Gb hard drive, the robot could
+navigate online approximately 110 hours without graph
+reduction before filling up the hard drive. When debug-
+ging data (not used for navigation) are not recorded in
+the database, this estimate would increase to approx-
+imately 33 days (800 hours). This means that if the
+robot is always visiting new locations at a mean velocity
+of 1.4 km/h (as in this experiment), it could travel up
+to 1120 km to map environments online. When graph
+reduction is used, debugging data are not saved and
+having the robot always revisiting the same areas like
+in this experiment, it could do SPLAM continuously for
+about 130 days before reaching the hard drive capacity.
+5 Discussion
+In terms of processing time, results show that SPLAM-
+MM is able to satisfy online processing requirements in-
+dependently of the size of the environment, by transfer-
+ring in LTM portions of the map which then cannot be
+used for loop closure detection, proximity detection and
+graph optimization. Results show also that path fol-
+lowing is still possible in such conditions by incremen-
+tally retrieving locations on the planned path. Thus, as
+shown in Section 4.3, the current hardware limitation
+of the system for long-term continuous SPLAM is hard
+drive capacity, not computation power.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+x 10
+4
+0
+500
+1000
+1500
+2000
+2500
+Time (ms)
+Node indexes
+ 
+ 
+Graph size
+0
+0.5
+1
+1.5
+2
+2.5
+x 10
+4
+Graph size (nodes)
+0
+1000
+2000
+3000
+4000
+0
+100
+200
+300
+Time (ms)
+Node indexes
+Fig. 18 Comparison of TPP planning time and LTM size,
+with (blue) and without (red) graph reduction. The peaks in
+the zoomed section show more precisely when a planning is
+done (when a waypoint is reached).
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+Time (h)
+Hard drive usage (MB)
+ 
+ 
+Raw data discarded
+Fig. 19 Comparison of hard drive usage with (blue) and
+without (red) graph reduction. The dashed curves represents
+results without saving in database the debugging data (i.e.,
+raw RGB and depth images).
+To successfully follow a path, results demonstrate
+the importance of adding loop closure and/or proxim-
+ity links with nodes on the planned path to localize the
+robot in the map. In our trials, the robot navigated in-
+door where static structures (e.g., walls) were most of
+the time visible using the laser rangefinder. However, in
+large empty spaces where the laser rangefinder would
+not be able to perceive nearby structures, it would be
+difficult for the robot to follow a path if appearance-
+based loop closure detection and visual proximity de-
+tection do not occur. A laser rangefinder with larger
+perceptual range or a 3D LIDAR sensor like the Velo-
+dyne could be used to increase perceptual range. For
+a lower cost solution, using a camera facing backward
+could be useful to allow the robot to detect similari-
+ties in images when traversing a path in opposite direc-
+tion (Carrera et al., 2011). Without adding new sensors,
+TPP could also stop sending new poses when no loop
+closure links or proximity links occur for a while. If no
+
+Long-Term Online Multi-Session Graph-Based SPLAM with Memory Management
+17
+loop closures were found over the next few meters, it
+would be possible to wait for the robot to rotate at
+this location so that it can look backward, increasing
+its chance to detect a loop closure to correct its po-
+sition on the planned path and then generate a new
+pose. A similar recovery approach is presented in (Mil-
+ford and Wyeth, 2010), where an exploration phase is
+triggered to re-localize the robot when failing to follow
+the planned path. Also, to be more robust to dynamic
+environments where there are cyclic changes over time,
+TPP could select nodes that match better the current
+time of the day rather than the most recent ones, to in-
+crease localization success as in (Krajn´ık et al., 2016).
+In comparison with large empty environments, those
+in which a lot of dynamic changes occur (e.g., navigat-
+ing through a crowd) would also make simultaneous
+planning and localization more difficult. For instance,
+mapping the area in session 1 without people walk-
+ing by helped the robot acquire the static structures
+of the environment since they were not hidden by peo-
+ple. These static structures facilitate localization when
+the robot comes back to these areas later one. If these
+static structures were previously occluded, they would
+be added to the map as the robot comes back to these
+areas (obviously if people are no longer in the robot’s
+field of view). If people partially occlude the robot’s
+sensors over a long distance, localization would still be
+possible but would occur less frequently.
+For online multi-session mapping with our memory
+management approach, the worst case is when all nodes
+of a previous map are transferred to LTM before a loop
+closure is detected (Labbe and Michaud, 2013). This
+results in definitely ignoring the previous map and dis-
+abling at the same time the ability to plan paths to
+a location in it. To avoid this problem, an additional
+heuristic could be to keep in WM at least one discrim-
+inative node for each map. However, if the number of
+mapping sessions becomes very high (e.g., thousands of
+sessions), these nodes would definitely have to be trans-
+ferred in LTM to satisfy online processing requirements.
+A strategy that makes the robot explore potential paths
+to link maps together would then be useful, and maps
+that could not be linked would eventually be unretriev-
+able.
+In the trials conducted, no invalid loop closures were
+detected, avoiding to corrupt the map with erroneous
+loop closure links. If this happens, graph optimization
+approaches such as (Latif et al., 2013; Sunderhauf and
+Protzel, 2012; Lee et al., 2013) deal with possible invalid
+matches, and could be used to increase robustness of
+SPLAM-MM. However, these approaches assume that
+the whole global map is available online, which is not
+the case here. They could be still used offline at the end
+of a session.
+As shown by Fig. 15, MPP in SPLAM-MM allows
+the robot to find an alternative path to reach the tar-
+geted pose when possible. However, if the alternative
+path is outside the local map, re-planning with TPP is
+required. Some paths may be also blocked temporary or
+permanently by some dynamic or new static obstacles.
+An approach similar to (Konolige et al., 2011) could be
+used to identify some links as blocked so that TPP can-
+not plan a path using them. The Patrol module could
+also manage waypoints that can and cannot be reached.
+Finally, the graph reduction approach can reduce
+significantly the number of nodes and links saved in
+LTM to reduce TPP planning time. However, because
+of dynamic events or the lack of features (e.g., Fig.
+10e), new nodes and links will inevitably be added to
+LTM over time when revisiting the same areas. As an
+improvement, nodes with featureless image could be
+merged through a maximum density threshold like in
+(Milford and Wyeth, 2010), as they cannot be used for
+loop closure detection. After applying graph reduction
+on the experimental data, there are still 3068 featureless
+nodes of 6059 nodes in the global graph, which would
+reduce by about 50% the remaining graph. However,
+even by limiting the rate at which the LTM grows, a
+continuous SLAM approach in unbounded dynamic en-
+vironments will always add new data over time. A com-
+plementary strategy would be to definitely forget some
+parts of the global map, at the cost of not being able
+to return to some locations.
+6 Conclusion
+By limiting the nodes of the map available online in
+WM for loop closure detection, proximity detection and
+graph optimization, results presented in this paper sug-
+gest that the proposed graph-based SPLAM-MM ap-
+proach is able to meet online processing requirements
+needed for simultaneous mapping, localizing and plan-
+ning in multi-session conditions. SPLAM-MM is tightly
+based on appearance-based loop closure detection, al-
+lowing it to naturally deal with the initial state prob-
+lem of multi-session mapping. To successfully localize
+on a planned path through areas previously transferred
+in LTM, memory management allows SPLAM-MM to
+deal with the necessity of retrieving upcoming nodes on
+the path in WM. Our code is open source and available
+at http://introlab.github.io/rtabmap.
+In future works, more robust failure recovery ap-
+proaches will be examined to test SPLAM-MM in dy-
+namic environments where the paths could often be
+blocked (temporally or permanently). We also plan to
+
+18
+Mathieu Labb´e, Fran¸cois Michaud
+study the impact of autonomous coverage and explo-
+ration strategies, especially how it can actively direct
+exploration based on nodes available for online map-
+ping. This could be also useful to conduct longer ex-
+periments at larger scale.
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+

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+POSTER V2: A simpler and stronger facial expression recognition network
+Jiawei Mao†
+Rui Xu†
+Xuesong Yin*
+Yuanqi Chang
+Binling Nie
+Aibin Huang∗
+School of Media and Design, Hangzhou Dianzi University, Hangzhou, China
+{jiaweima0,211330017,yinxs,yuanqichang,binlingnie,huangaibin}@hdu.edu.cn
+Abstract
+Facial expression recognition (FER) plays an impor-
+tant role in a variety of real-world applications such as
+human-computer interaction.
+POSTER V1 achieves the
+state-of-the-art (SOTA) performance in FER by effectively
+combining facial landmark and image features through
+two-stream pyramid cross-fusion design.
+However, the
+architecture of POSTER V1 is undoubtedly complex.
+It
+causes expensive computational costs. In order to relieve
+the computational pressure of POSTER V1, in this pa-
+per, we propose POSTER V2.
+It improves POSTER V1
+in three directions: cross-fusion, two-stream, and multi-
+scale feature extraction. In cross-fusion, we use window-
+based cross-attention mechanism replacing vanilla cross-
+attention mechanism. We remove the image-to-landmark
+branch in the two-stream design. For multi-scale feature
+extraction, POSTER V2 combines images with landmark’s
+multi-scale features to replace POSTER V1’s pyramid de-
+sign. Extensive experiments on several standard datasets
+show that our POSTER V2 achieves the SOTA FER perfor-
+mance with the minimum computational cost. For exam-
+ple, POSTER V2 reached 92.21% on RAF-DB, 67.49% on
+AffectNet (7 cls) and 63.77% on AffectNet (8 cls), respec-
+tively, using only 8.4G floating point operations (FLOPs)
+and 43.7M parameters (Param). This demonstrates the ef-
+fectiveness of our improvements. The code and models are
+available at https://github.com/Talented-Q/
+POSTER_V2.
+1. Introduction
+With the continuous development of technology and
+the continuous improvement of automation, the need
+for human-computer interaction is becoming increasingly
+strong.
+Facial expression recognition (FER) helps ma-
+chines to understand human emotions from facial expres-
+sions. This makes it as a core task for human-computer in-
+teraction. Besides, with its powerful expression understand-
+*Corresponding author.†Equal contribution.
+Figure 1. POSTER V2 results on RAF-DB. We compare POSTER
+V2 with three variants of POSTER V1 and other FER algorithms.
+The results indicate that POSTER V2 weighs the number of pa-
+rameters and accuracy better than other FER methods on RAF-
+DB.
+ing ability, FER has great potential applications in psychol-
+ogy, intelligent robotics, intelligent surveillance, virtual re-
+ality and synthetic animation. Therefore, research on FER
+is very necessary.
+Due to the increasing attention of FER, it has been
+able to develop rapidly in recent years. Early FER works
+[55, 59, 33, 20] used manual features [6, 34, 23] for the anal-
+ysis of facial expressions. However, FER algorithms based
+on manual features are often only applicable to specific FER
+tasks. When applied to real world scenarios, it is difficult for
+these algorithms to achieve the same results as in the experi-
+mental setting. With the development of deep learning, con-
+volutional neural networks (CNNs) are introduced to FER
+for improving the robustness of the network. Savchenko et
+al. [38] first verified the effectiveness of CNNs such as Mo-
+bileNet [19], EfficientNet [41] and RexNet [15] for FER.
+Zhao et al. proposed an efficient and robust FER network
+EfficientFace [57] for the analysis of facial expressions in
+the wild. Nevertheless, convolution-based FER algorithms
+cannot consider the global information of the image due to
+the limitation of convolutional local receptive field. Influ-
+enced by the vision transformer, Xue et al. [51] designed the
+first transformer-based FER network to model long-range
+arXiv:2301.12149v1  [cs.CV]  28 Jan 2023
+
+93
+POSTER V2
+POSTER V1
+92
+POSTER V1-S
+POSTER V1-T
+★
+Acc
+TransFER
+91
+RAF-DB Top-1 
+90
+DMUE
+89
+VTFF
+88
+87
+40
+45
+50
+55
+60
+65
+70
+75
+80
+Param(M)dependencies for FER. Kim et al. [24] improved the vision
+transformer (ViT) to combine both global and local features
+so that ViT can be adapted to FER task.
+Among many excellent FER works, POSTER V1
+[58] stands out with state-of-the-art (SOTA) performance.
+POSTER V1 mainly solves three key issues of FER at the
+same time: inter-class similarity, intra-class discrepancy
+and scale sensitivity. POSTER V1 cleverly combines facial
+landmark with image features through a network design of
+two-stream pyramidal cross-fusion transformer. With the
+difference and sparsity of landmark, POSTER V1 success-
+fully solves the issue of inter-class similarity and intra-class
+discrepancy in FER. The network design of pyramid archi-
+tecture introduces multi-scale features for POSTER V1 to
+solve the scale sensitivity problem. Along with the solution
+of the three main issues of FER, POSTER V1 shows the
+amazing expression analysis ability.
+Although POSTER V1 works so well on FER, the huge
+number of parameters and expensive computational cost
+brought by its network architecture affects the efficiency
+of FER. To address this issue, we revisit the network de-
+sign of POSTER V1 and improve it to obtain POSTER
+V2. We mainly improve POSTER V1 in three directions:
+two-stream, cross-fusion and multi-scale feature extrac-
+tion.
+POSTER V1 contains two main branches: image-
+to-landmark and landmark-to-image. Landmark-to-image
+branch is essential as the core of POSTER V1 to solve inter-
+class similarity and intra-class discrepancy.
+The image-
+to-landmark branch is only used to provide information to
+landmark that it fails to take into account. This does not
+contribute to solving the three main issues of FER. There-
+fore, in POSTER V2, we remove the image-to-landmark
+branch from the two-stream design. This greatly reduces
+the computational cost on POSTER V1. For cross-fusion,
+we use a window-based cross-attention mechanism instead
+of the vanilla cross-attention mechanism in POSTER V1.
+The window-based cross-attention mechanism not only pro-
+vides linear computational complexity for POSTER V2 but
+also enhances the local modeling capability of the network.
+In addition, POSTER V2 no longer uses an additional pyra-
+mid architecture for multi-scale feature extraction. We per-
+form multi-scale feature extraction directly from the image
+backbone as well as from the facial landmark detector. For
+the extracted multi-scale features, we use a vision trans-
+former network consisting of only two layers of transformer
+modules for integration. Based on the above designs, our
+POSTER V2 becomes a simpler and more powerful facial
+expression recognition network. It achieves SOTA perfor-
+mance on several standard FER datasets with only 8.4G
+floating point operations (FLOPs) and 43.7M parameters
+(Param). Figure 1 demonstrates the superiority of POSTER
+V2.
+Specially, POSTER V2 reached 92.21% on RAF-DB
+[29], 67.49% on AffectNet [32] (7 cls) and 63.77% on Af-
+fecNet (8 cls), respectively. This is better than POSTER
+V1 (RAF-DB with 92.05%, AffectNet (7 cls) with 67.31%
+and AffectNet (8 cls) with 63.34%). And POSTER V2 of-
+fers a smaller Param (43.7M vs. 71.8M) and FLOPs (8.4G
+vs. 15.7G). We hope that our work could contribute to the
+design of future FER models.
+In general, we summarize the contributions of this paper
+as follows:
+1) We design POSTER V2 by modifying POSTER V1
+from three perspectives: two-stream, cross-fusion and
+feature extraction.
+Compared with POSTER V1,
+POSTER V2 is simpler and stronger.
+2) POSTER V2 shows state-of-the-art performance on
+several standard FER datasets such as RAF-DB, Affec-
+Net and CAER-S. This shows the powerful expression
+analysis capability of POSTER V2.
+3) POSTER V2 greatly reduces the FLOPs and Param
+of POSTER V1. Specifically, POSTER V2 reduces
+28.1M of Param and 7.3G of FLOPs. This greatly im-
+proves the computational efficiency of the model.
+2. Related Work
+2.1. Facial Expression Recognition
+The study of FER has become very popular in re-
+cent years as more and more researchers focus on human-
+computer interaction. Zhao et al. [55] used the manual fea-
+ture LBP [34] for the research of FER with good results.
+Zhong et al. [59] proposed a two-stage multitask sparse
+learning framework (MTSL) for the FER task by explor-
+ing some common and specific information among differ-
+ent expressions. Savchenko et al. [38] studied lightweight
+convolutional neural networks for FER task learning and
+verified the effectiveness of CNNs for FER. Sang et al. [37]
+focused on reducing intra-class variation in facial expres-
+sion depth features and introduced a dense convolutional
+network [21] for the FER task. PSR [45] solves the prac-
+tical issues associated with individual wild images in FER
+in terms of pose, orientation and input resolution with its
+super-resolution pyramidal network architecture. Zhang et
+al. [54] proposed an erasing attention consistency method to
+handle the noise-labeled facial expression recognition task
+that is more challenging than the conventional FER.
+With the rise of transformer in the field of computer vi-
+sion, many FER methods combined with transformer have
+emerged. The vision transformer was first used for the study
+of FER by Xue et al. [51] and achieved state-of-the-art per-
+formance. VTFF [31] excels in dealing with facial expres-
+sion recognition tasks in the wild by virtue of its feature fu-
+sion. Huang et al. designed the teacher-student model PID-
+ViT [22] for modeling the probability distribution of frontal
+
+Figure 2. Pipeline of POSTER V1. POSTER V1 mainly contains facial landmark detector, image backbone, cross-fusion transformer
+encoders and pyramid network.
+and multi-pose facial expressions, and solved the problem
+of pose change and occlusion in FER. Zhao et al. [9] com-
+bined global and local attention in order to address the two
+key issues of occlusion and pose change in FER. POSTER
+V1 [58] solves the intra-class discrepancy, inter-class sim-
+ilarity and scale sensitivity issues of FER in the same time
+by integrating image features with facial landmark features
+through two-stream, cross-fusion and pyramid design.
+However, the huge computational cost of POSTER V1
+has prevented researchers from investigating further im-
+provements in FER. To solve this issue, we improved the
+architecture of POSTER V1 and proposed POSTER V2,
+which is simpler and more powerful for FER tasks.
+2.2. Vision Transformer
+Recently vision transformer has been widely used for
+computer vision tasks on large scale datasets with its ex-
+cellent ability to model long distance dependencies.
+Dosovitskiy et al. [8] pioneered the introduction of trans-
+former from the field of natural language processing to com-
+puter vision. Touvron et al. [42] used a teacher-student
+strategy to accelerate the training of transformer by distill-
+ing tokens. Zhou et al. [60] found that the reason why
+the transformer quickly saturates at deeper levels is that
+the attention map becomes increasingly similar as the trans-
+former goes deeper. Based on this observation, they pro-
+posed the Re-attention model to regenerate the attention
+map in order to enhance the diversity among layers at a
+small computational cost. Touvron et al. also designed CaiT
+[43], a deep vision transformer for optimal image classifi-
+cation. To solve the issue that ViT is inferior to traditional
+ResNet [17] on datasets without huge data size, Yuan et al.
+proposed T2T-ViT [52]. Besides, Hassani et al. proposed
+CCT [16] which uses convolution rather than patch em-
+bedding layer for self-attention processing. This introduces
+convolutional inductive bias for the transformer. Chen et
+al. proposed CrossViT [4], which combines image patches
+of different sizes by dual branches to produce stronger im-
+age features. Heo et al. [18] also verified whether pooling
+layers bring advantages to ViT as they do in convolutional
+neural networks (CNNs). Liu et al. [30] reduced the atten-
+tion mechanism from quadratic computational complexity
+to linear by window attention and the design of a shift win-
+dow scheme. Graham et al. grafted CNN with Transformer
+to obtain LeViT [13] with higher accuracy and faster speed.
+Wu et al. have designed a new architecture called convo-
+lutional visual transformer CVT [50], which improves the
+performance and efficiency of ViT by introducing convolu-
+tion into vision transformer to produce the better results of
+both designs. Chen et al. proposed a new architecture with
+a pyramidal structure and a novel region-to-local-attention
+vision transformer, RegionViT [3]. Wang et al. [48] intro-
+duced ViT into a CNN-like pyramid structure for intensive
+prediction tasks such as object detection and semantic seg-
+mentation.
+The architectural design of these vision transformer ef-
+forts inspires our improvements for POSTER V1.
+This
+leads to a better trade-off between accuracy and computa-
+tional complexity in FER with our POSTER V2.
+3. Method
+In this section, we first review the POSTER V1 process.
+We then describe the overall architecture of POSTER V2
+and discuss the specific details of POSTER V2 in three di-
+rections: two-stream, cross-fusion, and multi-scale feature
+extraction.
+3.1. A brief review of POSTER V1
+POSTER V1 contains four main core designs: facial
+landmark detector, image backbone, cross-fusion trans-
+former encoders and pyramid network. Given the input im-
+age X ∈ RH×W ×3, POSTER V1 obtain the image features
+Ximg and landmark features Xlm by facial landmark detec-
+tor and image backbone, respectively.
+The image features Ximg ∈ RN×D as well as the land-
+mark features Xlm ∈ RN×D are mapped into three ma-
+trices respectively: image query matrix Qimg, image key
+matrix Kimg, image value matrix Vimg and landmark query
+
+Cross-fusion
+Transform
+Landmark Feature
+Encoders
+Input Image
+Landmark
+Dector
+Cross-fusion
+Transform
+head
+Encoders
+Concat
+Image
+Backbone
+Cross-fusion
+Transform
+Image Feature
+EncodersFigure 3. The overview of POSTER V2 architecture. LMFi and IMFi denotes facial landmark features and image features at the i-th
+level of POSTER V2 respectively.
+matrix Qlm, landmark key matrix Klm, landmark value ma-
+trix Vlm in the cross-fusion transformer encoder. Specifi-
+cally expressed as:
+Qimg = XimgWq1, Qlm = XlmWq2,
+Kimg = XimgWk1, Klm = XlmWk2,
+Vimg = XimgWv1, Vlm = XlmWv2,
+(1)
+where Wq1, Wq2, Wk1, Wk2, Wv1 and Wv2 ∈ RD×D are
+the mapping matrix.
+The cross-fusion transformer encoder uses the vanilla
+cross-attention mechanism to interact image features and
+landmark features respectively. It is defined as follows:
+Attention(img) = softmax(QlmKT
+img/
+√
+d)Vimg,
+Attention(lm) = softmax(QimgKT
+lm/
+√
+d)Vlm,
+(2)
+where softmax(·) is softmax [1] activation function and
+1
+√
+d is an appropriately normalized scaling factor used to
+prevent the gradient from being too small.
+In summary cross-fusion transformer encoder can be de-
+noted as:
+X’img = Attention(img) + Ximg,
+Ximg o = MLP(Norm(X’img)) + X’img,
+X’lm = Attention(lm) + Xlm,
+Xlm o = MLP(Norm(X’lm)) + X’lm,
+(3)
+where MLP (·) is multi-layer perceptron and Norm (·)
+denotes the normalization operation.
+Finally, POSTER V1 extracts and integrates multi-scale
+features of images and landmarks by the pyramid network
+design. The specific details are shown in Figure 2.
+3.2. Architecture
+Figure 3 shows the pipeline for POSTER V2.
+The
+POSTER V2 keeps the facial landmark detector and im-
+age backbone in POSTER V1. In difference, we remove
+the POSTER V1 pyramid architecture and the image-to-
+landmark branch of the two-stream design. Meanwhile, we
+perform multi-scale feature extraction directly from the fa-
+cial landmark detector and image backbone. And we in-
+troduce a small vision transformer consisting of only two
+layers of vanilla tranformer blocks in POSTER V2 to in-
+tegrate multi-scale features. Moreover, we design the new
+cross-fusion transformer encoder with window-based cross-
+attention mechanism. Next, we discuss the detailed modifi-
+cations to POSTER V2.
+3.3. Two-stream
+Methods
+RAF-DB
+AffectNet
+Baseline
+91
+65.06
+POSTER V1
+92.05
+67.31
+POSTER w/o image to landmark branch
+91.82
+65.96
+POSTER w/o landmark to image branch
+91.62
+65.28
+Table 1. Ablation study of two branches in cross-fusion of
+POSTER V1. The baseline in the table keeps the baseline setting
+in POSTER V1.
+Although two-stream is central to the design of POSTER
+V1, POSTER V1 does not explore which branch of two-
+stream actually plays a major role. Thus, in this section, we
+first perform an ablation study of the two-stream to learn the
+contribution of the two branches to the FER. Table 1 shows
+the ablation results. We see that on the RAF-DB dataset, the
+accuracy of POSTER V1 slips by 0.23 after missing the im-
+age to landmark branch. If the landmark-to-image branch
+is missing, the accuracy of POSTER V1 on RAF-DB is re-
+duced by 0.43. Meanwhile, we observe a similar situation
+on the AffectNet dataset. This indicates that although the
+image-to-landmark branch contributes to the POSTER V1
+FER performance, it is the landmark-to-image branch that
+plays a decisive role in POSTER V1. Next, we analyze the
+above results at the theoretical level.
+Discussion.
+The two-stream design in POSTER V1 is
+mainly used to solve the issues of intra-class discrepancy
+and inter-class similarity in FER. It includes landmark-to-
+
+2nd POSTER-V2 level
+1st POSTER-V2 level
+ 3rd POSTER-V2 level
+Landmark Stage 2
+ Stage
+LMFi
+LMF2
+..
+Landmark
+Input Image
+LMFs
+LMF2
+ViT Model
+LMFi
+head
+IMFi
+IMF2
+IMFs
+Image Stage 2
+Image Stage 3
+IMFi
+IMF2
+Low-Level Feature Extraction (LFE)
+High-Level Feature Extraction (HFE)
+Multi-Level Feature Integration (MFI)Figure 4. Input images (row 1), facial landmark images (row 2),
+landmark-to-image branching attention visualization results (row
+3). We visualize the attention map belonging to the last layer of the
+landmarks to image branching for high-level features in POSTER
+V1. We can observe that with the help of landmark features, the
+attention map focuses more on the outstanding areas of face and
+less on the areas common to face.
+image and image-to-landmark branches.
+We revisit the
+influence of the two branches on POSTER V1.
+In the
+landmark-to-image branch, the landmark features inter-
+act with the image features as queries Qlm in the cross-
+attention mechanism. Image features are guided by land-
+mark features to more easily represent salient regions of fa-
+cial expressions when dealing with intra-class discrepancy
+issue. Also benefiting from the sparsity of landmark fea-
+tures, image features guided by landmark features reduce
+the focus on regions where faces are prevalent. This helps
+to reduce the impact of inter-class similarity in FER. The
+results of the visualization of landmark-to-image branch
+attention in Figure 4 also validate the above statements.
+Therefore, the landmark-to-image branch in the two-stream
+is essential and needs to be retained.
+For the image-to-
+landmark branch, the image features interact with the land-
+mark features as query Qimg to compensate for the lack
+of landmark features. Although this also benefits the FER
+task to some extent, it does not contribute to solving the
+issues of inter-class similarity and intra-class discrepancy
+as well as comes with a huge computational cost. This is
+consistent with the results we observed in the ablation ex-
+periments of Table 1. Thus, by making a trade-off between
+computational cost and accuracy, we eventually remove the
+image-to-landmark branch in the two-stream design.
+3.4. Cross-fusion
+In POSTER V2 we use window-based cross-attention
+mechanism instead of vanilla cross-attention mechanism in
+POSTER V1 for the purpose of linear computation. Fig-
+ure 5 illustrates the detailed differences between the two
+cross-attention mechanisms. For image features Ximg ∈
+RN×D, we first divide them into several non-overlapping
+windows zimg
+∈ RM×D, where zimg contains M to-
+Figure 5. Window-based cross attention mechanism and vanilla
+cross attention mechanism.
+kens. For the landmark feature Xlm ∈ RC×H×W , we first
+down-sample it to the window size zlm ∈ Rc×h×w, where
+c = D, M = h × w. Then we reshape it according to
+the shape of Zimg. At this point, the cross-attention with I
+heads in a local window can be formulated as:
+q = zlmwq, k = zimgwk, v = zimgwv,
+o(i) = θ(q(i)k(i)T/
+√
+d + b)v(i), i = 1,...,I,
+o = [o(1), . . . , o(I)]wo,
+(4)
+where wq, wk, wv, wo are the mapping matrix, respectively.
+θ (·) is the softmax function. [·] denotes the merge operation
+and b ∈ RI×I is the relative position bias.
+We perform the above cross-attention calculation for
+all windows. We refer to this cross-attention mechanism
+as window-based multi-head cross-attention (W-MCSA).
+Thus the cross-fusion transformer encoder in POSTER V2
+can be expressed as follows:
+X’img = W-MCSA(img) + Ximg,
+Ximg o = MLP(Norm(X’img)) + X’img,
+(5)
+Computational Complexity Analysis. Since the query in
+the two types of cross-attention computation keeps the same
+shape as the key, value, we can use the multi-head self-
+attention and the window-based multi-head self-attention
+complexity to represent their computational complexity.
+This can be indicated as follows:
+Ω(MCSA) = 4ND2 + 2N2D,
+Ω(W-MCSA) = 4ND2 + 2M2ND,
+(6)
+
+Attention Query
+Window-based Cross Attention Mechanism
+Vanilla Cross Attention MechanismAccording to Eqn 6, we can find that the window-based
+cross-attention mechanism we use successfully reduces the
+computational complexity of cross-fusion in POSTER V1
+from square level to linear level. This further improves the
+computational efficiency of POSTER V2.
+3.5. Multi-scale feature extraction
+From Figure 3, we can observe that POSTER V2 re-
+moves the pyramid design from POSTER V1. Moreover, in
+POSTER V2, we extract multi-scale features directly from
+facial landmark detector and image backbone. And we also
+add a small vision transformer network to POSTER V2 for
+the integration of multi-scale features.
+For the obtained
+multi-scale features o1, o2, o3, we directly merge in the to-
+ken dimension and using the vanilla transformer blocks for
+processing. This process is specifically described as:
+o = [o1, o2, o3],
+o’ = MSA(o) + o,
+oout = MLP(Norm(o’)) + o’,
+(7)
+where MSA (·) represents multi-head self-attention mech-
+anism. For above design we discuss as follows.
+Discussion. POSTER V1 adopts the pyramid structure to
+solve the scale sensitivity problem in FER. However, we
+consider that the pyramid structure design is only an up-
+sampling and down-sampling operation on the basis of the
+same scale feature map. Although it provides multi-scale
+information to some extent, we believe that it is not as good
+as multi-scale feature extraction directly from the network.
+The method analysis in section 4.3 also proves our point.
+For the integration of multi-scale features, we believe that
+the vanilla transformer block is sufficient for this task. We
+combine the tokens of all scale feature maps together, and
+the attention mechanism can model long-range dependen-
+cies for all scale tokens. Thus, different scales of token in-
+formation are delivered in the transformer block.
+4. Experiments
+We verify the effectiveness of POSTER V2 on several
+standard FER datasets such as RAF-DB [29], AffectNet
+[32] and CAER-S [27]. In the following, we first compare
+POSTER V2 with SOTA methods. We then conduct a se-
+ries of method analysis and ablation studies on POSTER
+V2. More detailed experimental setup, more experimen-
+tal results and visualization results are detailed in the Ap-
+pendix.
+4.1. Experiment Setup
+Datasets. We evaluat the FER performance of POSTER
+V2 on the widely used RAF-DB, AffectNet and CAER-S
+Dataset
+Train size
+Test size
+Classes
+RAF-DB
+12271
+3068
+7
+AffectNet (7 cls)
+280401
+3500
+7
+AffectNet (8 cls)
+283501
+4000
+8
+CAER-S
+44996
+20987
+7
+Table 2. Detailed size of the experimental dataset.
+datasets. The Real-world Affective Faces Database (RAF-
+DB) is a large-scale database of facial expressions, anno-
+tated by 315 staff members (students and faculty members
+of the University). For the selection of expressions, RAF-
+DB selected six basic emotions as well as neutral emotions
+from a range of expressions (e.g., smile, cackle, cry, anger,
+fear, dread, fear, shock, surprise, disgust, and no expres-
+sion), for a total of seven expressions for expression anno-
+tation. It mainly contains 12,271 training images as well
+as 3,068 test images. AffectNet is currently the largest pub-
+licly available dataset in the FER field. It contains about 1M
+images of faces associated with emotional words. It mainly
+contains 8 categories of primary emotions (neutral, happy,
+angry, sad, fear, surprise, disgust,contempt). We mainly use
+AffectNet settings based on class 7 (excluding contempt) as
+well as class 8. AffectNet (7 cls) consists of 280K training
+images and 3.500 validation images (500 images per cat-
+egory). AffectNet (8 cls) consists of 283K training images
+and 4.000 validation images (500 images per category). The
+CAER-S dataset was obtained from the CAER dataset con-
+taining 65,983 images. It is mainly divided into 7 types of
+expressions: neutral, happy, sad, surprised, fear, disgust and
+anger. In the FER task we used 44996 images for training
+and 20987 images for testing. The specific dataset configu-
+ration is shown in Table 2.
+Methods
+Year
+RAF-DB
+AffectNet (7 cls)
+AffectNet (8 cls)
+SCN [46]
+CVPR 2020
+87.03
+-
+60.23
+PSR [45]
+CVPR 2020
+88.98
+63.77
+60.68
+LDL-ALSG [5]
+CVPR 2020
+85.53
+59.35
+-
+RAN [47]
+TIP 2020
+86.9
+-
+-
+DACL [11]
+WACV 2020
+87.78
+65.2
+-
+KTN [28]
+TIP 2021
+88.07
+63.97
+-
+DMUE [39]
+CVPR 2021
+89.42
+63.11
+-
+FDRL [36]
+CVPR 2021
+89.47
+-
+-
+VTFF [31]
+TAC 2021
+88.14
+61.85
+-
+ARM [40]
+2021
+90.42
+65.2
+61.33
+TransFER [51]
+ICCV 2021
+90.91
+66.23
+-
+DAN [49]
+2021
+89.7
+65.69
+62.09
+EfficientFace [57]
+AAAI 2021
+88.36
+63.7
+60.23
+MA-Net [56]
+TIP 2021
+88.42
+64.53
+60.29
+Meta-Face2Exp [53]
+CVPR 2022
+88.54
+64.23
+-
+EAC [54]
+ECCV 2022
+90.35
+65.32
+-
+POSTER V1 [58]
+2022
+92.05
+67.31
+63.34
+POSTER V2
+-
+92.21
+67.49
+63.77
+Table 3. Comparison results with SOTA FER algorithm on RAF-
+DB and AffectNet.
+Settings. Similar to POSTER V1 [58], we also use the ir50
+[7] network pre-trained on the Ms-Celeb-1M [14] dataset as
+the image backbone. And MobileFaceNet [2] with frozen
+weights is used as our facial landmark detector. We employ
+
+Dataset
+Method
+Neutral
+Happy
+Sad
+Surprise
+Fear
+Disgust
+Anger
+Contempt
+mean Acc
+RAF-DB
+POSTER V1
+92.35
+96.96
+91.21
+90.27
+67.57
+75
+88.89
+-
+86.04
+RAF-DB
+POSTER V2
+92.06
+97.22
+92.89
+90.58
+68.92
+71.88
+88.27
+-
+85.97
+AffectNet (7 cls)
+POSTER V1
+67.2
+89
+67
+64
+64.8
+56
+62.6
+-
+67.23
+AffectNet (7 cls)
+POSTER V2
+65.4
+89.4
+68
+66
+64.2
+54.4
+65
+-
+67.45
+AffectNet (8 cls)
+POSTER V1
+59.4
+80.2
+66.6
+63.6
+63.6
+59.8
+58.8
+54.71
+63.34
+AffectNet (8 cls)
+POSTER V2
+60.6
+76.4
+66.8
+65.6
+63
+58
+60.2
+59.52
+63.76
+Table 4. Class-wise accuracy of POSTER V1 and POSTER V2 on RAF-DB, AffectNet (7 cls), and AffectNet (8 cls) datasets. Green, blue
+and red mark the highest value of single category in RAF-DB, AffectNet (7 cls) and AffectNet (8 cls) respectively.
+the Adam [25] optimizer for 200 epochs training. A train-
+ing scheme with a batch size of 144, a learning rate of 3.5e-4
+and a weight decay of 1e-4 was used. We use random hor-
+izontal flipping and random erasing as our data augmenta-
+tion methods. For the loss function, we choose the standard
+cross-entropy loss. We eventually realized POSTER V2 on
+a single NVIDIA RTX 3090 via Pytorch.
+4.2. Comparison with SOTA FER Methods
+Results on RAF-DB. We compare POSTER V2 with the
+SOTA FER algorithms in recent years on the RAF-DB
+datasets in Table 3.
+The experimental results show that
+POSTER V2 exhibits SOTA performance on RAF-DB.
+Compared with POSTER V1 (92.05), POSTER V2 im-
+proved by 0.16. +1.86 for POSTER V2 over EAC (90.35),
+and +1.3 for POSTER V2 over TransFER (90.91). This
+shows the superiority of PSTER V2 on RAF-DB. Table 4
+shows the comparison of POSTER V2 with POSTER V1
+for RAF-DB individual classes and average accuracy. Al-
+though POSTER V2 outperformed POSTER V1 in sev-
+eral categories, the average accuracy was slightly inferior
+to POSTER V1.
+Results on AffectNet. In Table 3, we also conduct FER ex-
+periments on AffectNet (7 cls) as well as AffectNet (8 cls).
+We observe that POSTER V2 exhibits SOTA FER effect
+in both AffectNet (7 cls) and AffectNet (8 cls). Compared
+with POSTER V1 (67.31, 63.34), POSTER V2 increases
+0.18, 0.43 on AffectNet (7 cls) and AffectNet (8 cls), re-
+spectively. On AffectNet (8 cls), POSTER V2 is higher than
+DAN (62.09) by 1.68. On AffectNet (7 cls), POSTER V2
+is greater than TransFER (66.23) with 1.26. This demon-
+strates that POSTER V2 can maintain excellent FER perfor-
+mance even on larger datasets. Table 4 shows that POSTER
+V2 exceeds POSTER V1 for the majority of individual class
+accuracies in both AffectNet (7 cls) and AffectNet (8 cls).
+As a result, POSTER V2 achieves better average accuracy
+than POSTER V1 on AffectNet.
+Results on CAER-S. We compare POSTER V2 with SOTA
+FER methods of recent years on the CAER-S dataset.
+Our POSTER V2 in Table 5 performs extremely well on
+the CAER-S dataset.
+Specifically, POSTER V2 scored
+92.98 on CAER-S. +0.27 for POSTER V2 over POSTER
+Methods
+Year
+CAER-S
+DSN [10]
+ICML 2018
+75.19
+CAER-Net-S [27]
+ICCV 2019
+73.51
+GRERN [12]
+IEEE Access 2020
+81.31
+EfficientFace [57]
+AAAI 2021
+85.87
+MA-Net [56]
+TIP 2021
+88.42
+GLAMOR-Net [26]
+NCA 2021
+89.88
+POSTER V1 [58]
+2022
+92.73
+POSTER V2
+-
+93
+Table 5. Comparison results with SOTA FER algorithm on CAER-
+S.
+V1 (92.73). +3.12 for POSTER V2 over GLAMOR-Net
+(89.88), and +4.58 for POSTER V2 over MA-Net (88.42).
++7.13 for POSTER V2 over EfficientFace (85.87).
+The
+excellent results on CAER-S prove that the success of
+POSTER V2 is no accident. It shows the powerful gen-
+eralization ability of POSTER V2.
+4.3. FLOPs and Param Comparison
+Methods
+#Param
+#FLOPs
+RAF-DB
+AffectNet
+POSTER V1-T
+52.2M
+13.6G
+91.36
+66.87
+POSTER V1-S
+62.0M
+14.7G
+91.54
+67.13
+POSTER V1
+71.8M
+15.7G
+92.05
+67.31
+POSTER V2
+43.7M
+8.4G
+92.21
+67.49
+Table 6. Comparison of Param and FLOPs with POSTER V1.
+From Table 6, we can see that POSTER V2 achieves
+better FER results with smaller Param and FLOPs than
+POSTER V1. Compared to POSTER V1-T, POSTER V2
+reduces 8.5M Param and 5.2G FLOPs, while increasing
+0.85% on RAF-DB and 0.62% on AffectNet. Compared
+to POSTER V1-S, POSTER V2 reduces 18.3M Param and
+6.3G FLOPs, while increasing 0.67% on RAF-DB and
+0.36% on AffectNet. Compared to POSTER V1, POSTER
+V2 reduces 28.1M Param and 7.3G FLOPs, while increas-
+ing 0.16% on RAF-DB and 0.18% on AffectNet. Therefore,
+POSTER V2 would be a better choice for the FER task.
+4.4. Method Analysis
+In this sub-section, we present a method analysis for the
+small ViT model we used in POSTER V2 on RAF-DB.
+
+Figure 6. Influence of different depth ViT models on POSTER V2
+for RAF-DB.
+Vit depth.
+Here, we investigate the impact of different
+depths for ViT on the FER performance of POSTER V2.
+In Figure 6, we show the influence of the ViT model with
+depth {2,4,6,8} on POSTER V2. We observe that for multi-
+scale integration we do not need to increase the depth of the
+ViT model. The ViT model with a depth of 2 is sufficient to
+handle the FER task. A deeper ViT model hurts the perfor-
+mance of POSTER V2 instead.
+ViT w/ pre-trained weights
+RAF-DB
+AffectNet
+
+92.21
+67.49
+
+91.49
+60.2
+Table 7. Impact of pre-trained ViT models for POSTER V2 on
+FER.
+Pre-trained Vit. We study the influence of the pre-trained
+ViT model on POSTER V2. We use the ViT pre-trained
+weights on ImagenNet-21K [35] for POSTER V2. Table 7
+shows that the performance of POSTER V2 on FER drops
+after using the pre-trained ViT model. We argue that this is
+mainly due to the fact that the pre-trained ViT model acts
+mainly on the feature extraction of the image-level inputs.
+However, in POSTER V2, ViT performs the multi-scale fea-
+ture integration task of feature-level inputs. The difference
+in input and task resulted in the pre-trained ViT not working
+on POSTER V2.
+4.5. Ablation Study
+Methods
+RAF-DB
+AffectNet
+POSTER V2
+92.21
+67.49
+w/o multi-scale feature extraction
+91.47
+66.51
+w/o ViT
+91.86
+66.92
+w/o W-MCSA
+91.56
+67.24
+w/o cross-fusion
+91.39
+66.35
+Table 8. Results of ablation experiments of key components of
+POSTER V2.
+We validate the effectiveness of our POSTER V1 im-
+provement component on the RAF-DB as well as on the
+AffectNet dataset.
+Multi-scale feature extraction. We first verify the effec-
+tiveness of extracting multi-scale features directly in the
+network. In this ablation experiment, we only use the im-
+age backbone as well as the last layer of feature maps from
+the facial landmark detector for cross-fusion. From Table 8
+we observe that POSTER V2 degrades significantly on the
+RAF-DB and AffectNet datasets when multi-scale feature
+extraction is not performed. This shows that our method
+of directly extracting multi-scale features can also solve the
+scale sensitivity issue of FER. Also, this indicates the im-
+portance of multi-scale features for FER.
+Vit. For the ViT used for multi-scale feature integration, we
+ablate it. We directly sum several different scale features
+for FER. According to the experimental results in Table 8,
+POSTER V2 decreases by 0.35 on RAF-DB and 0.57 on
+AffectNet when multi-scale feature integration is not per-
+formed by ViT. This suggests that ViT facilitates multi-scale
+feature integration.
+W-MCSA. We validate the effectiveness of W-MCSA for
+cross-fusion by ablation experiments. In this experiment,
+we use the vanilla cross-attention mechanism to replace our
+window-based cross-attention mechanism.
+We observed
+that POSTER V2 degraded on both RAF-DB and Affect-
+Net datasets. This shows that the W-MCSA we use both
+improves the FER accuracy and reduces the computational
+complexity of POSTER V1. Thus, W-MCSA is essential
+for POSTER V2.
+Cross-fusion. This experiment mainly verifies the role of
+landmark-to-image branch for POSTER V2. In the abla-
+tion experiments on cross-fusion, we merge the extracted
+image multi-scale features and landmark multi-scale fea-
+tures directly and integrate them by ViT. Table 8 shows that
+the effectiveness of POSTER V2 on RAF-DB as well as
+AffectNet drops sharply when cross-fusion is not applied.
+This shows that cross-fusion is the key for POSTER V2
+to achieve SOTA FER. Also, this indicates that addressing
+inter-class similarity and intra-class discrepancy are partic-
+ularly important for FER task.
+5. Conclusion
+In this paper, we improve POSTER V1 from three direc-
+tions: two-stream, cross-fusion, and multi-scale feature ex-
+traction to obtain a simpler and stronger vision transformer
+for FER, POSTER V2. Extensive FER experimental results
+show that POSTER V2 achieves the state-of-the-art FER
+performance while greatly reducing the Param and FLOPs
+of POSTER V1. This suggests that POSTER V2 achieves a
+better trade-off between accuracy and computational com-
+plexity. Therefore, POSTER V2 is a better choice for the
+FER task.
+
+92.4
+92.2
+RAF-DB Top-1 Accuracy (%)
+92
+91.8
+91.6
+91.4
+91.2
+2
+4
+6
+8
+ViT DepthAcknowledge
+This work was supported by Public-welfare Technology
+Application Research of Zhejiang Province in China un-
+der Grant LGG22F020032, and Key Research and Devel-
+opment Project of Zhejiang Province in China under Grant
+2021C03137.
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+
+Appendix
+A. Implementation Details
+For POSTER V2, we conduct FER experiments on
+the RAF-DB, AffectNet, and CAER-S datasets, respec-
+tively. For different datasets, we adopt different detail set-
+tings. Specifically, for different datasets, we exploit differ-
+ent learning rates for training according to the settings of
+POSTER V1. Moreover, for AffectNet (8 cls), POSTER
+V2 uses a classification head with a category number of 8
+for prediction. The rest of the settings are consistent with
+the experimental sections in the main text.
+config
+value
+optimizer
+Adam
+base learning rate
+3.50E-05
+weight decay
+1.00E-04
+batch size
+144
+training epochs
+200
+learning rate schedule
+ExponentialLR (gamma=0.98)
+augmentation
+RandomHorizontalFlip(),
+RandomErasing(scale=(0.02, 0.1)).
+drop path
+linspace(0, 0.5, 5)
+num classes
+7
+Table 9. Supervised training POSTER V2 from scratch on RAF-
+DB.
+RAF-DB Settings.
+We use the Adam optimizer with a
+learning rate of 3.5e-5 for 200 epochs training. The batch
+size is maintained at 144 and the weight decay remains at
+1e-4. The learning rate schedule uses an exponential decay
+with a gamma of 0.98. Data augmentation includes random
+horizontal flipping and random erasure. The specific set-
+tings are shown in Table 9.
+config
+value
+optimizer
+Adam
+base learning rate
+1.00E-06
+weight decay
+1.00E-04
+batch size
+144
+training epochs
+200
+learning rate schedule
+ExponentialLR (gamma=0.98)
+augmentation
+RandomHorizontalFlip(),
+RandomErasing(p=1, scale=(0.05, 0.05)).
+drop path
+linspace(0, 0.5, 5)
+num classes
+7
+Table 10. Supervised training POSTER V2 from scratch on Af-
+fectNet (7 cls).
+AffectNet (7 cls) Settings. On the AffcetNet (7 cls) dataset,
+we adjust the learning rate to 1e-6. The training epochs re-
+mains at 200. The batch size is maintained at 144 and the
+weight decay remains at 1e-4. The learning rate schedule
+uses an exponential decay with a gamma of 0.98. Data aug-
+mentation includes random horizontal flipping and random
+erasure. The detailed settings are shown in Table 10.
+config
+value
+optimizer
+Adam
+base learning rate
+1.00E-06
+weight decay
+1.00E-04
+batch size
+144
+training epochs
+200
+learning rate schedule
+ExponentialLR (gamma=0.98)
+augmentation
+RandomHorizontalFlip(),
+RandomErasing(p=1, scale=(0.05, 0.05)).
+drop path
+linspace(0, 0.5, 5)
+num classes
+8
+Table 11. Supervised training POSTER V2 from scratch on Af-
+fectNet (8 cls).
+AffectNet (8 cls) Settings. We use the Adam optimizer
+with a learning rate of 1e-6 for 200 epochs training. The
+batch size is maintained at 144 and the weight decay re-
+mains at 1e-4. The learning rate schedule uses an expo-
+nential decay with a gamma of 0.98. Data augmentation
+includes random horizontal flipping and random erasure. In
+addition, we set the number of categories to 8. Table 11
+shows the specific experimental settings.
+config
+value
+optimizer
+Adam
+base learning rate
+4.00E-05
+weight decay
+1.00E-04
+batch size
+144
+training epochs
+200
+learning rate schedule
+ExponentialLR (gamma=0.98)
+augmentation
+RandomHorizontalFlip(),
+RandomErasing(p=1, scale=(0.05, 0.05)).
+drop path
+linspace(0, 0.5, 5)
+num classes
+7
+Table 12. Supervised training POSTER V2 from scratch on
+CAER-S.
+CAER-S Settings. On the CAER-S dataset, we employ the
+Adam optimizer with a learning rate of 4e-5 for 200 epochs
+of training. The batch size is maintained at 144 and the
+weight decay remains at 1e-4. The learning rate schedule
+uses an exponential decay with a gamma of 0.98. Data aug-
+mentation includes random horizontal flipping and random
+erasure. The specific settings are shown in Table 12.
+B. Detailed Experimental Results
+In this section, we show more detailed experimental re-
+sults of POSTER V2 on each dataset. And we also show
+the confusion matrix of POSTER V2 in each dataset in Fig-
+ure 7.
+RAF-DB Results. Figure 8 shows the specific training pro-
+cess of POSTER V2 on RAF-DB. We observe that the train-
+ing loss and validation loss of POSTER V2 decrease un-
+til saturation during the training process. Furthermore, the
+training accuracy and validation accuracy of POSTER V2
+
+Figure 7. The confusion matrix of POSTER V2 on each dataset.
+Figure 8. The specific training process of POSTER V2 on RAF-
+DB.
+continue to increase until a small fluctuation.
+Figure 9. The detailed training process of POSTER V2 on Affect-
+Net (7 cls).
+AffectNet (7 cls) Results. We show in Figure 9 the detailed
+training of POSTER V2 on AffectNet (7 cls). POSTER V2
+achieves the best training results on AffectNet (7 cls) at an
+early stage. At this point, POSTER V2 achieves the highest
+accuracy on AffectNet (7 cls) for both the training and test
+sets. Therefore, we stop training in time to save training
+costs.
+AffectNet (8 cls) Results. Figure 10 shows the exact per-
+formance of POSTER V2 on AffectNet (8 cls). We observe
+a similar phenomenon on AffectNet (8 cls) as POSTER V2
+did on AffectNet (7 cls). POSTER V2 also reach saturation
+in the early stages of AffectNet (8 cls). POSTER V2 train-
+ing loss continues to show a decreasing trend, yet there is a
+small increase in validation loss. Nevertheless, the training
+Figure 10. The detailed training process of POSTER V2 on Af-
+fectNet (8 cls).
+accuracy of POSTER V2 on AffectNet (8 cls) continues to
+increase, and the validation accuracy has largely been op-
+timal and remains constant. Therefore, we take the same
+early end operation for POSTER V2 on AffectNet (8 cls) as
+we do for AffectNet (7 cls).
+Figure 11. The detailed training process of POSTER V2 on
+CAER-S.
+CAER-S Results. We show the specific training perfor-
+mance of POSTER V2 on CAER-S in Figure 11. Compared
+with other datasets, POSTER V2 has a relatively long sat-
+uration time on the CAER-S dataset. During the training
+process, the loss on the POSTER V2 training and validation
+sets decreases and saturates at a late stage. Meanwhile, the
+accuracy of POSTER V2 on both the training and validation
+sets has been increasing.
+
+the accuracy/loss curve of train/val
+100
+95
+90
+85 
+80
+75 
+70 
+65
+60
+accuracy
+55
+45
+40
+35
+30 
++..
+ 25 
+20 
+15 
+10 
+train-accuracy
+valid-accuracy
+5 
+valid-loss-x30
+0
+10
+15
+20
+25
+OE
+35
+40
+55
+90
+95
+100
+105
+110
+115
+180
+185
+195
+200
+the training epochthe accuracy/loss curve of train/val
+100
+95
+90 
+85
+80 
+75 
+70 
+65
+60 
+45
+35
+OE
+25
+20 
+15 
+10 
+.++,+++++++++.+
+++++*++*++
+train-accuracy
+valid-accuracy
+5 
+train-loss-x30
+valid-loss-x30
+0
+10
+15
+20
+25
+30 
+35
+40 
+45
+50
+55
+60
+65
+70
+75
+08
+85
+90
+95
+100
+105
+110
+115
+130
+135
+140
+145
+160
+180
+185
+190
+195
+200
+the training epochRAF-DB
+AffectNet (7 cls)
+AffectNet (8 cls)
+CAER-S
+0.0182 0.0122 0.0182 0.000 0.0122 0.0334
+0.0660 0.0940 0.0960 0.0120 0.0140 0.0640
+Neutral 
+0. 6060
+0.0200.0960.0840.0120.0140.000.1080
+Surprise
+0.0020 0.0060 0. 0177 0. 0073 0.0067 0.0210
+0. 8
+0.0020 0.0360 0.0020 0.0140 0.00800.1461
+0.00000.02700.10810.01350.0135
+Happy J 0. 0460
+0.00200.03800.00200.01200.0060
+ 0. 7
+ 0.6
+Fear J0.0000
+0.00030.0010
+0.8
+Sad Jo.1260 0.0120 0.668
+0. 0563 0. 0938 0.0437 0.0688
+0.6
+0380 0.0260 0.0420 0.0660 0.0220
+Disgust J0.0125 0.0030.7188
+Sad /0.1320 0. 0180
+0.03600.03000.03600.0680
+Disgust J0. 0043 0. 0003
+0.984
+0.00300.00300.0013
+0. 6
+0.0680 0.0620 0.0320
+0. 0034 0.0008 0.0042
+0.9722
+0.0034 0.0017 0. 0143
+ Surprise -0.0780 0.0760 0. 0260
+0.12000.02400.0160
+.0050~0.0070
+label
+Happy 
+0.01230.0407
+True
+0.9289
+0.00000.0439
+0. 4
+0.04800.0280
+0.0050 0.0080
+0060 0.0127
+ 0.4
+0.16200.0420
+0. 0185 0.0062
+0. 0309
+.0740 0.0240 0. 05800.5440
+Anger
+0.950
+0. 2
+0. 1
+0.9200
+0.03100.0440
+Conte
+Predicted label
+Predicted label
+Predicted labelthe accuracy/loss curve of train/val
+100
+95
+90
+85 
+80 
+75
+10
+65 
+60 
+55
+45 
+40 
+35 
+30 卡
+25 
+20 
+15 
+.
+10 -
+train-accuracy
+valid-accuracy
+5 
+train-loss-x30
+.
+valid-loss-x30
+0
+10
+15
+20
+25
+OE
+35
+40
+45
+55
+60
+65
+70
+80
+90
+95
+100
+105
+110
+120125
+130
+135140
+145
+150155
+160
+170175
+180
+185
+190
+195
+200
+the training epochthe accuracy/loss curve of train/val
+100
+95
+90
+85 
+80
+75 
+70
+65
+60
+ 55 
+ 50 
+45
+35
+30 -
+ 25 
+20 
+15 
+train-accuracy
+valid-accuracy
+5 
+valid-loss-x30
+0
+15
+20
+25
+30
+35404550
+55
+90
+95
+100
+105
+110
+115
+180
+185
+195
+200
+the training epochFigure 12. Comparison of POSTER V2 and POSTER V1 high-dimensional space t-SNE visualization results. POSTER V1 t-SNE visual-
+ization results (first row), POSTER V2 t-SNE visualization results (second row).
+Figure 13. POSTER V2 cross-fusion stage attention visualization results. For each triplet, we show the input image (left), the landmark
+image (middle), and attention map (right).
+C. Visualization
+T-SNE Visualization. We visualized the high-dimensional
+features of POSTER V1 and POSTER V2 using t-SNE
+[44]. As can be seen in Figure 12, both POSTER V2 and
+POSTER V1 present good t-SNE visualization results on
+RAF-DB and CAER-S datasets. There is almost no signif-
+icant difference between the t-SNE visualization results of
+POSTER V1 and POSTER V2 on CAER-S. POSTER V2
+has a closer intra-class distance than POSTER V1 on RAF-
+DB. Although POSTER V1 and POSTER V2 have poor t-
+SNE visualization results on AffectNet (7 cls) and Affect-
+Net (8 cls). But the inter-class distance between clusters in
+POSTER V2 is further than POSTER V1. Above results
+indicates that POSTER V2 is better than POSTER V1 in al-
+leviating the issues of inter-class similarity and intra-class
+discrepancy in FER.
+Attention Visualization. We visualize the attention map of
+the highest-level features of the POSTER V2 cross-fusion
+stage. From Figure 13, we observe that POSTER V2 suc-
+cessfully captures important facial expression features with
+the help of facial landmark features.
+
+RAF-DB
+AffectNet (7 cls)
+AffectNet (8 cls)
+CAER-SNeutral
+Happy
+Sad
+Surprise
+Fear
+Disgust
+Angry
+Contempt

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+Impact Invariant Trajectory Optimization of 5-Link Biped Robot 
+Using Hybrid Optimization 
+Aref Amiri, Hasan Salarieh1 
+Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran 
+ 
+Abstract 
+Bipedal robots have received much attention because of the variety of motion maneuvers that they can produce, and the many 
+applications they have in various areas including rehabilitation. One of these motion maneuvers is walking. In this study, we 
+presented a framework for the trajectory optimization of a 5-link (planar) Biped Robot using hybrid optimization. The walking is 
+modeled with two phases of single-stance (support) phase and the collision phase. The dynamic equations of the robot in each 
+phase are extracted by the Lagrange method. It is assumed that the robot heel strike to the ground is full plastic. The gait is optimized 
+with a method called hybrid optimization. The objective function of this problem is considered to be the integral of torque-squared 
+along the trajectory, and also various constraints such as zero dynamics are satisfied without any approximation. Furthermore, in a 
+new framework, there is presented a constraint called impact invariance, which ensures the periodicity of the time-varying 
+trajectories. On the other hand, other constraints provide better and more human-like movement..  
+Keywords: Trajectory optimization, bipedal robots, walking robots, zero dynamics; 
+ 
+ 
+  1. Introduction 
+The mechanism of movement and transfer of objects has always been one of the most important and active areas of 
+human research. Due to the limitations of moving with a wheel, replacing it with feet is an attractive but difficult 
+option, so this field is a hot topic in today's robotic world. With the advancement of robotics science and the usefulness 
+of this issue, a lot of research has been done on the design, optimization, and control of legged robots [1-6]. As the 
+science of bipedal robots has advanced in recent years, there have been significant efforts to improve the performance 
+of these robots in important maneuvers, such as walking and running, but research is still ongoing to find ideal answers 
+[7,8]. Designing reference trajectories for human walking cycles is very important. Several techniques have been 
+adopted to define reference trajectories. So far, many researchers have studied low-energy (or low input torques) paths 
+for bipedal robots [7,9]. We are looking for a periodic path that meets a specific goal in terms of speed and minimizes 
+the torque required to produce the gate. In general, this open and non-trivial problem is solved by finding numerical 
+answers. Various parameters can be considered to optimize the problem, for example, torques, Cartesian coordinate 
+or joint coordinates constraints can be used[10-12]. Many authors have used polynomial functions for Cartesian 
+coordinates of swing leg’s foot, hip, and trunk angle [13,14]. Polynomial functions are used for the coordinates of the 
+joints to limit the number of optimization parameters [15]. The optimal path for each coordinate of joints is usually 
+written in the form of polynomials with unknown coefficients. The coefficients should be obtained through the 
+optimization process [15]. For all bipedal robots, it is important to define optimal periodic motions despite the fact 
+that the number of actuators is less than the degree of freedom of the system, and also zero dynamics problem there 
+exists which should be satisfied during optimization. 
+In this paper, a new method is presented to produce a periodic path for the walking of bipedal robots which satisfies 
+the impact invariance constraint. Also, in order to achieve the feasible trajectory, the zero dynamics constraint is 
+satisfied without any approximation. In addition, by considering some other kinematic and dynamic constraints, and 
+                                                           
+1P.O.B. 11155-9567, Tehran, Iran 
+  [email protected] 
+
+using the hybrid optimization method, an optimal reference trajectory for human-like walking of bipedal robots has 
+been presented.  
+Section 2 presents the dynamics and kinematics of a model of the biped robot. Section 3 is devoted to the formulation 
+of the optimization variables. The constraints are defined in section 4 and the optimization method is also described 
+in section 5. Finally, in sections 5 and 6, the results and conclusion are given. 
+2. Dynamics and kinematics 
+Bipedal robots have different dynamics depending on their movement maneuvers. For example, a running robot with 
+5 links and without an ankle actuator in the flight phase has 7 degrees of freedom and only 4 actuators, so the system 
+is 3 degrees under-actuated. Here a bipedal walking robot will be examined. We assume that the robot is completely 
+on the ground and does not slip while walking. 
+On the other hand, during the single support phase, the other leg rises from the ground when the swing leg hits the 
+ground.  During the single support phase, the model has 5 degrees of freedom and needs at least 5 generalized 
+coordinates to identify the system. On the other hand, the robot has only 4 actuation, so the system has a degree of 
+under-actuation. In under-actuated systems, some parts of the dynamics are not affected by the actuator called the zero 
+dynamics. Here, zero dynamics is affected only by the earth's gravity. The robot's model can be modeled with absolute 
+or relative angles, if relative angles are used, zero dynamics can be easily separated from the main total dynamics. 
+Figure 1 shows the absolute and relative coordinates of a 5-link robot with point feet. 
+ 
+Figure 1 Relative and absolute angles 
+ 
+The general hybrid walking gait model is obtained by combining the single support phase model and the impact model: 
+Σ: {
+𝑥̇ = 𝑓(𝑥) + 𝑔(𝑥)𝑢    𝑥− ∉ Γ
+𝑥+ =  (𝑥−)    
+𝑥− ∈ Γ
+                                                                                                                                   (1) 
+where   is a mapping that transforms the states just before the contact to the states just after the contact. 𝑥: =
+(𝑞𝑇, 𝑞̇ 𝑇)𝑇 is the state vector that contains 𝑞: = (𝑞1, 𝑞2, … , 𝑞𝑛)
+𝑇 which is the vector of joint coordinates and 𝑞̇: =
+(𝑞̇ 1, 𝑞̇ 2, … , 𝑞̇ 𝑛)
+𝑇 is the vector of angular velocities, and 𝑥+ denotes the state vector just after the impact and 𝑥− shows 
+just before this event. 
+The switching set is shown as, 
+  
+  
+  
+  
+  
+𝑞 
+𝑞 
+𝑞 
+𝑞 
+𝑞 
+𝑞 : relative coordinates 
+: absolute coordinates 
+  
+: swing leg's foot
+1
+1
+2 : stance leg's foot
+2
+
+Γ = {(𝑞, 𝑞̇) ∈ 𝑥 ∣ 𝑃 
+𝑣(𝑞) = 0, 𝑃 
+ℎ(𝑞) > 0}                                                                                                                                 (2) 
+ 𝑃 
+𝑣(𝑞) and 𝑃 
+ℎ(𝑞) indicate the vertical and horizontal position of the swing leg, respectively. Now if we model the 
+single support phase alone, we have: 
+𝑀(𝑞)𝑞̈ + 𝑐(𝑞, 𝑞̇)𝑞̇ + 𝐺(𝑞) = (0, 𝑈𝑇)𝑇                                                                                                                          (3) 
+where 𝑀(𝑞) ∈ ℜ𝑛×𝑛 (𝑛 = 5) is the inertia matrix, 𝑐(𝑞, 𝑞̇) ∈ ℜ𝑛×𝑛 is the Coriolis matrix, and 𝐺(𝑞) ∈ ℜ𝑛 is the gravity 
+vector. As shown in Figure 2, the robot does not have any actuators (torques) on the feet, i.e. the robot has not the 
+ankle joint actuator, so the robot is under-actuated which adds a zero dynamic constraint to the problem as mentioned 
+in [16]. The vector 𝑈 ∈ ℜ𝑛−   is as follows: 
+𝑈 = [𝜏 , 𝜏 , 𝜏 , 𝜏 ]𝜏                                                                                                                                                        (4) 
+which represents 4 actuators (torques) on the robot. 2 actuators (torques) on the pelvis (hip) and 2 on the knee of each 
+leg. By separating the equations of (3) the first equation which produces the zero dynamics is written as:  
+∑
+ 
+ 
+𝑗= (𝑀 ,𝑗𝑞̈𝑗 + 𝑐 ,𝑗𝑞̇𝑗) + 𝐺 = 0                                                                                                                                (5) 
+which is called zero-hybrid dynamics and: 
+∑
+ 
+ 
+𝑗= (𝑀 ,𝑗𝑞̈𝑗 + 𝑐 ,𝑗𝑞̇𝑗) + 𝐺 = 𝜏 −                                                                                                                              (6) 
+are other rows of equation (3) (i = 2,…, 5). 
+ 
+Figure 2 Robot configuration and control torques 
+The trunk angle is assumed independent from other links with a separate actuator, In other words, one actuator is 
+responsible for moving the trunk. So if we temporarily separate the trunk from the other components, we are faced 
+with 4 degrees of freedom system. By determining the swing leg's foot (link number 5 in figure 2), the system still has 
+2 degrees of freedom, so the inverse kinematics has infinite answers. Therefore, By determining the position of the 
+hip, 2 more degrees of freedom are determined from the system, in this case, the inverse kinematic robot has 4 answers. 
+Among these 4 answers, the only acceptable answers are the one that satisfies the condition of not breaking the knee. 
+It is important to note that in order to find a suitable periodic answer, we assume that the initial configuration is the 
+same as the final one. 
+3. Optimization variables 
+One convenient way is to select the angles of each link based on a polynomial function of time with a series of 
+unknown coefficients. This choice enables us to have a smooth function with time. Here it is assumed that each angle 
+is a polynomial function of degree 4. It should be noted that the initial and final configuration of the system in each 
+step affects determining two parameters of the polynomial coefficients, the impact invariance constraint is also 
+1
+5
+4
+2
+3
+  =0
+  
+  
+  
+  
+
+effective on another coefficient. Therefore, in order to have at least 2 optimization parameters for each angle, we 
+consider a fourth-order polynomial function with unknown coefficients for the trajectories of each angle. 
+ 
+𝑞𝑘(𝑡) = ∑
+ 
+𝑛= 
+ =0 𝛼𝑘, 𝑡  (𝑘 = 1, … , 5)                                                                                                                                           (7)                       
+4. Definition of constraints 
+These constraints are to find the right trajectory to walk. It makes the shapes of the joint trajectories, the links 
+orientations, and the required torques for walking be within a reasonable range. 
+The constraints are defined as follows: 
+1) Constraints on the initial and final configuration: Initial and final configurations of the robot must be specified. 
+Since the robot moves in a periodic pattern, its initial and final configuration must coincide. 
+𝑞(@𝑡=0)=𝑞𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ,   𝑞(@𝑡=𝑇)=𝑞𝑓𝑖𝑛𝑎𝑙 ,                                                                                                                                       (8) 
+2) Knee movement constraints: In order to have human-like movement, the robot's knees should not be opened and 
+closed excessively ( 𝑚  and  𝑚  are two pre-especified upper bounds in Eq. (9)). 
+   𝑚 ≥ 𝑞 (𝑡) ≥ 0 ,  𝑚 ≥ 𝑞 (𝑡) ≥ 0,                                                                                                                             (9)
+ 
+ 
+3) Swing leg's foot constraint: The swing leg's foot should not collide with the ground except at the beginning and end 
+of the phase. 
+ 𝑝(0)
+𝑣
+=  𝑝(𝑇)
+𝑣
+= 0  𝑝(𝑡)
+𝑣 >0 for 0 < 𝑡 < 𝑇                                                                                                                                   (10)
+ 
+                                                                                                                                   
+ 
+4) Limitation of torques: In order to the physical limitations of the motors, the actuator torques have a certain limit. 
+|𝜏 − (𝑡)| ≤ 𝜏𝑚𝑎𝑥      𝑖 = 2, … ,5                                                                                                                                      (11) 
+5) Limitation of angular velocities: In order to the physical limitations of the motors, the actuator velocities have a 
+certain limit. 
+|𝑞̇ (𝑡)| ≤ 𝑞̇𝑚𝑎𝑥      𝑖 = 1, … ,5                                                                                                                                         (12) 
+6) Limitation of friction coefficient: The reaction of the heels, which is the result of the acceleration of the various 
+members of the robot, must observe a certain ratio. This ratio should be less than the coefficient of friction between 
+the heels and the ground. 
+−𝜇 ≤ |
+𝐹𝑥
+𝐹𝑦| ≤ 𝜇                                                                                                                                                              (13) 
+In the above equation, 𝜇 is the coefficient of friction, and 𝐹𝑥 and 𝐹𝑦 are sequentially the horizontal and vertical ground 
+reactions. 
+7) Zero dynamic constraint: the satisfaction of this constraint is important in two ways. First, if this constraint is not 
+satisfied, the problem of optimizing the input torques is practically ambiguous, because these torques are not really 
+applicable to the problem. Although it may lead to a feasible kinematic equation (kinematically possible), it is not 
+feasible in terms of control, or in other words, it is not dynamically possible.  
+8) Impact invariance constraint: this constraint means that in order to produce a periodic motion, in addition to the 
+configuration, the initial velocities at the beginning point of each cycle should be exactly the same as its previous 
+cycle. Since the velocities after the collision are dependent on the velocities before the collision, by satisfying this 
+constraint, the velocities before the collision are adjusted in such a way as to guarantee the periodicity of the motion. 
+Through the following formulae, this purpose is achieved. At first, the impact mapping formula is written as, 
+
+𝑞̇ + = Δ̃(𝑞−)𝑞̇ −                                                                                                                                                               (14) 
+Δ̃(𝑞−) ∈ ℜ ×  is the impact mapping which maps the angular rates of the leg before contact to the angular rates of 
+that leg after contact. The inverse of Δ̃ is denoted by, 
+𝜂̃(𝑞−) = (Δ̃(𝑞−))
+− 
+                                                                                                                                                   (15) 
+So 𝑞̇ − can be found as :  
+𝑞̇ − = 𝜂̃(𝑞−)𝑞̇ +                                                                                                                                                            (16) 
+The mathematical formulation of this mapping is obtained from the governing differential equations of the system. 
+After the swing leg's foot hits the ground, the positions do not change but the angular velocities change, which can be 
+achieved as following (see [17] for more information), 
+ 
+Δ𝑞̇ = 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− ⋅ Δ𝑣𝑒                                                                                                                       (17) 
+ 
+where 𝑣  is the velocity vector of the end of the swing leg and 𝑀 ∈ ℜ𝑛×𝑛 is the inertia matrix as mentioned in (3), the 
+matrix𝐽 ∈ ℜ𝑚×𝑛  (𝑚 = 2 for planar motions) is also obtained as:  
+𝐽 =
+∂𝑝𝑒
+∂𝑞                                                                                                                                                                             (18) 
+𝑝𝑒 is the position of the end of the swing leg. Assuming that the swing leg sticks to the ground after impact, the velocity 
+of the swing leg's foot after impact is zero, so 
+𝑞̇ + = 𝑞̇ − + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− ⋅ (−𝑣𝑒)                                                                                                                        (19) 
+We know that due to the placement of a leg on the ground, we can write: 
+𝑣𝑒 = 𝛼(𝑞)𝑞̇                                                                                                                                                                  (20) 
+where 𝛼(𝑞) is: 
+𝛼(𝑞) =
+∂𝑣𝑒
+∂𝑞̇                                                                                                                                                                    (21) 
+Finally, by placing )20( into )19( and separating 𝑞̇ −, the pre-impact angular velocity is obtained as follows: 
+𝑞̇ − = (𝐼 + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− − 𝛼(𝑞))
+− 𝑞̇ +                                                                                                  (22) 
+where  𝐼 ∈ ℜ𝑛×𝑛 is the identity matrix. In the above relation, both velocity vectors are written in the same coordinate 
+system, which requires a coordinate conversion, because the coordinate changes after the collision due to the change 
+in the role of the legs. For this purpose, consider the following mapping that converts the relative angles and angular 
+velocities to absolute ones: 
+1𝑟𝑒𝑙κ= 𝐻  𝑎𝑏𝑠𝜿                                                                                                                                                               (23) 
+where 𝜿 ∈ ℜ𝑛 can be the angles vector, the angular velocities vector or the angular accelerations vector. Superscripts  
+1𝑟𝑒𝑙 and 1𝑎𝑏𝑠  represent relative and absolute coordinates in which the vectors are defined, and also 𝐻 ∈ ℜ𝑛×𝑛 is 
+a square matrix. On the other hand, we have a mapping that converts old and new coordinates to each other. This 
+mapping can just be defined for an absolute angular coordinate. If we define the absolute coordinates in this way, we 
+have: 
+1 𝜓 = Γ 𝜓                                                                                                                                                                    (24) 
+
+where indices 1 and 2 indicate the coordinate system before and after the impact, 𝜓 ∈ ℜ𝑛×𝑛 can be velocity vector or 
+angular acceleration vector, and  Γ ∈ ℜ𝑛×𝑛 is the mapping matrix. Finally, with the above transformations, the 
+coordinate systems can be connected suitably as: 
+1 𝑞̇ + = 𝐻Γ𝐻− 1 𝑞̇ +                                                                                                                                                        (25) 
+So the invariancy of the impact during walking is written as it follows, 
+𝑞̇ − = (𝐼 + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− − 𝛼(𝑞))
+− 𝐻Γ𝐻− 𝑞̇ +                                                                                        (26)                                               
+As a result, according to Equation (26), the impact invariance constraint is obtained. In this way, by satisfying this 
+equality constraint, the velocity after impact will be similar to the initial velocity in the previous cycle. 
+5. Optimization 
+According to figure 3, optimization is performed using a hybrid method. This means that first, with the penalty method, 
+the constrained problem becomes unconstrained. Then, using the genetic algorithm, the first level of optimization is 
+applied. Finally, in the second level, the outputs of the first level are used as the input of a gradient-based method and 
+the problem is solved. The objective function is the Euclidean norm of input torques: 
+𝐽(𝛼) = ∫
+ 
+𝑇(𝜁−)
+0
+∥∥𝑈𝛼(𝑡)∥∥ 
+ 𝑑𝑡 = ∫
+ 
+𝑇(𝜁−)
+0
+⟨𝜏, 𝜏⟩𝑑𝑡                                                                                                             (27) 
+where 𝑇(𝜁−) corresponds to the step duration, 𝑈𝛼(𝑡) is the resulting torque obtained from (3) along the periodic 
+solution of the hybrid zero dynamics. To solve the problem more easily and accurately, we tried to satisfy 
+configuration constraints in the problem itself. Therefore, 2 coefficients of each coordinate and a total of 10 parameters 
+of equation (7) are determined by the configuration constraints. 
+ 
+ 
+Figure 3 optimization diagram
+According to equation (7), the number of unknown coefficients for a polynomial of order 4 is equal to 5. On the other 
+hand, due to the existence of 5 independent angles, the number of unknown coefficients in the problem is 25. By 
+Dynamics
+Kinematics
+Setting
+Initializing
+Barrier/Penalty
+Method
+Genetic 
+Algorithm
+Gradient
+Based
+Method
+Physically 
+constraints
+constraints
+kinematically
+constraints
+Cost
+Function
+1
+3
+2
+1: Setting: Type of optimization variables – Desired velocity – Initial and final configuration
+2: Initialization reduces the number of variables and simplifies optimization.
+3: Using penalty/barrier functions, the constrained problem becomes unconstrained.
+F(x, r) = f (x) + P(h(x), g(x), r)
+where f (x) is the cost function h(x) is the vector of equalities constraint, g(x) is the vector of
+inequalities constraint, r is a vector of penalty parameters and P is a real-valued function whose
+action of imposing the penalty on the cost function is controlled by r.
+
+determining the initial and final configuration of the robot, the number of optimization variables for this problem is 
+reduced to 15 (by initializing). 
+6. Results 
+The simulation is based on the specifications of the RABBIT robot (Table 1). As a review, the nonlinear and 
+constrained optimization problem is first converted to a non-constrained problem by the penalty method, then with 
+the values and parameters in Tables 2 and 3, the first layer optimization problem is solved using the genetic algorithm. 
+Next, the outputs of the first layer of optimization are considered as the start point (initial condition) of the second 
+layer of optimization. The maximum violation of the constraints will be equal to .01 and the maximum iteration of the 
+interior-point algorithm is equal to 20. The initial and final configuration of the system as well as other specifications 
+and constraint bounds are given in Tables 3 and 4, respectively. 
+Table 1 RABBIT parameters[18] 
+Symbol 
+Value 
+Name 
+m1, m5 
+3.2 kg 
+mass of lower leg 
+m2, m4 
+6.8 kg 
+mass of upper leg 
+m3 
+20 kg 
+mass of trunk 
+I1, I5 
+0.93 kg-m2 
+rotational inertia of lower leg, about its center of mass 
+I2, I4 
+1.08 kg-m2 
+rotational inertia of upper leg, about its center of mass 
+I3 
+2.22 kg-m2 
+rotational inertia of trunk, about its center of mass 
+l1, l5 
+0.4 m 
+length of lower leg 
+l2, l4 
+0.4 m   
+length of femur 
+l3 
+0.625 m 
+length of trunk 
+d1, d5 
+0.128 m 
+distance from lower leg center of mass to knee 
+d2, d4 
+0.163 m 
+distance from upper leg center of mass to hip 
+d3 
+0.2 m 
+distance from trunk center of mass to hip 
+ 
+Table 2 Quantities and specifications of genetic algorithms 
+Population size 
+300 
+Initial range 
+[-12,12] 
+Elite count 
+15 
+Crossover fraction 
+.8 
+Migration fraction 
+.2 
+Stall generation 
+50 
+Function count 
+10401 
+ 
+Table 3 Problem physical parameters and constraints 
+Maximum angular rate 
+5 rad/s 
+Maximum actuator torque 
+150 N.m 
+Step length 
+0.5 m 
+Velocity 
+1m/s 
+Maximum Friction coefficient                     
+0.7 
+
+ 
+ 
+Table 4 Initial and final configuration 
+Relative angles 
+Initial value@(t=0) 
+Final value@(t=T) 
+q1 
+-0.1681 
+0.4754 
+q2 
+0.3073 
+0.3073 
+q3 
+-0.6499 
+-0.0064 
+q4 
+0.0064 
+0.6499 
+q5 
+0.3073 
+0.3073 
+ 
+  
+Figure 4 The phase plots of joint angles vs. Joint angular rates 
+As can be seen from the results of Figure 4, simulation results show that optimization by considering zero-dynamics 
+constraint can produce an ideal limit cycle in walking of the biped. It is clear that angular velocities, like angles, are 
+quite smooth and without fractures or discontinuities. They are also a long distance from their saturation limit (5 
+radians per second). 
+
+ 
+Figure 5 Force reactions and Friction coefficient 
+It is also clear from Figure 5 that the ground reaction force is also a positive value to ensure that the robot does not 
+rise completely from the ground and the static friction coefficient required between the heels and the ground. As it is 
+known, the coefficient of friction has desirable values that do not reach the upper bound [19]. 
+ 
+Figure 6 Input torques 
+As can be seen from figure 6, the torques are without fractures and are also far from their saturation limits. 
+
+ 
+        Figure 7 Walking motion 
+ 
+ 
+Figure 8  Position of the swing leg's foot 
+As shown in Figures 7 and 8, the swing leg does not collide with the ground except at the beginning and end of the 
+phase.  
+7. Conclusion 
+This paper proposes a two-layer framework for generating optimal time-varying trajectories for bipedal robots. The 
+novelties of the proposed work are presenting and satisfying the impact invariance constraint in a new way to ensure 
+the periodicity of the gait in each phase and satisfying the hybrid zero dynamics simultaneously without any 
+
+approximation. Also to find a better optimal solution, a hybrid optimization is used. On the other hand, various 
+constraints were considered for a better motion of the robot. According to the simulation results, the accuracy of the 
+proposed method and the obtained optimal solution were confirmed. 
+ 
+References 
+[1]Shi, F., Homberger, T., Lee, J., Miki, T., Zhao, M., Farshidian, F., ... & Hutter, M. (2020). Circus ANYmal: A Quadruped Learning Dexterous 
+Manipulation with Its Limbs. arXiv preprint arXiv:2011.08811.Strunk, W., Jr., & White, E. B. (1979).The elements of style. (3rd ed.).New 
+York: Macmillan, (Chapter 4).  
+[2]Grizzle, J. W., Hurst, J., Morris, B., Park, H. W., & Sreenath, K. (2009, June). MABEL, a new robotic bipedal walker and runner. In 2009 
+American Control Conference (pp. 2030-2036). IEEE. 
+[3]Kakaei, M. M., & Salarieh, H. (2020). New Robust Control Method Applied to the Locomotion of a 5-Link Biped Robot. Robotica, 38(11), 
+2023-2038.Van der Geer, J., Hanraads, J. A. J., & Lupton R. A. (2000). The art of writing a scientific article. Journal of Scientific 
+Communications, 163, 51-59.  
+[4]Meghdari, Ali, et al. "A novel method of gait synthesis for bipedal fast locomotion." Journal of Intelligent and Robotic Systems 53.2 (2008): 
+101-118. 
+[5]Wright, Joe, and Ivan Jordanov. "Intelligent approaches in locomotion-a review." Journal of Intelligent & Robotic Systems 80.2 (2015): 255-
+277. 
+[6]Tzafestas, Spyros G., Thanassis E. Krikochoritis, and Costas S. Tzafestas. "Robust sliding-mode control of nine-link biped robot 
+walking." Journal of Intelligent and Robotic Systems 20.2 (1997): 375-402. 
+[7]Khan, Ameer Tamoor, Shuai Li, and Xuefeng Zhou. "Trajectory optimization of 5-link biped robot using beetle antennae search." IEEE 
+Transactions on Circuits and Systems II: Express Briefs 68.10 (2021): 3276-3280. 
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+Robotic Systems 105.1 (2022): 1-16. 
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+solids, 17(1), 24-35. 
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+Robotica (2021): 1-21. 
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+automatic control 48.1 (2003): 42-56. 
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+in Nonlinear Science and Numerical Simulation 80 (2020): 104949. 
+[13]Sarkar, Abhishek, and Ashish Dutta. "Optimal trajectory generation and design of an 8-dof compliant biped robot for walk on inclined 
+ground." Journal of Intelligent & Robotic Systems 94.3 (2019): 583-602. 
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+2.3 (1985): 289-307. 
+[18] Chevallereau, Christine, et al. "Rabbit: A testbed for advanced control theory." IEEE Control Systems Magazine 23.5 (2003): 57-79. 
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+165-172. 
+ 
+ 
+

C9AyT4oBgHgl3EQfSPck/content/tmp_files/load_file.txt

@@ -0,0 +1,316 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf,len=315
+page_content='Impact Invariant Trajectory Optimization of 5-Link Biped Robot Using Hybrid Optimization Aref Amiri, Hasan Salarieh1 Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran  Abstract Bipedal robots have received much attention because of the variety of motion maneuvers that they can produce, and the many applications they have in various areas including rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' One of these motion maneuvers is walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In this study, we presented a framework for the trajectory optimization of a 5-link (planar) Biped Robot using hybrid optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The walking is modeled with two phases of single-stance (support) phase and the collision phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The dynamic equations of the robot in each phase are extracted by the Lagrange method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' It is assumed that the robot heel strike to the ground is full plastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The gait is optimized with a method called hybrid optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The objective function of this problem is considered to be the integral of torque-squared along the trajectory, and also various constraints such as zero dynamics are satisfied without any approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Furthermore, in a new framework, there is presented a constraint called impact invariance, which ensures the periodicity of the time-varying trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, other constraints provide better and more human-like movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='. Keywords: Trajectory optimization, bipedal robots, walking robots, zero dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Introduction The mechanism of movement and transfer of objects has always been one of the most important and active areas of human research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" Due to the limitations of moving with a wheel, replacing it with feet is an attractive but difficult option, so this field is a hot topic in today's robotic world." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' With the advancement of robotics science and the usefulness of this issue, a lot of research has been done on the design, optimization, and control of legged robots [1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' As the science of bipedal robots has advanced in recent years, there have been significant efforts to improve the performance of these robots in important maneuvers, such as walking and running, but research is still ongoing to find ideal answers [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Designing reference trajectories for human walking cycles is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Several techniques have been adopted to define reference trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' So far, many researchers have studied low-energy (or low input torques) paths for bipedal robots [7,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' We are looking for a periodic path that meets a specific goal in terms of speed and minimizes the torque required to produce the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In general, this open and non-trivial problem is solved by finding numerical answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Various parameters can be considered to optimize the problem, for example, torques, Cartesian coordinate or joint coordinates constraints can be used[10-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Many authors have used polynomial functions for Cartesian coordinates of swing leg’s foot, hip, and trunk angle [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Polynomial functions are used for the coordinates of the joints to limit the number of optimization parameters [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The optimal path for each coordinate of joints is usually written in the form of polynomials with unknown coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The coefficients should be obtained through the optimization process [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' For all bipedal robots, it is important to define optimal periodic motions despite the fact that the number of actuators is less than the degree of freedom of the system, and also zero dynamics problem there exists which should be satisfied during optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In this paper, a new method is presented to produce a periodic path for the walking of bipedal robots which satisfies the impact invariance constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Also, in order to achieve the feasible trajectory, the zero dynamics constraint is satisfied without any approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In addition, by considering some other kinematic and dynamic constraints, and  1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 11155 9567, Tehran, Iran  salarieh@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='edu  using the hybrid optimization method, an optimal reference trajectory for human-like walking of bipedal robots has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Section 2 presents the dynamics and kinematics of a model of the biped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Section 3 is devoted to the formulation of the optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The constraints are defined in section 4 and the optimization method is also described in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Finally, in sections 5 and 6, the results and conclusion are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Dynamics and kinematics Bipedal robots have different dynamics depending on their movement maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' For example, a running robot with 5 links and without an ankle actuator in the flight phase has 7 degrees of freedom and only 4 actuators, so the system is 3 degrees under-actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Here a bipedal walking robot will be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' We assume that the robot is completely on the ground and does not slip while walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, during the single support phase, the other leg rises from the ground when the swing leg hits the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' During the single support phase, the model has 5 degrees of freedom and needs at least 5 generalized coordinates to identify the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, the robot has only 4 actuation, so the system has a degree of under-actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In under-actuated systems, some parts of the dynamics are not affected by the actuator called the zero dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" Here, zero dynamics is affected only by the earth's gravity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content="  The robot's model can be modeled with absolute or relative angles, if relative angles are used, zero dynamics can be easily separated from the main total dynamics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Figure 1 shows the absolute and relative coordinates of a 5-link robot with point feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Figure 1 Relative and absolute angles  The general hybrid walking gait model is obtained by combining the single support phase model and the impact model: Σ: { 𝑥̇ = 𝑓(𝑥) + 𝑔(𝑥)𝑢    𝑥− ∉ Γ 𝑥+ = \uf044 (𝑥−) 𝑥− ∈ Γ (1) where \uf044  is a mapping that transforms the states just before the contact to the states just after the contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 𝑥: = (𝑞𝑇, 𝑞̇ 𝑇)𝑇 is the state vector that contains 𝑞: = (𝑞1, 𝑞2, … , 𝑞𝑛) 𝑇 which is the vector of joint coordinates and 𝑞̇: = (𝑞̇ 1, 𝑞̇ 2, … , 𝑞̇ 𝑛) 𝑇 is the vector of angular velocities, and 𝑥+ denotes the state vector just after the impact and 𝑥− shows just before this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" The switching set is shown as,  𝑞  𝑞  𝑞  𝑞  𝑞  𝑞 : relative coordinates  : absolute coordinates  : swing leg's foot 1 1 2 : stance leg's foot 2  Γ = {(𝑞, 𝑞̇) ∈ 𝑥 ∣ 𝑃 𝑣(𝑞) = 0, 𝑃 ℎ(𝑞) > 0}                                                                                                                                 (2) 𝑃 𝑣(𝑞) and 𝑃 ℎ(𝑞) indicate the vertical and horizontal position of the swing leg, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Now if we model the single support phase alone, we have: 𝑀(𝑞)𝑞̈ + 𝑐(𝑞, 𝑞̇)𝑞̇ + 𝐺(𝑞) = (0, 𝑈𝑇)𝑇                                                                                                                          (3) where 𝑀(𝑞) ∈ ℜ𝑛×𝑛 (𝑛 = 5) is the inertia matrix, 𝑐(𝑞, 𝑞̇) ∈ ℜ𝑛×𝑛 is the Coriolis matrix, and 𝐺(𝑞) ∈ ℜ𝑛 is the gravity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' As shown in Figure 2, the robot does not have any actuators (torques) on the feet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the robot has not the ankle joint actuator, so the robot is under-actuated which adds a zero dynamic constraint to the problem as mentioned in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The vector 𝑈 ∈ ℜ𝑛−   is as follows: 𝑈 = [𝜏 , 𝜏 , 𝜏 , 𝜏 ]𝜏                                                                                                                                                        (4) which represents 4 actuators (torques) on the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 2 actuators (torques) on the pelvis (hip) and 2 on the knee of each leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' By separating the equations of (3) the first equation which produces the zero dynamics is written as: ∑  𝑗= (𝑀 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='𝑗𝑞̈𝑗 + 𝑐 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='𝑗𝑞̇𝑗) + 𝐺 = 0                                                                                                                                (5) which is called zero-hybrid dynamics and: ∑  𝑗= (𝑀 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='𝑗𝑞̈𝑗 + 𝑐 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='𝑗𝑞̇𝑗) + 𝐺 = 𝜏 −                                                                                                                              (6) are other rows of equation (3) (i = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Figure 2 Robot configuration and control torques The trunk angle is assumed independent from other links with a separate actuator, In other words, one actuator is responsible for moving the trunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' So if we temporarily separate the trunk from the other components, we are faced with 4 degrees of freedom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" By determining the swing leg's foot (link number 5 in figure 2), the system still has 2 degrees of freedom, so the inverse kinematics has infinite answers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Therefore, By determining the position of the hip, 2 more degrees of freedom are determined from the system, in this case, the inverse kinematic robot has 4 answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Among these 4 answers, the only acceptable answers are the one that satisfies the condition of not breaking the knee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' It is important to note that in order to find a suitable periodic answer, we assume that the initial configuration is the same as the final one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Optimization variables One convenient way is to select the angles of each link based on a polynomial function of time with a series of unknown coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' This choice enables us to have a smooth function with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Here it is assumed that each angle is a polynomial function of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' It should be noted that the initial and final configuration of the system in each step affects determining two parameters of the polynomial coefficients, the impact invariance constraint is also 1 5 4 2 3 =0  effective on another coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Therefore, in order to have at least 2 optimization parameters for each angle, we consider a fourth-order polynomial function with unknown coefficients for the trajectories of each angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  𝑞𝑘(𝑡) = ∑  𝑛= =0 𝛼𝑘, 𝑡  (𝑘 = 1, … , 5)                                                                                                                                           (7) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Definition of constraints These constraints are to find the right trajectory to walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' It makes the shapes of the joint trajectories, the links orientations, and the required torques for walking be within a reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The constraints are defined as follows: 1) Constraints on the initial and final configuration: Initial and final configurations of the robot must be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Since the robot moves in a periodic pattern, its initial and final configuration must coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" 𝑞(@𝑡=0)=𝑞𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ,   𝑞(@𝑡=𝑇)=𝑞𝑓𝑖𝑛𝑎𝑙 ,                                                                                                                                       (8) 2) Knee movement constraints: In order to have human-like movement, the robot's knees should not be opened and closed excessively ( 𝑚  and  𝑚  are two pre-especified upper bounds in Eq." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" 𝑚 ≥ 𝑞 (𝑡) ≥ 0 ,  𝑚 ≥ 𝑞 (𝑡) ≥ 0,                                                                                                                             (9)  3) Swing leg's foot constraint: The swing leg's foot should not collide with the ground except at the beginning and end of the phase." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 𝑝(0) 𝑣 =  𝑝(𝑇) 𝑣 = 0  𝑝(𝑡) 𝑣 >0 for 0 < 𝑡 < 𝑇                                                                                                                                   (10)  4) Limitation of torques: In order to the physical limitations of the motors, the actuator torques have a certain limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' |𝜏 − (𝑡)| ≤ 𝜏𝑚𝑎𝑥      𝑖 = 2, … ,5                                                                                                                                      (11) 5) Limitation of angular velocities: In order to the physical limitations of the motors, the actuator velocities have a certain limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' |𝑞̇ (𝑡)| ≤ 𝑞̇𝑚𝑎𝑥      𝑖 = 1, … ,5                                                                                                                                         (12) 6) Limitation of friction coefficient: The reaction of the heels, which is the result of the acceleration of the various members of the robot, must observe a certain ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' This ratio should be less than the coefficient of friction between the heels and the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' −𝜇 ≤ | 𝐹𝑥 𝐹𝑦| ≤ 𝜇                                                                                                                                                              (13) In the above equation, 𝜇 is the coefficient of friction, and 𝐹𝑥 and 𝐹𝑦 are sequentially the horizontal and vertical ground reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 7) Zero dynamic constraint: the satisfaction of this constraint is important in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' First, if this constraint is not satisfied, the problem of optimizing the input torques is practically ambiguous, because these torques are not really applicable to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Although it may lead to a feasible kinematic equation (kinematically possible), it is not feasible in terms of control, or in other words, it is not dynamically possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 8) Impact invariance constraint: this constraint means that in order to produce a periodic motion, in addition to the configuration, the initial velocities at the beginning point of each cycle should be exactly the same as its previous cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Since the velocities after the collision are dependent on the velocities before the collision, by satisfying this constraint, the velocities before the collision are adjusted in such a way as to guarantee the periodicity of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Through the following formulae, this purpose is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' At first, the impact mapping formula is written as,  𝑞̇ + = Δ̃(𝑞−)𝑞̇ −                                                                                                                                                               (14) Δ̃(𝑞−) ∈ ℜ ×  is the impact mapping which maps the angular rates of the leg before contact to the angular rates of that leg after contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The inverse of Δ̃ is denoted by, 𝜂̃(𝑞−) = (Δ̃(𝑞−)) − (15) So 𝑞̇ − can be found as : 𝑞̇ − = 𝜂̃(𝑞−)𝑞̇ +                                                                                                                                                            (16) The mathematical formulation of this mapping is obtained from the governing differential equations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" After the swing leg's foot hits the ground," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the positions do not change but the angular velocities change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' which can be achieved as following (see [17] for more information),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Δ𝑞̇ = 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− ⋅ Δ𝑣𝑒                                                                                                                       (17)  where 𝑣  is the velocity vector of the end of the swing leg and 𝑀 ∈ ℜ𝑛×𝑛 is the inertia matrix as mentioned in (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the matrix𝐽 ∈ ℜ𝑚×𝑛  (𝑚 = 2 for planar motions) is also obtained as: 𝐽 = ∂𝑝𝑒 ∂𝑞                                                                                                                                                                             (18) 𝑝𝑒 is the position of the end of the swing leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Assuming that the swing leg sticks to the ground after impact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=" the velocity of the swing leg's foot after impact is zero," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' so 𝑞̇ + = 𝑞̇ − + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− ⋅ (−𝑣𝑒)                                                                                                                        (19) We know that due to the placement of a leg on the ground,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' we can write: 𝑣𝑒 = 𝛼(𝑞)𝑞̇                                                                                                                                                                  (20) where 𝛼(����) is: 𝛼(𝑞) = ∂𝑣𝑒 ∂𝑞̇                                                                                                                                                                    (21) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' by placing )20( into )19( and separating 𝑞̇ −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the pre-impact angular velocity is obtained as follows: 𝑞̇ − = (𝐼 + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− − 𝛼(𝑞)) − 𝑞̇ +                                                                                                  (22) where  𝐼 ∈ ℜ𝑛×𝑛 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  In the above relation, both velocity vectors are written in the same coordinate system, which requires a coordinate conversion, because the coordinate changes after the collision due to the change in the role of the legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' For this purpose, consider the following mapping that converts the relative angles and angular velocities to absolute ones: 1𝑟𝑒𝑙κ= 𝐻  𝑎𝑏𝑠𝜿                                                                                                                                                               (23) where 𝜿 ∈ ℜ𝑛 can be the angles vector, the angular velocities vector or the angular accelerations vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Superscripts 1𝑟𝑒𝑙\uf0a3 and 1𝑎𝑏𝑠\uf0a3  represent relative and absolute coordinates in which the vectors are defined, and also 𝐻 ∈ ℜ𝑛×𝑛 is a square matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, we have a mapping that converts old and new coordinates to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' This mapping can just be defined for an absolute angular coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' If we define the absolute coordinates in this way, we have: 1 𝜓 = Γ 𝜓                                                                                                                                                                    (24)  where indices 1 and 2 indicate the coordinate system before and after the impact, 𝜓 ∈ ℜ𝑛×𝑛 can be velocity vector or angular acceleration vector, and  Γ ∈ ℜ𝑛×𝑛 is the mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' with the above transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the coordinate systems can be connected suitably as: 1 𝑞̇ + = 𝐻Γ𝐻− 1 𝑞̇ +                                                                                                                                                        (25) So the invariancy of the impact during walking is written as it follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 𝑞̇ − = (𝐼 + 𝑀− ⋅ 𝐽𝑇 ⋅ (𝐽 ⋅ 𝑀− ⋅ 𝐽𝑇)− − 𝛼(𝑞)) − 𝐻Γ𝐻− 𝑞̇ +                                                                                        (26) As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' according to Equation (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' the impact invariance constraint is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' In this way, by satisfying this equality constraint, the velocity after impact will be similar to the initial velocity in the previous cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Optimization According to figure 3, optimization is performed using a hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' This means that first, with the penalty method, the constrained problem becomes unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Then, using the genetic algorithm, the first level of optimization is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Finally, in the second level, the outputs of the first level are used as the input of a gradient-based method and the problem is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The objective function is the Euclidean norm of input torques: 𝐽(𝛼) = ∫  𝑇(𝜁−)  0  ∥∥𝑈𝛼(𝑡)∥∥  𝑑𝑡 = ∫  𝑇(𝜁−) 0 ⟨𝜏, 𝜏⟩𝑑𝑡                                                                                                             (27) where 𝑇(𝜁−) corresponds to the step duration, 𝑈𝛼(𝑡) is the resulting torque obtained from (3) along the periodic solution of the hybrid zero dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' To solve the problem more easily and accurately, we tried to satisfy configuration constraints in the problem itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Therefore, 2 coefficients of each coordinate and a total of 10 parameters of equation (7) are determined by the configuration constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Figure 3 optimization diagram According to equation (7), the number of unknown coefficients for a polynomial of order 4 is equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, due to the existence of 5 independent angles, the number of unknown coefficients in the problem is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' By Dynamics Kinematics Setting Initializing Barrier/Penalty Method Genetic Algorithm Gradient Based Method Physically constraints constraints kinematically constraints Cost Function 1 3 2 1: Setting: Type of optimization variables – Desired velocity – Initial and final configuration 2: Initialization reduces the number of variables and simplifies optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 3: Using penalty/barrier functions, the constrained problem becomes unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' F(x, r) = f (x) + P(h(x), g(x), r) where f (x) is the cost function h(x) is the vector of equalities constraint, g(x) is the vector of inequalities constraint, r is a vector of penalty parameters and P is a real-valued function whose action of imposing the penalty on the cost function is controlled by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  determining the initial and final configuration of the robot, the number of optimization variables for this problem is reduced to 15 (by initializing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Results The simulation is based on the specifications of the RABBIT robot (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' As a review, the nonlinear and constrained optimization problem is first converted to a non-constrained problem by the penalty method, then with the values and parameters in Tables 2 and 3, the first layer optimization problem is solved using the genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Next, the outputs of the first layer of optimization are considered as the start point (initial condition) of the second layer of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The maximum violation of the constraints will be equal to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='01 and the maximum iteration of the interior-point algorithm is equal to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The initial and final configuration of the system as well as other specifications and constraint bounds are given in Tables 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Table 1 RABBIT parameters[18] Symbol Value Name m1, m5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='2 kg mass of lower leg m2, m4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='8 kg mass of upper leg m3 20 kg mass of trunk I1, I5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='93 kg-m2 rotational inertia of lower leg, about its center of mass I2, I4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='08 kg-m2 rotational inertia of upper leg, about its center of mass I3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='22 kg-m2 rotational inertia of trunk, about its center of mass l1, l5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='4 m length of lower leg l2, l4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='4 m length of femur l3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='625 m length of trunk d1, d5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='128 m distance from lower leg center of mass to knee d2, d4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='163 m distance from upper leg center of mass to hip d3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='2 m distance from trunk center of mass to hip  Table 2 Quantities and specifications of genetic algorithms Population size 300 Initial range [-12,12] Elite count 15 Crossover fraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='8 Migration fraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='2 Stall generation 50 Function count 10401  Table 3 Problem physical parameters and constraints Maximum angular rate 5 rad/s Maximum actuator torque 150 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='m Step length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='5 m Velocity 1m/s Maximum Friction coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='7  Table 4 Initial and final configuration Relative angles Initial value@(t=0) Final value@(t=T) q1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='1681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='4754 q2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='3073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='3073 q3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='6499 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='0064 q4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='0064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='6499 q5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='3073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='3073  Figure 4 The phase plots of joint angles vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Joint angular rates As can be seen from the results of Figure 4, simulation results show that optimization by considering zero-dynamics constraint can produce an ideal limit cycle in walking of the biped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' It is clear that angular velocities, like angles, are quite smooth and without fractures or discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' They are also a long distance from their saturation limit (5 radians per second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Figure 5 Force reactions and Friction coefficient It is also clear from Figure 5 that the ground reaction force is also a positive value to ensure that the robot does not rise completely from the ground and the static friction coefficient required between the heels and the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' As it is known, the coefficient of friction has desirable values that do not reach the upper bound [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content='  Figure 6 Input torques As can be seen from figure 6, the torques are without fractures and are also far from their saturation limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content="  Figure 7 Walking motion  Figure 8  Position of the swing leg's foot As shown in Figures 7 and 8, the swing leg does not collide with the ground except at the beginning and end of the phase." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Conclusion This paper proposes a two-layer framework for generating optimal time-varying trajectories for bipedal robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' The novelties of the proposed work are presenting and satisfying the impact invariance constraint in a new way to ensure the periodicity of the gait in each phase and satisfying the hybrid zero dynamics simultaneously without any  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' Also to find a better optimal solution, a hybrid optimization is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' On the other hand, various constraints were considered for a better motion of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
+page_content=' According to the simulation results, the accuracy of the proposed method and the obtained optimal solution were confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf'}
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+ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training
+for E-Commerce Product Search
+Xuange Cui
+[email protected]
+JD.com
+Beijing, China
+Wei Xiong
+[email protected]
+JD.com
+Beijing, China
+Songlin Wang
+[email protected]
+JD.com
+Beijing, China
+ABSTRACT
+In this paper, we propose a robust multilingual model to improve the
+quality of search results. Our model not only leverage the processed
+class-balanced dataset, but also benefit from multitask pre-training
+that leads to more general representations. In pre-training stage,
+we adopt mlm task, classification task and contrastive learning task
+to achieve considerably performance. In fine-tuning stage, we use
+confident learning, exponential moving average method (EMA), ad-
+versarial training (FGM) and regularized dropout strategy (R-Drop)
+to improve the model’s generalization and robustness. Moreover,
+we use a multi-granular semantic unit to discover the queries and
+products textual metadata for enhancing the representation of the
+model. Our approach obtained competitive results and ranked top-8
+in three tasks. We release the source code and pre-trained models
+associated with this work1.
+CCS CONCEPTS
+• Information systems → Retrieval models and ranking.
+KEYWORDS
+search relevance, e-commerce, semantic matching, multilingual
+1
+INTRODUCTION
+With the rapid growth of e-Commerce, online product search has
+emerged as a popular and effective paradigm for customers to find
+desired products and engage in online shopping [7, 9, 11]. It is very
+challenging to accurately find and display relevant products. This
+is because the customer queries are ambiguous and implicit [12].
+For example, many users search for "iPhone" to find and purchase
+an "iPhone charger". However, the traditional binary classification
+model is difficult to clearly characterize this relationship. The Ama-
+zon KDD Cup 2022 presents a novel multilingual dataset [17] across
+English, Japanese and Spanish, and consists of three different sub-
+tasks to evaluate the model’s abilities of ranking and classifying.
+In this paper, our contributions can be summarized as follows:
+1) Data Augmentation. We use the translation model to convert
+Spanish to English for expanding the dataset. Through splitting
+the complement and irrelevant product text information, we could
+get a bigger dataset with balanced labels. We use confident learn-
+ing [14, 15] to find the potential label errors and remove ∼4% data
+from the training dataset. 2) MultiTask Pre-training. In pre-training
+stage, we use MLM task, classification task and contrastive learn-
+ing task for improving the model’s performance. 3) In fine-tuning
+stage, we use a multi-granular semantic unit to discover the queries
+and products textual metadata for enhancing the representation
+1https://github.com/cuixuage/KDDCup2022-ESCI
+SubTask
+Train Dataset
+Test dataset
+Languages
+Task1
+781K
+48K
+Spanish
+Task2
+1834K
+277K
+& English
+Task3
+1834K
+277K
+& Japanese
+Table 1: The statistics of datasets.
+of the model. And we observe that exponential moving average
+method(EMA) [6], adversarial training(FGM) [5] and regularized
+dropout strategy(R-Drop) [10] could improve the model’s general-
+ization and robustness.
+Our team participated in all tasks, and achieved considerably
+performance gain over the baseline solution. Specifically, our ap-
+proach ranked 5th in task1, ranked 7th in task2 and ranked 8th in
+task3.
+2
+BACKGROUND
+The Amazon KDD Cup 2022 [17] provides three subtasks. The
+task1 consists of a query-product ranking task aimed at ranking the
+results list. The Normalized Discounted Cumulative Gain(nDCG)
+[18] will be used to evaluate the model’s abilities of ranking.
+The task2 and task3 are classification tasks which require the
+model to classify the query/product pairs into correct categories.
+These tasks are designed to test the model’s ability of classifying.
+The micro-F1 [16] will be used as an evaluation metric. Moreover,
+the task2 consists of a multi-class product classification task aimed
+at classifying each product as being an Exact, Substitute, Comple-
+ment, or Irrelevant match for the query. The task3 will measure the
+model’s abilities of identifying the substitute products in the list of
+results for a given query.
+The statistics of the corpus are shown in Table 1. In this challenge,
+the organizers provide two different versions of the data set. One
+for task 1 which is reduced version in terms of number of examples
+and ones for tasks 2 and 3 which is a larger [17]. It is noted that the
+reduced version of the data set has more difficult samples. Our team
+participated in all subtasks, and the next section will introduce an
+overview of our system.
+3
+SYSTEM OVERVIEW
+3.1
+Multi-Task Pre-Training
+We compare several pre-trained multilingual language models from
+the XTREME Leaderboard2, and then we use the "microsoft/infoxlm-
+large3" as text encoder.
+2https://sites.research.google/xtreme
+3https://huggingface.co/microsoft/infoxlm-large
+arXiv:2301.13455v1  [cs.CL]  31 Jan 2023
+
+KDDCup ’22, August 17, 2022, Washington, DC, USA
+Xuange Cui, Wei Xiong, and Songlin Wang
+The InfoXLM𝑙𝑎𝑟𝑔𝑒 model [1] containing 94 languages and pre-
+trained with CCNet dataset, and has the same configurations of
+XLM-R [2] and a shared vocabulary size of 250002. Figure 1 shows
+a high-level overview of our proposed pretext tasks.
+Figure 1: A schematic overview of our novel pre-training
+tasks. These tasks encourage the encoded representations to
+be more general.
+MLM Task, is widely used for learning text representations [3].
+MLM trains a model to predict a random sample of input tokens
+that have been replaced by a [MASK] placeholder in a multi-class
+setting over the entire vocabulary [20]. We adopt MLM-Task on the
+multilingual product-catalogue dataset.
+Classification Task, contains three classification subtasks. One
+of them is Product2Query-Task, this task is a binary classification
+task. Based on the Poisson distribution, a piece of text is intercepted
+from commodity text information as the faked query. The Parame-
+ters passed to the Poisson distribution and more details can be found
+in appendix A.1. Product2Brand-Task and Product2Color-Task are
+multi-class classification that using product text information to
+predict the brand and the color of current item.
+Contrastive Learning Task, is mainly inspired by SimCSE [4]
+and EsimCSE [19]. During training, each data point is trained to
+find out its counterpart among (𝑁 − 1) from in-batch negative
+samples and the queue of data samples. The samples in the queue
+are progressively replaced.
+− log
+𝑒sim(h𝑖,h+
+𝑖 )/𝜏
+�𝑁
+𝑗=1 𝑒sim
+�
+h𝑖,h+
+𝑗
+�
+/𝜏 + �𝑄
+𝑞=1 𝑒sim
+�
+h𝑖,h+𝑞
+�
+/𝜏
+(1)
+The ℎ∗ is the sentence representation, where ℎ𝑖 and ℎ+
+𝑖 are se-
+mantically related. The ℎ+𝑞 denotes a sentence embedding in the
+momentum-updated queue. And the 𝑄 is the size of the queue,
+𝑠𝑖𝑚(ℎ1,ℎ2) is the cosine similarity scores of sentence representa-
+tions, 𝜏 is a temperature hyperparameter. In the end, we average
+the all N Li losses to calculate the contrastive loss Lcon.
+Algorithm 1: Training a MultiTask model.
+Input: DataSet D =
+�
+(𝑥,𝑦,𝑧)𝑖
+� |D |
+𝑖=1
+1 Initialize model parameters Θ randomly ;
+2 Model trainer 𝑇 that takes batches of training data as input
+to optimize the model parameters Θ ;
+3 Set the max number of epoch: 𝑒𝑝𝑜𝑐ℎmax ;
+4 for epoch in 1, 2, ...,𝑒𝑝𝑜𝑐ℎmax do
+5
+Shuffle D by mixing data from different tasks ;
+6
+for B in D do
+7
+// B is a mini-batch of pre-training task ;
+8
+Compute loss : 𝐿(Θ) ;
+9
+1. 𝐿(Θ) = Mask LM Loss ;
+10
+2. 𝐿(Θ) += Classification Loss ;
+11
+3. 𝐿(Θ) += Contrastive Learning Loss ;
+12
+Optimize the model using 𝐿(Θ) ;
+13
+end
+14 end
+Output: Pre-trained Model Θ
+3.2
+Fine-Tuning Methods
+After pre-training, we remove the classifiers for pre-training multi-
+task and concatenate some embeddings with an extra MLP classifier.
+The embeddings consist of three sets of representations. One of
+them is done by concatenating the queries’ 3-gram mean-pooling,
+bullet points’ 3-gram mean-pooling and descriptions’ 3-gram mean-
+pooling embeddings. The others consist of country embedding,
+brand embedding and color embedding, as shown in Figure 2.
+Exponential Moving Average Our model uses EMA [6] to
+smooth the trained parameters. Evaluations that use averaged pa-
+rameters sometimes produce significantly better results than the
+final trained values. Formally, we define the smoothed variables
+and trained variables as 𝜃𝑠 and 𝜃𝑡, EMA decay weight as: 𝜂. After
+each training step, we update 𝜃𝑠 by:
+𝜃𝑠 ← 𝜂𝜃𝑠 + (1 − 𝜂)𝜃𝑡
+(2)
+Adversarial Training Recently, adversarial attack has been
+widely applied in computer vision and natural language processing
+[5, 8, 13, 21]. Many works use it during fine-tuning, we explore the
+influence of adversarial training strategies and compare the FGSM,
+PGD, FREELB and SMART methods. The adversarial attack works
+by augmenting the input with a small perturbation that maximizes
+the adversarial loss:
+min
+𝜃
+E(𝑥,𝑦)∼D
+�
+max
+Δ𝑥 ∈Ω 𝐿(𝑥 + Δ𝑥,𝑦;𝜃)
+�
+(3)
+where the D is dataset, 𝑥 is input, 𝑦 is the gold label, 𝜃 is the model
+parameters, 𝐿(𝑥,𝑦;𝜃) is the loss function and Δ𝑥 is the perturbation.
+In our experiments, we adopt FGSM method in all tasks which based
+on the actual performances.
+R-Drop is proved to be an effective regularization method based
+on dropout, by minimizing the KL-divergence of the output distri-
+butions of every two sub-models generated via dropout in model
+training.
+L𝐾𝐿 = 𝛼 · [D𝐾𝐿 (𝐿𝑜𝑔𝑖𝑡1, 𝐿𝑜𝑔𝑖𝑡2) + D𝐾𝐿 (𝐿𝑜𝑔𝑖𝑡2, 𝐿𝑜𝑔𝑖𝑡1)]
+(4)
+
+ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training for E-Commerce Product Search
+KDDCup ’22, August 17, 2022, Washington, DC, USA
+Figure 2: In fine-tuning stage, we concatenate the multi-granular semantic units, the [CLS] embedding from XLM encoder and
+the IDs’ embeddings.
+We use the origin logits of model’s output as 𝐿𝑜𝑔𝑖𝑡1, and the logits
+after adversarial attack as 𝐿𝑜𝑔𝑖𝑡2.
+Embedding Mixup is widely used data augmentation method
+through linearly interpolating inputs and modeling targets of ran-
+dom samples. We use the contextual embedding vector of [CLS]
+and the corresponding label to generate synthetic examples for
+training. Such training has been shown to act as an effective model
+regularization strategy for text classification task. In conclusion, we
+present the self-supervised multitask pre-training tasks and the sev-
+eral fine-tuning methods for improving the models’ generalization
+and robustness.
+4
+EXPERIMENTS
+4.1
+Settings
+We use InfoXLM𝑙𝑎𝑟𝑔𝑒 as the text encoder, the EMA decay weight is
+set to 0.999. And our learning rate is set to 1e-5 with warm-up ratio
+over 10%, batch size is 32 and gradient clip norm threshold is set to
+1. In pre-training stage, the maximum number of epochs was set to
+10. And in the fine-tuning stage, the maximum number of epochs
+was set to 5. During adversarial training, we set 𝜀 to 1.0 in FGM that
+means calculate only one step in the adversarial attack. We find
+that the dataset has imbalanced label and use some data processing
+steps. Through splitting the complement and irrelevant product text
+information, we could get more pairs which have the same label
+and get a more balanced dataset. We use confident learning to find
+the potential label errors and remove ∼4% data from the training
+dataset. As presented in appendix A.1, the median of Spanish and
+English queries is 14 which satisfies the Poisson distribution of 𝜇 is
+set to 4. And the median of the Japanese query is 31 which satisfies
+the Poisson distribution with 𝜇 is set to 8.
+4.2
+Main Results
+Our approach achieved considerably performance gain over the
+baseline solution, and ranked top-8 in three tasks. The main results
+are shown in Table 2. In task1, we calculated the mean of all model
+outputs as the final ranking results. In task2 and task3, we almost
+used the same network structure except the number of neurons
+in the classifier. Finally, Our approach ranked 5th, 7th and 8th,
+respectively.
+SubTask
+Model
+Metric
+Ranking
+task1
+6 large models
+ndcg=0.9025
+5th
+task2
+only 1 large model
+micro f1=0.8194
+7th
+task3
+only 1 large model
+micro f1=0.8686
+8th
+Table 2: Performance of our approach on the private leader-
+board. In task1, we used six InfoXLM𝑙𝑎𝑟𝑔𝑒 models that fine-
+tuned by different datasets or methods. In task2 and task3,
+we used only one InfoXLM𝑙𝑎𝑟𝑔𝑒 model with the same net-
+work structure, as shown in Figure 2.
+Pre-Training Task
+CV-MLM Loss
+CV-Micro F1
+Mask LM
+1.966
+74.97
++Product2Query
+1.969
+75.05
+++Product2Brand
+1.978
+75.08
++++Contrastive Learning
+2.047
+75.08
+Table 3: The effect of different pre-training tasks and keep
+accumulating from top to bottom. We report the cross vali-
+dation MLM-Loss and Micro-F1 Score × 100 in the task2 set-
+ting.
+4.3
+Ablation Studies
+We investigate the impact of adopting different pre-training task
+in the task2 setting. In Table 3, we show the Mask-LM losses after
+5 epochs of pre-training and Micro-F1 scores after 2 epochs of
+fine-tuning. We find that the Product2Query task achieves an 0.008
+improvement compared to the baseline, and the contrastive learning
+task doesn’t get a significant gain.
+As shown in Table 4, we compare several loss functions, and we
+adopt Poly1 loss function in task2 and task3 which based on the
+actual performances. We observe that the Focal loss function and
+GHM loss function have worse performance than the cross-entropy
+loss function in the task2 setting.
+In this subsection, we explore several methods for further im-
+proving the model’s performance in fine-tuning stage. As presented
+
+KDDCup ’22, August 17, 2022, Washington, DC, USA
+Xuange Cui, Wei Xiong, and Songlin Wang
+Classification Loss
+CV-Micro F1
+CE Loss
+75.08
+Focal Loss
+74.73
+GHM Loss
+74.85
+Poly1 Loss
+75.21
+Table 4: The effect of different losses in the task2 setting. We
+report the cross validation Micro-F1 Score × 100.
+Methods
+CV-Micro F1
++EMA
+75.19
+++FGM
+75.30
++++R-Drop
+75.43
+++++Embedding Mixup
+75.43
+Table 5: The effect of different strategies and keep accumu-
+lating from top to bottom. We report the cross validation
+Micro-F1 Score × 100 in the task2 setting.
+Confident Learning
+CV-Metric
+with-in-task1
+NDCG, +0.005
+with-in-task2
+Micro-F1, -0.003
+with-in-task3
+Micro-F1, -0.002
+Table 6: The effect of removing 4% noisy labels.
+in Table 5, we adopt all of these methods to improve the model’s gen-
+eralization and robustness. We observe that the exponential moving
+average method(EMA), adversarial training(FGM) and regularized
+dropout strategy(R-Drop) could improve the model’s generalization
+and robustness. But the Embedding Mixup strategy doesn’t get a
+significant gain.
+As shown in Table 7, we consider using smaller datasets with
+removing ∼4% noisy labels. We used the smaller dataset to achieve
+an 0.005 improvement in task1, but we get worse results in tash2
+and task3. It could be explained that since task1 contains more
+difficult samples, the manually annotated data contains more label
+errors.
+5
+CONCLUSION AND FUTURE WORK
+In this work, we provide an overview of the combined approach to
+improve the quality of search results. We use data augmentation,
+multitask pretraining strategy and several fine-tuning methods to
+achieve considerably performance. Specifically, we use mlm task,
+classification task and contrastive learning task in pre-training
+stage. And we use exponential moving average method(EMA), ad-
+versarial training(FGM) and regularized dropout strategy(R-Drop)
+to improve the model’s generalization and robustness in fine-tuning
+stage. Moreover, we use a multi-granular semantic unit to discover
+the queries and products textual metadata for enhancing the repre-
+sentation of the model. Future work of our system includes: 1) Com-
+paring with other pre-trained language models, such as deborta𝑙𝑎𝑟𝑔𝑒.
+2) Using other training strategies, such as self-distillation.
+REFERENCES
+[1] Zewen Chi, Li Dong, Furu Wei, Nan Yang, Saksham Singhal, Wenhui Wang,
+Xia Song, Xian-Ling Mao, Heyan Huang, and Ming Zhou. 2020.
+InfoXLM:
+An Information-Theoretic Framework for Cross-Lingual Language Model Pre-
+Training. CoRR abs/2007.07834 (2020). arXiv:2007.07834 https://arxiv.org/abs/
+2007.07834
+[2] Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guil-
+laume Wenzek, Francisco Guzmán, Edouard Grave, Myle Ott, Luke Zettle-
+moyer, and Veselin Stoyanov. 2019. Unsupervised Cross-lingual Representa-
+tion Learning at Scale. CoRR abs/1911.02116 (2019). arXiv:1911.02116 http:
+//arxiv.org/abs/1911.02116
+[3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. BERT:
+Pre-training of Deep Bidirectional Transformers for Language Understanding.
+CoRR abs/1810.04805 (2018). arXiv:1810.04805 http://arxiv.org/abs/1810.04805
+[4] Tianyu Gao, Xingcheng Yao, and Danqi Chen. 2021. SimCSE: Simple Contrastive
+Learning of Sentence Embeddings. In Empirical Methods in Natural Language
+Processing (EMNLP).
+[5] Ian J. Goodfellow, Jonathon Shlens, and Christian Szegedy. 2015. Explaining and
+Harnessing Adversarial Examples. arXiv:1412.6572 [stat.ML]
+[6] Seng Hansun. 2013. A new approach of moving average method in time series
+analysis. In 2013 Conference on New Media Studies (CoNMedia). 1–4.
+https:
+//doi.org/10.1109/CoNMedia.2013.6708545
+[7] Rahul Radhakrishnan Iyer, Rohan Kohli, and Shrimai Prabhumoye. 2020. Mod-
+eling Product Search Relevance in e-Commerce. CoRR abs/2001.04980 (2020).
+arXiv:2001.04980 https://arxiv.org/abs/2001.04980
+[8] Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, and
+Tuo Zhao. 2020.
+SMART: Robust and Efficient Fine-Tuning for Pre-trained
+Natural Language Models through Principled Regularized Optimization. In Pro-
+ceedings of the 58th Annual Meeting of the Association for Computational Lin-
+guistics. Association for Computational Linguistics, Online, 2177–2190. https:
+//doi.org/10.18653/v1/2020.acl-main.197
+[9] Sen Li, Fuyu Lv, Taiwei Jin, Guli Lin, Keping Yang, Xiaoyi Zeng, Xiao-Ming Wu,
+and Qianli Ma. 2021. Embedding-based Product Retrieval in Taobao Search. CoRR
+abs/2106.09297 (2021). arXiv:2106.09297 https://arxiv.org/abs/2106.09297
+[10] Xiaobo Liang, Lijun Wu, Juntao Li, Yue Wang, Qi Meng, Tao Qin, Wei Chen, Min
+Zhang, and Tie-Yan Liu. 2021. R-Drop: Regularized Dropout for Neural Networks.
+CoRR abs/2106.14448 (2021). arXiv:2106.14448 https://arxiv.org/abs/2106.14448
+[11] Yiqun Liu, Kaushik Rangadurai, Yunzhong He, Siddarth Malreddy, Xunlong Gui,
+Xiaoyi Liu, and Fedor Borisyuk. 2021. Que2Search: Fast and Accurate Query and
+Document Understanding for Search at Facebook. Proceedings of the 27th ACM
+SIGKDD Conference on Knowledge Discovery & Data Mining (2021).
+[12] Hanqing Lu, Youna Hu, Tong Zhao, Tony Wu, Yiwei Song, and Bing Yin. 2021.
+Graph-based Multilingual Product Retrieval in E-commerce Search.
+CoRR
+abs/2105.02978 (2021). arXiv:2105.02978 https://arxiv.org/abs/2105.02978
+[13] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and
+Adrian Vladu. 2019. Towards Deep Learning Models Resistant to Adversarial
+Attacks. arXiv:1706.06083 [stat.ML]
+[14] Curtis G. Northcutt, Lu Jiang, and Isaac L. Chuang. 2021. Confident Learning:
+Estimating Uncertainty in Dataset Labels. Journal of Artificial Intelligence Research
+(JAIR) 70 (2021), 1373–1411.
+[15] Curtis G. Northcutt, Tailin Wu, and Isaac L. Chuang. 2017.
+Learning with
+Confident Examples: Rank Pruning for Robust Classification with Noisy La-
+bels. In Proceedings of the Thirty-Third Conference on Uncertainty in Artifi-
+cial Intelligence (Sydney, Australia) (UAI’17). AUAI Press, 10 pages.
+http:
+//auai.org/uai2017/proceedings/papers/35.pdf
+[16] Juri Opitz and Sebastian Burst. 2019. Macro F1 and Macro F1. CoRR abs/1911.03347
+(2019). arXiv:1911.03347 http://arxiv.org/abs/1911.03347
+[17] Chandan K. Reddy, Lluís Màrquez, Fran Valero, Nikhil Rao, Hugo Zaragoza,
+Sambaran Bandyopadhyay, Arnab Biswas, Anlu Xing, and Karthik Subbian. 2022.
+Shopping Queries Dataset: A Large-Scale ESCI Benchmark for Improving Product
+Search. arXiv:2206.06588
+[18] Yining Wang, Liwei Wang, Yuanzhi Li, Di He, Tie-Yan Liu, and Wei Chen. 2013.
+A Theoretical Analysis of NDCG Type Ranking Measures. CoRR abs/1304.6480
+(2013). arXiv:1304.6480 http://arxiv.org/abs/1304.6480
+[19] Xing Wu, Chaochen Gao, Liangjun Zang, Jizhong Han, Zhongyuan Wang, and
+Songlin Hu. 2021. ESimCSE: Enhanced Sample Building Method for Contrastive
+Learning of Unsupervised Sentence Embedding. CoRR abs/2109.04380 (2021).
+arXiv:2109.04380 https://arxiv.org/abs/2109.04380
+[20] Atsuki Yamaguchi, George Chrysostomou, Katerina Margatina, and Nikolaos
+Aletras. 2021. Frustratingly Simple Pretraining Alternatives to Masked Language
+
+ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training for E-Commerce Product Search
+KDDCup ’22, August 17, 2022, Washington, DC, USA
+Methods
+CV-Micro F1
+Random♦
+-
+Word2vec♣
+85.33
+Freeze♥
+85.29
+Table 7: The performance of different initialization methods
+of the multi-granular semantic unit. We report the cross val-
+idation Micro-F1 Score × 100 in the task3 setting.
+Modeling. CoRR abs/2109.01819 (2021). arXiv:2109.01819 https://arxiv.org/abs/
+2109.01819
+[21] Chen Zhu, Yu Cheng, Zhe Gan, Siqi Sun, Tom Goldstein, and Jingjing Liu. 2020.
+FreeLB: Enhanced Adversarial Training for Natural Language Understanding. In
+International Conference on Learning Representations. https://openreview.net/
+forum?id=BygzbyHFvB
+A
+APPENDIX
+A.1
+Poisson Distribution
+Figure 3: The length distribution of queries in different lan-
+guages.
+As presented in Figure 3, the median of Spanish and English
+queries is 14 which satisfies the Poisson distribution of 𝜇 is set to
+4. And the median of the Japanese query is 31 which satisfies the
+Poisson distribution with 𝜇 is set to 8.
+A.2
+EmbeddingBag Initialization
+The multi-granular semantic unit implemented by Embedding-
+Bag4. As presented in Table 7, the way of random initialization
+converges slowly, so we don’t record the final result. And when the
+Embedding-Bag is initialized by Word2vec, our approach obtain
+the best performance.
+4https://pytorch.org/docs/stable/generated/torch.nn.EmbeddingBag.html
+
+query_len distribution
+0.35
+geometric_p=0.2
+poisson_miu=4
+0.30
+es
+sn
+0.25
+Jp
+0.20
+Y-axis
+0.15
+0.10
+0.05
+0.00
+0
+5
+10
+15
+20
+25
+30
+X-axis

CNFQT4oBgHgl3EQf-jfx/content/tmp_files/load_file.txt

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+page_content='ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training  for E-Commerce Product Search  Xuange Cui  cuixuange@jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  Beijing, China  Wei Xiong  xiongwei9@jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  Beijing, China  Songlin Wang  wangsonglin3@jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com  Beijing, China  ABSTRACT  In this paper, we propose a robust multilingual model to improve the  quality of search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Our model not only leverage the processed  class-balanced dataset, but also benefit from multitask pre-training  that leads to more general representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In pre-training stage,  we adopt mlm task, classification task and contrastive learning task  to achieve considerably performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In fine-tuning stage, we use  confident learning, exponential moving average method (EMA), ad-  versarial training (FGM) and regularized dropout strategy (R-Drop)  to improve the model’s generalization and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Moreover,  we use a multi-granular semantic unit to discover the queries and  products textual metadata for enhancing the representation of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Our approach obtained competitive results and ranked top-8  in three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We release the source code and pre-trained models  associated with this work1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  CCS CONCEPTS  Information systems → Retrieval models and ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  KEYWORDS  search relevance, e-commerce, semantic matching, multilingual  1  INTRODUCTION  With the rapid growth of e-Commerce, online product search has  emerged as a popular and effective paradigm for customers to find  desired products and engage in online shopping [7, 9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' It is very  challenging to accurately find and display relevant products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' This  is because the customer queries are ambiguous and implicit [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  For example, many users search for "iPhone" to find and purchase  an "iPhone charger".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' However, the traditional binary classification  model is difficult to clearly characterize this relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The Ama-  zon KDD Cup 2022 presents a novel multilingual dataset [17] across  English, Japanese and Spanish, and consists of three different sub-  tasks to evaluate the model’s abilities of ranking and classifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  In this paper, our contributions can be summarized as follows:  1) Data Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We use the translation model to convert  Spanish to English for expanding the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Through splitting  the complement and irrelevant product text information, we could  get a bigger dataset with balanced labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We use confident learn-  ing [14, 15] to find the potential label errors and remove ∼4% data  from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2) MultiTask Pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In pre-training  stage, we use MLM task, classification task and contrastive learn-  ing task for improving the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 3) In fine-tuning  stage, we use a multi-granular semantic unit to discover the queries  and products textual metadata for enhancing the representation  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='com/cuixuage/KDDCup2022-ESCI  SubTask  Train Dataset  Test dataset  Languages  Task1  781K  48K  Spanish  Task2  1834K  277K  & English  Task3  1834K  277K  & Japanese  Table 1: The statistics of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And we observe that exponential moving average  method(EMA) [6], adversarial training(FGM) [5] and regularized  dropout strategy(R-Drop) [10] could improve the model’s general-  ization and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Our team participated in all tasks, and achieved considerably  performance gain over the baseline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Specifically, our ap-  proach ranked 5th in task1, ranked 7th in task2 and ranked 8th in  task3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  2  BACKGROUND  The Amazon KDD Cup 2022 [17] provides three subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The  task1 consists of a query-product ranking task aimed at ranking the  results list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The Normalized Discounted Cumulative Gain(nDCG)  [18] will be used to evaluate the model’s abilities of ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  The task2 and task3 are classification tasks which require the  model to classify the query/product pairs into correct categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  These tasks are designed to test the model’s ability of classifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  The micro-F1 [16] will be used as an evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Moreover,  the task2 consists of a multi-class product classification task aimed  at classifying each product as being an Exact, Substitute, Comple-  ment, or Irrelevant match for the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The task3 will measure the  model’s abilities of identifying the substitute products in the list of  results for a given query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  The statistics of the corpus are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In this challenge,  the organizers provide two different versions of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' One  for task 1 which is reduced version in terms of number of examples  and ones for tasks 2 and 3 which is a larger [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' It is noted that the  reduced version of the data set has more difficult samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Our team  participated in all subtasks, and the next section will introduce an  overview of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  3  SYSTEM OVERVIEW  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1  Multi-Task Pre-Training  We compare several pre-trained multilingual language models from  the XTREME Leaderboard2, and then we use the "microsoft/infoxlm-  large3" as text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  2https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='google/xtreme  3https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='co/microsoft/infoxlm-large  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='13455v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='CL]  31 Jan 2023  KDDCup ’22, August 17, 2022, Washington, DC, USA  Xuange Cui, Wei Xiong, and Songlin Wang  The InfoXLM𝑙𝑎𝑟𝑔𝑒 model [1] containing 94 languages and pre-  trained with CCNet dataset, and has the same configurations of  XLM-R [2] and a shared vocabulary size of 250002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Figure 1 shows  a high-level overview of our proposed pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Figure 1: A schematic overview of our novel pre-training  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' These tasks encourage the encoded representations to  be more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  MLM Task, is widely used for learning text representations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  MLM trains a model to predict a random sample of input tokens  that have been replaced by a [MASK] placeholder in a multi-class  setting over the entire vocabulary [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We adopt MLM-Task on the  multilingual product-catalogue dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Classification Task, contains three classification subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' One  of them is Product2Query-Task, this task is a binary classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Based on the Poisson distribution, a piece of text is intercepted  from commodity text information as the faked query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The Parame-  ters passed to the Poisson distribution and more details can be found  in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Product2Brand-Task and Product2Color-Task are  multi-class classification that using product text information to  predict the brand and the color of current item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Contrastive Learning Task, is mainly inspired by SimCSE [4]  and EsimCSE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' During training, each data point is trained to  find out its counterpart among (𝑁 − 1) from in-batch negative  samples and the queue of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The samples in the queue  are progressively replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  − log  𝑒sim(h𝑖,h+  𝑖 )/𝜏  �𝑁  𝑗=1 𝑒sim  �  h𝑖,h+  𝑗  �  /𝜏 + �𝑄  𝑞=1 𝑒sim  �  h𝑖,h+𝑞  �  /𝜏  (1)  The ℎ∗ is the sentence representation, where ℎ𝑖 and ℎ+  𝑖 are se-  mantically related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The ℎ+𝑞 denotes a sentence embedding in the  momentum-updated queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And the 𝑄 is the size of the queue,  𝑠𝑖𝑚(ℎ1,ℎ2) is the cosine similarity scores of sentence representa-  tions, 𝜏 is a temperature hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In the end, we average  the all N Li losses to calculate the contrastive loss Lcon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Algorithm 1: Training a MultiTask model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Input: DataSet D =  �  (𝑥,𝑦,𝑧)𝑖  � |D |  𝑖=1  1 Initialize model parameters Θ randomly ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  2 Model trainer 𝑇 that takes batches of training data as input  to optimize the model parameters Θ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  3 Set the max number of epoch: 𝑒𝑝𝑜𝑐ℎmax ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  4 for epoch in 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=',𝑒𝑝𝑜𝑐ℎmax do  5  Shuffle D by mixing data from different tasks ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  6  for B in D do  7  // B is a mini-batch of pre-training task ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  8  Compute loss : 𝐿(Θ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 𝐿(Θ) = Mask LM Loss ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 𝐿(Θ) += Classification Loss ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 𝐿(Θ) += Contrastive Learning Loss ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  12  Optimize the model using 𝐿(Θ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  13  end  14 end  Output: Pre-trained Model Θ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='2  Fine-Tuning Methods  After pre-training, we remove the classifiers for pre-training multi-  task and concatenate some embeddings with an extra MLP classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  The embeddings consist of three sets of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' One of  them is done by concatenating the queries’ 3-gram mean-pooling,  bullet points’ 3-gram mean-pooling and descriptions’ 3-gram mean-  pooling embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The others consist of country embedding,  brand embedding and color embedding, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Exponential Moving Average Our model uses EMA [6] to  smooth the trained parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Evaluations that use averaged pa-  rameters sometimes produce significantly better results than the  final trained values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Formally, we define the smoothed variables  and trained variables as 𝜃𝑠 and 𝜃𝑡, EMA decay weight as: 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' After  each training step, we update 𝜃𝑠 by:  𝜃𝑠 ← 𝜂𝜃𝑠 + (1 − 𝜂)𝜃𝑡  (2)  Adversarial Training Recently, adversarial attack has been  widely applied in computer vision and natural language processing  [5, 8, 13, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Many works use it during fine-tuning, we explore the  influence of adversarial training strategies and compare the FGSM,  PGD, FREELB and SMART methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The adversarial attack works  by augmenting the input with a small perturbation that maximizes  the adversarial loss:  min  𝜃  E(𝑥,𝑦)∼D  �  max  Δ𝑥 ∈Ω 𝐿(𝑥 + Δ𝑥,𝑦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='𝜃)  �  (3)  where the D is dataset, 𝑥 is input, 𝑦 is the gold label, 𝜃 is the model  parameters, 𝐿(𝑥,𝑦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='𝜃) is the loss function and Δ𝑥 is the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  In our experiments, we adopt FGSM method in all tasks which based  on the actual performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  R-Drop is proved to be an effective regularization method based  on dropout, by minimizing the KL-divergence of the output distri-  butions of every two sub-models generated via dropout in model  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  L𝐾𝐿 = 𝛼 · [D𝐾𝐿 (𝐿𝑜𝑔𝑖𝑡1, 𝐿𝑜𝑔𝑖𝑡2) + D𝐾𝐿 (𝐿𝑜𝑔𝑖𝑡2, 𝐿𝑜𝑔𝑖𝑡1)]  (4)  ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training for E-Commerce Product Search  KDDCup ’22, August 17, 2022, Washington, DC, USA  Figure 2: In fine-tuning stage, we concatenate the multi-granular semantic units, the [CLS] embedding from XLM encoder and  the IDs’ embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  We use the origin logits of model’s output as 𝐿𝑜𝑔𝑖𝑡1, and the logits  after adversarial attack as 𝐿𝑜𝑔𝑖𝑡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Embedding Mixup is widely used data augmentation method  through linearly interpolating inputs and modeling targets of ran-  dom samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We use the contextual embedding vector of [CLS]  and the corresponding label to generate synthetic examples for  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Such training has been shown to act as an effective model  regularization strategy for text classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In conclusion, we  present the self-supervised multitask pre-training tasks and the sev-  eral fine-tuning methods for improving the models’ generalization  and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  4  EXPERIMENTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1  Settings  We use InfoXLM𝑙𝑎𝑟𝑔𝑒 as the text encoder, the EMA decay weight is  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And our learning rate is set to 1e-5 with warm-up ratio  over 10%, batch size is 32 and gradient clip norm threshold is set to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In pre-training stage, the maximum number of epochs was set to  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And in the fine-tuning stage, the maximum number of epochs  was set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' During adversarial training, we set 𝜀 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='0 in FGM that  means calculate only one step in the adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We find  that the dataset has imbalanced label and use some data processing  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Through splitting the complement and irrelevant product text  information, we could get more pairs which have the same label  and get a more balanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We use confident learning to find  the potential label errors and remove ∼4% data from the training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' As presented in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1, the median of Spanish and  English queries is 14 which satisfies the Poisson distribution of 𝜇 is  set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And the median of the Japanese query is 31 which satisfies  the Poisson distribution with 𝜇 is set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='2  Main Results  Our approach achieved considerably performance gain over the  baseline solution, and ranked top-8 in three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' The main results  are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In task1, we calculated the mean of all model  outputs as the final ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In task2 and task3, we almost  used the same network structure except the number of neurons  in the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Finally, Our approach ranked 5th, 7th and 8th,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  SubTask  Model  Metric  Ranking  task1  6 large models  ndcg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='9025  5th  task2  only 1 large model  micro f1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='8194  7th  task3  only 1 large model  micro f1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='8686  8th  Table 2: Performance of our approach on the private leader-  board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In task1, we used six InfoXLM𝑙𝑎𝑟𝑔𝑒 models that fine-  tuned by different datasets or methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In task2 and task3,  we used only one InfoXLM𝑙𝑎𝑟𝑔𝑒 model with the same net-  work structure, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Pre-Training Task  CV-MLM Loss  CV-Micro F1  Mask LM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='966  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='97  +Product2Query  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='969  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='05  ++Product2Brand  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='978  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='08  +++Contrastive Learning  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='047  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='08  Table 3: The effect of different pre-training tasks and keep  accumulating from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We report the cross vali-  dation MLM-Loss and Micro-F1 Score × 100 in the task2 set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='3  Ablation Studies  We investigate the impact of adopting different pre-training task  in the task2 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In Table 3, we show the Mask-LM losses after  5 epochs of pre-training and Micro-F1 scores after 2 epochs of  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We find that the Product2Query task achieves an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='008  improvement compared to the baseline, and the contrastive learning  task doesn’t get a significant gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  As shown in Table 4, we compare several loss functions, and we  adopt Poly1 loss function in task2 and task3 which based on the  actual performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We observe that the Focal loss function and  GHM loss function have worse performance than the cross-entropy  loss function in the task2 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  In this subsection, we explore several methods for further im-  proving the model’s performance in fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' As presented  KDDCup ’22, August 17, 2022, Washington, DC, USA  Xuange Cui, Wei Xiong, and Songlin Wang  Classification Loss  CV-Micro F1  CE Loss  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='08  Focal Loss  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='73  GHM Loss  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='85  Poly1 Loss  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='21  Table 4: The effect of different losses in the task2 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We  report the cross validation Micro-F1 Score × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Methods  CV-Micro F1  +EMA  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='19  ++FGM  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='30  +++R-Drop  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='43  ++++Embedding Mixup  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='43  Table 5: The effect of different strategies and keep accumu-  lating from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We report the cross validation  Micro-F1 Score × 100 in the task2 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Confident Learning  CV-Metric  with-in-task1  NDCG, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='005  with-in-task2  Micro-F1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='003  with-in-task3  Micro-F1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='002  Table 6: The effect of removing 4% noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  in Table 5, we adopt all of these methods to improve the model’s gen-  eralization and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We observe that the exponential moving  average method(EMA), adversarial training(FGM) and regularized  dropout strategy(R-Drop) could improve the model’s generalization  and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' But the Embedding Mixup strategy doesn’t get a  significant gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  As shown in Table 7, we consider using smaller datasets with  removing ∼4% noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We used the smaller dataset to achieve  an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='005 improvement in task1, but we get worse results in tash2  and task3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' It could be explained that since task1 contains more  difficult samples, the manually annotated data contains more label  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  5  CONCLUSION AND FUTURE WORK  In this work, we provide an overview of the combined approach to  improve the quality of search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We use data augmentation,  multitask pretraining strategy and several fine-tuning methods to  achieve considerably performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Specifically, we use mlm task,  classification task and contrastive learning task in pre-training  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And we use exponential moving average method(EMA), ad-  versarial training(FGM) and regularized dropout strategy(R-Drop)  to improve the model’s generalization and robustness in fine-tuning  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Moreover, we use a multi-granular semantic unit to discover  the queries and products textual metadata for enhancing the repre-  sentation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Future work of our system includes: 1) Com-  paring with other pre-trained language models, such as deborta𝑙𝑎𝑟𝑔𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  2) Using other training strategies, such as self-distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  REFERENCES  [1] Zewen Chi, Li Dong, Furu Wei, Nan Yang, Saksham Singhal, Wenhui Wang,  Xia Song, Xian-Ling Mao, Heyan Huang, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content='04805 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1109/CoNMedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Mod-  eling Product Search Relevance in e-Commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR abs/2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content='  arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='04980 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  SMART: Robust and Efficient Fine-Tuning for Pre-trained  Natural Language Models through Principled Regularized Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In Pro-  ceedings of the 58th Annual Meeting of the Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Association for Computational Linguistics, Online, 2177–2190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Embedding-based Product Retrieval in Taobao Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR  abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='09297 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='09297  [10] Xiaobo Liang, Lijun Wu, Juntao Li, Yue Wang, Qi Meng, Tao Qin, Wei Chen, Min  Zhang, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' R-Drop: Regularized Dropout for Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  CoRR abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='14448 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='14448 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='14448  [11] Yiqun Liu, Kaushik Rangadurai, Yunzhong He, Siddarth Malreddy, Xunlong Gui,  Xiaoyi Liu, and Fedor Borisyuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' Que2Search: Fast and Accurate Query and  Document Understanding for Search at Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Proceedings of the 27th ACM  SIGKDD Conference on Knowledge Discovery & Data Mining (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  [12] Hanqing Lu, Youna Hu, Tong Zhao, Tony Wu, Yiwei Song, and Bing Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Graph-based Multilingual Product Retrieval in E-commerce Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  CoRR  abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='02978 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='02978 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='02978  [13] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and  Adrian Vladu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='06083 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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+page_content=' Confident Learning:  Estimating Uncertainty in Dataset Labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Journal of Artificial Intelligence Research  (JAIR) 70 (2021), 1373–1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  [15] Curtis G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Northcutt, Tailin Wu, and Isaac L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Learning with  Confident Examples: Rank Pruning for Robust Classification with Noisy La-  bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In Proceedings of the Thirty-Third Conference on Uncertainty in Artifi-  cial Intelligence (Sydney, Australia) (UAI’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' AUAI Press, 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  http:  //auai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/uai2017/proceedings/papers/35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='pdf  [16] Juri Opitz and Sebastian Burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Macro F1 and Macro F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR abs/1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='03347  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='03347 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='03347  [17] Chandan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Reddy, Lluís Màrquez, Fran Valero, Nikhil Rao, Hugo Zaragoza,  Sambaran Bandyopadhyay, Arnab Biswas, Anlu Xing, and Karthik Subbian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Shopping Queries Dataset: A Large-Scale ESCI Benchmark for Improving Product  Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='06588  [18] Yining Wang, Liwei Wang, Yuanzhi Li, Di He, Tie-Yan Liu, and Wei Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  A Theoretical Analysis of NDCG Type Ranking Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR abs/1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='6480  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='6480 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='6480  [19] Xing Wu, Chaochen Gao, Liangjun Zang, Jizhong Han, Zhongyuan Wang, and  Songlin Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' ESimCSE: Enhanced Sample Building Method for Contrastive  Learning of Unsupervised Sentence Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR abs/2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='04380 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='04380 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='04380  [20] Atsuki Yamaguchi, George Chrysostomou, Katerina Margatina, and Nikolaos  Aletras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' Frustratingly Simple Pretraining Alternatives to Masked Language  ZhichunRoad at Amazon KDD Cup 2022: MultiTask Pre-Training for E-Commerce Product Search  KDDCup ’22, August 17, 2022, Washington, DC, USA  Methods  CV-Micro F1  Random♦ Word2vec♣  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='33  Freeze♥  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='29  Table 7: The performance of different initialization methods  of the multi-granular semantic unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' We report the cross val-  idation Micro-F1 Score × 100 in the task3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' CoRR abs/2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='01819 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='01819 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/abs/  2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='01819  [21] Chen Zhu, Yu Cheng, Zhe Gan, Siqi Sun, Tom Goldstein, and Jingjing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  FreeLB: Enhanced Adversarial Training for Natural Language Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' In  International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='net/  forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='id=BygzbyHFvB  A  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='1  Poisson Distribution  Figure 3: The length distribution of queries in different lan-  guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  As presented in Figure 3, the median of Spanish and English  queries is 14 which satisfies the Poisson distribution of 𝜇 is set to  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And the median of the Japanese query is 31 which satisfies the  Poisson distribution with 𝜇 is set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='2  EmbeddingBag Initialization  The multi-granular semantic unit implemented by Embedding-  Bag4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' As presented in Table 7, the way of random initialization  converges slowly, so we don’t record the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content=' And when the  Embedding-Bag is initialized by Word2vec, our approach obtain  the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='  4https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='EmbeddingBag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='html  query_len distribution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='35  geometric_p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='2  poisson_miu=4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
+page_content='30  es  sn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQf-jfx/content/2301.13455v1.pdf'}
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