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+On-chip Hong-Ou-Mandel interference from separate quantum dot emitters in an
+integrated circuit
+�Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1
+1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+University of W¨urzburg, Physikalisches Institut and Wilhelm-Conrad-R¨ontgen-Research
+Center for Complex Material Systems, Am Hubland, D-97074 W¨urzburg, Germany
+2currently at: Department of Electrical and Computer Engineering,
+Princeton University, Princeton, NJ 08544, USA
+3Institute of Physics, University of Oldenburg, D-26129 Oldenburg, Germany
+(Dated: January 5, 2023)
+Scalable quantum photonic technologies require low-loss integration of many identical single-
+photon sources with photonic circuitry on a chip. Relatively complex quantum photonic circuits
+have already been demonstrated; however, sources used so far relied on parametric-down-conversion.
+Hence, the efficiency and scalability are intrinsically limited by the probabilistic nature of the sources.
+Quantum emitter-based single-photon sources are free of this limitation, but frequency matching of
+multiple emitters within a single circuit remains a challenge. In this work, we demonstrate a key
+component in this regard in the form of a fully monolithic GaAs circuit combing two frequency-
+matched quantum dot single-photon sources interconnected with a low-loss on-chip beamsplitter
+connected via single-mode ridge waveguides. This device enabled us to perform a two-photon inter-
+ference experiment on-chip with visibility reaching 66%, limited by the coherence of the emitters.
+Our device could be further scaled up, providing a clear path to increase the complexity of quantum
+circuits toward fully scalable integrated quantum technologies.
+Optical quantum computing and communication ap-
+plications with single photons and linear optics rely crit-
+ically on the quantum interference of two photons on
+a beamsplitter [1].
+This process, known as Hong-Ou-
+Mandel (HOM) effect, occurs when two identical single
+photons enter a 50:50 beamsplitter, one in each input
+port. When the photons are indistinguishable, they will
+coalesce into a two-photon Fock state [2], in which they
+exit the same but random output port. This process un-
+derlines the simplest non-trivial path-entangled NOON
+state generation and introduces an optical non-linearity
+which is the base for the implementation of more-complex
+photonic gates and protocols.
+Consequently, scalable optical quantum information
+technologies will require integrating many identical in-
+distinguishable single-photon sources with reliable pho-
+tonic circuits consisting of beamsplitters. Utilizing well-
+developed integrated photonics technology is particularly
+appealing in this regard, as it dramatically reduces the
+footprint of quantum devices. Furthermore, it allows con-
+trolling photon states with high fidelity due to the in-
+trinsic sub-wavelength stability of the path-lengths, low-
+losses, and near-perfect mode overlap at an integrated
+beamsplitter for high-fidelity quantum interference [3–5].
+Advances in integrated photonic technology allowed
+already realizations of relatively complex quantum cir-
+cuits demonstrating CNOT-gate operation [6, 7], boson
+sampling [7, 8], quantum walks [9], some simple quan-
+tum algorithms [7, 10] and chip-to-chip quantum tele-
+portation [11]. A combination of integrated photonic cir-
+cuits with spontaneous four-wave mixing photon sources
+has also been achieved [11–13].
+However, due to the
+used sources’ probabilistic nature, their efficiency and
+scalability are intrinsically limited. Quantum emitters-
+based single-photon sources are free of this limitation
+and recently have been shown to outperform spontaneous
+four-wave mixing and down-conversion photon sources
+in simultaneously reaching high levels of photons indis-
+tinguishability and brightness [14–18].
+Moreover, re-
+mote interference between two quantum emitters was al-
+ready demonstrated using trapped ions [19, 20], quan-
+tum dots [21–30], organic molecules [31, 32] or vacancy
+centers in diamond [33–36]. The vast majority of those
+experiments have been performed in free space as proof-
+of-principle demonstrations.
+Performing similar exper-
+iments on-chip, taking advantage of the photonic cir-
+cuit consisting of fully integrated quantum emitters and
+beamsplitter, has not been performed yet, and it is still
+a missing component towards scaling-up aforementioned
+quantum technologies.
+In this work, we demonstrate a crucial component in
+this regard in the form of a fully monolithic GaAs cir-
+cuit combing two frequency-matched quantum dot single-
+photon sources interconnected with on-chip beamsplitter
+via single-mode ridge waveguides. This device enabled
+performing two-photon interference experiments on-chip
+with visibility limited by the coherence of our emitters.
+Our semiconductor photonic device is schematically
+presented in Fig. 1a and b. It is based on InAs/GaAs
+distributed Bragg-reflector ridge waveguides, which have
+been proven to facilitate high optical quality quantum
+dot single-photon sources [37]. The central part of the de-
+vice consists of the single-mode directional coupler (DC),
+which is the integrated optical analog of the bulk beam-
+arXiv:2301.01706v1  [quant-ph]  4 Jan 2023
+
+2
+QD1
+QD2
+DC
+Output Arm 1
+Output Arm 2
+a
+c
+QD1
+QD1
+QD2
+QD2
+Input Arm 1
+Input Arm 2
+QDs
+5x DBR
+24x DBR
+0.6 µm
+1.3 µm
+λ/2 cav.
+b
+…
+1k
+2k
+3k
+4k
+5k
+1.3930
+1.3935
+1k
+2k
+3k
+4k
+5k
+Output Arm 2
+QD2
+PL intensity (arb. units)
+QD1
+1.3930
+1.3935
+Energy (eV)
+Output Arm 1
+FIG. 1. On-chip two-photon interference circuit and beam splitting operation. a, Schematic representation of the
+photonic circuit based on a directional coupler (DC) interconnected with two input waveguides with coupled quantum dots and
+two output arms with inverted tapers for photons collection. b, Ridge waveguide cross-section with marked layer structure.
+c, Demonstration of beam splitting operation for fabricated DC. The photoluminescence signal from QD1 and QD2 is recorded
+for Output Arms 1 and 2, respectively. QD1 and QD2 are frequency matched with a precision of 5 µeV.
+splitter. In the two input arms of the DC, two frequency-
+matched quantum dots (QDs) are located. For single-
+photon generation, the QDs are excited non-resonantly
+from the top using two separated picosecond pulsed laser
+beams.
+Photons interfered on the DC are finally col-
+lected off the chip using inverse taper out-couplers. Spec-
+tral filtering and detection are performed off-chip using a
+monochromator and two superconducting single-photon
+detectors.
+To find two quantum dots with matching optical tran-
+sition energies, the position of the excitation beam spot
+on each input arm of the DC was scanned using an au-
+tomatized translation stage.
+Within such a scanning
+routine, we localized two matching emission lines at
+1.3931 eV energy originating from the QDs located in
+two individual input arms of the DC and separated spa-
+tially by around 200 µm. In Fig. 1b, photoluminescence
+(PL) spectra from QD1 and QD2 recorded from DC out-
+put arms 1 and 2 are presented at a temperature of 4.5 K.
+Single well-resolved emission lines matching within 5 µeV
+fit precision are visible. Comparing amplitudes of QD1
+and QD2 emission peaks visible within both output arms,
+a beam splitting ratio of 48:52 is derived (including un-
+even transmission through out-coupling arms - more de-
+tails in Supplementary Section 8).
+To show that optical excitation of our QDs leads to
+the generation of single photons, we analyzed the photon
+emission statistics of separate QDs by performing sec-
+ond order-correlation experiments in Hanbury Brown and
+Twiss (HBT) configuration. For that purpose, QDs have
+been excited non-resonantly from the top by an 813 nm
+wavelength train of picosecond pulses at a repetition rate
+of 76 HMz. Photons emitted by the QDs were then cou-
+pled into the circuit input arm waveguides and guided
+into the directional coupler, where the signal was divided
+between two output arms. Next, photons were collected
+off-chip from the side of the sample using out-couplers
+and subsequently filtered spectrally by a monochroma-
+tor (70 µeV width) and coupled into two single-mode
+fibres connected with superconducting single-photon de-
+tectors (SSPD). Finally, the photon correlation statistics
+were acquired by a multichannel picosecond event timer.
+Data have been recorded under excitation powers corre-
+sponding to half of the QD saturation intensity.
+Fig. 2a and c present the second-order autocorrelation
+function g(2)
+HBT (τ) measurement recorded for each QD in-
+dividually. In the case of both QDs, a clear suppression of
+the central peak counts is visible, proving single-photon
+emission. To quantitatively evaluate the probability of
+multi-photon emission, g(2)
+HBT (0) values were calculated
+by integrating residual counts of the zero delay peak
+with respect to the neighboring six peaks, resulting in
+g(2)
+HBT (0) = 0.35 ± 0.08 and g(2)
+HBT (0) = 0.15 ± 0.02 for
+QD1 and QD2, respectively.
+In Fig. 2b and d, time-
+resolved photoluminescence traces of the QD1 and QD2
+emission are shown. In this case, the repetition rate of
+the laser was reduced to 19 MHz using a pulse picker.
+Clear bi-exponential signal decays are visible, with a fast
+and slow time constant of 720±5 ps and 12±1 ns for QD1
+and 600±5 ps and 22±1 ns for QD2. We attribute the
+fast decay to the spontaneous recombination of electron-
+hole pairs in QD (T1) and the slow one, which corre-
+sponds to about 2% (1.2%) of the total QD1 (QD2) line
+intensity, is tentatively interpreted as the recapturing of
+the carriers by the QD. Using fit parameters obtained
+from the time-resolved experiments, g(2)
+HBT (τ) correla-
+tion histograms have been fitted with double-sided bi-
+exponential decay convoluted with 80 ps width Gaussian
+
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+ experiment
+ fit
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Raw coincidences
+0
+5
+10 15 20
+101
+102
+103
+PL (coincidences)
+Time (ns)
+Afast/Aslow = 85
+-45 -30 -15
+0
+15 30 45
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+g(2)
+HBT(t)
+Delay time (ns)
+20
+40
+60
+80
+100
+120
+140
+160
+101
+102
+103
+ experiment
+ bi-exp fit
+Afast/Aslow = 50  
+QD1
+QD2
+a
+b
+c
+d
+FIG. 2.
+Single-photon generation and emission dy-
+namics. a,c Second order auto-correlation histograms of QD1
+and QD2 emission under pulsed 76 MHz repetition rate excita-
+tion. Data have been recorded in HBT configuration using an
+on-chip beamsplitter. b,d Time-resolved PL traces revealing
+bi-exponential decays with fast (slow) time constant of 720 ps
+(12 ns) and 600 ps (22 ns) for QD1 and QD2, respectively.
+instrumental response function (black dashed lines).
+To demonstrate on-chip two-photon interference, the
+single QDs in both input arms of the DC are excited us-
+ing the picosecond pulses. For that, the laser beam is di-
+vided into two independently controllable optical excita-
+tion axis and synchronized in advance to ensure optimal
+temporal overlap of emitted photons on the DC. It is per-
+formed by sending the emission from each QD separately
+through the on-chip DC and using time-resolved detec-
+tion to eliminate the time delay difference between inde-
+pendently generated single photons. The same technique
+is used to introduce an intentional 0.5 ns time delay for
+reference measurements. The excitation laser powers for
+each QD are adjusted such that their emission intensities
+are the same (around half of QD1 saturated intensity).
+As we utilize on-chip beam splitting operation using DC
+with single-mode inputs and outputs, we expect a very
+high spatial mode overlap of our interferometer. To test
+this, we send the continuous wave laser simultaneously
+into both DC input arms and record classical interference
+fringes with 98±1% visibility. In earlier experiments per-
+formed on ridge waveguide structures, we observed that
+the QD emission couples into the well-defined transverse-
+electric mode of the WG with close to unity degree of
+linear polarization [37]. In the case of the investigated
+device, the polarization of the emitted photons was an-
+alyzed after passing the whole circuit consisting of bend
+regions, DC itself, and out-couplers (more details in Sup-
+plementary Section 6). We found that for both QDs, the
+degree of polarization is above 95%, suggesting optimal
+polarization alignment for interference experiments.
+Within the above-mentioned prerequisites, the two-
+photon interference should be mainly limited by the co-
+herence of our single-photon emitters. To get access to
+the coherence times of our QDs, we performed a high-
+resolution measurement of the emission linewidths us-
+ing a scanning Fabry-Perot interferometer. We extract
+the full-width at half-maximum of 13.5±2.5 µeV and
+3.0±0.2 µeV by Lorentzian fit for QD1 and QD2, re-
+spectively (see Supplementary Section 7).
+The coher-
+ence times calculated based on observed broadenings are
+T QD1
+2
+= 100±20 ps and T QD2
+2
+= 440±30 ps. As the mea-
+surements are performed on the tens of seconds timescale,
+we speculate that recorded coherence times might be lim-
+ited by charge and spin noise [38, 39]. Following Ref. [40],
+we calculated the expected interference visibility of our
+two independent emitters and derived the theoretical vis-
+ibility in the range of Vtheory = 10-15%.
+Figure 3a shows two-photon interference data in the
+form of second-order HOM cross-correlation between
+photons exiting the two output arms of the on-chip beam-
+splitter. The height of the central peak is clearly below
+the half intensity of the neighboring peaks, proving that
+photons emitted by two separate QDs indeed interfere
+on the DC. Another interference signature is the pres-
+ence of the coincidences dip superimposed on the central
+peak around zero time delay. The depth of this dip con-
+stitutes to the interference events where photons arrive
+simultaneously at the DC, giving rise to the narrow-time
+window post-selected coalescence. In our case, the exact
+value of g(2)
+HOM(τ) at τ = 0 is equal to 0.17 in the case
+of background-corrected data and 0.31 for as-measured
+data. The same type of time post-selected interference
+can be observed for cw HOM correlations.
+Figure 3b
+shows the non-corrected cw, and pulsed HOM interfer-
+ence histograms overlapped on each other (correspond-
+ing cw g(2)
+HBT (τ) graphs are shown in Supplementary Sec-
+tion 10). Similar to the pulsed case, the cw correlation
+shows clear suppression of coincident counts at zero time
+delay, with time post-selected g(2)
+HOM(0) of 0.35, close to
+the pulsed as-measured value of 0.31.
+To evaluate photons full wave-packet interference
+probability (non-post-selected), we calculate the pulsed
+HOM correlation central peak area normalized by
+the
+average
+area
+of
+the
+neighboring
+six
+peaks.
+For
+integration
+window
+∆t
+of
+3
+ns,
+we
+obtain
+g(2)
+HOM(0, ∆t) = 0.459±0.002 for background corrected
+data and g(2)
+HOM(0, ∆t) = 0.587±0.002 for raw data,
+where uncertainty is based on the standard deviation
+of non-central peaks areas. In the case of background-
+corrected data, we reach a value below the 0.5 classical
+limit. It needs to be noted that derived g(2)
+HOM(0, ∆t) and
+g(2)
+HOM(0) values are partially influenced by the non-zero
+multi-photon emission extend observed in HBT measure-
+ments.
+
+4
+-45
+-30
+-15
+0
+15
+30
+45
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+g(2)
+HOM(t) 
+Delay time (ns)
+100
+200
+300
+400
+500
+600
+Raw coincidences
+-5 -4 -3 -2 -1
+0
+1
+2
+3
+4
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+HOM peak area - g(2)
+HOM(t,Dt) 
+Peak number
+ indist. (sync.) 
+ dist. (0.5 ns delay)
+Dtint = 3 ns
+V = 17.8±0.7%
+g(2)
+HOM(0,Dt) = 0.459±0.002 
+-4
+-2
+0
+2
+4
+0.0
+0.2
+0.4
+Dtint
+-6
+-4
+-2
+0
+2
+4
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+ cw
+ pulsed
+g(2)
+HOM(t) 
+Delay time (ns)
+a
+b
+c
+QD1
+QD2
+FIG. 3.
+On-chip two-photon interference from separate quantum emitters.
+a, Two-photon Hong-Ou-Mandel
+interference measurement between QD1 and QD2 showing the normalized HOM coincidences versus the delay time.
+The
+central peak area is suppressed with respect to neighboring peaks. Inset: Magnified view of the central peak area. b, Raw
+HOM interference measurement recorded under cw (red points) and pulsed (blue points) excitation. c, Integrated counts of
+the central eleven peaks (∆t = 3 ns integration window) of the HOM correlation in case of synchronized (blue bars) and 0.5 ns
+delayed (red bars) photons from QD1 and QD2. All presented data are recorded using an on-chip beamsplitter.
+As it has been recently pointed out in Ref. [41, 42],
+to estimate two-photon interference visibility for remote
+emitters correctly, it is necessary to perform reference
+HOM measurements for distinguishable photons, due to
+the possible blinking effect. Since within the fabricated
+circuit polarization rotation is impossible to unambigu-
+ously confirm the two-photon interference and properly
+evaluate visibility, photons were made distinguishable by
+introducing a 0.5 ns time delay between excitation pulses.
+Such delay should be sufficient to lose the temporal pho-
+tons overlap on the DC within the emitters coherence
+times and record reference data.
+Figure 3c demonstrates the normalized histogram of
+the central eleven peaks areas (∆t = 3 ns) of the
+HOM second-order cross-correlation in case of synchro-
+nized - indistinguishable (red bars) and 0.5 ns de-
+layed - distinguishable (grey bars) photons.
+The cen-
+tral peak area in the case of unsynchronized pho-
+tons is equal to g(2)
+HOMd(0, ∆t) = 0.558±0.002, which
+is slightly above the theoretically expected 0.5 value.
+We relate this discrepancy with non-zero multi-photon
+emission extend.
+Finally, we calculate remote sources
+two-photon interference visibility V
+following V
+=
+[g(2)
+HOMd(0, ∆t) − g(2)
+HOM(0, ∆t)]/g(2)
+HOMd(0, ∆t), resulting
+in V = 17.8±0.7% for background corrected data. This
+value is relatively close to the theoretically expected vis-
+ibility and even partially exceeds it, suggesting that it is
+limited solely by the coherence of the emitters (more de-
+tails Supplementary Section 9). Using background cor-
+rected data for the pulsed case, give post-selected visi-
+bility of V ′
+p = 66%. The probability of the time post-
+selected interference is known to depend on the ratio of
+the emitters coherence times to the setup timing reso-
+lution [21, 22, 43], thus possibly even higher V ′ values
+could be potentially achieved with faster detectors.
+While our results provide clear scientific evidence for
+on-chip generation and interference of on-demand sin-
+gle photons in a circuit, the recorded visibility values
+need to be improved for future practical applications. In
+the current device architecture, the indistinguishability
+of interfered photons is limited by T2/(2T1) of particu-
+lar QDs. We propose a few strategies for improving this
+ratio. Firstly, the QD charge environment could be stabi-
+lized via passivation [44], weak optical illumination [45],
+or gating [15, 46]. At the same time, by embedding QDs
+into optical cavities, the Purcell effect might be used to
+enhance the radiative emission rate 1/T1 [14–16, 46, 47].
+Recently, we demonstrated a QD circuit with ring cav-
+ities allowing to significantly increase the QD coupling
+efficiency into the WG mode and decrease the T1 be-
+low 200 ps [48].
+Finally, by applying a resonant exci-
+tation, the photons emission time-jitter could be mini-
+mized, and strong suppression of multi-photon emission
+events achieved [14–16, 37, 46, 48].
+Within such cir-
+cuit and excitation improvements, two-photon interfer-
+ence with near-unity visibility seems to be within reach.
+To realize circuits combining multiple QD sources cou-
+pled to cavities, deterministic fabrication technologies
+such as in-situ electron-beam lithography or imaging will
+be required. This will allow to preselect emitters with
+identical spectral characteristics, build cavities around
+them and combine them within a single functional pho-
+tonic circuit. In principle, since QD imaging could be
+performed in an automatized manner, a very large num-
+ber of emitters could be combined on a single chip. At
+such a stage of complexity, separate control over QD
+emission energies might also be desired.
+This could
+be directly implemented by a local laser drive via AC
+Stark effect [49] or adapting the circuit for the electric
+field [21, 46] and strain [22, 50–52] control. Ultimately, a
+practical quantum photonic chip will require the presence
+of additional functionalities such as single-photon detec-
+tors and phase-shifter.
+Fortunately, the GaAs circuits
+are compatible with superconducting detectors technol-
+ogy [53–55] and thanks to the large χ2 nonlinear coeffi-
+
+5
+DC
+Circuit
+Ring 
+cavity
+QD emitters
+Phase shifter
+Detectors
+FIG. 4.
+Envisioned fully integrated quantum photonic circuit.
+Draft of the possible circuit design with multiple
+quantum dot-based single photon sources coupled to ring cavities, interconnected with ridge waveguides, directional couplers,
+phase shifters, and superconducting detectors.
+cient of the GaAs, electro-optical phase shifters have al-
+ready been demonstrated [56]. Such an envisioned fully
+functional QDs-GaAs circuit is shown schematically in
+Fig. 4.
+In conclusion, we have shown that two identical QD
+single-photon sources can be integrated monolithically in
+a waveguide circuit and made to interfere with visibility
+limited by the coherence of those sources. We pointed
+out the potential strategies to improve the QDs perfor-
+mance by employing deterministic fabrication and cavity
+enhancement. The implemented integrated system could
+be potentially further extended to facilitate more com-
+plex circuits and fully on-chip operation. Results shown
+in this article, along with a clearly outlined path for fu-
+ture improvements, take us one step closer to scalable
+integrated quantum circuits based on quantum emitters
+capable of generating and manipulating large photonic
+states.
+Methods
+Sample description.
+To fabricate our integrated
+single-photon source waveguide device, we use a semi-
+conductor sample that contains self-assembled In(Ga)As
+QDs grown by the Stranski-Krastanow method at the
+center of a planar GaAs microcavity.
+The lower
+and upper cavity mirrors contain 24 and 5 pairs of
+Al0.9Ga0.1As/GaAs λ/4-layers, respectively, yielding a
+quality factor of ∼200.
+A δ-doping layer of Si donors
+with a surface density of roughly ∼1010 cm−2 was grown
+10 nm below the layer of QDs to dope them probabilis-
+tically.
+To fabricate ridge waveguides devices, the top
+mirror layer along with the GaAs cavity is etched down,
+forming the ridge with a width of ∼0.6 µm and a height
+of ∼1.3 µm.
+The cross-section of the WG with layer
+structure is shown in Figure 1b (see also Supplementary
+Section 1). Ridges have been defined by e-beam lithog-
+raphy and reactive ion etching.
+After processing, the
+sample was cleaved perpendicularly to the WGs, around
+30 µm away from the tapered out-coupler edges to get
+clear side access.
+Integrated circuit design We designed and fabricated
+GaAs directional couplers with different coupling lengths
+and gaps. A directional coupler with a near 50:50 cou-
+pling ratio at around 1.3931 eV was obtained when the
+gap distance was set to 120 nm and the coupling length to
+30 µm (see Supplementary Section 2 for layout scheme).
+The total length of the device was about 1 mm, including
+four S-bends with a radius of 60 µm and the input/output
+waveguides.
+Experimental setup.
+For all experiments, the sam-
+ple is kept in a low-vibrations closed-cycle cryostat (at-
+toDry800) at temperatures of ∼4.5 K. The cryostat is
+equipped with two optical windows allowing for access
+from the side and the top of the sample.
+A spectro-
+scopic setup consisting of two independent perpendicu-
+larly aligned optical paths is employed (see Supplemen-
+tary Section 3 for more details).
+QDs embedded into
+WGs are excited from the top through a first microscope
+objective with NA = 0.26, while the emission signal is
+detected from a side facet of the WG with a second ob-
+jective with NA = 0.4. The photoluminescence signal,
+simultaneously collected from both output arms of the
+DC, is then passed through a Dove prim to rotate the
+sample image plane from a horizontal into a vertical di-
+rection to fit the monochromator slit orientation.
+For
+PL analysis, the signal is then spectrally dispersed by a
+75 cm focal length monochromator and focused on a low-
+noise liquid-nitrogen-cooled CCD camera (around 40 µeV
+spectral resolution), allowing to resolve signal from both
+DC output arms spatially. For HBT and HOM experi-
+ments, the monochromator serves as a spectral filter with
+70 µeV width, and the signal from both DC outputs is
+introduced into separate single-mode optical fibres con-
+nected with superconducting single-photon counting de-
+tectors (30 ps time-response).
+Integrated beamsplitter visibility. To test the clas-
+sical visibility of the DC device, we simultaneously send
+the continuous wave laser light tuned to the energy of
+QDs transitions, using circular reflectors placed on the
+ends of the input arms waveguides (see Supplementary
+Section 8).
+The power of the laser coupled into both
+
+O6
+arms was adjusted such that the intensity from both in-
+put arms was the same. Next, we focused on the signal
+passing through DC and out coupled from one output
+arm. We observed intensity modulation in the function of
+time, related to small path-length difference fluctuation,
+allowing us to see interference pattern and calculate the
+interferometer visibility. In the case of the investigated
+device, the classical visibility of 98±1% was extracted.
+Correlation histograms analysis. For the time post-
+selected visibility V ′ analysis, we assume that for distin-
+guishable photons g(2)
+HOMd(0) is equal to 0.5, as reference
+measurement is not possible. It allows to calculate V ′ ac-
+cording to V ′ = [0.5 − g(2)
+HOM(0)]/0.5. Data from Fig.3b
+lead to raw visibility of V ′
+cw = 30% and V ′
+p = 38% for
+cw and pulsed excitation, respectively. For g(2)
+HBT (τ) and
+g(2)
+HOM(τ) correlation functions evaluation we take into
+account a presence of the time-independent background
+offset in recorded histograms (it constitutes to around
+15-20% of the coincidences), which we relate to the dark
+counts of the SSPDs (100-500 cps).
+Non-background-
+corrected HOM graphs can be found in Fig. 3c and the
+Supplementary Section 9.
+The authors thank Silke Kuhn for fabricating the
+structures.
+�L.D. acknowledges the financial support
+from the Alexander von Humboldt Foundation. We ac-
+knowledge financial support by the German Ministry
+of Education and Research (BMBF) within the project
+”Q.Link.X” (FKZ: 16KIS0871).
+We are furthermore
+grateful for the support by the State of Bavaria.
+∗ [email protected]
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+
+Supporting Information:
+On-chip Hong-Ou-Mandel interference from separate quantum
+dot emitters in an integrated circuit
+�Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1
+1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+University of W¨urzburg, Physikalisches Institut and
+Wilhelm-Conrad-R¨ontgen-Research Center for Complex Material Systems,
+Am Hubland, D-97074 W¨urzburg, Germany
+2currently at: Department of Electrical and Computer Engineering,
+Princeton University, Princeton, NJ 08544, USA
+3Institute of Physics, University of Oldenburg, D-26129 Oldenburg, Germany
+(Dated: January 5, 2023)
+1
+arXiv:2301.01706v1  [quant-ph]  4 Jan 2023
+
+S1: SAMPLE STRUCTURE
+The full layer structure of the sample is shown in Figure S1.
+etched
+etched
+600 nm
+1250 nm
+Bottom DBR mirror:
+24x Al0.9Ga0.1As/GaAs
+Top DBR mirror:
+5x Al0.9Ga0.1As/GaAs
+GaAs substrate
+In(Ga)As QDs and WL
+GaAs
+λ-cavity
+1 µm
+FIG. S1. Planar sample scanning electron microscope cross-section image with visible layers and
+schematically marked areas for etching.
+The quantum dot layer is placed inside a center of a
+λ cavity sandwiched between two distributed Bragg Reflectors consisting of the 5/24 alternating
+λ/4-thick layers of Al0.9Ga0.1As and GaAs.
+S2: INTEGRATED CIRCUIT LAYOUT
+Figure S2 shows the layout scheme of the fabricated GaAs device. It is based on 600 nm
+width single-mode ridge waveguides. The central part of the circuit is a directional cou-
+pler (DC) formed by two WGs separated by a 120 nm gap along a 30 µm long coupling
+region. WGs were brought together using two circular bend regions with a radius of 60 µm.
+Waveguides on the right end of the circuit are terminated by inverse taper (30 µm length)
+out-couplers, minimizing reflection and optimized for better light extraction out of the chip.
+The left side of the DC consists of 1.2 mm long straight WG sections designated for search-
+ing two quantum dots with the same transition frequencies. WGs on the left side of the
+circuit are terminated with circular Bragg grating mirrors optimized for increased reflectiv-
+ity (around 80% expected at 900 nm). Figure S3 shows the scanning electron microscope
+images of the fabricated integrated circuits.
+2
+
+350nm
+30µm
+30µm
+Gap=120nm
+W=600nm
+70µm
+R=60µm
+R=60µm
+1200µm
+730nm
+FIG. S2. Integrated circuit layout. Scheme of the fabricated GaAs directional coupler including
+circular Bragg reflectors located at the ends of the input WG arms and inverse taper out-couplers
+at the end of the output arms. Bending regions are based on circular profiles with a radius of
+60 µm.
+a
+b
+c
+d
+e
+f
+120 µm
+60 µm
+30 µm
+3 µm
+3 µm
+3 µm
+FIG. S3.
+Scanning electron microscope images of the fabricated devices.
+a,b, The DCs with
+different coupling lengths. c,d, The DC with 30 µm length coupling region. e, An inverse taper
+outcoupler. f, A circular Bragg grating reflector.
+S3: OPTICAL SET-UP
+For all experiments, the sample is kept in a low-vibrations closed-cycle cryostat (at-
+toDry800) at temperatures of ∼4.5 K. The cryostat is equipped with two optical windows
+allowing for access from both side and top of the sample. A spectroscopic setup consisting of
+two independent perpendicularly aligned optical paths is employed as shown schematically
+3
+
+SSPD1
+SSPD2
+Ti:Si pulsed 
+laser 813nm
+76 MHz
+Pulse picker 
+19MHz 
+(for TRPL)
+BS 50:50
+CW 660nm laser
+Obj.
+x10 
+BS 92:8
+Dove prism
+Obj. x20
+L1
+L2
+L3
+CCD
+Slit
+Slit
+Knife edge 
+mirror
+Monochromator
+Sample in cryostat
+Time-tagger
+L4
+SM fiber
+SM fiber
+SM fiber
+0.5 ns delay 
+fiber
+ND filter
+excitation
+XYZ position 
+control
+HWP+LP
+90 deg image rotation
+Dove prism
+Mirror
+FIG. S4. Optical setup. Scheme of the experimental configuration used for top excitation (blue
+path) and side detection (red path) photoluminescence and resonance fluorescence measurements.
+In the case of two-photon interference experiments, a QD was excited twice every laser pulse cycle
+with a delay of 3 ns, and the subsequently emitted photons, spatially and temporally overlapped
+in an unbalanced Mach-Zehnder interferometer (dashed lines) utilizing polarization maintaining
+(PM) fibers and beam-splitters (BS). For signal detection, two avalanche photo-diodes (APD) with
+350 ps response time were used. For polarization control in free space, a half-wave-plate (HWP)
+combined with a linear polarizer (LP) was used, while for polarization rotation (PR) in the fiber-
+based HOM interferometer ceramic sleeve connectors between two fiber facets were used allowing
+to align fast and slow axis at the desired angle.
+in Figure S4. Additionally, the excitation path allows for the separate routing of two laser
+beams for the simultaneous excitation of two spots on the sample. For HOM and HBT
+experiments, a tunable Ti:Si picosecond pulsed laser is used. QDs embedded into the two
+input arms of the DC are excited from the top through a first microscope objective with
+x10 magnification and NA = 0.26, while the emission signal from both DC output arms
+is detected simultaneously from the side facet of the sample with a second objective with
+x20 magnification and NA = 0.4. Photoluminescence signal from both arms is then passed
+through a spatial filter (lenses L1 and L2) and polarization optics. For light polarization
+analysis, a half-wave plate (HWP) combined with a linear polarizer (LP) is used. The sam-
+ple image plane is rotated from a horizontal into a vertical direction using a Dove prism,
+which allows simultaneous coupling signals from both DC output arms into the monochro-
+4
+
+mator. The collected light is analyzed by a high-resolution monochromator equipped with
+a liquid nitrogen-cooled low-noise charge-coupled device detector (CCD), featuring a spec-
+tral resolution of ∼40 µeV. Taking advantage of the spatial separation of DC output arms,
+the spectrum from both WG arms is resolved spatially on the CCD camera.
+For HBT
+and HOM experiments, the monochromator is used as a spectral filter with 70 µeV width.
+Next, the signal from both arms is separated spatially using a knife-edge mirror and cou-
+pled into single-mode fibres interconnected with superconducting single-photon detectors
+(SSPD). The time-correlated measurements are acquired using a stand-alone time-tagger.
+S4: POWER-RESOLVED PL
+In Fig. S5 QD1 and QD2 emission intensity as a function of excitation power is shown.
+An almost linear dependence of the emission intensity on excitation power suggests that the
+analyzed lines originate from the recombination of neutral or charged excitonic complexes.
+a
+b
+0.01
+0.1
+1
+0.01
+0.1
+1
+PL intensity (arb. units)
+Power (P/Psat)
+QD1
+I~ P0.90±0.05
+QD2
+I~ P0.93±0.05
+PL intensity (arb. units)
+Power (P/Psat)
+FIG. S5.
+Photoluminescence intensity vs incident excitation power. Solid red/blue curve: fit with
+a power function revealing linear dependence of the emission intensity on excitation power.
+5
+
+S5: WAVEGUIDE TRANSMISSION LOSSES
+To estimate the quality of the etched ridge waveguides, the optical WG transmission
+losses were determined. For that purpose, the sample was excited with very high pumping
+power, allowing us to observe spectrally broad QD ensemble emission. The beam spot was
+scanned along the DC input arm 1/2, and emission was detected from the side through the
+waveguide arm 1. Figure S6a and b show the corresponding attenuation of the measured
+intensities at 890 nm plotted as a function of the distance to the DC bends for input arms 1
+and 2, respectively. Input arm 1 exhibits transmission losses on the level of 6.5±0.5 dB/mm
+and arm 2 of 5.0±0.6 dB/mm. Waveguide transmission characteristics are limited by ridge
+sidewall imperfections, which could be potentially further improved by optimizing the etch-
+ing process.
+a
+b
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+8
+7
+6
+5
+4
+3
+2
+1
+0
+-1
+0.0
+0.2
+0.4
+0.6
+0.8
+5
+4
+3
+2
+1
+0
+-1
+Input Arm 1
+Losses: 6.5±0.5 dB/mm
+ experiment
+ fit
+Attenuation (dB)
+Distance (mm)
+Input Arm 2
+Losses: 5.0±0.6 dB/mm
+ experiment
+ fit
+Attenuation (dB)
+Distance (mm)
+FIG. S6. Waveguides transmission losses. Attenuation of the side detected ensemble PL signal in
+function of the distance from DC bend regions.
+6
+
+S6: POLARIZATION-RESOLVED PL
+Side detected emission from both studied emission lines show a high degree of linear
+polarization (DOLP) of around 95%, oriented in the sample plane as shown in Fig. S7. A
+high DOLP and its direction are related to the QDs dipole moments, which are mainly
+in-plane oriented and thus emitted photons mostly couple to and propagate in the TE
+waveguide mode. It needs to be noted that high DOLP is maintained after passing the
+whole circuits consisting of bends, DC, and out-coupler, which could potentially spoil the
+detected polarization contrast. The same polarization level is observed for both output arms
+of the DC.
+QD1
+QD2
+0
+30
+60
+90
+120
+150
+180
+210
+240
+270
+300
+330
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+30
+60
+90
+120
+150
+180
+210
+240
+270
+300
+330
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0
+30
+60
+90
+120
+150
+180
+210
+240
+270
+300
+330
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0
+30
+60
+90
+120
+150
+180
+210
+240
+270
+300
+330
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Norm. PL intensity (arb. units)
+ experiment
+ Sine fit
+DOLP = 95±2%
+DOLP = 95±2%
+DOLP = 94±2%
+DOLP = 95±2%
+Output Arm 1
+Output Arm 2
+FIG. S7. Polarization characteristics of the QD1 and QD2 emission coupled to input arms of the
+DC and detected from side facet output arms 1 and 2. Both QDs PL emission is strongly linearly
+polarized with around 95% degree of linear polarization.
+7
+
+S7: EMITTERS TRANSITION LINEWIDTHS
+Figure S8 shows a high-resolution spectrum of the QD1 and QD2 emission recorded under
+cw 660 nm excitation. Measurements are performed using a scanning Fabry-Perot interfer-
+ometer with a 3 µeV Lorentzian profile linewidth. Both spectra are fit using Lorentzian
+functions with full-width at half-maximum (FWHM) of 16.5±2.5 µeV and 6.0±0.2 µeV, for
+QD1 and QD2, respectively (error related to fit precision). It can be shown that the convo-
+lution of two Lorentzian profiles with FWHM1 and FWHM2 is also a Lorentzian profile with
+broadening of FWHM1+FWHM2. Following the above, we can correct the recorded optical
+linewidths for the finite resolution of our setup by simply subtracting its linewidth. The
+deconvoluted linewidths are 13.5±2.5 µeV and 3.0±0.2 µeV for QD1 and QD2, respectively.
+a
+b
+-60
+-40
+-20
+0
+20
+40
+60
+80
+-60
+-40
+-20
+0
+20
+40
+60
+QD1
+FWHMfit= 16.5±1.5 meV
+FWHMdec.= 13.5±1.5 meV
+ experiment
+ Lorentz fit
+Intensity (arb. units)
+Detuning (meV)
+QD2
+FWHMfit = 6.0±0.2 meV
+FWHMdec.= 3.0±0.2 meV
+ experiment
+ Lorentz fit
+Intensity (arb. units)
+Detuning (meV)
+FIG. S8. A high-resolution PL spectrum of QD1 and QD2, obtained using a home-built Fabry-
+Perot scanning cavity with a resolution of 3 µeV (FWHM), and free spectral range of 140 µeV at
+890 nm. Solid lines are fits with the Lorentz function.
+S8: DIRECTIONAL COUPLER CHARACTERISTICS
+To extract the DC splitting ratio r : t accounting for the different performance of both
+output arms due to the fabrication imperfections, the following procedure has been used.
+First, QD located in input arm 1 was excited, and PL signal from output arm 1 II1
+O1 and
+arm 2 II1
+O2 collected. Next, the same measurement has been repeated on QD located in arm
+2, revealing II2
+O1 and II2
+O2. If the uneven outcoupling efficiency between the two output arms
+8
+
+is quantified as a constant x (transmission ratio between two arms), the following set of
+equations applies
+r + t = 1,
+xr
+t = II1
+O1
+II1
+O2
+,
+x t
+r = II2
+O1
+II2
+O2
+.
+(1)
+Based on that, the DC splitting ratio r : t can be derived, accounting for the imbalanced
+outcoupling
+r : t =
+�
+II1
+O1
+II1
+O2
+II2
+O2
+II2
+O1
+.
+(2)
+In the case of the investigated DC with II1
+O1/II1
+O2 of 51:49 and II2
+O2/II2
+O1 of 46:54 we ended up
+with the r:t ratio of 48:52, which is very close to the desired 50:50.
+0
+50
+100
+150
+200
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Intensity (cps)
+Time frame
+laser coupled to:
+ Input Arm 1
+ Input Arm 2
+ Arm 1+2 (interference)
+classical visibility at 890 nm
+V = (Imax-Imin)/(Imax+Imin) = 98±1%
+FIG. S9. Intensity fluctuations of the cw laser light transmitted simultaneously through two input
+arms of the DC. Fluctuations in time are related to the off-chip path-length difference fluctuations
+of signal interfered on the directional coupler.
+To test the classical visibility of the DC device, we simultaneously send cw laser light
+tuned to the energy of QDs transitions (890 nm), using circular reflectors placed on the
+ends of the input arms waveguides. The power of the laser coupled into both arms was
+adjusted, so the intensity from both input arms was the same. Next, we focused on the
+9
+
+signal passing through the DC and out-coupled from one of the output arms. We observed
+intensity modulation in the function of time, related to the small path-length difference
+fluctuation, allowing us to record interference pattern as shown in Fig. S8. Classical DC
+visibility was obtained by calculating the intensity contrast
+VDC = Imax − Imin
+Imax + Imin
+.
+(3)
+In the case of the investigated DC device, a classical visibility of 98±1% was extracted.
+10
+
+S9: RAW HOM CORRELATION DATA
+Figure S10a shows a non-corrected two-photon Hong-Ou-Mandel interference experiment
+result between QD1 and QD2 performed under 76MHz and 19MHz pulsed laser repetition
+rate. In the case of both graphs, the same time-independent background offset is visible,
+corresponding to around 15% of the normalized peak intensity. Since the background level
+is the same for the HOM graphs at different laser repetition rates, we exclude the emit-
+ters long-decay contribution to the observable background. We attribute the observed cw
+offset to the SSPDs dark counts (100-500 cps). Figure S10b shows the non-corrected nor-
+malized histogram of the central eleven peaks areas (∆ = 3 ns integration window) of the
+HOM second-order cross-correlation in case of synchronized - indistinguishable (red bars)
+and 0.5 ns delayed - distinguishable (grey bars) photons. The non-corrected central peak
+area in case of synchronized photons is equal to g(2)
+HOM(0, ∆t) = 0.587±0.002, while in case
+of unsynchronized photons g(2)
+HOMd(0, ∆t) = 0.680±0.002. In this case, the non-corrected
+two-photon interference visibility yields Vraw = 12.1±0.3% in correspondence to 17.8±0.7%
+visibility obtained for background-corrected graphs (see Fig.3c in the main text).
+a
+b
+-50
+-25
+0
+25
+50
+75
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Raw data
+Raw norm. HOM counts - g(2)
+HOM(t)
+Delay time (ns)
+ 19 MHz
+ 76 MHz
+cw bck
+-5 -4 -3 -2 -1
+0
+1
+2
+3
+4
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6 Dtint = 3 ns
+Vraw = 12.1±0.3%
+g(2)
+HOM(0,Dt) = 0.587±0.002 
+Raw norm. HOM 
+peak area - g(2)
+HOM(t,Dt)
+Peak number
+ indist. (sync.) 
+ dist. (0.5 ns delay)
+bck
+FIG. S10.
+a, Two-photon Hong-Ou-Mandel interference measurement between QD1 and QD2
+performed using on-chip beamsplitter showing the normalized coincidences versus the delay time
+with no background counts correction. An experiment performed under 76MHz and 19MHz pulsed
+laser repetition rate yields the same cw background. b, Non-corrected integrated counts of the
+central eleven peaks (3 ns integration window) of the HOM correlation in case of synchronized (blue
+bars) and 0.5 ns delayed (red bars) photons from QD1 and QD2 under pulsed 76MHz excitation.
+11
+
+S10: SINGLE PHOTON EMISSION UNDER CW EXCITATION
+In Figure S11, HBT second-order correlation histograms recorded under cw (660 nm)
+excitation for QD1 and QD2 are shown. The data in Fig. S11 are fit with the function
+g(2)
+HBT(τ) = (1 − g(2)
+HBT(0))exp(−|τ|/τd) convoluted with 80 ps width Gaussian instrumental
+response function, where τ is the delay time between detection events, g(2)
+HBT(0) is the prob-
+ability of two-photon emission events, τd is decay time constant corresponding to the sum
+of the spontaneous emission rate 1/T1 and pump rate G of the source.
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+g(2)
+HBT(t)
+Delay time (ns)
+0
+10
+20
+30
+40
+50
+60
+70
+Raw coincidences
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+g(2)(0) = 0.08±0.01
+g(2)
+HBT(t)
+Delay time (ns)
+g(2)(0) = 0.16±0.01
+0
+10
+20
+30
+40
+50
+60
+70
+Raw coincidences
+a
+b
+QD1
+QD2
+FIG. S11. Second order auto-correlation histograms of a QD1 and b QD2 emission under non-
+resonant (660 nm) cw excitation. Data have been recorded in HBT configuration using an on-chip
+beamsplitter. Presented data are shown as measured with no background subtraction or other
+corrections.
+12
+

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@@ -0,0 +1,1527 @@
+Sparse Coding in a Dual Memory System for Lifelong Learning
+Fahad Sarfraz*1, Elahe Arani*1,2, Bahram Zonooz1,2
+1Advanced Research Lab, NavInfo Europe, The Netherlands
+2Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
+[email protected], [email protected], [email protected]
+Abstract
+Efficient continual learning in humans is enabled by a rich set
+of neurophysiological mechanisms and interactions between
+multiple memory systems. The brain efficiently encodes in-
+formation in non-overlapping sparse codes, which facilitates
+the learning of new associations faster with controlled inter-
+ference with previous associations. To mimic sparse coding
+in DNNs, we enforce activation sparsity along with a dropout
+mechanism which encourages the model to activate simi-
+lar units for semantically similar inputs and have less over-
+lap with activation patterns of semantically dissimilar inputs.
+This provides us with an efficient mechanism for balancing
+the reusability and interference of features, depending on the
+similarity of classes across tasks. Furthermore, we employ
+sparse coding in a multiple-memory replay mechanism. Our
+method maintains an additional long-term semantic memory
+that aggregates and consolidates information encoded in the
+synaptic weights of the working model. Our extensive eval-
+uation and characteristics analysis show that equipped with
+these biologically inspired mechanisms, the model can fur-
+ther mitigate forgetting1.
+1
+Introduction
+The ability to continually acquire, consolidate, and retain
+knowledge is a hallmark of intelligence. Particularly, as we
+look to deploy deep neural networks (DNNs) in the real
+world, it is essential that learning agents continuously inter-
+act and adapt to the ever-changing environment. However,
+standard DNNs are not designed for lifelong learning and
+exhibit catastrophic forgetting of previously learned knowl-
+edge (McCloskey and Cohen 1989) when required to learn
+tasks sequentially from a stream of data (McCloskey and
+Cohen 1989).
+The core challenge in continual learning (CL) in DNNs
+is to maintain an optimal balance between plasticity and
+the stability of the model. Ideally, the model should be sta-
+ble enough to retain previous knowledge while also plastic
+enough to acquire and consolidate new knowledge. Catas-
+trophic forgetting in DNNs can be attributed to the lack of
+stability, and multiple approaches have been proposed to ad-
+dress it. Among them, Rehearsal-based methods, (Riemer
+*These authors contributed equally.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1Code available at https://github.com/NeurAI-Lab/SCoMMER
+et al. 2018; Aljundi et al. 2019b) which aim to reduce for-
+getting by continual rehearsal of previously seen tasks, have
+proven to be an effective approach in challenging CL tasks
+(Farquhar and Gal 2018). They attempt to approximate the
+joint distribution of all the observed tasks by saving samples
+from previous tasks in a memory buffer and intertwine the
+training of the new task with samples from memory. How-
+ever, due to the limited buffer size, it is difficult to approx-
+imate the joint distribution with the samples alone. There
+is an inherent imbalance between the samples of previous
+tasks and the current task. This results in the network update
+being biased towards the current task, leading to forgetting
+and recency bias in predictions. Therefore, more informa-
+tion from the previous state of the model is needed to better
+approximate the joint distribution and constrain the update
+of the model to preserve the learned knowledge. However, it
+is still an open question what the optimal information is for
+replay and how to extract and preserve it.
+The human brain provides an existence proof for success-
+ful CL in complex dynamic environments without intransi-
+gence or forgetting. Therefore, it can provide insight into
+the design principles and mechanisms that can enable CL
+in DNNs. The human brain maintains a delicate balance
+between stability and plasticity through a complex set of
+neurophysiological mechanisms (Parisi et al. 2019; Zenke
+et al. 2017) and the effective use of multiple memory sys-
+tems (Hassabis et al. 2017). In particular, evidence suggests
+that the brain employs Sparse Coding, that the neural code is
+characterized by strong activations of a relatively small set
+of neurons. The efficient utilization of sparsity for informa-
+tion representation enables new associations to be learned
+faster with controlled interference with previous associa-
+tions while maintaining sufficient representation capacity.
+Furthermore, complementary learning systems (CLS) the-
+ory posits that effective learning requires two complemen-
+tary learning systems. The hippocampus rapidly encodes
+episodic information into non-overlapping representations,
+which are then gradually consolidated into the structural
+knowledge representation in the neocortex through the re-
+play of neural activities.
+Inspired by these mechanisms in the brain, we hypothe-
+size that employing a mechanism to encourage sparse cod-
+ing in DNNs and mimic the interplay of multiple memory
+systems can be effective in maintaining a balance between
+arXiv:2301.05058v1  [cs.NE]  28 Dec 2022
+
+Long-term  
+Memory 
+ 
+ 
+ 
+ 
+Working 
+Memory 
+Episodic 
+Memory
+Consolidation
+Data Stream
+Rehearsal
+Layer 
+ c:  1      2       3      4
+Activation Count
+Semantic Dropout
+Knowledge Retrieval
+Activation
+k-WTA
+Figure 1: SCoMMER employs sparse coding in a multi-memory experience replay mechanism. In addition to the instance-based
+episodic memory, we maintain a long-term memory that consolidates the learned knowledge in the working memory throughout
+training. The long-term memory interacts with the episodic memory to enforce consistency in the functional space of working
+memory through the knowledge retrieval loss. To mimic sparse coding in the brain, we enforce activation sparsity along with
+semantic dropout, whereby the model tracks the class-wise activations during training and utilizes them to enforce sparse code,
+which encourages the model to activate similar units for semantically similar inputs. Schematic shows how the activations from
+layer l are propagated to the next layer. Darker shades indicate higher values. Given a sample from class 4, semantic dropout
+retains the units with higher activation counts for the class, and top-k remaining (here 2) units with higher activations are
+propagated to the next layer. This enables the network to form semantically conditioned subnetworks and mitigate forgetting.
+stability and plasticity. To this end, we propose a multi-
+memory experience replay mechanism that employs sparse
+coding, SCoMMER. We enforce activation sparsity along
+with a complementary dropout mechanism, which encour-
+ages the model to activate similar units for semantically sim-
+ilar inputs while reducing the overlap with activation pat-
+terns of semantically dissimilar inputs. The proposed se-
+mantic dropout provides us with an efficient mechanism to
+balance the reusability and interference of features depend-
+ing on the similarity of classes across tasks. Furthermore,
+we maintain additional long-term semantic memory that ag-
+gregates the information encoded in the synaptic weights
+of the working memory. Long-term memory interacts with
+episodic memory to retrieve structural knowledge from pre-
+vious tasks and facilitates information consolidation by en-
+forcing consistency in functional space.
+Our empirical evaluation on challenging CL settings and
+characteristic analysis show that equipping the model with
+these biologically inspired mechanisms can further mitigate
+forgetting and effectively consolidate information across the
+tasks. Furthermore, sparse activations in conjunction with
+semantic dropout in SCoMMER leads to the emergence of
+subnetworks, enables efficient utilization of semantic mem-
+ory, and reduces the bias towards recent tasks.
+2
+Related Work
+The different approaches to address the problem of catas-
+trophic forgetting in CL can be broadly divided into three
+categories: Regularization-based methods regularize the up-
+date of the model in the parameter space (Farajtabar et al.
+2020; Kirkpatrick et al. 2017; Ritter et al. 2018; Zenke et al.
+2017) or the functional space (Rannen et al. 2017; Li and
+Hoiem 2017), Dynamic architecture expands the network
+to dedicate a distinct set of parameters to each task, and
+Rehearsal-based methods (Riemer et al. 2018; Aljundi et al.
+2019b) mitigate forgetting by maintaining an episodic mem-
+ory buffer and continual rehearsal of samples from previous
+tasks. Among these, our method focuses on rehearsal-based
+methods, as it has proven to be an effective approach in
+challenging continual learning scenarios (Farquhar and Gal
+2018). The base method, Experience Replay (ER) (Riemer
+et al. 2018) interleaves the training of the current task with
+the memory sample to train the model on the approximate
+joint distribution of tasks. Several studies focus on the differ-
+ent aspects of rehearsal: memory sample selection (Lopez-
+Paz and Ranzato 2017; Isele and Cosgun 2018), sample re-
+trieval from memory (Aljundi et al. 2019a) and what infor-
+mation to extract and replay from the previous model (Li and
+Hoiem 2017; Ebrahimi et al. 2020; Bhat et al. 2022).
+Dark Experience Replay (DER++) samples the output
+logits along with the samples in the memory buffer through-
+out the training trajectory and applies a consistency loss on
+the update of the model. Recently, CLS theory has inspired
+a number of approaches that utilize multiple memory sys-
+tems (Wang et al. 2022a,b; Pham et al. 2021) and show the
+benefits of multiple systems in CL. CLS-ER (Arani et al.
+2022) mimics the interplay between fast and slow learning
+systems by maintaining two additional semantic memories
+that aggregate the weights of the working model at differ-
+ent timescales using an exponential moving average. Our
+method enforces sparse coding for efficient representation
+and utilization of multiple memories.
+3
+Methodology
+We first provide an overview of motivation from biologi-
+cal systems before formally introducing the different com-
+ponents of the proposed approach.
+
+3.1
+Continual Learning in the Biological System
+Effective CL in the brain is facilitated by a complex set of
+mechanisms and multiple memory systems. Information in
+the brain is represented by neural activation patterns, which
+form a neural code (Foldiak and Endres 2008). Specifically,
+evidence suggests that the brain employs Sparse Coding, in
+which sensory events are represented by strong activations
+of a relatively small set of neurons. A different subset of
+neurons is used for each stimulus (Foldiak 2003; Barth and
+Poulet 2012). There is a correlation between these sparse
+codes (Lehky et al. 2021) that could capture the similar-
+ity between different stimuli. Sparse codes provide several
+advantages: they enable faster learning of new associations
+with controlled interference with previous associations and
+allow efficient maintenance of associative memory while re-
+taining sufficient representational capacity.
+Another salient feature of the brain is the strong differ-
+entiation and specialization of the nervous systems (Had-
+sell et al. 2020). There is evidence for modularity in bio-
+logical systems, which supports functional specialization of
+brain regions (Kelkar and Medaglia 2018) and reduces in-
+terference between different tasks. Furthermore, the brain
+is believed to utilize multiple memory systems (Atkinson
+and Shiffrin 1968; McClelland et al. 1995). Complementary
+learning systems (CLS) theory states that efficient learning
+requires at least two complementary systems. The instance-
+based hippocampal system rapidly encodes new episodic
+events into non-overlapping representations, which are then
+gradually consolidated into the structured knowledge repre-
+sentation in the parametric neocortical system. Consolida-
+tion of information is accompanied by replay of the neural
+activities that accompanied the learning event.
+The encoding of information into efficient sparse codes,
+the modular and dynamic processing of information, and
+the interplay of multiple memory systems might play a cru-
+cial role in enabling effective CL in the brain. Therefore, our
+method aims to incorporate these components in ANNs.
+3.2
+Sparse coding in DNNs
+The sparse neural codes in the brain are in stark contrast
+to the highly dense connections and overlapping representa-
+tions in standard DNNs which are prone to interference. In
+particular, for CL, sparse representations can reduce the in-
+terference between different tasks and therefore result in less
+forgetting, as there will be fewer task-sensitive parameters
+or fewer effective changes to the parameters (Abbasi et al.
+2022; Iyer et al. 2021). Activation sparsity can also lead to
+the natural emergence of modules without explicitly impos-
+ing architectural constraints (Hadsell et al. 2020). Therefore,
+to mimic sparse coding in DNNs, we enforce activation spar-
+sity along with a complementary semantic dropout mecha-
+nism which encourages the model to activate similar units
+for semantically similar samples.
+Sparse Activations:
+To enforce the sparsity in activations,
+we employ the k-winner-take-all (k-WTA) activation func-
+tion (Maass 2000). k-WTA only retains the top-k largest val-
+ues of an N × 1 input vector and sets all the others to zero
+before propagating the vector to the next layer of the net-
+work. Importantly, we deviate from the common implemen-
+tation of k-WTA in convolutional neural networks (CNNs)
+whereby the activation map of a layer (C × H × W ten-
+sor where C is the number of channels and H and W are
+the spatial dimensions) is flattened into a long CHW × 1
+vector input and the k-WTA activation is applied similar
+to the fully connected network (Xiao et al. 2019; Ahmad
+and Scheinkman 2019). We believe that this implementation
+does not take into account the functional integrity of an in-
+dividual convolution filter as an independent feature extrac-
+tor and does not lend itself to the formation of task-specific
+subnetworks with specialized feature extractors. Instead, we
+assign an activation score to each filter in the layer by taking
+the absolute sum of the corresponding activation map and
+select the top-k filters to propagate to the next layer.
+Given the activation map, we flatten the last two dimen-
+sions and assign a score to each filter by taking the absolute
+sum of the activations. Based on the sparsity ratio for each
+layer, the activation maps of the filters with higher scores are
+propagated to the next layers, and the others are set to zero.
+This enforces global sparsity, whereby each stimulus is pro-
+cessed by only a selected set of convolution filters in each
+layer, which can be considered as a subnetwork. We also
+consider each layer’s role when setting the sparsity ratio.
+The earlier layers have a lower sparsity ratio as they learn
+general features, which can enable higher reusability, and
+forward transfer to subsequent tasks use a higher sparsity for
+later layers to reduce the interference between task-specific
+features.
+Semantic Dropout:
+While the k-WTA activation function
+enforces the sparsity of activation for each stimulus, it does
+not encourage semantically similar inputs to have similar ac-
+tivation patterns and reduce overlap with semantically dis-
+similar inputs. To this end, we employ a complementary
+Semantic Dropout mechanism, which controls the degree
+of overlap between neural activations between samples be-
+longing to different tasks while also encouraging the sam-
+ples belonging to the same class to utilize a similar set of
+units. We utilize two sets of activation trackers: global ac-
+tivity counter, Ag ∈ RN, counts the number of times each
+unit has been activated throughout training, whereas class-
+wise activity counter, As ∈ RC×N, tracks the number of
+times each unit has been active for samples belonging to a
+particular class. N and C denote the total number of units
+and classes, respectively. For each subsequent task, we first
+employ Heterogeneous Dropout (Abbasi et al. 2022) to en-
+courage the model to learn the new classes by using neu-
+rons that have been less active for previously seen classes by
+setting the probability of a neuron being dropped to be in-
+versely proportional to its activation counts. Concretely, let
+[Al
+g]j denote the number of times that the unit j in layer l has
+been activated after learning t sequential tasks. For learning
+the new classes in task t+1, the probability of retaining this
+unit is given by:
+[P l
+h]j = exp(
+−[Al
+g]j
+maxi [Alg]i
+πh)
+(1)
+
+Algorithm 1: SCoMMER Algorithm for Sparse Coding in Multiple-Memory Experience Replay System
+Input: data stream D; learning rate η; consistency weight γ; update rate r and decay parameter α, dropout rates πh and πs
+Initialize: θs = θw
+M ←− {}
+1: for Dt ∈ D do
+2:
+while Training do
+3:
+Sample training data: (xt, yt) ∼ Dt and (xm, ym) ∼ M, and interleave x ← (xt, xm)
+4:
+Retrieve structural knowledge: Zs ← f(xm; θs)
+5:
+Evaluate overall loss loss: L = Lce(f(x; θw), y) + γLkr(f(xm; θw), Zs) (Eq. 4)
+6:
+Update working memory: θw ←− θw − η∇θwL
+7:
+Aggregate knowledge: θs ← αθs + (1 − α) θw,
+if r > a ∼ U(0, 1) (Eq. 3)
+8:
+Update episodic memory: M ←− Reservoir(M, (xt, yt))
+9:
+After Eh epochs, update semantic dropout probabilities at the end of each epoch: Ps
+(Eq. 2)
+10:
+Update heterogeneous dropout probabilities: Ph
+(Eq. 1)
+return θs
+where πh controls the strength of dropout with larger val-
+ues leading to less overlap between representations. We then
+allow the network to learn with the new task with hetero-
+geneous dropout in place of a fixed number of epochs, Eh.
+During this period, we let the class-wise activations emerge
+and then employ Semantic Dropout. It encourages the model
+to utilize the same set of units by setting the probability of
+retention of a unit for each class c as proportional to the
+number of times it has been activated for that class so far:
+[P l
+s]c,j = 1 − exp(
+−[Al
+s]c,j
+maxi [Als]c,i
+πs)
+(2)
+where πs controls the strength of dropout. The probabilities
+for semantic dropout are updated at the end of each epoch
+to enforce the emerging pattern. This provides us with an
+efficient mechanism for controlling the degree of overlap
+in representations as well as enabling context-specific pro-
+cessing of information which facilitates the formation of se-
+mantically conditioned subnetworks. Activation sparsity, to-
+gether with semantic dropout, also provides us with an effi-
+cient mechanism for balancing the reusability and interfer-
+ence of features depending on the similarity of classes across
+the tasks.
+3.3
+Multiple Memory Systems
+Inspired by the interaction of multiple memory systems in
+the brain, in addition to a fixed-size instance-based episodic
+memory, our method builds a long-term memory that aggre-
+gates the learned information in the working memory.
+Episodic Memory:
+Information consolidation in the brain
+is facilitated by replaying the neural activation patterns that
+accompanied the learning event. To mimic this mechanism,
+we employ a fixed-size episodic memory buffer, which can
+be thought of as a very primitive hippocampus. The memory
+buffer is maintained with Reservoir Sampling, (Vitter 1985)
+which aims to match the distribution of the data stream by
+assigning an equal probability to each incoming sample.
+Long-Term Memory:
+We aim to build a long-term
+semantic memory that can consolidate and accumulate
+the structural knowledge learned in the working memory
+throughout the training trajectory. The knowledge acquired
+in DNNs resides in the learned synaptic weights (Krishnan
+et al. 2019). Hence, progressively aggregating the weights
+of the working memory (θw) as it sequentially learns tasks
+allows us to consolidate the information efficiently. To this
+end, we build long-term memory (θs) by taking the expo-
+nential moving average of the working memory weights
+in a stochastic manner (which is more biologically plausi-
+ble (Arani et al. 2021)), similar to (Arani et al. 2022):
+θs ← αθs + (1 − α) θw,
+if r > a ∼ U(0, 1)
+(3)
+where α is the decay parameter and r is the update rate.
+Long-term memory builds structural representations for
+generalization and mimics the slow acquisition of struc-
+tured knowledge in the neocortex, which can generalize well
+across tasks. The long-term memory then interacts with the
+instance-level episodic memory to retrieve structural rela-
+tional knowledge (Sarfraz et al. 2021) for the previous tasks
+encoded in the output logits. Consolidated logits are then
+utilized to enforce consistency in the functional space of the
+working model. This facilitates the consolidation of infor-
+mation by encouraging the acquisition of new knowledge
+while maintaining the functional relation of previous knowl-
+edge and aligning the decision boundary of working mem-
+ory with long-term memory.
+3.4
+Overall Formulation
+Given a continuous data stream D containing a sequence of
+tasks (D1, D2, .., DT ), the CL task is to learn the joint dis-
+tribution of all the observed tasks without the availability of
+task labels at test time. Our proposed method, SCoMMER,
+involves training a working memory θw, and maintains an
+additional long-term memory θs and an episodic memory
+M. The long-term memory is initialized with the same pa-
+rameters as the working memory and has the same spar-
+sity constraints. Therefore, long-term memory aggregates
+the weights of working memory. We initialize heterogeneous
+dropout probabilities πh randomly to set the probability of
+retention of a fraction of units to 1 and others to 0 so that the
+first task is learned using a few, but sufficient units and the
+remaining can be utilized to learn subsequent tasks.
+
+Table 1: Comparison on different CL settings. The baseline results for S-CIFAR100 and GCIL are from (Arani et al. 2022).
+Buffer
+Method
+S-CIFAR10
+S-CIFAR100
+GCIL
+Class-IL
+Task-IL
+Class-IL
+Task-IL
+Unif
+Longtail
+–
+JOINT
+92.20±0.15
+98.31±0.12
+70.62±0.64
+86.19±0.43
+58.36±1.02
+56.94±1.56
+SGD
+19.62±0.05
+61.02±3.33
+17.58±0.04
+40.46±0.99
+12.67±0.24
+22.88±0.53
+200
+ER
+44.79±1.86
+91.19±0.94
+21.40±0.22
+61.36±0.39
+16.40±0.37
+19.27±0.77
+DER++
+64.88±1.17
+91.92±0.60
+29.60±1.14
+62.49±0.78
+18.84±0.60
+26.94±1.27
+CLS-ER
+66.19±0.75
+93.90±0.60
+35.23±0.86
+67.34±0.79
+25.06±0.81
+28.54±0.87
+SCoMMER
+69.19±0.61
+93.20±0.10
+40.25±0.05
+69.39±0.43
+30.84±0.80
+29.08±0.31
+500
+ER
+57.74±0.27
+93.61±0.27
+28.02±0.31
+68.23±0.16
+28.21±0.69
+20.30±0.63
+DER++
+72.70±1.36
+93.88±0.50
+41.40±0.96
+70.61±0.11
+32.92±0.74
+25.82±0.83
+CLS-ER
+75.22±0.71
+94.94±0.53
+47.63±0.61
+73.78±0.86
+36.34±0.59
+28.63±0.68
+SCoMMER
+74.97±1.05
+94.36±0.06
+49.63±1.43
+75.49±0.43
+36.87±0.36
+35.20±0.21
+T1
+T2
+T3
+T4
+T5
+After T1
+After T2
+After T3
+After T4
+After T5
+98.0
+88.3
+85.4
+86.2
+38.3
+88.5
+81.5
+29.9
+42.0
+96.8
+47.4
+45.0
+55.5
+70.0
+94.9
+Working Memory
+T1
+T2
+T3
+T4
+T5
+98.6
+92.0
+84.8
+87.7
+57.5
+79.7
+85.5
+49.0
+64.5
+86.7
+70.0
+52.0
+60.8
+79.2
+86.5
+Long-Term Memory
+Figure 2: Task-wise performance of working memory and
+the long-term memory. The long-term memory effectively
+aggregates knowledge encoded in the working memory and
+generalizes well across the tasks.
+During each training step, we interleave the batch of sam-
+ples from the current task xt ∼ Dt, with a random batch of
+exemplars from episodic memory xm ∼ M. Working mem-
+ory is trained with a combination of cross-entropy loss on
+the interleaved batch x ← (xt, xb), and knowledge retrieval
+loss on the exemplars. Thus, the overall loss is given by:
+L = Lce(f(x; θw), y) + γLkr(f(xm; θw), f(xm; θs)) (4)
+where γ controls the strength of the enforcement of con-
+sistency, and mean-squared error loss is used for Lkr. The
+training step is followed by stochastically updating the long-
+term memory (Eq. 3). The semantic dropout and heteroge-
+neous dropout probabilities are updated at the end of each
+epoch and task, respectively (using Eqs. 1 and 3). We use
+long-term memory for inference, as it aggregates knowledge
+and generalizes well across tasks (cf. Figure 2). Agorithm 1
+provides further training details.
+4
+Evaluation Protocol
+To gauge the effectiveness of SCoMMER in tackling the dif-
+ferent challenges faced by a lifelong learning agent, we con-
+sider multiple CL settings that test different aspects of the
+model.
+Class-IL presents a challenging CL scenario where each
+task presents a new set of disjoint classes, and the model
+must learn to distinguish between all the classes seen so
+far without the availability of task labels at the test time.
+It requires the model to effectively consolidate information
+across tasks and learn generalizable features that can be
+reused to acquire new knowledge. Generalized Class-IL
+(GCIL) (Mi et al. 2020) extends the Class-IL setting to more
+realistic scenarios where the agent has to learn an object over
+multiple recurrences spread across tasks and tackle the chal-
+lenges of class imbalance and a varying number of classes
+in each task. GCIL utilizes probabilistic modeling to sam-
+ple the number of classes, the appearing classes, and their
+sample sizes. Details of the datasets used in each setting are
+provided in the Appendix. Though our method does not uti-
+lize separate classification heads or subnets, for completion,
+we also evaluate the performance under the Task-IL setting,
+where the model has access to the task labels at inference.
+In this setting, we use the task label to select the subset of
+output logits to select from.
+5
+Empirical Evaluation
+We compare SCoMMER with state-of-the-art rehearsal-
+based methods across different CL settings under uniform
+experimental settings (details provided in Appendix). SGD
+provides the lower bound with standard training on sequen-
+tial tasks, and JOINT gives the upper bound on performance
+when the model is trained on the joint distribution.
+Table 1 shows that SCoMMER provides performance
+gains in the majority of the cases and demonstrates the ef-
+fectiveness of our approach under varying challenging CL
+settings. In particular, it provides considerable improve-
+ment under low buffer size settings, which suggests that our
+method is able to mitigate forgetting with fewer samples
+from previous tasks. The performance gains over CLS-ER,
+which employs two semantic memories, show that sparse
+coding in our method enables the effective utilization of a
+single semantic memory. In particular, the gains in the GCIL
+setting, where the agent has to face the challenges of class
+imbalance and learn over multiple occurrences of objects,
+alludes to several advantages of our method. Our proposed
+semantic dropout in conjunction with sparse activations en-
+ables the model to reuse the sparse code associated with the
+
+T1
+T2
+T3
+T4
+T5
+After T1
+After T2
+After T3
+After T4
+After T5
+98.8
+67.0
+92.1
+54.0
+16.9
+95.8
+55.9
+13.1
+22.9
+98.7
+15.8
+15.1
+36.0
+73.8
+98.2
+ER
+T1
+T2
+T3
+T4
+T5
+98.2
+89.2
+87.3
+82.0
+50.0
+90.0
+79.3
+33.4
+63.0
+94.8
+58.4
+29.5
+67.5
+81.1
+95.7
+DER++
+T1
+T2
+T3
+T4
+T5
+98.7
+89.0
+89.5
+78.2
+53.5
+89.0
+81.2
+42.4
+76.3
+87.5
+69.2
+41.5
+76.8
+83.3
+41.1
+CLS-ER
+T1
+T2
+T3
+T4
+T5
+98.6
+92.0
+84.8
+87.7
+57.5
+79.7
+85.5
+49.0
+64.5
+86.7
+70.0
+52.0
+60.8
+79.2
+86.5
+SCoMMER
+Figure 3: Task-wise performance of different methods. The heatmaps provide the test set of each task (x-axis) evaluated at the
+end of each sequential learning task (y-axis). SCoMMER retains the performance of earlier tasks better without compromising
+on the current task.
+Table 2: Ablation Study: Effect of systematically removing
+different components of SCoMMER on the performance of
+the models on S-CIFAR10. All components contribute to the
+performance gain.
+Sparse
+Long-Term
+Semantic
+Accuracy
+Activations
+Memory
+Dropout
+
+
+
+69.19±0.61
+
+
+
+67.38±1.51
+
+
+
+61.88±2.43
+
+
+
+49.44±5.43
+
+
+
+44.79±1.86
+recurring object and learn better representations with the ad-
+ditional samples by adapting the corresponding subset of fil-
+ters. Furthermore, compared to the dense activations in CLS-
+ER, the sparse coding in SCoMMER leads to the emergence
+of subnetworks that provide modularity and protection to
+other parts of the network since the entire network is not
+updated for each input image. This increases the robustness
+of the model to the class imbalance.
+Overall, our method provides an effective approach to em-
+ploy sparse coding in DNN and enables better utilization of
+long-term memory, which can effectively consolidate infor-
+mation across tasks and further mitigate forgetting.
+6
+Ablation Study
+To gain further insight into the contribution of each com-
+ponent of our method, we systematically remove them and
+evaluate the performance of the model in Table 2. The results
+show that all components of SCoMMER contribute to the
+performance gains. The drop in performance from remov-
+ing semantic dropout suggests that it is effective in enforc-
+ing sparse coding on the representations of the model, which
+reduces the interference between tasks and allows semanti-
+cally similar classes to share information. We also observe
+the benefits of multiple memory systems in CL. Additional
+long-term memory provides considerable performance im-
+provement and suggests that the EMA of the learned synap-
+tic weights can effectively consolidate knowledge across
+tasks. Furthermore, we observe that sparsity is a critical
+component for enabling CL in DNNs. Sparse activations
+alone significantly improve ER performance and also en-
+able efficient utilization of semantic memory. We highlight
+that these individual components complement each other
+and that the combined effect leads to the observed perfor-
+mance improvement in our method.
+7
+Characteristics Analysis
+We look at different characteristics of the model to under-
+stand what enables the performance gains in our method.
+We analyze the models trained on S-CIFAR100 with a buffer
+size of 200.
+7.1
+Stability-Plasticity Dilemma
+To better understand how well different methods maintain
+a balance between stability and plasticity, we look at how
+task-wise performance evolves as the model learns tasks se-
+quentially. The diagonal of the heatmap shows the plastic-
+ity of the model as it learns the new task, whereas the dif-
+ference between the accuracy of the task when it was first
+learned and at the end of the training indicates the stabil-
+ity of the model. Figure 3 shows that SCoMMER is able to
+maintain a better balance and provides a more uniform per-
+formance on tasks compared to baselines. While CLS-ER
+provides better stability than DER++, it comes at the cost of
+the model’s performance on the last task, which could be due
+to the lower update rate of the stable model. SCoMMER, on
+the other hand, retains performance on the earlier tasks (T1
+and T2) and provides good performance on the recent task.
+We also compare the long-term semantic and working mem-
+ory performance in Figure 2. Long-term memory effectively
+aggregates the learned knowledge into the synaptic weights
+of working memory and generalizes well across tasks.
+7.2
+Emergence of Subnetworks
+To evaluate the effectiveness of activation sparsity and se-
+mantic dropout in enforcing sparse coding in the model, we
+look at the average activity of the units in the penultimate
+layer. The emerging sparse code for each class is tracked
+during training using the class-wise activity counter and en-
+forced using semantic dropout probabilities (Equation 2).
+
+Figure 4: Class-wise activation counts of the filters in the penultimate layer of the model trained on S-CIFAR10 with 200 buffer
+size. Comparison of the activation counts on the test set with the learned class-wise probabilities, Ps, during training shows the
+effectiveness of semantic dropout in enforcing sparse coding. Right plot shows the cosine similarities between the activation
+counts of different classes. Semantically similar classes have higher correlation in activations. Darker color shows higher values.
+Given a test sample from class c, ideally, we would want
+the model to use the subset of neurons that had higher activ-
+ity for the training samples from class c without providing
+any task information. Concretely, we track the class-wise
+activity on the test set and plot the normalized activation
+counts for a set of neurons next to their class-wise proba-
+bilities at the end of training. Figure 4 shows a high correla-
+tion between the test set activation counts and the semantic
+dropout probabilities at the end of training, particularly for
+recent classes. The activation counts also hint at the natu-
+ral emergence of semantically conditioned subnets, as the
+model utilizes a different set of units for different classes.
+Furthermore, we observe that semantically similar classes
+have a higher degree of correlation between their activation
+patterns. For instance, cat and dog share the most active neu-
+rons, a similar pattern is observed between horse and deer,
+and car and truck. The cosine similarities between the ac-
+tivation counts of the different classes further supports the
+observation. This is even more remarkable given that these
+classes are observed in different tasks, particularly for cars
+and trucks, which are observed in the first and last tasks.
+7.3
+Task Recency Bias
+A major challenge in CL is the recency bias, in which the
+update of the model on new task samples biases its predic-
+tions toward the current task (Wu et al. 2019). This leads to
+considerable forgetting of earlier tasks. To compare the de-
+gree to which SCoMMER tackles this issue, we evaluate the
+probabilities of predicting each task by aggregating the soft-
+max output of samples from the test set of all seen tasks and
+averaging the probabilities of classes in each task. Figure
+5 shows that SCoMMER provides more uniform probabili-
+ties to predict each task. CLS-ER is able to mitigate the bias
+towards the last task, which can be attributed to the aggrega-
+tion of knowledge in the semantic memories; however, CLS-
+ER reduces the probability of predicting the last task, which
+explains the low performance. SCoMMER effectively mit-
+igates recency bias and provides uniform prediction proba-
+ER
+DER++
+CLS-ER
+SCoMMER
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Task Probability
+Task 1
+Task 2
+Task 3
+Task 4
+Task 5
+Figure 5: Average probabilities of predicting classes from
+each tasks at the end of training. SCoMMER provides more
+uniform probabilities across the tasks.
+bilities across tasks without any explicit regularization.
+8
+Conclusion
+Motivated by the mechanisms for information representation
+and utilization of multiple memory systems in the brain, we
+proposed a novel approach to employ sparse coding in mul-
+tiple memory systems. SCoMMER enforces activation spar-
+sity along with a complementary semantic dropout mecha-
+nism, which encourages the model to activate similar units
+for semantically similar inputs and reduce the overlap with
+dissimilar inputs. Additionally, it maintains long-term mem-
+ory, which consolidates the learned knowledge in working
+memory. Our empirical evaluation shows the effectiveness
+of the approach in mitigating forgetting in challenging CL
+scenarios. Furthermore, sparse coding enables efficient con-
+solidation of knowledge in the long-term memory, reduces
+the bias towards recent tasks, and leads to the emergence
+of semantically conditioned subnetworks. We hope that our
+study inspires further research in this promising direction.
+
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+
+A Appendix
+B
+Experimental Setting
+For a fair comparison with different CL methods in uni-
+form experimental settings, we extended the Mammoth
+framework (Buzzega et al. 2020). To disentangle the per-
+formance improvement of the algorithm from the training
+regimes (Mirzadeh et al. 2020), we use the same network
+(ResNet-18), optimizer (SGD), batch size for task data and
+memory buffer (32), data augmentations (random crop and
+random horizontal flip), and the number of epochs (50) for
+all our experiments.
+For hyperparameter tuning, we use a small held-out val-
+idation set and perform a grip search on activation sparsity,
+γ, dropout strengths, πh and πs, and the update frequency
+for long-term memory r. Table S1 provides the selected hy-
+perparameters for each setting. Note that our method does
+not require an extensive hyperparameter search for differ-
+ent buffer sizes, and sensitivity to hyperparameters section
+shows that the different parameters are complementary in
+nature and the model performs well for a number of differ-
+ent combinations. Therefore, majority of parameters can be
+fixed, which reduces the search space of hyperparameters
+significantly. We report the average and one standard devia-
+tion of three different seeds.
+C
+Continual Learning Datasets
+We consider the Class-IL and Generalized Class-IL setting
+for our empirical evaluation to extensively assess the ver-
+satility of our approach. Here, we provide details of the
+datasets used in each of the settings.
+C.1
+Class Incremental Learning (Class-IL)
+Class-IL (van de Ven and Tolias 2019) requires the agent
+to learn a new disjoint set of classes with each task, and
+the agent has to distinguish between all the classes seen so
+far without the availability of task labels at the test time.
+We consider the split variants of the benchmark datasets S-
+CIFAR10 and S-CIFAR100 where the classes are split into
+5 tasks with 2 and 20 classes each, respectively. The order
+of the classes in the experiments remains fixed, whereby for
+CIFAR10 the first task includes the first two classes, and so
+forth.
+C.2
+Generalized Class Incremental Learning
+(GCIL)
+GCIL (Mi et al. 2020) extends the Class-IL setting to more
+realistic scenarios. In addition to avoiding forgetting, the
+model has to tackle the challenges of class imbalance, learn-
+ing an object over multiple recurrences. GCIL utilizes prob-
+abilistic modeling to sample three characteristics of a task:
+the number of classes, the classes that appear, and their sam-
+ple sizes. Similarly to (Arani et al. 2022), we consider GCIL
+on the CIFAR100 dataset with 20 tasks, each with 1000 sam-
+ples, and the maximum number of classes in a single task set
+to 50. To disentangle the effect of class imbalance from
+the ability of the model to learn from recurring classes un-
+der non-uniform task lengths, we evaluate the model on uni-
+form (Unif) and longtail data distributions. we set the GCIL
+dataset seed to 1993 for all the experiments.
+D
+Implementation Details
+Here, we provide more details on the implementation of
+k-WTA activation for CNNs and the proposed semantic
+dropout mechanism.
+E
+k-WTA for Convolutional Neural
+Networks
+The common implementation of k-WTA in convolutional
+neural networks involves flattening the activation map into a
+long CHW ×1 vector and applying the activation of k-WTA
+in a way similar to that of the fully connected network (Xiao
+et al. 2019; Ahmad and Scheinkman 2019). This translates to
+setting some spatial dimensions of a filter to zero while prop-
+agating others. However, this implementation does not take
+into account the functional integrity of an individual con-
+volution filter as an independent feature extractor and does
+not enable the formation of task-specific subnetworks with
+specialized feature extractors. Different tasks cannot utilize
+a different subset of filters, and we cannot track the activity
+of an individual filter.
+Our implementation, on the other hand, assigns an activa-
+tion score to each filter in the layer by taking the absolute
+sum of the corresponding activation map. Given the activa-
+tion map of the layer l, Al, (C × W × H) where C is the
+number of filters, W and H are the width and height, we
+flatten the spatial dimensions, C × WH), and the activation
+score for each filter j is given by the absolute sum of its ac-
+tivations, [Cscore]j = �HW
+i=1 |[Al]j,i|. We then find the value
+k for the layer using the activation sparsity (% of the active
+filters in the layer), k ← %k × N l
+filters where N l
+filters is
+the number of filters in the layer l. The kth highest value of
+the filter activation scores vector, Cscore ∈ RC×1 gives the
+threshold value used to apply a mask to the input activation
+map, which only propagates the activations of filters with a
+score above threshold by setting the others to zero. Finally,
+the ReLU activation function is applied to the masked acti-
+vations. Algorithm 2 provides more details.
+For the ResNet-18 network in our method, we set the ac-
+tivation sparsity for each ResNet block, for example % k =
+[0.9, 0.9, 0.9, 0.8] enforces the activation sparsity of 0.9 in
+the first three ResNet blocks, that is 90% of the filters in each
+convolutional layer are active for a given stimulus and 80%
+in the convolutional layers of the last ResNet block.
+
+Table S1: Selected parameters for SCoMMER. For each of our experiments, we apply Heterogeneous and Semantic dropout
+only on the output of the last residual block in ResNet-18, the decay parameter for long-term memory is set to 0.999, the batch
+size of 32 is used for both the current task and the memory buffer, and the models are train for 50 epochs. For the first three
+ResNet blocks, we use an activation sparsity of 0.9 and vary the sparsity ratio for the last block.
+Dataset
+Buffer
+size
+Activation
+Sparsity
+η
+πh
+πs
+γ
+r
+S-CIFAR10
+200
+0.8
+0.1
+0.5
+2.0
+0.15
+0.5
+500
+0.8
+0.1
+0.5
+2.0
+0.15
+0.7
+S-CIFAR100
+200
+0.9
+0.1
+0.5
+3.0
+0.15
+0.1
+500
+0.9
+0.1
+0.5
+3.0
+0.15
+0.1
+GCIL - Unif
+200
+0.9
+0.05
+0.5
+3.0
+0.2
+0.6
+500
+0.9
+0.05
+0.5
+3.0
+0.2
+0.6
+GCIL - Longtail
+200
+0.9
+0.05
+0.5
+2.0
+0.2
+0.5
+500
+0.9
+0.05
+0.5
+3.0
+0.2
+0.6
+Algorithm 2: Global k-WTA for CNNs
+Input: Activation map A; activation ratio %k; number
+of filters Nfilters
+Evaluate activation scores:
+1: Flatten the spatial dimensions:
+2: Cscore ← Reshape(Cscore, C × HW)
+3: Assign score to each filter:
+4: Cscore = abs sum(Cscore, dim=1)
+Calculate threshold:
+5: Get k value for the layer:
+6: k ← %k × Nfilters
+7: Return kth largest value
+8: Cthresh = kth value(Cscore, k)
+Mask Activation Map:
+9: Initialize mask with zeros
+10: M ← Zeros(C × H × W)
+11: Set filter mask with score above threshold to 1
+12: M[Cscore > Cthresh] = 1
+13: Apply mask
+14: A ← M · A
+Apply ReLU activation function:
+15: A ← ReLU(A)
+return A
+E.1
+Semantic Dropout
+At the beginning of training, we initialize the heterogeneous
+dropout probabilities Ph so that for each layer l the prob-
+ability of (1.1 × %kl × N l
+filters) filters is set to 1 and the
+remaining set to 0. This is done to ensure that the learning
+of the first task does not utilize all filters while having the
+flexibility to learn a different subset of units for the classes
+in the first task. The semantic dropout probabilities Ps are
+updated at the end of each epoch, once the epoch num-
+ber e for the task is higher than the heterogeneous dropout
+warm-up period Eh to allow the emergence of class-wise ac-
+tivity patterns before it is explicitly enforced with seman-
+tic dropout. Note that to ensure that we have enough ac-
+tive filters before applying k-WTA activation, when apply-
+ing heterogeneous, we use the probabilities Ph to sample the
+(1.1 × %kl × N l
+filters) filters for the given layer before ap-
+plying k-WTA activation. The 1.1 factor is arbitrarily chosen
+and works well in practice; however, a different value can be
+selected. Further details of the method are provided in Algo-
+rithm 3.
+Importantly, we disable the dropout activation counter up-
+date for the buffer samples so that the sparse code is learned
+during task training. Also, dropout is applied only to work-
+ing memory as it is learned with gradient decent, whereas
+the long-term memory aggregates the weights of working
+memory. Our analysis shows that the learned sparse coding
+is effectively transferred to long-term memory through ema.
+For the ResNet-18 model used in our experiments, we ap-
+ply dropout at the output of each ResNet block. Although
+our method provides the flexibility to apply different dropout
+strengths for each block, we observe empirically that it
+works better if applied only at the output of the last ResNet
+block. This allows the model to learn features in the earlier
+layers that can generalize well across the tasks and to learn
+specialized features for the classes in later layers.
+F
+Performance of working memory
+To gain a better understanding of the performance of the
+different memories, Table S2 provides the performance of
+both working memory and long-term memory in the settings
+considered. Long-term memory consistently provides better
+generalization across tasks, especially in the Class-IL set-
+ting. This shows the benefits of using multiple memory sys-
+tems in CL. Furthermore, it demonstrates the effectiveness
+of the exponential moving average of the working memory
+weights as an efficient approach for aggregating the learned
+knowledge.
+
+Algorithm 3: Semantic Dropout
+Input: Activation map A; class labels y; activation ratio %k; number of filters Nfilters; dropout probabilities Ph and Ps
+Get the Heterogeneous Dropout Mask:
+1: Initialize Heterogeneous dropout mask with zeros
+2: Hmask ← Zeros(C × H × W)
+3: Calculate the sampling probabilities so that they sum to zero
+Psample = Ph / sum(Ph)
+4: Get the indices of retained filters
+Nretain = 1.1 × %k × Nfilters
+idx = Sample(range=Nfilters, #samples=Nretain, prob=Psample, replace=False)
+5: Set the mask at retained indices to 1
+Hmask[idx] = 1
+Get the Heterogeneous Dropout Mask:
+6: Initialize Semantic dropout mask with zeros
+7: Smask ← Zeros(C × H × W)
+8: Use the semantic dropout probabilities to select units
+retain = N ∼ U(0, 1) ≤ Ps
+9: Set the mask at retained indices to 1
+Smask[retain] = 1
+Select the mask for each input sample
+10: For each sample, select Semantic dropout mask if available for the class label, otherwise use Heterogeneous dropout mask:
+11: M = Smask if Ps[y] > 0, otherwise Hmask
+Mask Activation Map:
+12: A ← M · A
+return A
+G
+Sensitivity to Hyperparameters
+SCoMMER employs sparse coding in a multiple-memory
+replay mechanism. Therefore, the setting of two sets of pa-
+rameters is required: sparse coding (activation sparsity %k
+and dropout strength πs and πh) and the aggregation of in-
+formation in long-term memory (r, α). We show the effect
+of different sets of hyperparameters in Table S3. We can
+see that the different components are complementary in na-
+ture and therefore different combinations of parameters can
+provide similar performance. Interestingly, we observe that
+increasing the semantic dropout strength considerably in-
+creases the performance of the working model, but the long-
+term memory performance remains quite stable. The method
+is not highly sensitive to a particular set of parameters, and
+often we can fix the majority of parameters and fine-tune
+only a few, which significantly reduces the search space.
+
+Table S2: Performance of working memory and long-term memory in different settings. Long-term memory consistently pro-
+vides better performance.
+Buffer
+Memory
+S-CIFAR10
+S-CIFAR100
+GCIL
+Class-IL
+Task-IL
+Class-IL
+Task-IL
+Unif
+Longtail
+200
+Working
+58.03±5.17
+92.58±0.56
+30.07±0.71
+67.18±0.16
+27.64±0.30
+27.06±0.97
+Long-Term
+69.19±0.61
+93.20±0.10
+40.25±0.05
+69.39±0.43
+30.84±0.80
+29.08±0.31
+500
+Working
+66.10±3.60
+93.59±0.09
+41.36±1.07
+73.52±0.37
+34.34±0.88
+33.39±0.74
+Long-Term
+74.97±1.05
+94.36±0.06
+49.63±1.43
+75.49±0.43
+36.87±0.36
+35.20±0.21
+Table S3: Sensitivity to different hyperparameters. We pro-
+vide the performance of Working memory and Long-term
+memory of models trained on S-CIFAR-10 with 200 buffer
+size. For all experiments γ = 0.15, lr = 0.1, decay parameter
+= 0.999, πh = 0.5, and the model is trained for 50 epochs.
+For the first three ResNet blocks, we use an activation spar-
+sity of 0.9 and vary the sparsity ratio for the last block (%k)
+r
+%k
+πs
+Memory
+Working
+Long-Term
+0.4
+0.7
+1.0
+56.65
+69.58
+2.0
+59.46
+68.30
+3.0
+59.89
+68.93
+0.8
+1.0
+50.25
+67.19
+2.0
+58.01
+69.89
+3.0
+56.91
+68.72
+0.9
+1.0
+51.26
+67.49
+2.0
+56.58
+68.32
+3.0
+56.87
+66.89
+0.5
+0.7
+1.0
+57.01
+66.80
+2.0
+59.61
+69.26
+3.0
+60.51
+69.00
+0.8
+1.0
+49.09
+67.36
+2.0
+58.03
+69.19
+3.0
+60.37
+67.99
+0.9
+1.0
+49.38
+66.27
+2.0
+60.47
+68.16
+3.0
+57.64
+67.88
+0.6
+0.7
+1.0
+56.91
+67.85
+2.0
+61.2
+67.64
+3.0
+62.44
+67.94
+0.8
+1.0
+51.11
+65.97
+2.0
+58.61
+66.55
+3.0
+61.01
+69.36
+0.9
+1.0
+49.26
+66.93
+2.0
+58.35
+67.44
+3.0
+60.18
+67.90
+

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+IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+1
+Small Moving Object Detection Algorithm Based
+on Motion Information
+Ziwei Sun, Zexi Hua, and Hengcao Li, Fellow, IEEE
+Abstract—A Samll Moving Object Detection algorithm Based
+on Motion Information (SMOD-BMI) was proposed to detect
+small moving objects with low Signal-to-Noise Ratio (SNR).
+Firstly, To capture suspicious moving objects, a ConvLSTM-
+SCM-PAN model structure was designed, in which the Convo-
+lutional Long and Short Time Memory (ConvLSTM) network
+fused temporal and spatial information, the Selective Concatenate
+Module (SCM) was selected to solve the problem of channel
+unbalance during feature fusion, and the Path Aggregation
+Network (PAN) located the suspicious moving objects. Then, an
+object tracking algorithm is used to track suspicious moving
+objects and calculate their Motion Range (MR). At the same time,
+according to the moving speed of the suspicious moving objects,
+the size of their MR is adjusted adaptively (To be specific, if the
+objects move slowly, we expand their MR according their speed
+to ensure the contextual environment information) to obtain their
+Adaptive Candidate Motion Range (ACMR), so as to ensure that
+the SNR of the moving object is improved while the necessary
+context information is retained adaptively. Finally, a LightWeight
+SCM U-Shape Net (LW-SCM-USN) based on ACMR with a
+SCM module is designed to classify and locate small moving
+objects accurately and quickly. In this paper, the moving bird in
+surveillance video is used as the experimental dataset to verify
+the performance of the algorithm. The experimental results show
+that the proposed small moving object detection method based
+on motion information can effectively reduce the missing rate
+and false detection rate, and its performance is better than the
+existing moving small object detection method of SOTA.
+Index Terms—Object Detection; Small Moving Objects; Mo-
+tion Information; Motion Range; Low Signal-to-Noise Ratio
+I. INTRODUCTION
+T
+HE intelligent video analysis technology can reduce the
+work intensity of the monitoring center staff and reduce
+the false positives and missing positives caused by manual
+monitoring. And moving object detection is one of the basic
+tasks of intelligent video analysis technology [1], [2]. Through
+moving object detection technology, information such as the
+category, location, size and motion speed of moving objects
+can be obtained, which can provide basic data support for in-
+telligent video analysis technology such as behavior prediction
+and trajectory tracking of moving objects in the next step.
+For the detection of small moving objects, there are two
+main challenges.
+• The object has a low SNR. For the general unattended
+monitoring scene, the monitoring area is usually a room
+or an outdoor area. If a mouse or bird intrudes into the
+Manuscript received January 4, 2023.
+Ziwei Sun, Zexi Hua and Hengcao Li are with the School of Information
+Science and Technology, Southwest JiaoTong University, chengdu 611756,
+China.
+monitoring area, the number of pixels is usually small,
+as shown by Bird A in Fig. 1.
+• The moving object may blur. Since most of the low-cost
+surveillance cameras do not have the ability of low-delay
+photography, the moving object captured has a certain
+trailing phenomenon, which may lead the moving object
+blur, as shown by Bird B in Fig. 1.
+Fig. 1: Small and blurred moving birds in the surveillance
+area. The Bird A is small but clear, the Bird B is small and
+blur.
+To solve the above problems, researchers mainly use the
+motion information (spatio-temporal features). Of course, like
+other vision tasks, the detection method of moving small
+objects has also experienced the development from traditional
+methods based on knowledge-driven to deep learning methods
+based on data-driven.
+At present, the knowledge-driven moving object detection
+algorithms mainly include frame difference method [3], back-
+ground difference method [4], robust principal component
+Analysis method [5] and optical flow method [6]. In the early
+stage, the frame difference method, background difference
+method, and robust principal component analysis method
+were only suitable for the situation that the background was
+static and there was no more complex interference (such as
+illumination change, branches and leaves swaggling, water
+waves and so on). The optical flow method was suitable for the
+situation of moving background, but it still could not overcome
+some interference such as illumination change, the object stop
+or slow motion. However, through the continuous efforts of
+researchers, the traditional methods can accurately extract the
+moving object to a certain extent [7], [8], [9]. However, the
+traditional methods can only extract the pixels of the moving
+object at most, can not obtain other attributes of the moving
+object, and can not distinguish the interesting and uninteresting
+moving objects.
+arXiv:2301.01917v1  [cs.CV]  5 Jan 2023
+
+Bird B
+oBirdAIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+2
+In the early stage, methods based on deep learning were
+mainly combined with traditional methods and object detection
+methods: traditional methods such as frame difference method,
+background difference method, principal component analysis
+method, and optical flow method were combined with object
+detection methods, in which the traditional method provided
+time-related motion information, and the object detection
+provided space-related positioning information [10], [11], [12],
+[13]. These traditional methods with object detection have
+made considerable progress in detecting moving objects. How-
+ever, the detection performance of these methods is affected
+by the motion information provided by the traditional methods
+to a certain extent. At present, some researchers gradually pay
+attention to the full deep learning to obtain the temporal infor-
+mation and spatial information of moving objects at the same
+time, such as using ConvLSTM(Convolution Long Short Term
+Memory) for moving object detection [14], [15]. Or moving
+object detection with input of consecutive multiple frames
+merged [16], [17]. The method based on deep learning has
+a certain improvement in effect and function compared with
+the traditional method. It can distinguish between interested
+and uninterested moving objects, and can obtain the category
+and location of moving objects. However, there are still many
+false detections and missed detections when detect the small
+moving objects. We further analyze and find that most of the
+missed detections occur because the object is small or similar
+to the environment, and most of the false detections that occur
+are caused by various tiny moving things or things whose
+appearance is similar to the object of interest. Therefore, the
+main reason for this problem is that most moving objects
+account for a small proportion of pixels in the whole video
+frame, and the problem of low SNR (unbalanced positive and
+negative samples) is not easy to be eliminated in the training
+process.
+In order to solve the above problems, this paper analyzes
+our human method of small moving object recognition in
+complex environments. Our human approach to identifying
+small moving objects is divided into two stages. In the first
+stage, we will find out where the object may exist according
+to the motion information. In the second stage, we will focus
+on the area where the object may exist and carefully observe,
+so as to remove more interference information. Therefore, we
+propose a Small Moving Object Detection algorithm Based
+on Motion Information (SMOD-BMI). Firstly, a moving ob-
+ject detection model ConvLSTM-SCM-PAN (coarse-detection
+model) is designed to fuse spatio-temporal information, which
+can capture suspicious moving objects according to the mo-
+tion information of moving objects. Then, the Motion Range
+(MR) of suspicious moving objects is extracted by using the
+object tracking algorithm. At the same time, according to
+the moving speed of the suspicious moving objects, the size
+of their MR is adjusted adaptively (To be specific, if the
+objects move slowly, we expand their MR according their
+speed to ensure the contextual environment information) to
+obtain their Adaptive Candidate Motion Range (ACMR), so
+as to ensure that the SNR of the moving object is improved
+while the necessary context information is retained adaptively.
+Finally, a lightweight moving object detection model LW-
+SCM-USN (Fine detection model) based on the ACMR of the
+moving object is designed to accurately classify and locate the
+moving object on the basis of ensuring real-time. The main
+contributions of this paper are as follows.
+• The ConvLSTM-SCM-PAN model structure is designed
+to capture the suspicious moving objects. Among them,
+Convolution Long Short-Term Memory Network (ConvL-
+STM) fuses spatio-temporal information, Selective Con-
+catenation Module (SCM) to solve the problem of chan-
+nel imbalance during feature fusion, and PAN locates
+suspicious moving objects.
+• An adaptive method of extracting ACMR based on the
+amount of motion of the moving object is proposed. By
+using the object tracking technology and the amount of
+motion of the moving object, the ACMR of the suspected
+moving object are extracted adaptively, which improves
+the SNR of the moving object and retains the necessary
+context information of the moving object.
+• A LightWeight U-Shaped Network with SCM module
+(LW-SCM-USN) model structure is designed, and the
+accurate classification and location of moving objects are
+realized by using the ACMR of suspected objects.
+The remainder of this paper is structured as follows: Section
+II is a survey of related work on moving object detection.
+Section III describes the proposed SMOD-BMI in detail.
+Section IV is devoted to ablation experiments and comparison
+experiments of the proposed algorithm. Section V concludes
+our work.
+II. RELATED WORK
+According to the use of different characteristics of the
+object, the methods of moving object detection can be mainly
+divided into three categories: methods based on appearance
+information, methods based on motion information and meth-
+ods based on deep learning for moving object detection. In
+this section we will review these three categories.
+A. Appearance-based Object Detection
+From traditional methods [18], [19], [20] to deep learning-
+based methods [21], [22], [23], [24], [25], [26], [27], [28],
+object detection technology has now made great progress,
+which can accurately determine the specific class of each
+object and give the bounding box of each object. However,
+since these object detection algorithms only rely on the
+appearance features of the object, the detection effect is not
+good for small moving objects with complex backgrounds and
+unobvious appearance features [11], [29].
+B. Moving Object Detection based on Motion Information
+Since the object detection algorithm based on appearance
+feature can not detect small moving objects in complex
+background well, researchers have proposed various moving
+object detection algorithms based on motion information. The
+main methods are frame difference, background subtraction,
+optical flow and robust principal component analysis.
+
+IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+3
+1) Frame Difference Method: Because the object is mov-
+ing, there is a certain displacement between the position of
+the historical frame and the position of the current frame. The
+changed pixel, which is the pixel of the moving object, can be
+extracted by subtracting the historical frame from the current
+frame. When the simple frame difference method obtains the
+moving object, it is easy to appear the hole or ghost phe-
+nomenon [30]. Therefore, researchers have proposed various
+complex frame difference methods to solve this problem [31],
+[32], which have achieved certain effect improvement, but the
+problem cannot be completely solved.
+2) Background Subtraction Method: The environment and
+moving object are regarded as background and foreground.
+The background remains static, while the moving object moves
+in front of the background as the foreground. The key point
+of this method is the background modeling. There are many
+methods of background modeling, which are widely used at
+present, such as multi-frame average background modeling,
+simple Gaussian modeling [33], Gaussian mixture modeling
+[34], ViBe algorithm [35], etc., although the modeling effect
+of these background modeling methods is getting better and
+better. However, it still cannot completely overcome various
+disturbances such as wind and water waves, resulting in more
+interference in the extracted foreground.
+3) Optical Flow Method: The moving object detection
+method based on optical flow method distinguishes the back-
+ground and moving object by using optical flow field according
+to the feature that the brightness of adjacent points in the image
+is similar [6]. The key technology of optical flow method
+is to solve the estimation of optical flow. At present, the
+main optical flow estimation algorithms include correlation
+method, energy method, discrete optimization method and
+phase method [6]. The optical flow method does not need
+prior information to detect moving objects and can be used in
+dynamic background. However, the calculation of optical flow
+field distribution is difficult due to the change of light source,
+shadow and occlusion.
+4) Robust Principal Component Analysis (RPCA): The
+background is considered as a low-rank matrix and the moving
+objects are sparse. Therefore, this method converts the detec-
+tion of moving objects into low-rank sparse decomposition
+of the matrix composed of multiple frames, so as to obtain
+sparse moving objects [36]. Since the original robust principal
+component analysis method is time-consuming, subsequent
+researchers have proposed some improved schemes, such as
+Faster RPCA [37], which greatly improves the decomposition
+speed. However, when the background moves or the back-
+ground changes complex, the background matrix loses its low
+rank property, and it is difficult to decompose the moving
+object at this time. Therefore, the robust principal component
+analysis method is mainly suitable for the situation that the
+background is static or the background changes simple.
+C. Moving Object Detection with Deep Learning
+In recent years, influenced by the great progress of deep
+learning technology in vision tasks, researchers have begun
+to use deep learning technology to detect moving objects.
+Researchers have used deep learning techniques in two differ-
+ent ways to investigate how to detect moving objects, but all
+related studies follow the same basic rule, that is, you need
+to consider both time-based motion information and space-
+based position information. The difference between these two
+methods lies in how to obtain time-based motion information.
+One way is to obtain the motion information by using the
+traditional moving object detection method, which is called
+the traditional plus deep learning method, and the other way
+is to obtain the motion information directly by using deep
+learning, which is called the full deep learning method.
+1) Traditional plus Deep Learning Method: The traditional
+and deep learning moving object detection methods are sum-
+marized into two categories. 1) Firstly, the motion information
+is used to extract the foreground, and then the foreground is
+used for moving object detection [10], [11], [12], [13]. For
+example, literature [10] introduces the Fast RPCA algorithm
+to separate the foreground, and then implements Faster R-CNN
+object detection on the foreground map to effectively detect
+the moving small object in the panoramic video. In literature
+[11], the frame difference method was used to obtain the
+moving foreground, and then the CNN classification network
+was used to screen the region of interest. Finally, the CNN
+regression network was used to perform coordinate regression
+on the region of interest to obtain the moving object. Literature
+[12] uses the ViBe background modeling method to extract the
+foreground, and uses this foreground as the candidate moving
+object area of Fast R-CNN to set ANCHORS, so as to realize
+the detection of moving objects. In reference [10], the motion
+region was obtained by frame difference method, and then
+the motion region was connected and expanded. Finally, Deep
+CNN was used to classify and position regression the object
+in the motion region. 2) Traditional methods are directly fused
+with object detection to detect moving objects [38], [39]. For
+example, literature [38] inputted the frame difference between
+the original image and the two frames into VGG16 for fusion,
+and then inputted the fused feature layer into Faster R-CNN
+for object detection. Literature [39] proposed a method based
+on deep learning combining RGB and optical flow to segment
+moving objects.
+2) Full Deep Learning Method: There are two main cat-
+egories of moving object detection methods based on full
+deep learning. 1) ConvLSTM is used to fuse temporal and
+spatial information to segment or detect moving objects [14],
+[15]. For example, reference [14] introduces the attention
+Long Short-Term Memory (attention ConvLSTM) model to
+simulate the change of pixels over time, and then uses a
+spatial Transformer and conditional random field (CRF) to
+segment moving objects. In reference [15], the Pyramid dilated
+convolution (PDC) module was designed to extract multi-
+scale spatial features, and then these spatial features were
+concatenated and fed into the Extended deep Bidirectional
+ConvLSTM (DB-ConvLSTM) to obtain spatio-temporal in-
+formation. Finally, the moving objects in the video are de-
+tected by using the spatio-temporal information. 2) Detecting
+moving objects by merging and fusing temporal and spatial
+information of consecutive frames [16], [17]. For example, the
+paper [16] propose regions of objects of interest (ROOBI) by
+
+IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+4
+using the region proposal network, which combines the spatio-
+temporal information by merging the input of consecutive
+frames. After getting the Propose regions, the exact position of
+the object is located again by merging the input of consecutive
+multiple frames. In literature [17], continuous multiple frames
+are merged into CNN for background estimation, and then a
+compact encoder-decoder network is used to segment moving
+objects.
+III. THE PROPOSED SMOD-BMI
+Fig. 2 shows the overview diagram of the proposed SMOD-
+BMI, which contains three parts. Firstly, ConvLSTM-SCM-
+PAN model structure was designed to capture the suspicious
+moving objects. Secondly, An object tracking algorithm is
+used to track suspicious moving objects and calculate their
+MR. At the same time, according to the moving speed of the
+suspicious moving objects, the size of their MR is adjusted
+adaptively to obtain their ACMR, so as to ensure that the
+SNR of the moving object is improved while the necessary
+context information is retained adaptively. Finally, LW-SCM-
+USN based on ACMR with a SCM module is designed to clas-
+sify and locate small moving objects accurately and quickly.
+Section III-A describes the ConvLSTM-based suspicious mov-
+ing object detection method. Section III-B ACMR extraction
+method of suspicious moving object based on object tracking
+technology and motion amount. Section III-C describes the
+moving object detection method based on ACMR.
+A. To Capture the Suspicious Moving Object
+In this paper, we perform two steps to capture suspicious
+moving objects (coarse-detection of moving objects) in con-
+secutive video images. Firstly, the spatio-temporal information
+of the moving object was fused. Secondly, the spatio-temporal
+information is used to locate the suspicious moving object by
+object detection. This subsection will introduce the acquisition
+of spatio-temporal information of moving objects (Section
+III-A1), and the localization of suspicious moving objects
+(Section III-A2), respectively.
+1) Fusion of Spatio-temporal Information for Small Moving
+Objects: Motion is mainly reflected in time and space, that
+is, at different times, according to the spatial location of
+the object can show motion. Therefore, to capture the Small
+moving object, it is necessary to fuse its temporal and spatial
+information.
+As we have introduced in Section II-C2, there are two main
+ways to fuse the spatio-temporal information of the object
+based on deep learning. One is based on the recurrent neural
+network ConvLSTM, and the other is based on the input
+merging of consecutive multiple frames. ConvLSTM(structure
+shown in Fig. 3) contains three gates, namely input gate,
+output gate and forget gate, which are used to control the
+input and output and what information needs to be forgotten
+and discarded. At the same time, the input gate and output
+gate can also be understood as controlling the writing and
+reading of the memory cell. Continuous multi-frame merging
+input is to simply Concatenate consecutive frames of video
+images together and then input into the neural network.
+The coarse-detection phase captures the suspicious moving
+object, and the input is the whole video, which has the char-
+acteristics of many background interference and redundant in-
+formation (different frames have many identical backgrounds).
+According to the characteristics of ConvLSTM structure, it
+can remove unimportant or redundant information while fusing
+spatio-temporal information. So, in the first stage, we use Con-
+vLSTM to extract and fuse the spatio-temporal information of
+moving objects. Specifically, given the input n consecutive
+frames of images Xt ∈ R(H×W×3)|t = (1, 2, · · · , n) (Where
+H and W are the height and width of the input image, and n is
+an odd number), the ConvLSTM network FConvLSTM is used to
+fuse and extract the spatio-temporal features Hn ∈ R(H×W×C)
+(Where C is the number of channels) of the n consecutive
+frames of images,
+Ht = FConvLSTM ([Xt, Ht−1] ; ΘConvLSTM) ,
+(1)
+Where, when t = 1, H0 = 0. ΘConvLSTM is the learnable
+parameter of the ConvLSTM network. The spatio-temporal
+features Hn of n consecutive frames of images are input
+into the subsequent classification and positioning module to
+determine the category and spatial location information of the
+suspicious moving object.
+2) Localization of Suspicious Moving Objects: In convolu-
+tional neural networks, deeper layers, which generally have
+smaller size, have better global semantic information, and
+can predict larger objects. The layers with shallower depth,
+which generally have larger size, have more delicate spatial
+information and can predict smaller objects. However, the
+large feature layer often does not have a relatively high
+degree of semantic information, and the small feature layer
+does not have fine spatial positioning information. Therefore,
+relevant researchers have proposed the structure of FPN [40] to
+combine the strong semantic information of the small feature
+layer and the strong spatial positioning information of the
+large feature layer. However, the researchers of PANet(Path
+Aggregation Network) [41] found that when FPN transmitted
+information, there was information loss due to the transfer
+distance when the information was transmitted to the low-level
+feature layer. Therefore, path-enhanced FPN, namely PANet
+structure, was proposed. The PANet structure opens up a green
+channel for low-level information transmission and avoids low-
+level information loss to a certain extent. At the same time,
+we find that the detection performance will be improved when
+Selective Concatenation Module (SCM) [42] is added to the
+model (reference [42] introduces that SCM can help to better
+fuse high and low layer information (refer to reference [42] for
+details)). We believe that SCM can not only balance the fusion
+of channel information in different layers, but also suppress
+unimportant information and highlight the information that the
+model needs to focus on. So, we introduce the SCM and design
+the feature extraction structure of SCM-PANet (see Fig. 4).
+The spatio-temporal features Hn of n consecutive frames
+are input into the SCM-PANet structure to extract the features
+of the suspicious moving object FMOn,
+FMOn = FSCM-PAN (Hn; ΘSCM-PAN) ,
+(2)
+
+IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+5
+Fig. 2: Overview of the proposed SMOD-BMI. (a) Capture the suspicious moving object, the blue box in the figure represents
+the detected suspicious moving object. (b) The ACMR of the suspicious moving object is obtained. In the figure, the green
+box represents the original MR of the moving object tracked by the tracking algorithm, and the red box represents the MR
+adaptively adjusted according to the motion amount of the moving object. (c) Classification and localization of moving objects.
+Fig. 3: Structure diagram of ConvLSTM
+Where, ΘSCM-PAN is the learnable parameter of the SCM-
+PANet network.
+When the distance between the moving object and the
+surveillance camera is different, the size of the moving object
+is also different, so the moving object to be detected has the
+multi-scale property. According to the multi-scale property of
+the moving object, this paper uses the MultiScale Detection
+Head (MS-D Head) to detect the suspicious moving object.
+The objects in the middle frame of n consecutive frames
+has symmetric contextual information, which can get more
+accurate results in prediction. Therefore, this paper predicts
+the suspicious object in the middle frame of n consecutive
+frames as the detection result of the Coarse-detection stage.
+Specifically, the feature FMOn of the moving object is input
+into the MS-D Head to obtain the output of the model,
+On = FMS-D (FMOn; ΘMS-D) ,
+(3)
+
+(a) To capture the suspicious Moving Object (MO)
+MS-D
+ConyLSTM
+SCM-PAN
+Head
+Suspicious MO (Blue box)
+Video Frame
+Coarse Detection Model
+Track and calculate the Motion Range (MR)
+(b) To obtain the Adaptive Candidate MR (ACMR)
+(n+4)th
+MR (Green box) of the Suspicious MO
+Adaptively resize and crop the MR
+(c) To classify and locate the MO
+BackGround
+SS-D
+LW-SCM-USN
+Head
+Bird
+crop
+crop
+crop
+ACMR (Red box) of The Suspicious MO
+Fine Detection Model
+BirdConvLSTM
+Conv
+Bias
+I+X
+write
+read
++Bc
+*
+tanh
+tanh
+C
+Xt
+W:*
++B:
+0
+Ct
+J
+Concat
++Bf
+W.*
+Ht-1
+Ot
++Bo
+W
+*
+0
+0
+Ht
++ Data flow
+Next iterationIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+6
+Fig. 4: Structure diagram of the SCM-PANet model
+where, ΘMS-D is the learnable parameter of the MS-D Head.
+Then post-processing operations such as Boxes Decoding and
+non-maximum suppression were performed on the output of
+the model to obtain the location of the suspicious object in
+the middle frame of n consecutive frames,
+{PID1, · · · , PIDk}
+frame( n+1
+2 ) = FP (On) ,
+(4)
+Where, {·}
+frame( n+1
+2 ) means the locations of the moving ob-
+jects in
+� n+1
+2
+�th frames. PIDk indicates the predicted position
+of the object with IDk (the object with IDk is taken as an
+example unless otherwise specified). FP (·) denotes the post-
+processing method.
+B. To Obtain the ACMR
+In this paper, the MR of the suspicious moving object on
+n consecutive frames is extracted to improve the SNR of the
+moving object. At the same time, in order to ensure the context
+information of the suspicious moving object, the size of the
+MR is adaptively adjusted according to the motion amount of
+the suspicious moving object, so that the subsequent detection
+results are more accurate. Specifically, we will divide into
+two steps to obtain the ACMR of suspicious moving objects.
+Respectively, the original MR of the suspicious moving object
+is extracted using the object tracking technology (Section
+III-B1) and the MR is adaptively adjusted using the motion
+amount of the suspicious moving object to obtain the ACMR
+(Section III-B2).
+1) Acquisition of the Original MR of the Suspicious Moving
+Object: From the
+� n+1
+2
+�th frame, there are detection results
+of the suspicious moving object, and we start to track the
+suspicious moving object from the
+� n+1
+2
++ 1
+�th frame. In some
+cases, the appearance characteristics of small moving objects
+are not obvious, so we only use their motion information when
+tracking them, and use a relatively simple SORT [43] object
+tracking algorithm to track suspicious moving objects,
+�
+{PIDk}frame(i), {PIDk}frame(i+1), · · ·
+�
+= Ftrack (IDk) ,
+(5)
+where,
+�
+{PIDk}frame(i), {PIDk}frame(i+1), · · ·
+�
+represents the po-
+sition on consecutive image frames of a suspicious moving
+object with ID number k, and Ftrack (·) represents the SORT
+object tracking method. After obtaining the position of the
+suspicious moving object on consecutive image frames, we
+can find the Motion Range (MR) of the suspicious moving
+object on n consecutive frames. Specifically, the minimum
+circumscribed rectangle RectIDk at n positions is calculated
+according to the position of the same object on n consecutive
+frames of images,
+RectIDk =
+FMinRect
+��
+{PIDk}frame(i+1), · · · , {PIDk}frame(i+n)
+��
+,
+(6)
+where, FMinRect (·) denotes the function to find the mini-
+mum circumscribed rectangle of n rectangular boxes. For
+example,
+to
+find
+the
+minimum
+circumscribed
+rectangle
+[(xmin, ymin) , (xmax, ymax)] (Using the horizontal and vertical
+coordinates of the top left and bottom right vertices of the rect-
+angle) of {box1, · · · , boxn}, the specific calculation method is
+as follows,
+xmin = min
+�
+x1box1 , · · · , x1boxn
+�
+,
+ymin = min
+�
+y1box1 , · · · , y1boxn
+�
+,
+xmax = max
+�
+x2box1 , · · · , x2boxn
+�
+,
+ymax = max
+�
+y2box1 , · · · , y2boxn
+�
+,
+(7)
+where,
+��
+x1boxn , y1boxn
+�
+,
+�
+x2boxn , y2boxn
+��
+denotes the hori-
+zontal and vertical coordinates of the upper left and lower
+right vertices of boxn in the image. The obtained minimum
+circumscribed rectangle RectIDk is the MR of the moving
+object in n consecutive frames. Fig. 5 illustrates the MR of
+the moving object on five consecutive frames of images.
+2) Adaptively Adjust the MR to Obtain ACMR Based on the
+Amount of Motion: We crop the MR of suspicious moving
+object in n consecutive frames to remove the interference of
+other background and negative samples, which can improve
+the SNR of the moving object. However, if the moving
+object moves too slowly, the clipped MR will lack contextual
+environmental information (see the Raw MR In Fig. 6),
+which is not conducive to the detection of moving objects. In
+order to balance the contradiction between SNR and context
+information, this paper proposes an ACMR extraction method
+based on the amount of motion of the moving object, which
+adaptively adjusts the size of the MR of the moving object
+according to the speed of the object motion. There are two
+steps.
+Firstly, the amount of motion of the moving object over n
+consecutive frames is calculated. For an object of the same
+size, if it moves fast on n consecutive frames, its MR is large;
+
+Backbone
+SCM
+SCM
+FPN
+PAN
+MS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+7
+Fig. 5: MR of the moving bird over 5 consecutive frames. The blue box shows the position of the bird in each frame; The
+green box represents the minimum bounding rectangle of the five blue boxes, which is the MR of the moving bird over five
+consecutive frames.
+Fig. 6: The left picture shows the original monitoring picture, the right picture shows the MR of the moving object on five
+consecutive frames in the dashed frame, and the ACMR of the moving object in the solid frame. It is obvious that the object
+in the original MR is difficult to be correctly recognized, and the object in the ACMR is easier to be recognized.
+otherwise, its MR is small. Therefore, we use the ratio of the
+area of the MR of the moving object on n consecutive frames
+to the area of the single frame image occupied by the moving
+object to define its motion amount on n consecutive frames,
+σmov = S (RectIDk)
+S (ObjIDk) ,
+(8)
+where, σmov is the motion amount, S (·) represents the func-
+tion to calculate the area, and ObjIDk represents the object with
+ID number k. The area of MR is then the area of the minimum
+circumscribed rectangle RectIDk. Since the area occupied by
+a moving object in a single image frame may vary due to its
+shape changes, and it is difficult to calculate accurately, we
+use the rectangular area of its bounding box to approximately
+represent its area in this paper.
+Then, according to the amount of motion of the moving
+object, the MR of the moving object is adaptively adjusted as
+the Adaptive Motion Range (AMR) ARectIDk of the moving
+object. Specifically, a motion hyperparameter γ is introduced.
+When the amount of motion of the moving object is less than
+γ, the MR of the moving object is expanded to make the
+amount of motion of the moving object reach γ. Therefore,
+the AMR of the moving object can be expressed as follows,
+ARectIDk =
+�
+ARectIDk,
+σmov ≥ γ
+γ × ObjIDk,
+otherwise.
+(9)
+The ARectIDk is used to crop the corresponding n consecutive
+frames of video image
+�
+frame(1), · · · , frame(n)�
+respectively,
+and the n frame screenshots obtained are the Adaptive Can-
+didate Moving Region (ACMR) (ACMRIDk) of the moving
+object,
+f(i)
+ARectIDk = Fcut
+�
+frame(i), ARectIDk
+�
+,
+(10)
+ACMRIDk =
+�
+f(i)
+ARectIDk|i ∈ (1, · · · , n)
+�
+.
+(11)
+C. Moving Object Detection based on ACMR
+After the previous processing, we improve the SNR of the
+moving object, retain its contextual environmental information,
+and obtain the ACMR of the moving object. In the fine-
+detection stage, we can use the ACMR of the moving object
+
+Raw MR
+ACMR
+Raw MR
+ACMRIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+8
+to classify and locate the moving object. Specifically, the fine-
+detection phase includes the fusion of spatio-temporal infor-
+mation (III-C1) and the classification and localization(III-C2)
+of moving objects.
+1) Fusion of Spatio-temporal Information of Moving Ob-
+jects: The input of the fine-detection model is the ACMR
+of the moving object extracted earlier. The coarse-detection
+model may detect multiple suspicious moving objects at one
+time, so there will be multiple ACMRs, and the fine detection
+model will detect each ACMR separately. So it’s possible to
+run a coarse-detection model once and a fine-detection model
+many times. Therefore, in order to balance accuracy and speed,
+the method of fusing the spatio-temporal information of the
+moving target in the fine-detection stage uses the way of
+merging consecutive multiple frames. At the same time, in
+order to reduce data redundancy, except the middle frame, the
+rest of the frames are input in the form of grayscale image
+single channel. Specifically, firstly, grayscale the screenshots
+of the ACMR of the moving object except for the middle
+screenshot,
+f(i)′
+ARectIDk =
+�
+�
+�
+f(i)
+ARectIDk,
+i = int
+� n
+2
+�
+FGray
+�
+f(i)
+ARectIDk
+�
+,
+otherwise.
+(12)
+where, FGray (·) is a function that finds the grayscale of a color
+image. Then, the processed screenshots of the ACMRs are
+Concatenate in the channel dimension as the input of the fine-
+detection stage,
+XSIDK = Fconcat
+��
+f(1)′
+ARectIDk, · · · , f(n)′
+ARectIDk
+�
+, 2
+�
+,
+(13)
+Where, the second argument of the Fconcat function indicates
+that the concatenation operation is performed in the third input
+dimension (height, width, channel). The length and width of
+XSIDK is equal to the length and width of rectangle ARectIDk,
+and the number of channels is n+2, which contains the motion
+information and appearance information of the moving object.
+It is input into the fine-detection model to accurately classify
+and locate the moving object.
+2) Classification and Localization of Moving objects: In
+order to further improve the speed of the whole moving object
+detection process, this paper uses a lightweight U-Shaped
+Network (USN) (in the experiment, we use MobilenetV2 [44]
+as the backbone network of the USN) as the feature extraction
+network of the moving object in the fine-detection stage. At
+the same time, in order to better fuse high and low layer infor-
+mation, similar to the network of the coarse-detection model,
+we introduce the SCM [42] module and design the lightweight
+LW-SCM-USN feature extraction network structure, as shown
+in Fig. 7.
+The XSIDK fused with the spatio-temporal information of
+the moving object is input into the LW-SCM-USN feature
+extraction network to obtain the moving object feature FIDK
+fused with the spatio-temporal information,
+FIDK = FLW-SCM-USN (XSIDK; ΘLW-SCM-USN) ,
+(14)
+where, ΘLW-SCM-USN is the learnable parameter of the LW-
+SCM-USN.
+Fig. 7: Structure diagram of LW-SCM-USN
+The ACMR of moving object may contain more than one
+object. And due to the interference of background and negative
+samples, the detection accuracy of the coarse-detection model
+is not satisfactory, there will be false detection and missed
+detection. So, the ACMR may contain no object, one object
+or multiple objects. Therefore, the detection model in the fine-
+detection stage should still have the ability of multi-object
+detection. However, since the ACMRs of moving objects are
+only a small area (relative to the input image) and cannot
+contain a large number of moving objects, the output of the
+fine-detection model need not be designed with a complex
+structure. In summary, the paper uses a relatively simple Single
+Scale Detection Head (SS-D Head) structure as the output
+structure of the fine-detection model (see Fig. 7). Specifically,
+FIDK is fed into the SS-D Head to obtain the output of the
+fine-detection model,
+OIDK = FSS-D (FIDK; ΘSS-D) ,
+(15)
+where, ΘSS-D is the learnable parameter of the SS-D Head.
+Then, post-processing operations such as Boxes Decoding and
+non-maximum suppression are performed on the output to
+obtain the final detection result of moving object,
+{ClassesIDK, BoxesIDK} = FP (OIDK) ,
+(16)
+where, ClassesIDK represents the category of the object in
+the ACMR ACMRIDk of the moving object, and BoxesIDK
+(in this paper, the position of the object in the middle frame
+of n consecutive frames is taken as the detection result) is the
+bounding box of the corresponding object in this region.
+Finally, the bounding box of the moving object in the
+ACMR is mapped to the original video image, that is, the
+final detection result of the moving object is obtained.
+IV. EXPERIMENT
+In this section, A series of experiments are conducted to
+quantitatively and qualitatively evaluate the proposed SMOD-
+BMI. Next, we will introduce datasets (IV-A), evaluation
+metrics (IV-B), experimental platforms (IV-C), implementa-
+tion details (IV-D), parameter analysis experiment (IV-F) and
+comparative analysis experiment (IV-E).
+
+SCM
+Backbone
+SS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+9
+Fig. 8: Size distribution of the moving birds in the datasets
+A. Datasets
+We collected and annotated 20 videos containing moving
+bird objects (the size of video images is 1280 × 720) in
+an unattended traction substation. We end up with 10,381
+continuous annotated images with 11,631 objects in total.
+From Fig. 8, we can see that the size of moving birds is
+mainly distributed between 0 × 0 and 80 × 80 pixels, and
+about more than 50% of them are below 40 × 40 pixels. So
+these birds can be called moving small objects.
+B. Evaluation Metrics
+In this paper, the widely used measures in object detection,
+precision (Prec), recall (Rec), and average precision (AP)
+are adopted to evaluate the proposed SMOD-BMI. More
+specifically, Prec50, Rec50, AP50 (The subscript 50 means that
+the detection result is regarded as the True Positive, when
+the IOU between the detection result and the ground truth is
+greater than or equal to 50%. That is, the IOU threshold is set
+50% ), Prec75, Rec75, AP75 (The subscript 75 has the Similar
+meaning with the subscript 50) and AP (Average Precision
+averaged over multiple thresholds, IOU threshold is set from
+50% to 95%, in intervals of 5%) are adopted.
+C. Experimental Platforms
+All the experiments are implemented on a desktop computer
+with an Intel Core i7-9700 CPU, 32 GB of memory, and a
+single NVIDIA GeForce RTX 3090 with 24 GB GPU memory.
+D. Implementation Details
+We implemented the proposed method based on YOLOV4
+[28] with modifications.
+Specifically, for the coarse-detection model, a ConvLSTM
+module is embedded between the second and third layers of
+CSPDarkNet53, the backbone network of YOLOV4 model,
+and a SCM [42] is added to its PANet structure. For the input
+size of the coarse-detection model, we set it to 640 × 384 to
+ensure the ratio of effective input pixels as much as possible
+and at the same time ensure the running speed. During training,
+the input is n consecutive frames of images, the label is the
+position of the object on the intermediate frame, and the loss
+function of the YOLOV4 algorithm is reused.
+For the fine-detection model, the lightweight MobilenetV2
+is used as the backbone network of the U-shaped network, and
+the SCM [42] is added to the upsampling structure of the U-
+shaped network. For the input size of the fine-detection model,
+we set it to 160160. For the training data, we used the coarse-
+detection model and the object tracking SORT algorithm to
+collect the Motion Region (MR) containing the moving object
+as the positive samples and the negative samples without the
+object. During training, the input is the screenshot of the
+MR of n consecutive frames, the label is the position of the
+object on the intermediate screenshot, and the loss function of
+YOLOV4 is reused.
+In this paper, all experiments are implemented under the
+Pytorch framework. All network models are trained on an
+NVIDIA GeForce RTX 3090 with 24G of video memory. For
+the batch size setting, it is set to 4 when training the coarse-
+detection model designed in this paper and other comparison
+models, and it is set to 8 when training the fine-detection
+model. All the experimental models were trained from scratch,
+and no pre-trained models were used. The trainable parameters
+of the network were randomly initialized using a normal
+distribution with mean 0 and variance 0.01. Adam was chosen
+as the optimizer for the model in this paper. The initial learning
+rate is set to 0.001. For each iteration, the learning rate is
+multiplied by 0.95 and the model is trained for a total of
+100 iterations. In the training phase, we used simple data
+augmentation including random horizontal flipping, random
+Gaussian noise, etc. to enhance the robustness of the model.
+E. Comparative Analysis Experiments
+In order to verify the advancement of the proposed moving
+object detection algorithm. We design a series of comparative
+experiments to compare the accuracy of different methods in
+detecting moving objects. We designed and implemented some
+deep learning-based methods following their main ideas. The
+methods mainly compared in this paper have the following
+categories.
+• Object Detection method based on still images. We chose
+YOLOV4 as the representative algorithm of this kind of
+methods.
+• Multi-frame input is used to fuse spatio-temporal features,
+and then the method of object detection is used to realize
+the detection or segmentation of moving objects. For
+this class of methods, we use Mutlti-Input+YOLOV4
+(MI YOLOV4) to represent.
+• ConvLSTM is used to fuse spatio-temporal features,
+and then the object detection method is used to realize
+
+Distribute of Object Size
+5000 
+4000 
+1654
+1000
+0-20 20-40 40-60 60-80
+100-120
+160-180
+220-240
+280-300
+340-360
+400-420
+460-480
+520-540
+580-600
+640-660
+700-720
+760-780
+Square Root of the Area(pixls)IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+10
+the detection or segmentation of moving objects. For
+this type of methods, we use ConvLSTM+YOLOV4
+(CL YOLOV4) to represent.
+By the way, the parameters of the above model are designed
+as follows. The inputs are all set to 640 × 384. The number of
+consecutive input frames for MI YOLOV4, CL YOLOV4 and
+SMOD-MBI are set to 5 frames. For SMOD-MBI, its motion
+amount parameter σmov is set to 4.0.
+In the qualitative comparison experiment, we choose
+YOLOV4 as the baseline for comparison. YOLOV4 algorithm
+only considers the appearance features of moving objects,
+while the method proposed in this paper makes full use
+of the motion cues of moving objects. By comparing the
+experimental results as shown in Fig. 9, it can be seen that
+when the appearance characteristics of the moving object are
+obvious, YOLOV4 can also achieve a certain effect. However,
+when the appearance characteristics of the moving object are
+not obvious, YOLOV4 will miss detection, and YOLOV4 is
+also prone to false detection. However, the proposed method
+can achieve good results regardless of whether the appearance
+characteristics of the moving object are obvious or not. There-
+fore, for the detection of moving object, its motion cues are
+particularly important.
+Other methods considering the motion information of the
+moving object and the method proposed in this paper have
+little difference in qualitative comparison, so this paper designs
+a quantitative comparison experiment to compare the method
+proposed in this paper with other algorithms.
+The results of quantitative comparison experiments are
+shown in TABLE I. For the same detection method, the AP
+decreases sharply with the increase of IOU threshold. The
+reason for this is that the smaller the object, the harder it is for
+the detection to match the ground truth exactly, because subtle
+deviations in the detection results will be more noticeable
+compared to the ground truth. Compared with different detec-
+tion methods, the moving object detection method YOLOV4
+based only on appearance has a poor effect on detecting
+moving small objects, with AP50 of 64.34%. MI YOLOV4
+fuses the spatio-temporal information of the moving object
+by merging the input of multiple frames, which can improve
+the AP50 by 17.13%. Therefore, for the dataset we collected,
+motion information is a more important clue for detecting
+small moving targets in complex environments. CL YOLOV4
+uses ConvLSTM to merge the spatio-temporal information
+of the moving object, and can obtain an AP50 increase of
+1.76%, which shows that ConvLSTM is more suitable for
+fusing the spatio-temporal information of the moving object
+than the multi-frame merged input, because ConvLSTM has
+some special structures to remove the influence of redundant
+information.
+On the basis of CL YOLOV4, the proposed method SMOD-
+BMI uses object tracking technology and combines the motion
+amount of the moving object to obtain the Adaptive Candidate
+Motion Range (ACMR) of the moving object, and then finely
+detects the moving object in the ACMR. We reduce the
+threshold for judging the moving objects in coarse-detection
+stage, which will cause some false detections but will improve
+the detection rate. At the same time, we increase the threshold
+that is judged as a moving object in the fine-detection stage
+to reject false detections. The experimental results show that
+the proposed method improves AP50 by 4.25%, and reaches
+to 87.46%.
+Through the qualitative and quantitative analysis of the
+experimental results, it can be concluded that the small moving
+object detection method proposed in this paper is advanced and
+effective.
+F. Parameter Analysis Experiments
+1) Effect of Different Number of Consecutive Input Frames
+on the Performance of the Algorithm: We design test exper-
+iments with different numbers of consecutive frame inputs to
+evaluate the impact on the detection accuracy and efficiency
+of the proposed method. Specifically, there are 3 consecutive
+frames of input, 5 consecutive frames of input, 7 consecutive
+frames of input, etc. In theory, with the increase of the
+number of consecutive frames, the motion information of the
+moving object will be gradually enriched, and the detection
+accuracy of the algorithm will be gradually improved, but its
+running time will also increase accordingly. The results of
+the detection performance test of the algorithm are shown in
+Table II (the motion amount parameter σmov is set to 4.0).
+The experimental results show that the running speed of the
+algorithm is the fastest when 3 consecutive frames are input,
+and the detection accuracy is the highest when 7 consecutive
+frames are input. When the input is five consecutive frames,
+the speed and accuracy can have a good trade-off (the AP50
+reaches to 87.46%, and the running time is 0.12s).
+2) Influence of Different Amount of Motion Parameter σmov
+on the Accuracy of the Algorithm: We obtain the Adaptive
+Candidate Motion Ranges (ACMRs) of different sizes of the
+moving object by setting different motion amount parameter
+σmov. If the MR is small, the context background information
+is less; if the MR is large, the SNR is large. Therefore, different
+sizes of MRs of the same moving object have different effects
+on the performance of the algorithm. Fig. 10 is the influence
+of different motion amount parameter σmov on the accuracy
+of the algorithm.
+It can be seen from Fig. 10 that when the motion amount
+parameter σmov is 1.0, the detection accuracy of the proposed
+method is lower than that of MI YOLOV4 and CL YOLOV4
+(When the motion amount parameter σmov is set to 1.0, it
+is equivalent to that the algorithm does not use the adaptive
+adjustment mechanism to adjust the MR of the moving object,
+because even if the moving object is still, it still satisfies
+the motion amount parameter σmov of 1.0, so there is no
+need to adjust the MR of the moving object according to the
+motion amount of the moving object). In other words, with the
+addition of the fine-detection stage, its detection accuracy is
+reduced instead. This proves that when the MR of the moving
+object is too small, it lacks enough context information, which
+leads to the decline of detection accuracy. When we increase
+the motion amount parameter σmov, the detection accuracy is
+rapidly improved. However, when it is greater than 5.0, the
+detection accuracy starts to slowly decrease again.
+As previously analyzed, when the MR is too small, it lacks
+contextual information, and when the MR is too large, it is
+
+IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+11
+Scenario 1
+Scenario 2
+Scenario 3
+(a) YOLOV4
+(b) SMOD-BMI
+Fig. 9: Detection comparisons of YOLOV4 and SMOB-BMI (green box: ground truth bounding box; red box: YOLOV4
+bounding box; blue box: proposed method bounding box).
+TABLE I: Comparison with other moving object detection methods
+Frame num
+Prec50
+Rec50
+AP50
+Prec75
+Rec75
+AP75
+AP
+YOLOV4
+0.3200
+0.7074
+0.6434
+0.0790
+0.1747
+0.0553
+0.2106
+MI YOLOV4
+0.8561
+0.8478
+0.8145
+0.4098
+0.3846
+0.2109
+0.3298
+CL YOLOV4
+0.8717
+0.8592
+0.8321
+0.4165
+0.3955
+0.2123
+0.3422
+SMOD-BMI(ours)
+0.9197
+0.9118
+0.8746
+0.4827
+0.4786
+0.2482
+0.3827
+TABLE II: Effect of continuous image input with different number of frames on detection performance
+Frame num
+Prec50
+Rec50
+AP50
+Prec75
+Rec75
+AP75
+AP
+Run Time
+3
+0.9226
+0.9162
+0.8737
+0.4817
+0.4784
+0.2349
+0.3701
+0.11s
+5
+0.9197
+0.9118
+0.8746
+0.4827
+0.4786
+0.2482
+0.3827
+0.12s
+7
+0.9341
+0.9109
+0.8808
+0.4745
+0.4627
+0.2412
+0.3838
+0.14s
+easy to introduce more noise (in an extreme case, when the
+MR is already consistent with the original input image, the
+previous processing will be meaningless, because the input
+of the fine-detection stage is directly the original image. At
+the same time, because the fine-detection model is relatively
+simple and the input size is small (160 × 160), the detection
+effect is bound to be poor), so whether the MR is too small or
+too large, it will affect the accuracy of the algorithm. Through
+experiments, we find that when the motion amount parameter
+σmov is 5.0, the detection performance of the algorithm is the
+
+多
+快网间8888D团多
+快网间IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023
+12
+Fig. 10: Influence of Different Amount of Motion Parameter
+σmov on the Accuracy of the Algorithm
+best, and its AP50 is reaching 87.85%.
+Through parameter analysis experiments, we conclude that
+when the number of consecutive input frames is 5, the
+algorithm can get a good balance between accuracy and
+speed. When the motion parameter is 5.0, the accuracy of the
+algorithm reaches the highest. So we suggest the following
+parameter setting scheme. The number of consecutive input
+frames is set to 5, and the motion amount parameter σmov is
+set to 5.0.
+V. CONCLUSION
+Aiming at the problem that the moving object is difficult to
+detect in complex background, this paper analyzes the reason.
+The reason is that the proportion of moving small object pixels
+is small in complex background, which leads to low SNR. To
+solve this problem, this paper proposes a Small Moving Object
+Detection algorithm Based on Motion Information (SMOD-
+BMI). Firstly, we use the ConvLSTM-SCM-PANet model to
+coarsely detect the whole frame of a continuous video frame
+and capture the suspicious moving object. Then, we used
+the method of object tracking to track the suspicious moving
+object to determine the MR of the suspicious moving object
+on n consecutive frames. At the same time, according to the
+moving speed of the suspicious moving objects, the size of
+their MR is adjusted adaptively (To be specific, if the objects
+move slowly, we expand their MR according their speed to
+ensure the contextual environment information) to obtain their
+Adaptive Candidate Motion Range (ACMR), so as to ensure
+that the SNR of the moving object is improved while the
+necessary context information is retained adaptively. After
+that, we use LW-SCM-USN model to accurately classify and
+locate the suspicious moving object by using the ACMR of the
+suspicious moving object. Finally, qualitative and quantitative
+experiments verify the effectiveness and advancement of the
+proposed moving object detection algorithm based on motion
+information.
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+Maintaining Triconnected Components under
+Node Expansion
+Simon D. Fink � �
+Faculty of Informatics and Mathematics, University of Passau, Germany
+Ignaz Rutter � �
+Faculty of Informatics and Mathematics, University of Passau, Germany
+Abstract
+SPQR-trees are a central component of graph drawing and are also important in many further
+areas of computer science. From their inception onwards, they have always had a strong relation
+to dynamic algorithms maintaining information, e.g., on planarity and triconnectivity, under edge
+insertion and, later on, also deletion. In this paper, we focus on a special kind of dynamic update,
+the expansion of vertices into arbitrary biconnected graphs, while maintaining the SPQR-tree and
+further information. This will also allow us to efficiently merge two SPQR-trees by identifying the
+edges incident to two vertices with each other. We do this working along an axiomatic definition
+lifting the SPQR-tree to a stand-alone data structure that can be modified independently from the
+graph it might have been derived from. Making changes to this structure, we can now observe how
+the graph represented by the SPQR-tree changes, instead of having to reason which updates to the
+SPQR-tree are necessary after a change to the represented graph.
+Using efficient expansions and merges allows us to improve the runtime of the Synchronized
+Planarity algorithm by Bläsius et al. [8] from O(m2) to O(m · ∆), where ∆ is the maximum
+pipe degree. This also reduces the time for solving several constrained planarity problems, e.g. for
+Clustered Planarity from O((n + d)2) to O(n + d · ∆), where d is the total number of crossings
+between cluster borders and edges and ∆ is the maximum number of edge crossings on a single
+cluster border.
+2012 ACM Subject Classification Mathematics of computing → Graph algorithms
+Keywords and phrases SPQR-Tree, Dynamic Algorithm, Cluster Planarity
+Funding Funded by DFG-grant RU-1903/3-1.
+arXiv:2301.03972v1  [cs.DS]  10 Jan 2023
+
+S. D. Fink and I. Rutter
+1
+1
+Introduction
+The SPQR-tree is a data structure that represents the decomposition of a graph at its
+separation pairs, that is the pairs of vertices whose removal disconnects the graph. The
+components obtained by this decomposition are called skeletons. SPQR-trees form a central
+component of many graph visualization techniques and are used for, e.g., planarity testing
+and variations thereof [13, 19, 29, 31, 39] and for computing embeddings and layouts [3, 7, 11,
+20, 28, 42]; see [37] for a survey of graph drawing applications. Outside of graph visualization
+they are used in the context of, e.g., minimum spanning trees [6, 17], triangulations [5], and
+crossing optimization [28, 42]. They also have multiple applications outside of graph theory
+and even computer science, e.g. for creating integrated circuits [14, 44], business processes
+modelling [40], electrical engineering [24], theoretical physics [41] and genomics [22].
+Initially, SPQR-trees were devised by Di Battista and Tamassia for incremental planarity
+testing [16, 19]. As such, even in their initial form, SPQR-trees already allowed dynamic
+updates in the form of edge addition. Their use was quickly expanded to other on-line
+problems [18, 17]. In addition to the applications mentioned above, this also sparked a series
+of further papers improving the runtime of the incremental data structure [38, 39, 43] and
+also extending it to be fully-dynamic, i.e., allowing insertion and deletion of vertices and
+edges, in O(√n) time [21, 27], where n is the number of vertices in the graph. Recently,
+Holm and Rotenberg described a fully-dynamic algorithm for maintaining planarity and
+triconnectivity information in O(log3 n) time per operation [31, 32] (see also there for a short
+history on dynamic SPQR-tree algorithms).
+In this paper, we consider an incremental setting where we allow a single operation that
+expands a vertex v into an arbitrary biconnected graph Gν. Using the approach of Holm
+and Rotenberg [31], this takes O((deg(v) + |Gν|) · log3 n) time by first removing v and its
+incident edges and then incrementally inserting Gν. We improve this to O(deg(v) + |Gν|)
+using an algorithm that is much simpler and thus also more likely to improve performance in
+practice. In addition, our approach also allows to efficiently merge two SPQR-trees as follows.
+Given two biconnected graphs G1, G2 containing vertices v1, v2, respectively, together with
+a bijection between their incident edges, we construct a new graph G by replacing v1 with
+G2 − v2 in G1, identifying edges using the given bijection. Given the SPQR-trees of G1 and
+G2, we show that the SPQR-tree of G can be found in O(deg(v1)) time. More specifically, we
+present a data structure that supports the following operations: InsertGraphSPQR expands
+a single vertex in time linear in the size of the expanded subgraph, MergeSPQR merges two
+SPQR-trees in time linear in the degree of the replaced vertices, IsPlanar indicates whether
+the currently represented graph is planar in constant time, and Rotation yields one of
+the two possible planar rotations of a vertex in a triconnected skeleton in constant time.
+Furthermore, our data structure can be adapted to yield consistent planar embeddings for
+all triconnected skeletons and to test for the existence of three distinct paths between two
+arbitrary vertices with an additional factor of α(n) for all operations, where α is the inverse
+Ackermann function.
+The main idea of our approach is that the subtree of the SPQR-tree affected by expanding
+a vertex v has size linear in the degree of v, but may contain arbitrarily large skeletons. In a
+“non-normalized” version of an SPQR-tree, the affected cycle (‘S’) skeletons can easily be
+split to have a constant size, while we develop a custom splitting operation to limit the size
+of triconnected ‘R’ skeletons. This limits the size of the affected structure to be linear in the
+degree of v and allows us to perform the expansion efficiently.
+In addition to the description of this data structure, the technical contribution of this
+
+2
+Maintaining Triconnected Components under Node Expansion
+Problem
+Running Times
+before [8]
+using [8]
+with this paper
+Atomic
+Embeddability
+/
+Synchronized Planarity
+O(m8) [26]
+O(m2)
+O(m · ∆)
+ClusterPlanarity
+O((n + d)8) [26]
+O((n + d)2)
+O(n + d · ∆)
+Connected SEFE
+O(n16) [26]
+O(n2)
+O(n · ∆)
+bicon: O(n2) [10]
+Partially
+PQ-Constrained
+Planarity
+bicon: O(m) [10]
+O(m2)
+O(m · ∆)
+Row-Column
+Independent
+NodeTrix Planarity
+bicon: O(n2) [35]
+O(n2)
+O(n · ∆)
+Strip Planarity
+O(n8) [4, 26]
+O(n2)
+O(n · ∆)
+fixed emb: poly [4]
+Table 1 The best known running times for various constrained planarity problems before Syn-
+chronized Planarity [8] was published; using it as described in [8]; and using it together with
+the speed-up from this paper. Running times prefixed with “bicon” only apply for certain problem
+instances which expose some form of biconnectivity. The variables n and m refer to the number
+of vertices and edges of the problem instance, respectively. The variable d refers to the number of
+edge-cluster boundary crossings in Clustered Planarity instances, while ∆ refers to the maximum
+pipe degree in the corresponding Synchronized Planarity instances. This is bounded by the
+maximum number of edges crossing a single cluster border or the maximum vertex degree in the
+input instance, depending on the problem.
+paper is twofold: First, we develop an axiomatic definition of the decomposition at separation
+pairs, putting the SPQR-tree as “mechanical” data structure into focus instead of relying on
+and working along a given graph structure. As a result, we can deduce the represented graph
+from the data structure instead of computing the data structure from the graph. This allows
+us to make more or less arbitrary changes to the data structure (respecting its consistency
+criteria) and observe how the graph changes, instead of having to reason which changes to
+the graph require which updates to the data structure.
+Second, we explain how our data structure can be used to improve the runtime of
+the algorithm by Bläsius et al. [8] for solving Synchronized Planarity from O(m2) to
+O(m · ∆), where ∆ is the maximum pipe degree (i.e. the maximum degree of a vertex with
+synchronization constraints that enforce its rotation to be the same as that of another vertex).
+Synchronized Planarity can be used to model and solve a vast class of different kinds of
+constrained planarity, see Table 1 for an overview of problems benefiting from this speedup.
+Among them is the notorious Clustered Planarity, whose complexity was open for 30
+years before Fulek and Tóth gave an algorithm with runtime O((n + d)8) in 2019 [26], where
+d is the total number of crossings between cluster borders and edges. Shortly thereafter,
+Bläsius et al. [8] gave a solution in O((n + d)2) time. We improve this to O(n + d · ∆), where
+∆ is the maximum number of edge crossings on a single cluster border.
+This work is structured as follows. Section 2 contains an overview of the definitions
+used in this work. In Section 3, we describe the skeleton decomposition and show how it
+relates to the SPQR-tree. Section 4 extends this data structure by the capability of splitting
+
+S. D. Fink and I. Rutter
+3
+triconnected components. In Section 5, we exploit this feature to ensure the affected part of
+the SPQR-tree is small when we replace a vertex with a new graph. Section 6 contains more
+details on the background of Synchronized and Clustered Planarity and shows how
+our results can be used to reduce the time required for solving them.
+2
+Preliminaries
+In the context of this work, G = (V, E) is a (usually biconnected and loop-free) multi-graph
+with n vertices V and m (possibly parallel) edges E. For a vertex v, we denote its open
+neighborhood (excluding v itself) by N(v). For a bijection or matching ϕ we call ϕ(x) the
+partner of an element x. We use A ·∪ B to denote the union of two disjoint sets A, B.
+A separating k-set is a set of k vertices whose removal increases the number of connected
+components. Separating 1-sets are called cutvertices, while separating 2-sets are called
+separation pairs. A connected graph is biconnected if it does not have a cutvertex. A
+biconnected graph is triconnected if it does not have a separation pair. Maximal biconnected
+subgraphs are called blocks. Each separation pair divides the graph into bridges, the maximal
+subgraphs which cannot be disconnected by removing or splitting the vertices of the separation
+pair. A bond is a graph that consists solely of two pole vertices connected by multiple parallel
+edges, a polygon is a simple cycle, while a rigid is any simple triconnected graph. A wheel is
+a cycle with an additional central vertex connected to all other vertices.
+Finally, the expansion that is central to this work is formally defined as follows. Let
+Gα, Gβ be two graphs where Gα contains a vertex u and Gβ contains |N(u)| marked vertices,
+together with a bijection ϕ between the neighbors of u and the marked vertices in Gβ. With
+Gα[u →ϕ Gβ] we denote the graph that is obtained from the disjoint union of Gα, Gβ by
+identifying each neighbor x of u with its respective marked vertex ϕ(x) in Gβ and removing
+u, i.e. the graph Gα where the vertex u was expanded into Gβ.
+3
+Skeleton Decompositions
+A skeleton structure S = (G, origV, origE, twinE) that represents a graph GS = (V, E)
+consists of a set G of disjoint skeleton graphs together with three total, surjective mappings
+twinE, origE, and origV that satisfy the following conditions:
+Each skeleton Gµ = (Vµ, Ereal
+µ
+·∪ Evirt
+µ
+) in G is a multi-graph where each edge is either in
+Ereal
+µ
+and thus called real or in Evirt
+µ
+and thus called virtual.
+Bijection twinE : Evirt → Evirt matches all virtual edges Evirt = �
+µ Evirt
+µ
+such that
+twinE(e) ̸= e and twinE2 = id.
+Surjection origV : �
+µ Vµ → V maps all skeleton vertices to graph vertices.
+Bijection origE : �
+µ Ereal
+µ
+→ E maps all real edges to the graph edge set E.
+Note that each vertex and each edge of each skeleton is in the domain of exactly one of the
+three mappings. As the mappings are surjective, V and E are exactly the images of origV
+and origE. For each vertex v ∈ GS, the skeletons that contain an allocation vertex v′ with
+origV(v′) = v are called the allocation skeletons of v. Furthermore, let TS be the graph
+where each node µ corresponds to a skeleton Gµ of G. Two nodes of TS are adjacent if their
+skeletons contain a pair of virtual edges matched with each other.
+We call a skeleton structure a skeleton decomposition if it satisfies the following conditions:
+1 (bicon) Each skeleton is biconnected.
+2 (tree) Graph TS is simple, loop-free, connected and acyclic, i.e., a tree.
+3 (orig-inj) For each skeleton Gµ, the restriction origV |Vµ is injective.
+
+4
+Maintaining Triconnected Components under Node Expansion
+u
+(a)
+(b)
+(d)
+(c)
+Figure 1 Different views on the skeleton decomposition S. (a) The graph GS with a vertex u
+marked in blue. (b) The skeletons of G. Virtual edges are drawn in gray with their matching twinE
+being shown in orange. The allocation vertices of u are marked in blue. (c) The tree TS. The
+allocation skeletons of u are marked in blue. (d) The embedding tree of vertex u as described in
+Section 6.2. P-nodes are shown as white disks, Q-nodes are shown as large rectangles. The leaves of
+the embedding tree correspond to the edges incident to u.
+4 (orig-real) For each real edge uv, the endpoints of origE(uv) are origV(u) and origV(v).
+5 (orig-virt) Let uv and u′v′ be two virtual edges with uv = twinE(u′v′). For their respective
+skeletons Gµ and G′
+µ (where µ and µ′ are adjacent in TS), it is origV(Vµ) ∩ origV(Vµ′) =
+origV({u, v}) = origV({u′, v′}).
+6 (subgraph) The allocation skeletons of any vertex of GS form a connected subgraph of TS.
+Figure 1 shows an example of S, GS, and TS. We call a skeleton decomposition with only
+one skeleton Gµ trivial. Note that in this case, Gµ is isomorphic to GS, and origE and origV
+are actually bijections between the edges and vertices of both graphs.
+To model the decomposition into triconnected components, we define the operations
+SplitSeparationPair and its converse, JoinSeparationPair, on a skeleton decomposition
+S = (G, origV, origE, twinE). For SplitSeparationPair, let u, v be a separation pair of
+skeleton Gµ and let (A, B) be a non-trivial bipartition of the bridges between u and v.1
+Applying SplitSeparationPair(S, (u, v), (A, B)) yields a skeleton decomposition S′ = (G′,
+origV′, origE′, twinE′) as follows. In G′, we replace Gµ by two skeletons Gα, Gβ, where Gα is
+obtained from Gµ[A] by adding a new virtual edge eα between u and v. The same respectively
+applies to Gβ with Gµ[B] and eβ. We set twinE′(eα) = eβ and twinE′(eβ) = eα. Note that
+origV maps the endpoints of eα and eβ to the same vertices. All other skeletons and the
+mappings defined on them remain unchanged.
+For JoinSeparationPair, consider virtual edges eα, eβ with twinE(eα) = eβ and let
+Gβ ̸= Gα be their respective skeletons.
+Applying JoinSeparationPair(S, eα) yields a
+skeleton decomposition S′ = (G′, origV′, origE′, twinE′) as follows. In G′, we merge Gα with
+Gβ to form a new skeleton Gµ by identifying the endpoints of eα and eβ that map to the
+same vertex of GS. Additionally, we remove eα and eβ. All other skeletons and the mappings
+defined on them remain unchanged.
+The main feature of both operations is that they leave the graph represented by the
+skeleton decomposition unaffected while splitting a node or contracting and edge in TS,
+which can be verified by checking the individual conditions.
+▶ Lemma 1. Applying SplitSeparationPair or JoinSeparationPair on a skeleton de-
+composition S = (G, origV, origE, twinE) yields a skeleton decomposition S′ = (G′, origV′,
+1 Note that a bridge might consist out of a single edge between u and v and that each bridge includes the
+vertices u and v.
+
+S. D. Fink and I. Rutter
+5
+origE′, twinE′) with an unchanged represented graph GS′ = GS.
+Proof. We first check that all conditions still hold in the skeleton decomposition S′ returned
+by SplitSeparationPair. As (A, B) is a non-trivial bipartition, each set contains at least one
+bridge. Together with eα (and eβ), this bridge ensures that Gα (and Gβ) remain biconnected,
+satisfying condition 1 (bicon). The operation splits a node µ of TS into two adjacent nodes
+α, β, whose neighbors are defined exactly by the virtual edges in A, B, respectively. Thus,
+condition 2 (tree) remains satisfied. The mappings origV′, origE′ and twinE′ obviously still
+satisfy conditions 3 (orig-inj) and 4 (orig-real). We duplicated exactly two nodes, u and v of
+adjacent skeletons Gα and Gβ. Because 3 (orig-inj) holds for Gµ, Gα and Gβ share no other
+vertices that map to the same vertex of GS′. Thus, condition 5 (orig-virt) remains satisfied.
+Condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the
+allocation skeletons of some vertex z ∈ GS′ was no longer connected. This could only happen
+if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further
+neighbor that is also an allocation skeleton of z. Assume without loss of generality that Gα
+and the neighbor Gν of Gβ, but not Gβ itself, were allocation skeletons of z. Because Gν and
+Gβ are adjacent in TS′ there are virtual edges xy = twinE′(x′y′) with xy ∈ Gβ and x′y′ ∈ Gν.
+The same virtual edges are also present in the input instance, only with the difference that
+xy ∈ Gµ and µ (instead of β) and ν are adjacent in TS. As the input instance satisfies
+condition 5 (orig-virt), it is z ∈ origV(Vν) ∩ origV(Vµ) = origV({x, y}) = origV({x′, y′}). As
+origV({x, y}) = origV′({x, y}), this is a contradiction to Gβ not being an allocation skeleton
+of z.
+Finally, the mapping origE remains unchanged and the only change to origV is to include
+two new vertices mapping to already existing vertices. Due to condition 4 (orig-real) holding
+for both the input and the output instance, this cannot affect the represented graph GS′.
+Now consider the skeleton decomposition S′ returned by JoinSeparationPair. Identify-
+ing distinct vertices of distinct connected components does not affect their biconnectivity,
+thus condition 1 (bicon) remains satisfied. The operation effectively contracts and removes
+an edge in TS, which does not affect TS′ being a tree satisfying condition 2 (tree). Note
+that condition 2 (tree) holding for the input instance also ensures that Gα and Gβ are two
+distinct skeletons. As the input instance also satisfies condition 5 (orig-virt), there are exactly
+two vertices in each of the two adjacent skeletons Gα and Gβ, where origV maps to the
+same vertex of GS. These two vertices must be part of the twinE pair making the two
+skeletons adjacent, thus they are exactly the two pairs of vertices we identify with each other.
+Thus, origV |Vµ is still injective, satisfying condition 3 (orig-inj). As we modify no real edges
+and no other virtual edges, the mappings origV′ and origE′ obviously still satisfy condition
+4 (orig-real). As the allocation skeletons of each graph vertex form a connected subgraph,
+joining two skeletons cannot change the intersection with any of their neighbors, leaving
+5 (orig-virt) satisfied. Finally, contracting a tree edge cannot lead to any of the subgraphs of
+6 (subgraph) becoming disconnected, thus the condition also remains satisfied. Again, no
+changes were made to origE, while condition 5 (orig-virt) makes sure that origV mapped the
+two pairs of merged vertices to the same vertex of GS. Thus, the represented graph GS′
+remains unchanged.
+◀
+This gives us a second way of finding the represented graph by exhaustively joining all
+skeletons until there is only one left, obtaining the unique trivial skeleton decomposition:
+▶ Lemma 2. Exhaustively applying JoinSeparationPair to a skeleton decomposition S =
+(G, origV, origE, twinE) yields a trivial skeleton decomposition S′ = (G′, origV′, origE′, twinE′)
+where origE′ and origV′ define an isomorphism between G′
+µ and GS′.
+
+6
+Maintaining Triconnected Components under Node Expansion
+Proof. As all virtual edges are matched, and the matched virtual edge always belongs to
+a different skeleton (condition 2 (tree) ensures that TS is loop-free), we can always apply
+JoinSeparationPair on a virtual edge until there are none left. As TS is connected, this
+means that the we always obtain a tree with a single node, that is an instance with only a
+single skeleton. As a single application of JoinSeparationPair preserves the represented
+graph, any chain of multiple applications also does. Note that origE′ is a bijection and the
+surjective origV′ is also injective on the single remaining skeleton due to condition 3 (orig-inj),
+thus it also globally is a bijection. Together with condition 4 (orig-real), this ensures that any
+two vertices u and v of G′
+µ are adjacent if and only if origV′(u) and origV′(v) are adjacent
+in GS′. Thus origV′ is an edge-preserving bijection, that is an isomorphism.
+◀
+A key point about the skeleton decomposition and especially the operation SplitSepa-
+rationPair now is that they model the decomposition of a graph at separation pairs. This
+decomposition was formalized as SPQR-tree by Di Battista and Tamassia [16] and is unique
+for a given graph [33, 36]; see also [28, 30]. Angelini et al. [1] describe a decomposition
+tree that is conceptually equivalent to our skeleton decomposition. They also present an
+alternative definition for the SPQR-tree as a decomposition tree satisfying further properties.
+We adopt this definition for our skeleton decompositions as follows, not requiring planarity
+of triconnected components and allowing virtual edges and real edges to appear within one
+skeleton (i.e., having leaf Q-nodes merged into their parents).
+▶ Definition 3. A skeleton decomposition S = (G, origV, origE, twinE) where any skeleton
+in G is either a polygon, a bond, or triconnected (“rigid”), and two skeletons adjacent in TS
+are never both polygons or both bonds, is the unique SPQR-tree of GS.
+The main difference between the well-known ideas behind decomposition trees and our
+skeleton decomposition is that the latter allow an axiomatic access to the decomposition at
+separation pairs. For the skeleton decomposition, we employ a purely functional, “mechanical”
+data structure instead of relying on and working along a given graph structure. In our
+case, the represented graph is deduced from the data structure (i.e. SPQR-tree) instead of
+computing the data structure from the graph.
+4
+Extended Skeleton Decompositions
+Note that most skeletons, especially polygons and bonds, can easily be decomposed into
+smaller parts. The only exception to this are triconnected skeletons which cannot be split
+further using the operations we defined up to now. This is a problem when modifying
+a vertex that occurs in triconnected skeletons that may be much bigger than the direct
+neighborhood of the vertex. To fix this, we define a further set of operations which allow
+us to isolate vertices out of arbitrary triconnected components by replacing them with a
+(“virtual”) placeholder vertex. This placeholder then points to a smaller component that
+contains the actual vertex, see Figure 2. Modification of the edges incident to the placeholder
+is disallowed, which is why we call them “occupied”.
+Formally, the structures needed to keep track of the components split in this way
+in an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) are defined as
+follows. Skeletons now have the form Gµ = (Vµ ·∪ V virt
+µ
+, Ereal
+µ
+·∪ Evirt
+µ
+·∪ Eocc
+µ ). Bijection
+twinV : V virt → V virt matches all virtual vertices V virt = �
+µ V virt
+µ
+, such that twinV(v) ̸= v,
+twinV2 = id. The edges incident to virtual vertices are contained in Eocc
+µ
+and thus considered
+occupied; see Figure 2b. Similar to the virtual edges matched by twinE, any two virtual
+vertices matched by twinV induce an edge between their skeletons in TS. Condition 2 (tree)
+
+S. D. Fink and I. Rutter
+7
+v
+Gµ
+u
+(a)
+v
+vα
+vβ
+Gα
+Gβ
+uα
+uβ
+(b)
+Figure 2 (a) A triconnected skeleton Gµ with a highlighted vertex v incident to two gray virtual
+edges. (b) The result of applying IsolateVertex to isolate v out of the skeleton. The red occupied
+edges in the old skeleton Gα form a star with center vα, while the red occupied edges in Gβ connect
+all neighbors of v to form a star with center vβ ̸= v. The centers vα and vβ are virtual and matched
+with each other. Neighbor u of v was split into vertices uα and uβ.
+also equally applies to those edges induced by twinV, which in particular ensures that there
+are no parallel twinE and twinV tree edges in TS. Similarly, the connected subgraphs of
+condition 6 (subgraph) can also contain tree edges induced by twinV. All other conditions
+remain unchanged, but we add two further conditions to ensure that twinV is consistent:
+7 (stars) For each vα, vβ with twinV(vα) = vβ, it is deg(vα) = deg(vβ). All edges incident
+to vα and vβ are occupied and have distinct endpoints (except for vα and vβ). Conversely,
+each occupied edge is adjacent to exactly one virtual vertex.
+8 (orig-stars) Let vα and vβ again be two virtual vertices matched with each other by twinV.
+For their respective skeletons Gα and Gβ (where α and β are adjacent in TS), it is
+origV(Vα) ∩ origV(Vβ) = origV(N(vα)) = origV(N(vβ)).
+Note that both conditions together yield a bijection γvαvβ between the neighbors of
+vα and vβ, as origV is injective when restricted to a single skeleton (condition 3 (orig-
+inj)) and deg(vα) = deg(vβ). Operations SplitSeparationPair and JoinSeparationPair
+can also be applied to an extended skeleton decomposition, yielding an extended skeleton
+decomposition without modifying twinV. To ensure that conditions 7 (stars) and 8 (orig-stars)
+remain unaffected by both operations, SplitSeparationPair cannot be applied if a vertex
+of the separation pair is virtual.
+The operations IsolateVertex and Integrate now allow us to isolate vertices out of
+triconnected components and integrate them back in, respectively. For IsolateVertex, let v
+be a non-virtual vertex of skeleton Gµ, such that v has no incident occupied edges. Applying
+IsolateVertex(S, v) on an extended skeleton decomposition S yields an extended skeleton
+decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows. Each neighbor u of v is
+split into two non-adjacent vertices uα and uβ, where uβ is incident to all edges connecting u
+with v, while uα keeps all other edges of u. We set origV′(uα) = origV′(uβ) = origV(u). This
+creates an independent, star-shaped component with center v, which we move to skeleton
+Gβ, while we rename skeleton Gµ to Gα. We connect all uα to a single new virtual vertex
+vα ∈ V virt
+α
+using occupied edges, and all uβ to a single new virtual vertex vβ ∈ V virt
+β
+using
+occupied edges; see Figure 2. Finally, we set twinV′(vα) = vβ, twinV′(vβ) = vα, and add Gβ
+to G′. All other mappings and skeletons remain unchanged.
+For Integrate, consider two virtual vertices vα, vβ with twinV(vα) = vβ and the bijec-
+tion γvαvβ between the neighbors of vα and vβ. An application of Integrate(S, (vα, vβ))
+yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows.
+We merge both skeletons into a skeleton Gµ (also replacing both in G′) by identifying the
+neighbors of vα and vβ according to γvαvβ. Furthermore, we remove vα and vβ together with
+their incident occupied edges. All other mappings and skeletons remain unchanged.
+
+8
+Maintaining Triconnected Components under Node Expansion
+▶ Lemma 4. Applying IsolateVertex or Integrate on an extended skeleton decomposition
+S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′,
+origV′, origE′, twinE′, twinV′) with GS′ = GS.
+Proof. We first check that all conditions still hold in the extended skeleton decomposition S′
+returned by IsolateVertex. Condition 1 (bicon) remains satisfied, as the structure of Gα
+remains unchanged compared to Gµ and the skeleton Gβ is a bond. As we are again splitting
+a node of TS, condition 2 (tree) also remains satisfied. Due to the neighbors of vβ and vα
+mapping to the same vertices of GS′, conditions 3 (orig-inj), 4 (orig-real), and 5 (orig-virt)
+remain satisfied. Conditions 7 (stars) and 8 (orig-stars) are satisfied by construction.
+Lastly, condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the
+allocation skeletons of some vertex z ∈ GS′ was no longer connected. This could only happen
+if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further
+neighbor Gν that is also an allocation skeleton of z. Note that in any case, ν is adjacent
+to µ in TS and µ must be an allocation skeleton of z, thus it is z ∈ origV(Gν) ∩ origV(Gµ).
+Depending on the adjacency of ν, it is either origV(Gν)∩origV(Gµ) = origV′(Gν)∩origV(Gα)
+or origV(Gν) ∩ origV(Gµ) = origV′(Gν) ∩ origV(Gβ), as ν is not modified by the operation
+and both S and S′ satisfy 5 (orig-virt) and 8 (orig-stars). This immediately contradicts the
+skeleton of {α, β}, that is adjacent to ν, not being an allocation skeleton of z.
+Finally, the mapping origE remains unchanged and the only change to origV is to include
+some duplicated vertices mapping to already existing vertices. Due to condition 4 (orig-real)
+holding for both the input and the output instance, this cannot affect the represented graph
+GS′.
+Now consider the extended skeleton decomposition S′ returned by Integrate. The
+merged skeleton is biconnected, as we are effectively replacing a single vertex by a connected
+subgraph, satisfying 1 (bicon). The operation effectively contracts and removes an edge in
+TS, which does not affect TS′ being a tree, satisfying condition 2 (tree). Note that condition
+2 (tree) holding for the input instance also ensures that vα and vβ belong to two distinct
+skeletons. As the input instance satisfies condition 5 (orig-virt), the vertices in each of the
+two adjacent skeletons where origV maps to the same vertex of GS are exactly the neighbors
+of the matched vα and vβ. Thus, origV |Vα is still injective, satisfying condition 3 (orig-inj).
+As we modify no real or virtual edges, the mappings origV′, origE′ and twinE′ obviously still
+satisfy conditions 4 (orig-real) and 5 (orig-virt). Finally, contracting a tree edge cannot lead to
+any of the subgraphs of 6 (subgraph) becoming disconnected, thus the condition also remains
+satisfied. Conditions 7 (stars) and 8 (orig-stars) also remain unaffected, as we simply remove
+an entry from twinV.
+Again, no changes were made to origE, while condition 8 (orig-stars) makes sure that
+origV mapped each pair of merged vertices to the same vertex of GS. Thus, the represented
+graph GS′ remains unchanged.
+◀
+Furthermore, as Integrate is the converse of IsolateVertex and has no preconditions,
+any changes made by IsolateVertex can be undone at any time to obtain a (non-extended)
+skeleton decomposition, and thus possibly the SPQR-tree of the represented graph.
+▶ Remark 5. Exhaustively applying Integrate to an extended skeleton decomposition
+S = (G, origV, origE, twinE, twinV) yields a extended skeleton decomposition S′ = (G′,
+origV′, origE′, twinE′, twinV′) where twinV′ = ∅. Thus, S′ is equivalent to a (non-extended)
+skeleton decomposition S′ = (G′, origV′, origE′, twinE′).
+
+S. D. Fink and I. Rutter
+9
+v
+Gµ
+(a)
+Gν
+(b)
+(c)
+Figure 3 Expanding a skeleton vertex v into a graph Gν in the SPQR-tree of Figure 4b. (a) The
+single allocation skeleton Gµ of u with the single allocation vertex v of u from Figure 4b. The
+neighbors of v are marked in orange. (b) The inserted graph Gν with orange marked vertices.
+Note that the graph is biconnected when all marked vertices are collapsed into a single vertex.
+(c) The result of applying InsertGraph(S, u, Gν, ϕ) followed by an application of Integrate on the
+generated virtual vertices v and v′.
+5
+Node Expansion in Extended Skeleton Decompositions
+We now introduce our first dynamic operation that allows us to actually change the represented
+graph by expanding a single vertex u into an arbitrary connected graph Gν. This is done
+by identifying |N(u)| marked vertices in Gν with the neighbors of u via a bijection ϕ and
+then removing u and its incident edges. We use the “occupied stars” from the previous
+section to model the identification of these vertices, allowing us to defer the actual insertion
+to an application of Integrate. We need to ensure that the inserted graph makes the same
+“guarantees” to the surrounding graph in terms of connectivity as the vertex it replaces,
+that is all neighbors of u (i.e. all marked vertices in Gν) need to be pairwise connected via
+paths in Gν not using any other neighbor of u (i.e. any other marked vertex). Without this
+requirement, a single vertex could e.g. also be split into two non-adjacent halves, which could
+easily break a triconnected component apart. Thus, we require Gν to be biconnected when
+all marked vertices are collapsed into a single vertex. Note that this also ensures that the
+old graph can be restored by contracting the vertices of the inserted graph. For the sake of
+simplicity, we require vertex u from the represented graph to have a single allocation vertex
+v ∈ Gµ with origV−1(u) = {v} so that we only need to change a single allocation skeleton
+Gµ in the skeleton decomposition. As we will make clear later on, this condition can be
+satisfied easily.
+Formally, let u ∈ GS be a vertex that only has a single allocation vertex v ∈ Gµ (and
+thus only a single allocation skeleton Gµ). Let Gν be an arbitrary, new graph containing
+|N(u)| marked vertices, together with a bijection ϕ between the marked vertices in Gν
+and the neighbors of v in Gµ. We require Gν to be biconnected when all marked vertices
+are collapsed into a single node. Operation InsertGraph(S, u, Gν, ϕ) yields an extended
+skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows, see also Figure 3.
+We interpret Gν as skeleton and add it to G′. For each marked vertex x in Gν, we set
+origV′(x) = origV(ϕ(x)). For all other vertices and edges in Gν, we set origV′ and origE′
+to point to new vertices and edges forming a copy of Gν in GS′. We connect every marked
+vertex in Gν to a new virtual vertex v′ ∈ Gν using occupied edges. We also convert v to a
+virtual vertex, converting its incident edges to occupied edges while removing parallel edges.
+Finally, we set twinV′(v) = v′ and twinV′(v′) = v.
+▶ Lemma 6. Applying InsertGraph(S, u, Gν, ϕ) on an extended skeleton decomposition
+S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′,
+origV′, origE′, twinE′, twinV′) with GS′ isomorphic to GS[u →ϕ Gν].
+
+10
+Maintaining Triconnected Components under Node Expansion
+Proof. Condition 1 (bicon) remains satisfied, as the structure of Gµ remains unchanged
+and the resulting Gν is biconnected by precondition. Regarding TS, we are attaching a
+degree-1 node ν to an existing node µ, thus condition 2 (tree) also remains satisfied. As
+all vertices of Gν except for the vertices in N(v′) got their new, unique copy assigned by
+origV′ and origV′(N(v′)) = origV(N(v)), condition 3 (orig-inj) is also satisfied for the new
+Gν. As we updated origE alongside origV and Gν contains no virtual edges, conditions
+4 (orig-real) and 5 (orig-virt) remain satisfied. As ν is a leaf of TS with µ being its only
+neighbor, origV′(N(v′)) ⊂ origV(Vµ), and Gν is the only allocation skeleton for all vertices
+in Gν \ N(v′), condition 6 (subgraph) remains satisfied. Conditions 7 (stars) and 8 (orig-stars)
+are satisfied by construction. Finally, the mappings origE′ and origV′ are by construction
+updated to correctly reproduce the structure of Gν in GS′.
+◀
+On its own, this operation is not of much use though, as graph vertices only rarely have
+a single allocation skeleton. Furthermore, our goal is to dynamically maintain SPQR-trees,
+while this operation on its own will in most cases not yield an SPQR-tree. To fix this, we
+introduce the full procedure InsertGraphSPQR(S, u, Gν, ϕ) that can be applied to any graph
+vertex u and that, given an SPQR-tree S, yields the SPQR-tree of GS[u →ϕ Gν]. It consists
+of three preparations steps, the insertion of Gν, and two further clean-up steps:
+1. We apply SplitSeparationPair to each polygon allocation skeleton of u with more than
+three vertices, using the neighbors of the allocation vertex of u as separation pair.
+2. For each rigid allocation skeleton of u, we move the contained allocation vertex v of u to
+its own skeleton by applying IsolateVertex(S, v).
+3. We exhaustively apply JoinSeparationPair to any pair of allocation skeletons of u that
+are adjacent in TS. Due to condition 6 (subgraph), this yields a single component Gµ that
+is the sole allocation skeleton of u with the single allocation vertex v of u. Furthermore,
+the size of Gµ is linear in deg(u).
+4. We apply InsertGraph to insert Gν as skeleton, followed by an application of Integrate
+to the virtual vertices {v, v′} introduced by the insertion, thus integrating Gν into Gµ.
+5. We apply SplitSeparationPair to all separation pairs in Gµ that do not involve a
+virtual vertex. These pairs can be found in linear time, e.g. by temporarily duplicating
+all virtual vertices and their incident edges and then computing the SPQR-tree.2
+6. Finally, we exhaustively apply Integrate and also apply JoinSeparationPair to any
+two adjacent polygons and to any two adjacent bonds to obtain the SPQR-tree of the
+updated graph.
+The basic idea behind the correctness of this procedure is that splitting the newly inserted
+component according to its SPQR-tree in step 5 yields biconnected components that are each
+either a polygon, a bond, or “almost” triconnected. The latter (and only those) might still
+contain virtual vertices and all their remaining separation pairs, which were not split in step 5,
+contain one of these virtual vertices. This, together with the fact that there still may be
+pairs of adjacent skeletons where both are polygons or both are bonds, prevents the instance
+from being an SPQR-tree. Both issues are resolved in step 6: The adjacent skeletons are
+obviously fixed by the JoinSeparationPair applications. To show that the virtual vertices
+are removed by the Integrate applications, making the remaining components triconnected,
+we need the following lemma.
+2 Note that the wheels replacing virtual vertices in the proof of Theorem 10 also ensure this.
+
+S. D. Fink and I. Rutter
+11
+(a)
+v
+(b)
+Figure 4 The preprocessing steps of InsertGraphSPQR being applied to the SPQR-tree of Figure 1b.
+(a) The state after step 2, after all allocation skeletons of u have been split. (b) The state after
+step 3, after all allocation skeletons of u have been merged into a single one.
+▶ Lemma 7. Let Gα be a triconnected skeleton containing a virtual vertex vα matched with
+a virtual vertex vβ of a biconnected skeleton Gβ. Furthermore, let P ⊆
+�V (Gβ)
+2
+�
+be the set
+of all separation pairs in Gβ. An application of Integrate(S, (vα, vβ)) yields a biconnected
+skeleton Gµ with separation pairs P ′ = {{u, v} ∈ P | vβ /∈ {u, v}}.
+Proof. We partition the vertices of Gµ into the sets A, B, and N depending on whether the
+vertex stems from Gα, Gβ, or both, respectively. The set N thus contains the neighbors of vα,
+which were identified with the neighbors of vβ. We will now show by contradiction that Gµ
+contains no separation pairs except for those in P ′. Thus, consider a separation pair u, v ∈ Gµ
+not in P ′. First, consider the case where u, v ∈ A∪N. Observe that removing u, v in this case
+leaves B connected. Thus, we can contract all vertices of B into a single vertex, reobtain Gα
+and see that u, v is a separation pair in Gα. This contradicts the precondition that Gα is
+triconnected. Now consider the case where u, v ∈ B ∪ N. Analogously to above, we find that
+u, v is a separation pair in Gβ that does not contain vβ, a contradiction to {u, v} /∈ P ′. Finally,
+consider the remaining case where, without loss of generality, u ∈ A, v ∈ B. Since {u, v}
+is a separation pair, u has two neighbors x, y that lie in different connected components of
+Gµ−{u, v} and therefore also in different components of (Gµ−{u, v})−B which is isomorphic
+to Gα − {u, vα}. This again contradicts the precondition that Gα is triconnected.
+◀
+▶ Theorem 8. Applying InsertGraphSPQR(S, u, Gν, ϕ) to an SPQR-tree S yields an SPQR-
+tree S′ in O(|Gν|) time with GS′ isomorphic to GS[u →ϕ Gν].
+Proof. As all operations that are applied leave the extended skeleton decomposition valid,
+the final extended skeleton decomposition S′ is also valid. Observe that the purpose of
+the preprocessing steps 1–3 is solely to ensure that the preconditions of InsertGraph are
+satisfied and the affected component is not too large. Note that all rigids split in step 2
+remain structurally unmodified in the sense that edges only changed their type, but the
+graph and especially its triconnectedness remains unchanged. Step 4 performs the actual
+insertion and yields the desired represented graph according to Lemma 6. It thus remains
+to show that the clean-up steps turn the obtained extended skeleton decomposition into
+an SPQR-tree.
+Applying Integrate exhaustively in step 6 ensures that the extended
+skeleton decomposition is equivalent to a non-extended one (Remark 5). Recall that a
+non-extended skeleton decomposition is an SPQR-tree if all skeletons are either polygons,
+bonds or triconnected and two adjacent skeletons are never both polygons or both bonds
+(Definition 3). Step 6 ensures that the second half holds, as joining two polygons (or two
+bonds) with JoinSeparationPair yields a bigger polygon (or bond, respectively). Before
+
+12
+Maintaining Triconnected Components under Node Expansion
+step 6, all skeletons that are not an allocation skeleton of u are still unmodified and thus
+already have a suitable structure, i.e., they are either polygons, bonds or triconnected.
+Furthermore, the allocation skeletons of u not containing virtual vertices also have a suitable
+structure, as their splits were made according to the SPQR-tree in step 5. It remains to
+show that the remaining skeletons, that is those resulting from the Integrate applications
+in step 6, are triconnected. Note that in these skeletons, step 5 ensures that every separation
+pair consists of at least one virtual vertex, as otherwise the computed SPQR-tree would
+have split the skeleton further. Further note that, for each of these virtual vertices, the
+matched partner vertex is part of a structurally unmodified triconnected skeleton that was
+split in step 2. Lemma 7 shows that applying Integrate does not introduce new separation
+pairs while removing two virtual vertices if one of the two sides is triconnected. We can
+thus exhaustively apply Integrate and thereby remove all virtual vertices and thus also all
+separation pairs, obtaining triconnected components. This shows that the criteria for being
+an SPQR-tree are satisfied and, as InsertGraph expanded u to Gν in the represented graph,
+we now have the unique SPQR-tree of GS[u →ϕ Gν].
+Note that all operations we used can be performed in time linear in the degree of the
+vertices they are applied on. For the bipartition of bridges input to SplitSeparationPair,
+it is sufficient to describe each bridge via its edges incident to the separation pair instead of
+explicitly enumerating all in vertices in the bridge. Thus, the applications of SplitSepara-
+tionPair and IsolateVertex in steps 1 and 2 touch every edge incident to u at most once
+and thus take O(deg(u)) time. Furthermore, they yield skeletons that have a size linear in
+the degree of their respective allocation vertex of u. As the subtree of u’s allocation skeletons
+has size at most deg(u), the JoinSeparationPair applications of step 3 also take at most
+O(deg(u)) time. It also follows that the resulting single allocation skeleton of u has size
+O(deg(u)). The applications of InsertGraph and Integrate in step 4 can be done in time
+linear in the number of identified neighbors, which is O(deg(u)). Generating the SPQR-tree
+of the inserted graph in step 5 (where all virtual vertices where replaced by wheels) can
+be done in time linear in the size of the inserted graph [30, 33], that is O(|Gν|). Applying
+SplitSeparationPair according to all separation pairs identified by this SPQR-tree can
+also be done in O(|Gν|) time in total. Note that there are at most deg(u) edges between
+the skeletons that existed before step 4 and those that were created or modified in steps 4
+and 5, and these are the only edges that might now connect two polygons or two bonds. As
+these tree edges have one endpoint in the single allocation skeleton of u, the applications of
+Integrate and JoinSeparationPair in step 6 run in O(deg(u)) time in total. Furthermore,
+they remove all pairs of adjacent polygons and all pairs of adjacent bonds. This shows that
+all steps take O(deg(u)) time, except for step 5, which takes O(|Gν|) time. As the inserted
+graph contains at least one vertex for each neighbor of u, the total runtime is in O(|Gν|).
+◀
+▶ Corollary 9. Let S1, S2 be two SPQR-trees together with vertices u1 ∈ GS1, u2 ∈ GS2, and
+let ϕ be a bijection between the edges incident to u1 and the edges incident to u2. Operation
+MergeSPQR(S1, S2, u1, u2, ϕ) yields the SPQR-tree of the graph GS1[u1 →ϕ GS2 − u2], i.e. the
+union of both graphs where the edges incident to u1, u2 were identified according to ϕ and
+u1, u2 removed, in time O(deg(u1)) = O(deg(u2)).
+Proof. Operation MergeSPQR works similar to the more general InsertGraphSPQR, although
+the running time is better because we already know the SPQR-tree for the graph being
+inserted. We apply the preprocessing steps 1–3 to ensure that both u1 and u2 have sole
+allocation vertices v1 and v2, respectively. To properly handle parallel edges, we subdivide
+all edges incident to u1, u2 (and thus also the corresponding real edges incident to v1, v2) and
+
+S. D. Fink and I. Rutter
+13
+then identify the subdivision vertices of each pair of edges matched by ϕ. By deleting vertices
+v1 and v2 and suppressing the subdivision vertices (that is, removing them and identifying
+each pair of incident edges) we obtain a skeleton Gµ that has size O(deg(u1)) = O(deg(u2)).
+Finally, we apply the clean-up steps 5 and 6 to Gµ to obtain the final SPQR-tree. Again,
+as the partner vertex of every virtual vertex in the allocation skeletons of u is part of a
+triconnected skeleton, applying Integrate exhaustively in step 6 yields triconnected skeletons.
+As previously discussed, the preprocessing and clean-up steps run in time linear in degree of
+the affected vertices, thus the overall runtime is O(deg(u1)) = O(deg(u2)) in this case.
+◀
+5.1
+Maintaining Planarity and Vertex Rotations
+Note that expanding a vertex of a planar graph using another planar graph using Insert-
+GraphSPQR (or merging two SPQR-trees of planar graphs using Corollary 9) might actually
+yield a non-planar graph. This is, e.g., because the rigids of both graphs might require
+incompatible orders for the neighbors of the replaced vertex. The aim of this section is to
+efficiently detect this case, that is a planar graph turning non-planar. To check a general
+graph for planarity, it suffices to check the rigids in its SPQR-tree for planarity and each rigid
+allows exactly two planar embeddings, where one is the reverse of the other [19]. Thus, if a
+graph becomes non-planar through an application of InsertGraphSPQR, this will be noticeable
+from the triconnected allocation skeletons of the replaced vertex. To be able to immediately
+report if the instance became non-planar, we need to maintain a rotation, that is a cyclic
+order of all incident edges, for each vertex in any triconnected skeleton. Note that we do not
+track the direction of the orders, that is we only store the order up to reversal. As discussed
+later, the exact orders can also be maintained with a slight overhead.
+▶ Theorem 10. SPQR-trees support the following operations:
+InsertGraphSPQR(S, u, Gν, ϕ): expansion of a single vertex u in time O(|Gν|),
+MergeSPQR(S1, S2, u1, u2, ϕ): merging of two SPQR-trees in time O(deg(u1)),
+IsPlanar: queries whether the represented graph is planar in time O(1), and
+Rotation(u): queries for one of the two possible rotations of vertices u in planar tricon-
+nected skeletons in time O(1).
+Proof. Note that the boolean flag IsPlanar together with the Rotation information can
+be computed in linear time when creating a new SPQR-tree and that expanding a vertex or
+merging two SPQR-trees cannot turn a non-planar graph planar. We make the following
+changes to the operations InsertGraphSPQR and MergeSPQR described in Theorem 8 and
+Corollary 9 to maintain the new information. After a triconnected component is split in
+step 2 we now introduce further structure to ensure that the embedding is maintained on both
+sides. The occupied edges generated around the split-off vertex v (and those around its copy
+v′) are subdivided and connected cyclically according to Rotation(v). Instead of “stars”, we
+thus now generate occupied “wheels” that encode the edge ordering in the embedding of the
+triconnected component. When generating the SPQR-tree of the modified subgraph in step 5,
+now containing occupied wheels instead of only stars, we also generate a planar embedding for
+all its triconnected skeletons. If no planar embedding can be found for at least one skeleton,
+we report that the resulting instance is non-planar by setting IsPlanar to false. Otherwise,
+after performing all splits indicated by the SPQR-tree, we assign Rotation by generating
+embeddings for all new rigids. Note that for all skeletons with virtual vertices, the generated
+embedding will be compatible with the one of the neighboring triconnected component, that
+is, the rotation of each virtual vertex will line up with that of its matched partner vertex,
+thanks to the inserted wheel. Finally, before applying Integrate in step 6, we contract each
+
+14
+Maintaining Triconnected Components under Node Expansion
+occupied wheel into a single vertex to re-obtain occupied stars. The creation and contraction
+of wheels adds an overhead that is at most linear in the degree of the expanded vertex and
+the generation of embeddings for the rigids can be done in time linear in the size of the rigid.
+Thus, this does not affect the asymptotic runtime of both operations.
+◀
+▶ Corollary 11. The data structure from Theorem 10 can be adapted to also provide the exact
+rotations with matching direction for every vertex in a rigid. Furthermore, it can support
+queries whether two vertices v1, v2 are connected by at least 3 different vertex-disjoint paths
+via 3Paths(v1, v2) in O((deg(v1)+deg(v2))·α(n)) time. These adaptions change the runtime
+of InsertGraphSPQR to O(deg(u) · α(n) + |Gν|), that of MergeSPQR to O(deg(u1) · α(n)), and
+that of Rotation(u) to O(α(n)).
+Proof. The exact rotation information for Rotation can be maintained by using union-find
+to keep track of the rigid a vertex belongs to and synchronizing the reversal of all vertices
+within one rigid when two rigids are merged by Integrate as follows. We create a union-find
+set for every vertex in a triconnected component and apply Union to all vertices in the same
+rigid. Next to the pointer indicating the representative in the union-find structure, we store
+a boolean flag indicating whether the rotation information for the current vertex is reversed
+with regard to rotation of its direct representative. To find whether a Rotation needs to
+be flipped, we accumulate all flags along the path to the actual representative of a vertex
+by using an exclusive-or. As Rotation(u) thus relies on the Find operation, its amortized
+runtime is O(α(n)). When merging two rigids with Integrate, we also perform a Union on
+their respective representatives (which we need to Find first), making Integrate(S, (vα, vβ))
+run in O(deg(vα) + α(n)). We also compare the Rotation of the replaced vertices and flip
+the flag stored with the vertex that does not end up as the representative if they do not
+match. In total, this makes InsertGraphSPQR run in O(deg(u) · α(n) + |Gν|) time as there
+can be up to deg(u) split rigids. Furthermore, MergeSPQR now runs in O(deg(u1) · α(n)) time.
+Maintaining the information in which rigid a skeleton vertex is contained in can then
+also be used to answer queries whether two arbitrary vertices are connected by three disjoint
+paths. This is exactly the case if they are part of the same rigid, appear as poles of the same
+bond or are connected by a virtual edge in a polygon. This can be checked by enumerating
+all allocation skeletons of both vertices, which can be done in time linear in their degree.
+As finding each of the skeletons may require a Find call, the total runtime for this is in
+O((deg(v1) + deg(v2)) · α(n)).
+◀
+6
+Application to Synchronized Planarity
+In this section, we will give some background on the historical development of and further
+details on the problems Clustered Planarity and Synchronized Planarity together
+with summary of the algorithm of Bläsius et al. for solving both problems. Furthermore,
+we will show how our and also previous work on dynamic SPQR-trees can be used in the
+context of both problems.
+6.1
+Background and Discussion
+Lengauer [34] first discussed Clustered Planarity under a different name in 1989, which is
+why it was later independently rediscovered by Feng et al. [23] in 1995. Both gave polynomial-
+time algorithms for the case where the subgraph induced by any cluster is connected. In
+contrast, the question whether the general problem with disconnected clusters allows an
+
+S. D. Fink and I. Rutter
+15
+Figure 5 Schematic representation of the three operations used by Bläsius et al. [8] for solving
+Synchronized Planarity. Matched vertices are shown as bigger disks, the matching is indicated
+by the orange dotted lines. Top: Two cut-vertices matched with each other (left), the result of
+encapsulating their incident blocks (middle) and the bipartite graph resulting from joining both
+cut-vertices (right). Middle: A matched non-cut-vertex with a non-trivial embedding tree (left)
+that is propagated to replace both the vertex and its partner (right). Bottom: Three different
+cases of matched vertices with trivial embedding trees (blue) and how their pipes can be removed or
+replaced (red).
+efficient solution remained open for 30 years. In that time, polynomial-time algorithms were
+found for many special-cases [2, 15, 25, 29] before Fulek and Tóth [26] found an O((n + d)8)
+solution in 2019.
+Shortly thereafter, Bläsius et al. [8] gave a solution with runtime in
+O((n + d)2) that also exposes the main concepts needed to solve Clustered Planarity.
+The solution works via a linear-time reduction to the problem Synchronized Planarity,
+for which Bläsius et al. gave a quadratic algorithm. We improve the runtime of the latter
+algorithm. As Synchronized Planarity can be used as a modeling tool for several other
+constrained planarity problems next to Clustered Planarity [8], this also improves the
+time needed for solving any constrained planarity problem that can be solved via a linear-time
+reduction to Synchronized Planarity; see Table 1.
+In Clustered Planarity, the embedding has to respect a laminar family of clusters [9,
+34], that is every vertex is part of some (hierarchically nested) cluster and an edge may
+only cross a cluster boundary if it connects a vertex from the inside with one from the
+outside. In Synchronized Planarity, we are given a matching on some of the vertices in
+the graph and seek an embedding such that the rotations matched vertices line up under a
+given bijection [8]. The synchronization constraints imposed by matching two vertices are
+also called pipe. The reduction from the former problem to the latter employs the CD-tree
+representation of Clustered Planarity [9], where each cluster is represented as individual
+skeleton in which adjacent clusters were collapsed into single “virtual vertices”. The order
+of the edges “leaving” one cluster via a virtual vertex now needs to line up with the order
+in which they “enter” an adjacent cluster via its corresponding virtual vertex (see also [8,
+Figure 6]).
+
+16
+Maintaining Triconnected Components under Node Expansion
+The algorithm for solving Synchronized Planarity works by removing an arbitrary
+pipe each step, using one of three operations depending on the graphs around the matched
+vertices, see Figure 5.
+EncapsulateAndJoin If both vertices of the pipe are cut-vertices, they are “encapsulated”
+by taking a copy of their respective components and then collapsing each incident block
+to a single vertex to obtain stars with matched centers that have multiple parallel edges
+connecting them to their ray vertices. The original cut-vertices are split up so that each
+incident block gets its own copy and these copies are synchronized with the respective
+vertex representing a collapsed block. Now the cut-vertices can be removed by “joining”
+both stars, that is identifying their incident edges according to the bijection that is given
+alongside the matching.
+PropagatePQ If one of the vertices is not a cut-vertex and has an embedding tree that
+not only consists of a single P-node, two copies of this embedding tree are inserted
+(“propagated”) in place of both matched vertices, respectively. The inner nodes of the
+embedding trees are synchronized by matching corresponding vertices.
+SimplifyMatching In the remaining case, one of the vertices is not a cut-vertex but has a
+trivial embedding tree, i.e., only appears in a single parallel skeleton and no rigid skeleton
+in the SPQR-tree. If the vertex (or, more precisely, the parallel that completely defines
+it rotation) can respect arbitrary rotations, we can simply remove the pipe. The only
+exception to this is when the other pole of the parallel is also matched, in which case we
+can “short-circuit” the matching across the parallel.
+To summarize, every operation removes a pipe from the matching, while potentially
+introducing new pipes with vertices that have a smaller degree. Using a potential function,
+it can be shown that the progress made by the removal always dominates overhead of the
+newly-introduced pipes, and that the operations needed to remove all pipes is limited by the
+total degree of all matched vertices. Furthermore, the resulting instance without pipes can
+be solved in linear time. All of the three operations run in time linear in the degree of the
+un-matched vertices if the embedding trees they depend on are available. The contribution of
+this paper is to efficiently provide the embedding trees, which would require processing entire
+connected components at each step when done naïvely. Using the fully-dynamic SPQR-tree
+by Holm and Rotenberg [31, 32], this can be achieved with a poly-log cost of O(∆ · log3 n)
+leading to an overall runtime of O(m · ∆ · log3 n). Using the node expansion from this paper,
+we can improve the runtime from spending time linear in the size of the input instance (O(m))
+for each of the linearly many operations, to only spending time linear in the maximum degree
+(O(∆)) on each operation. The reduction from Clustered Planarity creates an instance
+of size O(n+d) in which the total degree of matched vertices is in O(d), corresponding to the
+total number of times an edge crosses a cluster boundary. Note that, while this means that
+O(d) operations are sufficient to reach a reduced instance, the number of crossings between
+edges and cluster boundaries can be quadratic in the number of vertices in a planar graph.
+We also note that while the improvement over using the Holm and Rotenberg approach is
+only poly-logarithmic, our datastructure has the additional benefit of being conceptually
+simpler and thus also more likely to improve performance in practice.
+6.2
+Using Node Expansion for Solving Synchronized Planarity
+We show how extended skeleton decompositions and their dynamic operation InsertGraphSPQR
+can be used to improve the runtime of the algorithm for solving Synchronized Planarity
+by Bläsius et al. [8] from O(m2) to O(m · ∆), where ∆ is the maximum pipe degree. As
+
+S. D. Fink and I. Rutter
+17
+already explained in the previous section, the algorithm spends a major part of its runtime on
+computing so-called embedding trees, which describe all possible rotations of a single vertex
+in a planar graph and are used to communicate embedding restrictions between vertices with
+synchronized rotation. Once the embedding trees are available, the at most O(m) executed
+operations run in time linear in the degree of the pipe/vertex they are applied on, that is
+in O(∆) [8]. Thus, being able to generate these embedding trees efficiently by maintaining
+the SPQR-trees they are derived from is our main contribution towards the speedup of the
+Synchronized Planarity algorithm.
+An embedding tree Tv for a vertex v of a biconnected graph G describes the possible
+cyclic orderings or rotations of the edges incident to v in all planar embeddings of G [12].
+The leaves of Tv are the edges incident to v, while its inner nodes are partitioned into two
+categories: Q-nodes define an up-to-reversal fixed rotation of their incident tree edges, while
+P-nodes allow arbitrary rotation; see Figure 1d. To generate the embedding tree we use
+the observation about the relationship of SPQR-trees and embedding trees described by
+Bläsius and Rutter [10, Section 2.5]: there is a bijection between the P- and Q-nodes in the
+embedding tree of v and the bond and triconnected allocation skeletons of v in the SPQR-tree
+of G, respectively.
+▶ Lemma 12. Let S be an SPQR-tree with a planar represented graph GS. The embedding
+tree for a vertex v ∈ GS can be found in time O(deg(v)).
+Proof. We use the rotation information from Theorem 10 and furthermore maintain an
+(arbitrary) allocation vertex for each vertex in GS. To compute the embedding tree of a
+vertex v starting at the allocation vertex u of v, we will explore the SPQR-tree by using
+twinE on one of the edges incident to u and then finding the next allocation vertex of v
+as one endpoint of the obtained edge. If u has degree 2, it is part of a polygon skeleton
+that does not induce a node in the embedding tree. We thus move on to its neighboring
+allocation skeletons and will also similarly skip over any other polygon skeleton we encounter.
+If u has degree 3 or greater, we inspect two arbitrary incident edges: if they lead to the
+same vertex, u is the pole of a bond, and we generate a P-node. Otherwise it is part of a
+triconnected component, and we generate a Q-node. We now iterate over the edges incident
+to u, in the case of a triconnected component using the order given by the rotation of u. For
+each real edge, we attach a corresponding leaf to the newly generated node. The graph edge
+corresponding to the leaf can be obtained from origE. For each virtual edge, we recurse on
+the respective neighboring skeleton and attach the recursively generated node to the current
+node. As u can only be part of deg(u) many skeletons, which form a subtree of TS, and the
+allocation vertices of u in total only have O(deg(u)) many virtual and real edges incident,
+this procedure yields the embedding tree of u in time linear in its degree.
+◀
+Our data structure can now be used to reduce the runtime of solving Synchronized
+Planarity by generating an SPQR-tree upfront, maintaining it throughout all applied
+operations, and deriving any needed embedding tree from the SPQR-tree.
+▶ Theorem 13. Synchronized Planarity can be solved in time in O(m · ∆), where m is
+the number of edges and ∆ is the maximum degree of a pipe.
+Proof. The algorithm works by splitting the pipes representing synchronization constraints
+until they are small enough to be trivial. It does so by exhaustively applying the three
+operations EncapsulateAndJoin, PropagatePQ and SimplifyMatching depending on the
+graph structure around the pairs of synchronized vertices. As mentioned by Bläsius et al.,
+all operations run in time linear in the degree of the pipe they are applied on if the used
+
+18
+Maintaining Triconnected Components under Node Expansion
+embedding trees are known, and O(m) operations are sufficient to solve a given instance [8].
+Our modification is that we maintain an SPQR-tree for each biconnected component and
+then generate the needed embedding trees on-demand in linear time using Lemma 12. See
+Section 6.1 for more background on the Synchronized Planarity operations modified in
+the following.
+Operation SimplifyMatching can be applied if the graph around a synchronized vertex
+v allows arbitrary rotations of v, that is the embedding tree of v is trivial. In this case, the
+pipe can be removed without modifying the graph structure. Thus, we can now easily check
+the preconditions of this operations without making any changes to the SPQR-tree.
+PropagatePQ takes the non-trivial embedding tree of one synchronized vertex v and inserts
+copies of the tree in place of v and its partner, respectively. Synchronization constraints on
+the inner vertices of the inserted trees are used to ensure that they are embedded in the
+same way. We use InsertGraphSPQR to also insert the embedding tree into the respective
+SPQR trees, representing Q-nodes using wheels. When propagating into a cutvertex we also
+need to check whether two or more incident blocks merge. We form equivalence classes on
+the incident blocks, where two blocks are in the same class if 1) the two subtrees induced by
+their respective edges share at least two nodes 2) both induced subtrees share a C-node that
+has degree at least 2 in both subtrees. Blocks in the same equivalence class will end up in the
+same biconnected component as follows: We construct the subtree induced by all edges in
+the equivalence class and add a single further node for each block in the class, connecting all
+leaves to the node of the block the edges they represent lead to. We calculate the SPQR-tree
+for this biconnected graph and then merge the SPQR-trees of the individual blocks into it by
+applying Corollary 9. As InsertGraphSPQR (and similarly all MergeSPQR applications) runs in
+time linear in the size of the inserted PQ-tree, which is limited by the degree of the vertex it
+represents, this does not negatively impact the running time of the operation.
+Operation EncapsulateAndJoin generates a new bipartite component representing how
+the edges of the blocks incident to two synchronized cutvertices are matched with each other.
+The size of this component is linear in the degree of the synchronized vertices. Thus, we can
+freshly compute the SPQR-tree for the generated component in linear time, which also does
+not negatively impact the running time.
+Furthermore, as we now no longer need to iterate over whole connected components to
+generate the embedding trees, we are also no longer required to ensure those components do
+not grow to big. We can thus also directly contract pipes between two distinct biconnected
+components using Corollary 9 instead of having to insert PQ-trees using PropagatePQ. This
+may improve the practical runtime, as PropagatePQ might require further operations to
+clean up the generated pipes, while the direct contraction entirely removes a pipe without
+generating new ones.
+◀
+▶ Corollary 14. Clustered Planarity can be solved in time in O(n + d · ∆), where d
+is the total number of crossings between cluster borders and edges and ∆ is the maximum
+number of edge crossings on a single cluster border.
+Proof. Note that for a graph not containing parallel edges to be planar, the number of
+edges has to be linear in the number of vertices. We apply the reduction from Clustered
+Planarity to Synchronized Planarity as described by Bläsius et al. [8]. Ignoring the
+parallel edges generated by the CD-tree, we can generate an SPQR-tree for every component
+of the resulting instance in O(n) time in total. The instance contains one pipe for every
+cluster boundary, where the degree of a pipe corresponds to the number of edges crossing the
+respective cluster boundary. Thus, the potential described by Bläsius et al. [8], which sums
+
+S. D. Fink and I. Rutter
+19
+up the degrees of all pipes with a constant factor depending on the endpoints of each pipe,
+is in O(d). Each operation applied when solving the Synchronized Planarity instance
+runs in time O(∆) (the maximum degree of a pipe) and reduces the potential by at least 1.
+Thus, a reduced instance without pipes, which can be solved in linear time, can be reached
+in O(d · ∆) time.
+◀
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+Topological Two-Dimensional Gravity
+on Surfaces with Boundary
+Jan Troostb
+b Laboratoire de Physique de l’´Ecole Normale Sup´erieure
+CNRS, ENS, Universit´e PSL, Sorbonne Universit´e,
+Universit´e de Paris F-75005 Paris, France
+E-mail:
+[email protected]
+Abstract
+We solve two-dimensional gravity on surfaces with boundary in terms of contact
+interactions and surface degenerations. The known solution of the bulk theory in terms
+of a contact algebra is generalized to include boundaries and an enlarged set of boundary
+operators. The latter allow for a linearization of the Virasoro constraints in terms of an
+extended integrable KdV hierarchy.
+arXiv:2301.01008v1  [hep-th]  3 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Open Topological Gravity
+2
+3
+The Virasoro Algebra Representations
+4
+3.1
+The Bulk Representation of the Virasoro Algebra
+. . . . . . . . . . . . . . . .
+4
+3.2
+The Extended Virasoro Representation . . . . . . . . . . . . . . . . . . . . . .
+5
+3.3
+The Recursion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+3.4
+The Generalized Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.5
+Amplitudes
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4
+The Extended Partition Function
+11
+4.1
+The Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.2
+A Few More Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5
+Conclusions
+13
+1
+Introduction
+Two-dimensional gravity on closed Riemann surfaces was solved in terms of matrix models
+[1–3], conformal field theory [4–6] and intersection theory [7,8]. While aspects of gravity on
+Riemann surfaces with boundary were partially understood in terms of matrix models early
+on [9, 10], a rigorous theory of topological gravity on Riemann surfaces with boundary was
+only recently established [11]. Since then, various perspectives on these theories have been
+developed [12–17]. The main approaches are through geometry and matrix models. The points
+of view provided by these methods on the resulting integrable KdV hierarchy are qualitatively
+distinct and usefully complementary.
+Two-dimensional gravity on closed Riemann surfaces was also understood in a conformal
+field theory approach closely related to string theory [18]. The theory was solved in terms of
+Virasoro recursion relations. These relations were derived from a contact algebra for vertex
+operators that carries all the topological information provided by the surface as well as the
+bundles on the moduli space of surfaces [7].
+Our goal in this paper is to extend the contact algebra approach [18] to topological gravity
+on Riemann surfaces with boundary. To that end, we study the contact algebra for operators
+in the presence of boundaries as well as how the bulk algebra is represented on an extended
+set of boundary vertex operators. Through representation theory and consistency conditions,
+we fix all constants in the extended open Virasoro algebra, and manage to derive the Virasoro
+recursion relation for the open and closed partition functions. Given a few initial correlators,
+this allows to solve the theory.
+The paper is structured as follows. In section 2 we review salient features of topological
+gravity on Riemann surfaces with boundary [11]. The extended representation of the bulk
+vertex operator contact algebra on the boundary vertex operators is constructed in section
+3 using consistency arguments. In section 4 the constraints are translated into a differential
+1
+
+Virasoro algebra that acts on the generating function of topological correlators. At that point,
+we make contact with the extended open string partition function [12] which is sufficient to
+prove that the solution to the Virasoro constraints indeed coincides with the known solution
+of open topological gravity. We conclude in section 5 with a summary and suggestions for
+future research.
+2
+Open Topological Gravity
+In this section, we recall features of the solution of open and closed topological gravity, respec-
+tively on Riemann surfaces with [11] or without boundary [7]. For open topological gravity,
+we indicate a few features of the rigorous geometric solution [11]. For topological gravity on
+Riemann surfaces (without boundary), we also briefly recall aspects of the solution in terms
+of a conformal field theory [18] with contact interactions. We then start out on the path to
+generalize that solution to Riemann surfaces with boundary.1
+Riemann Surfaces and Carriers of Curvature
+Topological gravity on Riemann surfaces (without boundary) [7] satisfies the ghost number
+conservation equation – or the dimension constraint on the integral over the moduli space of
+surfaces –:
+3g − 3 + nc =
+nc
+�
+i=1
+nc
+i .
+(2.1)
+The genus of the Riemann surface is g. The number of bulk vertex operator insertions is
+nc and nc
+i are the labels of the bulk vertex operators referring to the power of the tangent
+line bundle at a point [7]. A central idea in [18] was to graft the curvature associated to the
+Riemann surface itself onto the bulk vertex operators such that all topological properties of
+the theory are captured by local operators – this in turn allows for the solution of the theory
+in terms of contact interactions. When we associate a curvature 2(nc
+i − 1)/3 to each bulk
+vertex operator τnc
+i of power nc
+i, then ghost number conservation implies that:
+The Integrated Curvature = 2g − 2 =
+nc
+�
+i=1
+2
+3(nc
+i − 1) ,
+(2.2)
+namely that the curvature of the surface is faithfully represented. The puncture operator τ0
+has the smallest curvature contribution equal to −2/3, while the dilaton operator τ1 carries
+no curvature at all. All other operators carry positive curvature (in this convention).
+Riemann Surfaces with Boundary
+The integration over the moduli space of Riemann surfaces with boundaries and with boundary
+and bulk insertions leads to the dimensionality constraint valid for non-zero open correlation
+functions [11]:
+3g′ − 3 + no + 2nc = 2
+nc
+�
+i=1
+nc
+i .
+(2.3)
+1See also [19] for an interesting alternative.
+2
+
+The doubled genus g′ is the genus of the Riemann surface that is obtained by gluing a given
+Riemann surface with at least one boundary to its reflection. We therefore have the relation
+g′ = 2g + b − 1 where b is the number of boundaries of the original surface and g its genus.
+The number of boundary operator insertions σ is no [11]. In terms of the ordinary genus g
+and number of boundaries b, we have:
+6g − 6 + 3b + 2nc + no = 2
+nc
+�
+i=1
+nc
+i ,
+(2.4)
+in which we recognize the constraint (2.1) as the special case without boundaries.
+Our first step in generalizing the solution of the closed theory in terms of contact interac-
+tions [18] is to appropriately distribute curvature in the presence of boundaries and boundary
+insertions. We continue to assign curvature to the bulk insertions as before [18] – see equation
+(2.2). For simplicity, we momentarily imagine a single boundary, with a non-zero number no
+of boundary insertions σ. The ghost number conservation equation (2.4) then suggests that
+we should assign curvature −1/3 to each basic boundary insertion σ, in such a manner that
+we find the equation:
+Boundary Curvature = 1 = −no
+3 ,
+(2.5)
+in accord with our assignment for bulk curvature as well as the ghost number conservation
+equation (2.4). The relative factor of a half compared to the basic bulk (puncture) operator
+τ0 is due to the fact that the boundary operator increases the dimension of the moduli space
+by real dimension one (compared to a bulk operator which increases the real dimension by
+two). This reasoning can be generalized to the case of multiple boundaries with insertions. It
+is sufficient to introduce an extra label corresponding to each boundary (with its associated
+boundary insertions). We conclude that the boundary operator σ carries curvature −1/3.
+Higher Powers
+To prepare for reasonings to come, it may be useful to interject a thought experiment at this
+point. Note that the closed string vertex operator τn can be thought of as a power of the
+vertex operator τ1 in an approximate sense. The curvature it carries is then interpreted as
+the curvature n × 2/3 from which we subtract 2/3. The curvature remains bounded from
+below such that the vertex operators do not cut out such a large part of the surface for it
+to disappear entirely.2 Similarly, if we were to attempt to define an arbitrary power of the
+boundary operator σ to which we attached curvature −1/3, the operator would not have well-
+defined correlation functions. A manner to remedy this obstruction is to add explicit powers
+of the string coupling u to the operator: ρn = un−1σn. Now, the powers of the string coupling
+are counted by the genus g and the number of boundaries b on the one hand, and the explicit
+powers of u on the other hand. Suppose we study a correlation function of operators ρno
+j and
+τnc
+i. It satisfies the equation:
+2g − 2 + b +
+no
+�
+j=1
+(no
+j − 1) =
+nc
+�
+i=1
+2(nc
+i − 1)
+3
++
+no
+�
+j=1
+(2no
+j
+3
+− 1) .
+(2.6)
+2This is dictated by geometry or can be interpreted as a Seiberg bound [20].
+3
+
+This is still the ghost conservation equation (2.4) but rewritten in such a way as to make the
+explicit string coupling contributions visible on the left hand side. We made use of the fact
+that the coupling u corresponds to the vacuum expectation value of the exponential of the
+dilaton operator τ1 which couples to curvature. The operator ρn still caries ghost number n,
+but it also carries curvature 2n/3 − 1, as we made manifest in our manner of writing equation
+(2.6).3 While the boundary operators that we will soon encounter are more intricate still,
+they share features with the operators ρn.
+3
+The Virasoro Algebra Representations
+In this subsection, we briefly remind the reader of an intuitive manner to solve topological
+gravity on closed Riemann surfaces using contact terms [18]. We extend the approach to
+include boundaries and boundary vertex operators which can be viewed as representing the
+contact algebra. This section heavily relies on background provided in [18] to which we do
+refer for more details.
+3.1
+The Bulk Representation of the Virasoro Algebra
+The method of [18] to solve topological gravity on closed Riemann surfaces is to represent all
+the topological data in terms of local operators in a conformal field theory. As an example, we
+already saw that the curvature (which codes the genus) was assigned to local bulk operator
+insertions. Intersection numbers are then represented as integrals over the moduli space of
+the Riemann surface of conformal field theory correlators.4 We denote the curvature carrying
+bulk local operator insertions τn. Due to the topological nature of the theory, the contact
+interactions between the local operators suffice to compute the intersection numbers on the
+moduli space of Riemann surfaces.
+The method of [18] to solve topological gravity uses the fact that the algebra of integrated
+vertex operators is represented on localized bulk vertex operators (or states) in the form [18]:
+�
+Dϵ
+τm|τn⟩ = An
+m|τn+m−1⟩ ,
+(3.1)
+where the localized vertex operator τn is assumed to lie in the disk Dϵ over which the vertex
+operator τm is integrated. The representation arises from the contact term between the oper-
+ators τm and τn. When we wish to compute the algebra of consecutive actions of the locally
+integrated bulk vertex operators in the representation, we need to take into account that the
+first integrated operator may enter into contact with the second integrated operator. To keep
+track of this term, it is useful to define a measure of the non-commutativity of the operation
+of localizing the vertex operator, and integrating over it [18]:
+�
+Dϵ
+τm|τn⟩ −
+�
+Dϵ
+τn|τm⟩ = Cnm|τn+m−1⟩ .
+(3.2)
+3For n ≥ 1 the boundary operator now has sufficient curvature to have well-defined correlation functions.
+4This is heavily reminiscent of string theory (see e.g. [21]) and we allow string theory nomenclature to creep
+into our language.
+4
+
+Then, when we consider the action of two integrated vertex operators on a localized operator,
+we find a consistency condition between the representation coefficients A and the measure of
+non-commutativity C [18]:
+Am+k−1
+n
+Ak
+m − An+k−1
+m
+Ak
+n + CnmAk
+m+n−1
+=
+0 .
+(3.3)
+The coefficient An
+m is calculated in [18] and it equals the curvature of the insertion plus one:
+An
+m = 2
+3(n − 1) + 1 ,
+(3.4)
+and we retain that we have the contact contribution
+�
+Dϵ
+τm|τn⟩ = 2n + 1
+3
+|τn+m−1⟩ .
+(3.5)
+In turn this implies that the measure of non-commutativity C is proportional to the difference
+in the curvature of the insertions:
+Cmn = 2
+3(m − n) .
+(3.6)
+Note that when we identify the coefficients An of the representation on the bulk vertex operator
+space with an operator Ln−1, then the commutation relation (3.3) shows that we have a
+representation of the Virasoro algebra:
+[Ln, Lm] = 2
+3(m − n)Lm+n .
+(3.7)
+Thus, the contact algebra is a Virasoro algebra, represented on the space of bulk operator
+insertions. This is an essential tool in the solution to the bulk topological gravity theory [18],
+and we wish to extend it to Riemann surfaces with boundary.
+3.2
+The Extended Virasoro Representation
+In the presence of a boundary, we first address the question what happens when a bulk vertex
+operator is integrated over a small ring Rϵ near an empty boundary. We propose that the
+integrated vertex operator in that case generates an operator on the boundary:
+�
+Rϵ
+τn| ⟩b = u c(n)|σb
+n−1⟩ .
+(3.8)
+We have introduced operators σb
+n that live on a boundary of the Riemann surface. We have
+stripped off one factor of the string coupling constant u on the right hand side – we think of the
+bulk vertex operators as carrying one power of the coupling constant more than the boundary
+operators.5 We have allowed for a representation coefficient c(n) that is undetermined for
+now. The curvature of the operator σb
+n equals the curvature of the bulk vertex operator minus
+one, to compensate for the string coupling constant prefactor. Therefore, the curvature of the
+5This is standard in string theory. Alternatively, it can be viewed as a consequence of the relative contri-
+bution of bulk and boundary vertex operators to the dimension of moduli space.
+5
+
+operator σb
+n−1 equals 2(n − 1)/3 − 1. We allow for operators with n ≥ 2 and set other terms
+to zero.
+Thus, we have introduced a new space parameterized by the operators σb
+n. Our next step
+is to assume that the integrated bulk vertex operators also act on this space and provide a
+new representation of the Virasoro algebra. We need to make sure that the resulting operator
+carries the sum of the curvatures of the operators on the left hand side, and we propose that
+the contact algebra coefficient is again fixed to equal the curvature of the operator plus one –
+see equation (3.4). We thus find:
+�
+Rϵ
+τm|σb
+n⟩ = 2n
+3 |σb
+m+n−1⟩ .
+(3.9)
+This natural proposal partially fixes the normalization of the boundary vertex operators. We
+still need to check whether the integrated vertex operators satisfy the Virasoro algebra. The
+action (3.9) is indeed a representation of the Virasoro algebra, as before. For the action (3.8)
+to also enter into a representation of the Virasoro algebra, the coefficient c(n) needs to be a
+linear function of n. Finally, we use a choice of overall normalization of the boundary vertex
+operators to set c(n) = n+a
+3
+where a is a constant to be determined. We will later argue that
+consistency requires a = 0 and we therefore find the action on an empty boundary:
+�
+Rϵ
+τn| ⟩b = u n
+3|σb
+n−1⟩ .
+(3.10)
+In summary, we have extended the space of boundary operators considerably, and we have
+represented the Virasoro contact algebra on that space.
+3.3
+The Recursion Relation
+For topological gravity on closed Riemann surfaces, the representation of the contact algebra
+was leveraged into a recursion relation for the topological correlators [18]. The integral over
+bulk vertex operators was split into an integral over small disks where other operators reside,
+neighbourhoods of nodes and uneventful regions. The fact that integrals of bulk operators
+over the whole Riemann surface should commute, combined with the contact algebra, gave
+rise to consistency conditions on the contributions of nodes which in turn provided a recursion
+relation for correlators. Our claim is that the same reasoning applies to the integrated bulk
+vertex operators on Riemann surfaces with boundary. We again need to take into account the
+possible development of nodes on the Riemann surface, as well as possible generalized contact
+terms with the boundary, which we described previously.
+To ease into the generalized recursion relation, let us recall the closed recursion relation
+first [18]6:
+⟨τn+1
+�
+i∈C
+τni⟩c
+=
+�
+j
+2nj + 1
+3
+⟨τn+nj
+�
+i̸=j
+τni⟩c
+(3.11)
++u2
+18(
+n−1
+�
+k=0
+(⟨τkτn−k−1
+�
+i∈C
+τni⟩c +
+�
+C=C1∪C2
+⟨τk
+�
+i∈C1
+τni⟩c⟨τk−i−1
+�
+j∈C2
+τnj⟩c) .
+6We normalize the bulk correlators as ⟨τ0τ0τ0⟩c = 1 and ⟨τ1⟩c = 1/24. We often set the string coupling u
+to one.
+6
+
+Figure 1: Two degenerations of Riemann surfaces are depicted. The left figure represents a
+surface splitting into two surfaces. The sum of genera is conserved. The right figure shows a
+genus two Riemann surface that turns into a genus one Riemann surface, lowering the genus
+by one.
+The set C is a set of bulk operator insertions. The first term on the right hand side arises
+from the bulk contact algebra representation (3.5) while the second line has its origins in the
+fact that a Riemann surface can develop nodes which give rise to a Riemann surface of one
+genus less, or which splits the Riemann surface into two closed Riemann surfaces. See Figure
+1 and reference [18].
+The generalization to the case of extended open correlators is:
+⟨τn+1
+�
+i∈C
+τni
+�
+l∈O
+σb
+nl⟩o,ext
+=
+�
+j
+2nj + 1
+3
+⟨τn+nj
+�
+i̸=j
+τni
+�
+l
+σnl⟩o,ext +
+�
+j
+2nj
+3 ⟨
+�
+i
+τniσn+nj
+�
+l̸=j
+σnl⟩o,ext
++un + 1
+3
+⟨σb
+n
+�
+i∈C
+τni
+�
+l∈O
+σb
+nl⟩o,ext
+(3.12)
++u2
+18(
+n−1
+�
+k=0
+(⟨τkτn−k−1
+�
+i,j∈CO
+τnk⟩o,ext
++
+�
+(e,f)
+�
+CO=CO1∪CO2
+⟨τk
+�
+i,j∈CO1
+τniσb
+nj⟩e⟨τk−i−1
+�
+l,m∈CO2
+τnlσb
+nm⟩f) .
+The first line corresponds to the fact that we are considering an integrated bulk operator τn+1.
+It gives rise to the contact terms in the second line from the bulk contact term (3.5) and the
+boundary contact term (3.9). The third line arises from the naked boundary term (3.10). The
+fourth line arises from pinching off a handle. The fifth line requires explanation. We sum
+over the sectors (e, f) which can be either (open,closed), (closed,open) or (open,open).7 The
+first two arise when we split the surface into a closed Riemann surface and a Riemann surface
+with boundary.8 In that case, the open string sector will contain all the boundary insertions,
+necessarily. The third value, (open,open) arises when a node splits the Riemann surface into
+two Riemann surfaces with boundary. The set CO indicates the set of all bulk and boundary
+insertions, and we sum over their possible distributions CO1 and CO2 on the two disjoint
+7We exclude the case with no boundaries from our definition of extended open correlators. See equation
+(3.11) for the purely closed correlators.
+8We effectively obtain a factor of two from these first two sectors.
+7
+
+surfaces.9
+Note that the second line in the right hand side contains a correlator that is of one order
+less in the string coupling constant, and the third line a correlator that is down by two orders
+in the string coupling constant u.
+3.4
+The Generalized Vertex Operators
+To make further progress, we must discuss the nature of the extended set of boundary vertex
+operators σb
+n in more detail. We recall that in the geometric open topological theory [11], we
+found a single boundary vertex operator σ of curvature −1/3 in section 2. This matches the
+curvature of σb
+1 and we will indeed identify the two operators: σ = σb
+1.10 The curvature of
+the general operator σb
+n is 2n/3 − 1. To make such operators on the boundary, we can use a
+power of the operator σ as well as the string coupling constant (effectively of curvature one).
+A natural guess is that there is a component ρn = u−1(uσ)n to the boundary vertex operator
+σb
+n (as previewed in section 2). However, we also need to allow for more drastic processes.
+Up to now, a number of complications were implicit in our extended boundary vertex
+operators. To start with, we concentrate on the simplest extended operator, namely σb
+2. It
+naively corresponds to an insertion of uσσ. However, to understand further possibilities, we
+need to study the boundary analogue of nodes.
+A strip (or open string propagator) can
+be squeezed near the boundary of the moduli space of open Riemann surfaces, in various
+manners. Either the number of boundaries can decrease as in an annulus to disk transition,
+or the number of boundaries can increase as in a disk to two disks transition.11 See figure
+2. When the integrated bulk vertex operator is close to these singular configurations, it can
+either give rise to boundary vertex operators that sit on a single boundary or it can give rise
+to boundary vertex operators that sit on two different boundaries of disconnected surfaces.
+The boundary vertex operator σb
+2 must capture both these possibilities. Thus, we propose the
+equation:
+⟨. . . σb
+2 . . . ⟩o,ext = b1u⟨. . . σσ⟩o,ext + b2u⟨. . . σ⟩⟨σ . . . ⟩o,ext .
+(3.13)
+This equation shows that the generalized vertex operator σb
+2 exhibits a non-local characteristic.
+We recall that in the case of a node degeneration (see Figure 1), there was a universality
+between losing a handle and splitting a surface – both terms have equal coefficient in the
+second line of equation (3.11). We propose a similar universality here for the two terms in
+which the boundary operators remain on the same boundary, or split – compare Figures 1 and
+2 – and set the two constants in the above equation equal, namely b1 = b = b2. To determine
+the overall constant b, we calculate an amplitude.
+9If one labels boundaries, and their associated boundary operators, a finer combinatorics and summation
+is necessary.
+10There is a possible normalization factor between these two operators.
+Our previous choice of overall
+normalization of the boundary operators makes sure that this identification is spot on in standard conventions.
+11There is a third degeneration process in which the genus drops by one.
+When one labels boundary
+components, it will play a role. See e.g. [22] for a discussion in open/closed string field theory.
+8
+
+Figure 2: Two degenerations of Riemann surfaces with boundary are drawn. The left figure
+represents a disk splitting into two disks. The right figure shows an annulus that turns into a
+disk.
+3.5
+Amplitudes
+To understand the content of the recursion relation further, we need initial conditions, which
+we take from the most basic geometric calculations [11]. We have that the boundary three-
+point function is the only non-zero disk amplitude with only boundary σ insertions, and
+normalize it to one:12
+⟨σσσ⟩o,ext = 1 .
+(3.14)
+The other initial condition is that the bulk-boundary one-point function on the disk equals:
+⟨τ0σ⟩o,ext = 1 .
+(3.15)
+To save on indices, we will drop the upper index on the correlator from now on – it should be
+clear from the context which correlator we have in mind.
+To understand the structure of the vertex operator σb
+m≥2, we can use the puncture equation,
+namely, the recursion relation (3.12) for n = −1:
+⟨τ0
+�
+i∈C
+τni
+�
+l∈O
+σb
+nl⟩ =
+�
+j
+2nj + 1
+3
+⟨τnj−1
+�
+i̸=j
+τni
+�
+l
+σnl⟩ +
+�
+j
+2nj
+3 ⟨
+�
+i
+τniσnj−1
+�
+l̸=j
+σnl⟩ .
+(3.16)
+Let us also be explicit about the dilaton equation:
+⟨τ1
+�
+i∈C
+τni
+�
+l∈O
+σb
+nl⟩ =
+�
+j
+2nj + 1
+3
+⟨
+�
+i
+τni
+�
+l
+σnl⟩ +
+�
+j
+2nj
+3 ⟨
+�
+i
+τni
+�
+l
+σnl⟩ .
+(3.17)
+We are ready to calculate a first amplitude in two manners, using either the puncture equation,
+or the factorization equation (3.13):
+⟨τ0σ2σσ⟩
+=
+4
+3⟨σσσ⟩
+=
+2b⟨τ0σ⟩⟨σσσ⟩ .
+(3.18)
+In the first line we used the puncture equation (3.16). In the second line, we used the ansatz
+(3.13) and allowed for the two possible ways in which the vertex operators can split over two
+12This is a disk amplitude. We have set u = 1 once more.
+9
+
+correlators to give a non-vanishing result.13 Note that in the second line a factor of the string
+coupling constant implicitly cancelled between the two disk amplitudes and the expression for
+the operator σb
+2. Using the normalization of the initial conditions, we find:
+b = 2
+3 .
+(3.19)
+This fixes our reading of the extended vertex operator σb
+2 once and for all.
+For the next
+extended operator σb
+3 we propose a similar universal ansatz consistent with curvature conser-
+vation and splitting off a single vertex operator σ:
+⟨. . . σb
+3 . . . ⟩ = c(u⟨. . . σσ2⟩ + u⟨. . . σ2⟩⟨σ . . . ⟩) .
+(3.20)
+We can again determine the constant c using either the puncture or the factorization equation
+to determine one and the same amplitude consistently:
+⟨τ0σ3σ4⟩
+=
+2⟨σ2σ4⟩ = 8⟨σ3⟩⟨σ3⟩
+=
+c⟨τ0σ⟩⟨σ2σ4⟩ + 6c⟨τ0σ2σ2⟩⟨σ3⟩
+=
+4c⟨τ0σ⟩⟨σ3⟩⟨σ3⟩ + 8c⟨σ3⟩⟨σ3⟩ ,
+(3.21)
+and find that again c = 2/3 – the constant is fixed once more in terms of the bulk-boundary
+one-point function ⟨τ0σ⟩. Continuing recursively in this manner, e.g. exploiting the correlation
+functions ⟨τ0σb
+nσ2(n−1)⟩, we find:
+⟨. . . σb
+n⟩ = u 2
+3(⟨. . . σσb
+n−1⟩ + ⟨. . . σb
+n−1⟩⟨σ . . . ⟩) .
+(3.22)
+Thus, we have determined the intricate nature of the extended boundary vertex operators σb
+n
+and how they recursively code the splitting of boundaries of open Riemann surfaces.
+Tying up a loose end: fixing the constant a
+We tie up a loose end at the hand of another amplitude. The amplitude illustrates a splitting
+of open Riemann surfaces involving two disk one-point functions. We calculate the amplitude
+⟨τ3τ0σσ⟩ in two manners. We can apply recursion to the operator τ3, or to the operator τ0
+first. In this calculation, we restore the possible constant a that we introduced in subsection
+3.2 and use an appropriately modified recursion relation. We demonstrate that the constant
+can be determined by consistency. Using the a-modified recursion relation, we find:
+⟨τ3τ0σ2⟩
+=
+7
+3⟨τ2σ2⟩
+=
+1
+3⟨τ2σ2⟩ + 2
+9⟨τ0σ⟩⟨τ1τ0σ⟩ + 4
+3⟨τ0σ3σ⟩ + 3 + a
+3
+⟨σ2τ0σ2⟩ .
+(3.23)
+This implies:
+⟨τ2σ2⟩
+=
+1
+9⟨τ0σ⟩⟨τ0σ⟩ + 4
+3⟨σ2σ⟩ + 3 + a
+3
+2
+3⟨σ3⟩ .
+(3.24)
+13Ghost number conservation applies to each factor separately.
+10
+
+We can compute the latter correlator in another manner, using the puncture equation and
+the modified recursion relation:
+⟨τ2σ2⟩
+=
+1
+9⟨τ0σ⟩⟨τ0σ⟩ + 4
+3⟨σ2σ⟩ + 2 + a
+3
+⟨σ3⟩ .
+(3.25)
+Using our previous results, we find full consistency if and only if a = 0. Thus, we tied up the
+loose end in subsection 3.2.
+4
+The Extended Partition Function
+In this section we introduce the generating function of extended open string correlators and
+prove that the recursion relations for the correlators imply Virasoro constraints on the gen-
+erating function.
+This allows us to make our results more rigorous by connecting to the
+mathematics literature on the integrable structure of the intersection theory on moduli spaces
+of Riemann surfaces with boundary [12]. We conclude the section with a few example ampli-
+tudes.
+4.1
+The Generating Function
+We recall the generating functions of closed as well as open topological gravity correlation
+functions [11]:
+F c
+=
+�
+g≥0,n≥1,2g−2+n>0
+u2g−2
+n!
+�
+ki≥0
+⟨τk1 . . . τkn⟩c
+gtk1 . . . tkn
+F o,geom
+=
+�
+g′,k,l≥0,2g′−2+k+2l>0
+�
+ai≥0
+ug′−1
+k!l! ⟨τa1 . . . τalσk⟩o
+g′sk
+l�
+i=1
+tai .
+(4.1)
+In view of our enlarged space of boundary vertex operators, we also introduce a generating
+function for extended open topological gravity correlation functions:
+F o,ext
+=
+�
+g′,k,l≥0,2g′−2+k+2l>0
+�
+ai,bi≥0
+ug′−1
+k!l! ⟨τa1 . . . τalσb
+b1 . . . σb
+bk⟩o,ext
+g
+�
+i
+tai
+�
+j
+sbj .
+(4.2)
+The Extended Virasoro Constraints
+We define Virasoro generators
+Ln
+=
+�
+i≥0
+2i + 1
+2
+ti∂ti+n − 3
+2∂tn+1 + u2
+12
+n−1
+�
+i=0
+∂ti∂tn−i−1 + 3
+4
+t2
+0
+u2δn,−1 + 1
+16δn,0
+(4.3)
+Lext
+n
+=
+Ln +
+�
+i≥0
+(i + 1)si+1∂sn+i+1 + un + 1
+2
+∂sn + 3
+2
+s1
+u δn,−1 + 3
+4δn,0
+(4.4)
+11
+
+for n ≥ −1. These are defined such that the recursion relation (3.11) on the closed as well as
+the recursion relation (3.12) on the extended open correlators leads to the constraints:
+Ln exp F c
+=
+0
+Lext
+n exp(F c + F o,ext)
+=
+0 .
+(4.5)
+The extra constants terms in the closed Virasoro algebra (4.3) are due to the initialization
+cases ⟨τ 3
+0 ⟩c = 1 = 24 ⟨τ1⟩ at genus zero and one respectively, while the initial conditions
+⟨σ3⟩ = 1 = ⟨τ0σ⟩ on the disk lead to the extra constants in the extended Virasoro algebra
+(4.4), which satisfies14
+[Lm, Ln] = (m − n)Lm+n .
+(4.6)
+At this stage, we are able to make contact with rigorous results – these constraints on an
+extended partition function of open topological correlators defined through an extended (or
+unconstrained) integrable KdV hierarchy were found to hold in [12].15 The relation between
+the operators σb
+n and σ as well as the string coupling constant is neatly captured by a relation
+between derivatives of the extended partition function:
+∂sn = (2u
+3 )n−1∂n
+s1 .
+(4.7)
+This equation was proven from the KdV integrable hierarchy perspective in [12]. Using this
+equation, and setting extended open times sn≥2 to zero, this relation between derivatives imply
+the higher order Virasoro constraints on the geometric open topological partition function,
+where the open Virasoro generators are [11]:
+Lo
+n = Ln + (2u
+3 )n∂n+1
+s1
++ n + 1
+2
+u(2u
+3 )n−1∂n
+s1 + δn,−1
+3
+2
+s1
+u + δn,0
+3
+4 .
+(4.8)
+The Virasoro constraints and the initialization condition are sufficient to determine the full
+partition function [11,12]. Through the generating function of extended correlators, we have
+connected our arguments with rigorous results on intersection theory on moduli spaces of
+Riemann surfaces with boundary [11,12].
+4.2
+A Few More Amplitudes
+For illustrative purposes, we calculate a few more amplitudes. They render the integrable
+hierarchy structure, the Virasoro constraints and how to solve them more concrete.
+4.2.1
+Amplitudes on The Disk
+We have already indicated that on the disk only the third power of the elementary boundary
+vertex operator σ has a non-zero correlation function and equals one, ⟨σ3⟩ = 1. The disk
+bulk-boundary one-point function ⟨τ0σ⟩ is also one by a choice of normalization. Amplitudes
+14These generators are rescaled by a factor of 2/3 compared to section 3 in order to reach a standard
+normalization for the Virasoro algebra.
+15 The translation of variables and normalizations is: Lthere,ext
+n
+= (3/2)nLext
+n , tthere
+n
+= 3−n(2n + 1)!!tn and
+sthere
+n−1 = (2/3)n−1n!sn.
+12
+
+involving extended boundary vertex operators are computed through the reduction formula
+(3.22). A non-trivial example is:
+⟨τ2σ5⟩
+=
+10
+3 ⟨σb
+2σ4⟩ = 40
+3 ,
+(4.9)
+where we used the recursion relations (3.12) and (3.22) as well as the 6 choices of factoriza-
+tion. After taking into account the different normalization in footnote 15, this agrees with a
+more generic formula in [11]. Another interesting correlation function is ⟨τ2τ0σσ⟩. It can be
+computed through the puncture equation (in the first line below) and/or the L1 constraint
+(in the second line below):
+⟨τ2τ0σσσ⟩
+=
+5
+3⟨τ1σσσ⟩ = 10
+3 ⟨σσσ⟩
+=
+1
+3⟨τ1σσσ⟩ + 2⟨τ0σ2σσ⟩
+=
+2
+3⟨σσσ⟩ + 2 × 8
+3⟨σσσ⟩ = 10
+3 ⟨σσσ⟩ .
+(4.10)
+The two ways of computing are in agreement.
+4.2.2
+Higher Order Amplitudes
+Amplitudes that are higher order in the string coupling exhibit qualitatively new phenomena.
+We illustrate a few. We first compute amplitudes corresponding to cylinder diagrams, with two
+boundaries and genus zero. An interesting amplitude that involves a closed-open factorization
+due to a node can once again be computed in two manners:
+⟨τ2τ0τ0σ⟩
+=
+5
+3⟨τ1τ0σ⟩ = 5
+3⟨τ0σ⟩
+=
+2
+3⟨τ1τ0σ⟩ + 1
+9⟨τ 3
+0 ⟩c⟨τ0σ⟩ + 2
+3⟨τ0τ0σ2⟩
+=
+2
+3⟨τ0σ⟩ + 1
+9⟨τ 3
+0 ⟩c⟨τ0σ⟩ + 8
+9⟨τ0σ⟩⟨τ0σ⟩ .
+(4.11)
+Both ways of computing the correlator lead to the same result, given the normalization of
+the closed three-point function ⟨τ 3
+0 ⟩c as well as the bulk-boundary one-point function ⟨τ0σ⟩.
+Finally, we compute an order O(u1) amplitude.
+It involves the one-loop closed one-point
+function ⟨τ1⟩c:
+⟨τ3σ⟩ = 2
+3⟨σ3⟩ + ⟨σ2σ⟩ + 1
+9(1 + ⟨τ1⟩)⟨τ0σ⟩
+=((2
+3)3 + 2
+3)⟨σ3⟩ + 1
+9(1 + ⟨τ1⟩)⟨τ0σ⟩ .
+(4.12)
+Needless to say, many more results can be generated, e.g. by computer. We provided a few
+telling illustrations that provide insight into the foundation of the integrable hierarchy.
+5
+Conclusions
+In the spirit of the solution of the bulk theory [18] and building on earlier mathematical
+work [11, 12], we have solved two-dimensional topological gravity on Riemann surfaces with
+13
+
+boundary. By making use of an extended set of boundary vertex operators, we rendered the
+representation of the contact algebra on the boundary linear. Only in a second step the more
+complicated degeneration of surfaces with boundary is taken into account and the non-linear
+realization of the (half) Virasoro algebra is found [12]. The picture in which the solution of
+the theory is provided through contact interactions is a welcome intuitive complement to the
+geometric and matrix model approaches.
+While we have provided a compelling global picture, there are many details that remain
+to be worked out. It would be good to find the geometric counterpart to the extended set of
+boundary operators. The link between (the expectation values of) the conformal field theory
+fields implicit in our analysis [18] and the sections of vector bundles of open topological
+gravity can be clarified (e.g. by exploiting references [15, 20]). The analysis of the contact
+terms in terms of an integration over a degeneration region of the moduli space of open
+Riemann surfaces would be interesting. It will also be instructive to compare our analysis
+to the geometric derivation of the topological recursion relation through closed and open
+factorization [11], intuitively reviewed in [15].
+Another research direction is to exploit the insights developed here and apply them to
+more general theories. The generalization to the extended closed theory [23]) comes to mind,
+but mostly to open spin r curves. Geometric [24], integrable [25, 26], matrix model [27, 28]
+and conformal field theory insights [29] could be complemented by the perspective developed
+in this paper.
+The study of these topological theories of gravity is worthwhile in its own right. It occasion-
+ally fruitfully interfaces with recent developments. For instance, the KdV integrable hierarchy
+governing topological gravity also permeates the two-dimensional JT-gravity holographic dual
+of a peculiar (SYK) one-dimensional quantum system – see e.g. [30] and references thereto.
+We believe that the further study of these elementary solvable systems, their integrable hierar-
+chy but also their various manifestations in superficially different mathematical structures like
+topology, matrices and conformal field theory is worthwhile, and may eventually contribute
+to our understanding of quantum gravity.
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+Model-Agnostic Hierarchical Attention for 3D Object Detection
+Manli Shu1*
+Le Xue2
+Ning Yu2
+Roberto Martín-Martín2,3
+Juan Carlos Niebles2
+Caiming Xiong2
+Ran Xu2
+1 University of Maryland, 2 Salesforce Research, 3 UT Austin
+Abstract
+Transformers as versatile network architectures have re-
+cently seen great success in 3D point cloud object detec-
+tion. However, the lack of hierarchy in a plain transformer
+makes it difficult to learn features at different scales and
+restrains its ability to extract localized features. Such limita-
+tion makes them have imbalanced performance on objects
+of different sizes, with inferior performance on smaller ones.
+In this work, we propose two novel attention mechanisms
+as modularized hierarchical designs for transformer-based
+3D detectors. To enable feature learning at different scales,
+we propose Simple Multi-Scale Attention that builds multi-
+scale tokens from a single-scale input feature. For localized
+feature aggregation, we propose Size-Adaptive Local At-
+tention with adaptive attention ranges for every bounding
+box proposal. Both of our attention modules are model-
+agnostic network layers that can be plugged into existing
+point cloud transformers for end-to-end training. We evalu-
+ate our method on two widely used indoor 3D point cloud
+object detection benchmarks. By plugging our proposed mod-
+ules into the state-of-the-art transformer-based 3D detector,
+we improve the previous best results on both benchmarks,
+with the largest improvement margin on small objects.1
+1. Introduction
+3D point cloud data provides accurate geometric and spa-
+tial information, which are important to computer vision ap-
+plications such as autonomous driving and augmented reality.
+Different from image data, which has a grid-like structure,
+point clouds consist of unordered irregular points. Due to
+such unique properties of point clouds, previous works have
+proposed various deep network architectures for point cloud
+understanding [7,8,22–25,34,37,48,50]. With the success
+of transformers in natural language processing [4, 26, 40]
+and 2D vision [5,14,39], attention-based architectures for
+point clouds [20,38,46,49,52,53,55] are explored in recent
+*Work done during an internship at Salesforce. [email protected].
+1The code and models will be available at https://github.com/
+salesforce/Hierarchical_Point_Attention.
+Groundtruths
+Predictions
+Attention weights 
+(high to low)
+Plain Attention
+Our Attention
+Groundtruth object
+Figure 1. Visualization of the attention weights. With our hierar-
+chical attentions, the object center has higher attention weights with
+points that belong to the object, and the predicted bounding box
+is better aligned with the groundtruth. Our multi-scale attention
+extracts feature at different scales, which helps distinguish object
+boundaries. Our size-adaptive local attention aggregates features at
+the object level and helps refine the bounding box proposals.
+works and have seen great success in 3D point cloud object
+detection [16, 19, 32, 45]. Several properties of transform-
+ers make them ideal for learning on raw point clouds. For
+example, their permutation-invariant property is necessary
+for modeling unordered sets like point clouds, and their at-
+tention mechanism can model long-range relationships that
+help capture the global context for point cloud learning.
+Despite the advantages of transformers for point clouds,
+we find the state-of-the-art transformer detector to have im-
+balanced performance across different object sizes, with the
+lowest average precision on small objects (see Section 4.3).
+We speculate the inferior performance on small objects can
+be due to two factors. Firstly, to make the computation feasi-
+ble, transformer detectors use point cloud features consisting
+of a small set of points compared to the original point cloud.
+1
+arXiv:2301.02650v1  [cs.CV]  6 Jan 2023
+
+The extensively downsampled point cloud loses geometric
+details, which has a larger impact on small objects. Secondly,
+plain transformers (e.g., Transformer [40], ViT [5]) extract
+features at the global scale throughout the network, which
+does not support explicit localized feature learning.
+Motivated by the above observations, we expect existing
+point cloud transformers to benefit from a hierarchical fea-
+ture learning strategy, which allows multi-scale feature learn-
+ing and supports localized feature aggregation. Nonetheless,
+considering the computation intensity of point cloud trans-
+formers, it is inefficient to use higher-resolution (i.e., higher
+point density) point cloud features throughout the network.
+Furthermore, due to the irregularity of point clouds, it is
+non-trivial to integrate hierarchical design and multi-scale
+features into transformers for point cloud object detection.
+Our approach.
+In this work, we aim to improve
+transformer-based 3D object detectors with modularized
+hierarchical designs. We propose two attention modules for
+multi-scale feature learning and size-adaptive local feature
+aggregation. Our attention modules are model-agnostic and
+can be plugged into existing point cloud transformers for
+end-to-end training.
+We first propose Simple Multi-Scale Attention (MS-A). It
+builds higher resolution point features from the single-scale
+input feature with a learnable upsampling strategy and use
+both features in the attention function. To reduce computa-
+tion and parameter overhead, we transform the multi-scale
+features into multi-scale tokens and perform multi-scale to-
+ken aggregation [30] within a multi-head attention module.
+The second module is Size-Adaptive Local Attention (Local-
+A), which learns localized object-level features for each
+object candidate. It assigns larger attention regions to ob-
+ject candidates with larger bounding box proposals. The
+local attention regions are defined by their corresponding
+intermediate bounding box proposals.
+We evaluate our method on two widely used indoor 3D
+object detection benchmarks: ScanNetV2 [3] and SUN RGB-
+D [36]. We plug our attention modules into the state-of-
+the-art transformer-based 3D detector and perform end-to-
+end training. Our method improves the previous best result
+by over 1% in [email protected] and over 2% in [email protected] on
+ScanNetV2. Furthermore, our size-aware evaluation shows
+we have the most performance gain among small objects
+with a 2.5% increase in mAPS. We summarize our main
+contributions as follows:
+• We propose Simple Multi-Scale Attention (MS-A) to en-
+able multi-scale feature learning on single-scale features.
+• We present Size-Adaptive Local Attention (Local-A) for
+local feature aggregation within bounding box proposals.
+• We conduct experiments on two widely used indoor 3D
+detection benchmarks and surpass the previous best results
+on both benchmarks.
+2. Related Work
+Network architectures for point cloud learning.
+Exist-
+ing network architectures for point cloud learning can be
+roughly divided into two categories based on their point
+cloud representation: grid-based and point-based, yet in
+between, there also exist hybrid architectures that operate on
+both representations [9,33,51,53,57]. Grid-based methods
+project the irregular point clouds into grid-like structures,
+such as 3D voxels. With the grid-like structure, existing
+works have proposed a variety of 3D-convolution-based ar-
+chitectures [7,18,31,42]. Point-based methods, on the other
+hand, directly learn features from the raw point cloud. Within
+this category, graph-based method [8, 35, 44, 47, 56] use a
+graph to model the relationships among the points. Another
+line of work models a point cloud as a set of points, and
+extracts features through set abstraction [17,23,25,41]. Re-
+cent works explore transformer architecture for point-based
+learning [16,19,20,53,55], where each point is fed into the
+transformer as a token and the attention mechanism learns
+point features at a global scale. While previous methods
+improve point cloud learning by developing new backbones
+and modifying the overall network architecture, our work
+focuses on the attention mechanism of the point cloud trans-
+former. Instead of proposing new architectures for point
+cloud learning, we aim to provide a model-agnostic solution.
+Point cloud object detection.
+One major challenge in
+point cloud object detection is extracting object features.
+In 2D object detection, a common practice for extracting ob-
+ject features is to use a region proposal network (RPN) [29]
+to generate dense bounding box proposals (i.e., object can-
+didate) in a top-down manner and then extract features for
+each object candidate. However, in 3D vision, generating
+dense 3D bounding box proposals for point cloud data is
+inefficient due to the irregularity and sparsity of point clouds.
+Previous work [2,57] addresses this issue by projecting point
+clouds into 2D bird’s-eye views or voxels and then applying
+RPN. However, such projection operations can result in the
+loss of geometric information or introduce quantization er-
+rors. Another line of work seeks to generate 3D proposals
+in a bottom-up manner (i.e., point-based) [16, 19, 21, 34].
+VoteNet [21] samples a set of points from a point cloud
+as the initial object candidates and then assigns points to
+each object candidate through voting. Object features of
+each candidate are learned by aggregating features within
+its corresponding vote cluster (i.e., group). Instead of voting
+and grouping, follow-up works [16, 19] propose to use a
+transformer to automatically model the relationship between
+the object candidates and the point cloud. Although point-
+based methods do not have quantization errors caused by
+voxelization, to make the computation feasible, a point cloud
+needs to be extensively downsampled at the beginning of the
+2
+
+model. Such downsampling also causes a loss of geomet-
+ric information, while it is important for object detection to
+have fine-grained features to make accurate predictions. Our
+work is based on point-based transformer detectors. We ad-
+dress the downsampling issue by building higher-resolution
+features without increasing the computation budget.
+Hierarchical designs for 2D and 3D vision transformers.
+Extensive work has been done to adapt transformers for
+vision recognition. One direction is to borrow the hierar-
+chical design and inductive biases from convolutional neu-
+ral networks (ConvNet) [10]. In the 2D vision, one line
+of ConvNet-based hierarchical design [6,11,43] produces
+multi-scale feature maps for 2D images by progressively
+decreasing the resolution and expanding feature channels.
+Swin Transformer [15] adopts the idea of weight-sharing of
+ConvNet and proposes efficient self-attention with shifted
+windows. Shunted self-attention [30] attends to features at
+different scales through multi-scale token aggregation. In the
+3D vision, hierarchical designs for point cloud transformers
+are explored in previous works, where self-attentions are ap-
+plied to local regions (specified by k nearest neighbors [55]
+or a given radius [20]), and downsampling operations are
+performed after every encoding stage following the hierar-
+chical design of PointNet++ [25]. Patchformer [53] proposes
+a multi-scale attention block that performs extracts features
+at multiple granularities, but it requires voxelization on the
+point cloud. Different from previous works, we pack our
+hierarchical design into model-agnostic attention modules
+that can be plugged into any existing architecture and enable
+both multi-scale and localized feature learning.
+3. Method
+In this section, we first discuss the background, including
+a brief introduction to the task of point cloud object detection,
+an overview of point-based 3D detection methods, and the
+attention mechanism. Next, we dive into the detailed designs
+of our proposed attention modules.
+3.1. Background
+Point cloud object detection.
+Given a point cloud Praw
+with a set of P points P = {pi}P
+i=1, each point pi ∈ R3
+is represented by its 3-dimensional coordinate. 3D object
+detection on point cloud aims to predict a set of bounding
+boxes for the objects in the scene, including their locations
+(as the center of the bounding box), size and orientation
+of the bounding box, and the semantic class of the corre-
+sponding object. Note that due to the computation limit, the
+point cloud is downsampled at the early stage of a model
+to a subset of Praw, which contains N (N << P) points.
+P = SA(Praw) = {pi}N
+i=1 contains the aggregated groups
+of points around N group centers, where SA (set abstraction)
+is the aggregation function, and the group centers are sam-
+pled from the raw point cloud using Furthest Point Sample
+(FPS) [23], a random sampling algorithm that provides good
+coverage of the entire point cloud.
+Point-based 3D object detectors.
+Our method is built on
+point-based 3D object detectors [16,21,34], which detect 3D
+objects in point clouds in a bottom-up manner. Compared to
+other 3D detectors that generate box proposals in a top-down
+manner on the bird’s-eye view or voxelized point clouds [2],
+point-based methods work directly on the irregular point
+cloud and do not cause loss of information or quantization
+errors. In addition, point-based methods are suitable for
+more efficient single-stage object detection [1,13,28].
+The feature representation of the input point cloud
+{zi}N
+i=1, zi ∈ Rd is first obtained using a backbone model
+(e.g., PointNet++ [25]), where d is the feature dimen-
+sion. Point-based detectors generate bounding box predic-
+tions starting with M (M < N) initial object candidates
+{qi}M
+i=1, qi ∈ RC, sampled from the point cloud as object
+centers. A common approach for sampling the candidates is
+Furthest Point Sample (FPS). Once get the initial candidates,
+the detector then extracts features for every object candidate.
+Attention-based methods [16] learn features by doing self-
+attention among the object candidates, and cross-attention be-
+tween the candidates (i.e., query) and point features {zi}N
+i=1.
+The learned features of the object candidates will then be
+passed to prediction heads, which predict the attributes of
+the bounding box for each object candidate. The attributes of
+a 3D bounding box include its location (box center) ˆc ∈ R3,
+size (H/W/D dimensions) ˆd ∈ R3, orientation (heading an-
+gles) ˆa ∈ R, and the semantic label of the object ˆs. With
+these parameterizations, we can represent a bounding box
+proposal as ˆb = {ˆc, ˆd, ˆa,ˆs}. The detailed parameterizations
+of a bounding box are included in Appendix A.2.
+Attention mechanism
+is the basic building block of trans-
+formers. The attention function takes in query (Q), key (K),
+and value (V ) as the input. The output of the attention func-
+tion is a weighted sum of the value with the attention weight
+being the scaled dot-product between the key and query:
+Attn(Q, K, V ) = softmax(QKT
+√dh
+)V,
+(1)
+where dh is the hidden dimension of the attention layer.
+For self-attention, Q ∈ Rdh, K ∈ Rdh and V ∈ Rdv are
+transformed from the input X ∈ Rd via linear projection
+with parameter matrix W Q
+i
+∈ Rd×dh, W K
+i
+∈ Rd×dh, and
+W V
+i
+∈ Rd×dv respectively. For cross-attention, Q, K, and
+V can have different sources.
+In practice, transformers adopt the multi-head attention
+design, where multiple attention functions are applied in
+3
+
+…
+Object Features
+Point Features 
+Upsampled Point Features 
+…
+…
+Q
+K1×, V1×
+K2×, V2×
+Learnable 
+Upsample
+Concat & Linear
+Attention head=0
+Attention head=h/2 
+Object Features
+Prediction 
+Head
+Bounding Box  
+Proposals
+Point Features 
+Sampled Point Features  
+{Q} (batch)
+{K, V} (batch)
+(a)
+Simple Multi-Scale Attention 
+(b)
+Size-Adaptive Local Attention 
+Multi-Head 
+Attention 
+Pad & 
+Truncate
+Figure 2. An illustration of our hierarchical attention modules. (a). Simple Multi-Scale Attention (MS-A) learns features at different
+scales within the multi-head cross-attention module. It constructs high resolution (i.e., point density) point features from the single-scale
+input point features and uses keys and values of both scales. (b). Size-Adaptive Local Attention (Local-A) extracts localized features for
+each object candidate by restricting the attention range to be inside its bounding box proposal. The attention range (the token lengths of key
+and value) is adaptive for each object candidate (query) and we perform padding or truncating to allow batch processing
+.
+parallel across different attention heads. The input of each
+attention head is a segment of the layer’s input. Specifically,
+the query, key, and value are split along the hidden dimension
+into (Qi, Ki, Vi)h
+i=1, with Qi ∈ Rdh/h, Ki ∈ Rdh/h, Vi ∈
+Rdv/h, where h is the number of attention heads. The final
+output of the multi-head attention layer is the projection of
+the concatenated outputs of all attention heads:
+MultiHead(Q,K, V ) = Concat({Attn(Q0, K0, V0);
+...; Attn(Qh−1, Kh−1, Vh−1)})W O,
+(2)
+where the first term denotes the concatenation of the output
+and W O is the output projection matrix.
+3.2. Simple Multi-Scale Attention
+When applying transformers to point-based 3D object
+detection, the cross-attention models the relationship be-
+tween object candidates and all other points within the point
+cloud. The intuition is that, for each object candidate, every
+point within the point cloud (i.e., scene) either belongs to
+the object or can provide context information for the object.
+Therefore, it makes sense to gather all point features for
+every object candidate, and the importance of a point to the
+object candidate can be determined by the attention weight.
+However, due to the computation overhead of the atten-
+tion function, the actual number of points (i.e., tokens) that a
+model is learned on is set as 1024 [16,19], whereas the raw
+point cloud usually contains tens of thousands points [3,36].
+Such extensive downsampling on the point cloud causes a
+loss of detailed geometric information and fine-grained fea-
+tures, which are important for dense prediction tasks like
+object detection.
+To this end, we propose Simple Multi-Scale Attention
+(MS-A), which builds higher-resolution (i.e., higher point
+density) feature maps from the single-scale feature input. It
+then uses features of both scales as the key and value in the
+cross-attention between object candidates and other points.
+the multi-scale feature aggregation is realized through multi-
+scale token aggregation, where we use the key and value of
+different scales in different subsets of attention heads. Our
+goal is to create a higher-resolution feature map that provides
+fine-grained geometric details of the point cloud.
+The first step of our multi-scale attention is to obtain a
+higher-resolution feature map from the single-scale input.
+We propose a learnable upsampling operation. Given the
+layer’s input point cloud feature {zi}N
+i=1, zi ∈ Rd, we want
+to create a feature map with 2N points. To get the locations
+(i.e., coordinates) of the 2N points, we use FPS to sample
+4
+
+2N points from the raw point cloud {pi}2N
+i=1, pi ∈ R3. Next,
+for each sampled point pi, we search for the top three of
+its nearest neighbors (in the euclidean distance) in the input
+feature map {zi}N
+i=1, denoted as {z0
+i , z1
+i , z2
+i }. Then we cal-
+culate a weighted interpolation of the three-point features,
+weighted by the inverse of their distance to the sample point.
+The interpolated feature is then projected into the feature rep-
+resentation of sampled point. The upsampled point feature
+map can be written as:
+{˜zi}2N
+i=1, ˜zi = Φθ(interpolate({z0
+i , z1
+i , z2
+i }))
+(3)
+Here, Φθ is learnable projection function parameterized by
+θ. We choose MLP as our projection function.
+After the upsampling, we have two sets of point features
+of different scale {zi}N
+i=1, {˜zi}2N
+i=1. To avoid computation
+increase, we perform multi-head cross-attention on both sets
+of point features in a single pass by using features of different
+scales on different attention heads. We divide attention heads
+evenly into two groups, and use zi}N
+i=1 to obtain K and V
+in the first group while using the other for the second group.
+Both groups share the same set of queries transformed from
+{qi}M
+i=1. Since the input and output of this module are the
+same as a plain attention module, we can plug MS-A into any
+attention-based model to enable feature learning at different
+scales. In practice, we apply MS-A only at the first layer
+of a transformer which makes minimal modifications to the
+network and introduces little computation overhead.
+3.3. Size-Adaptive Local Attention
+Although the attention mechanism can model the relation-
+ship between every point pair, it is not guaranteed the learned
+model will pay more attention to points that are important
+to an object (e.g., those belonging to the object) than the
+ones that are not. The lack of hierarchy in transformers, on
+the other hand, does not support explicit localized feature
+extraction. Different from existing local attentions that are
+performed within a fixed region, we propose Size-Adaptive
+Local Attention (Local-A) that defines local regions based
+on the size of bounding box proposals.
+We first generate intermediate bounding box proposals
+{ˆbi}M
+i=1 with the features of object candidates ({qi}M
+i=1).
+We then perform cross-attention between every candidate qi
+and the points sampled from within its corresponding box
+proposal ˆbi. Therefore, we have customized size-adaptive
+local regions for every query point. For every input object
+candidate qil ∈ Rd, it is updated Local-A as:
+qi
+l+1 = Attn(Ql
+i, Ki, Vi), where
+(4)
+Ql
+i = qi
+lW Q, Ki = ZiW K, Vi = ZiW V with
+(5)
+Zi = {zk
+i | pos(zk
+i) in ˆbi}, ˆbi = Predl
+box(qi
+l).
+(6)
+In the Eq.( 6), we use pos(·) to denote the coordinate of a
+point in the 3D space, and Zi is a set of points inside box
+ˆbi. Note that the point features {zi}N
+i=1 are extracted by the
+backbone network and are not updated during the feature
+learning of object candidates. Predl
+box is the prediction head
+at layer l that generate intermediate box predictions.
+Since object candidates (i.e., query) will have different
+sets of keys and values depending on the size of their bound-
+ing box proposals, the number of K and V tokens also
+differs for each object candidate. To allow batch computa-
+tion, we set a maximum number of points (Nlocal) for the
+sampling process and use Nlocal as a fixed token length for
+every query point. For bounding boxes that contain less than
+Nlocal points, we pad the point sequence with an unused
+token to Nlocal and mask the unused tokens out in the cross-
+attention function; for those containing more than Nlocal
+points, we randomly discard them and truncate the sequence
+to have Nlocal points as keys and values. Lastly, in the case
+where the bounding box is empty, we perform ball query [23]
+around the object candidate to sample Nlocal points.
+Same as MS-A, Local-A does not pose additional require-
+ments on modules input, therefore we can apply it at any
+layer of a transformer. Specifically, we apply Local-A at the
+end of a transformer where bounding box proposals are in
+general more accurate.
+4. Experiments
+In this section, we first evaluate our method on two widely
+used indoor point cloud detection datasets, ScanNetV2 and
+SUN RGB-D. Next, we provide qualitative and quantita-
+tive analyses of our method, including visualizations of the
+bounding box predictions and attention weights, and eval-
+uations using our proposed size-aware metrics. Lastly, we
+include ablation studies on the design choices of our atten-
+tion modules. We include more experiments and ablation
+studies in Appendix A.1, including analyses on the infer-
+ence speed and the number of parameters of each individual
+attention module.
+4.1. Main Results
+Datasets.
+ScanNetV2 [3] consists of 1513 reconstructed
+meshes of hundreds of indoor scenes. It contains rich anno-
+tations for various 3D scene understanding tasks, including
+object classification, semantic segmentation, and object de-
+tection. For point cloud object detection, it provides axis-
+aligned bounding boxes with 18 object categories. We follow
+the official dataset split by using 1201 samples for training
+and 312 samples for testing. SUN RGB-D [36] is a single-
+view RGB-D dataset with 10335 samples. For 3D object
+detection, it provides oriented bounding box annotations
+with 37 object categories, while we follow the standard eval-
+uation protocol [21] and only use the 10 common categories.
+The training split contains 5285 samples and the testing set
+contains 5050 samples.
+5
+
+Methods
+#Params
+Backbone
+ScanNet V2
+[email protected]
+[email protected]
+VoteNet [21]
+-
+PointNet++
+62.9
+39.9
+H3DNet [54]
+-
+PointNet++
+64.4
+43.4
+H3DNet [54]
+-
+4×PointNet++
+67.2
+48.1
+3DETR [19]
+-
+transformer
+65.0
+47.0
+Pointformer [20]
+-
+transformer
+64.1
+42.6
+Group-Free6,256 [16]
+13.0M
+PointNet++
+67.3 (66.3)
+48.9 (48.5)
+w/ MS + Local (Ours)
+15.0M
+PointNet++
+67.9 (67.1) (↑ 0.6)
+51.4 (49.8) (↑ 2.5)
+RepSurf-U6,256 [27]
+13.1M
+PointNet++
+68.8 ( - )
+50.5 ( - )
+RepSurf-U6,256 (reproduce)
+13.1M
+PointNet++
+68.0 (67.4)
+50.2 (48.7)
+w/ MS + Local (Ours)
+15.1M
+PointNet++
+69.5 (68.8) (↑ 1.5)
+52.5 (51.1) (↑ 2.3)
+Group-Free12,512 [16]
+26.9M
+PointNet++w2x
+69.1 (68.6)
+52.8 (51.8)
+w/ MS + Local (Ours)
+28.9M
+PointNet++w2x
+70.3 (69.2) (↑ 1.2)
+54.6 (53.2) (↑ 1.8)
+RepSurf-U12,512 [27]
+27.1M
+PointNet++w2x
+71.2 ( - )
+54.8 ( - )
+RepSurf-U12,512 (reproduce)
+27.1M
+PointNet++w2x
+70.8 (70.2)
+54.4 (53.6)
+w/ MS + Local (Ours)
+29.1M
+PointNet++w2x
+71.7 (71.0) (↑ 0.9)
+56.5 (54.8) (↑ 2.1)
+Table 1. Performance of object detection on ScanNetV2. We follow the standard protocol [21] by reporting the best results over 5 × 5
+trials (5 trainings, each with 5 testings) and including the averaged results in the bracket. Group-FreeL,O denotes the variant with L decoder
+layers and O object candidates. The same notation applies to RepSurf-U. The detection code of RepSurf is not published, so we implement
+our version of RepSurf-U and apply our method to it. We include the results of our implementation of RepSurf-U.
+Methods
+[email protected]
+[email protected]
+VoteNet [21]
+59.1
+35.8
+H3DNet [54]
+-
+-
+H3DNet [54]
+60.1
+39.0
+3DETR [19]
+59.1
+32.7
+Pointformer [20]
+61.1
+36.6
+Group-Free6,256 [16]
+63.0 (62.6)
+45.2 (44.4)
+w/ MS + Local (Ours)
+63.8 (63.2) (↑ 0.8)
+46.6 (45.7) (↑ 1.4)
+RepSurf-U6,256 [27]
+64.3 ( - )
+45.9 ( - )
+RepSurf-U6,256 (repd.)
+64.0 (63.3)
+45.7 (45.2)
+w/ MS + Local (Ours)
+64.5 (63.8) (↑ 0.5)
+47.5 (46.1) (↑ 1.8)
+Table 2.
+Performance of object detection on SUN RGB-D.
+“repd." stands for the reproduced results of our implementation.
+“-" means the official result is not available.
+Evaluation metrics.
+For both datasets, we follow the stan-
+dard evaluation protocol [21] and use the mean Average Pre-
+cision (mAP) as the evaluation metric. We report mAP scores
+under two different Intersection over Union (IoU) thresholds:
+[email protected] and [email protected]. In addition, in Section 4.3, to
+evaluate model performance across different object sizes, we
+follow the practice in 2D vision [12] and implement our own
+size-aware metrics that measure the mAP on small, medium,
+and large objects respectively. On account of the randomness
+of point cloud training and inference, we train a model 5
+times and test each model 5 times. We report both the best
+and the average results among the 25 trials.
+Baselines.
+We validate our method by applying it to ex-
+isting transformer point cloud detectors. Group-Free [16]
+extracts features for object candidates using a transformer
+decoder with plain attention. We include two configura-
+tions of Group-Free in our comparison: Group-Free6,256
+samples a total of 256 object candidates for feature learning
+and bounding box prediction, using a transformer decoder
+with 6 layers; Group-Free12,512 is the largest configuration,
+which has 12 transformer layers and 512 object candidates.
+RepSurf-U [27] proposes a novel multi-surface (umbrella
+curvature) representation of point clouds that can explicitly
+describe the local geometry. For object detection, RepSurf-U
+adopts the transformer decoder of Group-Free and replaces
+its backbone with one that extracts features on both point
+clouds and the surface representations. The official imple-
+mentation and the averaged results of RepSurf-U for object
+detection are not publicly available, so we include the results
+of our own implementation of RepSurf-U.
+We also include the performance of previous point-based
+3D detectors for comparison. VoteNet [21] aggregates fea-
+tures for object candidates through end-to-end optimizable
+Hough Voting. H3DNet [54] proposes a hybrid set of ge-
+ometric primitives for object detection and trains multiple
+individual backbones for each primitive. 3DETR [19] solves
+point cloud object detection as a set-to-set problem using
+a transformer encoder-decoder network. Pointformer [20]
+proposes a hierarchical transformer-based point cloud back-
+bone and adopts the voting algorithm of VoteNet for object
+detection.
+Implementation details.
+For a baseline model with L
+transformer layers, we enable multi-scale feature learning
+by replacing the cross-attention of the 1-st layer with MS-A.
+After the L-th layer, we append an additional transformer
+6
+
+layer to perform local feature aggregation, which consists
+of Local-A and a feedforward layer. We follow the original
+training settings of the baseline models [16,27]. The detailed
+hyperparameter settings can be found in Appendix A.2.
+Results.
+From Table 1, on ScanNetV2, we observe consis-
+tent improvements in point cloud transformer detectors when
+equipped with our attention modules. By applying MS-A
+and Local-A to Group-Free, we achieve on-par performance
+with the state-of-the-art RepSurf-U detector. In addition, we
+can further improve RepSurf-U by over 1% in [email protected]
+and over 2% in [email protected] on varying model configurations.
+Table 2 shows a similar trend on SUN RGB-D, where our
+attention modules boost the [email protected] of group-Free to sur-
+pass RepSurf-U, and can further improve the state-of-the-art
+method by 0.5% in [email protected] and 1.8% in [email protected].
+4.2. Qualitative Results
+In Figure 3, we provide qualitative results on both datasets.
+The visualized results are of our methods applied to the
+Group-Free detectors. The qualitative results suggest that
+our model is able to detect and classify objects of different
+scales even in complex scenarios containing more than 10
+objects (e.g., the example in the bottom row). By looking
+into cross-attention weights in the transformer detector, we
+find that object candidates tend to have higher correlations
+with points that belong to their corresponding objects.
+4.3. Performance on objects of different sizes.
+In addition to the standard evaluation metrics, we are in-
+terested in examining models’ performance across different
+object sizes. Inspired by the size-aware metrics in 2D de-
+tection [12], we implement our own version of size-aware
+metrics for 3D detection. We conduct this analysis on Scan-
+NetV2, on which we calculate the volume for all the objects
+in all samples. We set the threshold for mAPS as the 30th
+percentile of the volume of all objects, and use the 70th
+percentile as the threshold for mAPL. More details about
+these metrics are included in Appendix A.2.
+MS-A
+Local-A
+mAPS
+mAPM
+mAPL
+-
+-
+63.1
+76.6
+83.2
+
+-
+65.0
+77.5
+83.9
+-
+
+65.2
+78.6
+83.9
+
+
+65.6 (↑ 2.5)
+79.0 (↑ 2.4)
+84.3 (↑ 1.1)
+Table 3.
+Performance on different size categories on Scan-
+NetV2. We define the S/M/L thresholds based on the statistics
+(volume distribution) of ScanNetV2 objects. The configuration in
+the first row denotes the Group-Free12,512 baseline.
+In Table 3, we evaluate our methods using size-aware
+metrics. We report the average result over 25 trials. The first
+row denotes the Group-Free12,512 baseline. Firstly, by com-
+paring the mAPS to mAPL, we notice that it has imbalanced
+performance across different object sizes. Looking at the
+improvement margins, we find our method to have the most
+performance gain on small and medium-sized objects. The
+result suggests that hierarchical designs can aid fine-grained
+and localized feature learning for point cloud transformer
+detectors and helps models detect smaller objects.
+4.4. Ablation Study
+In this subsection, we first conduct an ablation study on
+the stand-alone effects of our multi-scale attention and size-
+adaptive local attention. Next, we include empirical analyses
+of the design choices of our attention modules. If not other-
+wise specified, experiments in this subsection are conducted
+on ScanNetV2 with the Group-Free12,512 baseline. With-
+out loss of generality, the results in this subsection are the
+averaged numbers over 25 trials.
+The stand-alone effects of MS-A and Local-A.
+Table 4
+shows the stand-alone performance of our proposed attention
+modules. Compared to the plain attention baseline, both of
+our attentions are proved to be effective. When combined
+together, we find the two modules to be complementary to
+each other and bring more significant performance gain.
+MS-A
+Local-A
+[email protected]
+[email protected]
+-
+-
+68.6
+51.8
+
+-
+68.9
+52.5
+-
+
+68.9
+52.9
+
+
+69.2
+53.2
+Table 4. The stand-alone effect of our attention modules. The
+configuration in the first row denotes the Group-Free12,512 baseline.
+The results are averaged over 25 trials.
+The maximum number of points (Nlocal) in Local-A.
+In Local-A, for each object candidate (i.e., query), we sam-
+ple a set of points within its corresponding bounding box
+proposal and use the point features as the key and value
+for this object candidate in the cross-attention function. As
+introduced in Section 3.3, we cap the number of sampled
+points with Nlocal to allow batch computation.
+We provide an empirical analysis of the effects of Nlocal
+on Local-A. From Table 5, we find that too little number
+of points (e.g., Nlocal = 8) for Local-A results in a per-
+formance drop. On the other hand, as Nlocal continues to
+increase, we do not observe a significant performance gain
+compared to Nlocal = 16. Intuitively, a small Nlocal means
+the points within each bounding box are sampled sparsely,
+which can be too sparse to provide enough information about
+7
+
+Scene
+Groundtruth
+Prediction
+Attention
+Figure 3. Qualitative results on SUN RGB-D (top) and ScanNetV2 (bottom). The color of a bounding box in the middle two columns
+stands for the semantic label of the object. In the last column, we draw both the groundtruth (in green) and the prediction (in blue) of the
+object. We highlight the points that belong to an object for better visualization. In the last column, we visualize the attention weight of the
+last transformer layer (before applying Local-A). We visualize the cross-attention weight between an object candidate and the point cloud.
+Nlocal
+[email protected]
+[email protected]
+mAPS
+mAPM
+mAPL
+8
+67.8
+51.1
+64.6
+78.0
+82.8
+16
+68.9
+52.9
+65.2
+78.6
+83.9
+24
+68.9
+53.0
+65.4
+78.5
+84.0
+32
+68.3
+52.1
+64.7
+77.8
+84.3
+Table 5. The effect of Nlocal in Local-A. When there are enough
+points, a larger Nlocal means the points are sampled more densely
+within each bounding box proposal.
+any object. This explains why Nlocal = 8 does not work
+well. However, on the other hand, a large Nlocal may only
+benefit large objects and has little effect on smaller objects,
+because the latter are padded with unused tokens.
+MS-A with different feature resolutions.
+In Section 3,
+we propose learnable upsampling for MS-A to build higher-
+resolution point features from the single-scale input. In the
+same spirit, a parameterized downsampling procedure can
+be realized through conventional set abstraction [23], which
+aggregated point features within local groups and produce
+a feature map with fewer points (i.e., lower resolution). In-
+tuitively, a higher point density of the feature map provides
+more fine-grained features. To study the effects of feature
+maps of different granularity, we conduct an empirical analy-
+sis on MS-A using different sets of multi-scale feature maps
+representing point clouds of varying granularity.
+In Table 6, we examined the performance of two multi-
+scale choices in comparison with the single-scale baseline.
+The result suggests that coarse features (s = 0.5×) do not
+benefit transformer detectors. This is expected because trans-
+formers do not have limited receptive fields and thus do not
+Feature Scales s
+[email protected]
+[email protected]
+[1×]
+68.6
+51.8
+[1×, 2×]
+68.9
+52.5
+[0.5×, 1×, 2×]
+67.9
+51.7
+Table 6. Simple Multi-Scale Attention with different feature
+scales. Feature scale = s× means the feature map contains s × N
+points, with N being the original number of points. A larger s
+denotes a feature map with higher point density (i.e., resolution)
+rely on a coarse-grained feature map to learn global context.
+5. Conclusion
+In this work, we present Simple Multi-Scale Attention and
+Size-Adaptive Local Attention, two model-agnostic modules
+that bring in hierarchical designs to existing transformer-
+based 3D detectors. We enable multi-scale feature learning
+and explicit localized feature aggregation through improved
+attention functions, which are generic modules that can be
+applied to any existing attention-based network for end-to-
+end training. We improve the state-of-the-art transformer
+detector on two challenging indoor 3D detection benchmarks,
+with the largest improvement margin on small objects.
+As our attention modules promote fine-grained feature
+learning, which is important to various dense prediction
+vision tasks, one direction for future work is to adapt our
+attention modules for other point cloud learning problems
+such as segmentation. Another direction is to introduce more
+efficient attention mechanisms to the multi-scale attention to
+further bring down the computation overhead.
+8
+
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+A. Appendix
+A.1. More Experiments
+The placement of Simple Multi-scale Attention.
+We de-
+sign simple multi-scale attention as a compact network layer
+to enable hierarchical feature learning. As it can be inserted
+at any place within a network, we are interested in finding out
+how the placement of the multi-scale attention layer affects
+a model’s performance.
+Layers
+[email protected]
+[email protected]
+[0]
+68.9
+52.5
+[0, 4, 8]
+68.8
+52.4
+[0, 3, 6, 9]
+68.9
+52.3
+[0, 2, 4, 6, 8, 10]
+68.7
+52.6
+Table 7. Different placements of the simple multi-scale atten-
+tion layer. Layers = [i] means we replace the ith layer of the
+transformer decoder with our MS-A layer. The best results are in
+bold, and the second-best results are underlined.
+We consider different strategies to place MS-A within
+the transformer decoder of Group-Free. We divide the 12-
+layer decoder into several stages and place MS-A at the
+first layer of each stage. Specifically, our default setting
+uses a single MS-A at the first decoder layer, and we try
+placing MS-A by dividing the decoder evenly into 3, 4, and
+6 stages. From the results in Table 7, we do not observe a
+significant benefit of using more than one multi-scale layer.
+We conjecture this is because the up-scaled feature map in
+our multi-scale attention is obtained through interpolation
+10
+
+and simple linear projection. The up-scale point feature
+obtained in this way may mainly provide more accurate
+geometric information with a higher point density, while may
+not have much semantic difference than the original input
+feature. We expect such fine-grained geometric information
+to be particularly helpful at the beginning of the decoding
+stage (i.e., Layer= 0), yet may be less useful as the object
+features go deeper in the decoder and become more abstract.
+Per-category mAP on ScanNetV2 and SUN RGB-D.
+We include the detailed per-category mAP on both datasets
+in Table 9, Table 10, Table 11, and Table 12. For the results
+in this paragraph, we follow the baselines [16,19,20,27] and
+report the result of the best trial.
+Inference Speed.
+We analyze the parameter and computa-
+tion overhead of each of our attention modules. We measure
+the inference speeds for all model configurations on the same
+machine with a single A100 GPU. In Table 8, we can see that
+replacing plain attention with MS-A results in little parame-
+ter increase. While applying Local-A leads to a larger param-
+eter increase, the Local-A module itself contains the same
+number of parameters as a plain cross-attention. The param-
+eter increase is mainly due to the additional feed-forward
+layer and learnable positional embeddings, etc. In terms of
+inference speed, we find MS-A to cause more substantial
+latency in inference. Such latency is caused by applying
+the attention function on the key/value with 2 times more
+tokens (from 1024 to 2048). A future direction is to incorpo-
+rate more efficient attention mechanisms into the multi-scale
+attention function.
+MS-A
+Local-A
+#Params
+Inference Speed
+(M)
+(ms/frame)
+-
+-
+26.9
+186
+
+-
+27.0 (+0.1)
+225
+-
+
+28.8 (+1.9)
+191
+
+
+28.9 (+2.0)
+232
+Table 8. Ablating the parameter and computation overhead of
+individual attention modules.
+A.2. Implementation Details.
+We include implementation details covering several as-
+pects in this paragraph. In addition, we include our source
+code in the supplementary material, containing the full im-
+plementation of our attention modules.
+Training Details.
+Group-Free baseline. When applying our method to this
+baseline, we follow the original training settings. Specifi-
+cally, on ScanNetV2, the models are trained for 400 epochs
+on 4 GPUs with a batchsize of 32 (8 on each GPU). We use
+the same optimizer with the same learning rates and weight
+decays as the baseline training. On SUN RGB-D, models
+are trained for 600 epochs on 4 GPUs with the same learning
+rate and weight decay as the baseline training on this dataset.
+RepSurf-U baseline. The official implementation and
+training details of this baseline are not published. We im-
+plement our own version of RepSurf-U detector, for which
+we mostly follow the training setup of Group-Free and have
+done a grid search for the hyperparameters. Different from
+Group-Free, we train RepSurf-U models on ScanNetV2 and
+SUN RGB-D using a weight decay of 0.01 for all model
+parameters, because we find it to achieve better performance
+on our reproduced RepSurf-U. The learning rate and other
+hyperparameters remain the same as Group-Free on both
+datasets. When applying our method to the reproduced
+model, we do not change the hyperparameter configurations.
+Bounding box parameterization.
+In this paragraph, we
+include a brief introduction to the bounding box parameteri-
+zation used in our baselines. First, the predicted box center
+ˆc for each object candidate q is obtained by adding an offset
+to the coordinate of q. In this way, by predicting the cen-
+ter, the actual prediction made by a detector is this offset
+value. The size ˆd of a box is the height, width, and depth
+dimension. One way for predicting ˆd is to directly predict
+the values of H, W, and D. Another way is to divide a range
+of sizes into several bins and make a classification prediction
+that determines which “bin" the object belongs to. The final
+size prediction is obtained by adding the quantized size (i.e.,
+the bin) with a “residual" term which is also predicted by
+the model with another prediction head. The bounding box
+orientation ˆa is also parameterized as the combination of a
+quantized value and a residual term. Lastly, the prediction of
+the semantic label is a common classification problem that
+parameterizes a semantic label as a one-hot vector.
+Size-Aware Evaluation Metrics.
+For a quantitative anal-
+ysis of the model’s performance on objects of different
+sizes. We implement our own size-aware evaluation met-
+rics, namely mAPS, mAPM and mAPL. For each metric,
+we only calculate the mAP score among objects that fall
+into the corresponding size category (i.e., small, medium, or
+large). We conduct the size-aware evaluation on ScanNetV2,
+where we determine the threshold for dividing object size
+categories based on the statistics of this dataset. Specifically,
+we take the 1201 training samples and record the volume
+(v = H × W × D) of every groundtruth bounding box
+of every sample (see Figure 4). Among a total of 15733
+goundtruth bounding boxes, we take the 30th (v = 0.155)
+and 70th (v = 0.526) percentile as the thresholds for divid-
+ing small and large objects.
+11
+
+00.155 0.526
+1
+2
+3
+4
+5
+Volume of the object bounding box
+0
+200
+400
+600
+800
+1000
+Number of objects
+Figure 4. Volume distribution of the object groundtruth bounding boxes in ScanNetV2. We highlight the threshold of small objects
+(v <= 0.155, the 30th percentile) and large objects (v > 0.526, the 70th percentile)
+methods
+backbone
+cab
+bed chair sofa tabl door wind bkshf
+pic
+cntr desk curt fridg showr toil
+sink bath ofurn mAP
+VoteNet [21]
+PointNet++
+47.7 88.7 89.5 89.3 62.1 54.1 40.8
+54.3
+12.0 63.9 69.4 52.0 52.5
+73.3
+95.9 52.0 92.5 42.4
+62.9
+H3DNet [54]
+4×PointNet++
+49.4 88.6 91.8 90.2 64.9 61.0 51.9
+54.9
+18.6 62.0 75.9 57.3 57.2
+75.3
+97.9 67.4 92.5 53.6
+67.2
+3DETR [19]
+transformer
+49.4 83.6 90.9 89.8 67.6 52.4 39.6
+56.4
+15.2 55.9 79.2 58.3 57.6
+67.6
+97.2 70.6 92.2 53.0
+65.0
+Pointformer [20]
+Pointformer
+46.7 88.4 90.5 88.7 65.7 55.0 47.7
+55.8
+18.0 63.8 69.1 55.4 48.5
+66.2
+98.9 61.5 86.7 47.4
+64.1
+GroupFree6,256
+PointNet++
+54.1 86.2 92.0 84.8 67.8 55.8 46.9
+48.5
+15.0 59.4 80.4 64.2 57.2
+76.3
+97.6 76.8 92.5 55.0
+67.3
+w/ MS + Local
+PointNet++
+55.9 88.6 93.6 90.8 68.2 59.0 44.2
+50.3
+14.6 63.0 85.0 62.8 58.5
+68.6
+97.6 73.2 92.4 56.4
+67.9
+RepSurf-U6,256
+PointNet++
+55.5 87.7 93.4 85.9 69.1 57.3 48.8
+50.0
+16.5 61.0 81.6 66.2 59.0
+77.5
+99.2 78.2 94.0 56.8
+68.8
+RepSurf-U6,256 (repd.)
+PointNet++
+57.4 89.6 93.2 87.4 70.2 58.8 46.6
+47.4
+18.1 63.4 78.2 70.4 46.5
+81.0
+99.8 69.4 90.8 55.5
+68.0
+w/ MS + Local
+PointNet++
+51.2 89.5 93.4 87.5 71.8 60.5 49.0
+57.7
+21.9 65.2 82.1 70.3 53.3
+80.2
+98.2 68.8 91.9 58.2
+69.5
+GroupFree12,512
+PointNet++w2x 52.1 91.9 93.6 88.0 70.7 60.7 53.7
+62.4
+16.1 58.5 80.9 67.9 47.0
+76.3
+99.6 72.0 95.3 56.4
+69.1
+w/ MS + Local
+PointNet++
+53.7 91.9 93.4 88.8 72.1 61.3 52.8
+58.6
+17.4 70.8 83.3 69.9 56.5
+75.6
+98.5 70.3 94.4 56.9
+70.3
+RepSurf-U12,512
+PointNet++w2x 54.6 94.0 96.2 90.5 73.2 62.7 55.7
+64.5
+18.6 60.9 83.1 69.9 49.4
+78.4
+99.4 74.5 97.6 58.3
+71.2
+RepSurf-U12,512 (repd.) PointNet++w2x 54.5 90.7 93.4 87.6 76.3 64.4 54.4
+61.4
+19.0 62.2 84.0 69.2 48.8
+79.2
+99.8 75.9 92.2 62.0
+70.8
+w/ MS + Local
+PointNet++w2x 58.0 89.3 94.1 86.5 74.3 62.4 60.2
+57.9
+21.7 67.9 85.3 74.4 53.5
+75.9
+99.6 74.6 91.6 63.7
+71.7
+Table 9. Performance of [email protected] in each category on ScanNetV2.
+methods
+backbone
+cab
+bed chair sofa tabl door wind bkshf
+pic
+cntr desk curt fridg showr toil
+sink bath ofurn mAP
+VoteNet [21]
+PointNet++
+14.6 77.8 73.1 80.5 46.5 25.1 16.0
+41.8
+2.5
+22.3 33.3 25.0 31.0
+17.6
+87.8 23.0 81.6 18.7
+39.9
+H3DNet [54]
+4×PointNet++
+20.5 79.7 80.1 79.6 56.2 29.0 21.3
+45.5
+4.2
+33.5 50.6 37.3 41.4
+37.0
+89.1 35.1 90.2 35.4
+48.1
+GroupFree6,256
+PointNet++
+23.0 78.4 78.9 68.7 55.1 35.3 23.6
+39.4
+7.5
+27.2 66.4 43.3 43.0
+41.2
+89.7 38.0 83.4 37.3
+48.9
+w/ MS + Local
+PointNet++
+27.3 80.8 83.3 85.3 60.2 39.7 21.7
+40.4
+7.6
+41.7 61.5 42.9 42.3
+26.2
+96.1 38.5 89.5 39.7
+51.4
+RepSurf-U6,256
+PointNet++
+24.9 79.6 80.1 70.4 56.4 36.7 25.5
+41.4
+8.8
+28.7 68.0 45.2 45.0
+42.7
+91.3 40.1 85.1 39.2
+50.5
+RepSurf-U6,256 (repd.)
+PointNet++ 1.
+24.3 82.6 82.6 71.3 55.9 38.3 18.6
+40.3
+11.2 44.0 60.7 45.1 35.7
+36.6
+97.1 34.6 84.6 39.8
+50.2
+w/ MS + Local
+PointNet++
+27.1 80.9 83.0 77.1 58.0 45.8 24.8
+50.8
+10.5 31.9 67.7 44.6 40.6
+34.9
+97.7 38.3 87.3 44.6
+52.5
+GroupFree12,512
+PointNet++w2x 26.0 81.3 82.9 70.7 62.2 41.7 26.5
+55.8
+7.8
+34.7 67.2 43.9 44.3
+44.1
+92.8 37.4 89.7 40.6
+52.8
+w/ MS + Local
+PointNet++
+31.0 81.0 85.0 79.4 61.1 44.5 27.9
+50.6
+10.1 45.0 61.2 54.1 39.5
+43.5
+91.7 45.9 89.3 42.4
+54.6
+RepSurf-U12,512
+PointNet++w2x 28.5 83.5 84.8 72.6 64.0 43.6 28.3
+57.8
+9.6
+37.0 69.7 45.9 46.4
+46.1
+94.9 39.1 92.1 42.6
+54.8
+RepSurf-U12,512 (repd.) PointNet++w2x 27.6 82.7 85.3 68.8 60.6 44.0 27.3
+56.7
+9.6
+39.6 63.7 53.8 43.0
+42.4
+99.8 38.8 88.7 47.3
+54.4
+w/ MS + Local
+PointNet++w2x 29.3 83.6 85.7 78.7 66.2 45.6 30.4
+59.8
+10.4 34.2 60.0 60.8 48.1
+45.3
+99.9 44.5 87.1 48.4
+56.5
+Table 10. Performance of [email protected] in each category on ScanNetV2.
+12
+
+methods
+backbone
+bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP
+VoteNet [21]
+PointNet++
+75.5
+85.6
+31.9
+77.4 24.8 27.9
+58.6
+67.4 51.1
+90.5
+59.1
+H3DNet [54]
+4×PointNet++
+73.8
+85.6
+31.0
+76.7 29.6 33.4
+65.5
+66.5 50.8
+88.2
+60.1
+3DETR [19]
+transformer
+69.8
+84.6
+28.5
+72.4 34.3 29.6
+61.4
+65.3 52.6
+91.0
+61.1
+Pointformer [20]
+Pointformer
+80.1
+84.3
+32.0
+76.2 27.0 37.4
+64.0
+64.9 51.5
+92.2
+61.1
+GroupFree6,256
+PointNet++
+80.0
+87.8
+32.5
+79.4 32.6 36.0
+66.7
+70.0 53.8
+91.1
+63.0
+w/ MS + Local
+PointNet++
+83.2
+86.7
+34.5
+79.0 31.9 39.3
+66.0
+70.6 55.6
+90.8
+63.8
+RepSurf-U6,256
+PointNet++
+81.1
+89.3
+34.4
+80.4 33.5 37.3
+68.1
+71.4 54.8
+92.3
+64.3
+RepSurf-U6,256 (repd.)
+PointNet++
+79.5
+87.5
+33.8
+79.4 32.7 40.2
+69.0
+70.3 55.4
+92.1
+64.0
+w/ MS + Local
+PointNet++
+79.9
+87.0
+36.8
+79.5 33.8 41.4
+67.4
+71.2 55.3
+92.4
+64.5
+Table 11. Performance of [email protected] in each category on SUN RGB-D.
+methods
+backbone
+bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP
+VoteNet [21]
+PointNet++
+45.4
+53.4
+6.8
+56.5
+5.9
+12.0
+38.6
+49.1 21.3
+68.5
+35.8
+H3DNet [54]
+4×PointNet++
+47.6
+52.9
+8.6
+60.1
+8.4
+20.6
+45.6
+50.4 27.1
+69.1
+39.0
+GroupFree6,256
+PointNet++
+64.0
+67.1
+12.4
+62.6 14.5 21.9
+49.8
+58.2 29.2
+72.2
+45.2
+w/ MS + Local
+PointNet++
+66.2
+67.4
+10.8
+63.6 15.0 24.7
+56.7
+56.1 30.8
+74.3
+46.6
+RepSurf-U6,256
+PointNet++
+65.2
+67.5
+13.2
+63.4 15.0 22.4
+50.9
+58.8 30.0
+72.7
+45.9
+RepSurf-U6,256 (repd.)
+PointNet++
+61.4
+66.8
+11.3
+64.0 14.8 24.2
+51.8
+59.0 31.6
+71.7
+45.7
+w/ MS + Local
+PointNet++
+62.2
+67.6
+16.6
+65.0 15.0 24.2
+57.0
+59.0 30.9
+77.7
+47.5
+Table 12. Performance of [email protected] in each category on SUN RGB-D.
+13
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+Moment of inertia of slowly rotating anisotropic neutron stars in f(R, T) gravity
+Juan M. Z. Pretel1, ∗
+1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud,
+150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil
+(Dated: January 10, 2023)
+Within the framework of f(R, T) theories of gravity, we investigate the hydrostatic equilibrium of
+anisotropic neutron stars with a physically relevant equation of state (EoS) for the radial pressure.
+In particular, we focus on the f(R, T) = R + 2βT model, where β is a minimal coupling constant.
+In the slowly rotating approximation, we derive the modified TOV equations and the expression for
+the relativistic moment of inertia. The main properties of neutron stars, such as radius, mass and
+moment of inertia, are studied in detail. Our results revel that the main consequence of the 2βT term
+is a substantial increase in the surface radius for low enough central densities. Nevertheless, such
+a term slightly modifies the total gravitational mass and moment of inertia of the slowly rotating
+stars. Furthermore, the changes are noticeable when anisotropy is incorporated into the stellar fluid,
+and it is possible to obtain higher masses that are consistent with the current observational data.
+I.
+INTRODUCTION
+Despite the great success of General Relativity (GR) in
+predicting various gravitational phenomena tested in the
+solar system [1] and in strong-field situations (such as the
+final stage of compact-object binaries [2, 3]), it could not
+help to identify the nature of dark energy and other puz-
+zles. In other words, there are still many open problems
+in modern cosmology and it is well known that GR is not
+the only theory of gravity [4]. Indeed, it has been shown
+that GR is not renormalizable as a quantum field theory
+unless higher-order curvature invariants are included in
+its action [5, 6]. Furthermore, GR requires modifications
+at small time and length scales or at energies comparable
+with the Planck energy scales. In that regard, it has been
+argued that the early-time inflation and the late-time ac-
+celerated expansion of the Universe can be an effect of
+the modification of the geometric theory formulated by
+Einstein [7–10].
+One of the simplest ways to modify GR is by re-
+placing the Ricci scalar R in the standard Einstein-
+Hilbert action by an arbitrary function of R, this is, the
+so-called f(R) theories of gravity [11, 12].
+Extensive
+and detailed reviews on the cosmological implications
+of such theories can be found in Refs. [13–16]. On the
+other hand, at astrophysical level, these theories basically
+change the Tolman-Oppenheimer-Volkoff (TOV) equa-
+tions and hence the astrophysical properties of compact
+stars, such as mass-radius relations, maximum masses, or
+moment of inertia are somehow altered. See Ref. [17] for
+a broad overview about relativistic and non-relativistic
+stars within the context of modified theories of gravity
+formulated in both metric and metric-affine approaches.
+In most of the works reported in the literature about
+internal structure of compact stars in GR and modified
+theories of gravity it is very common to assume that such
+stars are made up of an isotropic perfect fluid. Never-
+∗ [email protected]
+theless, there are strong arguments indicating that the
+impact of anisotropy (this is, unequal radial and tangen-
+tial pressures) cannot be neglected when we deal with
+nuclear matter at very high densities and pressures, for
+instance, see Refs. [18–24] and references therein. In that
+regard, it has been shown that the presence of anisotropy
+can lead to significant changes in the main characteris-
+tics of compact stars [21–23, 25–31]. Within the frame-
+work of extended theories of gravity, it is also important
+to mention that non-rotating anisotropic compact stars
+have been recently studied by some authors in Refs. [32–
+50]. In addition, in the context of scalar-tensor theory
+of gravity, slowly rotating anisotropic neutron stars have
+been investigated in Ref. [51].
+Harko and collaborators [52] have proposed a gener-
+alization of f(R) modified theories of gravity in order
+to introduce a coupling between geometry and matter,
+namely f(R, T) gravity, where T denotes the trace of the
+energy-momentum tensor. Indeed, the simplest and most
+studied model involving a minimal matter-gravity cou-
+pling is given by f(R, T) = R+2βT gravity. The cosmo-
+logical aspects of this model have been recently explored
+in Refs. [53–57], while other authors have investigated
+the astrophysical consequences of the 2βT term on the
+equilibrium structure of isotropic [58–65] and anisotropic
+[37–42] compact stars. A characteristic of this model is
+that R = 0 outside a compact star, and hence the ex-
+terior spacetime is still described by the Schwarzschild
+exterior solution.
+As a result, it has been shown that
+for high enough central densities the contributions of the
+2βT term are irrelevant, whereas below a certain cen-
+tral density value the radius of an isotropic compact star
+undergoes substantial deviations from GR [62, 63].
+To determine the equilibrium configurations and mo-
+ment of inertia of slowly rotating anisotropic stars up to
+first order in the angular velocity, we will employ a phys-
+ically motivated functional relation σ (defined as the dif-
+ference between radial and tangential pressure) for the
+anisotropy profile known in the literature as quasi-local
+ansatz [25]. Moreover, we will follow a procedure anal-
+ogous to that carried out by Hartle in GR [66] in order
+arXiv:2301.02881v1  [gr-qc]  7 Jan 2023
+
+2
+to obtain the modified version of the differential equation
+which governs the difference between the angular velocity
+of the star and the angular velocity of the local inertial
+frames.
+To achieve our results, the present work is organized
+as follows: In Sec. II we briefly review f(R, T) gravity
+and we present the corresponding relativistic equations
+for the f(R, T) = R + 2βT model. In Sec. III we de-
+rive the modified TOV equations for anisotropic stellar
+configurations by adopting a non-rotating and slowly ro-
+tating metric. Section IV presents a well-known EoS to
+describe neutron stars as well as the anisotropy ansatz.
+In Sec. V we discuss our numerical results, and finally,
+our conclusions are presented in Sec. VI. In this paper
+we will use a geometric unit system and the sign conven-
+tion (−, +, +, +). However, our results will be given in
+physical units.
+II.
+BASIC FORMALISM OF f(R, T) GRAVITY
+A more general formulation of f(R) modified theories
+of gravity consists in the inclusion of an explicit gravity-
+matter coupling by means of an arbitrary function of the
+Ricci scalar R and the trace of the energy-momentum
+tensor T. Thus, the modified Einstein-Hilbert action in
+f(R, T) gravity is given by [52]
+S =
+1
+16π
+�
+f(R, T)√−gd4x +
+�
+Lm
+√−gd4x,
+(1)
+where g is the determinant of the spacetime metric gµν
+and Lm denotes the Lagrangian density for matter fields.
+The corresponding field equations in f(R, T) gravity can
+be obtained from the variation of the action (1) with
+respect to the metric:
+fR(R, T)Rµν − 1
+2f(R, T)gµν + [gµν□ − ∇µ∇ν]fR(R, T)
+= 8πTµν − (Tµν + Θµν)fT (R, T),
+(2)
+where Rµν is the Ricci tensor, Tµν the energy-momentum
+tensor, fR ≡ ∂f/∂R, fT ≡ ∂f/∂T, □ ≡ ∇µ∇µ is the
+d’Alembertian operator with ∇µ standing for the covari-
+ant derivative, and the tensor Θµν is defined in terms of
+the variation of Tµν with respect to the metric, namely
+Θµν ≡ gαβ δTαβ
+δgµν
+= −2Tµν + gµνLm − 2gαβ
+∂2Lm
+∂gµν∂gαβ .
+(3)
+Just as in f(R) gravity [11, 12], in f(R, T) theories the
+Ricci scalar is also a dynamical entity which is described
+by a differential equation obtained by taking the trace of
+the field equations (2), this is
+3□fR(R, T) + RfR(R, T) − 2f(R, T)
+= 8πT − (T + Θ)fT (R, T),
+(4)
+where we have denoted Θ = Θ µ
+µ . In addition, the four-
+divergence of Eq. (2) yields [67]
+∇µTµν =
+fT (R, T)
+8π − fT (R, T)
+�
+(Tµν + Θµν)∇µ ln fT (R, T)
++ ∇µΘµν − 1
+2gµν∇µT
+�
+.
+(5)
+In order to obtain numerical solutions that describe
+compact stars, one has to specify the particular model of
+f(R, T) gravity. In that regard, we consider the simplest
+model involving a minimal matter-gravity coupling pro-
+posed by Harko et al. [52], i.e. f(R, T) = R + 2βT grav-
+ity, which has been the most studied model of f(R, T)
+gravity at both astrophysical and cosmological scale. As
+a consequence, Eqs. (2), (4) and (5) can be written as
+follows
+Gµν = 8πTµν + βTgµν − 2β(Tµν + Θµν),
+(6)
+R = −8πT − 2β(T − Θ),
+(7)
+∇µTµν =
+2β
+8π − 2β
+�
+∇µΘµν − 1
+2gµν∇µT
+�
+,
+(8)
+where Gµν is the Einstein tensor.
+III.
+MODIFIED TOV EQUATIONS
+A.
+Non-rotating stars
+We shall assume that the matter source is described
+by an anisotropic perfect fluid with energy density ρ, ra-
+dial pressure pr and tangential pressure pt. Under theses
+assumptions, the energy-momentum tensor is given by
+Tµν = (ρ + pt)uµuν + ptgµν − σkµkν,
+(9)
+with uµ being the four-velocity of the fluid and which
+satisfies the normalization property uµuµ = −1, kµ is a
+unit radial four-vector so that kµkµ = 1, and σ ≡ pt − pr
+is the anisotropy factor.
+In addition, we consider that the interior spacetime
+of the spherically symmetric stellar configuration is de-
+scribed by the standard line element
+ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2),
+(10)
+where xµ = (t, r, θ, φ) are the Schwarzschild-like coordi-
+nates, and the metric potentials ψ and λ are functions
+only of the radial coordinate in a hydrostatic equilib-
+rium situation. Consequently, we can write uµ = e−ψδµ
+0 ,
+kµ = e−λδµ
+1 and the trace of the energy-momentum ten-
+sor (9) takes the form T = −ρ + 3pr + 2σ.
+Within the context of anisotropic fluids in f(R, T)
+gravity, the most adopted choice in the literature for
+the matter Lagrangian density is given by Lm = P,
+where P ≡ (pr + 2pt)/3.
+For more details about this
+
+3
+choice, see Refs. [37–40, 42]. Under this consideration,
+Θµν = −2Tµν + Pgµν and Eqs. (6), (7) and (8) become
+Gµν = 8πTµν + βTgµν + 2β(Tµν − Pgµν),
+(11)
+R = −8πT − 2β(3T − 4P),
+(12)
+∇µTµν =
+2β
+8π + 2β ∂ν
+�
+P − 1
+2T
+�
+.
+(13)
+For the metric (10) and energy-momentum tensor (9),
+the non-zero components of the field equations (11) are
+explicitly given by
+1
+r2
+d
+dr(re−2λ) − 1
+r2 = −8πρ + β
+�
+−3ρ + pr + 2
+3σ
+�
+,
+(14)
+e−2λ
+�2
+r ψ′ + 1
+r2
+�
+− 1
+r2 = 8πpr + β
+�
+−ρ + 3pr + 2
+3σ
+�
+,
+(15)
+e−2λ
+�
+ψ′′ + ψ′2 − ψ′λ′ + 1
+r (ψ′ − λ′)
+�
+= 8π(pr + σ) + β
+�
+−ρ + 3pr + 8
+3σ
+�
+,
+(16)
+where the prime represents differentiation with respect
+to the radial coordinate. Moreover, Eq. (13) implies that
+dpr
+dr = − (ρ + pr)ψ′ + 2
+r σ
++
+β
+8π + 2β
+d
+dr
+�
+ρ − pr − 2
+3σ
+�
+.
+(17)
+Eq. (14) leads to
+re−2λ = r −
+�
+r2
+�
+8πρ + β
+�
+3ρ − pr − 2
+3σ
+��
+dr,
+(18)
+or alternatively,
+e−2λ = 1 − 2m
+r ,
+(19)
+where m(r) represents the gravitational mass within a
+sphere of radius r, given by
+m(r) = 4π
+� r
+0
+¯r2ρ(¯r)d¯r
++ β
+2
+� r
+0
+¯r2
+�
+3ρ(¯r) − pr(¯r) − 2
+3σ(¯r)
+�
+d¯r.
+(20)
+At the surface, where the radial pressure vanishes,
+M ≡ m(rsur) is the total mass of the anisotropic compact
+star. From our anisotropic version (20), here we can see
+that by making σ = 0 one recovers the mass function for
+the isotropic case given in Ref. [63]. In view of Eq. (19),
+from Eq. (15) we obtain
+ψ′ =
+�m
+r2 + 4πrpr + βr
+2
+�
+−ρ + 3pr + 2
+3σ
+��
+×
+�
+1 − 2m
+r
+�−1
+,
+(21)
+and hence the relativistic structure of an anisotropic com-
+pact star within the context of f(R, T) = R+2βT gravity
+is described by the modified TOV equations:
+dm
+dr = 4πr2ρ + βr2
+2
+�
+3ρ − pr − 2
+3σ
+�
+,
+(22)
+dpr
+dr = − ρ + pr
+1 + a
+�m
+r2 + 4πrpr + βr
+2
+�
+3pr − ρ + 2
+3σ
+��
+×
+�
+1 − 2m
+r
+�−1
++
+a
+1 + a
+dρ
+dr
++
+2
+1 + a
+�σ
+r − a
+3
+dσ
+dr
+�
+,
+(23)
+dψ
+dr =
+1
+ρ + pr
+�
+−(1 + a)dpr
+dr + adρ
+dr + 2
+�σ
+r − a
+3
+dσ
+dr
+��
+,
+(24)
+where we have defined a ≡ β/(8π + 2β). As expected,
+the modified TOV equations in the isotropic scenario are
+retrieved when pr = pt [63]. Furthermore, when the min-
+imal coupling constant vanishes (this is, β = 0), we can
+recover the standard TOV equations for anisotropic stars
+in GR [23].
+Given an EoS for the radial pressure pr = pr(ρ) and
+an anisotropy relation for σ, Eqs. (22) and (23) can be
+integrated by guaranteeing regularity at the center of the
+star and for a given value of central energy density. In
+addition, according to Eq. (12), we notice that R = 0 in
+the outer region of the star. This means that we can still
+use the Schwarzschild vacuum solution to describe the
+exterior spacetime so that the interior solution is matched
+at the boundary r = rsur to the exterior Schwarzschild
+solution. Thus, the system of equations (22)-(24) can be
+solved by imposing the following boundary conditions
+m(0) = 0,
+ρ(0) = ρc,
+ψ(rsur) = 1
+2 ln
+�
+1 − 2M
+rsur
+�
+.
+(25)
+B.
+Slowly rotating stars
+In the slowly rotating approximation [66], i.e., when
+rotational corrections appear at first order in the angu-
+lar velocity of the stars Ω, the spacetime metric (10) is
+replaced by its slowly rotating counterpart [66, 68]
+ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2)
+− 2ω(r, θ)r2 sin2 θdtdφ,
+(26)
+where ω(r, θ) stands for the angular velocity of the lo-
+cal inertial frames dragged by the stellar rotation.
+In
+other words, if a particle is dropped from rest at a great
+distance from the rotating star, the particle would expe-
+rience an ever increasing drag in the direction of rotation
+of the star as it approaches. In fact, here it is convenient
+to define the difference ϖ ≡ Ω − ω as the coordinate an-
+gular velocity of the fluid element at (r, θ) seen by the
+freely falling observer [66].
+
+4
+Since Ω is the angular velocity of the fluid as seen by an
+observer at rest at some spacetime point (t, r, θ, φ), one
+finds that the four-velocity up to linear terms in Ω is given
+by uµ = (e−ψ, 0, 0, Ωe−ψ). To this order, the spherical
+symmetry is still preserved and it is possible to extend
+the validity of the TOV equations (22)-(24). Neverthe-
+less, the 03-component of the field equations contributes
+an additional differential equation for angular velocity
+ω(r, θ). By retaining only first-order terms in the angu-
+lar velocity, we have T03 = −[ϖ(ρ + pt) + ωpt]r2 sin2 θ
+and hence Eq. (11) gives the following expression
+G03 = −
+�
+2(4π + β)(ρ + pt)ϖ + 8πωpt
++β
+�
+−ρ + 1
+3pr + 8
+3pt
+�
+ω
+�
+r2 sin2 θ,
+(27)
+or alternatively,
+eψ−λ
+r4
+∂
+∂r
+�
+e−(ψ+λ)r4 ∂ϖ
+∂r
+�
++
+1
+r2 sin3 θ
+∂
+∂θ
+�
+sin3 θ∂ϖ
+∂θ
+�
+= 4(4π + β)(ρ + pt)ϖ.
+(28)
+Following the procedure carried out by Hartle in GR
+[66] and Staykov et al. in R2-gravity [68], we expand ϖ
+in the form
+ϖ(r, θ) =
+∞
+�
+l=1
+ϖl(r)
+� −1
+sin θ
+dPl
+dθ
+�
+,
+(29)
+where Pl are Legendre polynomials. In view of Eq. (29),
+we can write
+∂
+∂θ
+�
+sin3 θ∂ϖ
+∂θ
+�
+=
+�
+l
+ϖl(r)
+�
+(cos2 θ − sin2 θ)dPl
+dθ
+− sin θ cos θd2Pl
+dθ2 − sin2 θd3Pl
+dθ3
+�
+=
+�
+l
+ϖl(r) [l(l + 1) − 2] sin2 θdPl
+dθ ,
+(30)
+where we have used the Legendre differential equation
+d2Pl
+dθ2 + cos θ
+sin θ
+dPl
+dθ + l(l + 1)Pl = 0.
+(31)
+Thus, after substituting Eqs. (29) and (30) into (28),
+we get
+eψ−λ
+r4
+d
+dr
+�
+e−(ψ+λ)r4 dϖl
+dr
+�
+− l(l + 1) − 2
+r2
+ϖl
+= 4(4π + β)(ρ + pt)ϖl.
+(32)
+At great distances from the stellar surface, where
+spacetime must be asymptotically flat, the solution of
+Eq. (32) assumes the form ϖl(r) → c1r−l−2 + c2rl−1.
+Furthermore, the dragging angular velocity is expected
+to be ω → 2J/r3 (or alternatively, ϖ → Ω − 2J/r3) for
+r → ∞, where J is the angular momentum carried out
+by the star (see Ref. [69] for more details). Therefore, by
+comparison we can see that all coefficients in the Legen-
+dre expansion vanish except for l = 1. This means that
+ϖ is a function of r only, and Eq. (32) reduces to
+eψ−λ
+r4
+d
+dr
+�
+e−(ψ+λ)r4 dϖ
+dr
+�
+= 4(4π + β)(ρ + pt)ϖ,
+(33)
+and taking into account that e−(ψ+λ) = 1 at the edge of
+the star and beyond, the last equation can be integrated
+to give
+�
+r4 dϖ
+dr
+�
+rsur
+= 4(4π + β)
+� rsur
+0
+(ρ + pt)r4eλ−ψϖdr. (34)
+From Eq. (34) we can obtain the relativistic moment
+of inertia of a slowly rotating anisotropic compact star in
+f(R, T) = R + 2βT gravity by means of expression
+I = 2
+3(4π + β)
+� rsur
+0
+(ρ + pr + σ)eλ−ψr4 �ϖ
+Ω
+�
+dr,
+(35)
+and hence the angular momentum J = IΩ can be written
+as
+J = 2
+3(4π + β)
+� rsur
+0
+ρ + pr + σ
+�
+1 − 2m/r
+(Ω − ω)e−ψr4dr. (36)
+It can be seen that the above result then reduces to the
+pure general relativistic expression when β = 0. Further-
+more, when both parameters β and σ vanish, Eq. (36)
+reduces to the expression given in Ref. [69] for isotropic
+compact stars in Einstein gravity. Analogously as in GR,
+the differential equation (33) will be integrated from the
+origin at r = 0 with an arbitrary choice of the central
+value ϖ(0) and with vanishing slope, i.e., dϖ/dr = 0.
+Once the solution for ϖ(r) is found, we can then com-
+pute the moment of inertia via the integral (35).
+IV.
+EQUATION OF STATE AND ANISOTROPY
+ANSATZ
+Just as the construction of anisotropic compact stars
+in GR, to close the system of Eqs. (22)-(24), one needs to
+specify a barotropic EoS (which relates the radial pres-
+sure to the mass density by means of equation pr = pr(ρ))
+and also assign an anisotropy function σ since there is
+now an extra degree of freedom pt. Alternatively, it is
+possible to assign an EoS for radial pressure and another
+for tangential pressure.
+For instance, an approach for
+the study of anisotropic fluids has been recently carried
+out within the context of Newtonian gravity in Ref. [70]
+and in conventional GR [71], where both the radial and
+tangential pressures satisfy a polytropic EoS.
+In this work, we will follow the first procedure de-
+scribed in the previous paragraph in order to deal
+
+5
+with anisotropic neutron stars within the framework of
+f(R, T) gravity. Indeed, for radial pressure we use a well-
+known and physically relevant EoS which is compatible
+with the constraints of the GW170817 event (the first de-
+tection of gravitational waves from a binary neutron star
+inspiral [72]), namely, the soft SLy EoS [73]. This EoS
+is based on the SLy effective nucleon-nucleon interaction,
+which is suitable for the description of strong interactions
+in the nucleon component of dense neutron-star matter.
+Such unified EoS describes both the neutron-star crust
+and the liquid core (which is assumed to be a “minimal”
+npeµ composition), and it can be represented by the fol-
+lowing analytical expression
+ζ(ξ) = a1 + a2ξ + a3ξ3
+1 + a4ξ
+f(a5(ξ − a6))
++ (a7 + a8ξ)f(a9(a10 − ξ))
++ (a11 + a12ξ)f(a13(a14 − ξ))
++ (a15 + a16ξ)f(a17(a18 − ξ)),
+(37)
+where ζ ≡ log(pr/dyn cm−2), ξ ≡ log(ρ/g cm−3), and
+f(x) ≡ 1/(ex + 1). The values ai are fitting parameters
+and can be found in Ref. [74].
+In addition, we adopt the anisotropy ansatz proposed
+by Horvat et al. [25] to model anisotropic matter inside
+compact stars, namely
+σ = αprµ = αpr(1 − e−2λ),
+(38)
+with µ(r) ≡ 2m/r being the compactness of the star. The
+advantage of this ansatz is that the stellar fluid becomes
+isotropic at the origin since µ ∼ r2 when r → 0. It is also
+commonly known as quasi-local ansatz in the literature
+[25], where α controls the amount of anisotropy inside
+the star and in principle can assume positive or negative
+values [23, 25, 33, 50, 51, 75, 76]. Note that in the Newto-
+nian limit, when the pressure contribution to the energy
+density is negligible, the effect of anisotropy vanishes in
+the hydrostatic equilibrium equation. Regardless of the
+particular functional form of the anisotropy model, here
+we must emphasize that physically relevant solutions cor-
+respond to pr, pt ≥ 0 for r ≤ rsur.
+V.
+NUMERICAL RESULTS AND DISCUSSION
+Given an EoS for the radial pressure, we numerically
+integrate the modified TOV equations (22)-(24) with
+boundary conditions (25) from the stellar center to the
+surface r = rsur where the radial pressure vanishes. In
+addition, we have to specify a particular value for the cou-
+pling constant β and for anisotropy parameter α which
+appears in Eq. (38). For instance, for a central mass den-
+sity ρc = 2.0×1018 kg/m3 with SLy EoS (37), Fig. 1 illus-
+trates the mass function and anisotropy factor as func-
+tions of the radial coordinate for β = −0.01 and several
+values of α. The left plot reveals an increase in gravita-
+tional mass and a decrease in radius as α increases. More-
+over, from the right plot we can see that the anisotropy
+vanishes at the center (which is a required condition in
+order to guarantee regularity), is more pronounced in the
+intermediate regions, and it vanishes again at the stellar
+surface.
+For the anisotropy function (38), the left panel of Fig. 2
+displays the mass-radius relations for anisotropic neutron
+stars with SLy EoS in f(R, T) = R+2βT gravity for three
+particular values of the coupling constant β and different
+values of α. Here the total gravitational mass of each
+configuration is given by M = m(rsur), and the isotropic
+case in Einstein gravity has been included for compari-
+son purposes by a black solid line. The mass-radius re-
+lation exhibits substantial deviations from GR mainly
+in the low-mass region. On the other hand, anisotropy
+introduces considerable changes only in the high-mass re-
+gion. We remark that the 2βT term together with the
+presence of anisotropies (with positive values of α) al-
+low us to obtain maximum masses bigger than 2.0 M⊙.
+As a consequence, the introduction of anisotropies in
+f(R, T) = R + 2βT gravity gives rise to massive neutron
+stars that are in good agreement with the millisecond
+pulsar observations [77, 78]. From NICER and XMM-
+Newton data [79], the radius measurement for a 1.4 M⊙
+neutron star is 12.45 ± 0.65 km and, according to the
+mass-radius diagram, our results consistently describe
+this star when β = −0.01 (see blue curves).
+Further-
+more, it should be noted that the parameter β = −0.01
+is the one that best fits the mass-radius constraint from
+the GW170817 event (see the filled cyan region). Nev-
+ertheless, the massive pulsar J0740+6620 (whose radius
+is 12.35 ± 0.75 km [79]) could be described only when
+β = −0.03 and α = 0.4.
+It is worth commenting that the value of the parame-
+ter α could be constrained, but that will depend on the
+particular compact star observed in the Universe. For in-
+stance, the range α ∈ [−0.4, −0.2] consistently describes
+the millisecond pulsar J1614-2230 regardless of the value
+of β. However, for highly massive neutron stars whose
+masses are greater than 2.0 M⊙, positive values of α will
+be required. For PSR J0740+6620, whose gravitational
+mass is 2.08 M⊙, the best value for α is 0.2. In fact, this
+constraint will depend not only on the modified theory
+of gravity but also on the equation of state adopted for
+the radial pressure.
+According to the right panel of Fig. 2, the parameter
+β slightly modifies the total gravitational mass, however,
+the effect of anisotropy introduces more relevant changes.
+To better analyze the effects that arise as a result of the
+modification of Einstein’s theory as well as the incorpo-
+ration of anisotropies, in Fig. 3 we show the behavior
+of the surface radius as a function of the central den-
+sity. From the left plot we can conclude that the radius
+is significantly altered due to the 2βT term in the low-
+central-density region, while anisotropy slightly modifies
+the radius of the stars. The right plot corresponds to the
+pure general relativistic case and it can be observed that
+the radius undergoes more significant modifications with
+respect to its isotropic counterpart if the values for |α|
+
+6
+0
+2
+4
+6
+8
+10
+0.0
+0.5
+1.0
+1.5
+2.0
+r [km]
+m [M⊙]
+-0.6 -0.3
+0
+0.3
+0.6
+10.85
+10.92
+10.99
+α
+rsur [km]
+α
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0
+2
+4
+6
+8
+10
+-4
+-2
+0
+2
+4
+6
+r [km]
+σ [1033 Pa]
+α
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+FIG. 1.
+Radial behaviour of the mass function (left panel) and the anisotropy factor (right panel) in the framework of
+f(R, T) = R + 2βT gravity for β = −0.01 and different values of α. SLy EoS (37) is valid from 1011 kg/m3 up to the maximum
+density reachable within neutron stars [73], and in these plots we have considered ρc = 2.0 × 1018 kg/m3. The isotropic case is
+recovered when the anisotropy parameter vanishes (this is, α = 0). We can observe that the gravitational mass increases and
+the radius decreases as α increases. In addition, the anisotropy is more pronounced in the intermediate regions and vanishes
+at the stellar center as expected.
+are larger than those considered in the left plot.
+Eq. (33) is first solved in the interior region from the
+center to the surface of the star by considering an arbi-
+trary value for ϖ and with vanishing slope at r = 0. Then
+the same equation is solved in exterior spacetime from the
+surface to a sufficiently far distance from the star where
+ϖ(r) → Ω. In Fig. 4 we display the radial profile of these
+solutions for the central mass density considered above.
+We observe that ϖ(r) is an increasing function of the ra-
+dial coordinate, whereas ω(r) is a decreasing function and
+hence the largest rate of dragging of local inertial frames
+always occurs at the stellar center. Furthermore, appre-
+ciable effects (mainly in the interior region of the stellar
+configuration) can be noted on frame-dragging angular
+velocity due to the inclusion of anisotropies.
+Once ϖ(r) is known for each stellar configuration, we
+can then determine the moment of inertia by means of
+Eq. (35). Figure 5 presents the moment of inertia as a
+function of the total gravitational mass in GR and within
+the context of f(R, T) = R + 2βT gravity for β = −0.01.
+It can be observed that the moment of inertia undergoes
+irrelevant changes from GR, however, it can change sig-
+nificantly due to anisotropies in the high-mass region.
+VI.
+CONCLUSIONS
+In this work we have investigated slowly rotating
+anisotropic neutron stars in f(R, T) = R + 2βT grav-
+ity, where the degree of modification with respect to GR
+is measured by the coupling constant β. The modified
+TOV equations and moment of inertia have been derived
+within the context of anisotropic fluids by retaining only
+first-order terms in the angular velocity as measured by
+a distant observer (Ω). Notice that, within this linear
+approximation, the moment of inertia can be calculated
+from the structure of a non-rotating configuration since
+the TOV equations describing the static background are
+still valid. In addition, we have adopted the anisotropy
+ansatz proposed by Horvat and collaborators [25], where
+appears a dimensionless parameter α which measures the
+degree of anisotropy within the neutron star.
+We have analyzed the consequences of the extra term
+2βT together with anisotropies on the properties of neu-
+tron stars such as radius, mass, frame-dragging angular
+velocity and moment of inertia. Indeed, our results re-
+veal that the radius deviates considerably from GR in
+the low-central-density region, however, the total gravi-
+tational mass and the moment of inertia undergo slight
+modifications due to the influence of the effects generated
+by the minimal matter-gravity coupling. Furthermore,
+the presence of anisotropy generates substantial changes
+both in the mass and in the moment of inertia with re-
+spect to the isotropic case. The appreciable effects due
+to the inclusion of anisotropy occur mainly in the higher-
+central-density region, this is, for large masses (near the
+maximum-mass configuration).
+ACKNOWLEDGMENTS
+JMZP acknowledges financial support from the PCI
+program of the Brazilian agency “Conselho Nacional de
+Desenvolvimento Cient´ıfico e Tecnol´ogico”–CNPq.
+
+7
+PSR J1614-2230
+PSR J0740+6620
+GW170817
+β = 0
+β = -0.01
+β = -0.02
+β = -0.03
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+10
+12
+14
+16
+0.5
+1.0
+1.5
+2.0
+rsur [km]
+M [M⊙]
+17.8
+18.0
+18.2
+18.4
+18.6
+18.8
+0.5
+1.0
+1.5
+2.0
+Log ρc [kg/m3]
+M [M⊙]
+1.32
+1.36
+1.4
+17.94
+17.98
+18.02
+FIG. 2.
+Mass-radius diagrams (left panel) and mass-central density relations (right panel) for anisotropic neutron stars with
+SLy EoS (37) in f(R, T) = R + 2βT gravity for β = −0.01 (blue curves), β = −0.02 (orange curves) and β = −0.03 (in green).
+The solid lines correspond to α = 0 (that is, isotropic solutions), and the pure GR case (β = 0) is shown in both plots as a
+benchmark by a black line. The magenta horizontal band stands for the observational measurement for the millisecond pulsar
+J1614-2230 reported in Ref. [77]. The filled cyan region is the mass-radius constraint from the GW170817 event. The Radius
+of PSR J0740+6620 from NICER and XMM-Newton Data [79] is indicated by the top brown dot with their respective error
+bars. Moreover, the bottom brown dot represents the radius estimate for a 1.4 M⊙ neutron star [79].
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+17.6
+17.8
+18.0
+18.2
+18.4
+18.6
+18.8
+10
+15
+20
+25
+30
+Log ρc [kg/m3]
+rsur [km]
+17.76
+17.82
+17.88
+13
+15
+17
+α = -1.0
+α = -0.5
+α = 0
+α = 0.5
+α = 1.0
+17.6
+17.8
+18.0
+18.2
+18.4
+18.6
+9
+10
+11
+12
+13
+14
+Log ρc [kg/m3]
+rsur [km]
+FIG. 3.
+Surface radius as a function of the central mass density. On the left panel, different styles and colors of the curves
+correspond to different values of the parameters β and α as in Fig. 2. The most substantial deviations from GR take place at
+low central densities, whereas for large central densities the changes are very slight due to the 2βT term. On the right panel
+we display the modifications of the radius due to the inclusion of anisotropies when β = 0, where we have considered larger
+values for |α| in order to appreciate the changes in radius as a consequence of anisotropy. We can mainly observe three regions
+where the radius can decrease or increase depending on the value of α.
+
+8
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0
+10
+20
+30
+40
+0.4
+0.6
+0.8
+1.0
+r [km]
+ϖ/Ω
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0
+10
+20
+30
+40
+0.0
+0.2
+0.4
+0.6
+r [km]
+ω/Ω
+FIG. 4.
+Left panel: Numerical solution of the differential equation (33) for a given central mass density ρc = 2.0 × 1018 kg/m3
+in f(R, T) = R + 2βT gravity with β = −0.01 and different values of the free parameter α. The dotted lines represent the
+solutions of the exterior region, and as expected ϖ → Ω at great distances from the stellar surface. Right panel: Ratio of
+frame-dragging angular velocity to the angular velocity of the stars, namely ω(r)/Ω = 1 − ϖ(r)/Ω. Notice that the solution of
+the exterior problem provides an asymptotic behavior of ω(r).
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0.5
+1.0
+1.5
+2.0
+0.5
+1.0
+1.5
+2.0
+M [M⊙]
+I [1038 kg.m2]
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+1.6
+1.7
+1.8
+1.9
+2.0
+1.7
+1.8
+1.9
+2.0
+2.1
+M [M⊙]
+I [1038 kg.m2]
+FIG. 5.
+Left panel: Moment of inertia of slowly rotating anisotropic neutron stars as a function of the total mass within the
+context of f(R, T) = R + 2βT gravity for β = −0.01 in blue. Different styles of the curves correspond to different values of the
+anisotropy parameter α. Results based on Einstein’s theory have been included for comparison purposes and are represented
+by the black curves. We can appreciate that the moment of inertia is modified very slightly by the 2βT term, however, the
+anisotropies introduce relevant changes in the large-mass region. The right plot is a magnification of the left one.
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+

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+Geometric quantum discord and coherence in a dipolar interacting magnetic system
+Clebson Cruz,1, ∗ Maron F. Anka,2, † Hamid-Reza Rastegar-Sedehi,3, ‡ and Cleidson Castro4, §
+1Grupo de Informa¸c˜ao Quˆantica e F´ısica Estat´ıstica, Centro de Ciˆencias Exatas e das Tecnologias,
+Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos. Rua Bertioga,
+892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil.
+2Instituto de F´ısica, Universidade Federal Fluminense,
+Av. Gal. Milton Tavares de Souza s/n, 24210-346 Niter´oi, Rio de Janeiro, Brasil.
+3Department of Physics, College of Sciences, Jahrom University, Jahrom 74135-111, Iran
+4Centro de Forma¸c˜ao de Professores, Universidade Federal do Recˆoncavo da Bahia,
+Avenida Nestor de Mello Pita, 535 Amargosa, Bahia, Brazil.
+(Dated: January 10, 2023)
+The study of low-dimensional metal complexes has revealed fascinating characteristics regarding the ground-
+state crossover shown by spin-gaped systems. In this context, this work explores the effect of the quantum-level
+crossing, induced by the magnetic anisotropies of dipolar interacting systems, on the quantum discord and
+coherence of the system. The analytical expressions for the quantum discord, based on Schatten 1-norm, and
+the l1 trace-norm quantum coherence for dinuclear spin-1/2 systems, are provided in terms of the magnetic
+anisotropies. The results show that, while the quantum discord has a clear signature of the quantum level-
+crossing, the basis dependence of the quantum coherence hides the crossover regarding the measured basis. In
+addition, the global quantum coherence is wholly stored within the correlations of the system, regardless of its
+reference basis.
+Keywords: Dipolar Interaction; Qunatum discord; Quantum Coherence; Quantum-level crossing.
+I.
+INTRODUCTION
+The study of the quantum properties of composite systems has led to a revolution in the development of emerging quantum
+technologies [1–3]. The new generation of quantum devices explores physical properties associated with quantum correlations
+between particles [4, 5] and superposition principle for the system states [1, 6, 7]. In this scenario, the characterization of the
+quantumness of the physical systems is of paramount importance since the existence of quantum correlations and coherence are
+a valuable resource for several quantum tasks [4, 8, 9].
+However, the characterization of quantum correlations is a rather complicated task from the theoretical [10] and experimental
+[11] point of view. This scenario is aggravated in Condensed Matter systems, where the number of interacting components in
+the system is usually on the order of the Avogadro number [12]. Nevertheless, there are a few exceptions, like low-dimensional
+metal complexes (LDMC), for which full knowledge about their quantum properties can be obtained through the corresponding
+analytical solutions [4, 6, 13–20]. In such solid-state systems, intra-molecular interactions are strong enough to suppress ex-
+trinsic and intermolecular interactions [4, 13, 14, 21]. Therefore, their quantum features exhibit high stability against external
+perturbations such as temperatures [4, 13, 14, 16, 21], magnetic fields [4, 6, 16, 22], and pressures [6, 15]. These characteris-
+tics make these systems promising platforms for the development of emerging quantum technologies [4, 23–26]. In regard to
+these possible applications, the study of dipolar interacting magnetic systems has received considerable attention in the quantum
+information literature [17, 27–31].
+In this work, we present a theoretical study of quantum correlations and coherence for a dipolar interacting magnetic system,
+exploring the effects of magnetic anisotropies on the quantumness of the system. As a result, this study provides to the literature
+analytical expressions, in terms of the magnetic anisotropies, for the quantum discord, based on Schatten 1-norm, and the l1
+trace-norm quantum coherence written in an arbitrary basis, defined by the co-latitude and longitude angles of the Bloch sphere
+representation. According to the findings, the behavior of the quantum discord carries a noteworthy signature of the quantum
+level-crossing, caused by population changes resulting from the alteration of Boltzmann weights arising from the change of the
+magnetic anisotropies of the system. On the other hand, the basis dependency of quantum coherence is detrimental in terms of
+recognizing this crossover. In this regard, the measurement of the average quantum coherence is numerically obtained in order
+to obtain a basis-independent perspective for this quantum resource. The results not only demonstrate that the average coherence
+∗Electronic address: [email protected]
+†Electronic address: [email protected]ff.br
+‡Electronic address: [email protected]
+§Electronic address: [email protected]
+arXiv:2301.02891v1  [quant-ph]  7 Jan 2023
+
+2
+is completely stored within the correlations of the system, yet they also demonstrate that it is possible to retrieve the signature of
+the energy-level crossover present on the quantum discord measurement. Furthermore, the findings show how dipolar interaction
+coupling magnetic anisotropies impact quantum correlations and coherence in a dinuclear spin-1/2 system. Thus, the dipolar
+interaction model is a viable foundation for quantum technologies based on quantum discord and coherence.
+II.
+DINUCLEAR METAL COMPLEX WITH DIPOLAR INTERACTION
+The class of dinuclear metal complexes undergoes several types of magnetic coupling [12]. Among these are Heisenberg
+exchange, which is isotropic under rotations in spin space [4, 6, 20, 21], and Dzyaloshinskii-Moriya (DM) interaction [32–
+34], which accounts for weak ferromagnetism in some antiferromagnetic materials [12]. A ubiquitous example of anisotropic
+coupling in LDMCs is the dipolar interaction [17, 27–31]. This coupling arises from the influence of a magnetic field yielded
+by one of the magnetic moments in the other ones [12]. In particular, for dinuclear metal complexes, the Hamiltonian which
+describes this interaction is given by:
+H = −1
+3
+⃗S T
+A ·
+↔D · ⃗S B ,
+(1)
+where ⃗S j = {S x
+j, S y
+j, S z
+j} are the spin operators and
+↔D = diag(∆ − 3ϵ, ∆ + 3ϵ, −2∆) is a diagonal tensor, with ϵ and ∆ being the
+rhombic and axial parameters, respectively, related to the magnetic anisotropies in the dipolar model [12]. In particular, ∆ is
+related to the zero-field splitting of the energy levels [35]. Considering the Hamiltonian, Eq. (1), written in the S (z) eigenbasis,
+∆ > 0 becomes a signature that the spins are on z-axis, while ∆ < 0 indicates that the spins will be on the x − y plane.
+Considering a dinuclear metal complex in with d9 electronic configuration, Eq. (1) can describe two coupled spin 1/2 particles
+in the corresponding S z eigenbasis {|00⟩, |01⟩, |10⟩, |11⟩}
+H = 1
+6
+��������������
+∆
+3ϵ
+−∆ −∆
+−∆ −∆
+3ϵ
+∆
+��������������
+,
+(2)
+The energy levels of the system from the coupling parameters are composed by the
+E1 = 0 ,
+E2 = −2∆ ,
+E3 = ∆ + 3ϵ ,
+E4 = ∆ − 3ϵ .
+(3)
+From the thermal equilibrium, the density matrix for the coupled system is described by the Gibbs form ρAB = Z−1e−H/kBT,
+where
+Z = Tr(e−H/kBT) = 2eβ∆/6 cosh
+�β∆
+6
+�
++ 2e−β∆/6 cosh
+�βϵ
+2
+�
+.
+(4)
+is the canonical the partition function, with kB representing the Boltzmann’s constant. Thus, the dinuclear density matrix at sites
+labeled by A and B can be written in the S z eigenbasis as the so-called X-shaped mixed state
+ρAB = e− β∆
+6
+Z
+�������������������
+cosh
+� βϵ
+2
+�
+− sinh
+� βϵ
+2
+�
+e
+2β∆
+6 cosh
+� β∆
+6
+�
+e
+2β∆
+6 sinh
+� β∆
+6
+�
+e
+2β∆
+6 sinh
+� β∆
+6
+�
+e
+2β∆
+6 cosh
+� β∆
+6
+�
+− sinh
+� βϵ
+2
+�
+cosh
+� βϵ
+2
+�
+�������������������
+.
+(5)
+The density matrix eigenvalues (population) and their corresponding eigenvectors can be written as:
+PΨ− =
+1
+1 + e
+β∆
+3 + 2e− β∆
+6 cosh
+� βϵ
+2
+� → |Ψ−⟩,
+(6)
+PΨ+ =
+1
+1 + e− β∆
+3 + 2e− β∆
+6 cosh
+� βϵ
+2
+� → |Ψ+⟩,
+(7)
+PΦ+ =
+1
+1 + eβϵ + eβ( ∆+ϵ
+2 ) + eβ( ∆+3ϵ
+6 ) → |Φ+⟩,
+(8)
+PΦ− =
+eβϵ
+1 + eβϵ + eβ( ∆+ϵ
+2 ) + eβ( ∆+3ϵ
+6 ) → |Φ−⟩,
+(9)
+
+3
+where
+|Ψ±⟩ =
+1√
+2
+(|01⟩ ± |10⟩) ,
+|Φ±⟩ =
+1√
+2
+(|00⟩ ± |11⟩) .
+(10)
+are the so-called Bell states, which represent the maximally entangled states for a bipartite system [36].
+The study of LDMC has attracted the attention of both theoretical and experimental condensed matter physics communities
+due to the fascinating properties of their ground states [6, 37]. In the presence of an external magnetic field, this systems
+typically show a quantum level-crossing between its ground state and the first excited one when the field reaches a critical value
+since the external magnetic field splits its energy levels, changing their corresponding populations. However, since the dipolar
+interaction arises from the influence of the magnetic field created by one of the magnetic moments in the other, the splitting in
+energy levels is ruled by the axial (∆) and rhombic (ϵ) parameters, as can be seen in Eq. (3). In this regard, Fig. 1 shows the
+populations as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT.
+- 20
+- 10
+0
+10
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵ/kBT
+Populations
+- 20
+- 10
+0
+10
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵ/kBT
+Populations
+(a)
+(b)
+- 20
+- 10
+0
+10
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Δ/kBT
+Populations
+- 20
+- 10
+0
+10
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Δ/kB
+Δ/kB
+T
+Populations
+PΨ-
+PΨ+
+PΦ+
+PΦ-
+ϵ/kB= 5K
+ϵ/kB=-5K
+= 5K
+Δ/kB=-5K
+FIG. 1: (Color online) Populations, Eqs. (6)–(9), as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT.
+(a) Axial dependence considering the rhombic parameter ϵ/kB = 5 K (left) and ϵ/kB = −5 K (right). (b) Rhombic dependence considering the
+axial parameter ∆/kB = 5 K (left) and ∆/kB = −5 K (right). The inset shows the magnetic anisotropy dependence on the energy levels.
+As can be seen, in agreement with Eq. (3), when the spins are in the x − y plane (∆ < 0) with positive rhombic parameter
+(ϵ > 0), the ground state is the given by the state |Φ−⟩ (with population PΦ−), and there is no quantum level crossing. Thus, the
+system remains in the ground state. However, changing the signal of the rhombic parameter induces a quantum level crossing
+between states |Φ−⟩ and |Φ+⟩ (with population PΦ+). Moreover, for the spins oriented in the z axis (∆ > 0), it is possible to
+observe a quantum level crossing between the state |Φ−⟩ (if ϵ > 0) or |Φ+⟩ (if ϵ < 0) and the state |Ψ−⟩ (with population PΨ+) by
+increasing the ratio ∆/kBT to the critical point ∆ = |ϵ|.
+In reference [29], the authors study the effect of the magnetic anisotropies, described by the axial and rhombic parameters,
+on the nonlocal correlations of a dipolar interacting system of two spins-1/2, identified by the Peres-Horodecki separability
+criterion [38, 39]. In addition, they explore the change in the ground state on the thermal entanglement for the teleportation
+process. However, although quantum entanglement provides one path toward the characterization of nonlocal correlations, it
+does not encompass all quantum correlations in the system [21, 40–51]. Therefore, in order to expand this result, the following
+section presents a study of the quantum correlations and coherence described by the Schatten 1-norm geometric quantum discord
+and the l1 trace-norm quantum coherence.
+
+30
+20
+10
+10
+20
+0
+0
+5
+10
+E20
+10
+0
+10
+20
+30
+10
+0
+5
+10
+E20
+10
+0
+-10
+20
+0
+5
+0
+5
+1020
+10
+ler
+0
+-10
+20
+0
+5
+0
+5
+1060
+40
+20
+20
+40
+60
+20
+-10
+0
+10
+2060
+40
+20
+0
+20
+40
+60
+20
+0
+10
+204
+III.
+QUANTUM DISCORD
+Quantum discord has been defined as a measurement of the quantumness of correlations in a quantum system. It has been first
+introduced as an entropic measurement of genuinely quantum correlations in a quantum state, defined as the difference between
+the total and the classical correlation [40] Q(ρAB) = I(ρA : ρB) − C(ρAB), where I(ρA : ρB) = S (ρA) + S (ρB) − S (ρAB) represents
+the mutual information between the subsystems A and B, and C(ρAB) is the classical correlation of the composite system ρAB
+defined as C(ρAB) = max{Bk}
+�S (ρA) − �
+k pkS (ρk)�, with the maximization taking over positive operator-valued measurements
+(POVM’s) {Bk} performed locally only on subsystem B. However, this analytical maximization over POVMs is an arduous task
+even for a two-qubit system [10, 11, 40, 41, 47, 51]. In this scenario, the class of entropic measurements of correlations, such as
+the entropic quantum discord, is defined as nondeterministic polynomial time (NP-complete) problems [10]. Consequently, only
+a few results for the analytical expression of entropic quantum discord, and only for certain classes of states are exact solutions
+known [41, 43, 45, 46, 49, 50, 52]. Due to this fact, alternative measurements of quantum correlations have been proposed
+[41, 43, 45–50, 52–60], especially quantifiers based on geometric arguments [47–49, 53, 57–60].
+Geometric approaches are widely used to characterize and quantify quantum resources in a wide variety of quantum systems
+[61]. In particular, the Schatten 1-norm quantum discord [4, 21, 47, 48, 62], is a reliable geometric-based quantifier of the
+amount of quantum correlations in metal complexes [4, 21, 62, 63]. The so-called geometric quantum discord can be defined in
+terms of the minimal distance between a set ω of closest classical-quantum states ρc [21, 47, 48], given by:
+ρc =
+�
+k
+pkΠ{A}
+k
+⊗ ρ{B}
+k ,
+(11)
+where 0 ≤ pk ≤ 1 and �
+k pk = 1; {Π{A}
+k } define a set of orthogonal projectors for a given subsystem A and ρ{B}
+k
+the reduced
+density matrix for the subsystem B [47, 48]. Therefore, the geometric quantum discord can be expressed as
+QG(ρAB) = min
+ω ∥ρAB − ρc∥ ,
+(12)
+where ∥M∥ = Tr
+� √
+M†M
+�
+is the so-called 1-norm, and ρAB is the given quantum state at thermal equilibrium, Eq. (5).
+Therefore, considering the given dinuclear magnetic system of spins-1/2 in a quantum spin-lattice, ruled by a dipolar Hamil-
+tonian H, Eq.(1), the invariance under π rotation around a given spin axis (Z2 symmetry) [12, 46] allow us to compute the
+geometric quantum discord, based on Schatten 1-norm, for the two-qubit X state, Eq. (5), as [53, 64]
+QG(ρAB) = 1
+2
+�
+φ2
+1max{φ2
+2, φ2
+3} − φ2
+2min{φ2
+1, φ2
+3}
+max{φ2
+2, φ2
+3} − min{φ2
+1, φ2
+3} + φ2
+1 − φ2
+2
+(13)
+where
+φ1 =
+e
+β∆
+6 ���−1 + eβ∆/3��� + 2
+����sinh
+� βϵ
+2
+�����
+����2 cosh
+� βϵ
+2
+�
++ eβ∆/6 + eβ∆/2
+����
+,
+(14)
+φ2 =
+e
+∆
+6 ���−1 + eβ∆/3��� − 2
+����sinh
+� βϵ
+2
+�����
+����2 cosh
+� βϵ
+2
+�
++ eβ∆/6 + eβ∆/2
+����
+,
+(15)
+φ3 =
+2
+eβ∆/3 cosh
+� β∆
+6
+�
+sech
+� βϵ
+2
+�
++ 1
+− 1 .
+(16)
+Considering the dipolar magnetic system in thermal equilibrium described by Eq. (5), it is possible to examine how the
+magnetic anisotropies, represented by the axial (∆) and rhombic (ϵ) coupling parameters, affects the thermal quantum discord
+in the system. Fig. 2 shows the geometric quantum discord, based on Schatten 1-norm, Eq. (13), as a function of the ratio
+∆/kBT and ϵ/kBT. As expected, the quantum discord reaches its maximum (saturated) value of 1/2 as T approaches zero. As
+the temperature rises, the value of quantum discord decreases inexorably and goes to zero when T ≫ |∆| and T ≫ |ϵ|. On the
+other hand, given the spins in the x − y plane (∆ < 0), it is sufficient that only T ≫ |ϵ| to the discord reaches its minimum value.
+However, if the spins are in the z-axis (∆ > 0), one can increase the quantum discord by increasing the axial parameter ∆ even
+when T ≫ |ϵ|.
+Furthermore, regarding the magnetic anisotropies, the quantum discord presents a signature of the quantum level crossing in
+the dipolar interacting system, highlighted on the solid white line in Fig. 2. Considering the spins oriented in the z-axis (∆ > 0),
+the zero-field splitting leads the system to a quantum level crossing in the critical boundary ∆ = |ϵ|, where it is possible to
+detect a crossover between the states |Ψ+⟩ and |Φ−⟩, if ϵ > 0, or |Φ+⟩, if ϵ < 0. Moreover, for the spins oriented in the x − y
+
+5
+plane (∆ < 0), it is possible to observe a quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in the critical boundary ϵ = 0.
+On the other hand, the degree of quantum discord in the system can be increased by gradient ascent of the function QG(ρAB),
+perpendicularly to the crossing boundary, which occurs for values in which |ϵ| ≫ kBT (for ∆ < 0), ∆ ≫ kBT (for |ϵ| ≪ ∆), and
+|ϵ| ≫ ∆), corresponding to the lightest region in Fig. 2.Therefore, by controlling the axial (∆) and rhombic (ϵ) anisotropies is
+possible to manage the degree of quantum discord in the dipolar interacting system.
+In addition, in order to compare quantum discord to the level of entanglement in the system under investigation, we use the
+concurrence measure. Typically, concurrence is used to assess entanglement in bipartite systems, and it can be easily computed
+for any two-qubit system. The thermal concurrence examines the resemblance between the considered quantum state in thermal
+equilibrium and its bit-flipped density matrix, ¯ρ = ρAB(σy ⊗ σy)ρ∗
+AB(σy ⊗ σy). In particular, for the X- shaped density matrix, Eq.
+(5), the concurrence is analytically defined as
+C(ρAB) := max{0, A, B},
+(17)
+where
+A = e− β∆
+6
+Z
+�������e
+2β∆
+6 sinh
+�β∆
+6
+������� − cosh
+�βϵ
+2
+��
+,
+(18)
+B = e− β∆
+6
+Z
+������sinh
+�βϵ
+2
+������ − e
+2β∆
+6 cosh
+�β∆
+6
+��
+.
+(19)
+Dashed green line in Fig. 2 denotes the boundary given by C(ρAB) = 0. Inside this region, the concurrence is zero, and the
+state of the system is separable. However, within the region where entanglement is absent, the quantum discord of the system
+is still considerably more than zero, ensuring the presence of quantum-correlated states even when the system is in a separable
+syaye. On the other hand, for low temperatures, the entanglement is zero in the quantum level crossing boundary alongside
+the quantum discord at the quantum level crossing boundary. In this scenario, the existence or absence of entanglement and,
+therefore, quantum correlations, is dependent on its ground state, which might vary in response to magnetic anisotropies. Thus,
+the variation of Boltzmann’s weights, Eqs. (6)-(9), associated with the occupancy of the energy levels, is the physical mechanism
+responsible for the abrupt change in the quantum correlations near the energy-level crossover.
+ϵ
+kB T
+Δ
+kB T
+PΨ+
+PΦ+
+PΦ-
+FIG. 2: (Color online) Quantum Discord, based on Schatten 1-norm, for a dipolar interacting magnetic system, Eq. (13), as a function of
+the ratios ∆/kBT and ϵ/kBT. The solid white line denotes the boundary between the quantum level crossings. The dashed green line is the
+boundary given by the concurrence, Eq. (17), C(ρAB) = 0, inside which the entanglement of the system is absent.
+IV.
+QUANTUM COHERENCE
+Similar to the approach proposed for the entanglement theory, where the quantum entanglement can be characterized by the
+distance between a state of interest (ρ) and a set of states closed under local operations, and classical communication (separable
+states) [38, 61, 65], Baumgratz et al. [65] provided the mathematical tools for quantifying the amount of quantum coherence
+in a quantum system. Considering a d-dimensional Hilbert space, quantum coherence can be obtained from the minimal value
+
+Outf· J=6
+of a distance measurement D(ρ, σ), between the considered quantum state ρ and a set {σ = �d
+k |k⟩⟨k| ∈ I} of incoherent states,
+where the reference basis {|k⟩}{k=1,...,d} can be adequately defined considering the physics of the problem under investigation or
+the task that requires this quantum resource [6, 65, 66]. In this scenario, since the non-vanishing off-diagonal terms of the
+density operator ρ, which characterizes the quantum state of the system of interest, constitute the superposition from the chosen
+reference basis [61, 65], the authors established a reliable measurement of quantum coherence through the l1 trace norm as [65]
+Cl1 = min
+σ∈I ∥ρ − σ∥l1 =
+�
+i�j
+|⟨i|ρ| j⟩| .
+(20)
+Since coherence is a quantity that is reliant on the basis on which it is measured, it is essential to choose a reference basis for
+the system within a metrology setting [61, 65]. In this scenario, the basis of an arbitrary quantum state can be altered by means
+of unitary operations [36, 61]. In particular, for two-level systems such as spin-1/2, any reference basis can be obtained from the
+unitary transformation
+U(θ, φ) =
+�������
+cos
+� θ
+2
+�
+−eiφ sin
+� θ
+2
+�
+e−iφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+������� ,
+(21)
+where the θ and φ angles are the spherical equivalents of the co-latitude with respect to the z-axis, and the longitude concerning
+the x-axis in a Bloch sphere representation, respectively [36, 67]. In this regard, the unitary transformation for the bipartite state
+given by Eq. (5) is given by ρ{θ,φ}
+AB = ˆUAB(θ, φ)ρAB ˆUAB(θ, φ) [67], where
+ˆUAB(θ, φ) = U(θ, φ) ⊗ U(θ, φ) =
+�������������������
+cos2 � θ
+2
+�
+−eiφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+−eiφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+e2iφ sin2 � θ
+2
+�
+e−iφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+cos2 � θ
+2
+�
+− sin2 � θ
+2
+�
+−eiφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+e−iφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+− sin2 � θ
+2
+�
+cos2 � θ
+2
+�
+−eiφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+e−2iφ sin2 � θ
+2
+�
+e−iφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+e−iφ sin
+� θ
+2
+�
+cos
+� θ
+2
+�
+cos2 � θ
+2
+�
+�������������������
+(22)
+By varying the co-latitude and longitude angles {θ, φ}, one can obtain the bipartite state ρAB, Eq. (5), in any reference basis.
+Using the unitary transformation for the bipartite states, Eq. (22), in Eq. (5), one can obtain the representation of the density
+operator for the dipolar interacting magnetic system of two spins-1/2 written in an arbitrary basis as
+ρ{θ,φ}
+AB = e− β∆
+6
+4Z
+��������������
+ϱ11
+ϱ12
+ϱ12
+ϱ14
+ϱ∗
+12
+ϱ22
+ϱ23
+−ϱ12
+ϱ∗
+12
+ϱ23
+ϱ22
+−ϱ12
+ϱ∗
+14 −ϱ∗
+12 −ϱ∗
+12
+ϱ11
+��������������
+,
+(23)
+where
+ϱ11 = 2 sin2(θ)
+�
+sinh
+�β∆
+2
+�
++ cosh
+�β∆
+2
+�
+− sinh
+�βϵ
+2
+�
+cos(2φ)
+�
++ cosh
+�βϵ
+2
+�
+(cos(2θ) + 3) ,
+(24)
+ϱ12 = e−3iφ sin(θ)
+�
+2e2iφ cos(θ)
+�
+cosh
+�βϵ
+2
+�
+− eβ∆/2�
++ e4iφ sinh
+�βϵ
+2
+�
+(cos(θ) + 1) + sinh
+�βϵ
+2
+�
+(cos(θ) − 1)
+�
+,
+(25)
+ϱ14 = 2e−2iφ sin2(θ)
+�
+cosh
+�βϵ
+2
+�
+− eβ∆/2�
+− 4e−4iφ sinh
+�βϵ
+2
+�
+sin4 �θ
+2
+�
+− 4 sinh
+�βϵ
+2
+�
+cos4 �θ
+2
+�
+,
+(26)
+ϱ22 = 2
+�
+eβ∆/2 cos2(θ) + eβ∆/6 + sin2(θ)
+�
+sinh
+�βϵ
+2
+�
+cos(2φ) + cosh
+�βϵ
+2
+���
+,
+(27)
+ϱ23 = 2eβ∆/2 cos2(θ) − 2eβ∆/6 + 2 sin2(θ)
+�
+sinh
+�βϵ
+2
+�
+cos(2φ) + cosh
+�βϵ
+2
+��
+.
+(28)
+The diagonal entries of Eq. (23) are real, and the trace is 1. In addition, to ensure real eigenvalues, hermiticity restricts
+off-diagonal elements to two complex numbers, i.e., ϱi j is the complex conjugate of ϱji.
+Thus, from Eqs. (20) and (23), it is possible to write an analytical expression for the normalized quantum coherence in an
+arbitrary basis, defined by the co-latitude and longitude angles {θ, φ}, as:
+C{θ,φ}
+l1
+= e− β∆
+6
+6 |Z|
+�4 |ϱ12| + |ϱ14| + |ϱ23|� .
+(29)
+In order to examine the relationship between quantum coherence and quantum correlations, a new metric known as corre-
+lated coherence was established recently [67–69]. Quantum correlated coherence is a measure of coherence in which all local
+
+7
+components have been eliminated, i.e., all coherence in the system is totally recorded in the quantum correlations. For any
+given quantum state ρ, the correlated contribution to quantum coherence may be calculated by subtracting the local coherence
+of subsystems ρA = TrB(ρ) and ρB = TrA(ρ) from the overall coherence [68, 69]. Thus, the definition of correlated coherence
+according to the l1-norm of coherence is:
+Ccc(ρ{θ,φ}
+AB ) := Cl1(ρ) − Cl1(ρA) − Cl1(ρB).
+(30)
+Considering the density matrix of the dipolar interacting magnetic system written in an arbitrary basis, Eq. (23), the reduced
+density matrices of local subsystems are ρA = ρB = I/2, the maximally mixed state. Thus, regardless of the basis, the local
+subsystems will remain in the maximally mixed state, since it is basis invariant [36]. Consequently, the local contribution for
+the quantum coherence in this dipolar interacting system is always null, and the global coherence of the system, Eq. (29),
+is totally recorded in the quantum correlations of the system, regardless of its reference basis. Therefore, for a number of
+different combinations of values for the co-latitude and longitude angles {θ, φ}, the unitary transformation, Eq. (22), gives a
+direct connection between the overall and the correlated degrees of coherence.
+A.
+Axial Coherence
+In particular, due to the rotation symmetry of the dipolar interaction, the density matrix will be invariant when rotated both
+spins by an angle π along any given spin axis. Thus, choosing the co-latitude angle as θ = nπ (n = {0, 1, 2, ...}), regardless of
+the longitude angle φ, one can obtain the density matrix in the X-shaped form as described in Eq. (5). On the other hand, by
+applying the unitary transformation for the bipartite states, Eq. (22), for {θ = π/2; φ = nπ}, and {θ = π/2; φ = nπ/2}, in Eq. (5)
+one can obtain the density matrix S (x) and S (y) eigenbasis, respectively.
+ρ{X,Y}
+AB
+= e− β∆
+6
+2Z
+����������������
+eβ∆/2 + e∓βϵ/2
+0
+0
+∓
+�
+eβ∆/2 − e∓βϵ/2�
+0
+eβ∆/6 + e±βϵ/2 e±βϵ/2 − eβ∆/6
+0
+0
+e±βϵ/2 − eβ∆/6 eβ∆/6 + e±βϵ/2
+0
+∓
+�
+eβ∆/2 − e∓βϵ/2�
+0
+0
+eβ∆/2 + e∓βϵ/2
+����������������
+.
+(31)
+As can be seen, due to the symmetry of the X-shaped density matrices [70], the X-structure of the operator is preserved.
+Therefore, from Eqs. (5), (20) and (31), one can obtain the analytical expressions for the normalized axial quantum coherences
+as
+C{Z}
+l1
+=
+2
+3 |Z|
+�
+eβ∆/6
+������sinh
+�β∆
+6
+������� + e−β∆/6
+�����sinh
+�βϵ
+2
+������
+�
+(32)
+C{X,Y}
+l1
+=
+e− β∆
+6
+3 |Z|
+����eβ∆/2 − e∓βϵ/2��� +
+���eβ∆/6 − e±βϵ/2���
+�
+(33)
+Fig. 3 shows the axial quantum coherence in S (i) spin eigenbasis, where i = {x, y, z}. Different from the behavior observed
+for quantum discord (see Fig. 2), the axial quantum coherence is not sensible to the quantum level crossing. The quantum
+coherence in each axis {x, y, z} is minimized in only one energy-level crossover. As can be seen, considering the spins oriented
+in the z-axis (∆ > 0), the axial coherence in the S (x) eigenbasis is minimized on the critical boundary ∆ = −ϵ (with ϵ < 0), where
+it is possible to detect a crossover between the states |Ψ+⟩ and |Φ+⟩, while the coherence in S (y) eigenbasis is minimized on the
+critical boundary ∆ = ϵ (with ϵ > 0), where it is possible to detect a crossover between the states |Ψ+⟩ and |Φ−⟩ (see Fig. 2). As
+expected from Eq. 33, if the rhombic parameter is null (ϵ = 0), C{X}
+l1
+= C{Y}
+l1 . On the other hand, for the spins oriented in the x − y
+plane, (∆ < 0), it is possible to observe that C{Z}
+l1 is minimized in the quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in
+the critical boundary ϵ = 0.
+As shown in Fig. 3, the basis dependence of the quantum coherence hides the energy-level crossover in this dipolar interacting
+system regarding the measured basis. Therefore, the basis dependence on the quantum coherence defined by Baumgratz et al.
+[65], can be unfavorable to recognizing the quantum level crossing caused by population changes resulting from the alteration
+of Boltzman weights, Eqs. (6)-(9), arising from the change of the magnetic anisotropies of the dipolar interacting system.
+B.
+Average Coherence
+Since the coherence formulated in the quantum resource theory is a basis-dependent measurement [8, 61, 66], it is natural to
+define a basis-independent measurement [71–75]. Recent research has shown, via the use of relative entropies, as distance mea-
+surements of quantum correlations, that basis-independent measurements of entropic quantum coherence are precisely identical
+
+8
+(a)                                                                             (b)                                                                             (c)
+- 10
+- 5
+0
+5
+10
+- 10
+- 5
+0
+5
+10
+Δ
+kB T
+ϵ
+kB T
+- 10
+- 5
+0
+5
+10
+- 10
+- 5
+0
+5
+10
+Δ
+kB T
+ϵ
+kB T
+- 10
+- 5
+0
+5
+10
+- 10
+- 5
+0
+5
+10
+Δ
+kB T
+ϵ
+kB T
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+SX eigenbasis
+SY
+SZ eigenbasis
+eigenbasis
+FIG. 3: (Color online) Axial quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios
+∆/kBT and ϵ/kBT. The dashed white line represents the minimum value for the axial quantum coherence.
+to entropic discord [74]. On the other hand, a possible basis-free measurement of quantum coherence for a quantum system
+can be obtained from a geometrical standpoint by averaging the coherence of a state across all reference bases [71–73, 75].
+From a theoretical point of view, this measurement corresponds to averaging the coherence on a standard basis across all equiv-
+alent states ρ{θ,φ}
+AB
+= ˆUAB(θ, φ)ρAB ˆUAB(θ, φ). Therefore, as any two-qubit reference base can be created by applying the unitary
+operation described in Eq. (22), the average quantum coherence can be obtained from Eq. (29) as
+⟨Cl1⟩ = 1
+4π
+2π
+�
+0
+π
+�
+0
+sin (θ)C{θ,φ}
+l1
+dθdφ .
+(34)
+It is worth mentioning that these integrals are not trivial to solve, and an analytical expression for the average coherence is not
+presented. However, it can be numerically integrated by any quadrature method [76]. In this scenario, Eq. (34) is estimated by
+using the Clenshaw-Curtis rule on adaptively refined subintervals of the integration area [76, 77] since the numerical integration
+algorithms are often equally efficient and effective as conventional algorithms for well-behaved integrands such as Eqs. (29) and
+(30) [76].
+Fig. 4 shows the average quantum coherence for the dipolar magnetic interacting system. The solid white line represents the
+threshold at which the quantum-level crossing, described in previous sections, actually occurs. As expected, based on Fig. 3,
+when the temperature rises reaching the threshold T ≫ |∆| and T ≫ |ϵ|, the value of coherence reaches its lowest point and
+will be equal to zero. However, the behavior of the average coherence is completely different from that observed in the axial
+(basis-dependent) coherence shown in Fig. 3.
+Moreover, besides unified frameworks from relative entropic measurements has shown that basis-independent entropic quan-
+tum coherence is equivalent to entropic discord [74], this is not true for this geometrical approach. However, although the
+contour lines of the average coherence are quite different from that shown in the discord presented in Fig. 2, it is still able to
+identify the signature of the energy-level crossing that was seen during the measurement of the quantum discord. This result is
+due to the fact that the global coherence is totally stored within the correlations of the system, and its average behavior is affected
+by the presence of genuine quantum correlations measured by the quantum discord.
+In addition, the entanglement of the system is absent within the area shown by the dashed green line that denotes the boundary
+supplied by the concurrence, which is denoted by Eq. (17), C(ρAB) = 0. Thus, as one would anticipate based on the observation
+of the quantum discord in Fig. 2, even in the absence of entanglement, the average coherence that is completely stored on the
+correlations of the system is noticeably distinct from zero.
+V.
+CONCLUSIONS
+In summary, this paper explored the influence of magnetic anisotropies on the quantumness of a dipolar interacting magnetic
+system via a theoretical examination of the geometric quantum discord, measured by Schatten 1-norm, and the l1 trace-norm
+quantum coherence. The analytical formulations for these quantum information quantifiers were obtained in terms of magnetic
+anisotropies. In this scenario, the effects of dipolar coupling constants on these quantifiers are highlighted. It is demonstrated
+that the presence of dipolar anisotropies increases the degree to which the system possesses quantum correlation and coherence.
+
+9
+0
+5
+10
+0
+5
+10
+- 10
+- 5
+- 10
+- 5
+Δ
+kB T
+ϵ
+kB T
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+PΨ+
+PΦ+
+PΦ-
+FIG. 4: (Color online) Average quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios
+∆/kBT and ϵ/kBT. The solid white line represents the boundary between the quantum level crossings. The dashed green line is the boundary
+given by the concurrence, Eq. (17), C(ρAB) = 0, inside which the entanglement of the system is absent.
+As another remarkable result, it is proved that the global coherence, expressed in an arbitrary reference basis, determined by
+the co-latitude and longitude angles of the Bloch sphere representation, is totally stored within the correlations of the system.
+Moreover, according to the results, the behavior of quantum discord contains a notable hallmark of quantum level-crossing in
+the system, in contrast to the basis-dependent axial quantum coherence, which hides the energy-level crossover regarding the
+measured basis.
+Therefore, the dependency of the base on the quantum coherence specified by Baumgratz might be deleterious in identifying
+the crossing of levels owing to population changes originating from the changing of Boltzman weights due to the modification
+of the magnetic anisotropies of the studied system. In this regard, the average quantum coherence was measured numerically
+obtained in order to gain a viewpoint independent of the reference basis, unraveling that the average coherence is able to extract
+the signature of the energy-level crossover present in the measurement of quantum discord.
+Finally, the findings that were given provide light on the ways in which magnetic anisotropies caused by the dipolar interaction
+coupling of a dinuclear spin-1/2 system influence quantum correlations and coherence. Therefore, the dipolar interaction model
+is an excellent option for usage as a platform for quantum technologies that are based on quantum resources such as quantum
+coherence and quantum discord.
+ACKNOWLEDGEMENTS
+C. Cruz gratefully acknowledges Mario Reis for the valuable discussions. M. F. Anka thanks FAPERJ for financial support.
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