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+Journal of the Physical Society of Japan
+LETTERS
+Inelastic Neutron Scattering Study of the Spin Dynamics
+in the Breathing Pyrochlore System LiGa0.95In0.05Cr4O8
+Yu Tanaka1 *, Rafal Wawrzy´nczak2, Manh Duc Le3, Tatiana Guidi3, Yoshihiko Okamoto4, Takeshi Yajima1,
+Zenji Hiroi1, Masashi Takigawa1, and Gøran J. Nilsen3
+1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
+2Institut Laue-Langevin, CS 20156, Cedex 9, 38042 Grenoble, France
+3ISIS Facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot OX11 0QX, United Kingdom
+4Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
+The A-site ordered chromate spinels LiGa1−xInxCr4O8 host a network of size-alternating spin-3/2 Cr3+ tetrahedra
+known as a “breathing” pyrochlore lattice. For the x = 0.05 composition, the complex magneto-structural ordering
+observed in the parent x = 0 material is replaced by a single transition at T f = 11 K, ascribed to the collinear nematic
+order caused by strong spin-lattice coupling. We present here an inelastic neutron scattering study of the spin dynamics
+in this composition. Above T f , the dynamical scattering function S (Q, E) is ungapped and quasi-elastic, similar to
+undoped LiGaCr4O8. Below T f , the spectral weight splits between a broad inelastic feature at 5.8 meV and toward the
+elastic line. The former feature can be ascribed to spin precessions within antiferromagnetic loops, lifted to finite energy
+by the effective biquadratic spin-lattice term in the spin Hamiltonian.
+When magnetic frustration is combined with strong spin-
+lattice coupling, a range of possible magneto-structural be-
+haviors result,1) including nematic transitions,2) magnetiza-
+tion plateaus, and localized spin excitations.3,4) An ideal
+arena for exploring this interplay is provided by the chro-
+mate spinels, A2+Cr3+
+2 O4. Here, the frustration originates from
+the corner-sharing pyrochlore network of Cr3+ (S = 3/2)
+tetrahedra on the B-site, while the strong spin-lattice cou-
+pling arises from the direct overlap of Cr3+ d-orbitals on ad-
+jacent sites. A common starting point to understand the low-
+temperature physics of the chromate spinels is the so-called
+bilinear-biquadratic model,5,6)
+H = J
+�
+i,j
+Si · Sj + b
+�
+i,j
+(Si · Sj)2
+(1)
+where the classical nearest-neighbor Heisenberg Hamiltonian
+is extended with an effective biquadratic term, b(Si · Sj)2
+(where b is a coupling constant, and Si,j are classical spins).
+This term, generated by spin-lattice coupling to local distor-
+tions, lifts some of the degeneracy of the ground state mani-
+fold by selecting collinear or coplanar spin configurations. Al-
+though the bilinear-biquadratic model ignores the long-range
+interactions which eventually cause magneto-structural order
+in many chromate spinels, it is able to successfully describe
+both the short-range spin correlations and the magnetic phase
+diagram of virtually every member of the family. It has not,
+however, yet been applied to the unusual magnetic excita-
+tion spectra in the ordered phases of MgCr2O4,7) HgCr2O4,4)
+and ZnCr2O4;8) these are characterized by weak spin-wave
+branches and sharp, non-dispersive inelastic bands, with wave
+vector dependences characteristic of small spin clusters.4,9)
+While bands assigned to both hexamer and heptamer clus-
+ters are observed in MgCr2O4 and HgCr2O4, only the for-
+mer are seen in ZnCr2O4. Due to the complexity of the low-
+temperature magneto-structural orders in these materials, the
+connection between the structure and the apparent spin cluster
+excitations is unclear.
+This link is more evident, at least at high temperature, in
+the so-called “breathing” pyrochlore spinels A+A′3+Cr4O8,
+where the A-site is now populated by an ordered arrangement
+of mono- and trivalent cations. Here, the term “breathing”
+refers to the alternation of Cr3+-Cr3+ distances and, hence,
+magnetic exchanges J and J′ between adjacent Cr3+
+4 tetrahe-
+dra as a consequence of the order on the A-site. The degree
+of magnetic alternation is quantified by the breathing factor
+Bf = J′/J, where Bf → 0 corresponds to isolated tetrahedra
+and Bf → 1 to the isotropic pyrochlore lattice. Starting from
+the former limit, the excitations at T ∼ J are localized non-
+dispersive triplets (and higher multiplets) separated by a spin
+gap ∆ from the singlet ground state. When Bf is increased, ∆
+is suppressed, and the excitations become qualitatively simi-
+lar to the isotropic case beyond Bf ∼ 0.25. This simple picture
+is again complicated by the influence of the spin-lattice cou-
+pling and long-ranged terms, which are responsible for the
+collective excitations in the low-temperature ordered phases.
+LiGaCr4O8 and LiInCr4O8 were rediscovered by Okamoto
+et. al.10) as breathing pyrochlore systems, which have A+ =
+Li+ and A′3+ = Ga3+/In3+. The nearest-neighbor magnetic in-
+teractions on the small and large tetrahedra are estimated to
+be J ∼ 50 K and J′ ∼ 30 K (Bf ∼ 0.6) for LiGaCr4O8,
+and J ∼ 60 K and J′ ∼ 6 K (Bf ∼ 0.1) for LiInCr4O8.10)
+For LiGaCr4O8, the upper magneto-structural transition at
+Tu ∼ 20 K results in phase separation into cubic paramag-
+netic and tetragonal collinear phases. The cubic phase then
+undergoes another transition at Tl = 13.8 K, into a second
+tetragonal phase, the structure of which has not yet been deter-
+mined. As in other chromate spinels, both transitions are first-
+order. However, the paramagnetic component shows a diver-
+gence in the nuclear spin-lattice relaxation rate 1/T1 extracted
+from 7Li-NMR, implying proximity to a tricritical point or to
+a second-order transition to another phase.11)
+In this letter, we describe inelastic neutron scattering mea-
+surements of the spin excitation spectrum of the “breath-
+1
+arXiv:2301.05064v1  [cond-mat.str-el]  12 Jan 2023
+
+J. Phys. Soc. Jpn.
+LETTERS
+152 K
+50.4 K
+(a)
+(b)
+(c)
+(d)
+12.4 K
+5.2 K
+S (Q, E) (arb. units)
+Fig. 1.
+(Color online) Temperature dependence of S (Q, E) for LiGa0.95In0.05Cr4O8 recorded with 16 meV incident energy. (a)-(c) are taken at T > T f and
+(d) below T f . Blank patches are due to gaps between detectors.
+ing” pyrochlore chromate spinel, LiGa0.95In0.05Cr4O8, where
+Bf ∼ 0.6. Previous diffraction measurements indicate that
+LiGa0.95In0.05Cr4O8 undergoes a possible second-order tran-
+sition to a nematic collinear ground state at T f = 11.1 K,
+in accordance with predictions from the bilinear-biquadratic
+model.2) The excitations at T > T f are gapless and Lorentzian
+in form, as is also the case for MgCr2O4, HgCr2O4, and
+ZnCr2O4. Below T f , the spectral weight shifts to the elastic
+line and an inelastic feature at ∼5.8 meV. We identify the lat-
+ter with spin precession within antiferromagnetic hexagonal
+spin clusters created by the nematic order, and lifted to finite
+energy by the biquadratic term. We thus provide, for the first
+time, a plausible link between the magnetic excitations and
+magnetic structure. The remaining spectral weight appears to
+be consistent with collective spin-wave-like excitations.
+The powder sample of LiGa0.95In0.05Cr4O8 was prepared
+by sintering a stoichiometric mixture of the two end-member
+compounds LiGaCr4O8 and LiInCr4O8.12) These were in turn
+prepared by the standard solid-state route,10) using starting
+materials enriched with 7Li to reduce neutron absorption. For
+our inelastic neutron scattering measurements, 8.1 g of pow-
+der was packed in an Al sachet, which was rolled into an annu-
+lus and loaded into an Al can with � = 45 mm. The measure-
+ments were performed on the MARI direct-geometry time-of-
+flight chopper spectrometer at the ISIS facility, UK, using in-
+cident energies Ei = 10, 16, 25, and 35 meV. For all values of
+Ei, the elastic energy resolution was close to ∆E/E ∼ 4.5%.
+Temperatures between 5 and 300 K were accessed using a
+closed-cycle refrigerator. The diffraction measurements re-
+ported in Ref. 2 were performed on the same sample.
+The dynamical structure factors S (Q, E = Ei − E f ) mea-
+sured at four selected temperatures between 5.2 K and 152 K
+are shown in Fig. 1. We begin with an analysis of the data
+taken above T f : at T = 152 K≫ T f , a quasi-elastic rod
+of scattering extending up to ∼15 meV is observed. This
+is characteristic of diffusive spin excitations in the corre-
+lated paramagnetic state,13) and resembles the S (Q, E) of both
+LiInCr4O814) and LiGaCr4O815) at similar temperatures. The
+intensity of the rod is centered around 1.6 Å−1, corresponding
+approximately to the reciprocal space position of the Cr-Cr
+nearest-neighbor distance. Upon cooling to 50.4 K, intensity
+builds up near the elastic line, again as in LiGaCr4O8, but in
+contrast to LiInCr4O8, where the scattering becomes inelas-
+tic.14) The modulation of the quasi-elastic scattering is also
+enhanced, indicating the development of longer-ranged spin-
+spin correlations ⟨S (0) · S (r)⟩, as may be seen in Fig. 2.
+To determine the spatial extent of the correlations, we
+fit the Q-dependence of the scattering integrated between 2
+and 7 meV (≃ S (Q) at high temperature) to a shell model
+[Fig. 2(a)]:
+S (Q) = f(Q)2 �
+i
+⟨S (0) · S (ri)⟩ Ni
+sin(Qri)
+Qri
+,
+(2)
+where f(Q) is the magnetic form factor for Cr3+, and Ni is
+the coordination number of the ith shell at radial distance ri.
+For simplicity, r1 is approximated as the mean of the r1 and r′
+1
+distances.
+The summation in the fitting function was extended to the
+third neighboring shell, at which point the fit quality did
+not increase. The extracted parameters reveal antiferromag-
+2
+
+MAR20556Reduced SQW
+T= 50.4 K
+Energy transfer (me V)
+10
+0
+1
+2
+3
+4
+5
+Q(A-1)MAR20554Reduced SQW
+T= 12.4 K
+Energy transfer (me V)
+10
+0
+1
+2
+3
+5
+Q(A-1)MAR20551Reduced SQW
+T= 5.2 K
+Energy transfer (me V)
+10
+0
+1
+2
+3
+4
+Q(A-1)MAR20560Reduced SQW
+T= 152 K
+Energy transfer (me V)
+.10
+0
+1
+2
+3
+4
+Q(A-1)J. Phys. Soc. Jpn.
+LETTERS
+1
+2
+3
+4
+Q ( ˚A−1)
+S(Q) (a.u.)
+152 K
+75 K
+50 K
+24 K
+18 K
+14 K
+12.4 K
+5.2 K
+(a)
+E = 2−7 meV
+0
+2
+4
+6
+8
+10
+12
+r ( ˚A)
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+⟨S(0)·S(r)⟩/S(S+1)
+(b)
+RMC
+1.5 K
+30 K
+5.2 K
+24 K
+152
+Hexamer cluster
+Hexamer cluster
+Fig. 2.
+(Color online) (a) Q dependence of the magnetic scattering I(Q),
+integrated over the energy range 2-7 meV at different temperatures. Dashed
+lines are fitting curves calculated from the shell model (Eq. (2)) with the first
+three nearest neighbors and flat backgrounds. Solid red lines show the results
+of structure factor calculations for hexagonal chromium rings at T < 20 K.
+(b) Real space spin-spin correlation functions ⟨S (0) · S (ri)⟩ versus r. Solid
+circles are obtained from the fits to the shell model in Fig. 2(a), and the open
+triangles are obtained by the reverse Monte Carlo (RMC) simulation on the
+magnetic diffuse scattering observed in the elastic ND measurement.2) Green
+stars mark the correlations for an isolated hexagonal antiferromagnetic loop
+(Fig. 4).
+netic nearest-neighbor spin-spin correlations, with progres-
+sively weaker alternating ferro- and antiferromagnetic cor-
+relations for the second and third nearest neighbors, respec-
+tively [Fig. 2(b)]. The extracted correlations are thus con-
+sistent with the reverse Monte Carlo results presented in
+Ref. 2, where energy-integrated data from a diffractometer
+were used; i.e., the true S (Q) was reflected. The temperature
+dependence of the parameters indicates smooth growth of the
+spin-spin correlations in the entire temperature range, as ex-
+pected.
+To analyze the temperature dependence of the quasi-elastic
+feature further, the imaginary part of the magnetic dynamic
+susceptibility χ′′ was calculated by applying the fluctuation
+dissipation theorem16) χ′′(Q, E) = π(1 − e−
+E
+kBT )S (Q, E) to the
+E dependence of the intensity integrated over the Q range
+1.1 − 1.9 Å−1 [Fig. 3(a)]. At T > T f , the contribution of elas-
+tic scattering is subtracted by approximating it with a sharp
+Gaussian centered around E = 0. The obtained χ′′(ω) are well
+fit by a quasi-elastic Lorentzian χ′′(ω) = χ′ωΓ/(ω2 + Γ2),
+20
+15
+10
+5
+0
+χ" (arb. unit)
+10
+8
+6
+4
+2
+0
+E transfer (meV)
+ 5.2 K
+ 12.4 K
+ 14 K
+ 18 K
+ 24 K
+ 50 K
+ 75 K
+ 152 K
+12
+10
+8
+6
+4
+2
+0
+Γ (meV)
+160
+120
+80
+40
+0
+T (K)
+30
+20
+10
+0
+ χ'  (arb. units)
+(a)
+(b)
+Fig. 3.
+(Color online) (a) Energy dependence of the dynamic susceptibil-
+ity χ′′(ω), integrated over the Q range 1.1-1.9 Å−1 for all measured tem-
+peratures. The elastic line was subtracted from each dataset. The solid lines
+are resolution-broadened quasi-elastic Lorentzian fits. (b) Temperature de-
+pendence of the inverse relaxation rate Γ and the static susceptibility χ’ as
+determined by quasi-elastic Lorentzian fitting. The solid and dotted lines are
+a linear and power-law curve fits to Γ.
+which is the time-Fourier transform of an exponential de-
+cay exp(−t/τ), with τ ∝ 1/Γ and χ′ the static susceptibil-
+ity. On cooling below 18 K, the Lorentzian fits become poor
+at E < 2 meV, indicating that the scattering is no longer de-
+scribed by a single relaxation process. This coincides with the
+appearance of a stretching exponent β < 1 in fits of the T1 re-
+laxation process,17) and thus is likely connected with the onset
+of critical fluctuations above T f .
+Figure 3(b) shows the temperature dependence of the in-
+verse relaxation time Γ and the static susceptibility χ′ ex-
+tracted from the fits described above. Γ decreases smoothly
+in the temperature range 18 K, and is well described by a
+power law Γ ∝ T γ with γ = 0.66 (dashed line). For Heisen-
+berg spins on the isotropic pyrochlore lattice, theory predicts
+Γ ∝ T (γ = 1);13,18,19) however, a linear fit to the data (solid
+line) is poor at high temperature, even permitting a nonzero
+intercept Γ0 = 1.09 meV. Although a similar reduction of γ
+has also been observed in ZnCr2O4 (γ = 0.81), the cause re-
+mains unclear.3) Aside from this, χ′ is consistent with the bulk
+susceptibility.
+Turning now to the form of S (Q, E) below T f shown in
+Fig. 1(d), most of the high-temperature quasi-elastic scatter-
+ing shifts either towards the elastic line or to an inelastic fea-
+ture centered around 5.8 meV. The latter is similar to the “res-
+onance” observed in LiGaCr4O815) and other spinels, but is
+considerably broader in energy. Like the resonance, however,
+its structure factor suggests local modes on small antiferro-
+magnetic spin loops. An analysis of the reverse Monte Carlo
+spin configurations derived from fits to S (Q) in our previ-
+ous publication2) identifies these with a large number of six-
+membered hexagonal antiferromagnetic spin loops, as well
+as a few with eight or more members. Indeed, the calculated
+structure factor for the hexagonal rings (Fig. 4) agrees almost
+perfectly with that of the energy-integrated data in Figure
+2(a), also accounting for the variation of ⟨S (0) · S (ri)⟩ versus
+r from the model-independent fits above. As shown in Fig. 4,
+hexagonal antiferromagnetic spin loops are only possible in
+the presence of three types (colors) of collinear state on the
+Cr3+ tetrahedra.5)
+By analogy with the coplanar nematic state in the kagome
+lattice antiferromagnet,20–22) collinear nematic states on the
+3
+
+J. Phys. Soc. Jpn.
+LETTERS
+Fig. 4.
+(Color online) Hexamer loop determined within the breathing py-
+rochlore lattice (cyan bonds). Spheres represent Cr3+ ions. RBG coloring of
+the bonds and vertices of the tetrahedra corresponds to the bond ordering de-
+scribed in Refs. 2 and 5. Cyan arrows represent antiferromagnetically coupled
+spins on the nodes of hexagonal cluster precessing around the easy direction
+of nematic phase (dashed black lines).
+pyrochlore lattice support two types of loop excitations: (i)
+loop flips, which invert the moment directions around the
+loop, hence transforming one nematic ground state configura-
+tion to another, and (ii) “weathervane” modes, small displace-
+ments of the moment direction about the equilibrium direction
+[Fig. 4]. The former, related to the diffusive high-temperature
+excitations, is expected to produce a quasi-elastic signal with
+a temperature-dependent width, and thus cannot account for
+the inelastic feature. As such, we tentatively assign the fea-
+ture to weathervane modes on the hexagonal loops. Consid-
+ering only the bilinear term, the ground state criterion of two
+spins up and two down on each tetrahedron results in a zero
+net exchange field for the spins around the hexagon, and the
+weathervane modes therefore carry no energy cost. When the
+biquadratic (magneto-elastic) and other long-ranged terms are
+included, however, they are lifted to finite energy. In particu-
+lar, inserting the bilinear-biquadratic Hamiltonian (1) into the
+classical equation of motion
+dSi(t)
+dt
+= −1
+ℏSi(t) × ∇Si(t)H
+(3)
+results in an energy gap ∆E ≃ 8bavS 3, where bav is the aver-
+age bilinear-biquadratic coupling constant between the small
+and large tetrahedra. In deriving this expression, we assumed
+that there is no coupling between the loops and S z
+i(t) ≃ S ; i.e.,
+the spin displacements are small. From Jav = (J + J′)/2 =
+45 K estimated from the magnetic susceptibility and the ex-
+perimental excitation energy, we obtain bav ∼ 0.05Jav, which
+is close to the b reported for related materials.23) In addi-
+tion, using T f ≃ bS 4 for the isotropic pyrochlore lattice,24)
+bav ∼ 0.05Jav yields T f ∼ 12 K, which is in excellent agree-
+ment with experiment.
+Now we address the large width of the feature relative to the
+much sharper features observed in other spinels: could this
+be due to the disorder inherent to the nematic state? Below
+T f , the Cr-Cr bond lengths, and hence the biquadratic bond
+energies, are expected to follow a Gaussian distribution (is
+indeed found for the d-spacings in [2]). The resulting spec-
+trum is then broadened by σ(bav), the FWHM of the Gaus-
+sian. The experimental feature at 5.8 meV is approximately
+Gaussian, with an FWHM of ∼2 meV. To reproduce this, the
+distribution of mean Cr-Cr bond lengths around a spin loop
+is required to be ∼0.1 Å wide, assuming a linear relationship
+between the exchange and the Cr-Cr distance. This is larger
+by approximately a factor of 4 than the distribution estimated
+from Rietveld refinements, which, however, ignore any local
+structure.
+The significant amount of inelastic and quasi-elastic spec-
+tral weight at energies above and below the 5.8 meV feature,
+may be associated with other excitations (also observed on
+the kagome lattice), including the loop flips mentioned above
+and longer-ranged spin-wave-like excitations (which may ex-
+tend to much higher energies), perhaps belonging to the short-
+range magnetic order superimposed on the nematic state. The
+long high-energy tail of the inelastic scattering, extending to
+∼15 meV, is certainly compatible with the latter. Loop flips,
+on the other hand, are expected to give a quasi-elastic signal of
+width ∝ 1/ exp(−b/T). Ultimately, single-crystal studies and
+spin dynamics simulations of the bilinear-biquadratic model
+with disorder on the present lattice will be required to disen-
+tangle all the contributions to the excitation spectrum in the
+nematic phase.
+Looking beyond the breathing pyrochlores, many features
+of the LiGa1−xInxCr4O8 series are shared with the undistorted
+ZnxCd1−xCr2O4 family.8,25) Starting with the x − T phase di-
+agrams, the introduction of bond disorder by even vestigial
+doping is found to lead to the suppression of the N´eel phase
+and adoption of a disordered frozen state at small x in both
+cases, as also observed in Monte Carlo simulations.24) The
+persistence of a sharp phase transition in the specific heat,
+despite glassy behavior in the magnetic susceptibility, is also
+common to both systems. These commonalities suggest the
+intriguing possibility that ZnxCd1−xCr2O4 with x < 0.1 and
+other similar systems also exhibit nematic transitions.24)
+Comparing the x = 0.05 compositions of both families,
+In0.05 and Cd0.05, the form of the scattering is at first glance
+nearly identical above and below the transitions at T f . How-
+ever, the dynamic susceptibility χ′′(E) of In0.05 is describable
+using only one relaxation rate down to 18 K ∼ 1.6T f , while
+that of Cd0.05 requires a distribution of relaxation rates already
+below 4T f .8) This is indicative of a stronger doping effect in
+the latter case. In regard to the gap in S (Q, E) at T < T f , ∆E is
+4.5 meV in Cd0.05 versus 5.8 meV in In0.05, giving a ratio close
+to that of the exchange couplings in the two systems. Given
+the similar b/J, this could point to a similar physical origin for
+the gap. On the other hand, non-collinearity or strong further
+neighbor couplings could also generate a nonzero exchange
+field around a hexagon, and the former is thought to be fa-
+vored by bond disorder.26) Indeed, flat features in the inelastic
+scattering are also observed in Y2Ru2O7 and ZnCr2O4, where
+non-collinear orders have been proposed.
+We finally note that although inelastic resonances have
+been interpreted as quantum two-level excitations in the
+past,4) they should not be considered as such in the present
+case. This is because the singlet-triplet gap is rapidly sup-
+pressed by both further neighbor couplings and a negative bi-
+quadratic exchange. In addition to this, none of the expected
+higher multiplets are observed at any temperature.
+We have presented an inelastic neutron scattering study
+of the spin dynamics in the classical spin nematic mate-
+rial LiGa0.95In0.05Cr4O8. The high-temperature dynamics are
+4
+
+J. Phys. Soc. Jpn.
+LETTERS
+quasi-elastic and resemble those observed in other pyrochlore
+systems, while the excitation spectrum below the transition at
+T f = 11 K is dominated by a broad, non-dispersive inelastic
+feature at 5.8 meV. A plausible origin for this feature mode is
+the so-called weathervane modes on hexagonal antiferromag-
+netic loops (abundant in the nematic state), which are lifted
+to finite energy by the biquadratic term that induces the ne-
+matic order. Possible collective excitations with a bandwidth
+of 15 meV are also observed. In order to verify this interpreta-
+tion, more detailed spin dynamics simulations of the bilinear-
+biquadratic model on the breathing pyrochlore lattice will be
+required.
+Acknowledgments
+We thank Y. Motome, H. Shinaoka and M. Gingras
+for fruitful discussions. This work was supported by JSPS KAKENHI (Grant
+Nos. 25287083 and 16J01077). Y.T. was supported by the JSPS through the
+Program for Leading Graduate Schools (MERIT).
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+at:
+https://userclub.ill.eu.
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+5
+

2NE4T4oBgHgl3EQfagwY/content/tmp_files/load_file.txt

@@ -0,0 +1,497 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf,len=496
+page_content='Journal of the Physical Society of Japan  LETTERS  Inelastic Neutron Scattering Study of the Spin Dynamics  in the Breathing Pyrochlore System LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8  Yu Tanaka1 *, Rafal Wawrzy´nczak2, Manh Duc Le3, Tatiana Guidi3, Yoshihiko Okamoto4, Takeshi Yajima1,  Zenji Hiroi1, Masashi Takigawa1, and Gøran J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Nilsen3  1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan  2Institut Laue-Langevin, CS 20156, Cedex 9, 38042 Grenoble, France  3ISIS Facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot OX11 0QX, United Kingdom  4Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan  The A-site ordered chromate spinels LiGa1−xInxCr4O8 host a network of size-alternating spin-3/2 Cr3+ tetrahedra  known as a “breathing” pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' For the x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 composition, the complex magneto-structural ordering  observed in the parent x = 0 material is replaced by a single transition at T f = 11 K, ascribed to the collinear nematic  order caused by strong spin-lattice coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' We present here an inelastic neutron scattering study of the spin dynamics  in this composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Above T f , the dynamical scattering function S (Q, E) is ungapped and quasi-elastic, similar to  undoped LiGaCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Below T f , the spectral weight splits between a broad inelastic feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV and toward the  elastic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The former feature can be ascribed to spin precessions within antiferromagnetic loops, lifted to finite energy  by the effective biquadratic spin-lattice term in the spin Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  When magnetic frustration is combined with strong spin-  lattice coupling, a range of possible magneto-structural be-  haviors result,1) including nematic transitions,2) magnetiza-  tion plateaus, and localized spin excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='3,4) An ideal  arena for exploring this interplay is provided by the chro-  mate spinels, A2+Cr3+  2 O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Here, the frustration originates from  the corner-sharing pyrochlore network of Cr3+ (S = 3/2)  tetrahedra on the B-site, while the strong spin-lattice cou-  pling arises from the direct overlap of Cr3+ d-orbitals on ad-  jacent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' A common starting point to understand the low-  temperature physics of the chromate spinels is the so-called  bilinear-biquadratic model,5,6)  H = J  �  i,j  Si · Sj + b  �  i,j  (Si · Sj)2  (1)  where the classical nearest-neighbor Heisenberg Hamiltonian  is extended with an effective biquadratic term, b(Si · Sj)2  (where b is a coupling constant, and Si,j are classical spins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  This term, generated by spin-lattice coupling to local distor-  tions, lifts some of the degeneracy of the ground state mani-  fold by selecting collinear or coplanar spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Al-  though the bilinear-biquadratic model ignores the long-range  interactions which eventually cause magneto-structural order  in many chromate spinels, it is able to successfully describe  both the short-range spin correlations and the magnetic phase  diagram of virtually every member of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' It has not,  however, yet been applied to the unusual magnetic excita-  tion spectra in the ordered phases of MgCr2O4,7) HgCr2O4,4)  and ZnCr2O4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8) these are characterized by weak spin-wave  branches and sharp, non-dispersive inelastic bands, with wave  vector dependences characteristic of small spin clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4,9)  While bands assigned to both hexamer and heptamer clus-  ters are observed in MgCr2O4 and HgCr2O4, only the for-  mer are seen in ZnCr2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Due to the complexity of the low-  temperature magneto-structural orders in these materials, the  connection between the structure and the apparent spin cluster  excitations is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  This link is more evident, at least at high temperature, in  the so-called “breathing” pyrochlore spinels A+A′3+Cr4O8,  where the A-site is now populated by an ordered arrangement  of mono- and trivalent cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Here, the term “breathing”  refers to the alternation of Cr3+-Cr3+ distances and, hence,  magnetic exchanges J and J′ between adjacent Cr3+  4 tetrahe-  dra as a consequence of the order on the A-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The degree  of magnetic alternation is quantified by the breathing factor  Bf = J′/J, where Bf → 0 corresponds to isolated tetrahedra  and Bf → 1 to the isotropic pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Starting from  the former limit, the excitations at T ∼ J are localized non-  dispersive triplets (and higher multiplets) separated by a spin  gap ∆ from the singlet ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' When Bf is increased, ∆  is suppressed, and the excitations become qualitatively simi-  lar to the isotropic case beyond Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This simple picture  is again complicated by the influence of the spin-lattice cou-  pling and long-ranged terms, which are responsible for the  collective excitations in the low-temperature ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  LiGaCr4O8 and LiInCr4O8 were rediscovered by Okamoto  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='10) as breathing pyrochlore systems, which have A+ =  Li+ and A′3+ = Ga3+/In3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The nearest-neighbor magnetic in-  teractions on the small and large tetrahedra are estimated to  be J ∼ 50 K and J′ ∼ 30 K (Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='6) for LiGaCr4O8,  and J ∼ 60 K and J′ ∼ 6 K (Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1) for LiInCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='10)  For LiGaCr4O8, the upper magneto-structural transition at  Tu ∼ 20 K results in phase separation into cubic paramag-  netic and tetragonal collinear phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The cubic phase then  undergoes another transition at Tl = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 K, into a second  tetragonal phase, the structure of which has not yet been deter-  mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' As in other chromate spinels, both transitions are first-  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' However, the paramagnetic component shows a diver-  gence in the nuclear spin-lattice relaxation rate 1/T1 extracted  from 7Li-NMR, implying proximity to a tricritical point or to  a second-order transition to another phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='11)  In this letter, we describe inelastic neutron scattering mea-  surements of the spin excitation spectrum of the “breath-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05064v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='str-el]  12 Jan 2023  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  LETTERS  152 K  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K  (a)  (b)  (c)  (d)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K  S (Q, E) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' units)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  (Color online) Temperature dependence of S (Q, E) for LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8 recorded with 16 meV incident energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' (a)-(c) are taken at T > T f and  (d) below T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Blank patches are due to gaps between detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  ing” pyrochlore chromate spinel, LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8, where  Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Previous diffraction measurements indicate that  LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8 undergoes a possible second-order tran-  sition to a nematic collinear ground state at T f = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1 K,  in accordance with predictions from the bilinear-biquadratic  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2) The excitations at T > T f are gapless and Lorentzian  in form, as is also the case for MgCr2O4, HgCr2O4, and  ZnCr2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Below T f , the spectral weight shifts to the elastic  line and an inelastic feature at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' We identify the lat-  ter with spin precession within antiferromagnetic hexagonal  spin clusters created by the nematic order, and lifted to finite  energy by the biquadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' We thus provide, for the first  time, a plausible link between the magnetic excitations and  magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The remaining spectral weight appears to  be consistent with collective spin-wave-like excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  The powder sample of LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8 was prepared  by sintering a stoichiometric mixture of the two end-member  compounds LiGaCr4O8 and LiInCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='12) These were in turn  prepared by the standard solid-state route,10) using starting  materials enriched with 7Li to reduce neutron absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' For  our inelastic neutron scattering measurements, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1 g of pow-  der was packed in an Al sachet, which was rolled into an annu-  lus and loaded into an Al can with � = 45 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The measure-  ments were performed on the MARI direct-geometry time-of-  flight chopper spectrometer at the ISIS facility, UK, using in-  cident energies Ei = 10, 16, 25, and 35 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' For all values of  Ei, the elastic energy resolution was close to ∆E/E ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Temperatures between 5 and 300 K were accessed using a  closed-cycle refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The diffraction measurements re-  ported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2 were performed on the same sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  The dynamical structure factors S (Q, E = Ei − E f ) mea-  sured at four selected temperatures between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K and 152 K  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' We begin with an analysis of the data  taken above T f : at T = 152 K≫ T f , a quasi-elastic rod  of scattering extending up to ∼15 meV is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This  is characteristic of diffusive spin excitations in the corre-  lated paramagnetic state,13) and resembles the S (Q, E) of both  LiInCr4O814) and LiGaCr4O815) at similar temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The  intensity of the rod is centered around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='6 Å−1, corresponding  approximately to the reciprocal space position of the Cr-Cr  nearest-neighbor distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Upon cooling to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K, intensity  builds up near the elastic line, again as in LiGaCr4O8, but in  contrast to LiInCr4O8, where the scattering becomes inelas-  tic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='14) The modulation of the quasi-elastic scattering is also  enhanced, indicating the development of longer-ranged spin-  spin correlations ⟨S (0) · S (r)⟩, as may be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  To determine the spatial extent of the correlations, we  fit the Q-dependence of the scattering integrated between 2  and 7 meV (≃ S (Q) at high temperature) to a shell model  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2(a)]:  S (Q) = f(Q)2 �  i  ⟨S (0) · S (ri)⟩ Ni  sin(Qri)  Qri  ,  (2)  where f(Q) is the magnetic form factor for Cr3+, and Ni is  the coordination number of the ith shell at radial distance ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  For simplicity, r1 is approximated as the mean of the r1 and r′  1  distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  The summation in the fitting function was extended to the  third neighboring shell, at which point the fit quality did  not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The extracted parameters reveal antiferromag-  2  MAR20556Reduced SQW  T= 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K  Energy transfer (me V)  10  0  1  2  3  4  5  Q(A-1)MAR20554Reduced SQW  T= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K  Energy transfer (me V)  10  0  1  2  3  5  Q(A-1)MAR20551Reduced SQW  T= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K  Energy transfer (me V)  10  0  1  2  3  4  Q(A-1)MAR20560Reduced SQW  T= 152 K  Energy transfer (me V)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='10  0  1  2  3  4  Q(A-1)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  LETTERS  1  2  3  4  Q ( ˚A−1)  S(Q) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=')  152 K  75 K  50 K  24 K  18 K  14 K  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='4 K  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K  (a)  E = 2−7 meV  0  2  4  6  8  10  12  r ( ˚A)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2  ⟨S(0)·S(r)⟩/S(S+1)  (b)  RMC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='5 K  30 K  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K  24 K  152  Hexamer cluster  Hexamer cluster  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  (Color online) (a) Q dependence of the magnetic scattering I(Q),  integrated over the energy range 2-7 meV at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Dashed  lines are fitting curves calculated from the shell model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' (2)) with the first  three nearest neighbors and flat backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Solid red lines show the results  of structure factor calculations for hexagonal chromium rings at T < 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  (b) Real space spin-spin correlation functions ⟨S (0) · S (ri)⟩ versus r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Solid  circles are obtained from the fits to the shell model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2(a), and the open  triangles are obtained by the reverse Monte Carlo (RMC) simulation on the  magnetic diffuse scattering observed in the elastic ND measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2) Green  stars mark the correlations for an isolated hexagonal antiferromagnetic loop  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  netic nearest-neighbor spin-spin correlations, with progres-  sively weaker alternating ferro- and antiferromagnetic cor-  relations for the second and third nearest neighbors, respec-  tively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The extracted correlations are thus con-  sistent with the reverse Monte Carlo results presented in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2, where energy-integrated data from a diffractometer  were used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=', the true S (Q) was reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The temperature  dependence of the parameters indicates smooth growth of the  spin-spin correlations in the entire temperature range, as ex-  pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  To analyze the temperature dependence of the quasi-elastic  feature further, the imaginary part of the magnetic dynamic  susceptibility χ′′ was calculated by applying the fluctuation  dissipation theorem16) χ′′(Q, E) = π(1 − e−  E  kBT )S (Q, E) to the  E dependence of the intensity integrated over the Q range  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='9 Å−1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' At T > T f , the contribution of elas-  tic scattering is subtracted by approximating it with a sharp  Gaussian centered around E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The obtained χ′′(ω) are well  fit by a quasi-elastic Lorentzian χ′′(ω) = χ′ωΓ/(ω2 + Γ2),  20  15  10  5  0  χ" (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' unit)  10  8  6  4  2  0  E transfer (meV)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='2 K  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content="4 K  14 K  18 K  24 K  50 K  75 K  152 K  12  10  8  6  4  2  0  Γ (meV)  160  120  80  40  0  T (K)  30  20  10  0  χ'  (arb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' units)  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  (Color online) (a) Energy dependence of the dynamic susceptibil-  ity χ′′(ω), integrated over the Q range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='9 Å−1 for all measured tem-  peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The elastic line was subtracted from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The solid lines  are resolution-broadened quasi-elastic Lorentzian fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' (b) Temperature de-  pendence of the inverse relaxation rate Γ and the static susceptibility χ’ as  determined by quasi-elastic Lorentzian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The solid and dotted lines are  a linear and power-law curve fits to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  which is the time-Fourier transform of an exponential de-  cay exp(−t/τ), with τ ∝ 1/Γ and χ′ the static susceptibil-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' On cooling below 18 K, the Lorentzian fits become poor  at E < 2 meV, indicating that the scattering is no longer de-  scribed by a single relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This coincides with the  appearance of a stretching exponent β < 1 in fits of the T1 re-  laxation process,17) and thus is likely connected with the onset  of critical fluctuations above T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Figure 3(b) shows the temperature dependence of the in-  verse relaxation time Γ and the static susceptibility χ′ ex-  tracted from the fits described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Γ decreases smoothly  in the temperature range 18 K, and is well described by a  power law Γ ∝ T γ with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='66 (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' For Heisen-  berg spins on the isotropic pyrochlore lattice, theory predicts  Γ ∝ T (γ = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='13,18,19) however, a linear fit to the data (solid  line) is poor at high temperature, even permitting a nonzero  intercept Γ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='09 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Although a similar reduction of γ  has also been observed in ZnCr2O4 (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='81), the cause re-  mains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='3) Aside from this, χ′ is consistent with the bulk  susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Turning now to the form of S (Q, E) below T f shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 1(d), most of the high-temperature quasi-elastic scatter-  ing shifts either towards the elastic line or to an inelastic fea-  ture centered around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The latter is similar to the “res-  onance” observed in LiGaCr4O815) and other spinels, but is  considerably broader in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Like the resonance, however,  its structure factor suggests local modes on small antiferro-  magnetic spin loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' An analysis of the reverse Monte Carlo  spin configurations derived from fits to S (Q) in our previ-  ous publication2) identifies these with a large number of six-  membered hexagonal antiferromagnetic spin loops, as well  as a few with eight or more members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Indeed, the calculated  structure factor for the hexagonal rings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 4) agrees almost  perfectly with that of the energy-integrated data in Figure  2(a), also accounting for the variation of ⟨S (0) · S (ri)⟩ versus  r from the model-independent fits above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 4,  hexagonal antiferromagnetic spin loops are only possible in  the presence of three types (colors) of collinear state on the  Cr3+ tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='5)  By analogy with the coplanar nematic state in the kagome  lattice antiferromagnet,20–22) collinear nematic states on the  3  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  LETTERS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  (Color online) Hexamer loop determined within the breathing py-  rochlore lattice (cyan bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Spheres represent Cr3+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' RBG coloring of  the bonds and vertices of the tetrahedra corresponds to the bond ordering de-  scribed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Cyan arrows represent antiferromagnetically coupled  spins on the nodes of hexagonal cluster precessing around the easy direction  of nematic phase (dashed black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  pyrochlore lattice support two types of loop excitations: (i)  loop flips, which invert the moment directions around the  loop, hence transforming one nematic ground state configura-  tion to another, and (ii) “weathervane” modes, small displace-  ments of the moment direction about the equilibrium direction  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The former, related to the diffusive high-temperature  excitations, is expected to produce a quasi-elastic signal with  a temperature-dependent width, and thus cannot account for  the inelastic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' As such, we tentatively assign the fea-  ture to weathervane modes on the hexagonal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Consid-  ering only the bilinear term, the ground state criterion of two  spins up and two down on each tetrahedron results in a zero  net exchange field for the spins around the hexagon, and the  weathervane modes therefore carry no energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' When the  biquadratic (magneto-elastic) and other long-ranged terms are  included, however, they are lifted to finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' In particu-  lar, inserting the bilinear-biquadratic Hamiltonian (1) into the  classical equation of motion  dSi(t)  dt  = −1  ℏSi(t) × ∇Si(t)H  (3)  results in an energy gap ∆E ≃ 8bavS 3, where bav is the aver-  age bilinear-biquadratic coupling constant between the small  and large tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' In deriving this expression, we assumed  that there is no coupling between the loops and S z  i(t) ≃ S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=',  the spin displacements are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' From Jav = (J + J′)/2 =  45 K estimated from the magnetic susceptibility and the ex-  perimental excitation energy, we obtain bav ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Jav, which  is close to the b reported for related materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='23) In addi-  tion, using T f ≃ bS 4 for the isotropic pyrochlore lattice,24)  bav ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Jav yields T f ∼ 12 K, which is in excellent agree-  ment with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Now we address the large width of the feature relative to the  much sharper features observed in other spinels: could this  be due to the disorder inherent to the nematic state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Below  T f , the Cr-Cr bond lengths, and hence the biquadratic bond  energies, are expected to follow a Gaussian distribution (is  indeed found for the d-spacings in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The resulting spec-  trum is then broadened by σ(bav), the FWHM of the Gaus-  sian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The experimental feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV is approximately  Gaussian, with an FWHM of ∼2 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' To reproduce this, the  distribution of mean Cr-Cr bond lengths around a spin loop  is required to be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1 Å wide, assuming a linear relationship  between the exchange and the Cr-Cr distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This is larger  by approximately a factor of 4 than the distribution estimated  from Rietveld refinements, which, however, ignore any local  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  The significant amount of inelastic and quasi-elastic spec-  tral weight at energies above and below the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV feature,  may be associated with other excitations (also observed on  the kagome lattice), including the loop flips mentioned above  and longer-ranged spin-wave-like excitations (which may ex-  tend to much higher energies), perhaps belonging to the short-  range magnetic order superimposed on the nematic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The  long high-energy tail of the inelastic scattering, extending to  ∼15 meV, is certainly compatible with the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Loop flips,  on the other hand, are expected to give a quasi-elastic signal of  width ∝ 1/ exp(−b/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Ultimately, single-crystal studies and  spin dynamics simulations of the bilinear-biquadratic model  with disorder on the present lattice will be required to disen-  tangle all the contributions to the excitation spectrum in the  nematic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Looking beyond the breathing pyrochlores, many features  of the LiGa1−xInxCr4O8 series are shared with the undistorted  ZnxCd1−xCr2O4 family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8,25) Starting with the x − T phase di-  agrams, the introduction of bond disorder by even vestigial  doping is found to lead to the suppression of the N´eel phase  and adoption of a disordered frozen state at small x in both  cases, as also observed in Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='24) The  persistence of a sharp phase transition in the specific heat,  despite glassy behavior in the magnetic susceptibility, is also  common to both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' These commonalities suggest the  intriguing possibility that ZnxCd1−xCr2O4 with x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='1 and  other similar systems also exhibit nematic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='24)  Comparing the x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 compositions of both families,  In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 and Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05, the form of the scattering is at first glance  nearly identical above and below the transitions at T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' How-  ever, the dynamic susceptibility χ′′(E) of In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 is describable  using only one relaxation rate down to 18 K ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='6T f , while  that of Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 requires a distribution of relaxation rates already  below 4T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8) This is indicative of a stronger doping effect in  the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' In regard to the gap in S (Q, E) at T < T f , ∆E is  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='5 meV in Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05 versus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV in In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05, giving a ratio close  to that of the exchange couplings in the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Given  the similar b/J, this could point to a similar physical origin for  the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' On the other hand, non-collinearity or strong further  neighbor couplings could also generate a nonzero exchange  field around a hexagon, and the former is thought to be fa-  vored by bond disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='26) Indeed, flat features in the inelastic  scattering are also observed in Y2Ru2O7 and ZnCr2O4, where  non-collinear orders have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  We finally note that although inelastic resonances have  been interpreted as quantum two-level excitations in the  past,4) they should not be considered as such in the present  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This is because the singlet-triplet gap is rapidly sup-  pressed by both further neighbor couplings and a negative bi-  quadratic exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' In addition to this, none of the expected  higher multiplets are observed at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  We have presented an inelastic neutron scattering study  of the spin dynamics in the classical spin nematic mate-  rial LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='05Cr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' The high-temperature dynamics are  4  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  LETTERS  quasi-elastic and resemble those observed in other pyrochlore  systems, while the excitation spectrum below the transition at  T f = 11 K is dominated by a broad, non-dispersive inelastic  feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' A plausible origin for this feature mode is  the so-called weathervane modes on hexagonal antiferromag-  netic loops (abundant in the nematic state), which are lifted  to finite energy by the biquadratic term that induces the ne-  matic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Possible collective excitations with a bandwidth  of 15 meV are also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' In order to verify this interpreta-  tion, more detailed spin dynamics simulations of the bilinear-  biquadratic model on the breathing pyrochlore lattice will be  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Acknowledgments  We thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Motome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Shinaoka and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Gingras  for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' This work was supported by JSPS KAKENHI (Grant  Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' 25287083 and 16J01077).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' was supported by the JSPS through the  Program for Leading Graduate Schools (MERIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content='  2) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Wawrzy´nczak, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Yoshida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Okamoto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Manuel,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Casati, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Hiroi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Takigawa, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Nilsen: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  119 (2017) 087201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Broholm, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Ratcliff, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Ueda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Matsuda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Yokoyama, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Tchernyshyov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Shannon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Yokobori, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Kousaka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Bewley, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Guidi,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Akimitsu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Yamada: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Broholm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Suzuki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Toki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Matsuura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Aso, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Yamada: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Okamoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Nilsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Attfield, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Hiroi: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  110 (2013) 097203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  11) Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Yoshida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Takigawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Okamoto, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Hiroi: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Okamoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Nakazano, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Chalker: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Mutka, and  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Hiroi,  ILL  Experimental  Report  5-31-2275,  Available  at:  https://userclub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content='eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Lovesey: Theory of neutron scattering from condensed matter  (Clarendon Press, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Chalker: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Moessner, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'}
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+arXiv:2301.13427v1  [math.OC]  31 Jan 2023
+Disciplined Saddle Programming
+Philipp Schiele∗1, Eric Luxenberg∗2, and Stephen Boyd2
+1Department of Statistics, Ludwig-Maximilians-Universit¨at M¨unchen
+2Department of Electrical Engineering, Stanford University
+February 1, 2023
+Abstract
+We consider convex-concave saddle point problems, and more generally convex opti-
+mization problems we refer to as saddle problems, which include the partial supremum
+or infimum of convex-concave saddle functions. Saddle problems arise in a wide range
+of applications, including game theory, machine learning, and finance. It is well known
+that a saddle problem can be reduced to a single convex optimization problem by dual-
+izing either the convex (min) or concave (max) objectives, reducing a min-max problem
+into a min-min (or max-max) problem. Carrying out this conversion by hand can be
+tedious and error prone. In this paper we introduce disciplined saddle programming
+(DSP), a domain specific language (DSL) for specifying saddle problems, for which the
+dualizing trick can be automated. The language and methods are based on recent work
+by Juditsky and Nemirovski [JN22], who developed the idea of conic-representable sad-
+dle point programs, and showed how to carry out the required dualization automatically
+using conic duality. Juditsky and Nemirovski’s conic representation of saddle problems
+extends Nesterov and Nemirovski’s earlier development of conic representable convex
+problems; DSP can be thought of as extending disciplined convex programming (DCP)
+to saddle problems. Just as DCP makes it easy for users to formulate and solve com-
+plex convex problems, DSP allows users to easily formulate and solve saddle problems.
+Our method is implemented in an open-source package, also called DSP.
+∗Equal contribution.
+1
+
+Contents
+1
+Introduction
+3
+1.1
+Previous and related work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.2
+Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Saddle programming
+5
+2.1
+Saddle functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.2
+Saddle point problems
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+Saddle extremum functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.4
+Saddle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.5
+Solving saddle problems
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.6
+Dual reduction
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3
+Applications
+10
+3.1
+Robust bond portfolio construction . . . . . . . . . . . . . . . . . . . . . . .
+11
+3.2
+Model fitting robust to data weights . . . . . . . . . . . . . . . . . . . . . . .
+11
+3.3
+Robust production problem with worst case prices . . . . . . . . . . . . . . .
+12
+3.4
+Robust Markowitz portfolio construction . . . . . . . . . . . . . . . . . . . .
+13
+4
+Disciplined saddle point programming
+14
+4.1
+Saddle function calculus
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+4.2
+Conically representable saddle functions
+. . . . . . . . . . . . . . . . . . . .
+14
+5
+Implementation
+16
+5.1
+Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+5.2
+Calculus rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+5.3
+Saddle point problems
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+5.4
+Saddle extremum functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+5.5
+Saddle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+6
+Examples
+23
+6.1
+Robust bond portfolio construction . . . . . . . . . . . . . . . . . . . . . . .
+23
+6.2
+Model fitting robust to data weights . . . . . . . . . . . . . . . . . . . . . . .
+25
+6.3
+Robust Markowitz portfolio construction . . . . . . . . . . . . . . . . . . . .
+27
+2
+
+1
+Introduction
+We consider saddle problems, by which we mean convex-concave saddle point problems or,
+more generally, convex optimization problems that include the partial supremum or infimum
+of convex-concave saddle functions. Saddle problems arise in various fields such as game
+theory, robust and minimax optimization, machine learning, and finance.
+While there are algorithms specifically designed to solve some types of saddle point or
+minimax problems, another approach is to convert them into standard convex optimization
+problems using a trick based on duality that can be traced back to at least the 1920s. The
+idea is to express the infima or suprema that appear in the saddle problem via their duals,
+which converts them to suprema or infima, respectively. Roughly speaking, this turns a min-
+max problem into a min-min (or max-max) problem, which can then be solved by standard
+methods. Specific cases of this trick are well known; the classical example is converting a
+matrix game, a specific saddle point problem, into a linear program (LP) [MVN53]. While
+the dualizing trick has been known and used for almost 100 years, it has always been done
+by hand, for specific problems. It can only be carried out by those who have a working
+knowledge of duality in convex optimization, and are aware of the trick.
+In this paper we propose an automated method for carrying out the dualizing trick. Our
+method is based on the theory of conic representation of saddle point problems, developed
+recently by Juditsky and Nemirovski [JN22]. Based on this development, we have designed
+a domain specific language (DSL) for describing saddle problems, which we refer to as dis-
+ciplined saddle programming (DSP). When a problem description complies with the syntax
+rules, i.e., is DSP-compliant, it is easy to verify that it is a valid saddle problem, and more
+importantly, automatically carry out the dualizing trick. We have implemented the DSL in
+an open source software package, also called DSP, which works with CVXPY [DB16], a DSL
+for specifying and solving convex optimization problems. DSP makes it easy to specify and
+solve saddle problems, without any expertise in (or even knowledge of) convex duality. Even
+for those with the required expertise to carry out the dualizing trick by hand, DSP is less
+tedious and error prone.
+DSP is disciplined, meaning it is based on a small number of syntax rules that, if followed,
+guarantee that the specified problem is a valid saddle problem. It is analogous to disciplined
+convex programming (DCP) [GBY06], which is a DSL for specifying convex optimization
+problems. When a problem specification follows these syntax rules, i.e., is DCP-compliant, it
+is a valid convex optimization problem, and more importantly can be automatically converted
+to an equivalent cone program, and then solved. As a practical matter, DCP allows a large
+number of users to specify and solve even complex convex optimization problems, with no
+knowledge of the reduction to cone form. Indeed, most DCP users are blissfully unaware
+of how their problems are solved, i.e., a reduction to cone form. DCP was based on the
+theory of conic representations of convex functions and problems, pioneered by Nesterov
+and Nemirovski [NN92]. Widely used implementations of DCP include CVXPY [DB16],
+Convex.jl [Ude+14], CVXR [FNB20], YALMIP [Lof04], and CVX [GB14]. Like DCP did for
+convex problems, DSP makes it easy to specify and solve saddle problems, with most users
+unaware of the dualization trick and reduction used to solve their problems.
+3
+
+1.1
+Previous and related work
+Saddle problems.
+Studying saddle problems is a long-standing area of research, resulting
+in many theoretical insights, numerous algorithms for specific classes of problems, and a
+large number of applications.
+Saddle problems are often studied in the context of minimax or maximin optimization
+[DM90; DP95], which, while dating back to the 1920s and the work of von Neumann and
+Morgenstern on game theory [MVN53], continue to be active areas of research, with many
+recent advancements for example in machine learning [Goo+14]. A variety of methods have
+been developed for solving saddle point problems, including interior point methods [HT03;
+Nem99], first-order methods [Kor76; Nem04; Nes07; NO09; CLO13], and second-order meth-
+ods [NP06; Nes08], where many of these methods are specialized to specific classes of saddle
+problems. Depending on the class of saddle problem, the methods differ in convergence rate.
+For example, for the subset of smooth minimax problems, an overview of rates for different
+curvature assumptions is given in [The+19]. Due to their close relation to Lagrange duality,
+saddle problems are commonly studied in the context of convex analysis (see, for example,
+[BV04, §5.4], [Roc70, §33–37], [RW09, §11.J], [BL06, §4.3]), with an analysis via monotone
+operators given in [RY22].
+The practical usefulness of saddle programming in many applications is also increas-
+ingly well known. Many applications of saddle programming are robust optimization prob-
+lems [BBC11; BTEGN09]. For example, in statistics, distributionally robust models can be
+used when the true distribution of the data generating process is not known [DA19]. Another
+common area of application is in finance, with [CPT18, §19.3–4] describing a range of finan-
+cial applications that can be characterized as saddle problems. Similarly, [Boy+17; GI03;
+LB00] describe variations of the classical portfolio optimization problem as saddle problems.
+Disciplined convex programming.
+DCP is a grammar for constructing optimization
+problems that are provably convex, meaning that they can be solved globally, efficiently
+and accurately.
+It is based on the rule that the convexity of a function f is preserved
+under composition if all inner expressions in arguments where f is nondecreasing are convex,
+and all expressions where f is nonincreasing are concave, and all other expressions are
+affine. A detailed description of the composition rule is given in [BV04, §3.2.4]. Using this
+rule, functions can be composed from a small set of primitives, called atoms, where each
+atom has known curvature, sign, and monotonicity. Every function that can be constructed
+from these atoms according to the composition rule is convex, but the converse is not true.
+The DCP framework has been implemented in many programming languages, including
+MATLAB [GB14; Lof04], Python [DB16], R [FNB20], and Julia [Ude+14], and is used by
+researchers and practitioners in a wide range of fields.
+Well-structured convex-concave saddle point problems.
+As mentioned earlier, dis-
+ciplined saddle programming is based on Juditsky and Nemirovski’s recent work on well-
+structured convex-concave saddle point problems [JN22].
+4
+
+1.2
+Outline
+In §2 we describe saddle programming, which includes the classical saddle point problem, as
+well as convex problems that include functions described via partial minimization or maxi-
+mization of a saddle function. We describe some typical applications of saddle programming
+in §3. In §4 we describe disciplined saddle programming, which is a way to specify saddle
+programs in such a way that validity is easy to verify, and the reduction to an equivalent
+cone program can be automated. We describe our implementation in §5, showing how sad-
+dle functions, saddle extremum functions, saddle point problems, and saddle problems are
+specified. We present numerical examples in §6.
+2
+Saddle programming
+2.1
+Saddle functions
+A saddle function (also referred to as a convex-concave saddle function) f : X × Y → R
+is one for which f(·, y) is convex for any fixed y ∈ Y, and f(x, ·) is concave for any fixed
+x ∈ X . The argument domains X ⊆ Rn and Y ⊆ Rm must be nonempty closed convex. We
+refer to x as the convex variable, and y as the concave variable, of the saddle function f.
+Examples.
+• Functions of x or y alone. A convex function of x, or a concave function of y, are
+trivial examples of saddle functions.
+• Lagrangian of a convex optimization problem. The convex optimization problem
+minimize
+f0(x)
+subject to
+Ax = b,
+fi(x) ≤ 0,
+i = 1, . . . , m,
+with variable x ∈ Rn, where f0, . . . , fm are convex and A ∈ Rp×n, has Lagrangian
+L(x, ν, λ) = f(x) + νT (Ax − b) + λ1f1(x) + · · · + λnfm(x),
+for λ ≥ 0 (elementwise). It is convex in x and affine (and therefore also concave) in
+y = (ν, λ), so it is a saddle function with
+X =
+�
+i=0,...,m
+dom fi,
+Y = Rm
++ × Rp,
+• Bi-affine function. The function f(x, y) = (Ax + b)T(Cy + d), with X = Rp and
+Y = Rq, is evidently a saddle function. The inner product xT y is a special case of
+a bi-affine function. For a bi-affine function, either variable can serve as the convex
+variable, with the other serving as the concave variable.
+5
+
+• Convex-concave inner product. The function f(x, y) = F(x)TG(y), where F : Rp →
+Rn is a nonnegative elementwise convex function and G : Rq → Rn is a nonnegative
+elementwise concave function.
+• Weighted ℓ2 norm. The function
+f(x, y) =
+� n
+�
+i=1
+yix2
+i
+�1/2
+,
+with X = Rn and Y = Rn
++, is a saddle function.
+• Weighted log-sum-exp. The function
+f(x, y) = log
+� n
+�
+i=1
+yi exp xi
+�
+,
+with X = Rn and Y = Rn
++, is a saddle function.
+• Weighted geometric mean. The function f(x, y) = �n
+i=1 yxi
+i , with X = Rn
++ and Y =
+Rn
++, is a saddle function.
+• Quadratic form with quasi-semidefinite matrix. The function
+f(x, y) =
+�
+x
+y
+�T �
+P
+S
+ST
+Q
+� �
+x
+y
+�
+,
+where the matrix is quasi-semidefinite, i.e., P ∈ Sn
++ (the set of symmetric positive
+semidefinite matrices) and −Q ∈ Sn
++.
+• Quadratic form. The function f(x, Y ) = xTY x, with X = Rn and Y = Sn
++ (the set of
+symmetric positive semidefinite n × n matrices), is a saddle function.
+• As more esoteric example, the function f(x, Y ) = xTY 1/2x, with X = Rn and Y = Sn
++,
+is a saddle function.
+Combination rules.
+Saddle functions can be combined in several ways to yield saddle
+functions. For example the sum of two saddle functions is a saddle function, provided the
+domains have nonempty intersection. A saddle function scaled by a nonnegative scalar is
+a saddle function. Scaling a saddle function with a nonpositive scalar, and swapping its
+arguments, yields a saddle function: g(x, y) = −f(y, x) is a saddle function provided f is.
+Saddle functions are preserved by pre-composition of the convex and concave variables with
+an affine function, i.e., if f is a saddle function, so is f(Ax+ b, Cx+ d). Indeed, the bi-affine
+function is just the inner product with an affine pre-composition for each of the convex and
+concave variables.
+6
+
+2.2
+Saddle point problems
+A saddle point (x⋆, y⋆) ∈ X × Y is any point that satisfies
+f(x⋆, y) ≤ f(x⋆, y⋆) ≤ f(x, y⋆) for all x ∈ X , y ∈ Y.
+(1)
+In other words, x⋆ minimizes f(x, y⋆) over x ∈ X , and y⋆ maximizes f(x⋆, y) over y ∈ Y.
+The basic saddle point problem is to find such a saddle point,
+find x⋆, y⋆ which satisfy (1).
+(2)
+The value of the saddle point problem is f(x⋆, y⋆).
+Existence of a saddle point for a saddle function is guaranteed, provided some technical
+conditions hold. For example, Sion’s theorem [Sio58] guarantees the existence of a saddle
+point when Y is compact. There are many other cases.
+Examples.
+• Matrix game. In a matrix game, player one chooses i ∈ {1, . . . , m}, and player two
+chooses j ∈ {1, . . . , n}, resulting in player one paying player two the amount Cij. Player
+one wants to minimize this payment, while player two wishes to maximize it. In a mixed
+strategy, player one makes choices at random, from probabilities given by x and player
+two makes independent choices with probabilities given by y. The expected payment
+from player one to player two is then f(x, y) = xT Cy. With X = {x | x ≥ 0, 1Tx = 1},
+and similarly for Y, a saddle point corresponds to an equilibrium, where no player can
+improve her position by changing (mixed) strategy. The saddle point problem consists
+of finding a stable equilibrium, i.e., an optimal mixed strategy for each player.
+• Lagrangian. A saddle point of a Lagrangian of a convex optimization problem is a
+primal-dual optimal pair for the convex optimization problem.
+2.3
+Saddle extremum functions
+Suppose f is a saddle function. The function G : X → R ∪ {∞} defined by
+G(x) = sup
+y∈Y
+f(x, y),
+x ∈ X ,
+(3)
+is called a saddle max function. Similarly, the function H : Y → R ∪ {−∞} defined by
+H(x) = inf
+x∈X f(x, y),
+y ∈ Y,
+(4)
+is called a saddle min function. Saddle max functions are convex, and saddle min functions
+are concave. We will use the term saddle extremum (SE) functions to refer to saddle max
+or saddle min functions. Which is meant is clear from context, i.e., whether it is defined by
+minimization (infimum) or maximization (supremum), or its curvature (convex or concave).
+Note that in SE functions, we always maximize (or take supremum) over the concave variable,
+and minimize (or take infimum) over the convex variable. This means that evaluating G(x)
+os H(y) involves solving a convex optimization problem.
+7
+
+Examples.
+• Dual function. Minimizing a Lagrangian L(x, ν, λ) over x gives the dual function of
+the original convex optimization problem.
+• Maximizing a Lagrangian L(x, ν, λ) over y = (ν, λ) gives the objective function re-
+stricted to the feasible set.
+• Conjugate of a convex function. Suppose f is convex. Then g(x, y) = f(x) − xTy is a
+saddle function, the Lagrangian of the problem of minimizing f subject to x = 0. Its
+saddle min is the negative conjugate function: infx g(x, y) = −f ∗(y).
+• Sum of k largest entries. Consider f(x, y) = xTy, with Y = {y | 0 ≤ y ≤ 1, 1Ty = k}.
+The associated saddle max function G is the sum of the k largest entries of x.
+Saddle points via SE functions.
+A pair (x⋆, y⋆) is a saddle point of a saddle function f
+if and only if x⋆ minimizes the convex SE function G in (3) over x ∈ X , and y⋆ maximizes
+the concave SE function H defined in (4) over y ∈ Y. This means that we can find saddle
+points, i.e., solve the saddle point problem (2), by solving the convex optimization problem
+minimize
+G(x)
+subject to
+x ∈ X ,
+(5)
+with variable x, and the convex optimization problem
+maximize
+H(y)
+subject to
+y ∈ Y,
+(6)
+with variable y. The problem (5) is called a minimax problem, since we are minimizing a
+function defined as the maximum over another variable. The problem (6) is called a maximin
+problem.
+While the minimax problem (5) and maximin problem (6) are convex, they cannot be
+directly solved by conventional methods, since the objectives themselves are defined by max-
+imization and minimization, respectively. There are solution methods specifically designed
+for minimax and maximin problems [LJJ20; MB09], but as we will see minimax problems
+involving SE functions can be transformed to equivalent forms that can be directly solved
+using conventional methods.
+2.4
+Saddle problems
+In this paper we consider convex optimization problems that include SE functions in the
+objective or constraints, which we refer to as saddle problems. The convex problems that
+solve the basic saddle point problem (5) and (6) are special cases, where the objective is an
+8
+
+SE function. As another example consider the problem of minimizing a convex function φ
+subject to the convex SE constraint H(y) ≤ 0, which can be expressed as
+minimize
+φ(x)
+subject to
+f(x, y) ≤ 0 for all y ∈ Y,
+(7)
+with variable x. The constraint here is called a semi-infinite constraint, since (when Y is not
+a singleton) it can be thought of as an infinite collection of convex constraints, one for each
+y ∈ Y [HK93].
+Saddle problems include the minimax and maximin problems (that can be used to solve
+the saddle point problem), and semi-infinite problems that involve SE functions. There are
+many other examples of saddle problems, where SE functions can appear in expressions that
+define the objective and constraints.
+Robust cost LP.
+As a more specific example of a saddle problem consider the linear
+program with robust cost,
+minimize
+supc∈C cTx
+subject to
+Ax = b,
+x ≥ 0,
+(8)
+with variable x ∈ Rn, with C = {c | Fc ≤ g}. This is an LP with worst case cost over the
+polyhedron C [BBC11; BTEGN09]. This is a saddle problem with convex variable x, concave
+variable y, and an objective which is a saddle max function.
+2.5
+Solving saddle problems
+Special cases with tractable analytical expressions.
+There are cases where an SE
+function can be worked out analytically. An example is the max of a linear function over a
+box,
+sup
+l≤y≤u
+yTx, = (1/2)(u + l)Tx + (1/2)(u − l)T|x|,
+where the absolute value is elementwise. We will see other cases in our examples.
+Subgradient methods.
+We can readily compute a subgradient of a saddle max function
+(or a supergradient of a saddle min function), by simply maximizing over the concave variable
+(minimizing over the convex variable), which is itself a convex optimization problem. We can
+then use any method to solve the saddle problem using these subgradients, e.g., subgradient-
+type methods, ellipsoid method, or localization methods such as the analytic center cutting
+plane method. In [MB09] such an approach is used for general minimax problems.
+Methods for specific forms.
+Many methods have been developed for finding saddle
+points of saddle functions with the special form
+f(x, y) = xTKy + φ(x) + ψ(y),
+9
+
+where φ is convex, ψ is concave, and K is a matrix [BS15; Con13; CP11; Nes05a; Nes05b;
+CP16]. Beyond this example, there are many other special forms of saddle functions, with
+different methods adapted to properties such as smoothness, separability, and strong-convex-
+strong-concavity.
+2.6
+Dual reduction
+A well-known trick can be used to transform a saddle point problem into an equivalent prob-
+lem that does not contain SE functions. This method of transforming an inner minimization
+is not new; it has been used since the 1950s when Von Neumann proved the minimax the-
+orem using strong duality in his work with Morgenstern on game theory [MVN53]. Using
+this observation, he showed that the minimax problem of a two player game is equivalent
+to an LP. Duality allows us to express the convex (concave) SE function as an infimum
+(supremum), which facilitates the use of standard convex optimization. We think of this as
+a reduction to an equivalent problem that removes the SE functions from the objective and
+constraints.
+Robust cost LP.
+We illustrate the dualization method for the robust cost LP (8). The
+key is to express the robust cost or saddle max function supc∈C cTx as an infimum. We first
+observe that this saddle max function is the optimal value of the LP
+maximize
+xT c
+subject to
+Fc ≤ g,
+with variable c. Its dual is
+minimize
+gTλ
+subject to
+F Tλ = x,
+λ ≥ 0,
+with variable λ. Assuming that C is nonempty, this dual problem has the same optimal value
+as the primal, i.e.,
+sup
+c∈C
+cTx =
+inf
+λ≥0, F T λ=x gTλ
+Substituting this into (8) we obtain the problem
+minimize
+gTλ
+subject to
+Ax = b,
+x ≥ 0,
+F Tλ = x,
+λ ≥ 0,
+(9)
+with variables x and λ. This simple LP is equivalent to the original robust LP (8), in the
+sense that if (x⋆, λ⋆) is a solution of (9), then x⋆ is a solution of the robust LP (8).
+We will see this dualization trick in a far more general setting in §4.
+3
+Applications
+In this section we describe a few applications of saddle programming.
+10
+
+3.1
+Robust bond portfolio construction
+We describe here a simplified version of the problem described in much more detail in
+[LSB22].
+Our goal is to construct a portfolio of n bonds, giving by its holdings vector
+h ∈ Rn
++, where hi is the number of bond i held in the portfolio. Each bond produces a cash
+flow, i.e., a sequence of payments to the portfolio holder, up to some period T. Let ci,t be
+the payment from bond i in time period t. Let y ∈ RT be the yield curve, which gives the
+time value of cash: A payment of one dollar at time t is worth exp(−tyt) current dollars,
+assuming continuously compounded returns. The bond portfolio value, which is the present
+value of the total cash flow, can be expressed as
+V (h, y) =
+n
+�
+i=1
+T
+�
+t=1
+hici,t exp(−tyt).
+This function is convex in the yields y and concave (in fact, linear) in the holdings vector h.
+Now suppose we do not know the yield curve, but instead have a convex set Y of possible
+values, with y ∈ Y. The worst case value of the bond portfolio, over this set of possible yield
+curves, is
+V wc(h) = inf
+y∈Y V (h, y).
+We recognize this as a saddle min function. (In this application, y is the convex variable
+of the saddle function V , whereas elsewhere in this paper we use y to denote the concave
+variable.)
+We consider a robust bond portfolio construction problem of the form
+minimize
+φ(h)
+subject to
+h ∈ H,
+V wc(h) ≥ V lim,
+(10)
+where φ is a convex objective, typically a measure of return and risk, H is a convex set
+of portfolio constraints (for example, imposing h ≥ 0 and a total budget), and V lim is a
+specified limit on worst case value of the portfolio over the yield curve set Y, which has a
+saddle min as a constraint.
+For some simple choices of Y the worst case value can be found analytically. One example
+is when Y has a maximum element.
+In this special case, the maximum element is the
+minimizer of the value over Y (since V is a monotone decreasing function of y). For other
+cases, however, we need to solve the saddle problem (10).
+3.2
+Model fitting robust to data weights
+We wish to fit a model parametrized by θ ∈ Θ ⊆ Rn to m observed data points. We do this
+by minimizing a weighted loss over the observed data, plus a regularizer,
+m
+�
+i=1
+wiℓi(θ) + r(θ),
+11
+
+where ℓi is the convex loss function for observed data point i, r is a convex regularizer
+function, and the weights wi are nonnegative. The weights can be used to adjust a data
+sample that was not representative, as in [BAB21], or to ignore some of the data points (by
+taking wi = 0), as in [BGM20]. Evidently the weighted loss is a saddle function, with convex
+variable θ and concave variable w.
+We consider the case when the weights are unknown, but lie in a convex set, w ∈ W. The
+robust fitting problem is to choose θ to minimize the worst case loss over the set of possible
+weights, plus the regularizer,
+max
+w∈W
+m
+�
+i=1
+wiℓi(θ) + r(θ).
+We recognize the first term, i.e., the worst case loss over the set of possible weights, as a
+saddle max function.
+For some simple choices of W the worst case loss can be expressed analytically. For
+example with
+W = {w | 0 ≤ w ≤ 1, 1Tw = k},
+(with k ∈ [0, n]), the worst case loss is given by
+max
+w∈W
+m
+�
+i=1
+wiℓi(θ) = φ(ℓ1, . . . , ℓm),
+where φ is the sum-of-k-largest entries [BV04, §3.2.3]. (Our choice of symbol k suggests that
+k is an integer, but it need not be.) In this case we judge the model parameter θ by its worst
+loss on any subset of k of data points. Put another way, we judge θ by dropping the m − k
+data points on which it does best (i.e., has the smallest loss) [BGM20].
+CVXPY directly supports the sum-of-k-largest function, so the robust fitting problem
+can be formulated and solved without using DSP. To support this function, CVXPY carries
+out a transformation very similar to the one that DSP does.
+The difference is that the
+transformation in CVXPY is specific to this one function, whereas the one carried out in
+DSP is general, and would work for other convex weight sets.
+3.3
+Robust production problem with worst case prices
+We consider the choice of a vector of quantities q ∈ Q ⊆ Rn. Positive entries indicate goods
+we buy, and negative quantities are goods we sell. The set of possible quantities Q is our
+production set, which is convex. In addition, we have a manufacturing cost associated with
+the choice q, given by φ(q), where φ is a convex function. The total cost is the manufacturing
+cost plus the cost of goods (which includes revenue), φ(q) + pTq, where p ∈ Rn is vector of
+prices.
+We consider the situation when we do not know the prices, but we have a convex set
+they lie in, p ∈ P. The worst case cost of the goods is maxp∈P pTq. The robust production
+problem is
+minimize
+φ(q) + maxp∈P pTq
+subject to
+q ∈ Q,
+(11)
+12
+
+with variable q. Here too we can work out analytical expressions for simple choices of P,
+such as a range for each component, in which case the worst case price is the upper limit
+for goods we buy, and the lower limit for goods we sell. In other cases, we solve the saddle
+problem (11).
+3.4
+Robust Markowitz portfolio construction
+Markowitz portfolio construction [Mar52] chooses a set of weights (the fraction of the total
+portfolio value held in each asset) by solving the convex problem
+maximize
+µTw − γwTΣw
+subject to
+1Tw = 1,
+w ∈ W,
+where the variable is the vector of portfolio weights w ∈ Rn, µ ∈ Rn is a forecast of the
+asset returns, γ > 0 is the risk aversion parameter, Σ ∈ Sn
+++ is a forecast of the asset return
+covariance matrix, and W is a convex set of feasible portfolios. The objective is called the
+risk adjusted (mean) return.
+Markowitz portfolio construction is known to be fairly sensitive to the (forecasts) µ and
+Σ, which have to be chosen with some care; see, e.g., [BL91]. One approach is to specify
+a convex uncertainty set U that (µ, Σ) must lie in, and replace the objective with its worst
+case (smallest) value over this uncertainty set. This gives the robust Markowitz portfolio
+construction problem
+maximize
+inf(µ,Σ)∈U
+�
+µTw − γwTΣw
+�
+subject to
+1T w = 1,
+w ∈ W,
+with variable w. This is described in, e.g., in [Boy+17; GI03; LB00]. We observe that this
+is directly a saddle problem, with a saddle min objective, i.e., a maximin problem.
+For some simple versions of the problem we can work out the saddle min function explic-
+itly. One example, given in [Boy+17], uses U = M × S, where
+M
+=
+{µ + δ | |δ| ≤ ρ},
+S
+=
+{Σ + ∆ | Σ + ∆ ⪰ 0, |∆ij| ≤ η(ΣiiΣjj)1/2, i, j = 1, . . . , n},
+where ρ > 0 is a vector of uncertainties in the forecast returns, and η ∈ (0, 1) is a parameter
+that scales the perturbation to the forecast covariance matrix. (We interpret δ and ∆ as
+perturbations of the nominal mean and covariance µ and Σ, respectively.) We can express
+the worst case risk adjusted return analytically as
+inf
+(µ,Σ)∈U
+�
+µTw − γwTΣw
+�
+= µTw − γwTΣw − ρT |w| − γη
+� n
+�
+i=1
+Σ1/2
+ii |wi|
+�2
+.
+The first two terms are the nominal risk adjusted return; the last two terms (which are
+nonpositive) represent the cost of uncertainty.
+13
+
+4
+Disciplined saddle point programming
+4.1
+Saddle function calculus
+We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function with concave
+variables x and convex variables y. The set of operations that, when performed on saddle
+functions, preserves the saddle property are called the saddle function calculus. The calculus
+is quite simple, and consists of the following operations:
+1. Conic combination of saddle functions. Let φi(xi, yi), i = 1, . . . , k be saddle functions.
+Let θi ≥ 0 for each i. Then the conic combination, φ(x, y) = �k
+i=1 θiφi(xi, yi), is a
+saddle function.
+2. Affine precomposition of saddle functions. Let φ(x, y) be a saddle function, with x ∈ Rn
+and y ∈ Rm. Let A ∈ Rn×q, b ∈ Rn, C ∈ Rm×p, and d ∈ Rm. Then, with u ∈ Rq and
+v ∈ Rp, the affine precomposition, φ(Au + b, Cv + d), is a saddle function.
+3. Precomposition of saddle functions. Let φ(x, y) : X × Y ⊆ Rn×m → R be a saddle
+function, with x ∈ Rn and y ∈ Rm. The precomposition with a function f : Rp → Rn,
+φ(f(u), y), is a saddle function if for each i = 1, . . . , n one of the following holds:
+• fi(u) is convex and φ is nondecreasing in xi for all y ∈ Y and all x ∈ X .
+• fi(u) is concave and φ is nonincreasing in xi for all y ∈ Y and all x ∈ X .
+Similarly, the precomposition with a function g : Rq → Rm, φ(x, g(v)), is a saddle
+function if for each j = 1, . . . , m one of the following holds:
+• gj(v) is convex and φ is nonincreasing in yj for all x ∈ X and all y ∈ Y.
+• gj(v) is concave and φ is nondecreasing in yj for all x ∈ X and all y ∈ Y.
+4.2
+Conically representable saddle functions
+Nemirovski and Juditsky propose a class of conic representable saddle functions which fa-
+cilitate the automated dualization of saddle problems [JN22]. We will first introduce some
+terminology and notation, and then describe the class of conic representable saddle functions.
+Notation.
+We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function
+which is convex in x and concave in y. Let Kx, Ky and K be members of a collection K
+of closed, convex, and pointed cones with nonempty interiors in Euclidean spaces such that
+K contains a nonnegative ray, is closed with respect to taking finite direct products of its
+members, and is closed with respect to passing from a cone to its dual. We denote conic
+membership z ∈ K by z ⪰K 0. We call a set X ⊆ Rn K-representable if there exist constant
+matrices A and B, a constant vector c, and a cone K ∈ K such that
+X = {x | ∃u : Ax + Bu ⪯K c}.
+14
+
+CVXPY [DB16] can implement a function f exactly when its epigraph {(x, u) | f(x) ≤ u}
+is K-representable.
+Conically representable saddle functions.
+Let X and Y be nonempty and possessing
+K-representations
+X = {x | ∃u : Ax + Bu ⪯K c},
+Y = {y | ∃v : Cy + Dv ⪯K e}.
+A saddle function φ(x, y) : X × Y → R is K-representable if there exist constant matrices
+P, Q, R, constant vectors p and s and a cone K ∈ K such that for each x ∈ X and y ∈ Y,
+φ(x, y) = inf{f Ty + t | Pf + tp + Qu + Rx ⪯K s}.
+This definition generalizes simple class of bilinear saddle functions. See [JN22] for much
+more detail.
+Automated dualization.
+Suppose we have a K-representable saddle function φ as above.
+The power of the conic form is that the saddle extremum
+Φ(x) = sup
+y∈Y
+φ(x, y)
+admits a tractable conic form, meaning that it can be implemented in a DSL like CVXPY.
+Specifically,
+Φ(x)
+=
+sup
+y∈Y
+φ(x, y)
+=
+sup
+y∈Y
+inf
+f,t,u
+�
+f Ty + t
+�� Pf + tp + Qu + Rx ⪯K s
+�
+=
+inf
+f,t,u
+�
+sup
+y∈Y
+�
+f Ty + t
+� ���� Pf + tp + Qu + Rx ⪯K s
+�
+(12)
+=
+inf
+f,t,u
+�
+sup
+y∈Y
+�
+f Ty
+�
++ t
+���� Pf + tp + Qu + Rx ⪯K s
+�
+=
+inf
+f,t,u
+�
+inf
+λ
+�
+λTe
+����
+CTλ = f, DTλ = 0
+λ ⪰K∗ 0
+� ���� Pf + tp + Qu + Rx ⪯K s
+�
+(13)
+where in (12) we use Sion’s minimax theorem [Sio58] to reverse the inf and sup, and in (13)
+we invoke strong duality to replace the supremum over y with an infimum over λ. The final
+line implies a conic representation of the epigraph of Φ(x),
+{(x, u) | Φ(x) ≤ u} =
+
+
+(x, u)
+������
+∃λ, f, t, u :
+λTe + t ≤ u
+CTλ = f, DTλ = 0, λ ⪰K∗ 0
+Pf + tp + Qu + Rx ⪯K s
+
+
+ ,
+which is tractable and can be implemented in a DSL like CVXPY.
+15
+
+A mathematical nuance.
+Switching the inf and sup in (12) requires Sion’s theorem to
+hold. A sufficient condition for Sion’s theorem to hold is that the set Y is compact. However,
+the min and max can be exchanged even if Y is not compact. Then, due to the max-min
+inequality
+max
+y∈Y min
+x∈X f(x, y) ≤ min
+x∈X max
+y∈Y f(x, y),
+the equality in (13) is replaced with a less than or equal to, and we obtain a convex restriction.
+Thus, if a user creates a problem involving an SE function (as opposed to a saddle point
+problem only containing saddle functions in the objective), then DSP guarantees that the
+problem generated is a restriction. This means that the variables returned are feasible and
+the returned optimal value is an upper bound on the optimal value for the user’s problem.
+In our implementation, a saddle problem is solved by applying the above automatic
+dualization to both the objective f and −f and then solving each resulting convex problem,
+with the latter having the role of convex and concave variables switched. We do so in order
+to obtain both the convex and concave components of the saddle point, since the dualization
+removes the concave variable. The saddle problem is only reported as solved if the optimal
+value of the problem with objective f is within a numerical tolerance of the negated optimal
+value of the problem with objective −f. If this holds, this actually implies that
+max
+y∈Y min
+x∈X f(x, y) = min
+x∈X max
+y∈Y f(x, y),
+i.e., (12) was valid, even if for example Y is not compact. Thus, a user need not concern
+themselves with the compactness of Y (or any other sufficient condition for Sion’s theorem)
+when using DSP to find a saddle point; if a saddle point problem is solved, then the saddle
+point is guaranteed to exist.
+5
+Implementation
+In this section we describe our open source Python implementation of the concepts and
+methods described in §4, which we also call DSP. DSP works with CVXPY [DB16], an
+implementation of a DSL for convex optimization based on DCP. We use the term DSP in
+two different ways. We use it to refer to the mathematical concept of disciplined saddle
+programming, and also our specific implementation; which is meant should be clear from
+the context. The term DSP-compliant refers to a function or expression that is constructed
+according to the DSP composition rules given in §5.2. It can also refer to a problem that
+is constructed according to these rules. In the code snippets below, we use the prefix cp
+to indicate functions and classes from CVXPY. (We give functions and classes from DSP
+without prefix, whereas they would likely have a prefix such as dsp in real code.)
+5.1
+Atoms
+Saddle functions in DSP are created from fundamental building blocks or atoms. These
+building blocks extend the atoms from CVXPY [DB16]. In CVXPY, atoms are either jointly
+16
+
+convex or concave in all their variables, but in DSP, atoms are (jointly) convex in a subset
+of the variables and (jointly) concave in the remaining variables. We describe some DSP
+atoms below.
+Inner product.
+The atom inner(x,y) represents the inner product xTy. Since either
+x or y could represent the convex variable, we adopt the convention in DSP that the first
+argument of inner is the convex variable. According to the DSP rules, both arguments to
+inner must be affine, and the variables they depend on must be disjoint.
+Saddle inner product.
+The atom saddle_inner(F, G) corresponds to the function
+F(x)TG(y), where F and G are vectors of nonnegative and respectively elementwise convex
+and concave functions. It is DSP-compliant if F is DCP convex and nonnegative and G is
+DCP concave. If the function G is not DCP nonnegative, then the DCP constraint G >= 0
+is attached to the expression. This is analogous to how the DCP constraint x >= 0 is added
+to the expression cp.log(x). As an example consider
+f = saddle_inner(cp.square(x), cp.log(y)).
+This represents the saddle function
+f(x, y) = x2 log y − I(y ≥ 1),
+where I is the {0, ∞} indicator function of its argument.
+Weighted ℓ2 norm.
+The weighted_norm2(x, y) atom represents the saddle function
+(�n
+i=1 yix2
+i )1/2, with y ≥ 0. It is DSP-compliant if x is either DCP affine or both convex and
+nonnegative, and y is DCP concave. Here too, the constraint y >= 0 is added if y is not
+DCP nonnegative.
+Weighted log-sum-exp.
+The weighted_log_sum_exp(x, y) atom represents the saddle
+function log (�n
+i=1 yi exp xi), with y ≥ 0. It is DSP-compliant if x is DCP convex, and y is
+DCP concave. The constraint y >= 0 is added if y is not DCP nonnegative.
+Quasi-semidefinite quadratic form.
+The quasidef_quad_form(x, y, P, Q, S) atom
+represents the function
+f(x, y) =
+� x
+y
+�T � P
+S
+ST
+Q
+� � x
+y
+�
+,
+where the matrix is quasi-semidefinite, i.e., P ∈ Sn
++ and −Q ∈ Sn
++. It is DSP-compliant if x
+is DCP affine and y is DCP affine.
+Quadratic form.
+The saddle_quad_form(x, Y) atom represents the function xTY x,
+where Y is a PSD matrix. It is DSP-compliant if x is DCP affine, and Y is DCP PSD.
+17
+
+5.2
+Calculus rules
+The atoms can be combined according to the calculus described below to form expressions
+that are DSP-compliant.
+For example, saddle functions can be added or scaled.
+DCP-
+compliant convex and concave expressions are promoted to saddle functions with no concave
+or convex variables, respectively. For example, with variables x, y, and z, the expression
+f = 2.5 * saddle_inner(cp.square(x), cp.log(y)) + cp.minimum(y,1) - z
+is DSP-compliant, with convex variable x, concave variable y, and affine variable z.
+Calling the is_dsp method of an expression returns True if the expression is DSP-
+compliant. The methods convex_variables, concave_variables, and affine_variables,
+list the convex, concave, and affine variables, respectively. The convex variables are those
+that could only be convex, and similarly for concave variables.
+We refer to the convex
+variables as the unambiguously convex variables, and similarly for the concave variables.
+The three lists of variables gives a partition of all the variables the expression depends on.
+For the expression above, f.is_dsp() evaluates as True, f.convex_variables() returns the
+list [x], f.concave_variables() returns the list [y], and f.affine_variables() returns
+the list [z]. Note that the role of z is ambiguous in the expression, since it could be either
+a convex or concave variable.
+No mixing variables rule.
+The DSP rules prohibit mixing of convex and concave vari-
+ables. For example if we add two saddle expressions, no variable can appear in both its
+convex and concave variable lists.
+DSP-compliance is sufficient but not necessary to be a saddle function.
+Re-
+call that if an expression is DCP convex (concave), then it is convex (concave), but the
+converse is false.
+For example, the expression cp.sqrt(1 + cp.square(x)) represents
+the convex function
+√
+1 + x2, but is not DCP. But we can express the same function as
+cp.norm2(cp.hstack([1, x])), which is DCP. The same holds for DSP and saddle func-
+tion: If an expression is DSP-compliant, then it represents a saddle function; but it can
+represent a saddle function and not be DSP-compliant. As with DCP, such an expression
+would need to be rewritten in DSP-compliant form, to use any of the other features of DSP
+(such as a solution method). As an example, the expression x.T @ C @ y represents the
+saddle function xT Cy, but is not DSP-compliant. The same function can be expressed as
+inner(x, C @ y), which is DSP-compliant.
+When there are affine variables in a DSP-compliant expression, it means that those
+variables could be considered either convex or concave; either way, the function is a saddle
+function.
+Example.
+The code below defines the bi-linear saddle function f(x, y) = xT Cy, the ob-
+jective of a matrix game, with x the convex variable and y the concave variable.
+18
+
+Creating a saddle function.
+1 from dsp import *
+# notational convenience
+2 import cvxpy as cp
+3 import numpy as np
+4
+5 x = cp.Variable(2)
+6 y = cp.Variable(2)
+7 C = np.array([[1, 2], [3, 1]])
+8
+9 f = inner(x, C @ y)
+10
+11 f.is_dsp()
+# True
+12
+13 f.convex_variables()
+# [x]
+14 f.concave_variables()
+# [y]
+15 f.affine_variables()
+# []
+Lines 1–3 import the necessary packages (which we will use but not show in the sequel).
+In lines 5–7, we create two CVXPY variables and a constant matrix. In line 9 we construct
+the saddle function f using the DSP atom inner. Both its arguments are affine, so this
+matches the DSP rules. In line 11 we check if saddle_function is DSP-compliant, which
+it is. In lines 13–15 we call functions that return lists of the convex, concave, and affine
+variables, respectively. The results of lines 13–15 might seem odd, but recall that inner
+marks its first argument as convex and its second as concave.
+5.3
+Saddle point problems
+Saddle point problem objective.
+To construct a saddle point problem, we first create
+an objective using
+obj = MinimizeMaximize(f),
+where f is a CVXPY expression.
+The objective obj is DSP-compliant if the expression
+f is DSP-compliant. This is analogous to the CVXPY contructors cp.Minimize(f) and
+cp.Maximize(f), which create objectives from expressions.
+Saddle point problem.
+A saddle point problem is constructed using
+prob = SaddlePointProblem(obj, constraints, cvx_vars, ccv_vars)
+Here, obj is a MinimizeMaximize objective, constraints is a list of constraints, cvx_vars
+is a list of convex variables and ccv_vars is a list of concave variables. The objective must
+be DSP-compliant for the problem to be DSP-compliant. We now describe the remaining
+conditions under which the constructed problem is DSP-compliant.
+19
+
+Each constraint in the list must be DCP, and can only involve convex variables or concave
+variables; convex and concave variables cannot both appear in any one constraint. The list
+of convex and concave variables partitions all the variables that appear in the objective or
+the constraints. In cases where the role of a variable is unambiguous, it is inferred, and does
+not need to be in either list. For example with the objective
+MinimizeMaximize(weighted_log_sum_exp(x, y) + cp.exp(u) + cp.log(v) + z),
+x and u must be convex variables, and y and v must be concave variables, and so do not need
+to appear in the lists used to construct a saddle point problem. The variable z, however,
+could be either a convex or concave variable, and so must appear in one of the lists.
+The role of a variable can also be inferred from the constraints: Any variable that appears
+in a constraint with convex (concave) variables must also be convex (concave). With the
+objective above, the constraint z + v <= 1 would serve to classify z as a concave variable.
+With this constraint, we could pass empty variable lists to the saddle point constructor, since
+the roles of all variables can be inferred. When the roles of all variables are unambiguous,
+the lists are optional.
+The roles of the variables in a saddle point problem prob can be found by calling
+prob.convex_variables() and prob.concave_variables(), which return lists of vari-
+ables, and is a partition of all the variables appearing in the objective or constraints. This is
+useful for debugging, to be sure that DSP agrees with you about the roles of all variables. A
+DSP-compliant saddle point problem must have an empty list of affine variables. (If it did
+not, the problem would be ambiguous.)
+Solving a saddle point problem.
+The solve() method of a SaddlePointProblem object
+solves the problem. The solve method returns the optimal saddle value, i.e., the value of
+the objective at the saddle point. As in CVXPY, the solve method has the side effect of
+writing all variables’ .value attribute.
+Example.
+Here we create and solve a matrix game, continuing the example above where
+f was defined. We do not need to pass in lists of variables since their roles can be inferred.
+Creating and solving a matrix game.
+1 obj = MinimizeMaximize(f)
+2 constraints = [x >= 0, cp.sum(x) == 1, y >= 0, cp.sum(y) == 1]
+3 prob = SaddlePointProblem(obj, constraints)
+4
+5 prob.is_dsp()
+# True
+6 prob.convex_variables()
+# [x]
+7 prob.concave_variables()
+# [y]
+8 prob.affine_variables()
+# []
+9
+10 prob.solve()
+# solves the problem
+20
+
+11 prob.value
+# 1.6666666666666667
+12 x.value
+# array([0.66666667, 0.33333333])
+13 y.value
+# array([0.33333333, 0.66666667])
+5.4
+Saddle extremum functions
+Local variables.
+An SE function has one of the forms
+G(x) = sup
+y∈Y
+f(x, y)
+or
+H(y) = inf
+x∈X f(x, y),
+where f is saddle function. Note that y in the definition of G, and x in the definition of
+H, are local or dummy variables, understood to have no connection to any other variable.
+Their scope extends only to the definition, and not beyond.
+To express this subtlety in DSP, we use the class LocalVariable to represent these
+dummy variables. The variables that are maximized over (in a saddle max function) or
+minimized over (in a saddle min function) must be declared using the LocalVariable()
+constructor. Any LocalVariable in an SE function cannot appear in any other SE function.
+Constructing SE functions.
+We construct SE functions in DSP using
+saddle_max(f, constraints)
+or
+saddle_min(f, constraints).
+Here, f is a CVXPY scalar expression, and constraints is a list of constraints. We now
+describe the rules for constructing a DSP-compliant SE function.
+If a saddle_max is being constructed, f must be DSP-compliant, and the function’s con-
+cave variables, and all variables appearing in the list of constraints, must be LocalVariables,
+while the function’s convex variables must all be regular Variables. A similar rule applies
+for saddle_min.
+The list of constraints is used to specify the set over which the sup or inf is taken. Each
+constraint must be DCP-compliant, and can only contain LocalVariables.
+With x a Variable, y_loc a LocalVariable, z_loc a LocalVariable, and z a Variable,
+consider the following two SE functions:
+1 f_1 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1])
+2 f_2 = saddle_max(inner(x, y_loc) + z_loc, [y_loc <= 1, z_loc <= 1])
+Both are DSP-compliant. For the first, calling f_1.convex_variables() would return
+[x, z], and calling f_1.concave_variables() would return [y_loc].
+For the second,
+calling f_2.convex_variables() would return [x], and f_2.concave_variables() return
+[y_loc, z_loc].
+Let y be a Variable. Both of the following are not DSP-compliant:
+1 f_3 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1, z <= 1])
+2 f_4 = saddle_max(inner(x, y) + z_loc, [y_loc <= 1, z_loc <= 1])
+21
+
+The first is not DSP-compliant because z is not a LocalVariable, but appears in the
+constraints.
+The second is not DSP-compliant because y is not a LocalVariable, but
+appears as a concave variable in the saddle function.
+SE functions are DCP.
+When they are DSP-compliant, a saddle_max is a convex func-
+tion, and a saddle_min is a concave function. They can be used anywhere in CVXPY that a
+convex or concave function is appropriate. You can add them, compose them (in appropriate
+ways), use them in the objective or either side of constraints (in appropriate ways).
+Examples.
+Now we provide full examples demonstrating construction of a saddle_max,
+which we can use to solve the matrix game described in §5.3 as a saddle problem involving
+an SE function.
+Creating a saddle max.
+1 # Creating variables
+2 x = cp.Variable(2)
+3
+4 # Creating local variables
+5 y_loc = LocalVariable(2)
+6
+7 # Convex in x, concave in y_loc
+8 f = saddle_inner(C @ x, y_loc)
+9
+10 # maximizes over y_loc
+11 G = saddle_max(f, [y_loc >= 0, cp.sum(y_loc) == 1])
+Note that G is a CVXPY expression. Constructing a saddle_min works exactly the same
+way.
+5.5
+Saddle problems
+A saddle problem is a convex problem that uses SE functions. To be DSP-compliant, the
+problem must be DCP (which implies all SE functions are DSP-compliant).
+When you
+call the solve method on a saddle problem involving SE functions, and the solve is suc-
+cessful, then all variables’ .value fields are overwritten with optimal values. This includes
+LocalVariables that the SE functions maximized or minimized over; they are assigned to
+the value of a particular maximizer or minimizer of the SE function at the value of the
+non-local variables, with no further guarantees.
+Example.
+We continue our example from §5.4 and solve the matrix game using either a
+saddle max.
+22
+
+Creating and solving a saddle problem using a saddle max to solve the matrix game.
+1 prob = cp.Problem(cp.Minimize(G), [x >= 0, cp.sum(x) == 1])
+2
+3 prob.is_dsp()
+# True
+4
+5 prob.solve()
+# solving the problem
+6 prob.value
+# 1.6666666666666667
+7 x.value
+# array([0.66666667, 0.33333333])
+6
+Examples
+In this section we give numerical examples, taken from §3, showing how to create DSP-
+compliant problems. The specific problem instances we take are small, since our main point
+is to show how easily the problems can be specified in DSP. But DSP will scale to far larger
+problem instances.
+6.1
+Robust bond portfolio construction
+Our first example is the robust bond portfolio construction problem described in §3.1. We
+consider portfolios of n = 20 bonds, over a period T = 60 half-years, i.e., 30 years. The
+bonds are taken as representative ones in a global investment grade bond portfolio; for more
+detail, see [LSB22]. The payments from the bonds are given by C ∈ R20×60, with cash flow
+of bond i in period t denoted ci,t. The portfolio constraint set H is given by
+H = {h | h ≥ 0, pTh = B},
+i.e., the investments must be nonnegative and have a total value (budget) B, which we take
+to be $100. Here p ∈ R20
++ denotes the price of the bonds on September 12, 2022. The
+portfolio objective is
+φ(h) = 1
+2∥(h − hmkt) ◦ p∥1,
+where hmkt is the market portfolio scaled to a value of $100, and ◦ denotes Hadamard or
+elementwise multiplication. This is called the turn-over distance, since it tells us how much
+we would need to buy and sell to convert our portfolio to the market portfolio.
+The yield curve set Y is described in terms of perturbations to the nominal or current
+yield curve ynom, which is the yield curve on September 12, 2022. We take
+Y =
+�
+ynom + δ
+����� ∥δ∥∞ ≤ δmax, ∥δ∥1 ≤ κ,
+T−1
+�
+t=1
+(δt+1 − δt)2 ≤ ω
+�
+.
+We interpret δ as a shock to the yield curve, which we limit elementwise, in absolute sum,
+and in smoothness. The specific parameter values are given by
+δmax = 0.02,
+κ = 0.9,
+ω = 10−6.
+23
+
+In the robust bond portfolio problem (10) we take V lim = 90, that is, the worst case value
+of the portfolio cannot drop below $90 for any y ∈ Y.
+We solve the problem using the following code, where we assume the cash flow matrix
+C, the price vector p, the nominal yield curve y_nom, and the market portfolio h_mkt are
+defined.
+Robust bond portfolio construction.
+1 # Constants and parameters
+2 n, T = C.shape
+3 delta_max, kappa, omega = 0.02, 0.9, 1e-6
+4 B = 100
+5 V_lim = 90
+6
+7 # Creating variables
+8 h = cp.Variable(n, nonneg=True)
+9
+10 delta = LocalVariable(T)
+11 y = y_nom + delta
+12
+13 # Objective
+14 phi = 0.5 * cp.norm1(cp.multiply(h, p) - cp.multiply(h_mkt, p))
+15
+16 # Creating saddle min function
+17 V = 0
+18 for i in range(n):
+19
+t_plus_1 = np.arange(T) + 1
+# Account for zero-indexing
+20
+V += saddle_inner(cp.exp(cp.multiply(-t_plus_1, y)), h[i] * C[i])
+21
+22 Y = [
+23
+cp.norm_inf(delta) <= delta_max,
+24
+cp.norm1(delta) <= kappa,
+25
+cp.sum_squares(delta[1:] - delta[:-1]) <= omega,
+26 ]
+27
+28 V_wc = saddle_min(V, Y)
+29
+30 # Creating and solving the problem
+31 problem = cp.Problem(cp.Minimize(phi), [h @ p == B, V_wc >= V_lim])
+32 problem.solve()
+# 15.32
+We first define the constants and parameters in lines 2–5, before creating the variable
+for the holdings h in line 8, and the LocalVariable delta, which gives the yield curve
+24
+
+Nominal portfolio
+Robust portfolio
+Turn-over distance
+$0.00
+$15.32
+Worst-case value
+$86.99
+$90.00
+Table 1:
+Turn-over distance and worst-case value for the nominal (market) portfolio and the
+robust portfolio. The nominal portfolio does not meet our requirement that the worst-case value
+be at least $90.
+perturbation, in line 10. In line 11 we define y as the sum of the current yield curve y_nom
+and the perturbation delta. The objective function is defined in line 14. Lines 17–20 define
+the saddle function V via the saddle_inner atom. The yield uncertainty set Y is defined in
+lines 22–26, and the worst case portfolio value is defined in line 25 using saddle_min. We use
+the concave expression saddle_min to create and solve a CVXPY problem in lines 31–32.
+Table 1 summarizes the results. The nominal portfolio is the market portfolio, which
+has zero turn-over distance to the market portfolio, i.e., zero objective value. This nominal
+portfolio, however, does not satisfy the worst-case portfolio value constraint, since there are
+yield curves in Y that cause the portfolio value to drop to around $87, less than our limit
+of $90. The solution of the robust problem has turn-over distance $15.32, and satisfies the
+constraint that the worst-case value be at least $90.
+6.2
+Model fitting robust to data weights
+We consider an instance of the model fitting problem described in §3.2. We use the well
+known Titanic data set [HC17], which gives several attributes for each passenger on the ill-
+fated Titanic voyage, including whether they survived. A classifier is fit to predict survival
+based on the features sex, age (binned into three groups, 0–26, 26–53, and 53–80), and class
+(1, 2, or 3). These features are encoded as a Boolean vector ai ∈ R7. The label yi = 1 means
+passenger i survived, and yi = −1 otherwise. There are 1046 examples, but we fit our model
+using only the m = 50 passengers who embarked from Queenstown, one of three ports of
+embarkation. This is a somewhat non-representative sample; for example, the survival rate
+among Queenstown departures is 26%, whereas the overall survival rate is 40.8%.
+We seek a linear classifier ˆyi = sign(aT
+i θ + β0), where θ ∈ R7 is the classifier parameter
+vector and β0 ∈ R is the bias. The hinge loss and ℓ2 regularization are used, given by
+ℓi(θ) = max(0, 1 − yiaT
+i θ),
+r(θ) = η∥θ∥2
+2,
+with η = 0.05.
+The data is weighted to partially correct for the different survival rates for our training
+set (26%) and the whole data set (40.8%). To do this we set wi = z1 when yi = 1 and
+wi = z2 when yi = −1. We require w ≥ 0 and 1Tw = 1, and
+0.408 − 0.05 ≤
+�
+yi=1
+wi ≤ 0.408 + 0.05.
+25
+
+Thus W consists of weights on the Queenstown departure samples that correct the survival
+rate to within 5% of the overall survival rate.
+The code shown below solves this problem, where we assume the data matrix is already
+defined as A_train (with rows aT
+i ), the survival label vector is defined as y_train, and the
+indicator of survival in the training set is defined as surv.
+Model fitting robust to data weights.
+1 # Constants and parameters
+2 m, n = A_train.shape
+3 inds_0 = surv == 0
+4 inds_1 = surv == 1
+5 eta = 0.05
+6
+7 # Creating variables
+8 theta = cp.Variable(n)
+9 beta_0 = cp.Variable()
+10 weights = cp.Variable(m, nonneg=True)
+11 surv_weight_0 = cp.Variable()
+12 surv_weight_1 = cp.Variable()
+13
+14 # Defining the loss function and the weight constraints
+15 y_hat = A_train @ theta + beta_0
+16 loss = cp.pos(1 - cp.multiply(y_train, y_hat))
+17 objective = MinimizeMaximize(saddle_inner(loss, weights)
+18
++ eta * cp.sum_squares(theta))
+19
+20 constraints = [
+21
+cp.sum(weights) == 1,
+22
+0.408 - 0.05 <= weights @ surv,
+23
+weights @ surv <= 0.408 + 0.05,
+24
+weights[inds_0] == surv_weight_0,
+25
+weights[inds_1] == surv_weight_1,
+26 ]
+27
+28 # Creating and solving the problem
+29 problem = SaddlePointProblem(objective, constraints)
+30 problem.solve()
+After defining the constants and parameters in lines 2–5, we specify the variables for the
+model coefficient and the weights in lines 8–9 and 10–12, respectively. The loss function
+and regularizer which make up the objective are defined next in lines 15–18. The weight
+constraints are defined in lines 20–26. The saddle point problem is created and solved in
+26
+
+Nominal classifier
+Robust classifier
+Train accuracy
+82.0%
+80.0%
+Test accuracy
+76.0%
+78.6%
+Table 2: Nominal and worst-case objective values for classification and robust classification models.
+lines 29 and 30.
+The results are shown in table 2. We report the test accuracy on all samples in the
+dataset with a different port of embarkation than Queenstown (996 samples). We see that
+while the robust classification model has slightly lower training accuracy than the nominal
+model, it achieves a higher test accuracy, generalizing from the non-representative training
+data better than the nominal classifier, which uses uniform weights.
+6.3
+Robust Markowitz portfolio construction
+We consider the robust Markowitz portfolio construction problem described in §3.4. We take
+n = 6 assets, which are the (five) Fama-French factors [FF15] plus a risk-free asset. The
+data is obtained from the Kenneth R. French data library [Fre22], with monthly return data
+available from July 1963 to October 2022. The nominal return and risk are the empirical
+mean and covariance of the returns. (These obviously involve look-ahead, but the point of
+the example is how to specify and solve the problem with DSP, not the construction of a
+real portfolio.) We take parameters ρ = 0.02, η = 0.2, and risk aversion parameter γ = 1.
+In the code, we use mu and Sigma for the mean and covariance estimates, respectively,
+and the parameters are denoted rho, eta, and gamma.
+Robust Markowitz portfolio construction.
+1 # Constants and parameters
+2 n = len(mu)
+3 rho, eta, gamma = 0.2, 0.2, 1
+4
+5 # Creating variables
+6 w = cp.Variable(n, nonneg=True)
+7
+8 delta_loc = LocalVariable(n)
+9 Sigma_perturbed = LocalVariable((n, n), PSD=True)
+10 Delta_loc = LocalVariable((n, n))
+11
+12 # Creating saddle min function
+13 f = w @ mu + saddle_inner(delta_loc, w) \
+14
+- gamma * saddle_quad_form(w, Sigma_perturbed)
+15
+16 Sigma_diag = Sigma.diagonal()
+27
+
+Nominal portfolio
+Robust portfolio
+Nominal objective
+.295
+.291
+Robust objective
+.065
+.076
+Table 3: Nominal and worst-case objective for the nominal and robust portfolios.
+17 local_constraints = [
+18
+cp.abs(delta_loc) <= rho, Sigma_perturbed == Sigma + Delta_loc,
+19
+cp.abs(Delta_loc) <= eta * np.sqrt(np.outer(Sigma_diag, Sigma_diag))
+20 ]
+21
+22 G = saddle_min(f, local_constraints)
+23
+24 # Creating and solving the problem
+25 problem = cp.Problem(cp.Maximize(G), [cp.sum(w) == 1])
+26 problem.solve()
+# 0.076
+We first define the constants and parameters, before creating the weights variable in
+line 6, and the local variables for the perturbations in lines 8–10. The saddle function for
+the objective is defined in line 13, followed by the constraints on the perturbations. Both
+are combined into the concave saddle min function, which is maximized over the portfolio
+constraints in lines 25–26.
+The results are shown in table 3. The robust portfolio yields a slightly lower risk adjusted
+return of 0.291 compared to the nominal optimal portfolio with 0.295.
+But the robust
+portfolio attains a higher worst-case risk adjusted return of 0.076, compared to the nominal
+optimal portfolio which attains 0.065.
+Acknowledgements
+P. Schiele is supported by a fellowship within the IFI program of the German Academic
+Exchange Service (DAAD). This research was partially supported by ACCESS (AI Chip
+Center for Emerging Smart Systems), sponsored by InnoHK funding, Hong Kong SAR, and
+by ONR N000142212121.
+28
+
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+GOVERNMENT LICENSE
+The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne
+National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science lab-
+oratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for
+itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in
+said article to reproduce, prepare derivative works, distribute copies to the public, and perform
+publicly and display publicly, by or on behalf of the Government. The Department of Energy will
+provide public access to these results of federally sponsored research in accordance with the DOE
+Public Access Plan. http://energy.gov/downloads/doe-public-access-plan.
+1
+arXiv:2301.05286v1  [physics.app-ph]  12 Jan 2023
+
+Demonstration of an AI-driven workflow for autonomous high-resolution scanning
+microscopy
+Saugat Kandel,1 Tao Zhou,2 Anakha V Babu,1 Zichao Di,3 Xinxin Li,2, 4 Xuedan Ma,2, 4
+Martin Holt,2 Antonino Miceli,1 Charudatta Phatak,5 and Mathew Cherukara1, a)
+1)Advanced Photon Source, Argonne National Laboratory, Lemont,
+IL 60439.
+2)Nanoscience and Technology Division, Argonne National Laboratory, Lemont,
+IL 60439.
+3)Mathematics and Computer Science, Argonne National Laboratory, Lemont,
+IL 60439.
+4)Consortium for Advanced Science and Engineering, University of Chicago, Chicago,
+Illinois 60637, USA
+5)Materials Science Division, Argonne National Laboratory, Lemont,
+IL 60439.
+(Dated: 16 January 2023)
+2
+
+With the continuing advances in scientific instrumentation, scanning microscopes are now
+able to image physical systems with up to sub-atomic-level spatial resolutions and sub-
+picosecond time resolutions. Commensurately, they are generating ever-increasing vol-
+umes of data, storing and analysis of which is becoming an increasingly difficult prospect.
+One approach to address this challenge is through self-driving experimentation techniques
+that can actively analyze the data being collected and use this information to make on-
+the-fly measurement choices, such that the data collected is sparse but representative of
+the sample and sufficiently informative. Here, we report the Fast Autonomous Scanning
+Toolkit (FAST) that combines a trained neural network, a route optimization technique,
+and efficient hardware control methods to enable a self-driving scanning microscopy ex-
+periment. The key features of our method are that: it does not require any prior information
+about the sample, it has a very low computational cost, and that it uses generic hardware
+controls with minimal experiment-specific wrapping. We test this toolkit in numerical ex-
+periments and a scanning dark-field x-ray microscopy experiment of a WSe2 thin film,
+where our experiments show that a FAST scan of <25% of the sample is sufficient to pro-
+duce both a high-fidelity image and a quantitative analysis of the surface distortions in the
+sample. We show that FAST can autonomously identify all features of interest in the sam-
+ple while significantly reducing the scan time, the volume of data acquired, and dose on
+the sample. The FAST toolkit is easy to apply for any scanning microscopy modalities and
+we anticipate adoption of this technique will empower broader multi-level studies of the
+evolution of physical phenomena with respect to time, temperature, or other experimental
+parameters.
+a)Electronic mail: [email protected], [email protected]
+3
+
+I.
+INTRODUCTION
+Scanning microscopes are versatile instruments that use photons, electrons, ions, neutrons, or
+mechanical probes to interrogate atomic-scale composition, topography, and functionality of ma-
+terials, with up to sub-atomic spatial resolution and sub-picosecond time resolution1–3. Notwith-
+standing the variation in the probe modalities, these instruments all rely on a scan of the sample
+to generate spatially resolved signals that are then collected to form an image of the sample. On-
+going advances in instrumentation, such as the development of next-generation x-ray and electron
+detectors4,5, has meant that scanning microscopes can now image faster, and at higher resolutions,
+than ever before. We can now envision a broad use of these instruments to study not only static
+systems, but also multi-level studies of dynamic evolution of materials with time, temperature, or
+other parameters, even in situ or operando6. Fine-resolution large-field-of-view scanning exper-
+iments, however, come with some significant drawbacks: the volume of data generated and the
+probe-induced damage to the sample can be prohibitively large. For example, it is now routinely
+possible to perform x-ray imaging of 1 mm3 volumes at ≈10 nm resolution, but this generates
+≈ 1015 voxels of data7,8 and requires a commensurately high probe dose9. Meanwhile, the in-
+formation of interest in these experiments is often concentrated in sparse regions that contain
+interfaces, defects, or other specific structural elements. Directing the scan to only these locations
+could greatly reduce the scan time and data volume, but it is difficult to obtain this information a
+priori. Addressing this challenge with a human-in-the-loop protocol, where an experienced user
+examines the data acquired to identify trends and guide the scan, can be tedious and prohibitively
+time consuming (in comparison to the experimental acquisition time). Given these factors, the
+development of autonomous acquisition techniques that can continuously analyze acquired data
+and drive the sampling specifically towards regions of interest is imperative so as to make full use
+of the potential of these scientific instruments.
+In parallel to the advances in scientific instrumentation, the last decade has also seen the rapid
+development of deep learning (DL) techniques and their applications in all domains of science
+and technology, including for the acceleration and enhancement of advanced microscopy meth-
+4
+
+ods10–13. These DL-based inversion methods are enabling real-time data analysis, which is in turn
+opening the door to self-driving techniques that make real-time acquisition decisions based on the
+real-time data streams. Such self-driving or autonomous experimentation methods14 are methods
+that combine automated experimental control with on-the-fly data-driven decision making so that
+an algorithm adaptively explores parameter spaces of interest and conducts new experiments until
+it achieves a pre-defined completion criterion15. These methods therefore have the potential to
+not only remove the need for constant human supervision and intervention in experiments, but
+also make optimal choices in parameter spaces that are too large for humans to easily contextual-
+ize. As such, they have the potential to revolutionize experimental design in many scientific fields
+including the field of imaging and materials characterization.
+In general, the use of data-driven priors to direct future experiments is a Bayesian search prob-
+lem, for which the use of off-the-shelf deep learning methods usually do not suffice16. Specific
+to microscopy, a popular Bayesian search approach is to use unsupervised (without pre-training)
+Gaussian Processes (GPs) that could continuously determine the spatial locations that we are most
+uncertain about, then direct the scanning to these locations17–22. While GPs are powerful tech-
+niques, their computational cost tends to scale cubically with the number of points acquired. The
+decision making time increases during the experiment and quickly exceeds the acquisition time
+for the measurement itself. The development of scalable GPs is a significant area of research, but
+these methods are not yet ready for application in large-scale imaging problems23. General super-
+vised alternatives such as reinforcement learning can be powerful and fast, but they often require
+costly pre-training and tend to ignore the global state of the parameter space in exchange for a
+local search; as such they have only found limited traction for scanning imaging modalities24.
+Specifically for scanning microscopy applications, Godaliyadda et al.25 have proposed to
+achieve computationally efficient autonomous sampling with the Supervised Learning Approach
+for Dynamic Sampling (SLADS) technique. The SLADS technique uses curated feature maps
+to quantify the current measurement state and predict the total image quality improvement ob-
+tained by measuring a given point, thereby informing the choice of which point to measure next.
+Variations of this technique have found applications in live steering for dose-efficient crystal posi-
+5
+
+tioning for crystallography26, and for imaging with transmission electron microscopy 27 and mass
+spectrometry28 methods. These works, however, either involve training with and reconstruction
+of binary images only26,27, or, require extensive training with images closely related to the sample
+under study28. As such, they are difficult to translate to imaging settings with more complex im-
+ages, particularly for imaging without any prior assumptions about the sample. Meanwhile, Zhang
+et al.29 have incorporated a neural network (NN) within the SLADS method (for the SLADS-Net
+method) and shown in numerical experiments that it is sufficient to train the method on only a
+generic image, eschewing any prior knowledge about the sample, to produce high-fidelity image
+with sparse sampling. However, this has not yet been demonstrated in experiment.
+In this work, we report the Fast Autonomous Scanning Toolkit (FAST) that combines the
+SLADS-Net method, a route optimization technique, and efficient and modular hardware controls
+to make on-the-fly sampling and scan path choices for synchrotron-based scanning microscopy.
+This method relies on sample-agnostic training to dynamically measure and reconstruct a com-
+plicated (non-binary) sample, distinguishing this toolkit from existing SLADS-based workflows.
+Moreover, its computational cost is negligible compared to the acquisition time even when run on
+a low-power edge computing device placed at a synchrotron beamline, which presents a signifi-
+cant advantage over more generic autonomous experimentation techniques. These characteristics
+enable the application of our workflow in the high-precision nanoscale scanning x-ray microscopy
+instrument present at the hard x-ray nanoprobe beamline at the Advanced Photon Source.
+We validate the FAST scheme through real time demonstration at the hard x-ray nanoprobe
+beamline at the APS30. A few-layer exfoliated two-dimensional WSe2 thin film was chosen as a
+representative example; the preparation process for the thin film often leaves microscopic air bub-
+bles trapped underneath the thin film, deforming the 2D material. We show that an adaptive scan
+of < 25% of the sample is sufficient to produce a high-fidelity reconstruction that identifies all the
+bubbles within the field of view, and even to acquire quantitative information about the film curva-
+ture induced by these bubbles. The scheme quickly identifies the deformed part of the 2D material
+and focuses its attention there, while ignoring regions of the film that are flat and homogeneous.
+Film curvature reconstructed from the adaptive scan (< 25% coverage) is consistent with that re-
+6
+
+FIG. 1. (Artist’s representation) The APS synchrotron produces a coherent x-ray beam that is focused using
+a zone plate setup. It strikes a WSe2 film (green) exfoliated onto a Si substrate (blue), which generates
+diffraction patterns that are collected by a two-dimensional detector. Above the bubbles, the lattice of the
+film rotates, shifting the diffracted intensities away from its nominal positions. The beam position as well
+as the detector acquisition are autonomously controlled by the FAST AI-based workflow.
+constructed from full-grid scan (100% coverage). Given these characteristics, the FAST scheme
+can be directly applied in other scanning techniques and instruments at the APS and elsewhere,
+and may underpin the development of many multi-level experimental studies.
+7
+
+II.
+RESULTS
+Figure 1 shows the experimental setup that scans a focused x-ray beam on a sample while ac-
+quiring a two-dimensional diffraction image at each point. The live demonstration was performed
+on a few-layer WSe2 sample with the detector placed along the 008 Bragg peak, with 2θ = 43.1°
+at 10.4 keV. The diffraction patterns were processed on the detector computer (see Methods) to
+generate the integrated intensities for use in the FAST workflow. The final output of the workflow
+is a dark-field image of the WSe2 sample.
+A.
+Self-driving scanning microscopy workflow
+Figure 2A broadly illustrates the FAST workflow for the experiments reported here. To ini-
+tiate the workflow, a low-discrepancy quasi-random selection (generated using the Hammersely
+sequence31) of sample position is measured corresponding to 1% of the total area of interest. The
+integrated intensities of the measurements are transferred to the edge device, an NVIDIA Jetson
+Xavier AGX32 located adjacent to the detector, which used Inverse Distance Weighted (IDW) in-
+terpolation to estimate the dark-field image. The estimated image serves as input for the decision-
+making step whereby the prospective measurement points are identified.
+This self-driving workflow adopts the Supervised Learning Approach for Dynamic Sampling
+using Deep Neural Networks (SLADS-Net) algorithm29 to find the prospective measurement
+points. In effect, the SLADS-Net algorithm uses the current measurements to identify the best
+unmeasured points that, when added to the existing dataset, would have the greatest effect on the
+quality of the reconstructed image. As illustrated in Figure 2B, this is accomplished by, first,
+representing each unmeasured point as a feature vector with elements that depend on the mea-
+surement state in the neighborhood of the point. These feature vectors are used as input for a
+pre-trained multi-layer perceptron. The neural network then predicts the expected reduction in
+distortion (ERD), a metric (loosely speaking) for the expected improvement in the reconstruction
+quality obtained from measuring this unmeasured point, individually for each unmeasured point.
+The original SLADS-Net algorithm simply uses the unmeasured point with the highest ERD for
+8
+
+COMPUTING CANDIDATES
+OPTIMIZING PATH
+INITIAL ESTIMATION
+INITIAL MEASUREMENTS
+AI @ EDGE
+FINAL RESULT
+NEW ESTIMATION
+NEW MEASUREMENT
+𝒓
+Generate
+Features
+A
+B
+FIG. 2. The FAST workflow: (A) A set of random initial measurements are transferred to the edge device
+which sequentially generates an initial sample estimate, computes the candidate points to be measured
+next, and calculates the travel path for the measurement. The new measurements are combined with the
+existing measurements and used to calculate a new estimate, and the process is repeated until it achieves
+a completion criterion. (B) The candidate computation starts by examining the local neighborhood (with
+radius r) of each unmeasured point P, with the highlighted points indicating points already measured, to
+generate a 6-dimensional feature vector. The feature vector is transformed to a 50-dimensional vector using
+the Radial Basis Function (RBF) kernel and used as input to a multi-layer NN. The NN then predicts the
+expected improvement in the image (ERD) from measuring the point P. A set of unmeasured pixels with
+the highest ERD are selected as candidates for the next measurement.
+9
+
+5 hidden layers
+inputs
+50 nodes per layer
+V1
+output
+ERD
+02
+V50nVIDIAthe next measurement, and repeats this procedure pointwise. In practice, if the measurement pro-
+cedure and the motor movements are fast, then the ERD calculation also has to be commensurately
+fast to reduce the dead-time in the experiment. In this work, we mitigate this requirement by in-
+stead selecting a batch of points that have the highest ERD, sorted in descending order—we found
+that a batch of 50 points adequately minimized the experimental dead-time while still ensuring
+that the overall measurement was adequately sparse.
+The coordinates of these 50 points are passed on to a route optimization algorithm, based
+on Google’s OR-Tools33, to generate the shortest path for the motors to visit all of the them.
+This path is appended to the look-up table in the EPICS34 scan record, which then kicks off the
+data acquisition. Henceforth, the scan is automatically paused after every 50 points, raising a
+flag which event triggers a callback function on the edge device. There, a new estimated dark
+field image of the sample is generated, and the coordinates for the next 50 prospective points are
+computed. The scan is resumed after the EPICS scan record receives the new coordinates for the
+optimized scanning path. The actual scanning of the focused x-ray beam is achieved by moving
+two piezoelectric linear translation motors in step mode. The detector exposure time is set to 0.5 s
+and comes with an overhead of 0.2 s.
+For the 200×40 pixels object described in Section II C, the workflow required ≈0.15 s to
+compute the new positions, ≈42 s to scan the set of 50 positions, and a total of ≈0.37 s to process
+the diffraction patterns and communicate the measurements. This represents an overhead of ⪅ 2%.
+The workflow is currently entirely CPU-bound, relying on the on-board 8-core ARM CPUs, and
+does not take advantage of the GPU bundled into the NVIDIA AGX device. In the future, we
+expect to perform the computation in a parallelized and asynchronous fashion, which would further
+reduce this overhead. These timing results showcase the rapid data-driven decision-making ability
+that is characteristic of the FAST workflow.
+We also note that, for all the results reported in this work, the underlying NN was trained on a
+single generic image with no relation to microscopy. For details about the SLADS-Net algorithm
+and the sample-agnostic training procedure, the reader is referred to the Methods section.
+10
+
+B.
+Numerical demonstration for scanning dark-field microscopy
+Reconstructions (10% scan )
+True
+Measured points
+Raster grid (RG)
+LD random (LDR)
+FAST
+A
+B
+C
+D
+E
+G
+H
+I
+F
+FIG. 3. Numerical comparison of sampling methods: (A) shows the ground truth with the color scale
+representing the normalized intensity, (B-D) show respectively the RG, LDR, and FAST reconstructions at
+10% scan coverage, and (G-I) show the actual scan points that produce these reconstructions. (E-F) show
+the evolution of the NRMSE (lower is better) and SSIM (higher is better) as a function of the scan coverage.
+The FAST reconstruction stabilizes at 27% coverage while the other techniques take significantly longer to
+reach the same quality.
+We first validated the performance of the proposed workflow through a numerical experiment
+on a set of pre-acquired dark-field microscopy data. Here, we compared the FAST sampling with
+three static sampling techniques:
+1. Raster grid (RG) For a test sampling percentage, we generated a equally spaced raster grid
+that provides a uniform coverage of the sample.
+2. Uniform random (UR) sampling The measurement pixels were drawn from a uniform
+11
+
+LDR
+UR
+RG
+FAST150 μm.
+·
+..
+:random distribution.
+3. Low-discrepancy (LDR) random sampling For each measurement percentage, we gener-
+ated a low-discrepancy sampling grid using the quasi-random Hammersly sequence.
+The test dataset is a dark field image of size 600×400 pixels which represents 240,000 possible
+measurement positions. This covers a physical area of 900 µm×600 µm and encloses multiple
+flakes of WSe2 with various thicknesses, with the thicker regions associated with regions of higher
+brightness in the image (Figure 3). At this spatial resolution, only medium and large sized bubbles
+(with diameter > 2 um) can be observed. As explained previously, the bubbles deform the surface
+and shift the Bragg peak of the 2D materials away from their theoretical (flat region) positions,
+resulting in regions of darker contrast. Finally, the image also contains flake-free regions that have
+zero integrated intensities.
+For this comparison, we first initialized the FAST sampling with a 1% measurement coverage
+(as described above), then successively measured 50 additional points at iteration. For each FAST
+measurement, we also generate RG, UR, and LDR measurement masks with the same number
+of scan points. In this fashion, we generate a sequence of sampling masks and the associated
+reconstructions until we achieve 100% sampling.
+We present the numerical results in Figure 3, where we show a comparison of the various meth-
+ods at 10% sampling. Note that while the proposed method internally uses the fast IDW algorithm
+for the inpainting, the final images presented here are calculated using the higher quality bihar-
+monic inpainting technique35. The uniform random scheme performs worse than the LD-random
+and raster grid schemes and is not shown in the figure. In Figure 3A-D, we can see that the FAST
+sampling is able to reproduce with high fidelity the flake boundaries, the bubbles, and the regions
+of transition between the varying levels of thicknesses. In contrast, the LDR and raster schemes
+produce much lower quality reconstructions of these features. Figure 3E shows an evolution of the
+normalized root mean squared error (NRMSE) and fig. 3F the structural similarity metric (SSIM)
+(which measures multiscale perceptual similarity) for the different sampling techniques. It is ev-
+ident that FAST produces high quality reconstructions at much lower measurement percentages
+12
+
+than the examined static sampling techniques. We note that the result could be further improved
+in the future by using a more sophisticated inpainting technique within the FAST method. To un-
+derstand how FAST outperforms the other methods under the same sampling condition, we show
+the actual measured positions of the various schemes at 10% coverage (Figure 3G-I). FAST pref-
+erentially samples the regions with significant heterogeneity over the homogeneous regions. This
+is particularly useful for sparse samples, where the time spent sampling from empty regions adds
+little additional information.
+C.
+Experimental demonstration
+Full scan image
+FAST reconstructions
+Measured points
+A
+C
+E
+G
+B
+D
+F
+Measurements between 15-20%
+H
+FIG. 4. Evolution of the FAST scan: (A, C, E) show the reconstruction at 5%, 15%, and 20% reconstructions
+respectively, (B, D, F) show the corresponding actual measurement points. (G) shows the image obtained
+through a full-grid pointwise scan. The color scale in (A-G) show the normalized intensities. (H) shows
+only the points sampled between 15% and 20% coverage.
+We next demonstrate the application of the FAST workflow in a live experiment at a syn-
+chrotron beamline. A video showing the sampling, recorded live during the actual experiment, is
+available here36. Other than starting the workflow scripts at the beginning, the entire experiment
+was unmanned and fully automated. In order to measure the deformed WSe2 flakes in details, a
+higher spatial resolution of 100 nm was chosen. This limits the field of view to 20 µm×4 µm for
+13
+
+5%:15%+++20%2 μma scan point density of 200×40 points.
+In Figure 4 we show the reconstructed dark field image (subplots A,C,E) and the measurement
+points (subplots B,D,F) from 5 % to 20 % coverage and compare them to that obtained from raster
+scanning the sample with 100% coverage(subplot G). We see that the FAST method identifies
+some of the regions of hetereogeneity — the edges of the bubbles — and starts to preferentially
+sample these regions within 5 % coverage of the sample. At 15 % coverage, these regions are
+extensively sampled. The reconstruction does not change significantly between 15 % to 20 %,
+indicating that the reconstruction has stabilized. Moreover, the 20 % reconstruction also contains
+sharp and accurate reproductions of all the major features present in the full scan image.
+A point of interest is that the partially scanned bubble at the bottom right corners of Figure 4E-
+G shows up only in the 20% scan, and not in the 15% scan. To explain this, we note that the
+5% scan, and therefore the initial 1% random sampling, does not contain any measurements in
+the neighborhood of this bubble. The FAST scheme favors exploitation of regions it knows to
+be heterogeneous over exploration of this fully unknown region, and therefore only explores this
+region much later in the measurement process (Figure 4H). This is, in fact, an instance of the
+general exploration-exploitation tradeoff that exists in all Bayesian search procedures37. Potential
+mitigation steps could be to sample more initially (say 5% random points), or to deliberately
+introduce diversity into each batch of measurement points.
+So far we have reduced the diffraction image measured at each point to one single quantity
+(integrated intensity) in order to guide the automated experiment. These images often need to
+be reprocessed after the experiment to extract additional physically relevant results. Notably, the
+intensity distribution in the diffraction patterns contains information about the strain as well as
+the rotation of the crystal lattice, and in this case, the curvature of the 2D materials due to the
+bubbles underneath. A simple center of mass calculation in the X direction (CoMx) would yield
+the magnitude of the film curved in the XZ plane. The curvature (deviation of the CoMx from its
+nominal value) is the smallest around the center of the bubble and the largest at the edge. It also
+changes sign going from the left side to the right side. Center of mass calculation in the Y direction
+yields the magnitude of the film curved in the YZ plane. The results look slightly different from
+14
+
+the CoMx calculations due to the way the shifted Bragg peak intersects with the Ewald’s sphere.
+Figure 5A and B shows respectively the CoMx and CoMy obtained from raster scan with 100%
+coverage on the area of interest. The unit is the number of pixel shift, relative to the center of the
+nominal diffraction pattern. Figure 5C and B shows respectively the CoMx and CoMy obtained
+with FAST. The curvature information of the film were faithfully reproduced despite scanning just
+20% of the entire area. For more information on the reconstruction of the CoM maps, he reader is
+referred to the Methods section.
+CoMx
+CoMy
+FAST
+Full
+FIG. 5. Comparison of the per measured point center of mass of the diffraction patterns between the FAST
+scan at 20% coverage and full-grid scan. Subplots (A) and (B) show the inpainted COMx and COMy,
+respectively, for the full-grid raster FAST scan, and subplots (C) and (D) for the FAST scan.
+III.
+DISCUSSION
+In this work, we have showcased the FAST workflow that combines a sparse sampling algo-
+rithm with route planning to drive a scanning diffraction microscopy experiment at a synchrotron
+beamline. In addition to being an effective alternative to a full pointwise scan to acquire a dark-
+field image of the sample, FAST also produces accurate quantitative measurements of its phys-
+ical properties. For our live demonstration of a 200 points×40 points with a measurement time
+of 0.5 s/point, the FAST decision-making time was negligible, leading to an overall saving of
+≈80 min (about ≈65 %) of the experiment time. This saving was facilitated by our choice to ac-
+quire a batch of 50 measurements between the selection of the prospective measurement points.
+This ensured that the communication time stayed negligible with no noticeable loss in the quality
+of points acquired when compared to a pointwise candidate selection scheme (see Supplemental
+15
+
+CDB
+2 μmMaterials, Fig. S1).
+The generalizability of the FAST method comes from the fact that the key NN-based compo-
+nent of this workflow is trained on just the standard cameraman image38, not on close analogues of
+a sample of interest. While this generalizability results in a slight loss of performance of the tech-
+nique , it still shows excellent sparsity performance for cases tested in previous research29,39 and
+in the current work. This has the benefit that we do not need a priori knowledge of the sample. As
+such, while general pre-training would be difficult to satisfy for new and expensive experiments,
+the FAST approach can be used directly. Furthermore, the batch prediction and route optimization
+approach we implement can also be directly applied in any application of choice. Moreover, the
+experimental application of our work uses an extensible edge device and the widely used EPICS
+platform for hardware control, both of which can be incorporated into any instrument even with
+the SLADS-Net replaced by any other sampling strategies. For example, we could just replace the
+dark-field detection procedure described here with a fluorescence counting setup and use exactly
+the FAST scheme for a fluorescence-based imaging of the sample. Alternatively, since all the in-
+struments at the APS rely on EPICS controls, one can perform transmission, surface scattering,
+or any other 2D scanning experiment in any applicable beamline with only minor changes to the
+FAST routine.
+The computations in the current workflow have a time complexity of O(2N logN +kM logN),
+where N is the number of measured points, M the number of unmeasured points, and k the num-
+ber of nearest neighboring measurements (k = 10 in our case) that we use for the feature vector
+calculations. Here, the first term accounts for the creation of the nearest neighbor K-d tree and
+the second term for the nearest neighbor calculation. The remainder of the algorithm has a linear
+time complexity and could be performed in parallel for the unmeasured points. We expect that it is
+possible to reduce this complexity using an approximate nearest neighbor search method instead
+of the K-d tree approach. As such, a GPU-based implementation that takes advantage of the par-
+allelization and the approximation would likely significantly reduce the computation time. This
+stands in stark contrast with the time complexity of O
+�
+N3�
+(for N measured points) for Gaussian
+Processes, a similarly training-free method that is widely used for autonomous experimentation.
+16
+
+For an illustrative example, Vasudevan et al20 report a GP-based scanning microscopy experiment
+where the calculation of each set of measurement candidates takes ≈6 s on an NVIDIA DGX-2
+GPU for a 50×50 image; our workflow performs an equivalent calculation for a larger 200×40
+image within ≈1.5 s in a low-power CPU. We note, however, that GPs remain a very powerful and
+generalizable approach with a bevy of applications beyond only scanning microscopy.
+We caution that our workflow suffers from three important challenges. First, it depends heav-
+ily on the initial 1% random sampling to discover regions of heterogeneity. If an isolated feature
+present in an otherwise homogeneous region is not partially sampled during this random sam-
+pling step, then such a feature can be missed until much later in the scanning experiment (see
+Figure 4H). A related second limitation is that this method produces sub-optimal reconstructions
+if the sample is sufficiently heterogeneous that the data in each pixel changes significantly from
+pixel to pixel throughout the image (Supplemental Material in Hujsak et al27). The third limi-
+tation, more practical in nature, is that the scan paths require significant motor movement, often
+including a retracing over points already measured. As such, there could exist scenarios in which
+the time required for the motor movement eclipses the time required for a single measurement.
+We expect to address these limitations by explicitly including a measurement-density-based term
+39 or a movement-time-based term in the candidate selection procedure40, or by using a line-based
+sampling technique41.
+Despite these challenges, we believe that the proposed FAST technique has great potential. It
+is an ideal tool for use cases with limited sampling or dosage budgets. It can be used to isolate
+regions of interest in sparse settings, to prepare for pointwise scanning in these regions. More
+generally, it can be used to guide any scanning microscopy experiment where we do not need
+full pointwise information. In the future, we expect to extend this method for 3D imaging, fly
+scans, ptychography, and other imaging applications. We expect that these developments will
+significantly enhance the efficacy of scanning microscopy experiments, bolstering their use for the
+study of dynamic physical phenomena.
+17
+
+IV.
+METHODS
+A.
+The SLADS-Net algorithm
+The SLADS-Net algorithm29 used within the FAST workflow is an adaptation of the Super-
+vised Learning Approach for Dynamic Sampling (SLADS) algorithm originally developed by
+Godaliyadda et al25, and the algorithms differ only in their training approaches ( Section IV B). To
+explain the SLADS algorithm, we first denote the object we want to measure as A ∈ RN, where N
+is the total number of pixels in the image. Further, we can denote the pixel at location 1 ≤ s ≤ N
+as as so that a measurement at the location s extracts the value as; each measurement is thus
+characterized by the pair (s, as). After k measurements, then, we get the k×2 measurement vector
+Y k =
+�
+�������
+s1
+as1
+s2
+as2
+...
+sk
+ask
+�
+�������
+(1)
+Using these k measurements, then, we can reconstruct (e.g. via interpolation) an estimate ˆAkof the
+true object A. The difference between A and ˆAk is denoted as the distortion D(A, ˆAk) and can be
+calculated using any chosen metric. In the current work, we define D(A, ˆAk) to be the L2 norm:
+D(A, ˆAk) = ||A− ˆAk||2.
+Given the measurement Y k and the reconstruction ˆAk, a new measurement at any location s will
+presumably reduce the distortion in the reconstruction. We can denote this reduction in distortion
+(RD) as
+Rk,s = D(A, ˆAk)−D(A, ˆAk,s)
+(2)
+where ˆAk,s is the reconstruction that includes the newly added measurement at s. The goal of
+the SLADS algorithm is then to identify the pixel location that would maximize this reduction in
+distortion:
+sk+1 = argmax
+s
+Rk,s
+(3)
+18
+
+Of course, since we cannot know the value of the measurement as or the ground truth A, SLADS
+bases its selection on the conditional expectation of reduction in distortion (ERD), which is defined
+as:
+Rk,s = E
+�
+Rk,s��Y k�
+so that
+sk+1 = argmax
+s
+Rk,s.
+(4)
+The algorithm assumes that we can compute the ERD at s based on just the measurement state Yk
+as
+Rk,s = g(vk,s)
+(5)
+where vk,s is a location-dependent feature vector calculated using the measurement state Yk. The
+goal of the SLADS training procedure is to estimate the function g.
+B.
+Training
+The training procedure for the SLADS/SLADS-Net algorithm is a supervised procedure in
+which we generate a large number of (vk,s,Rk,s) pairs and use these to estimate g. Note that this
+is a pixelwise computation that is performed independently for each measurement location s; for
+each measurement s we have to calculate a reconstruction ˆAk,s before we can calculate the RD Rk,s.
+To make this computationally tractable, the Godaliyadda et al25 use approximations that ensure
+that the RD of each pixel only depends on its local neighborhood. Correspondingly, instead of
+working with the full measurement state Y k, the training procedure uses carefully designed feature
+vectors that capture the local neighborhood of the pixel at location s. As shown in Figure 2B, the
+feature vector for the pixel P consists of six features: (i) ∇x and ∇y are the spatial gradients at
+P, (ii) σ1,r and σ2,r measure the deviation of the estimated value for P from the nearby measured
+values (highlighted in red), and (iii) L (which is the distance of P from the closest measured point)
+and ρr measure the density of measurements around P.
+The original SLADS algorithm assumes that this feature vector is linearly related to the RD,
+and the training therefore is a linear regression procedure. The SLADS-Net adaptation first uses
+an radial basis function (RBF) kernelization to transform the 6-dimensional feature vector to a
+19
+
+50-dimensional vector, then replaces the linear predictor with a nonlinear fully-connected neural
+network that contains 5 hidden layers with 50 nodes each.
+In this work, we train the SLADS-Net neural network on only the standard cameraman image,
+without using any a priori information about the sample. For the training, we generate a mea-
+surement state Y k by randomly choosing a fixed number number of measurement locations, then
+calculate the feature vector vk,s and the RD Rk,s for each unmeasured pixel. We generate such
+sets of training pairs for 10 different sample coverage percentages between 1% and 80%. This
+overall comprises our training dataset. We use this data to train the neural network for 100 epochs
+using the Adam optimizer with the learning rate 0.001. We use this trained model for all the simu-
+lated and experimental measurements. We provide an example of a training measurement set—the
+measured points, the interpolated reconstruction, and the corresponding RD for the unmeasured
+points—in the supplemental materials (Fig. S2)
+C.
+Experimental measurements
+At each point of the measurement, a tight region of interest (RoI) around the expected position
+of the thin film Bragg peak was extracted from the corresponding diffraction image. Integrated
+intensities of the RoI were used to guide the NN prediction. For the flat region, the integrated
+intensity is high, showing up as brighter contrast on the dark field image. For the deformed region,
+the integrated intensity is low (darker contrast on the dark field image) as the illuminated film
+diffraction partially exits the selected RoI (see Supplemental Materials, Fig. S3).
+For the FAST experiment, the predicted ERD and the dark-field reconstruction served as visual
+guides to inform when to stop the experiment.. During the experiment, we noted that the ERD and
+the reconstruction had stabilized by ≈20 % scan coverage, but we let the experiment run to ≈35 %
+coverage to ensure that this behavior persisted (see Supplemental Materials, Fig. S4). While we
+used this visual criterion for our exploratory experiment, it is straightforward to design a numerical
+stopping criterion based on the absolute or relative convergence of the ERD, or on the per-iteration
+change in the reconstructed image.
+20
+
+DATA AND CODE AVAILABILITY
+The data and code will be made available at https://github.com/saugatkandel/fast_
+smart_scanning
+ACKNOWLEDGMENTS
+Work performed at the Center for Nanoscale Materials and Advanced Photon Source, both
+U.S. Department of Energy Office of Science User Facilities, was supported by the U.S. DOE,
+Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We also acknowl-
+edge support from Argonne LDRD 2021-0090 – AutoPtycho: Autonomous, Sparse-sampled Pty-
+chographic Imaging. We gratefully acknowledge the computing resources provided on Bebop, a
+high-performance computing cluster operated by the Laboratory Computing Resource Center at
+Argonne National Laboratory. X.L. acknowledges support from the National Science Foundation
+CBET Program under the award no. 2025214.
+COMPETING INTERESTS
+The authors declare that they have no competing financial interests.
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+25
+
+Supplemental material
+January 16, 2023
+1
+arXiv:2301.05286v1  [physics.app-ph]  12 Jan 2023
+
+A
+B
+Figure S1: Comparison of the FAST reconstructions for scan batch size of 1
+(FAST-1) and 50 (FAST-50) as a function of the scan coverage for the numerical
+simulation described in Section II.B. We observe that FAST-1 initially performs
+better, with lower NRMSE and higher SSIM, than FAST-50, but this advantage
+erodes quickly. We ended the FAST-1 experiment at ≈8.2 % sampling.due due to
+simulation time limitations.
+2
+
+A
+B
+C
+Figure S2: Example of training data. (A) shows a set of randomly selected mea-
+surement points. (B)shows the reconstruction calculated by interpolating these
+measurements. (c) shows the ERDs calculated for the unmeasured points, with the
+ERD highest at regions of hetereogeneity. The location of the measured points,
+the measured values, and the reconstruction are used to generate feature vectors
+for the training, and the ERDs are used as the training labels.
+3
+
+CoMx
+CoMy
+Figure S3: Example of ROI selection and change in diffraction patterns around
+the bubbles. (A) and (B) respectively show the CoMx and CoMy calcualted from
+the FAST scan with 20% covergae, as discussed in Section II.C. The diffraction
+patterns for the points marked with the ×, �, and + are shown in the bottom row.
+The × point is in a region without a bubble and has the diffraction pattern at the
+Bragg angle. The points marked with � and + are located at the top and bottom
+edges of the bubble, and therefore show additional anomalous diffraction spots.
+The dashed square boxes in the diffraction pattern figures indicate the ROI used
+for the dark-field image reconstructions (shown in Figure 4 in the main paper).
+The CoM calculations use the regions outside the dashed square boxes as the RoI.
+4
+
+AB
+XOX0+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+ERD
+1e6
+0
+10
+20
+30
+40
+50
+60
+Scan iteration
+0
+5
+10
+15
+20
+25
+30
+35
+Scan coverage (%)
+20% coverage
+Figure S4: Evolution in the ERD for the experimental demonstration. The ERD
+initially decreases rapidly, during which point the each batch of 50 points signifi-
+cantly improves the sample reconstruction. At per-iteration change in the ERD is
+much smaller at 20% coverage.
+5
+

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+arXiv:2301.02162v1  [stat.ME]  5 Jan 2023
+Improve Efficiency of Doubly Robust Estimator when
+Propensity Score is Misspecified
+Liangbo Lv∗ and Molei Liu†
+Abstract
+Doubly robust (DR) estimation is a crucial technique in causal inference and miss-
+ing data problems. We propose a novel Propensity score Augmentved Doubly robust
+(PAD) estimator to enhance the commonly used DR estimator for average treatment
+effect on the treated (ATT), or equivalently, the mean of the outcome under covariate
+shift. Our proposed estimator attains a lower asymptotic variance than the conven-
+tional DR estimator when the propensity score (PS) model is misspecified and the
+outcome regression (OR) model is correct while maintaining the double robustness
+property that it is valid when either the PS or OR model is correct. These are realized
+by introducing some properly calibrated adjustment covariates to linearly augment the
+PS model and solving a restricted weighted least square (RWLS) problem to minimize
+the variance of the augmented estimator. Both the asymptotic analysis and simula-
+tion studies demonstrate that PAD can significantly reduce the estimation variance
+compared to the standard DR estimator when the PS model is wrong and the OR is
+correct, and maintain close performance to DR when the PS model is correct. We
+further applied our method to study the effects of eligibility for 401(k) plan on the
+improvement of net total financial assets using data from the Survey of Income and
+Program Participation of 1991.
+Keywords: Causal inference; Covariate shift correction; Propensity score; Outcome regres-
+sion; Double robustness; Intrinsic efficiency.
+∗Liangbo Lv is an undergraduate student from the School of Statistics, Renmin University of China.
+†Molei Liu is an assistant professor at Columbia University Mailman School of Public Health.
+1
+
+1
+Introduction
+1.1
+Background
+Doubly robust (DR) estimation has attracted extensive interest in the literature on semipara-
+metric theory and causal inference and is frequently used in biomedical science, economics,
+and policy science studies. It incorporates two nuisance models, a propensity score (PS)
+model, and an outcome regression (OR) model to characterize distributions of the expo-
+sure and outcome against the adjustment covariates respectively, and draws valid inferences
+when either one of them is correctly specified. It has been well-established that when both
+the PS and OR models are correct, the DR estimator is semiparametric efficient and its
+asymptotic variance does not really depend on the estimating equations for the nuisance
+models (Tsiatis, 2006, e.g.). Nevertheless, there still remains an intriguing question on how
+to improve the asymptotic efficiency of the DR estimator when one nuisance model is mis-
+specified. For the scenario with correct PS and wrong OR models, there is a track of work
+(Cao et al., 2009; Tan, 2010, e.g.) proposing the so-called intrinsic efficient estimator that
+will be reviewed in Section 1.3. This type of estimator preserves the double robustness prop-
+erty and achieves improved efficiency over the standard DR estimator when the PS model
+is correct and the OR is wrong. Interestingly, we notice that the dual problem of this, i.e.,
+improving the (intrinsic) efficiency of the DR estimator under wrong PS and correct OR,
+is supposed to be equally important but has not been handled yet due to certain technical
+reasons that will be discussed later. Aimed in this paper, filling this methodological blank
+can effectively complement the existing tools for DR and semiparametric inference.
+1.2
+Problem Setup
+To make our idea easier to understand, we focus on a specific missing data problem: trans-
+fer estimation of the outcome’s mean in the presence of covariate shift (Huang et al., 2007,
+e.g.). This is also equivalent to estimating the average treatment effect on the treated (ATT)
+(Hahn, 2004, e.g.) in the context of causal inference and matching-adjusted indirect com-
+parison frequently conducted in biomedical studies (Signorovitch et al., 2010). Our method
+could be generalized to other settings such as estimating the average treatment effect (ATE)
+and transfer learning of regression models (Liu et al., 2020).
+Suppose there are n labeled samples with observed outcome Y and covariates X ∈
+Rd, and N unlabeled samples only observed on X. Let ∆ = 1 indicate that the sample
+is labeled and ∆ = 0 otherwise.
+The labeled observations (Yi, Xi) are collected from a
+source population S with ∆i = 1 for i = 1, 2, . . . , n. Assume (Yi, Xi) ∼ pS(x)q(y|x) for
+2
+
+i = 1, 2, . . . , n where pS(x) and q(y|x) represent the density of X on S and the conditional
+density of Y given X = x respectively. Meanwhile, there are unlabeled samples from a target
+population T indicated by ∆i = 0 and only observed on covariates Xi for i = n+1, . . . , N+n.
+Assume that on T , (Yi, Xi) ∼ pT (x)q(y|x) with pT (x) representing the density of X on
+T and the distribution of Y | X remaining to be the same as that on S. Our goal is to
+estimate µ0 = ET Y , the marginal mean of Y on T . In the absence of observed Y on the
+target samples, two simple strategies to estimate µ0 are introduced below.
+(PS) Define the propensity score (PS) or density ratio between the two populations as
+r0(x) = pT (x)/pS(x). Estimate r0(x) with some �r(x) and average the observed Yi
+weighted by �r(Xi) over i = 1, 2, . . . , n from S.
+(OR) Define the outcome regression (OR) or imputation model for Y as m0(x) = E[Y | X =
+x]. Estimate m0(x) with some �m(x) obtained using the labeled samples and average
+�m(Xi) over i = n + 1, . . . , n + N from T .
+Both the PS and OR strategies are built upon the assumption that the distribution of Y | X
+is the same between S and T so the knowledge of Y on S is transferable to T . This is in the
+same spirit as the no unmeasured confounding assumption in the context of causal inference.
+1.3
+Related literature
+Our work is based on the doubly robust (DR) inference framework that has been fre-
+quently studied and applied in the past years (Robins et al., 1994; Bang and Robins, 2005;
+Kang and Schafer, 2007; Tan, 2010; Vermeulen and Vansteelandt, 2015, e.g.). It combines
+the PS and OR models introduced in Section 1.2 to construct an estimator that is valid
+when at least one of the two nuisance models are correct and, thus, regarded as a more ro-
+bust statistical inference procedure than the simple PS and OR strategies. Early work in DR
+inference (Bang and Robins, 2005; Kang and Schafer, 2007, e.g.) mainly used working low-
+dimensional parametric regression to construct the PS and OR models. Recent progress has
+been made to accommodate the use of high-dimensional regression or complex machine learn-
+ing methods in estimating the nuisance models (Chernozhukov et al., 2018; Tan, 2020, e.g.),
+which is less prone to model misspecification. We focus the scope of this paper on the low-
+dimensional parametric setting that is technically less involved but more user-friendly and
+less sensitive to over-fitting in practice. It is also possible and valuable to generalize our work
+to the settings of high-dimensional parametric (Tan, 2020; Dukes and Vansteelandt, 2020,
+e.g.) or semi-non-parametric (Liu et al., 2020) nuisance models, in which model misspecifi-
+cation is still an important concern.
+3
+
+There has risen great interest in studying and improving the asymptotic efficiency of the
+DR estimator. One track of literature studied the local efficiency of the DR estimator, i.e., if
+it is semiparametric efficient when both the PS and OR models are known or correctly spec-
+ified. While it was shown that the standard DR estimator for the ATE (Robins et al., 1994)
+achieves such local efficiency (Hahn, 1998; Tsiatis, 2006). This result cannot be directly ap-
+plied to the ATT estimator because unlike ATE, the PS model of ATT is informative (or
+non-ancillary) (Hahn, 1998, 2004). Shu and Tan (2018) further studied this subtle issue and
+proposed locally efficient DR estimators for ATT based on its influence function.
+Meanwhile, another track of literature focuses on improving the efficiency of the DR es-
+timator in the presence of correct PS and potentially wrong OR models and, thus, is more
+relevant to our work that also aims at automatic variance reduction under model misspec-
+ification.
+A class of intrinsic efficient DR estimator has been proposed for the efficient
+estimation of ATE (Cao et al., 2009; Tan, 2010), ATT (Shu and Tan, 2018), casual regres-
+sion model (Rotnitzky et al., 2012), longitudinal data (Han, 2016), individual treatment rule
+(Pan and Zhao, 2021), etc. This type of estimator is (i) valid when either nuisance model is
+correct; (ii) equivalent with the standard DR estimator when both models are correct; and
+(iii) of the minimum variance under correct PS and wrong OR, among all the DR estimators
+with the same parametric specification of the OR model, and, consequently, more efficient
+than the standard DR estimator. In addition, it was shown that including more prognostic
+covariates or auxiliary basis in the PS model can always help to reduce the variance of the
+ATE estimator (Hahn, 2004; Tsiatis, 2006). Motivated by this, Cheng et al. (2020) proposed
+a double-index PS estimator for ATE that smooths the treatment over the parametric PS
+and OR models to achieve the DR property as well as variance reduction under correct PS
+and wrong OR. Nevertheless, such a strategy may also incur over-fitting issues and cause
+poor performance in finite or small sample studies (Gronsbell et al., 2022).
+Although the correct PS and wrong OR setting has been frequently studied, there is still a
+paucity of solutions to its dual problem, i.e., enhancing the DR estimator under the wrong PS
+and correct OR. Some early work like Kang and Schafer (2007) and Cao et al. (2009) argued
+that the simple OR strategy is an ideal choice when one knows the PS model is wrong since
+it is free of PS weighting that may decrease the effective sample size. However, since there
+are no perfect ways to examine model correctness without any additional assumptions, this
+strategy can never be as robust as the DR estimator to misspecification of the OR model.
+We also notice a large body of work in statistical learning and causal inference that aims
+at leveraging some auxiliary data or information to boost the asymptotic efficiency of certain
+estimators using the idea of augmentation. For example, Kawakita and Kanamori (2013),
+Chakrabortty et al. (2018) and Azriel et al. (2021) proposed different semi-supervised learn-
+4
+
+ing methods that improve estimation efficiency of the linear model leveraging large unlabeled
+data drawn from the same distribution as the labeled samples. Methods like Chen and Chen (2000)
+and Yang and Ding (2019) utilized external data with error-prone outcomes or covariates to
+construct control variate for variance reduction. These methods, as well as other examples,
+rely on some auxiliary data to construct estimators that always converge to zero and are
+asymptotically correlated with the target estimator. These zero estimators are then used
+to augment the target estimator properly for variance reduction. Our work also adapts the
+high-level idea of augmentation. But different from these methods, ours does not leverage
+any auxiliary samples or knowledge and additionally cares about the need of prioritizing va-
+lidity (double robustness) over statistical power. Consequently, the asymptotic behavior of
+our augmented estimator actually varies according to the correctness of the nuisance models
+and is more technically involved in to study.
+1.4
+Our contribution
+To estimate µ0 introduced in Section 1.2 efficiently, we propose a novel Propensity score
+Augmented Doubly robust (PAD) estimation method that enhances the standard DR es-
+timator of µ0 by linearly augmenting the PS model with some functions of X. Both the
+augmentation functions and their linear coefficients are wisely and carefully constructed such
+that the augmentation term always reduces the variance of the DR estimator if the PS is
+wrong and the OR is correct while it automatically converges to zero if the PS is correct,
+in order to avoid bias and ensure double robustness. Also, when both models are correct,
+our PAD estimator becomes asymptotically equivalent to the standard DR estimator. To
+our best knowledge, the proposed estimator is the first one to simultaneously have the DR
+property and a smaller variance than the standard DR estimator under wrong PS and correct
+OR models. Thus, our work serves as an important complement to existing DR inference
+approaches, especially to the intrinsically efficient DR estimators proposed to work for the
+setting with correct PS and wrong OR (Cao et al., 2009; Tan, 2010, e.g.).
+2
+Method
+2.1
+Doubly robust estimator
+As a prerequisite of our proposal, we first introduce the standard DR estimator for µ0 un-
+der the setup described in Section 1.2, which has been studied for years (Hahn, 1998, 2004;
+Shu and Tan, 2018, e.g.). Following a common strategy (Bang and Robins, 2005; Shu and Tan, 2018;
+Liu et al., 2020, e.g.), we form the PS and OR models as r(x) = exp(xTγ) and m(x) =
+5
+
+g(xTα) where γ and α are model coefficients and g(·) is a known and differentiable link
+function. We say that the PS (or OR) model is correct if there exists γ0 (or α0) such that
+the true r0(x) = exp(xTγ0) (or m0(x) = g(xTα0)). Denote the empirical mean operator on
+S and T as �ES and �ET such that
+�ESa(X, Y ) = n−1
+n
+�
+i=1
+a(Xi, Yi),
+�ET a(X, Y ) = N−1
+n+N
+�
+i=n+1
+a(Xi, Yi)
+for any function a(·). Suppose the two nuisance estimators �γ and �α are obtained respectively
+by solving the estimating equations:
+�ESX exp(X
+Tγ) = �ET X,
+�ESX{Y − g(X
+Tα)} = 0.
+(1)
+The estimating equations for γ in (1) is usually referred as covariate balancing (Imai and Ratkovic, 2014;
+Zhao and Percival, 2017), and those for α correspond to the ordinary least square regression
+when g(a) = a and the logistic regression when Y is binary and g(a) = expit(a) = ea/(1+ea).
+Note that one can use alternative estimation procedures to obtain γ and α, e.g., running a
+logistic regression on ∆ against X to estimate γ, and our proposed method could naturally
+adapt to different choices on this.
+Based on �γ and �α, the PS and OR estimators introduced in Section 1.2 can be specified as
+�µPS = �ESY exp(X
+T�γ) and �µOR = �ET g(X
+T�α) respectively. Then the standard DR estimator
+is constructed by augmenting one of them with another nuisance model:
+�µDR = �ES{Y − g(X
+T�α)} exp(X
+T�γ) + �ET g(X
+T�α).
+(2)
+When the PS model is correct and �γ converges to γ0, �ET g(X
+T�α) − �ESg(X
+T�α) exp(X
+T�γ)
+converges to zero and the remainder term �ESY exp(X
+T�γ) is exactly the PS estimator con-
+verging to µ0. Similarly, when OR is correct, we can show that �ES{Y − g(X
+T�α)} exp(X
+T�γ)
+converges to zero and �ET g(X
+T�α) converges to µ0. Thus �µDR is doubly robust in the sense
+that it is consistent when either the PS or OR model is correctly and consistently estimated.
+2.2
+Expansion of DR estimator under correct OR model
+To help the readers understand our method more intuitively, we now heuristically derive and
+analyze the asymptotic expansion of �µDR when the OR model is correctly specified. Suppose
+that �γ and �α converge to some ¯γ and ¯α defined as the solutions to the population-level
+estimating equations ESX exp(X
+Tγ) = ET X and ESX{Y − g(X
+Tα)} = 0, respectively.
+Let �r(x) = exp(X
+T�γ), ¯r(x) = exp(X
+T¯γ), and S(α) = S(Y, X, α) = X{Y − g(X
+Tα)}.
+Suppose that the OR model is correct, i.e., m0(x) = g(X
+Tα0) and α0 = ¯α, and n1/2(�α −
+6
+
+¯α, �γ − ¯γ) is asymptotically normal with mean zero following the standard M-estimation
+theory (Van der Vaart, 2000). Then we have
+�ES{Y − g(X
+T�α)}{�r(X) − ¯r(X)} = op(n−1/2)
+due to Neyman orthogonality (Neyman, 1959), which, as will be strictly proved in Section
+3, implies that �µDR defined in (2) is asymptotically equivalent with
+�µDR =�ES{Y − g(X
+T ¯α)}¯r(X) + �ET g(X
+T ¯α)
++
+�
+�ES{g(X
+T ¯α) − g(X
+T�α)}¯r(X) + �ET {g(X
+T�α) − g(X
+T ¯α)}
+�
+≈�ES{Y − g(X
+T ¯α)}¯r(X) + �ET g(X
+T ¯α) + L
+T�ESX{Y − g(X
+T ¯α)},
+where L = − ¯H−1 {ESX ˙g(X
+T ¯α)¯r(X) − ET X ˙g(X
+T ¯α)}, ¯H = ESXX
+T ˙g(X
+T ¯α), and ˙g(a) is
+the derivative of g(a). To derive the above result, we use the standard asymptotic expansion
+of �α given by our Lemma B3 in Appendix, and the symbol “≈” indicates that the difference
+between the two lines is up to op(n−1/2) and, thus, asymptotically negligible. So when OR
+is correct, the asymptotic variance of n1/2(�µDR − µ0) is equal to that of n1/2(�µDR − µ0), which
+can be expressed as
+aVar{n1/2(�µDR − µ0)} = ES{¯r(X)}2v(X) + 2L
+TESX¯r(X)v(X) + C,
+(3)
+where v(x) = Var(Y | X) and C is some positive constant free of ¯r(·) and, thus, needs not
+to be considered in the following derivation. Note that when the PS model also is correct,
+i.e., ¯r(·) = r0(·), we further have L = 0.
+Empirically, term L in (3) can be estimated by
+�L = − �H−1 �
+�ESX ˙g(X
+T�α) exp(X
+T�γ) − �ET X ˙g(X
+T�α)
+�
+,
+(4)
+where �H = �ESXX
+T ˙g(X
+T�α). Estimation of v(x) relies on our working assumption on the
+form of Var(Y | X). For example, one may assume Y = m0(X) + ǫ where ǫ ∼ N(0, σ2)
+so v(x) is invariant of x and can be simply imputed with the moment estimator of σ2.
+Also, for the common Poisson model Y ∼ Poisson{exp(X
+Tα0)} and logistic model Y ∼
+Bernoulli{expit(X
+Tα0)}, one can naturally estimate v(x) by exp(xT�α) and expit(xT�α){1 −
+expit(xT�α)} respectively. To preserve generality, we introduce a working model vθ(x) for
+v(x) with some nuisance parameter θ to be estimated as �θ that could be partially or fully
+determined by �α. Suppose that �θ converges to some ¯θ. As will be shown in Section 3,
+violation of this conditional variance model, i.e., v(x) ̸= v¯θ(x) does not impact the double
+robustness of our proposed estimator but only affects its efficiency gain when PS is wrong
+and OR is correct.
+7
+
+2.3
+PAD estimator
+Now we formally introduce the propensity score augmented doubly robust (PAD) estimator.
+Our central idea is to augment the PS model ¯r(X) = exp(X
+T¯γ) as ¯raug(X; β) = exp(X
+T¯γ)+
+Ψ
+Tβ and use ¯raug(·) to replace ¯r(·) in the DR estimator. Here Ψ is some properly constructed
+basis function of X and β is some loading coefficient vector to be estimated. We first describe
+the empirical construction procedures for PAD in Algorithm 1 and then discuss the reason
+and intuition of the key steps in this algorithm.
+Algorithm 1 Propensity score Augmented Doubly robust (PAD) estimation
+[Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional
+variance estimator as �θ.
+[Step 2] Specify Φ = φ(X) of larger dimensionality than X using any basis function φ(·),
+and take �Ψ = Φ − �ET [Φv�θ(X)]/�ET v�θ(X).
+[Step 3] Solve the restricted weighted least square (RWLS) problem:
+�β = argminβ �Vµ(β),
+s.t.
+�ESX ˙g(X
+T�α) �Ψ
+Tβ = 0,
+(5)
+where
+�Vµ(β) = �ES{exp(X
+T�γ) + �Ψ
+Tβ}2v�θ(X) + 2�L
+T�ESX{exp(X
+T�γ) + �Ψ
+Tβ}v�θ(X),
+(6)
+and �L is as defined in equation (4).
+[Step 4] Obtain the PAD estimator through
+�µPAD = �ES{Y − g(X
+T�α)}{exp(X
+T�γ) + �Ψ
+T�β} + �ET g(X
+T�α).
+For heuristic analysis, suppose that all estimators used in (6) converge to their limiting
+values. Then let Ψ = Φ −ET [Φv¯θ(X)]/ET vθ(X) be the limits of �Ψ, ¯β the limits of �β, with
+its specific form given by Lemma B1 in Appendix, and
+Vµ(β) = ES{exp(X
+T¯γ) + Ψ
+Tβ}2v¯θ(X) + 2L
+TESX{exp(X
+T¯γ) + Ψ
+Tβ}v¯θ(X)
+the limiting function of �Vµ(β) specified in Algorithm 1. We shall consider two scenarios
+separately to demonstrate that our proposed PAD estimator not only maintains double
+robustness property but also has a lower asymptotic variance than �µDR when the OR model
+is correctly specified and PS is wrong. Rigorous justification for these results will be provided
+in Section 3.
+8
+
+Correct PS model.
+When the PS model is correct, we easily have L = 0 as stated in
+Section 2.2 so Vµ(β) = ES{exp(X
+T¯γ) + Ψ
+Tβ}2v¯θ(X), and
+∂Vµ(β)
+∂β
+= ESΨ exp(X
+T¯γ)v¯θ(X) = ET Ψv¯θ(X).
+By definition of Ψ, we have ET Ψv¯θ(X) = 0, as ensured by the mean shift of Φ in Step
+2 of Algorithm 1. Thus, β = 0 minimizes Vµ(β) and consequently, is the solution of the
+population-level version of the RWLS problem (5) since the linear constraints in (5) is trivially
+satisfied by β = 0. This implies that as long as the PS model is correct, �β converges to 0
+so the augmented PS estimator exp(X
+T�γ) + �Ψ
+T�β converges to the correct PS model, which
+ensures �µPAD to converge to the true µ0. Meanwhile, it is clear that the augmentation of PS
+does not change the OR model at all. Therefore, �µPAD preserves the same DR property as
+�µDR, i.e., being (root-n) consistent whenever the PS or the OR model is correctly specified.
+Correct OR and wrong PS.
+Note that �µPAD = �µDR + �ES �Ψ
+T�β{Y − g(X
+T�α)} and when
+the OR model is correct,
+�ES �Ψ
+T�β{Y − g(X
+T�α)} =�ES �Ψ
+T�β{Y − g(X
+Tα0)} + �ES �Ψ
+T�β{g(X
+Tα0) − g(X
+T�α)}
+≈�ESΨ
+T¯β{Y − g(X
+Tα0)} + �ES(α0 − �α)
+TX ˙g(X
+T�α) �Ψ
+T�β,
+(7)
+in which we use the orthogonality between �Ψ
+T�β − Ψ
+T¯β and Y − g(X
+Tα0) on the first term,
+as well as expansion on g(X
+Tα0) − g(X
+T�α) in the second term of the first line, to derive
+the “≈” relation shown in the second line. Here, “≈” in (7) again means that the difference
+between the first and second line is up to op(n−1/2) and, thus, becomes asymptotically
+negligible.
+In addition, according to the moment constraint in the RWLS problem (5),
+�ESX ˙g(X
+T�α) �Ψ
+T�β converges to 0.
+So the second term in the second line of (7) is also
+negligible and �µPAD ≈ �µDR + �ESΨ
+T¯β{Y − g(X
+Tα0)}. Combining this with equation (3) as
+well as the asymptotic equivalence between �µDR and �µDR discussed in Section 2.2, we have
+aVar{n1/2(�µPAD − µ0)} = ES{¯r(X) + Ψ
+T¯β}2v(X) + 2L
+TESX{¯r(X) + Ψ
+T¯β}v(X) + C, (8)
+which, after dropping the invariant C, is equal to Vµ(¯β), the limiting value of the minimized
+objective function �Vµ(�β) in the RWLS problem (5). Note that β = 0 is always feasible to
+the linear constraint in (5) and if we simply replace ¯β with 0 in the right-hand side of (8),
+it reduces to the asymptotic variance of n1/2(�µDR − µ0) derived in (3). Meanwhile, when the
+PS model is wrong, ∂Vµ(β)/∂β is typically not 0 at β = 0 so the population-level minimizer
+¯β ̸= 0. Thus, aVar{n1/2(�µPAD − µ0)} ≤ aVar{n1/2(�µDR − µ0)} when the OR model is correct
+and the strict “<” will hold in general when the PS model is wrong.
+9
+
+3
+Asymptotic analysis
+In this section, we rigorously present the asymptotic properties of the proposed PAD estima-
+tor and compare PAD with the standard DR estimator. We first introduce some mild and
+common regularity assumptions. Without loss of generality, we assume that n/N = O(1) so
+the desirable parametric rate of the DR estimators will be O(n−1/2).
+Assumption 1. The supports of X and Φ are compact and EY 4 < ∞.
+Assumption 2. The link function g(·) is differentiable with derivative ˙g(·) and there exists
+a constant L such that |˙g(x1) − ˙g(x2)| < L|x1 − x2| for all x1, x2 ∈ R.
+Assumption 3. The dimension of Ψ is larger than that of X. Matrices ES{ΨΨ
+Tv¯θ(X)},
+ES{XX
+T exp(X
+T¯γ)}, ES{XX
+T ˙g(X
+T ¯α)} and ES{ΨX
+T ˙g(X
+T ¯α)} have all their eigenvalues
+bounded and staying away from zero.
+Assumption 4. The conditional variance function vθ(x) is differentiable on θ with a bounded
+partial derivative ∂θvθ(x). The estimator �θ converges to some ¯θ in probability and satisfies
+that n1/2(�θ − ¯θ) is asymptotic normal with mean zero.
+Remark 1. Assumptions 1–3 are all mild, standard, and commonly used to justify the
+asymptotic properties of M-estimation (Van der Vaart, 2000). Note that in Assumption 3,
+we take Ψ to have larger dimension than X and make regularity conditions on ES{XX
+T ˙g(X
+T ¯α)}
+and ES{ΨX
+T ˙g(X
+T ¯α)}. These are to ensure that �β is not zero and properly converges to ¯β.
+Assumption 4 constrains the way of specifying vθ(x) and estimating θ. Under Assumptions
+1–3, this assumption is satisfied when either θ is fully determined by α, e.g., in a Pois-
+son or logistic model for Y against X, or when θ is estimated by additionally fitting some
+parametric model of Var(Y | X) against X.
+Now we present the main results about the robustness and efficiency of our proposed PAD
+estimator in Theorem 1 with its proof given in Section B of the Appendix. Some important
+heuristics of this theorem has already been discussed in Section 2.3.
+Theorem 1. Under Assumptions 1–4, it holds that
+(i) Double robustness. When either the PS or the OR model is correctly specified, i.e.,
+r0(x) = exp(xTγ0) for some γ0 or m0(x) = g(xTα0) for some α0, �µPAD
+p−→ µ0 and
+n1/2(�µPAD − µ0) weakly converges to some normal distribution with mean zero.
+(ii) Variance reduction under wrong PS. When the OR model is correct while the
+PS model may be misspecified, the asymptotic variance of n1/2(�µPAD − µ0) is always
+10
+
+not larger than that of n1/2(�µDR − µ0). Further when ¯β ̸= 0 (the explicit form of ¯β is
+given in Lemma B1), n1/2(�µPAD − µ0) has a strictly smaller asymptotic variance than
+n1/2(�µDR − µ0).
+(iii) Equivalence under correct PS and OR. When both the PS and OR models are
+correct, n1/2(�µPAD − µ0) and n1/2(�µDR − µ0) are asymptotically equivalent and have the
+same asymptotic variance.
+4
+Simulation study
+We conducted simulation studies to evaluate our proposed estimator and compare it with
+the standard DR estimator.
+In our studies, we generate covariates X = (X1, X2, X3)T
+from N(0, Σ) with Σ = (σij) ∈ R3×3 and σij = 0.3|i−j|. For generation of the population
+assignment ∆ and outcome Y , we consider six settings, namely:
+(G1) Gassuian Y , Correct PS, Correct OR. Pr(∆ = 1 | X)} = expit(X1 − 2X2 + X3)
+and Y = 0.5X1 + 0.5X2 + X3 + ǫ where ǫ | X ∼ N(0, 1).
+(G2) Gassuian Y , Correct PS, Wrong OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3)
+and Y = 0.5X1 + 0.5X2 + sin(X2 + 0.5X3) + ǫ.
+(G3) Gassuian Y , Wrong PS, Correct OR. Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 −
+1.5|X1| − 1.5|X2| − |X3|) and Y = 0.5X1 + 0.5X2 + X3 + ǫ.
+(L1) Binary Y , Correct PS, Correct OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and
+Pr(Y = 1 | X) = expit(0.5X1 + 0.5X2 + X3).
+(L2) Binary Y , Correct PS, Wrong OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and
+Pr(Y = 1 | X)} = expit(0.5X1 + 0.5X2 + sin(X2 + 0.5X3))
+(L3) Binary Y , Wrong PS, Correct OR. Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 −
+1.5|X1| − 1.5|X2| − |X3|) and Pr(Y = 1 | X) = expit(0.5X1 + 0.5X2 + X3).
+In Settings (G1)–(G3), Y is a gaussian variable and we fit linear models for Y ∼ X with
+vθ(x) = 1. While in Settings (L1)–(L3), we fit logistic models for the binary Y against
+X with vθ(x) = expit(X
+Tα){1 − expit(X
+Tα)}. We consider different scenarios about the
+correctness of the PS and OR models to examine the robustness and efficiency of PAD.
+Bootstrap is used for estimating the asymptotic variance and constructing the confidence
+interval (CI). For effective variance reduction on PAD when PS is wrong, i.e. under Settings
+11
+
+(G3) and (L3), we include in the augmentation covariates Φ a decent amount of X’s basis
+functions including Xj, exp(Xj), |Xj|, exp(−Xj1 − Xj2), and exp(−X1 − X2 − X3) for all j
+and j1 ̸= j2 ∈ {1, 2, 3}. We set N = n = 500 or N = n = 1000 separately and generate 1000
+realizations for each setting.
+Table 1 reports the absolute average bias (Bias), standard error (SE), and coverage
+probability (CP) of the 95% CI of the DR and PAD estimators. When at least one nuisance
+models are correct, DR and PAD attain very close bias, which is much smaller compared to
+their SE and, thus, grants their CPs to be close to the nominal level. This indicates that
+PAD achieves the double robustness property just like the standard DR estimator under
+finite samples. To compare PAD and DR in terms of their estimation variance and efficiency,
+we present in Table 2 their relative efficiency (RE) defined as Var(�µDR)/ Var(�µPAD). Under
+Settings (G1), (G2), (L1), and (L2) where the PS model is correct, the two estimators show
+nearly identical variance, with their REs located between 1 ±0.04. Under Settings (G3) and
+(L3) with misspecified PS and correct OR models, our proposed PAD estimator shows 20%
+to 40% smaller variance than the standard DR estimator. All these results demonstrate that
+conclusions in Theorem 1 also apply well for finite samples. In specific, PAD performs very
+closely to the standard DR when the PS model is correct and is potentially better than DR
+in the presence of wrong PS models.
+5
+Real example
+The effects of the 401(k) program have been investigated for a long time (Abadie, 2003;
+Chernozhukov et al., 2018, e.g.). Different from other plans like Individual Retirement Ac-
+counts (IRAs), eligibility for 401(k) is completely decided by employers. Therefore, unob-
+served personal preferences for savings may make little difference in 401(k) eligibility. How-
+ever, there may be some other confounders affecting the causal studies of 401(k), such as job
+choice, income, and age. To address this problem, (Abadie, 2003) and (Chernozhukov et al., 2018)
+proposed to adjust for certain covariates related to job choice so that 401(k) eligibility can
+be regarded exogenous.
+Whether 401(k) eligibility contributes to the improvement of people’s net total financial
+assets is an important topic studied in existing literature like Abadie (2003) and Chernozhukov et al. (2018).
+However, whether 401(k) can improve the financial assets of those actually not eligible for
+401(k) is still an open and interesting problem. To investigate this problem, we analyze the
+data from the Survey of Income and Program Participation of 1991. The data set consists of
+n + N = 9275 observations. The outcome of our interests, Y is defined as the indication of
+having positive net total financial assets. There are 9 adjustment covariates in X, including
+12
+
+Table 1:
+The absolute average bias (Bias), standard error (SE), and coverage probability (CP) of the 95%
+confidence intervals of the DR and PAD estimators under the settings described in Section 4. All results are
+produced based on 1000 repetitions.
+n = N = 500
+n = N = 1000
+Setting
+Method
+Bias
+SE
+CP
+bias
+SE
+CP
+(G1)
+DR
+0.006
+0.145
+0.94
+0.005
+0.106
+0.92
+PAD
+0.005
+0.142
+0.93
+0.004
+0.105
+0.92
+(G2)
+DR
+0.007
+0.152
+0.92
+0.008
+0.111
+0.92
+PAD
+0.005
+0.149
+0.92
+0.007
+0.112
+0.92
+(G3)
+DR
+0.010
+0.162
+0.93
+0.001
+0.121
+0.92
+PAD
+0.005
+0.136
+0.93
+0.001
+0.105
+0.93
+(L1)
+DR
+0.000
+0.055
+0.92
+0.001
+0.040
+0.92
+PAD
+0.001
+0.054
+0.93
+0.001
+0.040
+0.93
+(L2)
+DR
+0.001
+0.054
+0.92
+0.004
+0.040
+0.92
+PAD
+0.001
+0.053
+0.92
+0.004
+0.040
+0.92
+(L3)
+DR
+0.005
+0.057
+0.91
+0.003
+0.038
+0.92
+PAD
+0.005
+0.052
+0.93
+0.002
+0.035
+0.93
+Table 2:
+Relative efficiency (RE) between DR and PAD, i.e., Var(�µDR)/ Var(�µPAD), under the settings de-
+scribed in Section 4.
+n, N
+(G1)
+(G2)
+(G3)
+(L1)
+(L2)
+(L3)
+500
+1.04
+1.04
+1.42
+1.04
+1.04
+1.20
+1000
+1.02
+0.98
+1.33
+1.00
+1.00
+1.18
+age, income, family size, years of education, benefit pension status, marriage, two-earner
+household status, individual participation in IRA plan, and home ownership status. The
+source (treated) samples S with ∆ = 1 are taken as those eligible for 401(k) and the target
+(untreated) samples T are those without 401(k) eligibility. We applied PAD and standard
+DR to estimate µ, the effect of 401(k) eligibility on improving the positive rate of net to-
+tal financial assets among people without 401(k) eligibility. The PS model is specified as
+exp(X
+Tγ) and the OR model is expit(X
+Tα). In our method, the augmentation covariates
+vector Φ consists of X, exp(−0.3Xj), |Xj|, and X2
+j for all Xj’s that are not binary. We
+again use bootstrap to estimate SEs and construct CIs.
+In Table 3, we report the point estimation, their estimated standard errors (ESE), and
+95% CIs for the treatment effect µ, obtained using the standard DR and our proposed PAD
+13
+
+methods. Outputs of both methods indicate that 401(k) eligibility has a significant effect on
+improving the rate of having positive net total financial assets among people who are actually
+not eligible for 401(k). The estimated treatment effect is 0.169 (95% CI: 0.142, 0.196) by the
+standard DR and 0.150 (95% CI: 0.126, 0.175) by PAD. Moreover, the ESE of our proposed
+PAD estimator is remarkably smaller than that of the standard DR estimator, with their
+estimated RE, i.e., Var(�µDR)/ Var(�µPAD) being around 1.25. This means our proposed PAD
+method can characterize the treatment effect µ more precisely than DR in this example.
+Table 3: The point estimation (PE), its estimated standard error (ESE), and 95% confidence interval (CI)
+for µ, the effect of 401(k) eligibility on improving the positive rate of net total financial assets among people
+without 401(k) eligibility, derived using the standard DR and the PAD methods.
+Method
+PE
+ESE
+CI
+DR
+0.169
+0.0140
+(0.142, 0.196)
+PAD
+0.150
+0.0125
+(0.126, 0.175)
+6
+Discussion
+In analogy to our PS model augmentation strategy, we also propose an OR model augmen-
+tation strategy (OAD) that augments the OR model with some bases of X satisfying certain
+moment conditions like Ψ in Algorithm 1. Description and discussion of this method are
+presented in Section A of the Appendix. Similar to Theorem 1, we are able to show that
+this OAD estimator is doubly robust, of a smaller variance than the standard DR estimator
+when the PS model is correct but the OR model is wrong, and equivalent with DR when
+both nuisance models are correct. Just like PAD, this OAD method is easy to implement
+and only requires convex optimization. We notice that some existing methods in intrinsic
+efficient DR estimation like Rotnitzky et al. (2012) and Gronsbell et al. (2022) rely on non-
+convex training to construct the OR model when it is not linear. This OAD strategy could
+mitigate this practical problem and still achieves the purpose of variance reduction in the
+presence of misspecified OR models.
+For ease of demonstration, we focus on covariate shift correction, or equivalently ATT
+estimation in this paper. Our proposed PAD estimation can be potentially generalized to
+address other causal or missing data problems like ATE estimation (Bang and Robins, 2005,
+e.g.), casual model estimation Rotnitzky et al. (2012), transfer learning of a regression model
+Liu et al. (2020), etc. Also, properly specifying the bases Φ is crucial for variance reduction
+in our method. The optimal choice of Φ for the most effective variance reduction is still
+14
+
+an open problem. Related to this, it may be useful and interesting to extend our current
+framework for high-dimensional sparse or sieve construction of the augmentation term Ψ
+Tβ.
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+17
+
+A
+Dual construction to augment OR
+In analogy to our PAD estimator, to improve the efficiency our the DR estimator under the
+correct PS and wrong OR models, we propose the Outcome regression Augmented Doubly
+robust (OAD) estimator in the following algorithm.
+Algorithm A1 Outcome regression Augmented Doubly robust (OAD) estimation
+[Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional
+variance estimator as �θ.
+[Step 2] Let Φ = φ(X) with function φ(·), �g(X
+T�α) = g(X
+T�α) − �ET g(X
+T�α) and
+�Ψ = Φ −
+�ET Φ�g(X
+T�α)
+�ET �g2(X
+T�α)
+�g(X
+T�α).
+[Step 3] Solve the restricted weighted least square (RWLS) problem:
+�β = argminβ �Vµ,OAD(β),
+s.t.
+�ESX �Ψ
+Tβ exp(X
+T�γ) = 0,
+(A1)
+where
+�Vµ,OAD(β) =n−1�
+VarS[{Y − g(X
+T�α) − �Ψ
+T�β} exp(X
+T�γ)] + N−1�
+VarT {g(X
+T�α) + �Ψ
+T�β}
++ 2�
+L∗
+T[N−1 �
+CovT (X, �Ψ
+T�β) + n−1 �
+CovS{X exp(X
+T�γ), �Ψ
+T�β exp(X
+T�γ)}],
+(A2)
+and �
+L∗ = {�ESX exp(X
+T�γ)X
+T}−1�ES{Y − g(X
+T�α)} exp(X
+T�γ)X.
+[Step 4] Obtain the OAD estimator:
+�µOAD = �ES{Y − g(X
+T�α) − �Ψ
+T�β} exp(X
+T�γ) + �ET {g(X
+T�α) + �Ψ
+T�β}.
+To demonstrate how Algorithm A1 works, we define that
+�µOAD =�ES{Y − g(X
+T ¯α) − Ψ
+T¯β} exp(X
+T¯γ) + �ET {g(X
+T ¯α) + Ψ
+T¯β}
++ ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X
+T{ESX exp(X
+T¯γ)X
+T}−1{�ET X − �ESX exp(X
+T¯γ)}.
+Then similar to our analysis in Section 2.2, when the PS model is correct, �µOAD is asymptot-
+ically equivalent to �µOAD, with its variance being:
+Vµ,OAD(β) =n−1 VarS[{Y − g(X
+T ¯α) − Ψ
+T¯β} exp(X
+T¯γ)] + N−1 VarT {g(X
+T ¯α) + Ψ
+T¯β}
++ 2L∗T[N−1CovT (X, Ψ
+T¯β) + n−1CovS{X exp(X
+T¯γ), Ψ
+T¯β exp(X
+T¯γ)}],
+1
+
+the limiting function of �Vµ,OAD(β) specified in Algorithm A1, where
+L∗ = {ESX exp(X
+T¯γ)X
+T}−1ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X.
+This corresponds to the objective function in equation (A2). Similar to the PAD construc-
+tion, when β = 0, Vµ,OAD(β) reduces to the asymptotic variance of the standard DR estimator.
+Thus, �µOAD has a smaller variance than the standard DR estimator when the PS model is
+correct and the OR model is wrong, under which we typically have ¯β ̸= 0.
+On the other hand, when OR is correctly specified, we have ¯α = α0, L∗ = 0, and thus
+∂Vµ,OAD(β)
+∂β
+|β=0 = CovT (g(X
+T ¯α), Ψ).
+By definition of Ψ, we have CovT (g(X
+T ¯α), Ψ) = 0. Hence, similar to the analysis in Section
+2.2, �µOAD preserved the same DR property as �µDR, i.e., being root-n consistent whenever the
+PS or the OR model is correctly specified.
+B
+Asymptotic justification
+B.1
+Technical lemma
+Lemma B1. Define a := ESΨ ∂g(XTα)
+∂αT
+|¯α, b := ESΨ exp(X
+T¯γ)v¯θ(X) + ESΨX
+Tv¯θ(X)LT,
+and Σ := ESΨΨ
+Tv¯θ(X), under Condition 2-5, the solution of the RWLS problem (5) is
+¯β = Σ−1a(a
+TΣ−1a)−1a
+TΣ−1b − Σ−1b.
+Proof. First we introduce Lagrange multiplier λ and write (5) as the Lagrange form:
+¯β = argminβ ES{exp(X
+T¯γ)+Ψ
+Tβ}2v¯θ(X)+2LESX{exp(X
+T¯γ)+Ψ
+Tβ}v¯θ(X)−λ
+TES
+∂g(X
+Tα)
+∂α
+|¯αΨ
+Tβ.
+Then we have the partial derivative of λ and β:
+ES
+∂g(X
+Tα)
+∂α
+|¯αΨ
+Tβ = 0,
+(B1)
+and
+2ESΨ{exp(X
+T¯γ) + Ψ
+Tβ}v¯θ(X) + 2ESΨX
+Tv¯θ(X)L
+T − ESΨ∂g(X
+Tα)
+∂αT
+|¯αλ = 0.
+(B2)
+From (B2) we have
+β = {2ESΨΨ
+Tv¯θ(X)}−1{ESΨ∂g(X
+Tα)
+∂αT
+|¯αλ−2ESΨ exp(X
+T¯γ)v¯θ(X)−2ESΨX
+Tv¯θ(X)L
+T},
+2
+
+together with (B1), we have
+ES
+∂g(X
+Tα)
+∂α
+|¯αΨ
+T{ESΨΨ
+Tv¯θ(X)}−1
+∗ {ESΨ∂g(X
+Tα)
+∂αT
+|¯αλ − 2ESΨ exp(X
+T¯γ)v¯θ(X) − 2ESΨX
+Tv¯θ(X)L
+T} = 0.
+this function can be simplified as
+a
+TΣ−1(aλ − 2b) = 0,
+and we further have
+λ = 2(a
+TΣ−1a)−1a
+TΣ−1b.
+Hence, we have
+¯β = Σ−1a(a
+TΣ−1a)−1a
+TΣ−1b − Σ−1b,
+and we can estimate it by
+�β = �Σ
+−1�a(�a
+T �Σ
+−1�a)−1�a
+T �Σ
+−1�b − �Σ
+−1�b
+for �a := �ES �Ψ∂g(XTα)
+∂αT
+|�α, �b := �ES �Ψ exp(X
+T�γ)v�θ(X)+�ES �ΨX
+Tv�θ(X)�LT, and �Σ := �ES �Ψ �Ψ
+Tv�θ(X)
+Lemma B2. Under Condition 3, 4 and 6, we have that �Ψ − Ψ = Op(n−1/2).
+Proof. By definition, we would have that
+�Ψ − Ψ =
+�ET {Φv�θ(X)}
+�ET v�θ(X)
+− ET {Φv¯θ(X)}
+ET v¯θ(X)
+.
+Under Condition 4 and 6, we have that
+�ET v�θ(X) − ET v¯θ(X) = �ET v¯θ(X) + �ET
+∂vθ(X)
+∂θ
+|�θ(�θ − ¯θ) − ET v¯θ(X) = Op(n−1/2)
+(B3)
+for �θ between �θ and ¯θ.
+By using the same techniques, we have that �ET {Φv�θ(X)} −
+ET {Φv¯θ(X)} = Op(n−1/2). And we have
+�Ψ − Ψ =
+�ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}�ET v�θ(X)
+�ET v�θ(X)ET v¯θ(X)
+=
+�ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}ET v¯θ(X) − [ET {Φv¯θ(X)}�ET v�θ(X) − ET {Φv¯θ(X)}ET v¯θ(X)]
+{ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X)
+= Op(n−1/2)ET v¯θ(X) − ET {Φv¯θ(X)}Op(n−1/2)
+{ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X)
+= Op(n−1/2).
+3
+
+Lemma B3. Under Condition 1, 2 and 4, 5, we have that �γ − ¯γ = Op(n−1/2) and �α − ¯α =
+Op(n−1/2).
+Proof. The estimation of γ has been given as
+�ESX exp(X
+T�γ) = �ET X,
+by applying Taylor series expansion, we have
+n−1
+n
+�
+i=1
+Xi exp(X
+T
+i ¯γ) + n−1
+n
+�
+i=1
+Xi exp(X
+T
+i �γ)X
+T
+i (�γ − ¯γ) = N−1
+n+N
+�
+i=n+1
+Xi,
+where �γ is some vector between �γ and ¯γ. According to (Van der Vaart, 2000), we have
+�γ − ¯γ = op(1). Let J represent matrix n−1 �n
+i=1 Xi exp(X
+T
+i �γ)X
+T
+i , and we have that
+J = n−1
+n
+�
+i=1
+Xi exp(X
+T
+i ¯γ)X
+T
+i +n−1
+n
+�
+i=1
+Xi exp(X
+T
+i γ∗)X
+T
+i Xi(�γ−¯γ) = ESX exp(X
+T¯γ)X
+T+op(1)
+for γ∗ between �γ and ¯γ. Hence, by central limit theorem and Slutsky theorem, we have
+that, under Condition 1 and 4
+�γ − ¯γ = J−1
+�
+N−1
+n+N
+�
+i=n+1
+Xi − n−1
+n
+�
+i=1
+Xi exp(X
+T
+i ¯γ)
+�
+=J−1
+�
+N−1
+n+N
+�
+i=n+1
+Xi − ET X + ESX exp(X
+T¯γ) − n−1
+n
+�
+i=1
+Xi exp(X
+T
+i ¯γ)
+�
+= Op(n−1/2).
+Furthermore, The estimation equation of �α is given by
+�ESS(�α) = �ESX{Y − g(X
+T�α)} = 0,
+by using Taylor series expansion, we have that
+�ESX{Y − g(X
+T ¯α)} + �ES
+∂S(α)
+∂αT
+����
+�α
+(�α − ¯α) = 0
+for �α between �α and ¯α, and we have
+�α − ¯α = −�ES
+�∂S(α)
+∂αT
+����
+�α
+�−1
+�ESX{Y − g(X
+T ¯α)}.
+By using the same techniques as those for obtaining the asymptotic properties of �γ, under
+Condition 2, 4 and 5, we have �α − ¯α = Op(n−1/2).
+4
+
+Lemma B4. Under Condition 1-6 and Lemma A1-A3, we can obtain that �β−¯β = Op(n−1/2).
+In addition, when the PS is correctly specified, we further have ¯β = 0 and �β = Op(n−1/2).
+Proof. By using the same techniques as (B3), under Condition 2-4, we first have that
+�a−a = �ES �Ψ∂g(X
+Tα)
+∂αT
+|�α−ESΨ∂g(X
+Tα)
+∂αT
+|¯α = �ESΨ∂g(X
+Tα)
+∂αT
+|¯α−ESΨ∂g(X
+Tα)
+∂αT
+|¯α+Op(n−1/2) = Op(n−1/2).
+In addition, we can have that �b − b = Op(n−1/2) and �Σ − Σ = Op(n−1/2). Furthermore, we
+can easily have that
+�Σ
+−1 − Σ−1 = Σ−1Σ{Σ + Op(n−1/2)}−1 − Σ−1
+= Σ−1[Σ{Σ + Op(n−1/2)}−1 − {Σ + Op(n−1/2)}{Σ + Op(n−1/2)}−1] = Op(n−1/2),
+based on which we can have (�aT �Σ
+−1�a)−1−(aTΣ−1a)−1 = Op(n−1/2). Let �Ω denote �Σ
+−1�a(�aT �Σ
+−1�a)−1�aT �Σ
+−1
+and Ω denote Σ−1a(aTΣ−1a)−1aTΣ−1. We can have that �Ω − Ω = Op(n−1/2), hence, we
+have that �β − ¯β = �Ω�b − Ωb = Op(n−1/2).
+On the other hand, when the PS is correctly specified, L = 0 and ESΨ exp(X
+T¯γ)v¯θ(X) =
+ET Ψv¯θ(X) = 0, which means
+¯β = Ωb = Ω{ESΨ exp(X
+T¯γ)v¯θ(X) + ESΨX
+Tv¯θ(X)L
+T} = Ω0 = 0.
+And at the same time, we have �β = Op(n−1/2).
+B.2
+Proof of Theorem 1
+Proof. Proof of Theorem 1 (i).
+When the OR is correctly specified, ¯α = α0. Consider �µOR where
+�µOR = �ES{Y − g(X
+T ¯α)}{exp(X
+T¯γ) + Ψ
+T¯β} + �ET g(X
+T ¯α)
++
+�
+ES
+∂g(X
+Tα)
+∂αT
+����
+¯α
+exp(X
+T¯γ) − ET
+∂g(X
+Tα)
+∂αT
+����
+¯α
+�
+ES
+�∂S(α)
+∂αT
+����
+¯α
+�−1
+�ESX{Y − g(X
+T ¯α)}.
+It is obvious that E�µOR = ET g(X
+T ¯α) = µ0. Hence, by using central limit theorem, we have
+that �µOR − µ0 = Op(n−1/2), n1/2(�µOR − µ0) weakly converges to gaussian distribution with
+mean 0. On the other hand, we have that
+�µP AD − �µOR = �ES{Y − g(X
+Tα0)}{exp(X
+T¯γ)X
+T(�γ − ¯γ) + Ψ
+T(�β − ¯β) + ( �Ψ − Ψ)
+T¯β}
+−
+�
+�ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+{exp(X
+T¯γ) + Ψ
+T¯β} − �ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+(�α − ¯α) + op(n−1/2)
+5
+
+−
+�
+ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+exp(X
+T¯γ) − ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+ES
+�∂S(α)
+∂αT
+����
+α0
+�−1
+�ESX{Y − g(X
+Tα0)},
+by using central limit theorem, along with Lemma A2-A4, we have that
+�ES{Y − g(X
+Tα0)}{exp(X
+T¯γ)X
+T(�γ − ¯γ) + Ψ
+T(�β − ¯β) + ( �Ψ − Ψ)
+T¯β}
+= [�ES{Y − g(X
+Tα0)} exp(X
+T¯γ)X
+T](�γ − ¯γ) + [�ES{Y − g(X
+Tα0)}Ψ
+T](�β − ¯β)
++ [�ES{Y − g(X
+Tα0)}¯β
+T]( �Ψ − Ψ) = Op(n−1/2)op(1) + Op(n−1/2)op(1) + Op(n−1/2)op(1) = op(n−1/2).
+On the other hand,
+−
+�
+�ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+{exp(X
+T¯γ) + Ψ
+T¯β} − �ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+(�α − ¯α)
+=
+�
+�ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+{exp(X
+T¯γ) + Ψ
+T¯β} − �ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+�ES
+�∂S(α)
+∂αT
+����
+¯α
+�−1
+�ESX{Y − g(X
+T ¯α)}
+=
+�
+ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+exp(X
+T¯γ) − ET
+∂g(X
+Tα)
+∂αT
+����
+α0
++ Op(n−1/2)
+�
+∗
+�
+ES
+�∂S(α)
+∂αT
+����
+α0
+�−1
++ Op(n−1/2)
+�
+�ESX{Y − g(X
+Tα0)}.
+(B4)
+Hence, we have that
+−
+�
+�ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+{exp(X
+T¯γ) + Ψ
+T¯β} − �ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+(�α − ¯α)
+−
+�
+ES
+∂g(X
+Tα)
+∂αT
+����
+α0
+exp(X
+T¯γ) − ET
+∂g(X
+Tα)
+∂αT
+����
+α0
+�
+ES
+�∂S(α)
+∂αT
+����
+α0
+�−1
+�ESX{Y − g(X
+Tα0)}
+= �ESX{Y − g(X
+Tα0)}Op(n−1/2) = op(n−1/2).
+Thus, from previous results, we have that �µP AD − �µOR = op(n−1/2). Together with Slutsky
+theorem, we futher have that �µP AD − µ0 = Op(n−1/2) and n1/2(�µP AD − µ0) weakly converges
+to gaussian distribution with mean 0.
+When the PS is correctly specified, ¯γ = γ0, we consider �µPS where
+�µPS = �ES{Y − g(X
+T ¯α)}{exp(X
+Tγ0) + Ψ
+T¯β} + �ET g(X
+T ¯α)
++ ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X
+T{ESX exp(X
+Tγ0)X
+T}−1{�ET X − �ESX exp(X
+Tγ0)}.
+Together with the results from Lemma A4, we have that E�µPS = ESY exp(X
+T¯γ) = ET Y =
+µ0. By using the central limit theorem, we have that �µPS − µ0 = Op(n−1/2), n1/2(�µPS − µ0)
+weakly converges to gaussian distribution with mean 0. On the other hand, we have that
+�µP AD − �µPS = �ES{Y − g(X
+T ¯α)}{exp(X
+T¯γ)X
+T(�γ − ¯γ) + Ψ
+T(�β − ¯β)}
+6
+
+− ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X
+T{ESX exp(X
+Tγ0)X
+T}−1{�ET X − �ESX exp(X
+Tγ0)} + op(n−1/2)
+By using the techniques from (B4), we would have
+�ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X
+T(�γ − ¯γ)
+− ES{Y − g(X
+T ¯α)} exp(X
+T¯γ)X
+T{ESX exp(X
+Tγ0)X
+T}−1{�ET X − �ESX exp(X
+Tγ0)} = op(n−1/2)
+And from Lemma A4, we have �β = Op(n−1/2). Thus, we have �µP AD − µ0 = Op(n−1/2). On
+the other hand, it is worth noticing that �β is the continuous function of �θ, �γ and �α, so
+under central limit theorem and Slutsky theorem, we would have the asymptotic normality
+of �β. Hence, we further have that n1/2(�µPS − µ0) weakly converges to gaussian distribution
+with mean 0.
+Proof. Proof of Theorem 1 (ii).
+First we denote U as
+U = VarT (E(Y |X)) + LESXX
+T Var(Y |X)L
+T
+When the OR is correctly specified, the asymptotic variance of �µP AD, Var{n−1/2(�µP AD−µ0)}
+is
+ES{exp(X
+T¯γ) + Ψ
+T¯β}2v¯θ(X) + 2LESX{exp(X
+T¯γ) + Ψ
+T¯β}v¯θ(X) + U,
+and ¯β contributes to minimizing this variance. When ¯β = 0, the function above is written
+as
+ES{exp(X
+T¯γ)}2v¯θ(X) + 2LESX{exp(X
+T¯γ)}v¯θ(X) + U,
+which is the same as the asymptotic variance of �µDR, Var{n−1/2(�µDR − µ0)}. Hence, when
+¯β ̸= 0, �µP AD has the smaller asymptotic variance than standard doubly robust estimator
+�µDR.
+Proof. Proof of Theorem 1 (iii).
+When both the PS and OR is correctly specified, consider �µB, where
+�µB = �ES{Y − g(X
+Tα0)} exp(X
+Tγ0) + �ET g(X
+Tα0).
+By using central limit theorem, n1/2(�µB −µ0) weakly converges to gaussian distribution with
+mean 0. On the other hand, by using Taylor series expansion, we would have �µP AD − �µB =
+op(n−1/2) and �µDR − �µB = op(n−1/2). Hence, they have the same asymptotic variance.
+7
+
+This figure "DMLmse.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.02162v1
+
+This figure "biasdml.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.02162v1
+

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+1
+An Enhanced Gradient-Tracking Bound for Distributed
+Online Stochastic Convex Optimization
+Sulaiman A. Alghunaim and Kun Yuan
+Abstract—Gradient-tracking
+(GT)
+based
+decentralized
+methods have emerged as an effective and viable alterna-
+tive method to decentralized (stochastic) gradient descent
+(DSGD) when solving distributed online stochastic opti-
+mization problems. Initial studies of GT methods implied
+that GT methods have worse network dependent rate than
+DSGD, contradicting experimental results. This dilemma
+has recently been resolved, and tighter rates for GT methods
+have been established, which improves upon DSGD.
+In this work, we establish more enhanced rates for GT
+methods under the online stochastic convex settings. We
+present an alternative approach for analyzing GT methods
+for convex problems and over static graphs. When compared
+to previous analyses, this approach allows us to establish
+enhanced network dependent rates.
+Index Terms—Distributed stochastic optimization, decen-
+tralized learning, gradient-tracking, adapt-then-combine.
+I. Introduction
+We consider the multi-agent consensus optimization prob-
+lem, in which n agents work together to solve the following
+stochastic optimization problem:
+minimize
+x∈Rd
+f(x) = 1
+n
+n
+�
+i=1
+fi(x)
+fi(x) ≜ E[Fi(x; ξi)].
+(1)
+Here, fi : Rd → R is the private cost function held by agent
+i, which is defined as the expected value of some loss function
+Fi(·, ξi) over local random variable ξi (e.g., data points). An
+algorithm that solves (1) is said to be a decentralized method
+if its implementation requires the agents to communicate only
+with agents who are directly connected to them (i.e., neighbors)
+based on the given network topology/graph.
+One of the most popular decentralized methods to solve prob-
+lem (1) is decentralized stochastic gradient descent (DSGD)
+[1]–[3]. While DSGD is communication efficient and simple to
+implement, it converges slowly when the local functions/data
+are heterogeneous across nodes. Furthermore, because data
+heterogeneity can be amplified by large and sparse network
+topologies [4], DSGD performance is significantly degraded
+with these topologies.
+In this work, we analyze the performance of the gradient-
+tracking method [5], [6], which is another well-known decentral-
+ized method that solves problem (1). To describe the algorithm,
+we let wij ≥ 0 denote the weight used by agent i to scale
+information received from agent j with wij = 0 if j /∈ Ni where
+Ni is the neighborhood of agent i. The adapt-then-combine
+gradient-tracking (ATC-GT) method [5] is described as follows:
+xk+1
+i
+=
+�
+j∈Ni
+wij(xk
+j − αgk
+j )
+(2a)
+S. A. Alghunaim ([email protected]) is with the
+Department of Electrical Engineering, Kuwait University, Kuwait. K.
+Yuan ([email protected]) is with the Center for Machine Learning
+Research, Peking University, China.
+gk+1
+i
+=
+�
+j∈Ni
+wij
+�
+gk
+j + ∇Fj(xk+1
+j
+; ξk+1
+j
+) − ∇Fj(xk
+j ; ξk
+j )�
+(2b)
+with initialization g0
+i = ∇Fi(x0
+i ; ξ0
+i ) and arbitrary x0
+i ∈ Rd.
+Here, ∇Fi(xk
+i ; ξk
+i ) is the stochastic gradient and ξk
+i is the data
+sampled by agent i at iteration k.
+Gradient-tracking can eliminate the impact of heterogeneity
+between local functions [5]–[8]. In massive numerical experi-
+ments reported in [9]–[12], GT can significantly outperform
+DSGD in the online stochastic setting. Initial studies on the
+convergence rate of GT methods are inadequate; they provide
+loose convergence rates that are more sensitive to network
+topology than vanilla DSGD. According to these findings, GT
+will converge slower than DSGD on large and sparse networks,
+which is counter-intuitive and contradicts numerical results
+published in the literature. Recent works [13], [14] establish
+the first convergence rates for GT that are faster than DSGD
+and more robust to sparse topologies under stochastic and non-
+convex settings. In this paper, we will provide additional en-
+hancements for GT under convex and strongly convex settings.
+A. Related works
+Gradient-tracking (GT) methods, which utilize dynamic
+tracking mechanisms [15] to approximate the globally averaged
+gradient, have emerged as an alternative to decentralized gradi-
+ent descent (DGD) [1]–[3], [16], [17] with exact convergence for
+deterministic problems [5]–[8]. Since their inception, numerous
+works have investigated GT methods in a variety of contexts [9],
+[10], [18]–[28]. However, all of these works provide convergence
+rates that can be worse than vanilla DSGD. In particular, these
+results indicate that GT is less robust to sparse topologies even
+if it can remove the influence of data heterogeneity. The work
+[14] established refined bounds for various methods including
+GT methods that improve upon DSGD under nonconvex set-
+tings. Improved network dependent bounds for GT methods in
+both convex and non-convex settings are also provided in [13].
+In this work, we provide additional improvements over previous
+works in convex and strongly convex settings – see Table I.
+It should be noted that there are other methods that are
+different from GT methods but have been shown to have com-
+parable or superior performance – see [14], [29] and references
+therein. In contrast to these other methods, GT methods have
+been shown to converge in a variety of scenarios, such as
+directed graphs and time-varying graphs [18], [19], [22]. We
+should also mention that there are modifications to GT ap-
+proaches that can improve the rate at the price of knowing addi-
+tional network information and/or more computation/memory
+[21]. However, the focus of this study is on basic vanilla GT
+methods.
+B. Contributions
+• We present an alternative approach for analyzing GT
+methods in convex and static graph settings, which may
+arXiv:2301.02855v1  [math.OC]  7 Jan 2023
+
+2
+TABLE I: Convergence rate to reach ϵ accuracy. The strongly convex (SC) and PL condition rates ignores iteration logarithmic factors.
+The quantity λ = ρ(W − 1
+n11T) ∈ (0, 1) is the mixing rate of the network where W is the network combination matrix. a0 = ∥¯x0 − x⋆∥2,
+ς2
+⋆ = 1
+n
+�n
+i=1 ∥∇fi(x⋆)∥2, ς2
+0 = 1
+n
+�n
+i=1 ∥∇fi(x0) − ∇f(x0)∥2, x0 is the initialization for all nodes, and x⋆ is an optimal solution of (1).
+Reference
+Iterations to ϵ accuracy
+Remark
+Convex
+[13]
+1
+nϵ2 +
+log(
+1
+1−λ )1/2
+(1−λ)1/2
+1
+ϵ3/2 +
+log(
+1
+1−λ )(a0+ς2
+0 )
+1−λ
+1
+ϵ
+Rate holds only when iteration number K >
+log(
+1
+1−λ )
+1−λ
+Convex
+Our work
+1
+nϵ2 +
+1
+(1−λ)1/2
+1
+ϵ3/2 + (a0+ς2
+⋆)
+(1−λ)
+1
+ϵ
+–
+SC
+[9]
+1
+nϵ +
+1
+(1−λ)3/2
+1
+√ϵ + C
+√ϵ
+C depends on 1/(1 − λ)
+PL∗
+[10]
+1
+nϵ +
+1
+(1−λ)3/2
+1
+√ϵ + ˜C log 1
+ϵ
+˜C depends on 1/(1 − λ)
+SC
+[13]
+1
+nϵ +
+log(
+1
+1−λ )1/2
+(1−λ)1/2
+1
+√ϵ +
+log(
+1
+1−λ )
+(1−λ)
+log
+�
+(a0+ς2
+0 )
+(1−λ)ϵ
+�
+Rate holds only when iteration number K >
+log(
+1
+1−λ )
+1−λ
+PL∗
+[14]
+1
+nϵ +
+�
+1
+(1−λ)1/2 +
+1
+(1−λ)√n
+� 1
+√ϵ +
+1
+1−λ log
+�
+(a0+ς2
+⋆)
+ϵ
+�
+Rate holds by tuning stepsize from [14, Theorem 2]
+SC
+Our work
+1
+nϵ +
+1
+(1−λ)1/2
+1
+√ϵ +
+1
+1 − λ log
+�
+(a0+ς2
+⋆)
+ϵ
+�
+–
+∗ The PL condition is weaker than SC and can hold for nonconvex functions; any SC function satisfies the PL condition.
+be useful for analyzing GT methods in other settings such
+as variance-reduced gradients.
+• In stochastic and convex environments, our convergence
+rate improve and tighten existing GT bounds. We show,
+in particular, that under convex settings, GT methods
+have better dependence on network topologies than in
+nonconvex settings [14]. Also, our bounds removes the
+network dependent log factors in [13] – See Table I.
+II. ATC-GT and Main Assumption
+In this section, we describe the GT algorithm (2) in network
+notation and list all necessary assumptions. We begin by defin-
+ing some network quantities.
+A. GT in network notation
+We define xk
+i ∈ Rd as the estimated value of x ∈ Rd at
+agent i and iteration (time) k, and we introduce the augmented
+network quantities:
+xk ≜ col{xk
+1, . . . , xk
+n} ∈ Rdn
+f(xk) ≜
+n
+�
+i=1
+fi(xk
+i )
+∇f(xk) ≜ col{∇f1(xk
+1), . . . , ∇fn(xk
+n)}
+∇F(xk) ≜ col{∇F1(xk
+1; ξk
+1), . . . , ∇Fn(xk
+n; ξk
+n)}
+gk ≜ col{gk
+1, . . . , gk
+n} ∈ Rdn.
+Here, col{·} is an operation to stack all vectors on top of each
+other. In addition, we define
+W ≜ [wij] ∈ Rn×n,
+W ≜ W ⊗ Id,
+(3)
+where W is the network weight (or combination, mixing, gossip)
+matrix with elements wij, and symbol ⊗ denotes the Kronecker
+product operation. Using the above quantities, the ATC-GT
+method (2) can be described as follows:
+xk+1 = W[xk − αgk]
+(4a)
+gk+1 = W[gk + ∇F(xk+1) − ∇F(xk)],
+(4b)
+with initialization g0 = ∇F(x0) and arbitrary x0.
+B. Assumptions
+Here, we list the assumptions used in our analyses. Our first
+assumption is on the network graph stated below.
+Assumption 1 (Weight matrix). The network graph is as-
+sumed to be static and, the weight matrix W to be doubly
+stochastic and primitive. We further assume W to be symmetric
+and positive semidefinite.
+■
+It is important to note that assuming W
+to be positive
+semidefinite is not restrictive; given any doubly stochastic and
+symmetric ˜
+W, we can easily construct a positive semidefinite
+weight matrix by W = (I + ˜
+W)/2. We also remark that, under
+Assumption 1, the mixing rate of the network is:
+λ ≜
+��W − 1
+n11T�� =
+max
+i∈{2,...,n} |λi| < 1.
+(5)
+The next assumption is on the objective function.
+Assumption 2 (Objective function). Each function fi :
+Rd → R is L-smooth
+∥∇fi(y) − ∇fi(z)∥ ≤ L∥y − z∥,
+∀ y, z ∈ Rd
+(6)
+and (µ-strongly) convex for some L ≥ µ ≥ 0. As a result, the
+aggregate function f(x) =
+1
+n
+�n
+i=1 fi(x) is also L-smooth and
+(µ-strongly) convex. (When µ = 0, then the objective functions
+are simply convex.)
+■
+We now state our final assumption related to the gradient
+noise.
+Assumption 3 (Gradient noise). For all {i}n
+i=1 and k =
+0, 1, . . ., we assume the following inequalities hold
+E �
+∇Fi(xk
+i ; ξk
+i ) − ∇fi(xk
+i ) | F k�
+= 0,
+(7a)
+E �
+∥∇Fi(xk
+i ; ξk
+i ) − ∇fi(xk
+i )∥2 | F k�
+≤ σ2,
+(7b)
+for some σ2 ≥ 0, where F k ≜ {x0, x2, . . . , xk} is the algorithm-
+generated filtration. We further assume that conditioned on F k,
+the random data {ξt
+i} are independent of one another for any
+{i}n
+i=1 and {t}t≤k.
+■
+III. Error Recursion
+To establish the convergence of (4), we will first derive
+an error recursion that will be key to our enhanced bounds.
+
+3
+Motivated by [14], the following result rewrites algorithm (4)
+in an equivalent manner.
+Lemma 1 (Equivalent GT form). Let x0 take any arbitrary
+value and z0 = 0. Then for static graphs, the update for xk in
+algorithm (4) is equivalent to following updates for k = 1, 2, . . .
+xk+1 = (2W − I)xk − αW2∇F(xk) − Bzk
+(8a)
+zk+1 = zk + Bxk
+(8b)
+with initialization x1 = W(x0 − α∇F(x0)) and z1 = Bx0, and
+B = I − W.
+Proof. Clearly with the above initialization, both x1 are iden-
+tical for the updates (4) and (8). Now, for k ≥ 1, it holds from
+(8a) that
+xk+1 − xk = (2W − I)(xk − xk−1) − B(zk − zk−1)
+− αW2(∇F(xk) − ∇F(xk−1)).
+Substituting zk − zk−1 = Bxk−1 ((8b)) and B = I − W into
+the above equation and rearranging the recursion gives
+xk+1 = 2Wxk − W2xk−1 − αW2(∇F(xk) − ∇F(xk−1)).
+Following the same approach, we can also describe the xk
+update for the GT algorithm (4) as above – see [14], [29]. Hence,
+both methods are equivalent for static graph W.
+Under Assumption 1, the fixed point of recursion (8), denoted
+by (x⋆, z⋆), satisfies:
+0 = αW2∇f(x⋆) + Bz⋆
+0 = Bx⋆.
+(9)
+where x⋆ = 1 ⊗ x⋆ and x⋆ is the optimal solution of (1). The
+existence of z⋆ can be shown by using similar arguments as in
+[30, Lemma 3.1] or [29, Lemma 1]. By introducing the notation
+˜xk ≜ xk − x⋆,
+˜z ≜ zk − z⋆,
+(10)
+using (8) and the fact (2W − I)x⋆ = x⋆, we can get the error
+recursion:
+�
+˜xk+1
+˜zk+1
+�
+=
+�
+2W − I
+−B
+B
+I
+� �
+˜xk
+˜zk
+�
+− α
+�
+W2�
+∇f(xk) − ∇f(x⋆) + vk�
+0
+�
+,
+(11)
+where vk ≜ ∇F(xk) − ∇f(xk).
+Remark 1 (Alternative analysis approach). By describing
+GT (4) in the alternative form (8), we are able to derive the
+error recursion from the fixed point (11). This is similar to the
+way Exact-diffusion/D2 is analyzed in [4], [12]. This alternative
+approach allows us to derive tighter bounds compared with
+existing GT works [9], [10], [13], [14].
+■
+Convergence analysis of (11) still remains difficult. We will
+exploit the properties of the matrix W to transform recursion
+(11) into a more suitable form for our analysis. To that end,
+the following quantities are introduced:
+¯xk ≜ 1
+n(1T
+n ⊗ Id)xk = 1
+n
+n
+�
+i=1
+xk
+i ,
+(12a)
+¯ek
+x ≜ 1
+n(1T
+n ⊗ Id)˜xk = ¯xk − x⋆,
+(12b)
+∇f(xk) ≜ 1
+n(1T
+n ⊗ Id)∇f(xk) = 1
+n
+n
+�
+i=1
+∇fi(xk
+i ),
+(12c)
+¯vk ≜ 1
+n(1T
+n ⊗ Id)vk.
+(12d)
+Under Assumption 1, the matrix W admits the following eigen-
+decomposition:
+W = UΣU−1 = �
+1 ⊗ Id
+ˆU
+�
+�
+��
+�
+U
+�
+Id
+0
+0
+Λ
+�
+�
+��
+�
+Σ
+� 1
+n1T ⊗ Id
+ˆUT
+�
+�
+��
+�
+U−1
+(13)
+where Λ is a diagonal matrix with eigenvalues strictly less than
+one and ˆU is an dn × d(n − 1) matrix that satisfies
+ˆUT ˆU = I,
+(1T ⊗ Id) ˆU = 0
+(14a)
+ˆU ˆUT = I − 1
+n11T ⊗ Id.
+(14b)
+Lemma 2 (Decomposed error recursion). Under Assump-
+tion 1, there exists matrices ˆV and Γ to transform the error
+recursion (11) into the following form:
+¯ek+1
+x
+= ¯ek
+x − α∇f(xk) + α¯vk,
+(15a)
+ˆxk+1 = Γˆxk − α ˆV−1
+l
+Λ2 ˆUT�
+∇f(xk) − ∇f(x⋆) + vk�
+,
+(15b)
+where
+ˆxk ≜ ˆV−1
+� ˆUT˜xk
+ˆUT˜zk
+�
+,
+(16)
+and ˆV−1
+l
+denotes the left block of ˆV−1 = [ ˆV−1
+l
+ˆV−1
+r ]. Moreover,
+the following bounds hold:
+∥ ˆV∥2 ≤ 3,
+∥ ˆV−1∥2 ≤ 9,
+∥Γ∥ ≤ 1+λ
+2 ,
+(17)
+where λ = maxi∈{2,...,n} λi.
+Proof. See Appendix A
+The preceding result will serve as the starting point for deriv-
+ing the bounds that will lead us to our conclusions. Specifically,
+we can derive the following bounds from the above result.
+Lemma 3 (Coupled error inequality). Suppose Assump-
+tions 1–2 hold. Then, if α <
+1
+4L, we have
+E ∥¯ek+1
+x
+∥2 ≤ (1 − µα) E ∥¯ek
+x∥2 − α�
+E f(¯xk) − f(x⋆)�
++ 3αc2
+1L
+2n
+E ∥ˆxk∥2 + α2σ2
+n
+,
+(18)
+and
+E ∥ˆxk+1∥2 ≤ γ E ∥ˆxk∥2 + α2c2
+2λ4
+(1 − γ) E ∥∇f(xk) − ∇f(x⋆)∥2
++ α2c2
+2λ4nσ2,
+(19)
+where γ ≜ ∥Γ∥, c1 ≜ ∥ ˆV∥, and c2 = ∥ ˆV−1∥.
+Proof. See Appendix B.
+IV. Convergence Results
+In this section, we present our main convergence results in
+Theorems 1 and 2. We then discuss our results and highlight
+the differences with existing bounds.
+Theorem 1 (Convex case). Suppose that Assumptions 1-2
+are satisfied. Then, there exists a constant stepsize α such that
+1
+K
+K−1
+�
+k=0
+�
+E[f(¯xk) − f ⋆] + L
+n E ∥xk − 1 ⊗ ¯xk∥2�
+≤ σ∥¯e0
+x∥
+√
+nK
++
+�
+Lλ4σ2
+1 − λ
+�1/3 �
+∥¯e0
+x∥2
+K
+� 2
+3
+
+4
++
+�
+Lλ2
+1 − λ∥¯e0
+x∥2 +
+ς2
+⋆
+L(1 − λ)
+�
+C
+K ,
+(20)
+where ¯e0
+x ≜ ¯x0 − x⋆, ς2
+⋆ ≜
+1
+n
+�n
+i=1 ∥∇fi(x⋆)∥2, and C is an
+absolute constant.
+Proof. See Appendix C.
+Theorem 2 (Strongly-convex case). Suppose that Assump-
+tions 1-2 are satisfied. Then, there exists a constant stepsize α
+such that
+E ∥¯eK
+x ∥2 + 1
+n∥xK − 1 ⊗ ¯xK∥2 ≤ ˜O
+�
+σ2
+nK +
+σ2
+(1 − λ)K2
+�
++ ˜O
+�
+σ2
+(1 − λ)2nK3 + (a0 + ς2
+⋆) exp [−(1 − λ)K]
+�
+,
+(21)
+where a0 ≜ ∥¯x0 − x⋆∥2, ς2
+⋆ ≜
+1
+n
+�n
+i=1 ∥∇fi(x⋆)∥2, and the
+notation ˜O(·) ignores logarithmic factors.
+Proof. See Appendix D.
+In comparison to [13], our results removes the log factor
+O(log(
+1
+1−λ)) and holds for any number of iteration K – see
+Table I. Moreover, observe that for the strongly-convex case,
+unlike [13], we do not have a network term 1/(1−λ) multiplying
+the highest order exponential term exp(·).
+Remark 2 (Improvement upon nonconvex GT rates). The
+GT rates for convex and strongly-convex settings provided in
+Theorems 1 and 2 improve upon the GT rates for non-convex
+[13], [14] and PL condition [14] settings. For example, observe
+from Table I that the GT rate under the PL condition [14] is
+1
+nϵ +
+�
+1
+(1−λ)1/2 +
+1
+(1−λ)√n
+�
+1
+√ϵ +
+1
+1−λ log
+�
+(a0+ς2
+⋆)
+ϵ
+�
+, which has
+an additional term
+1
+(1−λ)√n
+1
+√ϵ compared to our strongly-convex
+rate.
+■
+Remark 3 (Comparison with Exact-diffusion/D2 [12]). For
+the convex case, the difference with Exact-diffusion/D2 [12] is in
+the highest order term. Exact-diffusion/D2 is
+�
+a0
+(1−λ) + ς2
+⋆
+�
+1
+K
+while GT is
+�
+a0
+(1−λ) +
+ς2
+⋆
+(1−λ)
+�
+1
+K where GT has 1/(1 − λ) mul-
+tiplied by ς2
+⋆, which is slightly worse than Exact-diffusion/D2.
+A similar conclusion can be reached for the strongly-convex
+scenario.
+■
+V. Simulation results
+This section will present several numerical simulations that
+compare Gradient-tracking with centralized SGD (CSGD) and
+decentralized SGD (DSGD).
+Linear regression. We consider solving a strongly-convex
+problem (1) with fi(x) =
+1
+2E(aT
+i x − bi)2 in which random
+variable ai ∼ N(0, Id), bi = aT
+i x⋆
+i + ni for some local so-
+lution x⋆
+i ∈ Rd and ni ∼ N(0, σ2
+n). The stochastic gradient
+is calculated as ∇Fi(x) = ai(aT
+i x − bi). Each local solution
+x⋆
+i = x⋆ +vi is generated using the formula x⋆
+i = x⋆ +vi, where
+x⋆ ∼ N(0, Id) is a randomly generated global solution while
+vi ∼ N(0, σ2
+vId) controls similarities between local solutions.
+Generally speaking, a large σ2
+v will result in local solutions
+{x⋆
+i }n
+i=1 that are vastly different from one another. We used
+d = 5, σ2
+n = 0.01, and σ2
+v = 1 in simulations. Experiments
+are carried out on ring and exponential graphs of size n = 30,
+respectively. Each algorithm’s stepsize (learning rate) is care-
+fully tuned so that they all converge to the same relative mean-
+square-error. Each simulation is run 30 times, with the solid line
+representing average performance and the shadow representing
+0
+100
+200
+300
+400
+iteration
+10
+4
+10
+3
+10
+2
+10
+1
+100
+relative error
+Exponential graph with 30 nodes
+CSGD
+DSGD
+Gradient-Tracking
+0
+250
+500
+750
+1000 1250 1500 1750 2000
+iteration
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+relative error
+Ring graph with 30 nodes
+CSGD
+DSGD
+Gradient-Tracking
+Fig. 1: Comparison between different algorithms over exponential
+and ring graphs when solving distributed linear regression with
+heterogeneous data distributions. The spectral gap 1 − λ is 0.33
+and 0.0146 for exponential and ring graphs, respectively.
+0
+200
+400
+600
+800
+iteration
+10
+4
+10
+3
+10
+2
+10
+1
+100
+relative error
+Exponential graph with 30 nodes
+CSGD
+DSGD
+Gradient-Tracking
+0
+200
+400
+600
+800
+1000 1200 1400 1600
+iteration
+10
+4
+10
+3
+10
+2
+10
+1
+100
+relative error
+Ring graph with 30 nodes
+CSGD
+DSGD
+Gradient-Tracking
+Fig. 2: Comparison between different algorithms over exponential
+and ring graphs when solving distributed logistic regression.
+standard deviation. The results are depicted in Fig. 1. The rela-
+tive error is shown on the y-axis as 1
+n
+�n
+i=1 E∥xk
+i −x⋆∥2/∥x⋆∥2.
+When running over the exponential graph which has a well-
+connected topology with 1 − λ = 0.33, it is observed that
+both DSGD and Gradient-tracking perform similarly to CSGD.
+However, when running over the ring graph which has a badly-
+connected topology with 1 − λ = 0.0146, DSGD gets far
+slower than CSGD due to its sensitivity to network topology.
+In contrast, Gradient-tracking just gets a little bit slower than
+CSGD and performs far better than DSGD. This phenomenon
+coincides with our established complexity bound in Table I
+showing that GT has a much weaker dependence on network
+topology (i.e., 1 − λ).
+Logistic regression. We next consider the logistic regres-
+sion problem, which has fi(x) = E ln(1 + exp(−yihT
+i x)) where
+(hi, yi) represents the training dataset stored in node i with
+hi ∈ Rd as the feature vector and yi ∈ −{1, +1} as the label.
+This is a convex but not strongly-convex problem. Similar to
+the linear regression experiments, we will first generate a local
+solution x⋆
+i based on x⋆
+i = x⋆ + vi using vi ∼ N(0, σ2
+vId). We
+can generate local data that follows distinct distributions using
+x⋆
+i . To this end, we generate each feature vector hi ∼ N(0, Id)
+at node i. To produce the corresponding label yi, we create a
+random variable zi ∼ U(0, 1). If zi ≤ 1 + exp(−yihT
+i x⋆
+i ), we
+set yi = 1; otherwise yi = −1. Clearly, solution x⋆
+i controls
+the distribution of the labels. By adjusting σ2
+v, we can easily
+control data heterogeneity. The remaining parameters are the
+same as in linear regression experiments. The performances
+of each algorithm in logistic regression depicted in Fig. 2 are
+consistent with that in linear regression, i.e., Gradient-tracking
+performs well for both graphs while DSGD has a significantly
+deteriorated performance over the ring graph due to its less
+robustness to network topology.
+Appendix A
+Decomposed Error Recursion
+
+5
+Prof of Lemma 2
+Using the decomposition (13) and B = I − W:
+W2 = UΣ2U−1 = �
+1 ⊗ Id
+ˆU
+� �
+Id
+0
+0
+Λ2
+� � 1
+n1T ⊗ Id
+ˆUT
+�
+(22a)
+B = U(I − Σ)U−1 = �
+1 ⊗ Id
+ˆU
+� �
+0
+0
+0
+I − Λ
+� � 1
+n1T ⊗ Id
+ˆUT
+�
+,
+(22b)
+with I − Λ > 0. Substituting (22) into (11) and multiplying
+both sides by blkdiag{U−1, U−1} on the left, we obtain
+�
+U−1˜xk+1
+U−1˜zk+1
+�
+=
+�
+2Σ2 − I
+−(I − Σ)
+I − Σ
+I
+� �
+U−1˜xk
+U−1˜zk
+�
+− α
+�
+Σ2U−1�
+∇f(xk) − ∇f(x⋆) + vk�
+0
+�
+.
+(23)
+Since ˜zk always lies in the range space of B, we have (1T
+n ⊗
+Id)˜zk = 0 for all k. Using, the structure of U from (13) and the
+definitions (12), we have
+U−1˜xk =
+�
+¯ek
+x
+ˆUT˜xk
+�
+,
+U−1˜zk =
+�
+0
+ˆUT˜zk
+�
+U−1∇f(x) =
+�
+∇f(xk)
+ˆUT∇f(x)
+�
+.
+Thus, by using the structure of Σ2 and Σ2
+b given in (22), we
+can rewrite (23) as
+¯ek+1
+x
+= ¯ek
+x − α�
+∇f(xk) − ∇f(x⋆)�
+(24a)
+� ˆUT˜xk+1
+ˆUT˜zk+1
+�
+=
+�
+2Λ − I
+−(I − Λ)
+I − Λ
+I
+� � ˆUT˜xk
+ˆUT˜zk
+�
+− α
+�
+Λ2 ˆUT�
+∇f(xk) − ∇f(x⋆)vk�
+0
+�
+.
+(24b)
+Let
+G ≜
+�
+2Λ − I
+−(I − Λ)
+I − Λ
+I
+�
+.
+(25)
+It is important to note that the matrix G is identical to the one
+studied in [14] (for nonconvex case). Therefore, following the
+same arguments used in [14, Appendix B], we can decompose it
+as G = ˆVΓ ˆV−1 for matrices ˆV and Γ satisfying the conditions
+in the lemma. Multiplying the second equation in (24) by ˆV−1,
+we arrive at (15).
+Appendix B
+Coupled Error Inequalities
+Proof of Lemma 3
+Proof of inequality (18)
+The proof adjusts the argument from [31, Lemma 8]. Using
+(15a) and Assumption 3, we have
+E[∥¯ek+1
+x
+∥2|F k]
+= ∥¯ek
+x − α
+n
+�n
+i=1(∇fi(xk
+i ) − ∇fi(x⋆))∥2 + α2 E[∥¯vk∥2|F k]
+≤ ∥¯ek
+x − α
+n
+�n
+i=1(∇fi(xk
+i ) − ∇fi(x⋆))∥2 + α2σ2
+n
+= ∥¯ek
+x∥2 + α2∥ 1
+n
+n�
+i=1
+(∇fi(xk
+i ) − ∇fi(x⋆))∥2
+− 2α
+n
+n�
+i=1
+�
+∇fi(xk
+i ), ¯ek
+x
+�
++ α2σ2
+n
+,
+(26)
+where we used �n
+i=1 ∇fi(x⋆) = 0. The second term on the right
+can be bounded as follows:
+α2∥ 1
+n
+n�
+i=1
+�
+∇fi(xk
+i ) − ∇fi(¯xk) + ∇fi(¯xk) − ∇fi(x⋆)�
+∥2
+≤ 2α2∥ 1
+n
+n�
+i=1
+(∇fi(xk
+i ) − ∇fi(¯xk))∥2
++ 2α2∥ 1
+n
+n�
+i=1
+(∇fi(¯xk) − ∇fi(x⋆))∥2
+≤ 2α2
+n
+n�
+i=1
+∥∇fi(xk
+i ) − ∇fi(¯xk)∥2
+(27)
++ 2α2∥∇f(¯xk) − ∇f(x⋆)∥2
+≤ 2α2L2
+n
+∥xk − 1 ⊗ ¯xk∥2 + 2α2∥∇f(¯xk) − ∇f(x⋆)∥2
+≤ 2α2L2
+n
+∥xk − 1 ⊗ ¯xk∥2 + 4Lα2(f(¯xk) − f(x⋆)),
+(28)
+where the first two inequalities follows from Jensen’s inequal-
+ity. The third inequality follows from the Lipschitz gradient
+assumption. In the last inequality, we used the L-smoothness
+property of the aggregate function [32]:
+∥∇f(¯xk) − ∇f(x⋆)∥2 ≤ 2L�
+f(¯xk) − f(x⋆)�
+.
+Note that for L-smooth and µ-strongly-convex function f, it
+holds that [32]:
+f(x) − f(y) − L
+2 ∥x − y∥2 ≤ ⟨∇f(y), (x − y)⟩
+(29a)
+f(x) − f(y) + µ
+2 ∥x − y∥2 ≤ ⟨∇f(x), (x − y)⟩.
+(29b)
+Using these inequalities, the cross term in (28) can be bounded
+by
+− 2α
+n
+n�
+i=1
+⟨∇fi(xk
+i ), ¯ek
+x⟩
+= 2α
+n
+n�
+i=1
+�
+− ⟨∇fi(xk
+i ), ¯xk − xk
+i ⟩ − ⟨∇fi(xk
+i ), xk
+i − x⋆⟩�
+≤ 2α
+n
+n�
+i=1
+�
+− fi(¯xk) + fi(xk
+i ) + L
+2 ∥¯xk − xk
+i ∥2
+− µ
+2 ∥xk
+i − x⋆∥2 − fi(xk
+i ) + fi(x⋆)
+�
+≤ −2α�
+f(¯xk) − f(x⋆)�
++ Lα
+n
+n�
+i=1
+∥¯xk − xk
+i ∥2 − µα∥¯xk − x⋆∥2
+= −2α�
+f(¯xk) − f(x⋆)�
++ Lα
+n ∥xk − 1 ⊗ ¯xk∥2 − µα∥¯ek
+x∥2,
+(30)
+where the last inequality holds due to − 1
+n
+�n
+i=1 ∥xk
+i − x⋆∥2 ≤
+−∥ 1
+n
+�n
+i=1(xk
+i −x⋆)∥2. Substituting (28) and (30) into (26) and
+taking expectation, we obtain:
+E ∥¯ek+1
+x
+∥2 ≤ (1 − µα) E ∥¯ek
+x∥2 − 2α(1 − 2Lα) E �
+f(¯xk) − f(x⋆)�
++ αL
+n (1 + 2αL) E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2
+n
+≤ (1 − µα) E ∥¯ek
+x∥2 − α�
+E f(¯xk) − f(x⋆)�
++ 3Lα
+2n E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2
+n
+,
+(31)
+where the last step uses α
+≤
+1
+4L. Using (14), we have
+∥ ˆUT˜xk∥2 = ∥ ˆUT ˆU˜xk∥2 = ∥xk − 1 ⊗ ¯xk∥2. Hence,
+∥xk − 1 ⊗ ¯xk∥2 (16)
+= ∥ ˆVˆxk∥2 − ∥ ˆUT˜zk∥2 ≤ ∥ ˆV∥2∥ˆxk∥2.
+(32)
+Substituting the above into (31) yields (18).
+
+6
+Proof of inequality (19)
+From (15b), we have
+E[∥ˆxk+1∥2|F k]
+= E
+���Γˆxk − α ˆV−1
+l
+Λ2 ˆUT�
+∇f(xk) − ∇f(x⋆) + vk�
+|F k���
+2
+(7a)
+=
+���Γˆxk − α ˆV−1
+l
+Λ2 ˆUT�
+∇f(xk) − ∇f(x⋆)����
+2
++ α2 E
+��� ˆV−1
+l
+Λ2 ˆUTvk��F k���
+2
+(7b)
+≤
+���Γˆxk − α ˆV−1
+l
+Λ2 ˆUT�
+∇f(xk) − ∇f(x⋆)����
+2
++ α2∥ ˆV−1
+l
+∥2∥Λ2∥2∥ ˆUT∥2nσ2.
+Now, for any vectors a and b, it holds from Jensen’s inequality
+that ∥a + b∥2 ≤ 1
+θ ∥a∥2 +
+1
+1−θ ∥b∥ for any θ ∈ (0, 1). Utilizing
+this bound with θ = γ ≜ ∥Γ∥ on the first term of the previous
+inequality, we get
+E[∥ˆxk+1∥2|F k]
+≤ γ∥ˆxk∥2 +
+α2∥ ˆ
+V−1
+l
+∥2∥Λ2∥2∥ ˆ
+UT∥2
+(1−γ)
+∥∇f(xk) − ∇f(x⋆)∥2
++ α2∥ ˆV−1
+l
+∥2∥Λ2∥2∥ ˆUT∥2nσ2.
+Taking expectation and using ∥ ˆUT∥ ≤ 1, ∥ ˆV−1
+l
+∥2 ≤ ∥ ˆV−1∥2,
+and ∥Λ2∥2 ≤ λ4 yield our result (19).
+Appendix C
+Proof of Theorem 1
+Using similar argument to (28) and (32), it holds that
+∥∇f(xk) − ∇f(x⋆)∥2
+≤ 2∥∇f(1 ⊗ ¯xk) − ∇f(x⋆)∥2 + 2∥∇f(xk) − ∇f(1 ⊗ ¯xk)∥2
+≤ 4nL[f(¯xk) − f(x⋆)] + 2c2
+1L2∥ˆxk∥2.
+Plugging the above bound into (19) gives
+E ∥ˆxk+1∥2 ≤
+�
+γ +
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+�
+E ∥ˆxk∥2
++
+4α2c2
+2Lλ4n
+(1−γ)
+E ˜f(¯xk) + α2c2
+2λ4nσ2
+≤ ¯γ E ∥ˆxk∥2 +
+4α2c2
+2Lλ4n
+(1−γ)
+E ˜f(¯xk) + α2c2
+2λ4nσ2,
+where ˜f(¯xk) ≜ f(¯xk)−f(x⋆), ¯γ ≜ 1+γ
+2 , and the last inequiality
+holds when γ +
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+≤ 1+γ
+2 , which is satisfied for
+α ≤
+1 − λ
+4c1c2Lλ2 .
+(33)
+Iterating the last recursion (for any k = 1, 2, . . . ) gives
+E ∥ˆxk∥2 ≤ ¯γk∥ˆx0∥2 +
+4α2c2
+2Lλ4n
+(1−γ)
+k−1
+�
+ℓ=0
+¯γk−1−ℓ E ˜f(¯xℓ)
++
+k−1
+�
+ℓ=0
+¯γk−1−ℓ �
+α2c2
+2λ4nσ2�
+≤ ¯γk∥ˆx0∥2 +
+4α2c2
+2Lλ4n
+(1−γ)
+k−1
+�
+ℓ=0
+¯γk−1−ℓ E ˜f(¯xℓ)
++
+α2c2
+2λ4nσ2
+1−¯γ
+.
+(34)
+In the last inequality we used �k−1
+ℓ=0 ¯γk−1−ℓ ≤
+1
+1−¯γ . Averaging
+over k = 1, 2 . . . , K and using ¯γ = 1+γ
+2 , it holds that
+1
+K
+K
+�
+k=1
+E ∥ˆxk∥2
+≤
+2∥ˆx0∥2
+(1−γ)K +
+4α2c2
+2Lλ4n
+(1−γ)K
+K
+�
+k=1
+k−1
+�
+ℓ=0
+� 1+γ
+2
+�k−1−ℓ E ˜f(¯xℓ) +
+2α2c2
+2λ4nσ2
+1−γ
+≤
+2∥ˆx0∥2
+(1−γ)K +
+8α2c2
+2Lλ4n
+(1−γ)2K
+K−1
+�
+k=0
+E ˜f(¯xk) +
+2α2c2
+2λ4nσ2
+1−γ
+.
+(35)
+It follows that
+1
+K
+K−1
+�
+k=0
+E ∥ˆxk∥2 ≤
+3∥ˆx0∥2
+(1 − γ)K +
+8α2c2
+2Lλ4n
+(1−γ)2K
+K−1
+�
+k=0
+E ˜f(¯xk)
++ 2α2c2
+2λ4nσ2
+1 − γ
+.
+(36)
+where we added
+∥ˆx0∥2
+(1−γ)K and used ∥ˆx0∥2
+K
+≤
+∥ˆx0∥2
+(1−γ)K . Now when
+µ = 0, we can rearrange (18) to get
+E(f(¯xk) − f(x⋆)) ≤ 1
+α
+�
+E ∥¯ek
+x∥2 − E ∥¯ek+1
+x
+∥2�
++ 3c2
+1L
+2n E ∥ˆxk∥2 + ασ2
+n .
+(37)
+Averaging over k = 0, . . . , K − 1 (K ≥ 1), it holds that
+1
+K
+K−1
+�
+k=0
+E ˜f(¯xk) ≤
+∥¯e0
+x∥2
+αK
++
+3c2
+1L
+2nK
+K−1
+�
+k=0
+E ∥ˆxk∥2 + ασ2
+n .
+(38)
+Multiplying inequality (36) by 2 ×
+3c2
+1L
+2n , adding to (38), and
+rearranging we obtain
+�
+1 −
+24α2c2
+1c2
+2L2λ4
+(1−γ)2
+� 1
+K
+K−1
+�
+k=0
+E ˜f(¯xk) +
+3c2
+1L
+2nK
+K−1
+�
+k=0
+E ∥ˆxk∥2
+≤ ∥¯e0
+x∥2
+αK
++ 9c2
+1L∥ˆx0∥2
+(1 − γ)nK + ασ2
+n
++ 6α2c2
+1c2
+2Lλ4σ2
+1 − γ
+.
+(39)
+Notice from (16) that
+∥ˆx0∥2 ≤ ∥ ˆV−1∥2 �
+∥ ˆUT˜x0∥2 + ∥ ˆUT˜z0∥2�
+.
+(40)
+If we start from consensual initialization x0 = 1 ⊗ x0 and use
+the fact z0 = 0, the above reduces to
+∥ˆx0∥2 ≤ ∥ ˆV−1∥2∥ ˆUTz⋆∥2 ≤ α2c2
+2λ4
+(1 − λ)2 ∥ ˆUT∇f(x⋆)∥2,
+(41)
+where the last step holds by using (9) and (22), which implies
+that ˆUTz⋆ = α(I − Λ)−1Λ2 ˆUT∇f(x⋆). Plugging the previous
+inequality into (39) and setting 1
+2 ≤ 1 −
+24α2c2
+1c2
+2L2λ4
+(1−γ)2
+, i.e.,
+α ≤
+1 − λ
+4
+√
+6c1c2Lλ2 ,
+(42)
+gives
+1
+K
+K−1
+�
+k=0
+Ek ≤ ∥¯e0
+x∥2
+αK
++ a1α + a2α2
+�
+��
+�
+≜ΨK
++a⋆α2
+K
+,
+(43)
+where we defined Ek ≜ 1
+2 E ˜f(¯xk) +
+3c2
+1L
+2n E ∥ˆxk∥2 and
+a⋆ ≜ 18c2
+1c2
+2Lλ4∥ ˆUT∇f(x⋆)∥2
+(1 − λ)3n
+(44a)
+a1 ≜ σ2
+n
+a2 ≜ 12c2
+1c2
+2Lλ4σ2
+1 − λ
+.
+(44b)
+We now select the stepsize α to arrive at our result in a manner
+similar to [31]. First note that the previous inequality holds for
+α ≤ 1
+α ≜ min
+�
+1
+4L,
+1 − λ
+4
+√
+6c1c2Lλ2
+�
+.
+(45)
+
+7
+Setting α = min
+��
+∥¯e0
+x∥2
+a1K
+� 1
+2 ,
+�
+∥¯e0
+x∥2
+a2K
+� 1
+3 , 1
+α
+�
+≤
+1
+α we have
+three cases: i) If α = 1
+α, which is smaller than both
+�
+∥¯e0
+x∥2
+a1K
+� 1
+2
+and
+�
+∥¯e0
+x∥2
+a2K
+� 1
+3 , then
+ΨK = α∥¯e0
+x∥2
+K
++ a1
+α + a2
+α2
+≤ α∥¯e0
+x∥2
+K
++
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2
++ a
+1
+3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
+;
+ii) If α =
+�
+∥¯e0
+x∥2
+a1K
+� 1
+2 <
+�
+∥¯e0
+x∥2
+a2K
+� 1
+3 , then
+ΨK ≤ 2
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2
++ a2
+�
+∥¯e0
+x∥2
+a1K
+�
+≤ 2
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2
++ a
+1
+3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
+;
+iii) If α =
+�
+∥¯e0
+x∥2
+a2K
+� 1
+3 <
+�
+∥¯e0
+x∥2
+a1K
+� 1
+2 , then
+ΨK ≤ 2a
+1
+3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
++ a1
+�
+∥¯e0
+x∥2
+a2K
+� 1
+3
+≤ 2a
+1
+3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
++
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2
+.
+Combining the above cases, we have
+ΨK ≤ 2
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2
++ 2a1/3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
++ α∥¯e0
+x∥2
+K
+.
+Therefore, substituting into (43) we conclude that
+1
+K
+K−1
+�
+k=0
+Ek ≤ 2
+�
+a1∥¯e0
+x∥2
+K
+� 1
+2 + 2a
+1
+3
+2
+�
+∥¯e0
+x∥2
+K
+� 2
+3
++
+(α∥¯e0
+x∥2 + a⋆
+α2 )
+K
+.
+Plugging the constants (44) and the upper bound for α in (45),
+and using ς2
+⋆ = 1
+n∥ ˆUT∇f(x⋆)∥2 = 1
+n
+�n
+i=1 ∥∇fi(x⋆)−∇f(x⋆)∥2
+yields our rate (20).
+Appendix D
+Proof of Theorem 2
+Substituting the bound
+∥∇f(xk) − ∇f(x⋆)∥2 ≤ L2∥xk − x⋆∥2
+≤ 2L2∥xk − 1 ⊗ ¯xk∥2 + 2L2∥1 ⊗ ¯xk − x⋆∥2
+≤ 2L2c2
+1∥ˆxk∥2 + 2nL2∥¯ek
+x∥2,
+into (19), we get
+E ∥ˆxk+1∥2
+≤
+�
+γ +
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+�
+E ∥ˆxk∥2 +
+2α2c2
+2L2λ4n
+(1−γ)
+∥¯ek
+x∥2 + α2c2
+2λ4nσ2
+≤
+�1 + γ
+2
+�
+E ∥ˆxk∥2 +
+2α2c2
+2L2λ4n
+(1−γ)
+∥¯ek
+x∥2 + α2c2
+2λ4nσ2,
+(46)
+where we used condition (33) in the last inequality. Using
+−α�
+E f(¯xk) − f(x⋆)�
+≤ 0 in (18) and combining with above,
+it holds that
+� E ∥¯ek+1
+x
+∥2
+c2
+1
+n E ∥ˆxk+1∥2
+�
+≤
+�
+1 − µα
+3
+2αL
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+1+γ
+2
+�
+�
+��
+�
+≜A
+� E ∥¯ek
+x∥2
+c2
+1
+n E ∥ˆxk∥2
+�
++
+�
+α2σ2
+n
+α2c2
+1c2
+2λ4σ2
+�
+�
+��
+�
+≜b
+.
+(47)
+The spectral radius of the matrix A can be upper bounded by:
+ρ(A) ≤ ∥A∥1 = max
+�
+1 − µα +
+2c2
+1c2
+2α2L2λ4
+(1−γ)
+,
+1+γ
+2
++ 3
+2Lα
+�
+≤ 1 − µα
+2 ,
+(48)
+where the last inequality holds under the stepsize condition:
+α ≤ min
+�
+µ(1 − γ)
+4c2
+1c2
+2L2λ4 , 1 − γ
+3L + µ
+�
+.
+(49)
+Since ρ(A) < 1, we can iterate inequality (47) to get
+� E ∥¯ek
+x∥2
+c2
+1
+n E ∥ˆxk∥2
+�
+≤ Ak
+� E ∥¯e0
+x∥2
+c2
+1
+n E ∥ˆx0∥2
+�
++
+k−1
+�
+ℓ=0
+Aℓb
+≤ Ak
+� E ∥¯e0
+x∥2
+c2
+1
+n E ∥ˆx0∥2
+�
++ (I − A)−1b.
+(50)
+Taking the (induced) 1-norm, using the sub-multiplicative
+properties of matrix induced norms, it holds that
+E ∥¯ek
+x∥2 +
+c2
+1
+n E ∥ˆxk∥2 ≤ ∥Ak∥1˜a0 +
+��(I − A)−1b
+��
+1
+≤ ∥A∥k
+1˜a0 +
+��(I − A)−1b
+��
+1 .
+(51)
+where ˜a0 = E ∥¯x0 − x⋆∥2 +
+c2
+1
+n E ∥ˆx0∥2. We now bound the last
+term by noting that
+(I − A)−1b
+=
+1
+det(I−A)
+�
+1−γ
+2
+3
+2αL
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+µα
+�
+b
+=
+1
+αµ(1 − γ)( 1
+2 −
+3α2c2
+1c2
+2L3λ4
+(1−γ)2µ
+)
+�
+1−γ
+2
+3
+2αL
+2α2c2
+1c2
+2L2λ4
+(1−γ)
+µα
+� �
+α2σ2
+n
+α2c2
+1c2
+2λ4σ2
+�
+≤
+4
+αµ(1 − γ)
+�
+�
+(1−γ)α2σ2
+2n
++ 3
+2c2
+1c2
+2α3Lλ4σ2
+2α4c2
+1c2
+2L2λ4σ2
+n(1−γ)
++ α3c2
+1c2
+2µλ4σ2
+�
+� ,
+where det(·) denotes the determinant operation. In the last step
+we used 1
+2 −
+3c2
+1c2
+2α2L3λ4
+(1−γ)2µ
+≥ 1
+4 or α ≤
+√µ(1−γ)
+2
+√
+3c1c2L3/2λ2 . Therefore,
+from (51)
+E ∥¯ek
+x∥2 +
+c2
+1
+n E ∥ˆxk∥2
+≤ (1 − αµ
+2 )k˜a0 +
+��(I − A)−1b
+��
+1
+≤ (1 − αµ
+2 )k˜a0 + 2σ2
+µn α
++
+6c2
+1c2
+2(L/µ)λ4σ2+4c2
+1c2
+2λ4σ2
+1−γ
+α2 +
+8c2
+1c2
+2L2λ4σ2
+µn(1−γ)2
+α3.
+(52)
+Using (1 − αµ
+2 )K ≤ exp(− αµ
+2 K) and (41), it holds that
+E ∥¯eK
+x ∥2 +
+c2
+1
+n E ∥ˆxK∥2
+≤ exp(− αµ
+2 K)(a0 + α2a⋆) + a1α + a2α2 + a3α3,
+(53)
+where
+a0 ≜ E ∥¯x0 − x⋆∥2,
+a⋆ ≜
+c2
+1c2
+2λ4
+(1−λ)2n∥ ˆUT∇f(x⋆)∥2
+(54a)
+a1 ≜ 2σ2
+µn ,
+a2 ≜ 10c2
+1c2
+2Lλ4σ2
+µ(1 − γ)
+(54b)
+a3 ≜ 8c2
+1c2
+2L2λ4σ2
+µn(1 − γ)2 .
+(54c)
+
+8
+Note that by combining all stepsize conditions, it is sufficient
+to require
+α ≤ 1
+α ≜ min
+�
+1 − λ
+8L , µ(1 − λ)
+8c2
+1c2
+2L2λ4 ,
+√µ(1 − λ)
+4
+√
+3c1c2L3/2λ2
+�
+.
+(55)
+We now select
+α = min
+�
+ln
+�
+max
+�
+2, µ2(a0 + a⋆
+α2 ) K
+a1
+��
+/µK, 1
+α
+�
+≤ 1
+α. (56)
+Under this choice the exponential term in (53) can be upper
+bounded as follows. i) If α =
+ln(max{1,µ2(a0+a⋆/α2)K/a1})
+µK
+≤ 1
+α
+then
+exp(− αµ
+2 K)(a0 + α2a⋆)
+≤ ˜O
+�
+(a0 + a⋆
+α2 ) exp
+�
+− ln
+�
+max
+�
+1, µ2(a0 + a⋆
+α2 )K/a1
+����
+= O
+�
+a1
+µK
+�
+;
+ii) Otherwise α = 1
+α ≤
+ln(max{1,µ2(a0+a⋆/α2)K/a1})
+µK
+and
+exp(− αµ
+2 K)(a0 + α2a⋆) = exp
+�
+− µK
+2α
+�
+(a0 + a⋆
+α2 ).
+Therefore, under the stepsize condition (56) it holds that
+E ∥¯eK
+x ∥2 +
+c2
+1
+n E ∥ˆxK∥2
+≤ exp(− αµ
+2 K)(a0 + α2a⋆) + a1α + a2α2 + a3α3
+≤ ˜O
+�
+a1
+µK +
+a2
+µ2K2 +
+a3
+µ3K3 + (a0 + a⋆
+α2 ) exp
+�
+− K
+α
+��
+.
+Plugging the constants (54) into the above inequality, using
+(55) and (32) yields our rate (21).
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+

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+Freeform Islamic Geometric Patterns
+Rebecca Lin1,2 and Craig S. Kaplan3
+1 Massachusetts Institute of Technology, Cambridge, MA, USA
+2 University of British Columbia, Vancouver, BC, Canada
+3 University of Waterloo, Waterloo, ON, Canada
+Abstract. Islamic geometric patterns are a rich and venerable orna-
+mental tradition. Many classic designs feature periodic arrangements of
+rosettes: star shapes surrounded by rings of hexagonal petals. We present
+a new technique for generating ‘freeform’ compositions of rosettes: finite
+designs that freely mix rosettes of unusual sizes while retaining the aes-
+thetics of traditional patterns. We use a circle packing as a scaffolding
+for developing a patch of polygons and fill each polygon with a motif
+based on established constructions from Islamic art.
+Keywords: Islamic geometric patterns · modular design · circle packing
+1
+Introduction
+Islamic geometric patterns are a rich and venerable ornamental tradition [6]. The
+most iconic of these patterns involve periodic arrangements of star shapes, with
+gaps between them filled by additional polygons with less symmetry (Fig. 2).
+Often, the concave corners of stars are filled with rings of petal-shaped hexagons,
+yielding composite shapes called rosettes [25] (Fig. 1).
+We refer to the number of arms in a star or rosette as its order. These motifs
+appear in patterns in several standard orders: multiples of three and four are the
+most common due to their compatibility with rotations in periodic symmetry
+groups. It is challenging to create patterns that incorporate unusual orders or
+unusual combinations of orders. For example, considerable geometric sleight-of-
+hand is required for orders such as 11 and 13 (Fig. 2b), which are incompatible
+with crystallographic symmetries [3, Pg. 484].
+Whether designers employ stars and rosettes of standard or unusual orders,
+they typically construct periodic compositions repeating in two directions in the
+plane. Repetition is one of the hallmarks of ornamentation: surface decoration on
+walls and floors, clothing, and objects should be appealing but not distracting.
+When presented with a periodic pattern, we visually ‘factor’ for a non-repeating
+kernel and a rule for filling the plane with copies of that kernel. Thus the eye
+may casually appreciate the pattern without being overwhelmed by it. Accord-
+ing to Gombrich [18], ‘aesthetic delight lies somewhere between boredom and
+confusion’, a sentiment echoed by many others [21,2,11]. In decorative contexts,
+a measure of boredom helps a pattern recede from conscious attention.
+arXiv:2301.01471v1  [cs.GR]  4 Jan 2023
+
+2
+R. Lin and C.S. Kaplan
+Fig. 1. Freeform designs composed of rosettes of many different orders.
+On the other hand, art benefits from a larger dose of confusion. An artwork
+like a painting is a finite composition that rewards careful study, and so every
+part of that composition can bear some measure of novelty. In contrast, an
+infinite Islamic pattern that pleases the eye when elaborated over a wall might
+lose its appeal if cropped, framed, and hung on that same wall. As an artwork,
+it would have no natural boundary—no broad composition to guide the eye.
+This article presents a technique for constructing ‘freeform’ Islamic geomet-
+ric patterns: finite, non-repetitive arrangements of rosettes intended as self-
+contained compositions rather than as ornamental textures. A few sample com-
+positions appear in Fig. 1. Our freeform designs give us significant flexibility to
+mix and match unusual rosette orders. We move along Gombrich’s continuum,
+away from the boredom of ornamentation and towards the confusion of art. The
+resulting visual experiment allows us to reimagine the canonical motifs of Islamic
+geometric patterns in a highly non-traditional setting.
+We construct motifs based on an initial polygonal patch using a hybrid of
+standard techniques (Sec. 3). We define the overall arrangement of rosettes from
+a circle packing derived from a triangulation (Sec. 4). We show how any circle
+packing can be converted into a patch of connected polygons (Sec. 4.3). We then
+inscribe a motif in each polygon and join the motifs together to form the final
+pattern (Sec. 4.4). This technique is robust over a wide range of rosette orders.
+The designer can control the final pattern by starting with a triangulation of
+their choosing. We support a few additional special effects via ‘gadgets’ that
+perform local surgery on the computed circle packing (Sec. 5). We also adapt
+our technique to periodic patterns via toroidal circle packings (Sec. 6).
+
+rosetteFreeform Islamic Geometric Patterns
+3
+2
+Related Work
+Artists and mathematicians use many strategies to disrupt the potential monoto-
+ny of ornamental Islamic patterns. For example, an artist often introduces mild
+variations in colours, decorative fills, or calligraphic inscriptions in periodic pat-
+terns of otherwise identical stars or rosettes. They also sometimes alter the geom-
+etry at the centres of selected rosettes while maintaining outward compatibility
+with the rest of the pattern.
+As the practice of Islamic geometric patterns grew in sophistication, artists
+sought to incorporate stars or rosettes of unusual orders into their work. The
+Topkapı scroll, a 15th-century visual guide to the drawing of Islamic ornament,
+included a number of patterns with unusual combinations of stars. Cromwell [12]
+analyzed these patterns and articulated rules for their construction. Later, he
+presented a robust method for assembling patterns from irregular stars with dif-
+ferent numbers of points [13]. That work demonstrated the wheel construction
+(Sec. 3), which we will detail and use in our method. More recently, Gailiunas [16]
+studied the amount of geometric error that accumulates when juxtaposing oth-
+erwise incompatible stars.
+Bonner [3] presented a comprehensive treatment of the modular construction
+of Islamic patterns. His polygonal technique, also known as polygons-in-contact
+(PIC) after Hankin [20], builds a motif in every tile of a polygonal tiling (Fig. 2).
+Bonner’s book includes a vast collection of patterns with different combinations
+of stars, including some ‘non-systematic’ patterns that feature stars or rosettes
+with unusual orders, such as 7, 9, 11, 13, and 14.
+Fig. 2. Illustration of polygons-in-contact, with examples of rosettes highlighted. (a) A
+classical Islamic geometric pattern derived from a tiling by regular decagons, pentagons,
+and barrel-shaped hexagons. (b) A non-systematic pattern with an underlying tiling
+by regular 11-gons, 13-gons, and irregular pentagons.
+Another means of achieving irregularity is to move away from the Euclidean
+plane. In Islamic architecture, domes are often decorated with specialized geo-
+
+(a)
+(b)4
+R. Lin and C.S. Kaplan
+metric patterns adjusted to varying curvature [6]. Kaplan and Salesin [24] demon-
+strated adapting PIC to produce patterns on the sphere and in the hyperbolic
+plane. While repetitive in the mathematical sense, hyperbolic patterns are nec-
+essarily distorted when projected into the Euclidean plane. Kaplan [22] later
+presented a more general method for mapping planar patterns with sufficient
+symmetry, including many Islamic patterns, onto arbitrary surfaces in 3D.
+A Moroccan zellij design typically features a large central star surrounded
+by radially symmetric constellations of smaller modules [7]. These modules are
+formed from a standard set of individual tile shapes derived from an 8-pointed
+star. The result is a monumental work containing substantial visual novelty
+and appeal. The puzzle of creating such designs is more combinatorial than
+geometric: the artist seeks new discrete configurations of a fixed set of shapes.
+Recently, Kaplan [23] presented an algorithm for the procedural generation of
+small zellij compositions, which shares some aesthetic goals with our work.
+Modern mathematics allows us to produce patterns that are orderly without
+being periodic. Many techniques have been proposed that use substitution tilings
+or quasiperiodicity to guide the placement of Islamic motifs [4,27,8,9,29]. Some
+researchers have even credited ancient designers with an explicit understanding
+of quasiperiodicity [1,26], though such claims are controversial [14]. Non-periodic
+patterns with long-range organization occupy an aesthetic sweet spot: they ad-
+vertise global structure, but the precise nature of that structure is not trivially
+unravelled by the eye.
+In the broader world of computer graphics, researchers have explored some
+interactive and automated techniques for laying out small motifs to create or-
+namental patterns [17]. Practical numerical algorithms for constructing circle
+packings are relatively new [10], so circle packings have not received much at-
+tention as an organizing tool for pattern design. A notable exception is the work
+of Hamekasi and Samavati [19], who use circle packings to guide the placement of
+motifs in Persian floral designs. Most recently, Brewer et al. derived circle pack-
+ings from k-uniform tilings and used them as a framework in which to inscribe
+Islamic motifs [5]. Their technique overlaps somewhat with ours, though they
+are restricted to arrangements that can arise naturally from the vertex types
+and polygon orders of the tilings they use as a starting point.
+3
+Modular Motif Construction
+Many standard techniques for constructing Islamic patterns are modular: they
+decompose the canvas into disjoint regions such as disks or polygons and define
+a procedure for filling every region with a motif. This section summarizes two
+motif construction techniques that will form the basis of our method.
+In the polygons-in-contact technique (PIC), the canvas is subdivided into
+polygons that meet edge-to-edge. We choose a contact angle θ ∈ (0, π/2). For
+every edge of a polygon P in the subdivision, we construct the two rays that grow
+from the edge’s midpoint towards the interior of P, rotated by ±θ relative to
+the edge. A motif is formed by truncating these rays where they meet rays from
+
+Freeform Islamic Geometric Patterns
+5
+other edges. In simple cases, we need only compute intersections with rays from
+neighbouring edges (Fig. 3a), or from two edges away (Fig. 3b). A more robust
+construction requires heuristics to decide how to truncate rays, such as minimiz-
+ing the total length of the motif’s line segments [3, Sec. 4.4.2]. Fig. 2 shows two
+patterns created by constructing motifs for every polygon in a subdivision.
+In the wheel construction, the modules are circles, each tangent to neigh-
+bouring circles in a larger pattern. The construction inscribes a star in every
+circle. Given a circle C of radius r, we first identify a set of points S on its
+boundary, including the points where C meets its neighbours. We also choose a
+smaller circle C′ with radius αr for a given α ∈ (0, 1), lying in the interior of
+C. Let p and q be two points in S. We construct the perpendicular bisector of
+chord pq and find the intersection of that bisector with C′. Then we draw line
+segments from p and q to the intersection. Fig. 3 shows stars that emerge when
+this process is repeated for all pairs of p and q in S that are consecutive (c) or
+non-consecutive (d). Here we can control the sharpness of the star by varying
+the scaling ratio α between the radii of the outer and inner circles.
+Fig. 3. Regular 9-pointed stars constructed using PIC (left) and the wheel construction
+(right). Using PIC, we truncate rays at their first (a) or second (b) intersections with a
+contact angle of 2π/5. Using the wheel construction, we draw zig-zag paths connecting
+consecutive points (c) or every other point (d) on the outer circle.
+Both of these constructions can produce symmetric n-pointed stars. For PIC,
+a symmetric star is produced when P is regular; for the wheel construction, we
+require the points in S to be distributed evenly around C, and for C and C′ to
+be concentric. We can convert between PIC’s θ and the wheel construction’s α
+in this case. Stars (a) and (c) in Fig. 3 are related by
+α = 1 −
+sin (π/n) sin θ
+sin (π(n + 2)/2n − θ)
+(1)
+and stars (b) and (d) are related by
+α = 1 − 2 sin (π/n) sin (π(n − 2)/2n) sin (θ − π/n)
+sin (π/2 + 2π/n − θ)
+.
+(2)
+Empirically, the wheel construction works well when forming a star whose
+points lie on a common circle: it degrades gracefully as the point distribution
+
+(a)
+(b)
+(d)6
+R. Lin and C.S. Kaplan
+becomes uneven. PIC is a better choice for small polygons whose irregularity
+is harder to characterize. We shall use both in our method. Note that neither
+of these techniques explicitly constructs rosettes. Although explicit rosette con-
+structions exist [25], we will allow rosettes to emerge as a by-product of the
+polygonal decompositions we use as a basis for motif construction, as in the
+examples of Fig. 2.
+4
+Freeform Designs
+In this section, we present the steps that make up our main technique for con-
+structing finite, freeform compositions of rosettes. The steps are visualized in
+Fig. 4. We begin with an arbitrary simplicial complex (a), which induces a circle
+packing (b). Based on the circle packing, we construct a polygonal patch (c),
+comprising large cyclic polygons separated by smaller irregular pentagons. Fi-
+nally, we use a combination of PIC and the wheel construction to define motifs
+for each polygon (d), and optionally render the design (e). In the following sub-
+sections, we describe each of these steps in detail.
+Fig. 4. Our method takes a complex (a) and computes a circle packing (b). Then it
+forms a freeform patch of polygons (c), from which it develops motifs that form a
+seamless constellation (d) that may be styled (e).
+
+(a)
+(b)
+(e)
+(d)
+cFreeform Islamic Geometric Patterns
+7
+4.1
+Complex
+The main input to our technique is a complex, more formally a planar, simply
+connected, pure simplicial 2-complex K. In simpler terms, we may regard K as a
+collection of non-overlapping triangles in the plane, meeting edge-to-edge. The
+union of the triangles defines a region the plane—a simple polygon. We will
+refer to the vertices, edges, and faces of the complex. We distinguish between its
+boundary vertices, which lie on the simple polygon, and interior vertices, which
+lie interior to the polygon.
+We may construct input complexes in numerous ways. It is easy to author
+them manually by placing and connecting vertices. They can also be generated
+procedurally, such as by computing the Delaunay triangulation of a point set.
+4.2
+Circle Packing
+Let K be a complex with n vertices. A circle packing for K is a collection of
+non-overlapping circles {C1, . . . , Cn} whose tangencies echo the combinatorial
+structure of K. Each circle Ci corresponds with vertex vi of the complex, and
+two circles Ci and Cj are externally tangent if and only if vi and vj are con-
+nected by an edge in K. The Discrete Uniformization Theorem guarantees that a
+circle packing exists for any given complex K [28]. Although the circle packing’s
+connectivity will be identical to that of its complex, they will generally not be
+equivalent geometrically: the locations and sizes of the circles are not directly
+related to the locations of the vertices in the complex, or to the shapes of its
+triangles.
+Collins and Stephenson [10] describe a simple numerical algorithm that com-
+putes circle packings through iterative adjustments of an initial assignment of
+radii to the Ci. The radii of boundary circles must be further constrained with
+additional boundary conditions. The simple Python script by Eppstein [15] ac-
+cepts explicit values for boundary radii. Given a boundary vertex of degree n, our
+implementation chooses a radius r for a circle that would be perfectly surrounded
+by 2n − 2 unit circles, giving r = (1 − sin φ)/ sin φ, where φ = π/(2n − 2).
+4.3
+Polygonal Patch
+A patch is a finite set of polygons with disjoint interiors whose union is a topo-
+logical disk. Given a circle packing, we construct a patch that has a large cyclic
+polygon (i.e., a polygon whose vertices lie on a common circle) associated with
+each circle, separated from other cyclic polygons by haloes of pentagonal ‘filler
+polygons’. By design, these polygons can serve as scaffolding for building motifs
+typical in Islamic geometric patterns.
+Let C be an interior circle in a circle packing, and let k be the degree of the
+vertex associated with C in the complex. As illustrated in Fig. 5a, we construct
+a cyclic 2k-gon P in the interior of C. To begin, we set the vertices of P to be the
+k points of tangency between C and its neighbours, together with the midpoints
+of the minor arcs of C connecting adjacent tangency points. Now let τ ∈ (0, 1)
+
+8
+R. Lin and C.S. Kaplan
+be a user-selected scaling factor. Scale P relative to the centre of C by a factor
+of τ, and add the scaled polygon to the patch. By default, we use τ = 0.8, a
+choice that we discuss in Sec. 7.
+The gaps between circles in the packing are triangular regions bounded by
+arcs of three mutually tangent circles. Let Ci, Cj, and Ck be one such trio
+of circles. We divide the space between their cyclic polygons into three new
+pentagons, as shown in Fig. 5b, by drawing edges connecting vertices of cyclic
+polygons. Three outer line segments pass through the pairwise tangencies of
+the circles. Three inner segments connect arc midpoints to a new point o, the
+incentre of the triangle formed from the centres of Ci, Cj, and Ck.
+Fig. 5. Constructing a polygonal patch from a circle packing: we create a cyclic polygon
+for every circle (a), and fill the gaps between three mutually tangent circles with trios
+of irregular pentagons (b).
+4.4
+Motif Construction
+The final step in our process is to construct a motif for every polygon in the
+patch produced in the previous step. Here we apply both the wheel construction
+and PIC, depending on the type of polygon being decorated. Our large cyclic
+polygons yield motifs that garner attention. We safeguard the quality of these
+motifs by exploiting the robustness of the wheel construction in their develop-
+ment (Fig. 6a). We then use PIC for the more unpredictable filler pentagons
+(Fig. 6b). Optionally, we remove motif segments around the boundary of the
+resulting composition, paring it down to a core of whole rosettes (Fig. 6c). Our
+construction depends on a single global contact angle θ, as described in Sec. 3.
+By default, we use θ = 2π/5, the angle for which PIC would inscribe a perfect
+pentacle in a regular pentagon.
+Let C be an interior circle in the packing with centre o and k points of
+tangency. Let P be the cyclic 2k-gon associated with C in the patch. We use the
+wheel construction to build a star centred at o whose outer points lie at the edge
+
+(a)
+(b)Freeform Islamic Geometric Patterns
+9
+Fig. 6. To construct a design from a patch, we use the wheel construction to create a
+star in every cyclic polygon (a) and apply PIC to build motifs for filler polygons (b).
+Optionally, we remove the outer layers of geometry to extract an arrangement of whole
+rosettes (c).
+midpoints of P. Generally, these midpoints do not lie on a common circle, but the
+wheel construction is tolerant of small deviations in their distances from o. Let rC
+be the radius of C, and define r to be rC cos(π/2k). The value r approximates the
+radius of an inscribed circle meeting P’s edge midpoints, an approximation that
+converges on the correct value when P is regular. Now compute α by plugging
+the user-supplied contact angle θ into Eqn. 2, and let C′ be a circle with center
+o and radius αr. The radius of C′ is chosen to ensure that the contact angles at
+the points of the star approximate θ. We apply the wheel construction using the
+edge midpoints of P and the inner circle C′, connecting every other star point
+as in Fig. 3d.
+It remains to build motifs for the filler pentagons. Let Q be one such pen-
+tagon. As in PIC, construct a pair of rays emanating from the midpoint of every
+edge of Q, and truncate them where they intersect rays growing from neigh-
+bouring edges. If an edge e of Q is adjacent to a cyclic polygon, then we choose
+contact angles that yield rays parallel to the star edges meeting across e (Fig. 6b,
+red). These angles may not be symmetric across the perpendicular bisector of e,
+but the discrepancy is small in practice. If e is adjacent to another pentagon, on
+the other hand, then we use θ as the contact angle for its rays (Fig. 6b, purple).
+In summary, our method uses a patch to construct a constellation of localized
+motifs that combine to form familiar visual elements: rosettes. By our application
+of the Discrete Uniformization Theorem, each rosette corresponds to a vertex in
+the triangulation K, and two rosettes are adjacent if and only if their vertices
+share an edge in K. The order of a rosette is twice the degree of its associated
+vertex.
+5
+Gadgets
+The basic technique of the previous section can produce a wide variety of freeform
+designs with combinations of rosettes of different orders. However, some config-
+urations found in traditional Islamic geometric patterns remain out of reach,
+
+(a)
+(b)
+C10
+R. Lin and C.S. Kaplan
+most obviously because we define only one way to fill the triangular gaps be-
+tween cyclic polygons. In this section, we introduce two gadgets that help us
+recover some of that variety, increasing the visual intrigue of our designs. Gad-
+gets are small subgraphs with labelled vertices that can be incorporated into a
+complex. These vertices then determine local clusters of polygons during patch
+construction, overriding the polygons of Sec. 4.3.
+A square gadget is a 5-vertex subgraph with a central vertex a of degree 4,
+as shown in Fig. 8a. Given a complex containing a copy of the square gadget,
+we obtain a circle packing containing a cluster of circles like the one shown in
+Fig. 8b, where circle A is associated with vertex a. When building the patch,
+we remove A from the circle packing and tile the hole left behind with four
+pentagons, as shown in Fig. 8c. The new point o is the mean of vertices i ,j, k,
+and ℓ. Our motif construction will produce a squarish region surrounded by four
+rosettes containing a central octagon.
+Fig. 7. The square gadget (a) produces a circle packing (b) from which we derive four
+filler pentagons (c).
+Fig. 8. The bowtie gadget (a) produces a circle packing (b) from which we derive a
+cluster (c) of four filler pentagons and a barrel-shaped hexagon.
+A bowtie gadget is a 6-vertex subgraph with two central vertices a and b
+of degree 4, as shown in Fig. 7a. As with the square gadget, we remove the
+corresponding circles A and B from the circle packing and fill the void with a
+new configuration of tiles. First, when constructing a cyclic polygon for circle D
+associated with vertex d, we divide the minor arc between the tangencies with
+C and E into three equal pieces instead of the usual two, yielding vertices j and
+
+C
+a
+A
+D
+B
+h
+d
+E
+(a)
+(b)
+Cd
+D
+10
+a
+E
+A
+B
+e
+F
+(a)
+(b)
+cFreeform Islamic Geometric Patterns
+11
+j′ in Fig. 7c. Similarly, we divide F’s arc into three, which gives us vertices ℓ
+and ℓ′. We then construct a bowtie-shaped arrangement of four pentagons and
+one barrel-shaped hexagon, as illustrated in Fig. 7c, where o is the mean of i, j,
+and ℓ and o′ is the mean of j′, k, and ℓ′.
+When the barrel-shaped hexagon in the centre of the
+bowtie gadget becomes too thin, it can produce a motif
+with a small region of overlap at its centre, as shown in red
+in the inset. When these overlaps occur, we replace the
+red segments with a perfect ‘X’, shown in blue. The blue
+segments alter the contact angles with the edges of the hexagon; we propagate
+any changes to the hexagon’s four neighbouring pentagons.
+Recall that without gadgets, our construction was limited to
+rosettes of even orders. But when a bowtie gadget appears in
+a complex, vertices d and f each contribute three edges to their
+corresponding cyclic polygons. Therefore, if a complex vertex acts
+as d or f in one such gadget, as in the central vertex of the
+subgraph in the inset, that vertex will yield a rosette of odd order. More generally,
+we may hang any odd number of suitably oriented bowtie gadgets from a vertex
+to obtain an odd-order rosette.
+Fig. 9 shows a freeform design constructed from a random arrangement of
+bowtie and square gadgets. In future work, we hope to explore gadgets beyond
+these two.
+Fig. 9. A composition based on a square grid, where every square is randomly subdi-
+vided with a diagonal, a square gadget, or a bowtie gadget.
+
+12
+R. Lin and C.S. Kaplan
+6
+Periodic Patterns
+While the focus of our technique is the creation of finite, freeform compositions,
+we have also examined its ability to produce more orderly designs. For example, a
+finite subset of a periodic arrangement of bowtie gadgets (Fig. 10, left) yields an
+approximation of the decagonal design in Fig. 2a (Fig. 10, right). Other periodic
+arrangements of triangles and gadgets can reproduce different classic designs.
+However, because of flexibility in the circle packing algorithm, these freeform
+designs could contain rosettes of continuously varying scales.
+We can extend our technique to generate truly periodic patterns by gener-
+alizing the Discrete Uniformization Theorem beyond the Euclidean plane. In
+particular, if K is embedded on a torus, then the theorem guarantees the ex-
+istence of a circle packing in the torus’s intrinsic metric [28, Ch. 9]. The circle
+packing algorithm is, in some sense, even simpler in this case because there is no
+longer any need for explicit boundary conditions: every circle is completely sur-
+rounded. The torus can then be cut open and unrolled, yielding a finite collection
+of circles that can be stamped out to produce an infinite periodic packing.
+Fig. 11 gives an example of a periodic pattern generated from a simplicial
+complex embedded on a torus. The light grey disks in Fig. 11b should be inter-
+preted as translated copies of the dark grey disks with the same indices. The
+numerical circle packing algorithm yields a layout that tiles the plane by trans-
+lation (Fig. 11c), from which we can create a periodic pattern with rosettes of
+orders 10, 12, 14, and 16.
+Future work could explore the analogous extensions of this technique to
+other spaces, such as the sphere and the Poincar´e disk model of the hyperbolic
+plane [24].
+Fig. 10. A periodic arrangement of bowtie gadgets (left) can be used to generate a
+freeform version of the pattern in Fig. 2a (right).
+
+Freeform Islamic Geometric Patterns
+13
+Fig. 11. A triangulation drawn on a square with periodic boundary conditions (a) is
+used to generate a circle packing (b) that covers the plane through translation (c). We
+construct motifs to obtain a periodic Islamic geometric pattern (d).
+7
+Discussion
+In this section, we discuss some of the details of our technique, including al-
+ternative approaches that we considered during its development. Some of these
+alternatives may offer opportunities for future work.
+Selecting interior circles. We typically do not generate a motif for every
+circle in the packing. Boundary circles, and circles adjacent to them, can dif-
+fer substantially in size from their neighbours. These variations can propagate
+through the rest of the construction and produce unacceptably distorted motifs,
+such as uneven rosette petals (Fig. 12). Future work could consider ways to op-
+timize the geometry of the circle packing to serve patch and motif construction.
+For now, we omit outer layers of circles in our final designs. Note that this ap-
+proach may separate the design into multiple connected components, in which
+case we simply keep the largest component.
+Beyond these technicalities, we can be selective for aesthetic reasons. Hav-
+ing the freedom to craft the shape of a design provides opportunities to create
+interesting compositions (Fig. 15).
+
+(b)
+a
+(d)14
+R. Lin and C.S. Kaplan
+Fig. 12. Circles near the boundary of a packing can lead to distorted stars and rosettes
+(right). We discard outer circles, which can sometimes partition a design into multiple
+connected components (left).
+Choosing a scale factor. Recall that the
+parameter τ controls the scale of each cyclic
+polygon relative to its circle, which in turn affects the shapes of filler pentagons.
+As shown in the inset, the quality of a motif generated within a filler pentagon
+decreases as that pentagon deviates from regularity. Thus we seek to choose τ
+to minimize the total deviation across a design.
+To gauge the deviation of a pentagon Q from regularity, we adopt a contin-
+uous symmetry measure by Zabrodsky et al. [30], which quantifies the minimal
+distance that the vertices of Q must travel to form a regular pentagon. Let the
+error of a freeform patch be the average deviation of its pentagons from regular-
+ity. We can compute this error for a range of closely-spaced τ values and choose
+the one with minimal error (Fig. 13a). Over a range of circle packings, we see
+significant deviation outside the range (0.7, 0.9) and find that τ = 0.8 produces
+satisfactory results, as shown throughout this work.
+Fig. 13. A patch with pentagons coloured by their deviations (red) from regularity
+(green). In (a), cyclic polygons are scaled by τ = 0.7, 0.8, and 0.9, showing that 0.8
+produces good quality overall. In (b), they are offset by a fixed amount, with less
+consistent results.
+
+a)
+b
+RFreeform Islamic Geometric Patterns
+15
+In the future, we hope to investigate other measurements of polygon quality
+in order to produce patches that are closer to ideal. For example, PIC can often
+produce a satisfactory motif in a polygon that has lower-order symmetries while
+not being fully regular.
+As an alternative to treating τ as a scaling factor, we also considered offsetting
+cyclic polygons by a constant inward distance τ. However, we found that this
+approach was not as successful in producing high-quality pentagons (Fig. 13b).
+With either interpretation of τ, the quality is the poorest for pentagons adjacent
+to two cyclic polygons of widely different radii. Hamekasi and Samavati note this
+issue as well [19], and mitigate it by avoiding complexes containing neighbouring
+vertices of widely varying degrees. In future work, we would like to develop a
+global optimization that chooses a different scaling factor for every cyclic polygon
+so as to maximize the overall quality of all filler pentagons.
+Cyclic vs. regular polygons. It is tempt-
+ing to construct regular polygons in place of
+cyclic polygons, as these would yield perfectly
+symmetric stars as motifs. Using the aforemen-
+tioned regularity measurement [30], we fit a reg-
+ular polygon ˆP to each cyclic polygon P gen-
+erated in Sec. 4.3, and centre ˆP on the circum-
+centre of P. The result for τ = 0.8 is shown in
+the inset. This approach prioritizes the quality of large, prominent stars. How-
+ever, it yields distorted pentagons whose motifs self-intersect. By choosing cyclic
+polygons rather than regular polygons, our algorithm sacrifices some quality in
+large stars for the sake of creating feasible connections between them.
+Fig. 14. Generative designs constructed from Delaunay triangulations of random points
+sets, without (left) and with (right) bowtie gadgets.
+
+orders
+18
+17
+16
+15
+14
+13
+12
+11
+10
+9
+816
+R. Lin and C.S. Kaplan
+8
+Results
+We demonstrate the versatility of our technique by presenting a range of freeform
+designs. For stylized results such as the filled composition in Fig. 14 and the
+interlaced design in Fig. 17, we adapt the rendering algorithms described by
+Kaplan in Bonner’s text [3, Sec. 4.5].
+Our method places no constraints on the input complex, giving users consid-
+erable control over the output design. Fully generative designs can be created
+using Delaunay triangulations of random point sets, leading to arrangements of
+rosettes with various orders (Fig. 14, left). We can further increase the number
+of possible charges and broaden the expressiveness of our technique by insert-
+ing random gadgets (Fig. 14, right). Of course, an artist may select a subset of
+rosettes in a generative design to craft a desired high-level composition (Fig. 15).
+Fig. 15. A generative design in which the user has manually chosen to keep a subset
+of rosettes from an initial arrangement, producing a more dynamic composition with
+an irregular boundary and internal voids.
+On the other hand, we can begin with a highly structured complex and obtain
+a repetitive final design (Figs. 10 and 17), or use a toroidal complex to produce a
+truly periodic pattern (Fig. 11d). In principle, these approaches could be used to
+produce exactly or approximately periodic drawings of many historical Islamic
+geometric patterns. However, we have not attempted to catalogue exactly which
+ones are possible because existing construction techniques are much better suited
+to the task of drawing them.
+In between the extremes of full control and generative randomness, we can in-
+sert carefully constructed subgraphs into a complex to create a single high-order
+rosette (Fig. 16a), or create appealing local arrangements of rosettes (Fig. 16b,c).
+
+Freeform Islamic Geometric Patterns
+17
+Another way to balance order and chaos is to place random gadgets within an
+otherwise ordered grid (Fig. 9).
+Fig. 16. A tuned design with a high-order rosette (a), for which τ = 0.96 and α = 0.75,
+and a composition (c) incorporating multiple instances of a web-like sub-complex (b).
+The high-order rosette in Fig. 16a is a special case, in that it requires hand-
+tuning. Recall that adjacent circles with widely varying radii can produce dis-
+torted motifs (Fig. 12). To produce a satisfactory large rosette, we manually set
+the τ and α values for its cyclic polygon for a better fit with the surrounding
+geometry.
+Close examination of many of our results reveals small geometric discrepan-
+cies of the kind illustrated starkly in Fig. 12. When rosettes have arms that vary
+too dramatically in width or length, they disrupt the elegance of a pattern and
+the feeling of ‘inevitability’ in its construction. There are several places in our
+work where we choose global constants like τ that produce acceptable results in
+general without adapting to the detailed geometry of local parts of individual
+designs. The large rosette in Fig. 16a gives one clear example of where local ad-
+justments can improve a design. In future work, we would like to explore more
+fine-grained constructions that can enhance the quality of every rosette based
+on the configuration of the circle packing in its immediate neighbourhood.
+9
+Conclusion
+We presented a robust method for constructing freeform Islamic geometric pat-
+terns comprising rosettes of unusual orders. Our technique relies on the theory
+of circle packings, giving us a principled geometric scaffolding from which to
+develop a polygonal patch and then motifs. The user controls the initial com-
+plex and any gadgets in it, allowing for significant creative freedom in the design
+
+(b)
+(a)18
+R. Lin and C.S. Kaplan
+process. Our results are more organic and less repetitive than existing patterns
+and suggest many ideas for further exploration. We believe they communicate
+the aesthetic of Islamic geometric patterns while also interpreting them in a non-
+traditional context. They still manage to convey the ‘aesthetic delight’ that Gom-
+brich discussed [18], but with slightly less boredom and more confusion, paving
+the way for more artistic applications of these designs. The work enhances our
+understanding of traditional patterns and reveals new opportunities—freeform
+or otherwise—for both ornamentation and art-making.
+Fig. 17. A highly structured composition (right) generated from a finite subset of a
+conceptually periodic complex (left). Although the circle packing is not constructed in
+a toroidal domain as in Sec. 6, the resulting composition appears close to periodic.
+Acknowledgements
+This research was supported by the Natural Sciences and Engineering Research
+Council of Canada and the Cheriton School of Computer Science at the Univer-
+sity of Waterloo.
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+

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+Synchronized states in dissipatively coupled harmonic oscillator networks
+Juan N. Moreno,1, ∗ Christopher W. W¨achtler,1, 2, † and Alexander Eisfeld1, 3, ‡
+1Max Planck Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany
+2Department of Physics, University of California, Berkeley, California 94720, USA
+3Universit¨at Potsdam, Institut f¨ur Physik und Astronomie,
+Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Deutschland
+The question under which conditions oscillators with slightly different frequencies synchronize
+appears in various settings.
+We show that synchronization can be achieved even for harmonic
+oscillators that are bilinearly coupled via a purely dissipative interaction. By appropriately tuned
+gain/loss stable dynamics may be achieved where for the cases studied in this work all oscillators
+are synchronized. These findings are interpreted using the complex eigenvalues and eigenvectors of
+the non-Hermitian matrix describing the dynamics of the system.
+I.
+INTRODUCTION
+Synchronization is a fascinating phenomenon, which
+can be interpreted as a display of cooperative behavior
+appearing in many complex systems [1, 2]. Since the first
+observation by Huygens in the late 1600s [3], it has been
+studied in diverse communities, where it plays an im-
+portant role in our understanding for example in electric
+networks in engineering, circadian rhythms in biology,
+pattern formation in statistical mechanics, and chemical
+reactions in chemistry [4–6]. By now, it is seen as a uni-
+versal phenomenon that is important both in fundamen-
+tal studies and in technical applications, ranging from
+laser networks [7], to phase-locked loops [8], Josephson
+junction arrays [9, 10], spin-torque resonators [11], and
+power grids [12].
+Even today, the originally observed
+phenomenon of clock synchronization remains a crucial
+application for modern communication networks [13, 14].
+Typically synchronization is viewed in terms of the ad-
+justment of rhythms of autonomous oscillators, which at-
+tain stable periodic orbits without active regulation from
+the outside [15] and thus require nonlinearities in the
+governing equations of motion. Far less common is the
+investigation of synchronization in models that are lin-
+ear in both the oscillators and the couplings. Without
+dissipation, coupled harmonic oscillators form collective
+eigenmodes, where the individual oscillators perform mo-
+tion with a fixed phase relation. However, a system not
+initialized in an eigenmode usually stays in a superposi-
+tion of several eigenmodes with different eigenfrequencies
+resulting in a beating pattern. Moreover, if the number
+of coupled oscillators is large, the system dynamics does
+not need to exhibit perfect revivals in general and syn-
+chronized motion is absent.
+Hence in a closed system
+of oscillators, only for an eigenmode as initial condition
+one obtains a time-independent phase relation between
+the oscillators. However if the system is not closed, but
+subject to gain and loss, the open system dynamics allow
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+for a situation where all eigenmodes but one are damped.
+Then, synchronization is possible as long as the respec-
+tive eigenstate is present in the initial state. However, in
+order to achieve a situation where all but one mode are
+damped, one needs to carefully balance gain and loss.
+In contrast to a self-sustained system where the non-
+linearity counteracts the dissipation (or gain) in order to
+stabilize periodic orbits, a single linear harmonic oscilla-
+tor only exhibits the following dynamics in the absence
+of periodic driving: Either the dissipation exceeds the
+gain, such that the amplitude of the dissipative systems
+shrinks and eventually reaches a single point in phase
+space, or in other way around, where the gain exceeds
+the dissipation, the oscillation amplitude infinitely grows.
+In the special case where both are equivalent the system
+is effectively described by closed system dynamics with
+infinitely many closed orbits in phase space depending
+on the initial energy of system. However, when coupling
+between linear oscillators are introduced, many more so-
+lutions are possible.
+Here, we investigate a network of linear harmonic os-
+cillators subject to gain and loss. Generally, one would
+consider each oscillator to couple to its own environment
+and direct coupling between two or more entities in the
+network.
+However, a purely dissipative coupling leads
+to intriguing phenomena also for self-sustained oscilla-
+tors like for example oscillator death [1]. In our model
+of linear oscillators, it allows for the emergence of dissi-
+pation free subspaces in parameter space. Within these
+subspaces we find periodic motion of all oscillators in the
+network, that is starting from an (nearly) arbitrary initial
+state the system reaches a regime during time propaga-
+tion in which all oscillators exhibit synchronized motion
+for a long time. At this point, let us specify the notion
+of synchronization we use throughout this work:
+— With ’long time’ we mean times long compared to the
+eigenfrequencies of the individual oscillators and we fo-
+cus on the case where all oscillators have small deviations
+from a common ’mean frequency’. In the ideal case they
+oscillate forever.
+— With ’synchronoized’ we mean that the oscillators
+have a fixed phase relation.
+Ideally we want that all
+oscillators have the same amplitude. If this is the case,
+arXiv:2301.13614v1  [nlin.CD]  30 Jan 2023
+
+2
+then we denote it as full synchronization. If the system
+is not in a fully synchronized state, we will characterize
+its degree of synchronization by a suitable measure.
+— With ’arbitrary’ initial state we mean that for most
+initial states synchronization is achieved, yet there exist
+some special initial conditions that do not lead to syn-
+chronization.
+We
+note
+that
+within
+the
+above
+definitions
+for
+uncoupled oscillators one only finds synchronization,
+when there is no gain and loss and all oscillators have
+the same frequency.
+The remainder of the paper is organized as follows: In
+Sec. II A we summarize some general considerations of
+synchronization for linearly coupled harmonic oscillators
+important for our work, followed by the specific model
+under investigation in Sec. II B. In the subsequent Sec. III
+we discuss our results, which includes the special case of
+two coupled oscillators in Sec. III A and the more general
+case of many oscillators in Sec. III B. Finally, we conclude
+in Sec. IV.
+II.
+MODEL AND BASIC FORMALISM
+A.
+General considerations of synchronization in
+linear oscillator models
+To introduce the basic concepts and notation, we con-
+sider N harmonic oscillators in a network, each labeled
+by a subscript n = 1,...,N. The motional state of each
+oscillator is characterized by a time dependent complex
+amplitude an(t) = ∣an(t)∣eiφn(t). If all oscillators in the
+network oscillate with a common real frequency ωsyn
+while their relative amplitudes remain constant, we will
+refer to it as synchronization. Using a vector notation
+⃗a(t) = [a1(t),...,aN(t)]⊺, such synchronized motion may
+be expressed as
+⃗a(t) = f(t)⃗asyne−iωsynt,
+(1)
+where f(t) is a real function that takes into account the
+possibility that the amplitudes decay (or grow) over time,
+which we will discuss in Sec. II B in more detail. In the
+case of f(t) = 1 the motion represents a periodic steady
+state, which we refer to as ideal synchronized motion.
+The above notion is not sufficient to fully characterize
+synchronized motion as for example a single oscillatory
+site in the network (while all other oscillators are at rest)
+also fulfills Eq. (1). It is thus necessary to also quan-
+tify the degree of synchronization of a vector ⃗a, which we
+denote by S(⃗a). To this end, we use the inverse partici-
+pation ratio [16]
+S(⃗a) =
+1
+∑N
+n=1 ∣an∣4 ,
+(2)
+which takes values between 1 and N. Here, a value of
+S = 1 corresponds to the aforementioned case of a single
+oscillator in motion, whereas a value of S = N indicates
+FIG. 1.
+Illustration of potentially attainable synchronized
+motion in a network of N = 3 oscillators. The inverse partici-
+pation ratio S(⃗a) increases from top to bottom in accordance
+with the transition from partially to fully synchronized mo-
+tion.
+fully synchronized motion, i.e. all nodes have the same
+amplitude (without phase). Values of S = ˜N < N cor-
+respond to partial synchronization of approximately ˜N
+oscillators. In Fig. 1, we illustrate different degrees of
+synchronization and their respective dynamics in a net-
+work of three oscillators.
+The time evolution of a linearly coupled network of
+harmonic oscillators in the presence of gain and loss is
+generally expressed as
+d
+dt⃗a = −iW ⃗a,
+(3)
+where we assume the non-Hermitian matrix W to be
+time-independent. Then, the state of the system at time
+t is simply given by
+⃗a(t) = e−iW t⃗a(0),
+(4)
+where ⃗a(0) denotes the initial state at time t = 0. Thus,
+the dynamics of the network is fully characterized by the
+matrix W, in particular by its eigenvalues and eigenvec-
+tors. Since W is (in general) non-Hermitian, there exist
+right and left eigenvectors defined via
+W ⃗cj =wj⃗cj
+and
+⃗z†
+jW = ⃗z†
+jwj.
+(5)
+
+0003
+Here, † indicates the complex conjugated and transpose,
+and the eigenvectors are normalized according to
+⃗c†
+j⃗cj = 1
+and
+⃗z†
+j′⃗cj = δj′j.
+(6)
+Note, that in general ⃗c†
+j ≠ ⃗z†
+j. The matrix W can now be
+expressed as W = ∑j wj⃗cj ⃗z†
+j, such that the time evolution
+of Eq. (4) is conveniently given by
+⃗a(t) = ∑
+j
+⃗cje−iwjt ⃗z†
+j⃗a(0),
+(7)
+where ⃗z†
+j⃗a(0) is the initial weight of the eigenstate j.
+While the real part of the complex eigenvalue wj deter-
+mines the oscillation frequency of eigenmode j, the imag-
+inary part Im[wj] determines whether the oscillatory mo-
+tion is damped (Im[wj] < 0), growing (Im[wj] > 0) or
+oscillates forever (Im[wj] = 0).
+In order to obtain a time evolution of the form of
+Eq. (1) with f(t) = 1 after some initial transient time, i.e.
+dynamically reach the eigenstate with Im[wsync] = 0, the
+initial state needs to have non-vanishing overlap with the
+synchronized eigenstate [⃗z†
+synca(0) ≠ 0]. Furthermore, all
+other eigenstates present in the initial state need to have
+Im[wj] < 0, such that they are damped. In the following,
+we will therefore search for conditions and parameters
+under which one eigenstate fulfills Im[wsync] = 0 while
+all other eigenstates fulfill Im[wj] < 0. Subsequently, we
+will characterize the degree of synchronization of the re-
+sulting state in terms of S; cf. Eq. (2).
+B.
+Linear oscillators with purely dissipative
+coupling
+After
+the general considerations of the previous
+Sec. II A, let us now specify the network of interest
+throughout the remainder of this work: The individual
+oscillators have frequencies Ωn ∈ R and are arranged on
+a ring. Each oscillator is subject to gain/loss mediated
+via the rate γ ∈ R and interacts with its two nearest
+neighbors via a purely dissipative coupling v ∈ R. For
+simplicity we assume that the coupling and dissipation
+is equal for all oscillators; we are interested in the pos-
+sibility of synchronization when the frequency of each
+oscillator is different, which corresponds to the notion of
+synchronization as an adjustment of rhythms due to the
+presence of interactions. The equation of motion of the
+n-th oscillator is then given by
+d
+dtan =(−iΩn − γ)an − v(an+1 + an−1),
+(8)
+with a0 ≡ aN and aN+1 ≡ a1 to fulfill periodic boundary
+conditions. Note that positive values of γ represent loss
+whereas negative values correspond to gain. To simplify
+notation we express all energies in units of v and take v
+to be positive (the case of negative v will be discussed
+later), i.e. ωn = Ωn/v, g = γ/v and τ = tv. Furthermore,
+we parameterize the frequencies as ωn = ¯ω + ∆n. Then,
+Eq. (8) becomes
+d
+dτ an =[−i(¯ω + ∆n) − g]an − (an+1 + an−1).
+(9)
+Our goal in the following is to determine the values of
+g for a given set of frequency differences ∆n, such that
+the oscillators perform synchronized motion in the sense
+discussed in Sec. II A.
+As the term (−i¯ω − g) is independent of the oscilla-
+tor index n, it only trivially contributes to the overall
+dynamics; specifically oscillations with frequency ¯ω and
+damping/growing with rate g. In matrix representation,
+Eq. (9) can be written in the form of Eq. (3) with t → τ
+and W = (¯ω − ig)I + M, where
+M =
+⎛
+⎜⎜⎜⎜⎜
+⎝
+∆1
+−i
+0
+...
+−i
+−i
+∆2
+−i
+...
+0
+0
+−i
+⋮
+−i
+0
+...
+−i ∆N
+⎞
+⎟⎟⎟⎟⎟
+⎠
+(10)
+Note, that the (left and right) eigensvectors of W and
+M are identical and their eigenvalues are simply shifted,
+i.e., if M⃗cj = λj⃗cj then W ⃗cj = wj⃗cj with
+wj = ¯ω + Re[λj] + i(−g + Im[λj]),
+v > 0.
+(11)
+Moreover, as M only depends on ∆n, the eigenvectors
+and thus the degree of synchronization S(⃗c) is indepen-
+dent of g.
+Let us summarize the general conditions of the pre-
+vious Sec. II A for synchronized motion tailored to the
+specifics of our system discussed here:
+(i) There exists a single eigenstate ⃗csync of W with
+purely real eigenvalue. This corresponds to a state
+⃗csync that fulfills −g+Im[λsync] = 0, where M⃗csync =
+λsync⃗csync.
+(ii) All other eigenstates of W have negative imaginary
+part for the set of parameters determined in (i).
+That corresponds to −g+Im[λj] < 0 for all j ≠ sync.
+(iii) The synchronization measure S(⃗csync) should be as
+large as possible. Ideally S(⃗csync) = N.
+So far, we have taken v to be positive. For negative
+values of v we define the scaled energies in terms of −v
+such that ωn = ¯ω + ∆n = −Ωn/v, g = −γn/v, and τ = −tv.
+Then, Eq. (9) becomes
+d
+dτ an = [−i(¯ω + ∆n) − g]an + (an+1 + an−1),
+(12)
+where the first term remains identical while the sign
+changes in front of the oscillator couplings. As a result,
+the eigenvalues of W [cf. Eqs. (10) and (11)] are given
+by
+wj = ¯ω + Re[λj] + i(−g − Im[λj]),
+v < 0.
+(13)
+
+4
+Here, the real part of the eigenvalues (as well as the cor-
+responding eigenstates and thus the measure S) remains
+unchanged, while the imaginary part simply changes its
+sign. Thus, eigenstates that are decaying for v > 0, are
+growing for v < 0 and vice versa.
+III.
+RESULTS
+In the following we first discuss the case of N = 2 in
+Sec. III A, which provides a clear picture of the basic
+mechanism underlying synchronization of linear oscilla-
+tors interacting via dissipative couplings. Subsequently
+in Sec. III B, we consider a ring of N > 2 oscillators and
+show that also in this case synchronized motion may be
+achieved and follows similar arguments as before.
+A.
+Two coupled oscillators (N = 2)
+Without loss of generality, we may choose the scaled
+frequency differences of the two oscillators to be ∆1 =
++∆ and ∆2 = −∆, such that matrix M governing the
+dynamics [cf. Eq. (10)] is given by
+M = (∆
+−i
+−i −∆)
+(14)
+Here, we have chosen v > 0. However, from the discussion
+in Sec. II B we know that a negative value of v simply
+results in a change of sign of the imaginary part of the
+eigenvalues. The two eigenvalues and corresponding right
+eigenvectors of M are given by
+λ± = ±
+√
+∆2 − 1
+(15)
+⃗c± =
+1
+√
+1 + ∣∆ ±
+√
+∆2 − 1∣2
+(i(∆ ±
+√
+∆2 − 1)
+1
+)
+(16)
+If ∣∆∣ < 1 ( ∣∆∣ > 1) the eigenvalues λ± are both purely
+imaginary (real) and non-degenerate.
+In contrast, for
+∆ = ±1 not only are the eigenstates degenerate but also
+the corresponding eigenvectors coalesce, i.e., these values
+of ∆ correspond to exceptional points. The impact of
+exceptional points on synchronization goes beyond the
+scope of the present work and we will focus in the follow-
+ing on the cases ∣∆∣ > 1 and ∣∆∣ < 1.
+a.
+Overview:
+As discussed in Sec. II B, the eigenen-
+ergies w± = ¯ω+Re[λ±]+i(−g+Im[λ±]) describe the overall
+possibility of long lasting synchronized motion in terms
+of oscillation frequency and damping, while S quantifies
+the degree of synchronization. Let us start by consid-
+ering the imaginary part of the eigenenergies w± given
+by Im[w±] = −g + Im[λ±], which determines the (expo-
+nential) damping or growing. In Figs. 2 (a) and (b) we
+show Im(w−) and Im(w+), respectively, as a function of
+the frequency difference ∆ and the dissipation strength
+g. Note, that ∆ as well as g can take on positive and
+FIG. 2. Top row: Density plots of the imaginary part Im(w±)
+as a function of the frequency difference ∆ and the dissipation
+strength g: (a) w− and (b) w+. Dissipation-free synchroniza-
+tion is found along the white line. Middle row: Correspond-
+ing real part (c) Re(w−) and (d) Re(w+) as a function of ∆,
+which corresponds to the oscillation frequency of the respec-
+tive eigenvector. Last row: Degree of synchronization S as
+function of ∆ of the eigenvalue (e) ⃗c− and (f) ⃗c+. The largest
+value is found for ∣∆∣ < 1 corresponding to fully synchronized
+motion.
+negative values. The red areas in Fig. 2(a) and (b) indi-
+cate positive values corresponding to amplitude growth
+whereas the blue areas indicate negative values and thus
+amplitude damping. The two regions are separated by
+a white region, where amplitudes neither increase nor
+decrease. We discuss this most relevant region for dissi-
+pation free synchronization in more detail below.
+As expected from the discussion above, quite different
+behavior of Im[w±] is observed depending on whether
+∣∆∣ > 1 or ∣∆∣ < 1.
+Similarly, a pronounced difference
+is found in the behavior of the real part Re[w±] = ¯ω +
+Re[λ±], which describes the oscillation frequency of the
+eigenmodes and is shown in Fig. 2(c) and (d). For ∣∆∣ < 1
+the frequency remains unchanged and both eigenstates
+oscillate with the mean frequency ¯ω. However, for ∣∆∣ > 1
+the frequency of the − state [cf. Fig. 2(c)] is decreasing,
+while that of the + state [cf. Fig. 2(d)] is increasing. Both
+follow the functional form of a square-root with opposite
+sign, cf. Eq. (15). Lastly, in Fig. 2(e) and (f) we show
+the degree of synchronization S as function of ∆, which
+
+-state
++state
+2
+2.0
+(a)
+(b)
+1
+1.0
+Im(W=)
+0
+0.0
+-1
+-1.0
+-2
+-2.0
+2
+(c)
+(d)
+0
+-2
++
+t
+2.5
+(e)
+(f)
+2.0
+1.5
+1.0
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+V
+V5
+is given by [cf. Eq. (16)]
+S(⃗c±,∆) = { 2
+, ∣∆∣ < 1
+2
+∆2
+2∆2−1
+, ∣∆∣ > 1 .
+(17)
+As expected, the maximum value lies within the range
+of ∣∆∣ < 1 and rapidly decreases as ∣∆∣ increases, indi-
+cating the absence of synchronization. After this broad
+overview we will in the following discuss in more detail
+the potential of synchronized motion in the system of
+N = 2 oscillators, focusing on the three criteria (i)–(iii)
+formulated in Sec. II B.
+b.
+Detailed discussion of the regime ∣∆∣ > 1:
+In this
+case, the eigenvalues λ± become purely real [cf. Eq. (15)],
+such that the eigenenergies take the simple form w± =
+(¯ω±
+√
+∆2 − 1)−ig. Most importantly, the imaginary part
+is solely given by −g for both states and is independent
+of ∆, which can also be seen in Figs. 2(a) and (b). Thus,
+both eigenstates show the same dynamical response to
+dissipation, i.e., either both are dissipation free (g = 0) or
+the amplitudes decay/increase with the same rate given
+by −g. Although there exists a dissipation free subspace
+for g = 0, and thus requirement (i) is fulfilled, require-
+ment (ii) cannot be fulfilled simultaneously. The reasons
+is that both states have different oscillation frequencies
+¯ω ±
+√
+∆2 − 1 and none of them is decaying, resulting in a
+beating pattern. We show an example of such a time evo-
+lution of the real amplitudes Re(an) governed by Eq. (9)
+in Fig. 3(a) for ∆ = 1.1 and g = 0.
+c.
+Detailed discussion of the regime ∣∆∣ < 1:
+Af-
+ter we have ruled out the possibility of synchroniza-
+tion [according to our conditions (i)–(iii)] in the previ-
+ous regime, we now discuss the case of ∣∆∣ < 1, where
+dissipation free synchronized motion is indeed possible.
+For ∣∆∣ < 1 the eigenvalues λ± are purely imaginary [cf.
+Eq. (15)] and dissipation free states are determined by
+0 = −g ±
+√
+∣1 − ∆2∣, such that condition (i) may be ful-
+filled. In contrast to the previous case, we need to differ-
+entiate between the two states: Dissipation vanishes for
+the + state if g = g+ ≡
+√
+∣1 − ∆2∣, and for the − state if
+g = g− ≡ −
+√
+∣1 − ∆2∣. Each of these solutions describes a
+half circle with radius one, cf. Figs. 2(a) and (b).
+We now examine whether condition (ii) is also ful-
+filled in this regime.
+When the − state is dissipation
+free, the amplitude of the + state is growing exponen-
+tially as Im[w+(g−)] = −g− +
+√
+1 − ∆2 = 2
+√
+1 − ∆2 > 0.
+This is also verified by Fig. 2: Along the white region
+in panel (a) within the regime ∣∆∣ < 1, the area in panel
+(b) is red. In contrast, along the white region in panel
+(b), the area in panel (a) is blue, i.e. while the + state
+is dissipation free, the − state is damped. Specifically,
+Im[w−(g+)] = −g+ −
+√
+1 − ∆2 = −2
+√
+1 − ∆2 < 0.
+Thus,
+synchronized motion for ∣∆∣ < 1 is found whenever the
+condition g =
+√
+1 − ∆2 is fulfilled. Moreover, this state
+has a degree of synchronization of S = 2 and is therefore
+fully synchronized for all ∣∆∣ < 1.
+In Fig. 3(b) we show the dynamics for the parameters
+∆ = 0.6 and g = 0.8 when starting in the initial state
+FIG. 3. Examples of different dissipation free dynamics found
+for the case of N = 2 oscillators. We plot the real amplitude
+Re(an(τ)) of the first oscillator in red (n = 1) and the second
+one in blue (n = 2).
+(a) For ∆ = 1.1 and g = 0, the pres-
+ence of two oscillation frequencies within the dissipation free
+subspace leads to beating. (b) For ∆ = 0.6 and g = 0.8, only
+a single eigenstate with its respective oscillation frequency is
+dissipation free, while the other is damped leading to a pe-
+riodic steady state of both oscillators, i.e., synchronization.
+Parameters:
+¯ω = 10, ⃗a(0) = (1, 0)⊺.
+These results are ob-
+tained by direct integration of the differential equation.
+It
+agrees perfectly with the results obtained via diagonalization.
+⃗a(0) = (1,0)⊺.
+As discussed previously, we expect to
+find synchronized motion for these parameters. Indeed,
+after a short transient time of τ ≳ 2 a stationary oscil-
+latory motion emerges where both oscillators have the
+same amplitude. Note the phase shift between the two
+oscillators, which may be understood as follows: Consid-
+ering the + state ⃗c+ [cf. Eq. (16)], the long time dynamics
+is given by ⃗async(t) = ⃗c+ exp[−iω+t]; cf. Eq. (7). Then,
+Re[⃗async(t)] =N (cos(ω+t + φ)
+cos(ω+t) ),
+(18)
+where the phase difference φ fulfills tan(φ) = −
+√
+1 − ∆2/∆
+and N = (1 + ∣∆ +
+√
+∆2 − 1∣2)−1/2 is the normalization
+constant from Eq. (16).
+B.
+Many coupled oscillators on a ring
+In this section, we generalize our results from the pre-
+vious Sec. III A for the case of two coupled oscillators to
+large numbers of oscillators arranged on a ring. Also for
+the case of N oscillators, the dynamics is governed by
+Eqs. (9)–(11). In the following we will first discuss the
+case of equal frequencies of all oscillators. Afterwards, we
+discuss the more relevant case of frequency differences.
+
+5.0
+(a)
+Re(ai)
+Re(a2)
+2.5
+Re(ai)
+0.0
+-2.5
+-5.0
+2
+(b)
+Re(ai)
+Re(a2)
+1
+Re(ai)
+188888888888888
+0
+-1
+-2
+0
+2
+4
+6
+8
+10
+26
+1.
+Identical frequencies of all oscillators
+To gain a basic understanding of the eigenstates and
+eigenvector structure we now consider the case when all
+frequencies are identical, i.e. ∆n = ∆. Then, the eigen-
+values and (right) eigenvectors of W are given by
+wj = (¯ω + ∆) − i(g ± 2cos(2πj
+N )),
+v ≷ 0,
+(19)
+⃗cj =
+1
+√
+N
+N
+∑
+n=1
+ei 2π
+N jn⃗en,
+(20)
+where ⃗en is the nth unit-vector. As all eigenstates are
+independent of ∆ or g. One sees that most eigenstates
+are degenerate.
+For even N only the eigenstates with
+j = N and j = N/2 are not degenerate; for odd N only
+the state with j = N is not degenerate. Moreover, the real
+part of the eigenenergies wj, i.e. the oscillation frequen-
+cies, is simply shifted by ∆ for all eigenstates. However,
+the imaginary part of wj, which dictates the dissipation
+and more importantly the possibility of dissipation free
+dynamics, requires a more careful analysis.
+a.
+Positive v:
+The imaginary part of the jth eigen-
+value Im[wj] = 0 if g = gj ≡ −2cos(2πj/N). Then, all
+other eigenvalues wj′ with j′ ≠ j have imaginary part
+given by
+Im[wj′(gj)] = 2cos(2πj
+N ) − 2cos(2πj′
+N ).
+(21)
+Furthermore, we need to distinguish the two cases of
+odd and even N:
+For an odd number of oscillators
+and j ≠ (N ± 1)/2 there is always at least one j′ with
+Im[wj′(gj)] > 0, and thus condition (ii) is not fulfilled.
+On the other hand, if j = (N ± 1)/2 all other eigenstates
+are damped except for j′ = j ∓ 1. Yet, this state is also
+dissipation free and condition (ii) cannot be fulfilled. For
+even N, however, there exists a non-degenerate eigen-
+state j = N/2 that fulfills (i) and (ii). Then, g = 2 and
+⃗csyn ≡ ⃗cN/2 =
+1
+√
+N (−1,1...,−1,1)⊺, which corresponds
+to anti-phase synchronization between nearest neighbors
+with the same frequency ¯ω + ∆.
+b.
+Negative v:
+In contrast to the previous case. the
+imaginary part of the jth eigenstate now is equal to zero
+if g = gj ≡ +2cos(2πj/N) and thus Eq. (21) becomes
+Im[wj′(gj)] = −2cos(2πj
+N ) + 2cos(2πj′
+N )
+(22)
+for all other eigenvalues wj′ with j′ ≠ j. Here, only if
+j = N are all other states damped and conditions (i)
+and (ii) fulfilled. The corresponding eigenstate is ⃗csyn ≡
+⃗cN =
+1
+√
+N (1,...,1)⊺, i.e., in-phase synchronization of all
+oscillators with frequency ¯ω + ∆.
+2.
+Oscillators with different frequencies
+In this section, we discuss the case of arbitrary fre-
+quency differences ∆n for each oscillator on the ring. In
+this case, the matrix M [cf.
+Eq. (10)] can no longer
+be diagonalized analytically. Therefore, we discuss the
+basic behavior along a few examples of ∆n and solve
+the eigenvalue problem numerically. Yet, these examples
+demonstrate that dissipation free synchronized motion
+also exists in such a general setup.
+A convenient way to investigate how the properties
+of synchronization are affected by changes of ∆n, is to
+parametrize the frequency difference according to
+∆n = sn∆,
+(23)
+and analyze the behavior of the eigenvalues and eigen-
+vectors of W as a function of ∆ for a given (and fixed)
+set of sn. Furthermore, we choose v to be negative, such
+that for ∆ = 0 there exists a fully synchronized eigenstate
+if g = 2 (see the discussion in Sec. III B 1b). Note that a
+negative value of v implies gj = Im[λj].
+In the following we consider as example the case of
+N = 5 oscillators and show in Fig. 4 the results of the nu-
+merical diagonalization of the matrix M for three differ-
+ent realizations of ⃗s = (s1,...,s5) (different columns). We
+choose the largest difference between neighboring values
+of sn to be equal to one, i.e. max[sn−sn+1] = 1. Then, for
+∆ < 1 all frequency differences between neighboring os-
+cillators are always smaller than the dissipative coupling
+between them (which has magnitude one).
+The case of N = 2 in our network of oscillators allows us
+to represent the full parameter space as shown in Fig. 2
+and identify the dissipation free subspaces and synchro-
+nization within. However, for larger system sizes (as con-
+sidered now) a representation similar to Fig. 2 becomes
+very space consuming. Yet, a dissipation free subspace is
+always necessary for synchronization, which corresponds
+to the white lines in Figs. 2(a) and (b).
+Thus, in or-
+der to determine whether conditions (i)–(iii) are fulfilled,
+it is sufficient to only search along the parameters for
+which each eigenstate becomes dissipation free. In par-
+ticular, the relevant information of Fig. 2(a) and (b) may
+be conveniently combined to contain only g± = Im[λ±] as
+function of ∆. Accordingly, the top row of Fig. 4 shows
+the imaginary part of all eigenvalues Im[λj] as function
+of the parameter ∆ and the middle row shows the re-
+spective real parts Re[λj].
+Lastly, in the bottom row
+we plot the degree of synchronization S of each eigen-
+vector also as function of ∆. The eigenvalues of M are
+sorted in descending order of their imaginary parts, i.e.
+Im[λ1] > Im[λ2] > ⋅⋅⋅ > Im[λN].
+In the following we discuss different regimes of ∆ and
+its impact on the possibility of synchronized motion in
+accordance with conditions (i)–(iii).
+We focus on the
+eigenstate ⃗c1 with largest imaginary part Im[λ1] (high-
+lighted as thick blue lines in Fig. 4). The reason is that
+for g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free
+while all other eigenstates are simultaneously damped. In
+contrast, if we would choose g such that another eigen-
+state ⃗cj≠1 would become dissipation free, there is at least
+one eigenstate that is exponentially growing. It is thus
+
+7
+2
+1
+0
+1
+2
+Im( i)
+1
+2
+3
+4
+5
+10
+5
+0
+5
+10
+Re( i)
+0
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+(ci)
+s=[0.14, 0.2, 1.2, -0.46, -1.1]
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+s=[-0.24, -0.07, 0.93, -0.08, -0.54]
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+s=[-0.04, 0.22, 1.22, -0.77, -0.63]
+FIG. 4. Examples of dissipation free and (fully) synchronized dynamics in a ring of N = 5 oscillators with random frequency
+disorder. The three different columns correspond to three different set of (scaled) frequency realizations ⃗s. The value of v is
+taken to be negative. In the top row we show the imaginary part Im[λj] of the eigenvalues λj of the matrix M as a function
+∆. The middle row shows the corresponding real part Re[λj] and the bottom row the degree of synchronization S(⃗cj) of the
+corresponding eigenstates ⃗cj. For all three considered realizations, there exists an eigenstate (blue) with the maximum value of
+S (bottom row) for small values of ∆ ≲ 1. This eigenstate also has the largest imaginary part of its associated eigenvalue (top
+row), which allows the tuning g in such a way that it becomes dissipation free while all other eigenstates are damped.
+sufficient to only analyze the possibility of synchroniza-
+tion of ⃗c1 in the following.
+a.
+No frequency difference (∆ = 0):
+This means
+that there are no variations in the oscillator frequen-
+cies and the situation is exactly the same as discussed
+in Sec. III B 1b. Consequently, the eigenvalues of W are
+given by Eq. (19). From the discussion in Sec. III B 1b,
+we know that if g = 2 = Im[λsyn] there exists a dissi-
+pation free synchronized state ⃗csyn ≡
+1
+√
+5(1,...,1)⊺ with
+associated real eigenvalue wsyn = ¯ω, i.e. all oscillators are
+in phase and oscillate with frequency ¯ω. This is exactly
+what we observe in Fig. 4: the eigenvalue with largest
+imaginary part has imaginary part Im[λ1] = 2 (blue thick
+lines in the top row). Note that Im[λ2] = Im[λ3] and
+Im[λ4] = Im[λ5]. Furthermore, Re[λj] = 0 (middle row)
+which implies an oscillation frequency of ¯ω.
+b.
+Small frequency differences (0 < ∆ < 1):
+In this
+regime, the disorder in the frequency differences between
+nearest neighboring oscillators always remains smaller
+than the coupling between them (which is 1). We thus
+expect that the degree of synchronization also remains
+large [S(⃗c1) ≈ N], i.e. the full delocalization of the eigen-
+state ⃗c1 persists. In the bottom row of Fig. 4 we observe
+exactly this behavior of the thick blue line correspond-
+ing to ⃗c1: For small values of ∆, S(⃗c1) is maximal and
+slowly decreases as ∆ approaches the value of 1. Thus,
+the synchronized state remains close to be fully synchro-
+nized within this regime [condition (iii)]. Note, that the
+values for which S(⃗c1) starts to decrease depends on the
+specific realization of disorder ⃗s.
+The imaginary part of the corresponding eigenvalue
+(top row) continues to be the largest value of all eigen-
+values (blue thick line), Im[λ1] > Im[λj≠1].
+Thus, for
+g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free
+while all other eigenstates are damped, i.e. conditions (i)
+and (ii) are fulfilled. As ∆ increases, Im[λ1] decreases
+resulting from the larger amount of frequency disorder.
+Simultaneously, the real part Re[λ1] remains close to 0
+such that the oscillation frequency of the synchronized
+state ⃗c1 also continues to be close ¯ω. Note, the value of
+Re[λ1] only affects the oscillation frequency.
+c.
+Large frequency differences (∆ ≥ 1):
+As ∆ is
+increased further, the frequency difference exceeds the
+nearest neighbor interaction such that – similar to (An-
+derson) localization in finite systems [17] – the degree of
+synchronization S(⃗c1) of the previously delocalized eigen-
+state ⃗c1 rapidly decreases as ∆ increases; see blue thick
+
+8
+lines in the bottom row of Fig. 4. Hence, only partial
+synchronization is possible in this regime and condition
+(iii) is not fulfilled.
+At the same time, the largest imaginary value Im[λ1]
+continues to decrease as function of ∆.
+Yet, close to
+∆ = 1 it remains well separated from the second largest
+imaginary value Im[λ2] such that a suitable choice of g
+still allows for dissipation free dynamics with a sinlge os-
+cillation frequency. However, Im[λ1] may coalesce with
+Im[λ2] for larger values of ∆ depending on the specific
+realization of ⃗s. An example of such a degeneracy is ob-
+served for ∆ ≈ 1.6 in the top right panel of Fig. 4. As a re-
+sult, both eigenstates would be dissipation free resulting
+in the beating pattern discussed previously in Sec. III A.
+However, as mentioned above, only partial synchroniza-
+tion is possible in this regime anyways.
+d.
+Very large frequency differences (∆ ≫ 1):
+In the
+regime of very large frequency differences, we expect that
+the degree of synchronization takes its minimum value
+S(⃗cj) = 1 for all eigenstates j since the scaling follows
+∆ ≫ v. This implies that the values ∆n = ∆sn are much
+larger than the dissipative coupling strength v. Then, M
+is approximately diagonal with eigenvectors ⃗cj nearly lo-
+calized. Note that in this limit there is no synchronized
+state.
+We have checked numerically that for ∆ larger
+than the smallest difference between the sn the synchro-
+nization measure of all eigenstates approaches one, as
+expected (not shown here).
+Lastly, to demonstrate that the dynamics of the sys-
+tem of oscillators is consistent with our discussion of
+the different regimes above (obtained from analyzing the
+eigenvectors and eigenfrequencies), we show in Fig. 5 ex-
+amples of Re[an(τ)] as a function of the scaled time τ
+for ⃗s = (1.14,0.20,1.20,−0.46,−1.1) (corresponding to the
+first column of Fig. 4) for three different values of ∆. In
+all cases, we choose the initial state ⃗a0 = (1,1,2,−1,−1).
+Panel (a) corresponds to the case of vanishing fre-
+quency difference, i.e. ∆ = 0. We choose the dissipation
+g = 2 such that only the eigenstate with largest imagi-
+nary part is dissipation free. As expected after a short
+transient time of τ ≈ 2.5 all oscillators are in-phase syn-
+chronized.
+In panel (b), we increase the frequency difference to
+be ∆ = 0.5. Hence, the synchronized state is dissipation
+free for g = 1.91.
+Analogues to the previous case (a),
+all oscillators are synchronized after a transient time of
+τ ≈ 2.5, yet with a small phase shift. Importantly, all
+oscillators have the same amplitude consistent with the
+finding of Fig. 4 that the degree of synchronization is
+maximal [S(⃗c1) = 5 for this value of ∆].
+Contrarily, in panel (c) where ∆ = 1.1 (and g = 1.51
+to match the condition of dissipation free dynamics) the
+amplitudes vary among the oscillators. This is in accor-
+dance with S(⃗c1) < 5. However, still only a single oscil-
+lation frequency is present (after some transient time).
+This is an example of partial synchronization.
+FIG. 5. Dynamical behavior of Re(ai(τ)) given by Eq.(14)
+for different values of the scaling factor ∆.
+In all three
+cases the mean frequency of the oscillators is ¯ω = 10 and
+the disorder is the same of the first panel of Fig. (4), namely
+⃗s = (1.14, 0.20, 1.20, −0.46, −1.1) The coupling strength v
+is taken to be negative and all frequencies are given in units
+of ∣v∣. The initial condition is ⃗a0 = (1, 1, 2, −1, −1). Panels (a)
+and (b) show fully synchronized motion, while panel (c) is an
+example of partial synchronization.
+IV.
+CONCLUSIONS
+In this work we have investigated the possibility of
+long-lived synchronized motion in networks of harmonic
+oscillators, which are subject to gain/loss and interact
+via nearest neighbor dissipative couplings. In this con-
+text, we refer to synchronization as the existence of a
+single eigenstate of the dynamical matrix, which is dis-
+sipation free.
+Furthermore, if it attains the maximum
+value of the (inverse) participation ratio we refer to it
+as ‘fully synchronized’. We find that in the case of only
+two coupled oscillators, synchronization may always be
+achieved by tuning the gain appropriately as long as the
+frequency difference between the two oscillators is smaller
+than their interaction strength.
+A similar behavior may be observed in larger net-
+works, i.e. many oscillators arranged on a ring with near-
+est neighbor interactions, yet the possibility of synchro-
+nization then depends on the specifics of the system at
+hand: If all oscillators are identical, synchronized col-
+
+1
+2
+3
+4
+5
+2
+(a)
+A=0 g=2
+Re(ai)
+2
+(b)
+A=0.5
+Re(
+3
+A=1.1 g=1.51
+2
+Re(ai)
+0
+-2
+二
+0
+2
+4
+6
+8
+10
+12
+149
+lective motion may be achieved for an even number of
+sites with repulsive dissipative couplings (v positive) or
+an odd number of sites with attractive dissipative in-
+teractions (v negative). For small frequency differences
+compared to the coupling between the oscillators, this
+behavior remains, which we show specifically for the case
+of N = 5, yet it should also hold for larger networks.
+However, as the number of coupled oscillators increases,
+it becomes increasingly difficult to achieve full synchro-
+nization and may only be observed for very small fre-
+quency differences. For larger frequency differences, the
+(inverse) participation ratio decreases significantly such
+that only partial synchronization may be achieved. This
+is in accordance with Anderson localization, where on-
+site disorder results in localized eigenstates.
+However,
+as the dynamical matrix in this work is non-Hermitian,
+Anderson localization is not directly applicable.
+Here,
+future work is needed to study the interplay of synchro-
+nization and localization, in particular in the thermody-
+namic limit and arbitrary small frequency perturbations.
+Synchronization as discussed in this work is intimately
+related to the existence of dissipation free dynamics and
+thus isolated points/submanifolds in parameter space.
+Hence, they require a very precise tuning of gain and
+loss in order to obtain periodic steady states.
+This is
+however hard to achieve in any realistic experiment and
+the synchronized state will experience some gain or loss.
+We can relax the condition Im[wj] = 0 by solely requir-
+ing ∣Im[wj]∣ ≪ ∣Re[wj]∣, which means that the change of
+amplitude of oscillation is small over many oscillations.
+In addition, we then require Im[wj] ≪ Im[wsync], which
+means that all other eigenstates decay much faster than
+the ’synchronized’ one. In principle, one may relax the
+condition even further and demand that there exists only
+one state with Im[wj] > 0, while all other states fulfill
+Im[wi] ≤ 0.
+Then the synchronized state would grow
+while all other states are exponentially damped.
+ACKNOWLEDGMENTS
+C.W.W. acknowledges support from the Max-Planck
+Gesellschaft via the MPI-PKS Next Step fellowship and is
+financially supported by the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) – Project
+No. 496502542 (WA 5170/1-1). A.E. acknowledges sup-
+port from the DFG via a Heisenberg fellowship (Grant
+No EI 872/10-1).
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+Synchronization: a universal concept in nonlinear sci-
+ences (Cambridge university press, Cambridge Univer-
+sity Press, 2003).
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+plications to physics, biology, chemistry, and engineering
+(CRC Press, 2018).
+[3] M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesen-
+feld, Proc. R. Soc. Lond. A 458, 563 (2002).
+[4] S. H. Strogatz and I. Stewart, Sci. Am. 269, 102 (1993).
+[5] M. Rosenblum and A. Pikovsky, Contemp. Phys. 44, 401
+(2003).
+[6] A. Arenas, A. D´ıaz-Guilera, J. Kurths, Y. Moreno, and
+C. Zhou, Physics reports 469, 93 (2008).
+[7] K. Thornburg, M. M¨oller, R. Roy, T. Carr, R.-D. Li, and
+T. Erneux, Phys. Rev. E 55, 3865 (1997).
+[8] J. J. Lynch and R. A. York, IEEE Microw. Guide Wave
+Lett. 5, 213 (1995).
+[9] A.
+Cawthorne,
+P.
+Barbara,
+S.
+Shitov,
+C.
+Lobb,
+K. Wiesenfeld, and A. Zangwill, Phys. Rev. B 60, 7575
+(1999).
+[10] R. Fazio and H. Van Der Zant, Phys. Rep. 355, 235
+(2001).
+[11] A. Slavin, Nature Nanotech. 4, 479 (2009).
+[12] T. Nishikawa and A. E. Motter, New J. Phys. 17, 015012
+(2015).
+[13] J. C. Bellamy, IEEE Commun. Mag. 33, 70 (1995).
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+[16] B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469
+(1993).
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+A. Eisfeld, arXiv:1404.4475 [cond-mat.dis-nn] (2014).
+

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+Emergence of complex network topologies from flow-weighted optimization of network
+efficiency
+Sebastiano Bontorin,1, 2 Giulia Cencetti,1 Riccardo Gallotti,1 Bruno Lepri,1 and Manlio De Domenico3, 4, ∗
+1Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (TN), Italy
+2Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy
+3University of Padua, Via Francesco Marzolo 8, 35131, Padua, Italy
+4Padua Center for Network Medicine, University of Padua
+(Dated: January 23, 2023)
+Transportation and distribution networks are a class of spatial networks that have been of interest
+in recent years. These networks are often characterized by the presence of complex structures such
+as central loops paired with peripheral branches, which can appear both in natural and man-made
+systems, such as subway and railway networks.
+In this study, we investigate the conditions for
+the emergence of these non-trivial topological structures in the context of human transportation
+in cities. We propose a minimal model for spatial networks generation, where a network lattice
+acts as a spatial substrate and edge velocities and distances define an effective temporal distance
+which quantifies the efficiency in exploring the urban space. Complex network topologies can be
+recovered from the optimization of joint network paths and we study how the interplay between a flow
+probability between two nodes in space and the associated travel cost influences the resulting optimal
+network. In the perspective of urban transportation we simulate these flows by means of human
+mobility models to obtain Origin-Destination matrices. We find that when using simple lattices, the
+obtained optimal topologies transition from tree-like structures to more regular networks, depending
+on the spatial range of flows. Remarkably, we find that branches paired to large loops structures
+appear as optimal structures when the network is optimized for an interplay between heterogeneous
+mobility patterns of small range travels and longer range ones typical of commuting. Finally, we
+show that our framework is able to recover the statistical spatial properties of the Greater London
+Area subway network.
+I.
+INTRODUCTION
+Cities represent one of the most fascinating man-made
+complex systems, exhibiting complex features ranging
+on different scales: from their structure and dynamical
+behavior, up to the scaling of socio-economic factors
+with their size [1–5]. These features represent a strong
+hint
+towards the existence of universal
+underlying
+mechanics behind apparently very different cities [6–8].
+Out of these structural properties, one of the most
+relevant, as it plays a fundamental role mediating the
+complex interplay between human dynamics [9, 10] and
+mobility in urban context, are transportation networks
+[11–15]. These networks are a class of spatial networks
+whose properties have been investigated in the literature
+during the last two decades [14, 16].
+In particular,
+they have been studied under the lens of optimality
+conditions and minimization of cost-based functionals
+[16], in order to identify specific features behind efficient
+networks.
+The concept of optimal networks [2] and
+energy-like minimization [17] has its natural under-
+standing in the physics language.
+States of a system
+which minimize a functional defining trade-offs between
+system’s observables (e.g., free energy) represent the
+most likely to be observed states of many real world
+systems. While in some complex systems, such as cities,
+these physical variables can not be derived from first
+∗ Corresponding author: [email protected]
+principles, these analogies and concepts can still offer
+a valid perspective and provide an embedding of these
+systems in a space where the interplay between their
+structure and dynamics can be unfolded and better
+understood. Simple laws have been studied [16, 18] to
+better understand the emergence of hierarchy and the
+role of traffic in the network state.
+Moreover, global
+and local optimization criteria lie in the evolution of
+man-made systems where policy makers and planners
+can adopt some of these criteria in their plans [14].
+Transportation networks are often characterized by
+the presence of complex structures [19–21] such as loops
+paired with branches [22], which can appear both in
+natural [23] and man-made systems [14], like railway
+and subway networks.
+These structures represent the
+key topological elements behind efficient public trans-
+portation systems [20]. In this study we investigate the
+conditions for the emergence of these non-trivial struc-
+tures [18, 24] in the context of human transportation in
+cities. We aim to reconstruct these topologies by means
+of an optimal configuration [25] of the network state.
+Under the assumption of a fixed total cost and a limited
+set of high-capacity connections (e.g., a constraint in
+the expenditure available on infrastructure), the optimal
+configuration is the assignation of connections’ velocities,
+or edges’ weights, such that the joint amount of time
+required to travel between two nodes is minimized for
+all pairs of nodes. Moreover, as these networks represent
+the arteries in urban exploration/navigation via public
+arXiv:2301.08661v1  [physics.soc-ph]  20 Jan 2023
+
+2
+transportation, we study the role of traffic [18, 26] in
+weighting the importance of specific set of connected
+edges (paths).
+We model the urban morphological
+structures which generate heterogeneous distributions of
+human mobility in space, biasing these optimal networks
+to converge towards specific topological features.
+We
+aim to explore the minimum requirements and the
+conditions for these optimization processes to reproduce
+the empirical structures aforementioned.
+At variance with the recent works on network effi-
+ciency, we adopt some fundamentally different model-
+ing choices. We evaluate the efficiency in terms of time
+necessary to explore the network, where edges’ weights
+act as travel speeds. We also weight path efficiency by
+the traffic probability between nodes.
+The underlying
+network lattice (as represented in its simplest form by
+the triangular lattice in the next sections) acts as a sub-
+strate that allows the network to evolve [27] and pos-
+sibly exhibit the network topological features typical to
+real world systems.
+On this framework, we show how
+introducing simple probabilities biasing the optimal effi-
+ciency between points in space force a transition between
+a tree-like topology and a network resembling a simple
+lattice.
+We show also that the modeling of traffic-like
+flows forces the emergence of preferential shared paths in
+space. The optimal configuration of these shared paths
+leads to complex topologies, which ultimately shows fea-
+tures seen in real systems. Features such as a bi-modality
+in the edges’ velocity distribution, characteristic of multi-
+layered transportation, and the central core with loops
+paired with branches typical of subway systems [20, 22]
+are recovered.
+We finally show an application of the
+model within the Greater London Area, finding similari-
+ties of the optimal model with its London Underground
+network.
+II.
+FRAMEWORK FOR URBAN SPATIAL
+MORPHOLOGY
+We introduce here a general framework for spatial net-
+works with the aim of recovering a minimal model for ur-
+ban morphologies that encode both transportation prop-
+erties and urban features such as population and density
+of points of interest (POIs). To this aim, we begin from
+the definition of a network substrate which acts as an
+effective discretization of the spatial dimension. Its sim-
+plest form can be found in an hexagonal 2-dimensional
+tiling [28] and its planar dual, the triangular lattice.
+More formally, in this network substrate each tile in space
+is represented by a node, connected to its set of neighbor-
+ing nodes (see Fig. 1). The existence of a physical edge
+between nodes/tiles i and j is encoded in the adjacency
+matrix A where Aij = 1 if the regions are neighbors in the
+lattice. Distances and metrics are therefore computed on
+top of this network substrate and are biased by nodes’
+features.
+Simulated 
+Annealing: 
+ 
+minimization 
+E({we})
+ {we}init : we = 1.0 ∀e
+ 
+ 
+{we}optimal
+we
+dij
+cΠ = ∑
+e∈Π
+de
+we
+de
+we
+Euclidean distance
+Network
+distance
+Urban 
+morphology
+Urban street
+network
+Minimal 
+model
+A
+B
+C
+FIG. 1.
+Spatial network model for urban mor-
+phology: A) Mapping population distribution and urban
+transportation network to a minimal spatial network where
+nodes encode urban features. Example with hexagonal tiling
+mapped to the triangular lattice. B) Network-based distance
+cΠij versus euclidean distance dij; edges weights/velocity we
+are depicted as widths. C) Edges weights of the lattice sub-
+strate are optimized via simulated annealing to unravel spatial
+features of the optimal transportation network.
+Nodes of this network can encode spatial features
+at the urban scale, such as population or amenities’
+density in a given node. We therefore have a minimal
+representation
+of
+a
+urban
+morphological
+structure
+(see Fig.
+1), and a network substrate that acts as a
+transportation system and can be optimized to generate
+optimal transportation networks [27].
+The path-based
+temporal distance on top of the transportation network
+acts as the fundamental metric we aim to minimize.
+The
+rationale
+behind
+a
+network-based
+distance
+is
+grounded on the assumption that in the context of
+public transportation, urban systems are not navigated
+by considering geographical distance but rather by eval-
+uating the travel-time between departure and arrival.
+More specifically, multi-layer transportation networks
+[11, 21] are characterized by layers having a hierarchical
+organization with different characteristic speeds [29].
+
+3
+Spatial probability 
+ of targets 
+from sample source s
+psj(L, β)
+β = 0.1
+β = 5.0
+β = 0.1
+β = 5.0
+source
+Optimized Network 
+States  
+ 
+G({we})
+A
+B
+pij ∝ e−β
+rij
+⟨r⟩
+spatial
+network 
+(HEX)
+pij ∝ e−β
+Lij
+⟨L⟩
+non-spatial 
+network 
+(ER)
+0.0
+1.0
+psj
+FIG. 2.
+Optimization of synthetic networks: The role of β is studied for two network models: the triangular lattice
+(HEX) and the non-spatial (ER) network. A) Heatmap of target nodes probabilities pij from source node (yellow) under two
+different β values: as the penalty parameter grows, farther nodes are more penalized and flows tend to stay close to the source.
+B) Samples of the associated optimized network states: when flows are not affected by distances (β = 0.1) source nodes target
+all the other nodes in the network with approximately equal probability, the optimal network converges to a tree-like structure.
+With larger β (β = 5.0), trip probabilities are more localized and the presence of loops appear in the optimal structure.
+Thus, an effective temporal distance becomes fundamen-
+tal in determining accessibility and efficiency in urban
+space exploration.
+In this model, we denote e as an edge in the network,
+and we as the associated edge’s weight which can be seen
+as a velocity of the edge in the transportation network.
+de is the euclidean distance of edge e between the nodes
+it is connecting; here edge weights are visually mapped
+as widths of the links. Information about edges distance
+in this framework can be relevant when generalizing to
+the case of random spatial networks where edges have
+different lengths.
+In the case of a general non-spatial
+network, where there is no notion of spatial distances,
+the model can be adapted by fixing de = 1. Finally, we
+define ΩΓij as the set of paths connecting the two nodes.
+We then maximize the efficiency of this underlying sub-
+strate. The transportation efficiency between two nodes
+i−j is computed as a cost in terms of time [30], and we do
+not account for congestion, which can be introduced in a
+possible extension of this framework. We find the path
+(a set of connected edges starting from source node i and
+ending in destination node j) with the smallest cumula-
+tive time, where the time delay introduced by choosing
+en edge e is measured as a ratio between its euclidean dis-
+tance and its weight, representing a proxy of speed. Refer
+to Fig. 1 for a graphic depiction. Here G({we}) is used
+to indicate the network configuration with the associated
+set of edges weights {we}. We therefore aim to find the
+assignment of weights {we} as a trade-off between net-
+β = 0.01
+β = 1.5
+β = 10
+A
+B
+FIG. 3.
+Loop Dimension vs β: Minimum cycle basis
+is used as a network’s observable to study the appearance of
+loops. For each point the median and its absolute deviation
+are shown. A) The average size (number of edges) of the loops
+that constitute the cycle basis. B) Cycle basis dimension as
+number of loops.
+The optimal network ranges from a tree
+structure to a lattice-like with small loops, as the probability
+of long range movement decreases (large β). A transition in
+the cycle basis property is observed at β ∼ 1.0 for the trian-
+gular lattice under study, where the optimal network results
+in an intermediate state with large loops.
+
+Average Loop Size vs β
+8 -
+Size [# edges]
+6
+2 
+0
+10-3
+10-2
+10-1
+100
+101
+102
+Cycle Basis Dimension vs β
+Dimension [# cycles]
+10
+TI
+8
+6
+4 
+2
+0
+10-3
+10-2
+10-1
+100
+101
+102
+β4
+work efficiency in transportation, by minimizing the set
+of costs {cij} of travelling between pair of nodes i − j
+where each element cij is defined as:
+cij({we}) =
+min
+Π∈ΩΓij
+�
+� �
+e∈Πij
+de
+we
+�
+� ,
+(1)
+and in absence of further information, the optimiza-
+tion procedure is the equivalent of minimizing the travel
+costs �
+ij cij. Here we add a novel ingredient, in which
+we couple the optimization of the network temporal dis-
+tances with a traffic flow or probability between pairs
+of nodes. Operationally, when dealing with real world
+Origin-Destination (OD) matrices, this probability can
+be then mapped to a traffic Tij between two points. Tij
+represents the probability of a person from node i to
+travel to node j, and a traffic can be recovered when in-
+formation about populations in source and target nodes
+is added. Tij effectively acts as a rank in the importance
+of a specific path in the network. As paths connecting
+different pairs of nodes may share common edges of the
+network substrate, complex topologies emerge from the
+shared paths jointly optimizing the network efficiency.
+The flow-weighted transportation efficiency therefore be-
+comes:
+E(G({we})) =
+1
+N(N − 1)
+N
+�
+i
+N
+�
+j̸=i
+Tij · cij({we})
+(2)
+We also require that the total network infrastructure
+cost, defined as the cumulative sum of edges weights
+per unit length, multiplied by edge distance CG
+=
+�
+e∈G dewe is conserved.
+This is a generalization of a
+standard optimization process, in the sense that when
+Tij = 1, ∀(i, j), the efficiency is optimized for all possible
+trip pairs (i, j) with equal importance, where the Mini-
+mum Spanning Tree often represents the optimal solution
+[14].
+Before tackling the problem of traffic-like (OD) flows,
+we study a simpler definition of Tij, which allows to un-
+derstand the role of distance in the optimization process,
+in absence of other nodes’ features:
+Tij ∝ e−βdij.
+(3)
+The coefficient β appearing in Eq. 3 is introduced as
+a penalizing parameter and determines how relevant is
+the pair-wise distance dij when computing probabilities.
+We can understand it as the inverse of a characteristic
+traveling distance for an agent on the network β ∼
+1
+d0 .
+While several alternatives on the integration of distance
+in spatial-dependent probabilities (such as power-laws
+Tij ∼ d−γ
+ij ) can be employed, we focus on the exponential
+dependence as it represents the foundational result from
+the maximum entropy derivation of gravity flows [31].
+The introduction of gravity-like flows will be discussed
+in Section IV.
+In the following, we introduce the application of the
+model on simple substrates to explore the role of β in
+absence of spatial urban features.
+III.
+OPTIMIZATION OF SIMPLE NETWORK
+SUBSTRATES
+In order to asses the role of the characteristic distance
+parameter β in the emergence of specific topologies, we
+compute networks statistics on a set of generative models
+for both spatial and non spatial networks. As hexagonal
+tiling of space is preferable when an analysis includes as-
+pects of connectivity [28], the first model we study is a
+triangular lattice. The reason behind this choice is that
+it represents the planar dual [14, 32] of the hexagonal lat-
+tice. Therefore, as space is discretized in hexagonal tiles,
+the spatial network connecting its centers is the triangu-
+lar lattice, which is isotropic and presents less equivalent
+degenerate paths of a rectangular lattice.
+As a direct
+reference to hexagonal tiling, we refer to this network
+as HEX (see Fig. 2). We also extend the analysis also
+to the case of a random network model where nodes are
+not embedded in a metric space. Specifically, we study
+an Erd˝os-R´enyi (ER) network topology, where the def-
+inition of distance between nodes Lij can be defined in
+terms of topological shortest path distance [33].
+As a first benchmark we simplify flows as a spatial
+probability Tij = pij that decays exponentially with dis-
+tance and does not consider nodes features, the resulting
+equation for pij is:
+pij =
+e−βdij/⟨d⟩
+�N−1
+k̸=i e−βdik/⟨d⟩
+(4)
+Where ⟨d⟩ =
+1
+N(N−1)
+�
+i̸=j dij is the average distance
+of points in the network and acts as a normalization fac-
+tor (euclidean distance ⟨r⟩ in case of a spatial network or
+topological ⟨L⟩ for the ER network).
+Therefore pij encodes how much of the nearby space is
+explored by a single source node. An illustration of the
+spatial dependence of target probabilities and samples of
+the resulting optimal topologies are presented in Fig. 2.
+For a range of β values the optimization process is
+performed on an ensemble of these models. To assess the
+emergence of complex structures we observe the number
+of loops that emerge in the optimal state.
+This mea-
+sure is relevant in the context of spatial networks, where
+loops break the symmetry introduced by optimal struc-
+tures such as trees. We compute the minimum cycle ba-
+sis set as a metric to observe the emergence of loops [36]:
+i.e. the minimum set of loops (where a single loop is en-
+coded in a set of edges that defines a closed path in the
+graph) such that any other closed path in the network can
+
+5
+Optimal networks and edges weights distribution P(we)
+Exponential
+Distribution 
+Morphology
+A
+B
+β = 3.0
+3-Points 
+Steiner Tree 
+Morphology
+Steiner node
+10.0
+0.0
+Wj
+β = 4.0
+β = 0.1
+G-KDE
+FIG. 4.
+Minimal models of urban morphology and attractiveness distributions under study (3-Points and
+Exponential decay): Morphology of POIs, where attractiveness Wj is mapped with color intensity (yellow being higher).
+Optimized edges weights distributions P({we}) are characterized by the bi-modal nature that reveals the multi-layered structure
+of the optimal transportation networks when close-range flows are paired with long-range traffic typical of commuting towards
+city centers (Insets P(Tij) with peaks on large-flows due to POIs). Gaussian KDE is shown in orange as a visual aid. A)
+3-points polycentric distribution of POIs, resembling the euclidean Steiner Tree problem [34, 35] for three points. The network
+is optimized with β = 0.1 and β = 4.0, and shows the appearance of branches connecting the POIs paired to large loops in
+the periphery. B) Optimal state and distribution of speeds with exponential decay of wj from the center and an exemplifying
+result with β = 3.0. The optimal topology is characterized by a central loop paired with branches.
+be reconstructed via combination of this cycle basis [36].
+Specifically, we investigate the cycle basis dimension (the
+number of loops that constitute this set) and the average
+loop size, against a range of β values. This metric allows
+to quantify the emergence of spatial topological features
+that differentiate the optimal state from a tree structure.
+Results for these synthetic systems are presented in Fig.
+3. Additional boxplots are shown in SM Figures 1-2. A
+tree-like topology is recovered when the flows probabil-
+ities are distributed uniformly across all nodes in space
+(when β → 0 and distance is therefore not a penalizing
+variable in Eq. 4), while loops emerge when farther tar-
+gets become less likely to be explored and the network
+is globally optimized for close-range trips. Notably, in
+Fig. 3 around β ≈ 1.0, we observe a sharp transition
+in the average loop size in the HEX lattice under analy-
+sis: connections appear between neighboring nodes which
+are far from the tree-root as it becomes more efficient to
+have a direct link. In this β regime the tree topology
+does not guarantee the most efficient configuration for
+peripheral nodes, which have their high probability tar-
+gets in their direct neighborhood (see Eq. 4). Thus in the
+optimization process edges appear between leaves nodes
+which are in separated branches: this ultimately breaks
+the tree structure and leads to the emergence of large-
+scale loops. Eventually the optimal network converges
+to a simpler structure with small loops as the network is
+optimized for nodes to target only direct neighbors in the
+lattice. Finally, in SM Section 2 we show an application
+on the case of a single target node in the perimeter of
+the lattice, where the model reproduces leaves venation
+patterns [14, 37].
+IV.
+SPATIAL ATTRACTIVENESS AND
+TRAFFIC-LIKE FLOWS
+In the context of urban systems, optimal transporta-
+tion networks need to be devised to accommodate traffic
+flows [26] towards specific areas of interest, e.g. due to
+high commercial and business land use density. Hence we
+extend the efficiency optimization framework in the case
+where we have more realistic traffic on top of the urban
+networks, as the presence of nodes with high attractive-
+ness (POIs) biases the flows towards them. In urban sce-
+narios we adopt spatial-interaction models to mimic more
+traffic-like flows. In these models, flows are obtained via a
+gravity-like equation: Tij ∝ pipj exp (−βdij) [10] which
+can be derived from first principles via entropy maxi-
+mization, thus representing the most likely set of flows
+to be observed. In the context of urban exploration, the
+gravity equation can be mapped to a model for spatial in-
+teraction [31, 38] where nodes with a given attractiveness
+Wj compete as possible targets for traffic:
+Tij ∝ 1
+Z PiWj exp (−βdij)
+(5)
+Normalization Z accounts for all possible trip al-
+ternatives �
+k Wk exp (−βdik).
+Pi is the population
+density in node i and Wj encodes a suitable definition
+of benefit/attractiveness of node j as a possible target
+[38].
+Tij is therefore the fraction of population in
+
+β: 0.1
+102
+("1)
+P
+101
+0.0
+0.1
+Tijbeta is:4.0
+100
+Gaussian KDE
+P(We)
+10-1
+2
+3
+4
+Weβ: 4.0
+101
+P(Tij)
+100
+10-1
+10-2
+0.0
+0.2
+0.4
+Tij.beta is:0.01
+100
+Gaussian KDE
+P(We)
+10-1
+10-2
+2
+4
+6
+8
+We6
+node i commuting/travelling on average to node j. To
+better understand the role of nodes’ attractiveness, we
+start with the simplest assumption of equal population
+distribution on all nodes: Pi = 1.0 ∀i; we will introduce
+more realistic population distribution in the next section
+with the London case study.
+We apply these models on the triangular lattice to
+unravel the optimal topologies that emerge when traf-
+fic probabilities are biased towards some nodes having
+high attractiveness (simulating POIs) and we study two
+spatial configurations for nodes’ Wj. In the first configu-
+ration high Wj is assigned to three nodes (POIs) placed
+at the vertices of an equilateral triangle. We study the 3-
+points distribution as it mimics a prototypical polycentric
+distribution of city-centers, and it can be linked to the
+solution of the euclidean Steiner Tree problem [34, 35].
+The Steiner Tree is a class of problems where given a set
+of N points in a plane the goal is to find the set of lines
+connecting the points with minimum cumulative length.
+In our case, the solution would lie in the central node of
+the lattice being the Fermat point [35] and the Steiner
+node, which connects the three vertices of the high Wj
+triangle, as illustrated in Fig. 4 panel A. The second case
+is a distribution of Wj that decays exponentially from the
+center, mimicking a more realistic morphology for a ur-
+ban monocentric structure.
+The two morphologies are
+depicted in Fig. 4.
+We find that due to nodes in the network biasing the
+traffic flows, as it can be seen in the insets of Fig. 4 A,
+the traffic flows get divided into two types: a close range
+paired to a long range set of flows, due to POI polariza-
+tion. We show in Fig. 4 optimal solutions for values of
+β = 0.1, 4.0. Interestingly, optimal solutions are char-
+acterized by three central lines branching from the cen-
+ter (which therefore acts as Steiner node) and connecting
+the three nodes with high attractiveness, therefore resem-
+bling the solution of the Steiner tree problem. Moreover,
+in the case of more localized flows (β = 4.0) these lines
+are also paired with large scale loops connecting farther
+nodes. We also find that the heterogeneity of traffic flows
+forces the appearance of a second mode in the distribu-
+tion of speeds we (see Fig. 4). The two peaks in the
+optimal P(we) can be interpreted as two different lev-
+els of speed, which suggests that the entire process can
+be decomposed in two distinct mechanisms which can
+be mapped as a bi-layer network: one layer at high ca-
+pacity with long-range/commuting trajectories and the
+other one at low velocity with short-range paths. These
+two layers can be ideally separated, hinting towards a
+possible extension of the model to multilayer networks.
+V.
+GREATER LONDON AREA: GENERATIVE
+MODEL FOR THE SUBWAY SYSTEM
+We extend in this section the application of the model
+by integrating data from a real urban structure. Specifi-
+cally, we model the morphology of Greater London Area
+(GLA) on top of our framework and apply the efficiency
+optimization process with the aim of understanding if
+the temporal efficiency optimization of the spatial sub-
+strate paired with realistic flows is sufficient to yield a
+transportation network with similar topological features
+(such as a central core paired with peripheral branches
+[22]) as the London subway system. To extend the model
+to real urban scenarios, we first obtain the distribution
+of amenities [39] from OpenStreetMap [40] and we use
+this density of points in space as a proxy to estimate
+the attractiveness Wj of a tile. Census data for Greater
+London Area yards from 2014 is used to recover popu-
+lation density Pi. These densities are then mapped to
+Uber’s H3 tiling to recover the spatial discretization in
+hexagonal tiles, such that we can have direct mapping to
+the nodes on a triangular lattice, as in the examples dis-
+cussed in previous sections. We thus have the ingredients
+to finally simulate the spatial interaction flows Tij in Eq.
+5. In Fig. 5 the integration of urban data describing the
+London’s morphology in the model is explained and we
+provide a depiction of the OD matrix that arises from
+the spatial interaction model.
+With the aim of repro-
+ducing real features, we impose an upper limit on edge
+weight, so that the distribution of weights is bounded
+during the optimization process: we ∈ (0, w∗). This bet-
+ter simulates the upper bound in speed of real multilayer
+systems. Further explanation of data recovery and inte-
+gration in the model is provided in SM Section 3. We
+find (see SM Figure 6) that {we} distribution displays a
+bi-modal shape, and this allows the analysis of the gen-
+erated network in a sub-graph defined by the set of high
+speed edges. In Fig. 5, panel C, we show a sample re-
+sult for β = 0.35 of this sub-graph. The characterization
+of the network into a central core paired with peripheral
+branches as the optimal state can be visually observed.
+The model’s subgraph of high speed edges is compared
+to the real tube network in the Greater London Area
+[21] to assess the similarities between the optimal struc-
+ture and the real subway system. We quantify this sim-
+ilarity by means of spatial scaling laws [22], these are
+convenient to highlight the recovery of the central core
+structure characterized by loops, paired with quasi mono-
+dimensional lines branching from the core. We investi-
+gate the distribution of nodes stations using the profile
+function N(r) that quantifies the total number of stations
+at a distance r from the network barycenter, computed
+as the average location of all station nodes [22]. Results
+of this scaling analysis for the real and simulated net-
+works are presented in Fig. 6. The two scaling regimes
+indicate the separation of core and branches: the scaling
+of r2 in the core center and a second trend due to mono-
+dimensional branches for r > rC, where rC is the radius
+of the core structure. The second trend can be computed
+analytically via an integral curve for N(r > rC) which
+can be approximated by a power law rγ (γ = 1.25±0.02,
+see SM Section 4), as in Ref. [22]. The curve of N(r)
+is consistent with the real network and confirms scaling
+
+7
+Efficiency optimized
+Network state on GLA
+Morphology
+Pi
+Wj
+POI Distribution 
+(OSM)
+Census Data
+Tij ∝ PiWα
+j exp (−βdij)
+OD fluxes
+Network optimization
+under realistic fluxes
+Greater
+London Area
++ 
+Metro network
+A
+B
+C
+FIG. 5.
+Optimal network model for Greater London Area subway system: Application of the efficiency optimization
+with realistic flows on the urban structure of the Greater London Area. A) Urban morphology data is recovered from Census
+and OSM and population and POI densities are mapped to the H3 tiling. B) Data is mapped to the triangular lattice, with
+nodes having features which allow the calculation of traffic-like flows, a sample OD matrix is shown where Tij are computed
+with β = 0.35. C) Optimal network state for the London model, where only edges and nodes corresponding to the second
+mode are shown (see SM Section 3). Central core structure with loops paired with peripheral branches can be visually seen.
+laws prediction from [22].
+core
+branches
+FIG. 6.
+Scaling properties of GLA Tube stations:
+Profile of the number of stations (nodes in the optimal dis-
+cretized network (see SM Section 3) reproducing GLA under-
+ground) versus the distance from the barycenter. The scaling
+of N(r) profile of the model is compared with the real net-
+work system. Scaling properties predicted in [22] are verified,
+finding the two different scaling regimes separated at
+r
+rC ∼ 1
+for core paired with branches systems, where rC is the core
+radius (characterized by r2 scaling), and NC is the number of
+stations in the core. The scaling exponent γ = 1.25 ± 0.02 is
+obtained as a linear fit of the integral curve [22] for r > rC
+(see SM Section 4 for more details).
+VI.
+DISCUSSION
+Starting from simple conditions on temporal efficiency
+on a spatial network substrate, we show that network op-
+timization paired with traffic-like flows weighting the im-
+portance of specific connections in space can reproduce
+complex networks features from man-made transporta-
+tion networks.
+Specifically, we devise a framework for
+spatial networks where nodes can encode features of ur-
+ban systems and can ultimately lead to the study of opti-
+mal topologies in real scenarios. A key novelty lies in the
+optimization process happening on a spatial substrate,
+such that edges of the resulting optimal network are op-
+timized to improve the efficiency of the shared space by
+all nodes in the network. We show how the probabili-
+ties of moving from one point to another in space force a
+transition between a tree-like and a lattice-like topology
+in the optimal network. Fixing certain target points in
+space with a higher attractiveness for flows can repro-
+duce theoretical results such as the Steiner tree solution
+or leaves venation patterns. We also show that extend-
+ing these probabilities using urban spatial information
+and traffic-like flows modeling forces the emergence of
+shared preferential paths that are organized as complex
+topologies, resulting from traffic weighted optimization
+of network time efficiency, which ultimately exhibits the
+characteristics seen in real systems. We recover features
+such as a bi-modality in the speed distribution of the
+edges of the network, characteristic of multilayer trans-
+portation. Or the appearance of a central core with loops
+coupled to branches typical of underground systems, as
+in the case of the London underground system. We find
+that branches paired to large loops structures appear as
+optimal structures when the network is optimized for an
+
+Tiling (HEX3-Res6) and POI Density
+OSM amenities distribution
+Mapping to the hex lattice model
+W; extraction - here log(density)
+London Greater Area
+52.0
+51.8
+Lat
+51.
+51.2
+51.0
+0.2
+0.0
+0.2
+Lon51.70
+51.65
+51.60
+51.55
+51.5
+51.45
+51.40
+51.35
+51.30
+0.4
+-0.2 
+0.D
+0.21.0
+0.9
+M
+0.8
+0.7
+Traffic Density
+0.6
+0.5
+0.4
+0.3
+0.21.0
+0.9
+0.8
+0.7
+Traffic Density
+0.6
+0.5
+0.4
+0.3
+0.2Rescaled number of stations at distance r from barycenter
+Power law r2
+101
+Model
+LGA Tube
+Powerlaw ry
+100
+10-1
+.OL
+10-1
+1008
+interplay of traffic flows mixed between small range trav-
+els and longer range ones typical of commuting.
+This
+novel framework for the optimization of spatial networks
+in urban contexts may show further improvements and
+extensions to better accommodate the concepts of multi-
+layer and shared space. It could be extended also to the
+case of inter-cities transportation, where specific nodes in
+the network substrate represent cities. To conclude, in
+this work the problem is addressed in a theoretical way
+with the aim of reproducing and understanding some fea-
+tures observed in real spatial networks, but future works
+can exploit this framework as a basis to understand how
+to generate optimal transportation networks in a urban
+planning scenario.
+Competing Financial Interests
+The authors declare no competing financial interests
+Data Availability
+The data used in this work are publicly available from
+the original references
+Code Availability
+The code to perform the analysis will be available upon
+request.
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+SUPPLEMENTARY MATERIAL FOR “EMERGENCE OF COMPLEX NETWORKS TOPOLOGIES
+FROM FLOW-WEIGHTED OPTIMIZATION OF NETWORK EFFICIENCY”
+S1.
+APPLICATION ON SIMPLE NETWORK SUBSTRATES
+In this Section we report more detailed results for the optimization of simple network substrates that was discussed
+in the Main text, specifically for the triangular (HEX) lattice (planar dual of the hexagonal tiling of space) and
+Erd˝os-R´enyi (ER) network. The aim is to explore the resulting topologies that emerge both in spatial and non spatial
+networks when simple probabilities are taken into consideration (see Main text).
+S1.1.
+HEX Lattice and ER Network
+Results of model application on the triangular (HEX) lattice and results of model application on an Erd˝os-R´enyi
+(ER) non-spatial network. For the non-spatial generative model, an ensemble of 20 networks with N = 30 nodes and
+edge probability ρ = 0.2 is generated. For the spatial case, the optimization process is repeated 20 times for each
+value of β on the triangular lattice with N = 37 nodes.
+SM Fig. 1 and SM Fig. 2 show boxplots for the distributions of metrics computed on the cycle basis in panels A
+on both figures. In panels B, samples of the optimized network states are shown for different values of β.
+
+11
+Hex Lattice
+N = 37
+A
+B
+Supplementary Figure S1.
+Cycle basis properties and samples - HEX lattice. Boxplot statistics for the cycle basis
+dimension and optimal network samples for triangular lattice across different β values.
+
+Average Loop Size vs β
+10 
+8 -
+edges]
+6
+#
+Size
+4 :
+2
+0
+0.0001 0.001
+0.01
+0.1
+0.5
+1.0
+1.5
+2.0
+3.0
+4.0
+5.0
+10.0
+15.0100.0
+Cycle Basis Dimension vs β
+Dimension [# cycles]
+10.0
+7.5
+5.0
+2.5
+0.0
+0.0001 0.001
+0.01
+0.1
+0.5
+1.0
+1.5
+2.0
+3.0
+4.0
+5.0
+10.0
+15.0
+100.0
+βB: 10.0
+Yβ: 0.001
+7β: 1.0
+YYβ: 4.0β: 0.1β: 2.0
+蒸β: 5.0
+茶β: 0.5β: 3.012
+ER Network
+N = 30, ρ = 0.2
+A
+B
+Supplementary Figure S2.
+Cycle basis properties and samples - ER network. Boxplot statistics for the cycle basis
+dimension and optimal network samples for an Erdos-Renyi (ER) network across different β values.
+
+Average Loop Size vs β
+8
+6
+5
+#1
+4 
+Size 
+S
+2
+1 
+0
+0.0001 0.001
+0.01
+0.1
+0.5
+1.0
+1.5
+2.0
+3.0
+4.0
+5.0
+10.0
+15.0
+100.0
+Cycle Basis Dimension vs β
+Dimension [# cycles]
+15
+10
+5
+0
+0.0001 0.001
+0.01
+0.1
+0.5
+1.0
+1.5
+2.0
+3.0
+4.0
+5.0
+10.0
+15.0
+100.0
+ββ: 100.0B: 0.0001β: 1.5β: 5.0β: 0.01β: 3.0β: 10.0β: 0.5β: 4.013
+S2.
+LEAVES PATTERNS
+Here we show an application of the model to reproduce leaf venation patterns [37]. A single attracting node (single
+target or sink) is considered at one of the perimeter nodes of the lattice, and the substrate is optimized using all the
+other nodes as sources.
+Supplementary Figure S3.
+Optimal networks resembling leaves’ veins patterns. Optimal state when a single target
+(orange node) in a single spatial extremity is considered. Efficiency is optimized for all nodes in space to reach the target. The
+resulting optimal state resembles tree-like patterns found in leaves (A), while the distribution of edges weights is shown in (B).
+
+B
+beta is:1.0
+100
+10-1
+P(We)
+102.
+100
+101
+We14
+S3.
+APPLICATION ON LONDON TUBE NETWORK
+In this section we discuss more in detail the application of the model on the Greater London Area urban morphology.
+We show that the London subway network spatial properties can be recovered by means of the optimal network state.
+In particular, to simulate a real transportation system, an upper bound on edges travel velocity is imposed.
+S3.1.
+Data Integration: Census and OpenStreetMap Data
+To extend the model to real urban scenarios, we gather data regarding the urban morphological structure from
+OpenStreetMap (OSM) [40] and Census.
+1.
+OSM Data
+To model the attractiveness of nodes in the lattice, we use the density of Points Of Interest in the urban space. We
+use amenities [39] points in OSM as a proxy for Points Of Interest (POIs), and a node j attractiveness (Wj) therefore
+encodes the density of amenities in space.
+The bounding box for Greater London Area (GLA) is obtained via OSM (https://wiki.openstreetmap.org/
+wiki/Bounding_Box) and POI densities are recovered inside this box. Specifically we use the following amenities
+sub-categories:
+’cafe’,’college’,’library’,’school’,’university’,’kindergarten’,’restaurant’,’pub’, ’fast
+food’,’bar’,’bank’,’dentist’,’pharmacy’,’hospital’,’clinic’,’doctor’,’arts
+centre’,’cinema’,’community centre’,’police’,’post office’,’marketplace’
+In Fig. S4 the amenities points recovered from OSM are plotted in the bounding box.
+Lat
+Lon
+51.72
+51.25
+-0.60
+0.35
+Map background is provided by OSM under Open Database License
+Supplementary Figure S4.
+OSM amenities query for GLA bounding box.
+Amenities retrieved from OSM
+(https://openstreetmap.org/copyright) for the bounding box defined by the following longitude [−0.6, 0.35] and latitude
+interval [51.25, 51.75].
+
+tMisisenden
+Chippin
+AONB
+Amersham
+Ingatestone
+ghwycombe
+Billericay
+consfielo
+wWickto
+Basildon
+arlow
+Maidenhead
+Stanford-le
+endon
+M4
+Windsol
+Bracknel
+M2
+ton
+M20
+ghtwater
+Snodland
+Biggir
+LHET
+WestMalling
+Ma
+nam
+eafletLDataby@OpenStreetMapunderODbl15
+2.
+Census Data
+London Wards - Census Data 2014 
+A
+B
+C
+Supplementary Figure S5.
+Census data retrieval. A) London’s wards data from Census 2014 (https://data.london.
+gov.uk/dataset/ward-profiles-and-atlas).
+B) Points are generated in space following wards polygons with a density
+proportional to Census data. C) Points are mapped to HEX3 tiles with associated densities of points. Finally, a restriction to
+a disc is used to enforce symmetry in the distribution of nodes and to ease the computational load.
+Points are mapped to density in space by using HEX3 tiling (https://eng.uber.com/h3/) with spatial resolution
+RES = 8. This spatial discretization process via tiling is of particular relevance to map this information to the
+HEX triangular lattice model which was described in the Main text. Both Fig. S5 and Fig. S4 show how model
+information Pi and Wj are recovered from data and then mapped to tiles as in shown in panel C in Fig. S5. Tiles
+covering the urban area are then mapped to the spatial network model as nodes in the triangular lattice (see Fig. 5
+in Main text).
+For computational simplicity and to preserve the isotropy of the lattice substrate from its central point, we restrict
+our analysis to a disc centered in the London’s region with highest attractiveness, that lies approximately in the City
+Of London district. Results are robust against integration of the remaining GLA region. As the optimal edges weights
+are influenced by the traffic on the substrate, the discarded regions in the Greater London Area do not add relevant
+contributions when compared to more central regions with higher population density.
+S3.2.
+Results with distributions for traffic and bounded weights distribution
+To simulate a real system we limit upper edges weights to a fixed value w∗ = 7 ∗ winit where winit is the initial
+edges weight assigned to the network state (winit = 1.0) before optimization process, such that �
+e∈G dewe,init = C.
+In this simulation we work with β = 0.35.
+
+'LS
+51.6
+51.5
+51.4
+51.3
+.4
+0.2
+0.D
+0.216
+A
+B
+Pi
+Wj
+Urban morphology mapped to
+The hexagonal lattice
+Optimal network and
+edges weights distribution
+Supplementary Figure S6.
+Mapping the data to Hex lattice + GLA optimized network. A) Urban morphology
+data mapped to the HEX tiling and then mapped to the triangular (HEX) lattice. At this step, OD matrix is generated and
+the network is then optimized. B) Resulting optimal network state with its distribution of edges weights showing a bi-modal
+shape. Only fast edges having weight larger than a threshold (we > 5) are kept to isolate the sub-graph constituted by a high
+velocity set of edges, such as a subway system.
+A
+London Metro network
+(Restricted to disc)
+Model output (pruned)
+B
+Supplementary Figure S7.
+Pruned network vs real tube. A) London subway network, restricted to the disc under
+study. B) GLA model output, limited to the nodes and edges which have a weight we > 5.0, where this value was chosen
+from the previously plotted {we} distribution as a threshold to separate the second “mode” with fast edges. This “discretized”
+network is the model on which statistical measures are performed.
+
+Model network0.5 .
+0.4 
+P(we)
+0.3 -
+0.2 
+0.1 
+0.0
+0
+1
+2
+4
+5
+6
+7
+WeCG Real network
+Model pruned (2nd mode)17
+S4.
+SCALING OF NETWORK STATIONS
+The spatial organization of the transportation network can be inspected by taking into consideration the number
+of nodes’ stations N(r) up to a distance r from the barycenter of stations. In the Main text we analyze the scaling
+regimes for the simulated and real London Tube network [21] following the analysis employed in [22]. Specifically,
+they obtain functional forms for the scaling properties of N(r) for different distances from the barycenter. We report
+here the scalings obtained in [22]:
+N(r) ∼
+�
+�
+�
+�
+�
+ρCπr2
+for r < rC
+ρCπr2
+C + NB
+� r
+rC
+dr
+∆(r)
+for
+rC < r < rmax
+N
+for r > rmax
+(S1)
+Specifically, they show that in the large distance regime (r > rC and r < rmax ) the number of stations can be
+approximated by adding the integral curve NB
+� r
+rC
+dr
+∆(r) to a constant term. In [22] it is also reported that the large
+distance behavior can be also, in general, approximated by a scaling law. Therefore we plot here the computation of
+the integral curve against rescaled values of r, and show that in that regime it can be approximated by a power law,
+and its exponent can be obtained via a linear fit. The N(r) curve in the Main text was computed on the real London
+Tube restricted to the disc area as presented in SM Fig. 7. The associated exponent γ = 1.25 ± 0.02 was used in
+the Main text to highlight the secondary scaling for r > rC. As discussed in Ref. [22], due to ∆(r) in SM Eq. S1
+being often noisy, this scaling property is not often well reproduced in empirical networks, and it is often restricted
+to a small region of r values. In Fig. 6 of the Main text, we see that the secondary scaling with the exponent γ
+approximates N(r) in a limited interval for r > rC. The value of the exponent in the branches region is expected to
+be γ < 2.0 [22] and our result is consistent with this.
+Supplementary Figure S8.
+Scaling of the integral curve. Log-Log plot of the integral curve and its linear fit. With
+the associated value of the γ exponent approximated, to highlight a secondary scaling region as mentioned in [22].
+
+Scaling of the integral curve - fit with y = 1.253
+Power law ry
+Integral Curve
+102
+6 × 101
+4 × 101
+3 × 101
+100
+2 × 100
+3 × 100
+r
+Tc18
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+IEEE INTERNET OF THINGS JOURNAL
+1
+Ancilia: Scalable Intelligent Video Surveillance for
+the Artificial Intelligence of Things
+Armin Danesh Pazho∗, Student Member, IEEE, Christopher Neff∗, Student Member, IEEE, Ghazal Alinezhad
+Noghre, Student Member, IEEE, Babak Rahimi Ardabili, Student Member, IEEE, Shanle Yao, Mohammadreza
+Baharani, Member, IEEE, Hamed Tabkhi, Member, IEEE
+Abstract—With the advancement of vision-based artificial in-
+telligence, the proliferation of the Internet of Things connected
+cameras, and the increasing societal need for rapid and eq-
+uitable security, the demand for accurate real-time intelligent
+surveillance has never been higher. This article presents Ancilia,
+an end-to-end scalable, intelligent video surveillance system for
+the Artificial Intelligence of Things. Ancilia brings state-of-the-
+art artificial intelligence to real-world surveillance applications
+while respecting ethical concerns and performing high-level
+cognitive tasks in real-time. Ancilia aims to revolutionize the
+surveillance landscape, to bring more effective, intelligent, and
+equitable security to the field, resulting in safer and more secure
+communities without requiring people to compromise their right
+to privacy.
+Index Terms—Surveillance, artificial intelligence, IoT, com-
+puter vision, application, real-world, real-time, edge, anomaly.
+I. INTRODUCTION
+There is a growing need for effective and efficient surveil-
+lance technologies that can be deployed to protect our cities,
+people, and infrastructure. For example, in Itaewon, South
+Korea, a holiday celebration left over 150 dead due to severe
+overcrowding, with many blaming the tragedy on careless
+government oversight [1]. In Moore County, North Carolina,
+directed attacks against two power substations left over 45,000
+residents without power for days as technicians rushed to
+restore power and authorities struggled to find the source of
+the attacks [2]. With enough forewarning through smart video
+surveillance, they could have been prevented.
+With the recent emergence of the Artificial Intelligence
+of Things (AIoT), some surveillance solution providers have
+started adding basic forms of artificial intelligence to their
+systems. However, their methods are still naive and unable
+to enhance security in a truly meaningful way [3]. This is
+because, while a lot of research is conducted on tasks that
+would benefit surveillance systems, most works focus on
+algorithmic improvements in a lab environment instead of
+paying attention to factors that are prevalent in real-world
+scenarios [4], [5]. Most research focuses on a single algorithm
+and how to tweak it to get the best possible results on readily
+available datasets that often do not reflect a real surveillance
+environment. Few works explore how different algorithms af-
+fect the performance of other downstream algorithms in multi-
+The authors are with the Electrical and Computer Engineering Department,
+The University of North Carolina at Charlotte, Charlotte, NC, 28223 USA.
+{adaneshp, cneff1, galinezh, brahimia, mbaharan, htabkhiv}@uncc.edu
+∗ Corresponding authors have equal contribution.
+algorithm systems. Few still explore the effects of noise (both
+data derived and the system produced) in end-to-end accuracy.
+Beyond this, real-world intelligent surveillance necessitates
+real-time performance. The cognitive abilities of advanced
+artificial intelligence are only helpful if they can be provided
+to security personnel quickly enough to take appropriate action
+before it is too late.
+EdgeN-1
+Camera(s)
+C0
+C1
+CN-1
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Global
+Node
+EdgeN-1
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+EdgeN-1
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+EdgeN-1
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local Node(s)
+L0
+L1
+LN-1
+Global Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Global
+Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Edge0
+Camera(s)
+C0
+C1
+CN-1
+Local
+Node(s)
+L0
+L1
+LN-1
+Global
+Node
+Cloud
+Service(s)
+User 
+Device(s)
+D0
+D1
+DN-1
+User 
+Device(s)
+D0
+D1
+DN-1
+Fig. 1. Conceptual overview of Ancilia.
+In this article, we present Ancilia, the first end-to-end
+scalable, intelligent video surveillance system able to perform
+high-level cognitive tasks in real-time while achieving state-
+of-the-art results. Ancilia takes advantage of the prevalence
+of cameras in the Internet of Things (IoT) and uses localized
+servers and existing cameras, facilitating processing on the
+edge without the need for additional infrastructure upgrades.
+Shown in Fig. 1, Ancilia exists within three logical and
+physical segments: the edge, the cloud, and user devices.
+The edge uses a plethora of advanced artificial intelligence
+algorithms processing data received from cameras to facilitate
+intelligent security. Using a single workstation to perform edge
+processing, Ancilia can monitor up to 4 cameras in real-time at
+30 FPS, or up to 8 cameras at 15 FPS, in scenarios with both
+medium and heavy crowd density. Ancilia performs high-level
+cognitive tasks (i.e. action recognition, anomaly detection)
+with ∼ 1% deviation in accuracy from current SotA.
+Ancilia is designed from the ground up to respect the pri-
+vacy of the people and communities being surveilled. Ancilia
+does not store any personally identifiable information in any
+databases and does not make use of invasive artificial intelli-
+gence techniques such as facial recognition or gait detection.
+Ancilia strictly provides a pose and locational information for
+high-level tasks (i.e. action recognition, anomaly detection),
+as opposed to identity information, which is common. Ancilla
+arXiv:2301.03561v1  [cs.CV]  9 Jan 2023
+
+0IEEE INTERNET OF THINGS JOURNAL
+2
+looks at what a person is doing, not who they are. This allows
+Ancilia to act as a buffer to help remove biases based on
+race, ethnicity, gender, age, and socio-economic factors, which
+can lead to a reduction in the unnecessary conflict between
+authorities and marginalized communities that has become
+increasingly problematic. After data is processed on edge and
+sent to the cloud for communication and service management
+with user devices. A mobile app allows user devices to receive
+data from the cloud, including alerts when potential security
+concerns arise.
+In summary, this article has the following contributions:
+• We present Ancilia, the first end-to-end scalable real-
+world intelligent video surveillance system capable of
+performing high-level cognitive tasks in real-time while
+achieving SotA accuracy.
+• We analyze the ethical concerns of intelligent video
+surveillance, both from a privacy and fairness perspective,
+and illustrate how Ancilia’s design is purpose-built to
+address them.
+• We perform an end-to-end empirical evaluation of Ancilia
+using two high-level cognitive tasks directly related to
+intelligent surveillance, action recognition, and anomaly
+detection, investigating the trade-off in accuracy required
+to achieve real-time performance.
+• We perform an exhaustive system-level evaluation of
+Ancilia’s real-time performance and scalability across
+different classes of hardware and increasing scenario
+intensities, displaying how Ancilia is able to meet real-
+time intelligent security needs in different contexts.
+II. RELATED WORK
+There has been a plethora of research regarding the use
+of artificial intelligence for video surveillance [4], [6]–[8].
+[9] proposes the use of region proposal based optical flow to
+suppress background noise and a bidirectional Bayesian state
+transition strategy to model motion uncertainty to enhance
+spatio-temporal feature representations for the detection of
+salient objects in surveillance videos. [10] proposes the use
+of a person detector, tracking algorithm, and mask classifier
+for tracking pedestrians through surveillance video streams.
+In [4], it is determined that in order to address the latency
+concerns of real-time video surveillance, a shift towards edge
+computing is needed. Nikouei et al. [11]–[13] explore the
+feasibility of using low-power edge devices to perform object
+detection and tracking in surveillance scenarios. They argue
+that in worst case 5 FPS is high enough throughput for tracking
+humans in surveillance applications, and as such computation
+can be pushed to the edge. However, their results show that
+even light weight convolutional neural networks can prove
+problematic for low-power devices, often reducing throughput
+below the 5 FPS threshold. [14] proposes a system using low-
+power embedded GPUs to perform detection, tracking, path
+prediction, pose estimation, and multi-camera re-identification
+in a surveillance environment, while placing a focus on real-
+time execution and the privacy of tracked pedestrians. [15]
+proposes a similar system, focusing solely on object detec-
+tion, tracking, and multi-camera re-identification to increase
+throughput. [16] proposes using a combination of lightweight
+object detection models on the edge and more computation-
+ally expensive models in the cloud, splitting computation
+between the two to provide real-time video surveillance in
+a construction site environment. [17] proposes the use of
+background detection, vehicle detection, and kalman filter [18]
+based tracking for parking lot surveillance and determining lot
+occupancy. [19] proposes a system that uses object detection,
+person tracking, scene segmentation, and joint trajectory and
+activity prediction for pedestrians in a surveillance setting.
+The future of intelligent surveillance is heading towards
+systems able to perform high-level cognitive tasks. A recent
+survey focusing on real-world video surveillance [4] asserts
+that while the domain of video surveillance is comprised
+of understanding stationary object, vehicles, individuals, and
+crowds, the ability to determine when anomalous events oc-
+cur is paramount for intelligent surveillance systems. Other
+research has supported this assertion [6]. [20] utilizes the
+Infinite Hidden Markov Model and Bayesian Nonparametric
+Factor Analysis to find patterns in video streams and detect
+abnormal events. [21] proposes active learning and fuzzy
+aggregation to learn what constitutes an anomaly continually
+over time, adapting the scenarios not seen in standard datasets.
+[22] proposes a system to detect suspicious behaviors in a
+mall surveillance setting, using lightweight algorithms such
+as segmentation, blob fusion, and kalman filter based tracking
+[18]. AnomalyNet [23] is a recently proposed recurrent neutral
+network with adaptive iterative hard-thresholding and long
+short-term memory that works directly off pixel information to
+eliminate background noise, capture motion, and learn sparse
+representation and dictionary to perform anomaly detection in
+video surveillance.
+III. ETHICAL CONCERNS
+Video surveillance has always been associated with social
+and ethical concerns, whether in traditional form or more
+recent intelligent formats. Respecting citizens’ privacy and
+autonomy while improving public safety and security are the
+most well-known and enduring ethical issues in this context
+[24]–[27]. Developing a successful smart video surveillance
+solution that addresses the public safety problem and engages
+the community up to a certain level is only possible by
+considering these concerns.
+There is rising attention among scholars to the issue of
+incorporating privacy concerns at the design level, referred to
+as ”privacy by design” [28]. The source of discrimination and
+privacy violation in many data-driven and AI-based systems,
+such as Smart video surveillance technology, is using Personal
+Identifiable Information (PII) [29], [30]. Using PII, such as
+actual footage of people’s daily activities at any stage of the
+technology, can increase the risk of privacy violation. There is
+a long-lasting debate on the ethical challenges of using facial
+recognition technologies in different sectors and how using
+this technology can result in privacy violation [31]–[34].
+Avoiding facial recognition technologies does not guarantee
+the system is entirely privacy persevering. Storing images of
+pedestrians is another source of ethical violation. From the
+
+IEEE INTERNET OF THINGS JOURNAL
+3
+Neural 
+Network
+Filter
+Match and 
+Combine
+Algorithm
+SQL 
+Database
+Node 
+Boundary
+Cloud
+Cloud
+(C) Cloud 
+Node
+SA
+Flow 
+within a 
+Node
+Flow 
+Between 
+Nodes
+Communication
+AR
+R
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node N-1
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node N-1
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+Database
+Global 
+Tracker
+Statistical 
+Analysis
+(B) Global Node
+FL
+FD
+IDG
+FL
+D
+I
+SA
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+High-level 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+BBP
+BBO
+IDL
+P
+P,IDL
+C
+P,IDL
+BBP
+FL
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+High-level 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+BBP
+BBO
+IDL
+P
+P,IDL
+C
+P,IDL
+BBP
+FL
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node N-1
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Video 
+Stream
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+Downstream 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+F
+PB
+OB
+T
+P
+PT
+C
+PT
+PB
+E
+D
+Database
+Global 
+Tracker
+Statistical 
+Analysis
+(B) Global Node
+FL
+FD
+IDG
+FL
+D
+I
+SA
+IoU
+Confidence
+Confidence
+Object 
+Detector
+Pedestrian
+Tracker
+Pose 
+Estimator
+Feature 
+Extractor
+High-level 
+Tasks
+Crop 
+Selection
+(A) Local Node 0
+BBP
+BBO
+IDL
+P
+P,IDL
+C
+P,IDL
+BBP
+FL
+D
+Edge 
+Boundary
+Video 
+Stream 
+from 
+Camera
+(D) User 
+Device(s)
+AR
+R
+User
+Device
+Fig. 2. Ancilia algorithmic details. N local nodes are connected to a single global node on the edge. The final analyses are transferred to the cloud node to
+feed the application on the user device. Multiple edges may be connected to the could, though this figure only shows one edge for clarity. BBP , BBO, IDL,
+P, C, FL, D, FD, IDG, I, SA, R, and AR refer to bounding boxes for pedestrians, bounding boxes of objects, local identities, poses, person crops that
+passed selection, features from the local node, data from the downstream tasks, features from the database, global identities, information from the database,
+completed statistical analysis, requests from users, and requested attributes respectively.
+discrimination perspective, using any form of PII can con-
+tribute to the issue of marginalization in policing systems [35].
+Therefore, an essential step in designing a non-discriminatory
+system is to ensure the system is not dependent on PII.
+This requires a specific approach toward the design of such
+technology in the choice of algorithm, the type of data used,,
+and the storing of such data.
+Ancilia addresses this by not storing any PII or sending any
+PII across the network. Such data is destroyed after it is used.
+Ancilia utilizes pose-based methods for all high-level cognitive
+tasks, ensuring no PII is ever used in such algorithms. This
+allows for such processing to occur without any potential
+for gender, ethnicity, or class-based discrimination. As such,
+Ancilia is able to address the ethical challenge of privacy in
+smart video surveillance systems while also addressing the
+ethical issue of discrimination.
+IV. ANCILIA ALGORITHMIC FRAMEWORK
+The algorithmic core of Ancilia is separated into two
+conceptual systems: the local nodes containing the algorithmic
+pipeline of each camera and the global node that handles
+all processing that requires understanding of multiple camera
+perspectives. These two systems make up the algorithmic core
+of Ancilia and are the basis on which all higher understanding
+is achieved. A visual representation of this algorithmic core
+can be seen in Fig. 2.
+A. Single Camera Vision Pipeline
+As seen in Fig. 2, the local algorithmic pipeline starts when
+an image is extracted from the camera. The image is first run
+through an object detector to locate people, vehicles, animals,
+and other important objects in the scene. This is important not
+only because it acts as the basis for the rest of the algorithmic
+pipeline but also because it can be used for basic situational
+awareness. Sometimes, just the presence of a certain object
+in a scene is noteworthy, like a person in an unauthorized
+location, a bag left unattended, or the presence of a firearm.
+Ancilia uses YOLOv5 [36] for this purpose (however, it can be
+any detector). The locational coordinates of persons are sent to
+a tracker, where tracklets are created, matching each person
+with their previous detections in prior images. Ancilia uses
+ByteTrack [37]. The tracking allows for understanding how
+a person moves throughout a scene, which is vital for many
+surveillance applications. It also allows Ancilia to understand
+which poses belong to which persons over time, which is
+vital for many high-level tasks that provide much-needed
+situational awareness. Image crops of the people detected in
+the image are also sent to a human pose estimator, where two-
+dimensional pose skeletons are created. Ancilia uses HRNet
+[38] for extracting 2D skeletons. Using pose data for higher-
+level tasks has two major benefits over simply using raw pixel
+data. First, pose data is of much lower dimensionality than
+pixel data, making it much less computationally expensive and
+allowing the Ancilia to function in real-time. Second, pose
+data contains absolutely zero identifiable information, making
+it impossible for high-level tasks to form unintended biases
+based on ethnicity, gender, age, or other identity-based metrics.
+B. Multi-Camera Person Re-identification
+While the tracker tracks people within a single camera,
+locational information cannot accurately re-identify a person
+across multiple cameras. For this, the same person crops that
+were sent to the human pose estimator are also sent to a person
+re-identification feature extractor, where an abstract feature
+representation is created for each person. Only one feature
+representation is created for each person over a period of 30
+frames, and only when the quality of the representation can
+be assured, as poor quality representations are detrimental to
+accurate multi-camera person re-identification. Ancilia uses a
+feature representation filtering algorithm to verify two qualities
+
+IEEE INTERNET OF THINGS JOURNAL
+4
+Pre-
+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
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+Extractor
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+Selection
+A
+Transfer
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+Detector
+Tracker
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+Estimator
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+Extractor
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+Selection
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+Detector
+Tracker
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+Estimator
+Task 0
+Task 1
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+Feature 
+Extractor
+Crop 
+Selection
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+Transfer
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+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
+Task N-1
+Feature 
+Extractor
+Crop 
+Selection
+A
+Transfer
+FL
+D0
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+DN-1
+∞ 
+∞ 
+β1 
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+Pre-
+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
+Task N-1
+Feature 
+Extractor
+Crop 
+Selection
+A
+Transfer
+FL
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+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
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+Feature 
+Extractor
+Crop 
+Selection
+A
+Transfer
+FL
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+Camera
+Neural 
+Network
+Algorithm
+Data
+Batched 
+Data
+Sequential 
+Data
+Frame 
+Batching
+Object 
+Batching
+Comm.
+Process
+IoT
+Flow within 
+a Node
+Transfer to 
+Global Node
+Frame 
+Unbatching
+De-identified
+Data
+Queue
+Comm.
+Process
+Pre-
+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
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+Extractor
+Crop 
+Selection
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+Transfer
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+Processor
+Object 
+Detector
+Tracker
+A
+Pose 
+Estimator
+Task 0
+Task 1
+Task N-1
+Feature 
+Extractor
+Crop 
+Selection
+A
+Transfer
+FL
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+∞ 
+β1 
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+β3 
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+δ1
+δN-1
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+δ1
+δN-1
+Fig. 3. A detailed view of system design in Ancilia’s local nodes. β and δ refer to different batch sizes. FL and D represent local features and data received
+from downstream tasks respectively.
+for person crops. First, a person crop must contain a high-
+quality view of the person. To this end, the filter algorithm uses
+the 2D pose skeleton and verifies that at least 9 keypoints were
+detected with at least 60% confidence. The filter algorithm
+looks at the overlap (i.e. Intersection of Union) of the bounding
+boxes generated by the object detector. An individual’s bound
+box must have an Intersection over Union (IoU) of no more
+than 0.1 with any other person. If those two conditions are met,
+the person crop is determined to be of high enough quality to
+produce an adequate feature representation. If more than one
+crop is deemed suitable for a single person during a 30 frame
+window, the one with the most confident pose is selected. The
+features created by the feature extractor are sent to the global
+node for multi-camera person re-identification. Ancilia uses
+OSNet [39] to extract feature representations.
+C. higher Level Tasks
+High-level tasks are executed on the local node, and have
+access to the object, tracking, and pose data generated in the
+previous steps. Since the decision of which high-level tasks are
+needed is highly application dependent, we do not consider
+these tasks to be part of the Ancilia algorithmic core, but
+instead an extension to be customized based on intended use.
+In this paper, we use action recognition and anomaly detection
+as two common examples of high-level tasks that are highly
+relevant to intelligent surveillance. For action recognition, we
+choose PoseConv3D [40] and CTR-CGN [41], two state-of-
+the-art networks that can utilize the 2D human pose skeletons
+provided by Ancilia. For anomaly detection, we use GEPC
+[42] and MPED-RNN [43], which are based on 2D human
+pose skeletons.
+V. SYSTEM DESIGN
+Beyond the algorithmic design, Ancilia can be analyzed
+from a system-level design and implementation perspective.
+The local node in particular has a complex system design, as
+seen in Fig. 3. The global node and cloud are much simpler,
+as shown in Fig. 2.
+A. Parallelism
+A key design objective of Ancilia is to achieve higher
+efficiency by balancing throughput and latency. Ancilia uses
+pipelining to take advantage of process parallelism. Each
+major task is implemented as a separate process, which
+executes concurrently with other processes. These processes
+communicate with each other using queues to utilize memory
+resources better and enable fast inter-process communication.
+While pipelining is a well-known technique for optimization,
+the overhead associated with its implementation means a
+balance needs to be found. Figure 3 shows a detailed view of
+the system design on the local node. Each pipeline stage is sep-
+arated by a queue with a size limit of λ1 elements, preventing
+any potential overflow from uneven execution speed between
+pipeline stages. By default, Ancilia uses a λ1 value of 4. As
+is common, Ancilia offloads highly parallel tasks that rely on
+neural networks (i.e. object detection, pose estimation, feature
+extraction, and many high-level tasks) to Graphics Processing
+Units (GPUs) for execution.
+B. Data Batching
+Batching is another technique Ancilia implement to better
+utilize hardware resources. Generally, batching is able to
+greatly increase the throughput of a system at the cost of end-
+to-end latency. However, many high-level tasks (e.g. action
+
+IEEE INTERNET OF THINGS JOURNAL
+5
+recognition, anomaly detection) require multiple video frames
+worth of input data (often called a window) before the can start
+processing, so the latency that would be incurred by batching
+input frames is already inherent in these high-level task, as
+long as the frame batch and high-level task window are of
+the same size. Further, as frame batching ultimately increases
+the throughput, the end-to-end latency is decreased when
+compared processing each frame sequentially. While object
+detection works on entire frames, all other neural networks in
+Ancilia work off individual objects. These objects are batched
+together before being input to the network, greatly increasing
+hardware utilization. There can be multiple object batches
+within a single frame batch, based on how many of the relevant
+objects are detected in the video.
+C. Local Node
+Once the local node receives the video stream from the
+camera, the proprocessor is responsible for all basic image
+processing necessary before sending the frames through the
+algorithmic core. After preprocessing, frames are batched in
+sequential segements of size β1. Ancilia sets β1 = 30 to match
+the window size of high-level tasks. This is also convenient
+as most modern security and IoT cameras record video at
+either 30 or 60 FPS. The batched frames are sent to the object
+detector, which outputs a list of objects with class labels and
+bounding box coordinates. Bounding boxes for pedestrians are
+sent to the tracker, while bounding boxes for other objects
+are passed through the system for use in high-level tasks
+and statistical analysis. A crop of each pedestrian from the
+original frame is passed through to the pose estimator. At
+the tracker, bounding boxes for pedestrians are unbatched
+to fit the tracker’s sequential operation. The tracker groups
+the pedestrians and either matches them with previously seen
+pedestrians or assigns them a unique local ID. Afterwards,
+the pedestrians are once again batched by frame and sent to
+the pose estimator. At the pose estimator, the object batching
+is performed on the person crops, with a batch size of
+β2 = 32. These batches are fed to the pose estimator, which
+outputs human pose skeletons for each person crop. Then the
+pedestrian bounding boxes, person crops, local IDs, and human
+pose skeletons are once again batched by frame and combined
+with the object bounding boxes from the object detector.
+The pedestrian bounding boxes, person crops, local IDs, and
+pose skeletons are sent to crop selection, while the pedestrian
+bounding boxes, object bounding boxes, local IDs, and pose
+skeletons are sent to each high-level task as necessary. While
+the person crops are necessary for the feature extraction that
+enables multi-camera re-identification, no identifiable data is
+sent to any of the high-level tasks, keeping in line with the
+ethical concerns mentioned in Sec. III.
+Crop selection filters out low-quality person crops based on
+bounding box overlap and keypoint confidence, as described
+in Sec. IV. The remaining crops are then batched, with size β3
+being dynamic based on the number of persons in the scene,
+and sent to the feature extractor. Once features are extracted,
+the are sent for transfer to the server. Each high-level task
+receives data at the granularity of a frame batch, and sends
+data to the server at whatever granularity that task requires.
+Each high-level task has its own process and works in parallel
+with other tasks as well as with crop selection and feature
+extraction. Communication is completely decoupled from the
+pipeline, so once the data is sent the local node pipeline
+continues to function as normal. Importantly, no identifiable
+information is ever sent to the global node, keeping in line
+with the privacy and ethical concerns mentioned in Sec. III
+D. Global Node
+All received data is stored in a relational database on the
+global node. The matching algorithm described in Sec. IV
+compares the received features with existing features in the
+database over the period λ5 and assigns a global ID based on
+the results. The default value for lλ5 is set to 1 hour, but this
+should be changed to suit the needs of the application. An
+assortment of algorithms performs statistical analysis using
+the relational database, as detailed in Sec. IV. The analysis
+is transmitted to the cloud node using APIs provided by the
+cloud service provider. By default, Ancilia uses Amazon Web
+Services, but this can be altered based on user/application
+needs. The cloud (e.g. Amazon Web Services (AWS)) receives
+analyzed data from the global node.
+VI. EXPERIMENTAL RESULTS
+A. Algorithmic Core
+The algorithmic core of Ancilia consist of multiple algo-
+rithms, each of which works off of data generated by the
+previous algorithms. As these algorithms leverage imperfect
+neural networks, they generate noise that accumulates through
+the system. To understand the source of this noise, we must
+first look at the accuracy of each of these core algorithms
+in isolation. Table I shows the accuracies of the algorithmic
+core’s four main tasks: object detection, pedestrian tracking,
+human pose estimation, and person re-identification. The table
+also shows the accuracies of the top SotA models in each
+task. These SotA methods are not suitable for intelligent
+surveillance applications, as their excessive computation and
+vast parameters make real-time execution impossible, but the
+comparison allows us to see the maximum potential allowable
+by current research and the accuracy loss incurred to keep
+Ancilia performing in real-time.
+TABLE I
+ACCURACY OF ANCILIA’S ALGORITHMIC CORE NETWORKS IN
+ISOLATION. ACCURACIES OF OBJECT DETECTION AND POSE ESTIMATION
+ARE USING THE COCO DATASET [44], TRACKING USING MOT20 [45],
+AND PERSON REID USING DUKEMTMC [46]. SOTA REPRESENT THE
+HIGHEST ACCURACIES CURRENTLY ACHIEVABLE WITH
+STATE-OF-THE-ART METHODS WHEN COMPUTATION AND LATENCY ARE
+NOT A CONCERN.
+Task
+Method
+Metric
+Accuracy
+SotA
+Object Detection
+YOLOv5 [36]
+mAP
+49.0
+65.0 [47]
+Tracking
+ByteTrack [37]
+MOTA
+77.8
+77.9 [48]
+Pose Estimation
+HRNet [38]
+AP
+75.1
+81.1 [49]
+Person ReID
+OSNet [39]
+Top-1
+88.6
+95.6 [50]
+Object detection sees the biggest hit to accuracy, with a 16%
+drop from SotA. This is intuitive, as YOLOv5 [36] is not only
+
+IEEE INTERNET OF THINGS JOURNAL
+6
+the largest model in the algorithmic core, but also the only
+one that operates on the raw camera stream. So while larger
+models are available and would be able to produce higher
+accuracy, even a slight increase in model size or computation
+would result in a noticeable decrease in throughput. Human
+pose estimation sees a decrease in accuracy for a similar
+reason, though much smaller in scale at only 6%. While
+HRNet [38] is not run on the raw camera stream, it is run
+individually for each person detected by the object detector.
+As such, maintaining a small model size is preferable. Person
+re-identification sees a slightly larger drop in accuracy than
+human pose estimation at 7%. While this is partly due to
+using a lightweight model, OSNet [39], the SotA model for
+person reID is also lightweight. However, the SotA uses a
+centroid based retrieval method not suitable for pen-set reID,
+of which most surveillance scenarios are. Pedestrian tracking
+sees almost no drop in accuracy, approximately 0.1%. This
+stems from the comparative ease of tracking pedestrians in
+a single camera, where a simple, lightweight algorithm like
+ByteTrack [37] see almost no performance difference from
+the top of the line SotA approaches.
+B. High-level Tasks
+To better understand how the noise generated by the al-
+gorithmic core effects overall performance, and thus how
+well Ancilia performs in the realm of real-world intelligent
+surveillance, we examine the performance of two high-level
+cognitive surveillance tasks when running on Ancilia. For
+Ancilia to be a benefit to intelligent surveillance tasks, we
+must ensure that excess false alarms or missed positive events
+do not occur. To assess this, we choose action recognition
+and anomaly detection, as these tasks can utilize the human
+pose information generated by the algorithmic core, resulting
+in faster and less biased inference. Since both these methods
+utilize temporal batches of human poses for each individual,
+these experiments will directly reflect the quality of the object
+detection, tracking, re-identification, and pose estimation data
+generated by Ancilia.
+1) High-level Task - Action Recognition: We select two
+state-of-the-art action recognition models, PoseConv3d [40]
+and CTR-GCN [41], and train them using data generated with
+Ancilia. For each model, we train and test with full (30 FPS)
+and half (15 FPS) throughput on NTU60-XSub [51]. Both
+models use a window size of 30 and are trained for 24 epochs
+using Stochastic Gradient Descent (SGD) with a momentum of
+0.9 and Cosine Annealing scheduling. PoseConv3d and CTR-
+GCN have weight decay of 3e−4 and 5e−4 and an initial
+learning rate of 0.4 and 0.2, respectively.
+The results of these experiments can be seen in Tab. II.
+We report the Top-1 and Top-5 accuracy and compare the
+results using data generated by Ancilia to the original data
+available through the PYSKL toolbox [52]. We can see that
+Ancilia is able to provide data of comparable quality to the
+original; action recognition as a high-level task in Ancilia sees
+around 1% drop in accuracy compared to the original data
+using PoseConv3D [40] at full throughput, and around 3% at
+half throughput. Using CTR-GCN [41], Ancilia sees a 2.5%
+TABLE II
+TOP-1 AND TOP-5 ACCURACIES ON NTU60-XSUB [51] IN FULL AND
+HALF THROUGHPUT MODES FOR POSECONV3D [40] AND CTR-GCN
+[41].
+Model
+Data
+FPS
+Top-1 (%)
+Top-5 (%)
+PoseConv3D [40]
+[52]
+15
+91.96
+99.47
+30
+92.76
+99.57
+Ours
+15
+88.79
+98.82
+30
+91.99
+99.28
+CTR-GCN [41]
+[52]
+15
+86.36
+98.46
+30
+83.07
+98.26
+Ours
+15
+81.58
+97.52
+30
+80.44
+97.2
+drop at full throughput and a 4.8% drop at half throughput,
+compared to the original data. From this we can infer that
+PoseConv3D is more robust to noise than CTR-GCN, however
+both performed reasonably well with data generated from
+Ancilia, demonstrating its efficacy for intelligent surveillance
+applications.
+Another interesting observation is that CTR-GCN [41]
+actually performed noticeably better at half throughput than
+at full throughput. This means that CTR-GCN is more suited
+to taking advantage of the higher temporal window allowed
+when using half throughput. This is something to consider
+when choosing an action recognition model when a real-time
+throughput of 30 FPS cannot be guaranteed.
+2) High-level Task - Anomaly Detection: Using the Shang-
+haiTech dataset [53] we train two state-of-the-art anomaly
+detection models, GEPC [42] and MPED-RNN [43], using
+both data generated by Ancilia and the data provided by the
+original authors. The same training strategy from Sec. VI-B1
+is used, with both models trained in full (20 FPS) and half (10
+FPS) modes. GEPC is trained for 25 epochs with a window
+size of 30 and stride of 20 using Adam optimizer with a
+learning rate of 1e-4, weight decay of 1e-5, and batch size
+of 512. MPED-RNN is trained with an input window size of
+30, a reconstruction window of 12, and a prediction window of
+6. The model is trained for 5 epochs using the Adam optimizer
+with a learning rate of 1e−3 and a batch size of 265.
+TABLE III
+AUC ROC, AUC PR, AND EER ON SHANGHAITECH DATASET [53] IN
+FULL AND HALF THROUGHPUT MODES FOR GEPC [42] AND MPED-RNN
+[43].
+Model
+Data
+FPS
+AUC ROC
+AUC PR
+EER
+GEPC [42]
+[42]
+10
+0.6906
+0.5951
+0.35
+20
+0.7372
+0.6427
+0.31
+Ours
+10
+0.6888
+0.5905
+0.35
+20
+0.7223
+0.6023
+0.32
+MPED-RNN [43]
+[43]
+10
+0.6645
+0.5733
+0.37
+20
+0.7023
+0.5869
+0.36
+Ours
+10
+0.6685
+0.5661
+0.37
+20
+0.6679
+0.5487
+0.37
+The results of this experiment can be seen in Tab. III.
+In line with current practices, we report Area Under the
+Receiver Operating Characteristic Cure (AUC ROC), Area
+
+IEEE INTERNET OF THINGS JOURNAL
+7
+1
+2
+3
+4
+5
+6
+7
+8
+0   
+10   
+20   
+30   
+40   
+50   
+60   
+Normal
+Heavy
+Extreme
+Nodes
+Throughput (FPS)
+(a) Server A
+1
+2
+3
+4
+5
+6
+7
+8
+15   
+20   
+25   
+30   
+Normal
+Heavy
+Extreme
+Nodes
+Throughput (FPS)
+(b) Server B
+1
+2
+3
+4
+5
+6
+7
+8
+0   
+10   
+20   
+30   
+40   
+50   
+60   
+Normal
+Heavy
+Extreme
+Nodes
+Throughput (FPS)
+(c) Workstation
+Fig. 4. Throughput of Ancilia across different crowd densities. Hardware details can be seen in Tab. IV.
+TABLE IV
+SYSTEM CONFIGURATIONS. STATS ARE PER CPU/GPU OF THE LISTED TYPE.
+Processor
+GPU
+Name
+Model
+Cores
+Clock Speed
+Model
+CUDA Cores
+VRAM
+Server A
+2× EPYC 7513
+32
+2.6 GHz
+4× V100
+5120
+32 GB
+Server B
+2× Xeon E5-2640 v4
+10
+2.4 GHz
+2× Titan V
+5120
+12 GB
+Workstation
+Threadripper Pro 3975WX
+32
+3.50 GHz
+3× A6000
+10752
+48 GB
+Under the Precision-recall Curve (AUC PR), and the Equal
+Error Rate (EER). With GEPC, we can see that Ancilia more
+than measures up to the task, with only a 1.5% drop in AUC
+ROC at full throughput and less than a 0.2% drop in AUC ROC
+at half throughput. AUC PR shows a more substantial drop
+of 4% at full throughput, but goes down to less than 0.5% at
+half throughput. Equal Error Rates are almost identical, seeing
+almost no change (less than 0.01) when using Ancilia. MPED-
+RNN, which displayed lower overall accuracy in all regards to
+begin with, sees a more significant drop in AUC ROC at full
+throughput, losing 3.5%. However, at half throughput the AUC
+ROC actually increases when using Ancilia, though only by
+0.5%. The AUC PR results mirror that of GEPC, dropping
+3.8% at full throughput and 0.7% at half throughput. The
+Equal Error Rates are once again nearly identical. Being able
+to perform a high-level task such as anomaly detection while
+maintaining accuracies so close to current SotA in research,
+demonstrates Ancilia’s ability to produce quality data, suitable
+for intelligent surveillance applications.
+C. Real-time System Performance
+Algorithmic accuracy is vital for ensuring the information
+provided by high-level cognitive tasks is beneficial for surveil-
+lance applications. However, Ancilia’s ability to perform in
+real-time is equally important. We conduct a series of ex-
+periments, evaluating the runtime performance of Ancilia on
+different hardware, with different scenario intensities, and for
+increasing number of local nodes per hardware device. We
+focus on the performance of the local node, as the global node
+is completely decoupled from the algorithmic pipeline and has
+no noticeable effect on throughput or latency.
+We choose three different hardware configurations for these
+experiments: a high-end server, a lower-end server, and a high-
+end workstation, as seen in Tab. IV. For our scenarios, we
+use the DukeMTMC-video dataset [46] and pick three scenes
+with different crowd densities: normal density, heavy density,
+and extreme density. The distribution of detection density
+in each scenario can be seen in Fig. 5. Note that what is
+considered ”normal density” will change based on application
+environment, which is why we report on such a wide range.
+Each video lasts for
+32k frames, with
+7k frames warm-up
+and cool-down. We test using 1, 2, 4, 6, and 8 local nodes on
+a single system, showing how throughput and latency scale
+in such cases. Each experiment is conducted three times, the
+throughput and latency averaged across runs. The results of
+these experiments can be seen in Tab. V and Fig. 4. The
+distribution of throughput in these scenes can be seen in Fig. 5.
+Under normal crowd density, Server A and Workstation
+are both able to achieve over 50 FPS with a single local
+node, with an end-to-end latency of 5.39 and 4.90 seconds
+respectively. This is well above the 30 and 20 FPS required
+by action recognition and anomaly detection algorithms at
+full throughput, and the latency is low enough to be suitable
+for most surveillance applications where the main concern is
+notify authorities in time for appropriate response. Server A
+is able to handle 6 local nodes in the normal scenario while
+maintaining above 30 FPS, while Workstation can do so with
+4 nodes. Server A is able to maintain above 26 FPS while
+running all 8 local nodes, while Workstation drops to just
+below 18 FPS at 8 local nodes. Server B falls just short of
+30 FPS even with a single node in normal crowd density.
+However, it is able to maintain above 20 FPS while handling
+two nodes simultaneously. Due to having only a single GPU
+and limited VRAM, Server B was unable to run 4 or more
+nodes concurrently.
+Heavy crowd density proves more challenging, with both
+Server A and Workstation only able to achieve above 30
+FPS with up to 4 nodes. The end-to-end latency is also
+longer than it was under normal crowd density, with Server
+A seeing almost double the latency and Workstation seeing
+around a 20% to 80% increase in most cases. Server A and
+Workstation are able to mainatin above 20 FPS at 8 and 6
+
+IEEE INTERNET OF THINGS JOURNAL
+8
+TABLE V
+AVERAGE THROUGHPUT AND LATENCY WHEN RUNNING MULTIPLE LOCAL NODES ON A SINGLE SERVER.
+Server A
+Server B
+Workstation
+Crowd Density
+Nodes
+FPS
+Latency (s)
+FPS
+Latency (s)
+FPS
+Latency (s)
+Normal
+(70 detections
+per second)
+1
+52.94
+5.39
+29.76
+9.73
+54.91
+4.90
+2
+48.99
+5.97
+22.07
+12.7
+45.06
+5.97
+4
+38.91
+7.29
+-
+-
+31.67
+8.53
+6
+31.35
+12.55
+-
+-
+23.10
+13.61
+8
+26.51
+17.94
+-
+-
+17.98
+24.95
+Heavy
+(216 detections
+per second)
+1
+40.16
+15.66
+26.48
+11.93
+48.52
+5.95
+2
+41.22
+10.87
+19.55
+14.51
+41.45
+7.10
+4
+34.54
+14.48
+-
+-
+30.01
+11.20
+6
+27.05
+22.42
+-
+-
+20.99
+30.65
+8
+20.02
+34.68
+-
+-
+15.77
+46.28
+Extreme
+(744 detections
+per second)
+1
+17.80
+36.04
+14.50
+43.81
+25.68
+24.78
+2
+20.90
+30.73
+13.40
+47.57
+23.52
+26.97
+4
+17.27
+38.15
+-
+-
+17.56
+39.43
+6
+9.56
+76.71
+-
+-
+8.94
+94.49
+8
+6.49
+130.08
+-
+-
+6.31
+134.82
+nodes respectively, while Workstation drops to just above 15
+FPS at 8 nodes. Interestingly, with heavy crowd density we
+start to see unusual behavior with Server A having worse
+performance with a singe node than it does with 2 nodes.
+This is caused by the abundance of CPU and GPU resources
+available to the server and a single node being unable to
+fully utilize them. As such, the behavior of Server A in the
+heavy and extreme crowd density scenarios does not start to
+match the expected behavior and mimic the other systems until
+multiple nodes are being run simultaneously. This behavior
+is not too concerning, considering it does not make sense
+to purchase such a high-end server class machine for only
+running a single local node, when a more latency focused
+workstation would be both cheaper and more effective. Server
+B behaves similarly to how it did with normal crowd density,
+except that it falls slightly below 20 FPS when running two
+nodes. Assuming only half throughput was needed for high-
+level tasks, Server B would still be suitable for running up to
+two nodes.
+With the extreme crowd density scenario, Ancilia begins
+to struggle. None of the systems are able to achieve above
+30 FPS even with a single camera, putting full throughput
+action recognition out of reach. Server A is able to achieve
+above 20 FPS with 2 nodes (but notably not with 1) and
+Workstation is able to do so with 1 or 2 nodes. Both Server
+A and Workstation can maintain above 15 FPS at 4 nodes,
+but both drop to around 9 and 6 FPS at 6 and 8 nodes,
+respectively. [54] argues that 5 FPS is suitable for tacking
+pedestrians, and while that is true, high-level tasks that rely
+on detailed human motion, such as action recognition and
+anomaly detection, often struggle for accuracy when running
+below 10 FPS. Another issue is with the increased latency.
+Running only 1 node, Server A and Workstation have latencies
+of 36 seconds and 25 seconds respectively, which is suitable
+for many surveillance applications, but might be too much
+for those that require sharper response times. The latency
+increase to over 2 minutes for both systems with 8 nodes.
+Combined with the low throughput, it becomes difficult to
+recommend running more than 4 nodes on a single system with
+Ancilia when operating under extreme crowd density, expect
+0   
+0.05   
+0.1   
+0.15   
+0.2   
+0.25   
+0.3   
+0.35   
+0.4   
+0.45   
+0
+5
+10
+15
+20
+Extreme
+Heavy
+Normal
+Average Number of Detections/Batch
+Probability
+0   
+0.02   
+0.04   
+0.06   
+0.08   
+0.1   
+0.12   
+0.14   
+0.16   
+20
+30
+40
+50
+60
+70
+80
+90
+Extreme
+Heavy
+Normal
+Throughput (FPS)
+Probability
+Fig. 5.
+The performance of Ancilia in terms of throughput and detection
+distribution.
+for applications where low throughput and high latency are
+not as much of a concern. Server B is unable to achieve 15
+FPS, but does stay above 10 FPS for both 1 and 2 nodes,
+making it suitable for half throughput in anomaly detection.
+However, the latencies of 44 and 48 seconds might be too
+much for some applications. This is the extreme scenario, and
+it understandably provides quite the challenge for real-time
+execution.
+Overall, Ancilia is able to meet the needs of high-level cog-
+nitive tasks while still achieving performance suitable for real-
+
+IEEE INTERNET OF THINGS JOURNAL
+9
+time intelligent surveillance applications. Exact performance is
+dependent on both the hardware used and the intensity of the
+scene, but these results show that even for the most extreme of
+scenarios, Ancilia can be used to provide intelligent assistance
+to surveillance applications.
+VII. CONCLUSION
+In this article we presented Ancilia, an end-to-end scal-
+able intelligent video surveillance system for the Artificial
+Intelligence of Things. Through empirical evaluation, Ancilia
+has demonstrated its ability to bring state-of-the-art artificial
+intelligence to real-world surveillance applications. Ancilia
+performs high-level cognitive tasks (i.e. action recognition and
+anomaly detection) in real-time, all while respecting ethical
+and privacy concerns common to surveillance applications.
+ACKNOWLEDGMENTS
+This research is supported by the National Science Foun-
+dation (NSF) under Award No. 1831795 and NSF Graduate
+Research Fellowship Award No. 1848727.
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+BIOGRAPHY
+Armin Danesh Pazho (S’22) is currently a Ph.D.
+student at the University of North Carolina at Char-
+lotte, NC, United States. With a focus on Artificial
+Intelligence, Computer Vision, and Deep Learning,
+his research delves into the realm of developing AI
+for practical, real-world applications and addressing
+the challenges and requirements inherent in these
+fields. Specifically, his research covers action recog-
+nition, anomaly detection, person re-identification,
+human pose estimation, and path prediction.
+Christopher Neff (S’18) is a National Science
+Foundation Graduate Research Fellow and Doctoral
+Candidate at the University of North Carolina at
+Charlotte. His dissertation focus is on tackling the
+challenges of bringing human-centric computer vi-
+sion to real-world applications. His previous work
+focuses on person re-identification, human pose es-
+timation, action recognition, real-time system devel-
+opment, lightweight algorithms, noisy data, domain
+shift, and real-world applications.
+Ghazal Alinezhad Noghre (S’22) is currently pur-
+suing her Ph.D. in Electrical and Computer Engi-
+neering at the University of North Carolina at Char-
+lotte, NC, United States. Her research concentrates
+on Artificial Intelligence, Machine Learning, and
+Computer Vision. She is particularly interested in the
+applications of anomaly detection, action recogni-
+tion, and path prediction in real-world environments,
+and the challenges associated with these fields.
+Babak Rahimi Ardabili is a Ph.D. student in the
+Public Policy Analysis program at the University
+of North Carolina at Charlotte, United States. His
+main research area is emerging technologies policy
+making. He mainly focuses on the intersection of
+Artificial Intelligence and policy from a privacy
+perspective and the challenges of bringing the tech-
+nology to the community.
+Shanle Yao is an Electrical Engineering Graduate
+student from the University of North Carolina at
+Charlotte. His dissertation focus is on optimization
+and application of Computer Vision pipeline perfor-
+mance and throughput. His areas of interest include
+object detection, multiple objects tracking, human
+pose estimation, semantic segmentation and real-
+world applications.
+Mohammedreza Baharani is an ML researcher and
+edge system deployment engineer at ForesightCares.
+He received his Ph.D. in computer engineering in
+2021 from the University of North Carolina at
+Charlotte, USA, and was a postdoctoral researcher
+at the TeCSAR Lab. His research focuses on the
+intersection of computer architecture engineering
+and machine learning, with the goal of enabling AI
+algorithms on edge devices to have a positive impact
+in fields such as healthcare.
+Hamed Tabkhi (S’07–M’14) is an Associate Pro-
+fessor in the Department of Electrical and Com-
+puter Engineering, University of North Carolina at
+Charlotte, USA. He was a post-doctoral research
+associate at Northeastern University. Hamed Tabkhi
+received his Ph.D. degree in 2014 from Northeast-
+ern University under the direction of Prof. Gunar
+Schirner. Overall, his research focuses on transfor-
+mative computer systems and architecture for cyber-
+physical, real-time streaming and emerging machine
+learning applications.
+
+Award1831795
+d PI), Shannon Reid, Dougl.
+er,Robert Phocas, Arun Ravi
+leta,ChristopherNeff,James
+&Integrative
+Communit
+oroach
+ervice of Public
+t"policing
+edge
+tiple
+co

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+This paper has been accepted for publication at 2023 IEEE International Conference on Robotics and
+Automation (ICRA 2023)
+Light-Weight Pointcloud Representation with Sparse Gaussian Process
+Mahmoud Ali and Lantao Liu
+Abstract— This paper presents a framework to represent
+high-fidelity pointcloud sensor observations for efficient com-
+munication and storage. The proposed approach exploits Sparse
+Gaussian Process to encode pointcloud into a compact form.
+Our approach represents both the free space and the occupied
+space using only one model (one 2D Sparse Gaussian Process)
+instead of the existing two-model framework (two 3D Gaussian
+Mixture Models). We achieve this by proposing a variance-
+based sampling technique that effectively discriminates between
+the free and occupied space. The new representation requires
+less memory footprint and can be transmitted across limited-
+bandwidth communication channels. The framework is exten-
+sively evaluated in simulation and it is also demonstrated using
+a real mobile robot equipped with a 3D LiDAR. Our method
+results in a 70∼100 times reduction in the communication rate
+compared to sending the raw pointcloud.
+I. INTRODUCTION
+With the rapid advancement of LiDAR technology, we
+now can build maps with remarkably high resolution. For
+example, each full scan of an only 16-channel 3D LiDAR
+can give us 57600 points in the pointcloud that represents
+the surrounding obstacles. However, a price for using the
+high resolution LiDAR is the computation, storage, and com-
+munication costs when mapping the environments. While
+one might be able to upgrade the computation and storage
+by using a high performance computer system, the com-
+munication usually becomes a bottleneck due to the low
+communication bandwidth available. In practice, the low
+bandwidth communication is considered as a major challenge
+for many robotics applications such as occupancy mapping
+of underwater and subterranean environments (caves, tunnels,
+mines, etc), search-and-rescue missions in disaster scenarios
+with a degraded communication infrastructure, and planetary
+exploration missions [1]. The low bandwidth can prevent
+a robot from real-time sharing its sensor observations, and
+this can significantly degrade the system responsiveness if
+the robot needs to follow or interact with external control
+or supervision platforms. This work tackles the problem
+of sharing high-fidelity 3D pointcloud through a limited
+bandwidth communication channel.
+The system we consider consists of a robot (the scout)
+equipped with a LiDAR and a communication apparatus, and
+deployed in a low-bandwidth environment. The scout sends
+1Mahmoud Ali and Lantao Liu are with the Luddy School of Informatics,
+Computing, and Engineering, Indiana University, Bloomington, IN 47408
+USA, {alimaa, lantao}@iu.edu
+Occupancy Surface
+VSGP
+variance-based sampling
+SGP
+OctoMap
+wifi
+Scout
+Base
+Fig. 1: System Overview.
+(a)
+(b)
+(c)
+Fig. 2: (a) Gazebo simulated mine tunnel; (b) Original pointcloud
+generated by a VLP16 LiDAR in red, and reconstructed pointcloud
+from the VSGP model in white; (d) Occupancy Map generated by
+OctoMap from the reconstructed pointcloud.
+the observations that it acquires to a base for building the
+occupancy map of the environment, see Fig. 1. Our approach
+exploits the Variational Sparse Gaussian Process (VSGP) [2]
+as a generative model to represent the pointcloud in a
+compact form. This lightweight representation is transmitted
+through low-bandwidth communication to the base where the
+original pointcloud is reconstructed. Extensive evaluations
+reveal that our approach results in a 70∼100 times reduction
+in the memory as well as the communication rate required to
+transmit pointcloud data. For example, Fig. 2a shows a scene
+of a simulated mine tunnel, where its raw pointcloud (shown
+in red, Fig. 2b) requires around 750 KB of memory. Our
+approach is able to represent the same observation using only
+6 KB of memory and transmit through limited-bandwidth
+communication. On the receiver side of the communication
+channel, the compact representation is used to reconstruct
+the original pointcloud (reconstructed pointcloud shown in
+white, Fig. 2b). An occupancy map of the scene can be built
+using the reconstructed pointcloud, see Fig. 2c.
+II. RELATED WORK
+Pointcloud compression algorithms have been investigated
+in recent years to cope with the demands to store and
+communicate the high-precision 3D points [3]. For example,
+the space partitioning trees approaches that exploit the 3D
+correlation between pointcloud points are widely used to
+arXiv:2301.11251v1  [cs.RO]  26 Jan 2023
+
+Velodynecompress the pointcloud data [4]–[9]. Recently, deep learning
+based approaches were also proposed to leverage data and
+learn or encode the pointcloud compression [10]–[12]. Dif-
+ferent from these frameworks, the probabilistic approaches
+exploit the compactness of the distributions to compress 3D
+sensor observation. For instance, Gaussian Mixture Models
+(GMM) [13]–[15] have been proposed as a generative model
+to encode 3D occupancy map. The GMM approach encodes
+the 3D data as a mixture of Gaussian densities to represent
+the occupied and free spaces around the robot.
+Gaussian Process (GP) has been proven to be an excellent
+framework to model spatial phenomena or features in a
+continuous domain [16]–[18]. Unfortunately, the standard
+GP has a cubic time complexity and this results in very
+limited scalability to large datasets. Methods for reducing the
+computing burdens of GPs have been previously investigated.
+For example, GP regressions can be done in a real-time
+fashion where the problem can be estimated locally with
+local data [19]. Sparse GPs (SGPs) [20]–[26] tackle the com-
+putational complexity of the normal GP through leveraging
+the Bayesian rule with a sequential construction of the most
+relevant subset of the data.
+We propose a new probabilistic pointcloud compression
+approach which is based on the VSGP [2] and inspired by
+the GMM approach. While the GMM shares the accumulated
+sensory information as a set of accumulated Gaussian den-
+sities which are sampled and used as an occupancy map of
+the environment, in contrast, the proposed approach relies on
+sharing of immediate sensor observation to be reconstructed
+on the other side of the communication channel for further
+processing based on the required task (e.g. 3D mapping,
+object recognition, tracking, etc).
+This proposed VSGP-based approach offers a few ad-
+vantages over the recent GMM approach: while the GMM
+approach uses two 3D GMMs to fit the occupied and free
+points [13]–[15], our approach uses only one 2D VSGP to
+fit all the occupancy surface, including both the occupied
+and free points. The primary reason that our approach uses
+one VSGP instead of two is that we are using the variance
+calculated by the VSGP at each sampled point during the
+reconstruction process to decide if it belongs to the occupied
+or the free space. Therefore, the proposed approach results
+in a more compact representation of the sensor observation,
+which requires less memory than the GMM approach and,
+as a consequence, leads to a lower communication rate.
+III. BACKGROUND
+GP is a non-parametric model described by a mean
+function m(x), and a co-variance function (kernel) k(x,x′),
+where x is the GP input [27]:
+f(x) ∼ GP
+�
+m(x),k
+�
+x,x′��
+.
+(1)
+Considering a data set D = {(xi,yi)}N
+i=1 with N training
+inputs x and their corresponding scalar outputs (observations)
+y. After training the GP using the data set D, the output y∗ for
+any new query x∗ can be estimated using the GP prediction:
+p(y∗|y) = N(y∗|my(x∗),ky(x∗,x∗)+σ2),
+(2)
+where my(x) and ky(x,x′) are the posterior mean and co-
+variance functions [2]. The GP prediction equation depends
+on the values of the hyperparameters (Θ,σ2) where Θ is the
+kernel parameters and σ2 is the noise variance.
+The computation complexity of a full GP is O(N3).
+In order to reduce the computation complexity, different
+approximation methods were proposed in the literature by
+considering only M input points to represent the entire
+training data [27]. These input points are called the inducing
+points Xm and their corresponding values of the underlying
+function f(x) are called the inducing variables fm. Replacing
+the entire data set with only the M-inducing inputs leads to
+the SGP which has a computational complexity of O(NM2).
+Titsias [2] proposed a variational learning framework to
+jointly estimate the kernel hyperparameters and the inducing
+points. Titsias’ framework approximates the true exact poste-
+rior of a GP p( f|y,Θ) by a variational posterior distribution
+q( f, fm),
+q(f, fm) = p(f|fm)φ( fm),
+(3)
+where φ( fm) is the free variational Gaussian distribution. The
+Kullback-Leibler (KL) divergence is used to describe the dis-
+crepancy between the approximated and the true posteriors.
+Minimizing the KL divergence between the approximated
+and the true posteriors KL[q( f, fm)||p( f|y,Θ)] is equivalent
+to maximizing the variational lower bound of the true log
+marginal likelihood:
+FV (Xm) = log
+�
+N
+�
+y | 0,σ2I +Qnn
+��
+−
+1
+2σ2 Tr( �K),
+Qnn = KnmK−1
+mmKmn,
+�K = Cov(f | fm) = Knn −KnmK−1
+mmKmn,
+(4)
+where FV (Xm) is the variational objective function, Tr( �K) is a
+regularization trace term, Knn is the original n×n co-variance
+matrix, Kmm is m × m co-variance matrix on the inducing
+inputs, Knm is n×m cross-covariance matrix between training
+and inducing points, and Knm = KT
+mn. More details on VSGP
+can be found in Titsias’s work [2].
+IV. METHODOLOGY
+The proposed approach exploits the VSGP as a generative
+model to encode 3D pointcloud. The VSGP is selected
+among different approximation approaches of GP due to
+the following reasons: i) The variational approximation dis-
+tinguishes between the inducing points M (as a variational
+parameter) and the kernel hyperparameters (Θ,σ). ii) The
+regularization term Tr( �K) in the variational objective func-
+tion (Eq. (4)) regularizes the hyperparameters to avoid over-
+fitting of the data. iii) The variational approximation offers
+a discrete optimization scheme for selecting the inducing
+inputs Xm from the original data1.
+A. VSGP as a generative model for the occupancy surface
+Inspired by [13], we project the occupied points ob-
+served by a ranging sensor, e.g., LiDAR, onto a circular
+surface around the sensor origin with a predefined radius
+1For more information about the inducing point selection, check [2]
+
+roc. This surface is called occupancy surface, see Fig. 3.
+In our approach, the sensor observation is defined in the
+spherical coordinate system, where any observed point xi
+is described by the tuple (θi,αi,ri) which represents the
+azimuth, elevation, and radius values, respectively. Also,
+any pointcloud data can be converted from the cartesian
+coordinates (xi,yi,zi) to the spherical coordinates (θi,αi,ri)
+using the following equations:
+ri =
+�
+x2
+i +y2
+i +z2
+i ,
+θi = tan−1(yi,xi),
+αi = cos−1(zi/ri).
+(5)
+All observed points that lie outside the circular occupancy
+surface (with a radius ri > roc) or on the surface (with a
+radius ri = roc) are neglected and considered as free space.
+The rest of the points that are inside the circular surface (with
+a radius ri < roc) are projected on the occupancy surface and
+called the occupied points. Therefore, the occupancy surface
+radius roc acts as the maximum range of the sensor. Each
+occupied point xi on the surface is defined by two attributes:
+the azimuth and elevation angles xi = (θi,αi), and assigned
+an occupancy value f(xi) that is a function of the point radius
+ri. The probability of occupancy f(xi) at each point on the
+occupancy surface is modeled by a VSGP:
+f(x) ∼ VSGP
+�
+m(x),k
+�
+x,x′��
+.
+(6)
+Considering noisy measurements, we add a white noise ε to
+the occupancy function f(x), so the observed occupancy is
+described as yi = f(xi)+ε where ε follows a Gaussian dis-
+tribution N
+�
+0,σ2
+n
+�
+. The final model of the occupancy surface
+is a 2D VSGP where the input is the azimuth and elevation
+angles, x ∈ {(θ,α)}n
+i=1, and the corresponding output is the
+expected occupancy yi. The three main components of the
+final VSGP are:
+1) Zero-Mean Function m(x): There are different for-
+mulas to describe the relationship between the occupancy
+of a point f(xi) on the occupancy surface and its radius
+ri [13]. For example, one candidate is f(xi) = 1/ri where ri
+is bounded by the minimum and the maximum range of the
+sensor rmin < ri < rmax = roc, where rmin > 0. Our approach
+relates the occupancy of a point f(xi) to its radius ri by the
+following equation f(xi) = roc − ri. This mapping between
+the occupancy and the radius of a point is compatible with
+the previous assumption that the occupancy surface radius
+roc represents the maximum range of the sensor. Moreover,
+this mapping is encoded in our VSGP model as a zero-mean
+function m(x) = 0 that sets the occupancy value of the non-
+observed points to zero. This mapping behavior mimics the
+mechanism of the LiDAR itself.
+2) Rational Quadratic (RQ) Kernel: The RQ kernel is
+selected because a GP prior with an RQ kernel is expected
+to have functions that vary across different length scales.
+This quality of the RQ kernel copes with the nature of the
+occupancy surface, specifically in unstructured environments
+where a range of diverse length scales is required, i.e.,
+kRQ
+�
+x,x′�
+= σ2
+�
+1+ (x−x′)2
+2αℓ2
+�−α
+,
+(7)
+(a)
+(b)
+(c)
+Fig. 3: (a) Gazebo scene of a robot in a tunnel (black); (b)
+The occupancy surface generated from the original pointcloud,
+where warmer colors reflect smaller f(xi) values (less occupancy);
+(c) The inner surface represents the original occupancy surface
+(same as in b), and the middle surface represents the reconstructed
+occupancy surface using the VSGP model. The outer grey-coded
+surface represents the variance associated with each point on the
+reconstructed occupancy surface where brighter colors reflect high
+uncertainty. Raw pointcloud is shown in red in (b) and (c).
+where σ2
+f is the signal variance, l is the length-scale, and α
+sets the relative weighting of large and small scale variations.
+The RQ co-variance function is more expressive in terms of
+modeling the occupancy surface than the most commonly
+used Squared Exponential (SE) co-variance function. This
+can be reasoned by the fact that the RQ kernel (when α
+and l > 0) is equivalent to a scale mixture of SE kernels
+with mixed characteristic length-scales [27]. In practice, we
+take into account the resolution of LiDAR along both the
+azimuth and elevation axes to initiate different length-scales
+along each axis to reflect the LiDAR resolution.
+3) Inducing Points Selection: The variational learning
+framework proposed in [2] jointly optimizes the variational
+parameters (inducing points) and the hyperparameters (Θ,σ)
+through a variational Expectation-Maximization (EM) algo-
+rithm. In general, the original discrete optimization frame-
+work [2] suggests having an incremental set of the inducing
+points, so that during the Expectation step (E-step) a point
+from the input data is added to the inducing points set to
+maximize the variational objective function FV and minimize
+the KL divergence between the true and approximated pos-
+teriors KL[q( f)||p( f|y,Θ)]. Then the hyperparameters are
+updated during the Maximization step (M-step).
+Since LiDAR’s field of view is limited within a certain
+range, the projection of the observed points on the circular
+surface leads to a limited input domain for the VSGP. In
+our case, the azimuth and the elevation axes are limited
+to (−π to π) and (−15◦ to 15◦), respectively. The limited
+input domain is used to initiate a fixed number of inducing
+points at evenly distributed locations on the occupied part of
+the occupancy surface. In this way, a different combination
+of the points is selected at each E-step to maximize the
+variational objective function FV and minimize the KL
+divergence. Then the hyperparameters are updated during the
+M-step. The number of the inducing points M is chosen to
+compromise the computational and memory complexity on
+one side and the accuracy of the reconstructed pointcloud
+on the other side. More inducing points result in higher
+computations complexity O(NM2), larger memory to store
+the encoded observation, and higher bandwidth to transfer it.
+However, more inducing points increase the accuracy of the
+
+reconstructed pointcloud. We chose M=500 inducing points
+to keep the average deviation between the reconstructed
+pointcloud and the original pointcloud under 15 cm, see
+Section V-A.2 and Fig. 5. After the training phase on the
+scout side is completed, the selected inducing points are
+combined together with the hyperparameters values of the
+VSGP and are transmitted from the scout to the base.
+B. Variance-based sampling
+On the base side, the inducing points and the values of
+the hyperparameters, which are received from the scout,
+are used to reconstruct the original occupancy surface. The
+reconstruction is done through a GP configured with the
+same kernel (RQ) and likelihood (Gaussian) as the VSGP
+on the scout side. The base GP is trained on the inducing
+points and has a computation complexity of O(M3) where
+M is the number of the inducing points, so we refer it as
+a sparse GP (SGP) and refer the reconstructed occupancy
+surface as the SGP occupancy surface. A grid of query points
+x∗ = {(θ,α)}K
+i=1 with the same resolution of the LiDAR
+along the azimuth and the elevation axes is generated to
+reconstruct the original pointcloud from the SGP occupancy
+surface – we refer the reconstructed pointcloud as the SGP
+pointcloud. If up-sampling of the pointcloud is required for
+any reason, a query grid with higher resolution can be used
+for the reconstruction process. The SGP occupancy surface
+is used to predict the occupancy f(xi) of each point xi of
+the query grid x∗. The occupancy is converted back to the
+spherical radius ri = roc − f(xi) to restore the 3D spherical
+coordinates of each point.
+One advantage of the GP and its variants over other
+modeling techniques is the uncertainty (variance) associated
+with the predicted value at any query point. Considering
+the VSGP model of the occupancy surface on the scout
+side, the variance associated with the occupied points is low
+compared to the variance related to the free points. Selecting
+the inducing points as a set from the original occupied
+points maintains low-variance values for the occupied part of
+the reconstructed SGP occupancy surface on the base side.
+Therefore, the variance value associated with any point on
+the reconstructed SGP occupancy surface is used to predict
+if that point belongs to the occupied or the free part of the
+occupancy surface, see Fig. 4. We use a variance threshold
+Vth as a judging criterion. In fact, the variance related to
+the occupancy surface is different from one observation to
+another, and it is affected by both the number of observed
+(occupied) points and their distribution over the occupancy
+surface. Therefore, we chose the variance threshold Vth as
+a variable that changes with the distribution of the variance
+over the occupied and free parts of the occupancy surface.
+Vth is defined as a linear combination of the variance mean
+vm and standard deviation vstd over the surface, i.e., Vth =
+Km ∗vm +Kstd ∗vstd where Km and Kstd are constants. These
+two constants are tuned by first setting Vth = vm (Km = 1 ,
+Kstd = 0), then we increase Kstd and decrease Km gradually
+till we get the values that give the highest accuracy for the
+reconstructed SGP pointcloud (considering a fixed number of
+(a)
+(b)
+(c)
+Fig. 4: Variance-based sampling. (a) Gazebo scene shows the
+entrance of the tunnel; (b) shows the original (inner), reconstructed
+(middle), and variance (outer) surfaces. It also shows the re-
+constructed pointcloud (in white) through reconstructing from all
+points (free and occupied) of the occupancy surface. (c) shows
+reconstructed SGP pointcloud after removing all points that most
+likely belong to the free part of the occupancy surface. Raw
+pointcloud is shown in red in (b) and (c).
+inducing points). Our sampling-based approach is capable
+of discriminating between the free points that most likely
+belong to the free part of the SGP occupancy surface and
+the occupied points that belong to the the occupied part of
+the SGP occupancy surface. After removing the free part
+of the SGP occupancy surface, the Cartesian coordinates of
+the occupied points are calculated using the inverse form of
+Eq. (5) to restore the original point cloud, see Fig. 4c.
+V. EXPERIMENTAL DESIGN AND RESULTS
+The proposed approach is implemented in Python3 on
+top of GPflow-v2 [28] and TensorFlow-v2.4 [29] under
+ROS framework [30]. Both real-time simulation and real-
+time demonstration were considered to evaluate the proposed
+approach. In both the simulation and the hardware experi-
+ments, a VLP-16 LiDAR was used with a maximum range
+of 10m, a frequency of 4Hz, and a resolution of (0.1◦,2◦)
+along the azimuth and the elevation axis, respectively. This
+configuration results in a maximum pointcloud size of 57600
+points. The query grid, which is used to sample the SGP
+occupancy surface on the base side, has the same resolution
+as the VLP-16 LiDAR. A 3D occupancy grid map with a
+resolution of 5cm is generated from the reconstructed SGP
+pointcloud through Octomap [31].
+We investigate the performance of our framework and
+compare it with the GMM approach [13]–[15]. While the
+GMM approach tackles the occupancy mapping problem as
+a whole, our approach focuses on compressing sensor obser-
+vations through limited-bandwidth communication channels.
+To be able to compare the two approaches, we implemented
+the GMM approach in such a way that it is used to encode
+one sensor observation at a time instead of generating an
+entire occupancy map. We compared our approach with two
+versions of the GMM approach: i) A CPU-based implemen-
+tation of GMM that follows the same guidelines of [13].
+ii) An upgraded GPU-based implementation of GMM. We
+implemented the GPU-GMM to have a fair computation
+comparison with our VSGP approach which runs on GPU.
+A. Simulation Experiments
+1) Simulation Setup:: The simulation setup consists of
+two machines that communicate to each other over WiFi:
+
+The first machine, where the scout and the environment are
+simulated, is an Intel® Core™ i7 NUC11 PC equipped with
+64 GB RAM and 6 GB Geforce RTX2060 GPU. The second
+machine, which acts as the base, is an Intel® Core™ i7
+Alienware Laptop equipped with 32 GB RAM and 8 GB
+Geforce RTX2080 GPU. Both are connected using a 2.4 GHz
+WiFi router. The network flow is monitored using the ifstat
+tool to evaluate the communication performance. The mine
+tunnel of the cpr inspection world, which is developed by
+ClearPath robotics, is used as our simulation environment.
+This environment is selected because it represents one of
+the targeted low-bandwidth subterranean environments. The
+mine tunnel part of the cpr inspection world fits in a rectan-
+gular area with an approximated area of 30×65m2, the tunnel
+length is around 135m. The ground elevation and the height
+of the tunnel are different from one place to another. The
+ClearPath Jackal robot is used as the scout. The proposed
+approach was evaluated through 20 real-time simulation
+trials. In each trial, the robot starts at the beginning of the
+cave and follows a predefined path along the mine using
+way-point based navigation.
+2) Simulation Results: We evaluate the performance of
+our approach based on the reduction in the memory and the
+communication rate required to transmit the sensor observa-
+tions between the scout and the base. The VSGP representa-
+tion requires only 1514 floating points (FP) to represent the
+entire pointcloud (3 FP for each inducing point (3x500) + 6
+FP for robot pose + 6 FP for the hyperparameters). This value
+is less than the memory needed by the GMM approach which
+requires ∼ 2000 FP (10 FP for each component (10x200)
+distributed as 6 FP for covariance + 3 FP for mean + 1
+FP for weight) [13]. We send the robot pose to the base
+because our approach encodes the observation relative to the
+robot body frame, while the GMM approach first transforms
+the observation from the robot body frame to a global frame
+using the robot current pose and then sends the encoded
+Gaussians densties with respect to the global frame.
+To quantify the accuracy of the reconstructed SGP point-
+cloud, we use the Root Mean Square Deviation (RMSD)
+between the radius predicted by our approach and the actual
+radius of each point on the occupancy surface.
+RMSD =
+�
+∑N
+i=1 (ri − ˆri)2
+N
+,
+(8)
+where N is the size of the pointcloud, ri is the actual radius at
+(θi,αi), and ˆri is the estimated radius value at the same point
+(θi,αi). Fig. 5a shows the mean and the standard deviation
+of the RMSD for each predicted point over 110 observations
+(each observation has around 10K to 50K points). Also,
+Fig. 5a implicitly reflects the memory required by VSGP and
+GMM to store one observation, as described before that the
+memory required to store one observation can be calculated
+by multiplying the number of inducing points (bottom x-axis)
+by 3 and multiplying the number of components (top x-axis)
+by 10. We match pairs of the VSGP and GMM models (in
+terms of the number of inducing points and components)
+based on the memory requirement and the accuracy of the
+(a)
+(b)
+(c)
+(d)
+Fig. 5: Performance comparisons. (a) shows the RMSD between
+the reconstructed and the original pointcloud for VSGP(vs #induc-
+ing points) and GMM(vs #components); (b) illustrates the training
+time against the pointcloud size (considering 500-inducing points
+VSGP, and equivalently, 200-components GMM); (c) represents
+the training time versus the #VSGP-inducing points and #GMM-
+components; (d) shows the prediction time versus the #VSGP-
+inducing points and #GMM-components.
+reconstructed pointcloud (reflected by the RMSD) for each
+pair, see table I. For example, 500-inducing points VSGP
+results in an average RMSD value for each point of 9 cm
+with a standard deviation of 10 cm. This corresponds to an
+average RMSD of 11 cm with a standard deviation of 25 cm
+for a 200-components GMM.
+TABLE I: VSGP vs GMM(ind: inducing, cps: components)
+VSGP
+GMM
+#
+Memory
+RMSD
+#
+Memory
+RMSD
+ind
+∼FPs
+∼cm
+cps
+∼FPs
+∼cm
+200
+600
+20±22
+50
+500
+27±50
+300
+900
+14±15
+100
+1000
+16±35
+400
+1200
+12±14
+150
+1500
+13±31
+500
+1500
+9±10
+200
+2000
+11±29
+600
+1800
+9±10
+250
+2500
+11±30
+Now we analyze the results in Fig. 5. Fig. 5a shows the
+RMSD values associated with VSGP have a smaller standard
+deviation than the GMM’s. It also shows that increasing the
+number of the VSGP-inducing points (bottom x-axis) or the
+number of the GMM-components (top x-axis) will result in
+smaller RMSD (higher accuracy).
+An intensive evaluation of the training and the prediction
+phases is presented in Figs. 5b-5d. The reduction in the
+training time versus the reduction in the size of the raw
+pointcloud is presented in Fig. 5b, where 0% removal percent
+means a pointcloud size of 57.6K points. Fig. 5c shows
+the increase in training time versus the number of inducing
+points and the number of components. We compare the
+training time of the VSGP, the GMM-CPU (considering the
+
+#Components
+50
+100
+150
+200
+250
+0.75 -
+GMM
+0.50
+VSGP
+RMSI
+0.25
+0.00
+0.25
+200
+300
+00f
+500
+600
+#Inducing Points103
+Su
+VSGP
+GNIM
+GNIM-GPU
+102
+3350606775
+Removal Components
+50
+100
+150
+200
+250
+103
+GMM
+GMM-GPU
+VSGP
+102
+200
+400
+600
+Inducing PointsComponents
+50
+100
+150
+200
+250
+VSGP
+[stu]
+GMM
+40
+Prediction
+20
+200
+400
+600
+Inducing Points(a)
+(b)
+(c)
+(d)
+Fig. 6: (a) shows the simulated mine environment in Gazebo;
+(b) shows the Octomap of the mine generated from the original
+pointcloud; (c) shows the Octomap generated from the recon-
+structed SGP pointcloud; (d) shows the communication rate and the
+accumulated data sent from the scout to the base in case of sending
+raw pointcloud PCL(1750KB/S, 840MB), GMM data(25.8KB/S,
+12.4MB), and VSGP data(18.2KB/S, 8.7MB). The y-axis is plotted
+in log-scale.
+default configuration of the GMM approach used in [13]),
+and the GMM-GPU implementation. The results show that
+our approach outperforms both the CPU and GPU imple-
+mentation of the GMM approach in terms of training time.
+Fig. 5d presents the variation of the prediction time of the
+VSGP versus the number of the inducing points, where the
+values shown in the figure represent the time required to
+predict the occupancy value associated with all the points of
+the grid query x (57600 points).
+Fig. 5d indicates that for a matching pair of GMM and
+VSGP (Table I), GMM has a less sampling time than
+the paired VSGP. However, the pointcloud reconstruction
+process of the VSGP is more convenient than the GMM
+approach because a fundamental difference between sam-
+pling the VSGP and the GMM is that: when we sample
+from a GMM, we get a sample (from a distribution) with
+random values (θs,αs,rs), so we do not have control over
+the location of the sample on the occupancy surface (θs,αs).
+In contrast, for the VSGP approach, we predict the radius
+value rs for a certain point on the occupancy surface defined
+by (θs,αs). So, we have control over the point location on
+the occupancy surface. While constructing the 3D octomap
+of the tunnel environment using the scout-base scheme, the
+average communication rate was 1750 KB/S, 25.8 KB/S,
+and 18.2 KB/S for sending raw point clouds, GMM encoded
+data, and VSGP encoded data respectively, see Fig. 6d. The
+accumulated data sent through the network is reduced from
+840 MB for sending raw pointcloud to 12.4 MB in case
+of GMM and 8.7 MB in case of VSGP. This indicates a
+(a)
+(b)
+(c)
+Fig. 7: Indoor demonstration. (a) shows octomap of the laboratory
+building generated from the original pointcloud. (b) shows octomap
+generated from the reconstructed SGP pointcloud. (c) shows the
+reduction in the communication rate and the accumulated data sent
+from the scout to the base, where log-scale is used for y-axis. PCL
+represents the raw pointcloud.
+compression ratio of ∼ 96 (840/8.7 ∼ 1750/18.2).
+B. Hardware Experiment
+A Jackal mobile robot, equipped with a VLP-16 LiDAR
+and NUC11 PC, was used as the scout, while the Alien-
+ware laptop was used as the base. The demonstration was
+conducted in an indoor environment, where the VSGP-
+encoded pointcloud data was sent from the scout to the
+base to generate a 3D Octomap [31] of the building from
+the SGP reconstructed pointcloud in real-time, see Fig. 7.
+Fig. 7c shows the reduction in the communication rate for
+the hardware experiment. The communication rate dropped
+from around 560 KB for transmitting raw pointcloud to
+around 8 KB for transmitting the encoded VSGP (this ratio
+is equivalent to 70 times smaller rate). The communication
+rate of the hardware experiment is low compared to the
+simulation experiment because the LiDAR resolution was
+halved during the hardware experiment. The total amount
+of data transmitted at the end of each trial was around 100
+MB for sending raw pointcloud and only around 1.4 MB for
+sending the VSGP encoded observation.
+VI. CONCLUSION
+In this paper, we introduce a lightweight representation
+for the 3D pointcloud using the VSGP. This representation
+allows high-fidelity observations to be efficiently stored
+and transmitted through limited-bandwidth communication
+channels. Based on the results of the simulation and hardware
+experiments, our approach results in around 70-100 times
+smaller size representation of the sensor observation. This
+compact representation can facilitate many of the robotics
+
+KB
+PCI
+GMM
+Rate
+102
+VSGP
+MB
+PCL
+101
+Data
+GMM
+VSGP
+0
+O0T
+200
+300
+400
+500
+Time [S]S/
+POL
+101
+VSGP
+Rate
+KB
+PCL
+Data
+102
+VSGP
+0
+25
+50
+75
+100
+125
+150
+175
+Time [S]applications which are limited by the communication band-
+width such as subterranean and underwater exploration,
+search and rescue missions, and planetary exploration. In
+addition, our approach can also be beneficial in the context
+of multi-robot collaboration where a number of robots are
+required to share high-volume information (3D pointcloud)
+through low-bandwidth channels.
+ACKNOWLEDGEMENT
+This work was supported by National Science Foundation
+with grant numbers 2006886 and 2047169.
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+

JNFIT4oBgHgl3EQfZCuh/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf,len=412
+page_content='This paper has been accepted for publication at 2023 IEEE International Conference on Robotics and  Automation (ICRA 2023)  Light-Weight Pointcloud Representation with Sparse Gaussian Process  Mahmoud Ali and Lantao Liu  Abstract— This paper presents a framework to represent  high-fidelity pointcloud sensor observations for efficient com-  munication and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The proposed approach exploits Sparse  Gaussian Process to encode pointcloud into a compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Our approach represents both the free space and the occupied  space using only one model (one 2D Sparse Gaussian Process)  instead of the existing two-model framework (two 3D Gaussian  Mixture Models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We achieve this by proposing a variance-  based sampling technique that effectively discriminates between  the free and occupied space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The new representation requires  less memory footprint and can be transmitted across limited-  bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The framework is exten-  sively evaluated in simulation and it is also demonstrated using  a real mobile robot equipped with a 3D LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Our method  results in a 70∼100 times reduction in the communication rate  compared to sending the raw pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' INTRODUCTION  With the rapid advancement of LiDAR technology, we  now can build maps with remarkably high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For  example, each full scan of an only 16-channel 3D LiDAR  can give us 57600 points in the pointcloud that represents  the surrounding obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' However, a price for using the  high resolution LiDAR is the computation, storage, and com-  munication costs when mapping the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' While  one might be able to upgrade the computation and storage  by using a high performance computer system, the com-  munication usually becomes a bottleneck due to the low  communication bandwidth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In practice, the low  bandwidth communication is considered as a major challenge  for many robotics applications such as occupancy mapping  of underwater and subterranean environments (caves, tunnels,  mines, etc), search-and-rescue missions in disaster scenarios  with a degraded communication infrastructure, and planetary  exploration missions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The low bandwidth can prevent  a robot from real-time sharing its sensor observations, and  this can significantly degrade the system responsiveness if  the robot needs to follow or interact with external control  or supervision platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This work tackles the problem  of sharing high-fidelity 3D pointcloud through a limited  bandwidth communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  The system we consider consists of a robot (the scout)  equipped with a LiDAR and a communication apparatus, and  deployed in a low-bandwidth environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The scout sends  1Mahmoud Ali and Lantao Liu are with the Luddy School of Informatics,  Computing, and Engineering, Indiana University, Bloomington, IN 47408  USA, {alimaa, lantao}@iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='edu  Occupancy Surface  VSGP  variance-based sampling  SGP  OctoMap  wifi  Scout  Base  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 1: System Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 2: (a) Gazebo simulated mine tunnel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (b) Original pointcloud  generated by a VLP16 LiDAR in red, and reconstructed pointcloud  from the VSGP model in white;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (d) Occupancy Map generated by  OctoMap from the reconstructed pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  the observations that it acquires to a base for building the  occupancy map of the environment, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Our approach  exploits the Variational Sparse Gaussian Process (VSGP) [2]  as a generative model to represent the pointcloud in a  compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This lightweight representation is transmitted  through low-bandwidth communication to the base where the  original pointcloud is reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Extensive evaluations  reveal that our approach results in a 70∼100 times reduction  in the memory as well as the communication rate required to  transmit pointcloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 2a shows a scene  of a simulated mine tunnel, where its raw pointcloud (shown  in red, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 2b) requires around 750 KB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Our  approach is able to represent the same observation using only  6 KB of memory and transmit through limited-bandwidth  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' On the receiver side of the communication  channel, the compact representation is used to reconstruct  the original pointcloud (reconstructed pointcloud shown in  white, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' An occupancy map of the scene can be built  using the reconstructed pointcloud, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' RELATED WORK  Pointcloud compression algorithms have been investigated  in recent years to cope with the demands to store and  communicate the high-precision 3D points [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For example,  the space partitioning trees approaches that exploit the 3D  correlation between pointcloud points are widely used to  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='11251v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='RO]  26 Jan 2023  Velodynecompress the pointcloud data [4]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Recently, deep learning  based approaches were also proposed to leverage data and  learn or encode the pointcloud compression [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Dif-  ferent from these frameworks, the probabilistic approaches  exploit the compactness of the distributions to compress 3D  sensor observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For instance, Gaussian Mixture Models  (GMM) [13]–[15] have been proposed as a generative model  to encode 3D occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The GMM approach encodes  the 3D data as a mixture of Gaussian densities to represent  the occupied and free spaces around the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Gaussian Process (GP) has been proven to be an excellent  framework to model spatial phenomena or features in a  continuous domain [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Unfortunately, the standard  GP has a cubic time complexity and this results in very  limited scalability to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Methods for reducing the  computing burdens of GPs have been previously investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  For example, GP regressions can be done in a real-time  fashion where the problem can be estimated locally with  local data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Sparse GPs (SGPs) [20]–[26] tackle the com-  putational complexity of the normal GP through leveraging  the Bayesian rule with a sequential construction of the most  relevant subset of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  We propose a new probabilistic pointcloud compression  approach which is based on the VSGP [2] and inspired by  the GMM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' While the GMM shares the accumulated  sensory information as a set of accumulated Gaussian den-  sities which are sampled and used as an occupancy map of  the environment, in contrast, the proposed approach relies on  sharing of immediate sensor observation to be reconstructed  on the other side of the communication channel for further  processing based on the required task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 3D mapping,  object recognition, tracking, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  This proposed VSGP-based approach offers a few ad-  vantages over the recent GMM approach: while the GMM  approach uses two 3D GMMs to fit the occupied and free  points [13]–[15], our approach uses only one 2D VSGP to  fit all the occupancy surface, including both the occupied  and free points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The primary reason that our approach uses  one VSGP instead of two is that we are using the variance  calculated by the VSGP at each sampled point during the  reconstruction process to decide if it belongs to the occupied  or the free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Therefore, the proposed approach results  in a more compact representation of the sensor observation,  which requires less memory than the GMM approach and,  as a consequence, leads to a lower communication rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' BACKGROUND  GP is a non-parametric model described by a mean  function m(x), and a co-variance function (kernel) k(x,x���),  where x is the GP input [27]:  f(x) ∼ GP  �  m(x),k  �  x,x′��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (1)  Considering a data set D = {(xi,yi)}N  i=1 with N training  inputs x and their corresponding scalar outputs (observations)  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' After training the GP using the data set D, the output y∗ for  any new query x∗ can be estimated using the GP prediction:  p(y∗|y) = N(y∗|my(x∗),ky(x∗,x∗)+σ2),  (2)  where my(x) and ky(x,x′) are the posterior mean and co-  variance functions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The GP prediction equation depends  on the values of the hyperparameters (Θ,σ2) where Θ is the  kernel parameters and σ2 is the noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  The computation complexity of a full GP is O(N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  In order to reduce the computation complexity, different  approximation methods were proposed in the literature by  considering only M input points to represent the entire  training data [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' These input points are called the inducing  points Xm and their corresponding values of the underlying  function f(x) are called the inducing variables fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Replacing  the entire data set with only the M-inducing inputs leads to  the SGP which has a computational complexity of O(NM2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Titsias [2] proposed a variational learning framework to  jointly estimate the kernel hyperparameters and the inducing  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Titsias’ framework approximates the true exact poste-  rior of a GP p( f|y,Θ) by a variational posterior distribution  q( f, fm),  q(f, fm) = p(f|fm)φ( fm),  (3)  where φ( fm) is the free variational Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The  Kullback-Leibler (KL) divergence is used to describe the dis-  crepancy between the approximated and the true posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Minimizing the KL divergence between the approximated  and the true posteriors KL[q( f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' fm)||p( f|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='Θ)] is equivalent  to maximizing the variational lower bound of the true log  marginal likelihood:  FV (Xm) = log  �  N  �  y | 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='σ2I +Qnn  ��  −  1  2σ2 Tr( �K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Qnn = KnmK−1  mmKmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  �K = Cov(f | fm) = Knn −KnmK−1  mmKmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (4)  where FV (Xm) is the variational objective function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Tr( �K) is a  regularization trace term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Knn is the original n×n co-variance  matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Kmm is m × m co-variance matrix on the inducing  inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Knm is n×m cross-covariance matrix between training  and inducing points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' and Knm = KT  mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' More details on VSGP  can be found in Titsias’s work [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' METHODOLOGY  The proposed approach exploits the VSGP as a generative  model to encode 3D pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The VSGP is selected  among different approximation approaches of GP due to  the following reasons: i) The variational approximation dis-  tinguishes between the inducing points M (as a variational  parameter) and the kernel hyperparameters (Θ,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' ii) The  regularization term Tr( �K) in the variational objective func-  tion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (4)) regularizes the hyperparameters to avoid over-  fitting of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' iii) The variational approximation offers  a discrete optimization scheme for selecting the inducing  inputs Xm from the original data1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' VSGP as a generative model for the occupancy surface  Inspired by [13], we project the occupied points ob-  served by a ranging sensor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=', LiDAR, onto a circular  surface around the sensor origin with a predefined radius  1For more information about the inducing point selection, check [2]  roc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This surface is called occupancy surface, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  In our approach, the sensor observation is defined in the  spherical coordinate system, where any observed point xi  is described by the tuple (θi,αi,ri) which represents the  azimuth, elevation, and radius values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Also,  any pointcloud data can be converted from the cartesian  coordinates (xi,yi,zi) to the spherical coordinates (θi,αi,ri)  using the following equations:  ri =  �  x2  i +y2  i +z2  i ,  θi = tan−1(yi,xi),  αi = cos−1(zi/ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (5)  All observed points that lie outside the circular occupancy  surface (with a radius ri > roc) or on the surface (with a  radius ri = roc) are neglected and considered as free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  The rest of the points that are inside the circular surface (with  a radius ri < roc) are projected on the occupancy surface and  called the occupied points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Therefore, the occupancy surface  radius roc acts as the maximum range of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Each  occupied point xi on the surface is defined by two attributes:  the azimuth and elevation angles xi = (θi,αi), and assigned  an occupancy value f(xi) that is a function of the point radius  ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The probability of occupancy f(xi) at each point on the  occupancy surface is modeled by a VSGP:  f(x) ∼ VSGP  �  m(x),k  �  x,x′��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (6)  Considering noisy measurements, we add a white noise ε to  the occupancy function f(x), so the observed occupancy is  described as yi = f(xi)+ε where ε follows a Gaussian dis-  tribution N  �  0,σ2  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The final model of the occupancy surface  is a 2D VSGP where the input is the azimuth and elevation  angles, x ∈ {(θ,α)}n  i=1, and the corresponding output is the  expected occupancy yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The three main components of the  final VSGP are:  1) Zero-Mean Function m(x): There are different for-  mulas to describe the relationship between the occupancy  of a point f(xi) on the occupancy surface and its radius  ri [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For example, one candidate is f(xi) = 1/ri where ri  is bounded by the minimum and the maximum range of the  sensor rmin < ri < rmax = roc, where rmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Our approach  relates the occupancy of a point f(xi) to its radius ri by the  following equation f(xi) = roc − ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This mapping between  the occupancy and the radius of a point is compatible with  the previous assumption that the occupancy surface radius  roc represents the maximum range of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Moreover,  this mapping is encoded in our VSGP model as a zero-mean  function m(x) = 0 that sets the occupancy value of the non-  observed points to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This mapping behavior mimics the  mechanism of the LiDAR itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  2) Rational Quadratic (RQ) Kernel: The RQ kernel is  selected because a GP prior with an RQ kernel is expected  to have functions that vary across different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  This quality of the RQ kernel copes with the nature of the  occupancy surface, specifically in unstructured environments  where a range of diverse length scales is required, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=',  kRQ  �  x,x′�  = σ2  �  1+ (x−x′)2  2αℓ2  �−α  ,  (7)  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 3: (a) Gazebo scene of a robot in a tunnel (black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (b)  The occupancy surface generated from the original pointcloud,  where warmer colors reflect smaller f(xi) values (less occupancy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (c) The inner surface represents the original occupancy surface  (same as in b), and the middle surface represents the reconstructed  occupancy surface using the VSGP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The outer grey-coded  surface represents the variance associated with each point on the  reconstructed occupancy surface where brighter colors reflect high  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Raw pointcloud is shown in red in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  where σ2  f is the signal variance, l is the length-scale, and α  sets the relative weighting of large and small scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  The RQ co-variance function is more expressive in terms of  modeling the occupancy surface than the most commonly  used Squared Exponential (SE) co-variance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This  can be reasoned by the fact that the RQ kernel (when α  and l > 0) is equivalent to a scale mixture of SE kernels  with mixed characteristic length-scales [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In practice, we  take into account the resolution of LiDAR along both the  azimuth and elevation axes to initiate different length-scales  along each axis to reflect the LiDAR resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  3) Inducing Points Selection: The variational learning  framework proposed in [2] jointly optimizes the variational  parameters (inducing points) and the hyperparameters (Θ,σ)  through a variational Expectation-Maximization (EM) algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In general, the original discrete optimization frame-  work [2] suggests having an incremental set of the inducing  points, so that during the Expectation step (E-step) a point  from the input data is added to the inducing points set to  maximize the variational objective function FV and minimize  the KL divergence between the true and approximated pos-  teriors KL[q( f)||p( f|y,Θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Then the hyperparameters are  updated during the Maximization step (M-step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Since LiDAR’s field of view is limited within a certain  range, the projection of the observed points on the circular  surface leads to a limited input domain for the VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In  our case, the azimuth and the elevation axes are limited  to (−π to π) and (−15◦ to 15◦), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The limited  input domain is used to initiate a fixed number of inducing  points at evenly distributed locations on the occupied part of  the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In this way, a different combination  of the points is selected at each E-step to maximize the  variational objective function FV and minimize the KL  divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Then the hyperparameters are updated during the  M-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The number of the inducing points M is chosen to  compromise the computational and memory complexity on  one side and the accuracy of the reconstructed pointcloud  on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' More inducing points result in higher  computations complexity O(NM2), larger memory to store  the encoded observation, and higher bandwidth to transfer it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  However, more inducing points increase the accuracy of the  reconstructed pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We chose M=500 inducing points  to keep the average deviation between the reconstructed  pointcloud and the original pointcloud under 15 cm, see  Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' After the training phase on the  scout side is completed, the selected inducing points are  combined together with the hyperparameters values of the  VSGP and are transmitted from the scout to the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Variance-based sampling  On the base side, the inducing points and the values of  the hyperparameters, which are received from the scout,  are used to reconstruct the original occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The  reconstruction is done through a GP configured with the  same kernel (RQ) and likelihood (Gaussian) as the VSGP  on the scout side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The base GP is trained on the inducing  points and has a computation complexity of O(M3) where  M is the number of the inducing points, so we refer it as  a sparse GP (SGP) and refer the reconstructed occupancy  surface as the SGP occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' A grid of query points  x∗ = {(θ,α)}K  i=1 with the same resolution of the LiDAR  along the azimuth and the elevation axes is generated to  reconstruct the original pointcloud from the SGP occupancy  surface – we refer the reconstructed pointcloud as the SGP  pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' If up-sampling of the pointcloud is required for  any reason, a query grid with higher resolution can be used  for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The SGP occupancy surface  is used to predict the occupancy f(xi) of each point xi of  the query grid x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The occupancy is converted back to the  spherical radius ri = roc − f(xi) to restore the 3D spherical  coordinates of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  One advantage of the GP and its variants over other  modeling techniques is the uncertainty (variance) associated  with the predicted value at any query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Considering  the VSGP model of the occupancy surface on the scout  side, the variance associated with the occupied points is low  compared to the variance related to the free points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Selecting  the inducing points as a set from the original occupied  points maintains low-variance values for the occupied part of  the reconstructed SGP occupancy surface on the base side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Therefore, the variance value associated with any point on  the reconstructed SGP occupancy surface is used to predict  if that point belongs to the occupied or the free part of the  occupancy surface, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We use a variance threshold  Vth as a judging criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In fact, the variance related to  the occupancy surface is different from one observation to  another, and it is affected by both the number of observed  (occupied) points and their distribution over the occupancy  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Therefore, we chose the variance threshold Vth as  a variable that changes with the distribution of the variance  over the occupied and free parts of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Vth is defined as a linear combination of the variance mean  vm and standard deviation vstd over the surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=', Vth =  Km ∗vm +Kstd ∗vstd where Km and Kstd are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' These  two constants are tuned by first setting Vth = vm (Km = 1 ,  Kstd = 0), then we increase Kstd and decrease Km gradually  till we get the values that give the highest accuracy for the  reconstructed SGP pointcloud (considering a fixed number of  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 4: Variance-based sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (a) Gazebo scene shows the  entrance of the tunnel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (b) shows the original (inner), reconstructed  (middle), and variance (outer) surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' It also shows the re-  constructed pointcloud (in white) through reconstructing from all  points (free and occupied) of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (c) shows  reconstructed SGP pointcloud after removing all points that most  likely belong to the free part of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Raw  pointcloud is shown in red in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  inducing points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Our sampling-based approach is capable  of discriminating between the free points that most likely  belong to the free part of the SGP occupancy surface and  the occupied points that belong to the the occupied part of  the SGP occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' After removing the free part  of the SGP occupancy surface, the Cartesian coordinates of  the occupied points are calculated using the inverse form of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (5) to restore the original point cloud, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' EXPERIMENTAL DESIGN AND RESULTS  The proposed approach is implemented in Python3 on  top of GPflow-v2 [28] and TensorFlow-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='4 [29] under  ROS framework [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Both real-time simulation and real-  time demonstration were considered to evaluate the proposed  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In both the simulation and the hardware experi-  ments, a VLP-16 LiDAR was used with a maximum range  of 10m, a frequency of 4Hz, and a resolution of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='1◦,2◦)  along the azimuth and the elevation axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This  configuration results in a maximum pointcloud size of 57600  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The query grid, which is used to sample the SGP  occupancy surface on the base side, has the same resolution  as the VLP-16 LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' A 3D occupancy grid map with a  resolution of 5cm is generated from the reconstructed SGP  pointcloud through Octomap [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  We investigate the performance of our framework and  compare it with the GMM approach [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' While the  GMM approach tackles the occupancy mapping problem as  a whole, our approach focuses on compressing sensor obser-  vations through limited-bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  To be able to compare the two approaches, we implemented  the GMM approach in such a way that it is used to encode  one sensor observation at a time instead of generating an  entire occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We compared our approach with two  versions of the GMM approach: i) A CPU-based implemen-  tation of GMM that follows the same guidelines of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  ii) An upgraded GPU-based implementation of GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We  implemented the GPU-GMM to have a fair computation  comparison with our VSGP approach which runs on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Simulation Experiments  1) Simulation Setup:: The simulation setup consists of  two machines that communicate to each other over WiFi:  The first machine, where the scout and the environment are  simulated, is an Intel® Core™ i7 NUC11 PC equipped with  64 GB RAM and 6 GB Geforce RTX2060 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The second  machine, which acts as the base, is an Intel® Core™ i7  Alienware Laptop equipped with 32 GB RAM and 8 GB  Geforce RTX2080 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Both are connected using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='4 GHz  WiFi router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The network flow is monitored using the ifstat  tool to evaluate the communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The mine  tunnel of the cpr inspection world, which is developed by  ClearPath robotics, is used as our simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  This environment is selected because it represents one of  the targeted low-bandwidth subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The  mine tunnel part of the cpr inspection world fits in a rectan-  gular area with an approximated area of 30×65m2, the tunnel  length is around 135m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The ground elevation and the height  of the tunnel are different from one place to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The  ClearPath Jackal robot is used as the scout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The proposed  approach was evaluated through 20 real-time simulation  trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In each trial, the robot starts at the beginning of the  cave and follows a predefined path along the mine using  way-point based navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  2) Simulation Results: We evaluate the performance of  our approach based on the reduction in the memory and the  communication rate required to transmit the sensor observa-  tions between the scout and the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The VSGP representa-  tion requires only 1514 floating points (FP) to represent the  entire pointcloud (3 FP for each inducing point (3x500) + 6  FP for robot pose + 6 FP for the hyperparameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This value  is less than the memory needed by the GMM approach which  requires ∼ 2000 FP (10 FP for each component (10x200)  distributed as 6 FP for covariance + 3 FP for mean + 1  FP for weight) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We send the robot pose to the base  because our approach encodes the observation relative to the  robot body frame, while the GMM approach first transforms  the observation from the robot body frame to a global frame  using the robot current pose and then sends the encoded  Gaussians densties with respect to the global frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  To quantify the accuracy of the reconstructed SGP point-  cloud, we use the Root Mean Square Deviation (RMSD)  between the radius predicted by our approach and the actual  radius of each point on the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  RMSD =  �  ∑N  i=1 (ri − ˆri)2  N  ,  (8)  where N is the size of the pointcloud, ri is the actual radius at  (θi,αi), and ˆri is the estimated radius value at the same point  (θi,αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5a shows the mean and the standard deviation  of the RMSD for each predicted point over 110 observations  (each observation has around 10K to 50K points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Also,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5a implicitly reflects the memory required by VSGP and  GMM to store one observation, as described before that the  memory required to store one observation can be calculated  by multiplying the number of inducing points (bottom x-axis)  by 3 and multiplying the number of components (top x-axis)  by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We match pairs of the VSGP and GMM models (in  terms of the number of inducing points and components)  based on the memory requirement and the accuracy of the  (a)  (b)  (c)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5: Performance comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (a) shows the RMSD between  the reconstructed and the original pointcloud for VSGP(vs #induc-  ing points) and GMM(vs #components);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (b) illustrates the training  time against the pointcloud size (considering 500-inducing points  VSGP, and equivalently, 200-components GMM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (c) represents  the training time versus the #VSGP-inducing points and #GMM-  components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (d) shows the prediction time versus the #VSGP-  inducing points and #GMM-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  reconstructed pointcloud (reflected by the RMSD) for each  pair, see table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' For example, 500-inducing points VSGP  results in an average RMSD value for each point of 9 cm  with a standard deviation of 10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This corresponds to an  average RMSD of 11 cm with a standard deviation of 25 cm  for a 200-components GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  TABLE I: VSGP vs GMM(ind: inducing, cps: components)  VSGP  GMM  #  Memory  RMSD  #  Memory  RMSD  ind  ∼FPs  ∼cm  cps  ∼FPs  ∼cm  200  600  20±22  50  500  27±50  300  900  14±15  100  1000  16±35  400  1200  12±14  150  1500  13±31  500  1500  9±10  200  2000  11±29  600  1800  9±10  250  2500  11±30  Now we analyze the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5a shows the  RMSD values associated with VSGP have a smaller standard  deviation than the GMM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' It also shows that increasing the  number of the VSGP-inducing points (bottom x-axis) or the  number of the GMM-components (top x-axis) will result in  smaller RMSD (higher accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  An intensive evaluation of the training and the prediction  phases is presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5b-5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The reduction in the  training time versus the reduction in the size of the raw  pointcloud is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5b, where 0% removal percent  means a pointcloud size of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='6K points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5c shows  the increase in training time versus the number of inducing  points and the number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' We compare the  training time of the VSGP, the GMM-CPU (considering the  #Components  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='75 -  GMM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='50  VSGP  RMSI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='25  200  300  00f  500  600  #Inducing Points103  Su  VSGP  GNIM  GNIM-GPU  102  3350606775  Removal Components  50  100  150  200  250  103  GMM  GMM-GPU  VSGP  102  200  400  600  Inducing PointsComponents  50  100  150  200  250  VSGP  [stu]  GMM  40  Prediction  20  200  400  600  Inducing Points(a)  (b)  (c)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 6: (a) shows the simulated mine environment in Gazebo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  (b) shows the Octomap of the mine generated from the original  pointcloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (c) shows the Octomap generated from the recon-  structed SGP pointcloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (d) shows the communication rate and the  accumulated data sent from the scout to the base in case of sending  raw pointcloud PCL(1750KB/S, 840MB), GMM data(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='8KB/S,  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='4MB), and VSGP data(18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='2KB/S, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='7MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The y-axis is plotted  in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  default configuration of the GMM approach used in [13]),  and the GMM-GPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The results show that  our approach outperforms both the CPU and GPU imple-  mentation of the GMM approach in terms of training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5d presents the variation of the prediction time of the  VSGP versus the number of the inducing points, where the  values shown in the figure represent the time required to  predict the occupancy value associated with all the points of  the grid query x (57600 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 5d indicates that for a matching pair of GMM and  VSGP (Table I), GMM has a less sampling time than  the paired VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' However, the pointcloud reconstruction  process of the VSGP is more convenient than the GMM  approach because a fundamental difference between sam-  pling the VSGP and the GMM is that: when we sample  from a GMM, we get a sample (from a distribution) with  random values (θs,αs,rs), so we do not have control over  the location of the sample on the occupancy surface (θs,αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  In contrast, for the VSGP approach, we predict the radius  value rs for a certain point on the occupancy surface defined  by (θs,αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' So, we have control over the point location on  the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' While constructing the 3D octomap  of the tunnel environment using the scout-base scheme, the  average communication rate was 1750 KB/S, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='8 KB/S,  and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='2 KB/S for sending raw point clouds, GMM encoded  data, and VSGP encoded data respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The  accumulated data sent through the network is reduced from  840 MB for sending raw pointcloud to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='4 MB in case  of GMM and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='7 MB in case of VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This indicates a  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 7: Indoor demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (a) shows octomap of the laboratory  building generated from the original pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (b) shows octomap  generated from the reconstructed SGP pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' (c) shows the  reduction in the communication rate and the accumulated data sent  from the scout to the base, where log-scale is used for y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' PCL  represents the raw pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  compression ratio of ∼ 96 (840/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='7 ∼ 1750/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Hardware Experiment  A Jackal mobile robot, equipped with a VLP-16 LiDAR  and NUC11 PC, was used as the scout, while the Alien-  ware laptop was used as the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The demonstration was  conducted in an indoor environment, where the VSGP-  encoded pointcloud data was sent from the scout to the  base to generate a 3D Octomap [31] of the building from  the SGP reconstructed pointcloud in real-time, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' 7c shows the reduction in the communication rate for  the hardware experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The communication rate dropped  from around 560 KB for transmitting raw pointcloud to  around 8 KB for transmitting the encoded VSGP (this ratio  is equivalent to 70 times smaller rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The communication  rate of the hardware experiment is low compared to the  simulation experiment because the LiDAR resolution was  halved during the hardware experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' The total amount  of data transmitted at the end of each trial was around 100  MB for sending raw pointcloud and only around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='4 MB for  sending the VSGP encoded observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' CONCLUSION  In this paper, we introduce a lightweight representation  for the 3D pointcloud using the VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This representation  allows high-fidelity observations to be efficiently stored  and transmitted through limited-bandwidth communication  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Based on the results of the simulation and hardware  experiments, our approach results in around 70-100 times  smaller size representation of the sensor observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' This  compact representation can facilitate many of the robotics  KB  PCI  GMM  Rate  102  VSGP  MB  PCL  101  Data  GMM  VSGP  0  O0T  200  300  400  500  Time [S]S/  POL  101  VSGP  Rate  KB  PCL  Data  102  VSGP  0  25  50  75  100  125  150  175  Time [S]applications which are limited by the communication band-  width such as subterranean and underwater exploration,  search and rescue missions, and planetary exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In  addition, our approach can also be beneficial in the context  of multi-robot collaboration where a number of robots are  required to share high-volume information (3D pointcloud)  through low-bandwidth channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This work was supported by National Science Foundation  with grant numbers 2006886 and 2047169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content=' Seeger, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Herbrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Fast sparse gaussian process  methods: The informative vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In 16th annual conference  on neural information processing systems, number CONF, pages 609–  616, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content=' Ghahramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Sparse gaussian processes using  pseudo-inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Advances in neural information processing systems,  18:1257, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  [25] Matthias Seeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Bayesian gaussian process models: Pac-bayesian  generalisation error bounds and sparse approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content='  [26] Rishit Sheth, Yuyang Wang, and Roni Khardon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content='  [27] Christopher K Williams and Carl Edward Rasmussen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Gaussian  processes for machine learning, volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  MIT press Cambridge,  MA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Gpflow: A gaussian process library using tensor-  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content='  Tensorflow: A system for large-scale machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In 12th {USENIX} symposium on operating systems design  and implementation ({OSDI} 16), pages 265–283, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  [30] Morgan Quigley, Ken Conley, Brian Gerkey, Josh Faust, Tully Foote,  Jeremy Leibs, Rob Wheeler, Andrew Y Ng, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Ros: an open-source  robot operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' In ICRA workshop on open source software,  volume 3, page 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content=' Kobe, Japan, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
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+page_content=' Octomap: An efficient probabilistic 3d mapping  framework based on octrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}
+page_content='  Autonomous robots, 34(3):189–206,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'}

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+WKB Across Caustics: The Screened-WKB Method
+Oscar P. Bruno∗
+Martin D. Maas∗
+Abstract
+We present a new methodology, based on the WKB approximation and Fast Fourier Trans-
+forms, for the evaluation of wave propagation through inhomogeneous media. This method can
+accurately resolve fields containing caustics, while still enjoying the computational advantages
+of the WKB approximation, namely, the ability to resolve arbitrarily high-frequency problems in
+computing times which are orders-of-magnitude shorter than those required by other algorithms
+presently available. For example, the proposed approach can simulate with high accuracy (with
+errors such as e.g. 0.1%–0.001%) the propagation of 5 cm radar signals across two-dimensional
+configurations resembling atmospheric ducting conditions, spanning hundreds of kilometers and
+millions of wavelengths in electrical size, in computing times of a few minutes in a single CPU
+core. [Preliminary version]
+1
+Introduction
+Computations of high-frequency wave propagation through inhomogeneous media play a pivotal
+roles in diverse fields such as telecommunications, remote sensing, seismics, quantum mechanics,
+and optics. A wide range of methodologies have been developed over the last century for the treat-
+ment of high-frequency volumetric-propagation problems. Given that direct numerical simulation
+of the configurations of interest, which comprise thousands to millions of wavelengths in acousti-
+cal/electrical size, is unfeasible in 2D and even more in 3D, the proposed approaches usually contain
+a combination of analytic and numerical approximations.
+The celebrated WKB approximation, also known as the Wentzel-Kramers-Brillouin approxima-
+tion [2,8], was the first method to obtain accurate solutions to problems involving propagation over
+large distances, and is based on the introduction of a system of ray-coordinates, over which the
+amplitude and phase of the solution exhibit slow variations. However, the WKB approximation
+can break down in certain situations, particularly when the ray mapping becomes singular (i.e.
+at caustics). Many approaches have been proposed over time to overcome this limitation, most
+notably the KMAH-index theory, according to which a correction can be incorporated after the
+caustic, of the form (−i)m, where the constant m depends on the number and type of caustics that
+the ray has traversed. This formulation still breaks down at caustics, is inaccurate near caustics,
+and, given its complexity, it is seldom used in practice.
+Another notable approach to solve these type of problems is provided by the parabolic approxi-
+mation introduced in [9], together with its many subsequent versions and improvements, including
+the wide-angle approximation [5]. The method of phase-screens [14], in turn, which assumes a con-
+stant refractivity profile along each vertical volumetric screen, is applicable for certain restricted
+sets of configurations. While, unlike the classical WKB approach, these methods are valid at and
+around caustics, their limitations arise from its computational cost. For example, mesh-sizes of
+∗Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA
+1
+arXiv:2301.03814v1  [physics.comp-ph]  10 Jan 2023
+
+the order of ∆z ≈ λ
+4 and ∆x ≈ 12.5λ are reported for propagation distances in the order of a
+few hundreds of wavelengths (see [7] and references therein). (Here z and x denote the vertical
+and range variables, respectively.) The parabolic equation methods are most often based on use
+of finite-difference approximations, which gives rise to associated dispersion errors, while Fourier
+expansions in the vertical axis are only applicable in the lowest-order parabolic approximations.
+The combined effect of large propagation distances, and the presence of dispersion error, and the
+requirement of fine spatial discretizations, can lead to extremely large computational cost, under
+reasonable error tolerances, for challenging configurations commonly arising in applications.
+Other notable approaches include the Gaussian beams formulation, with contributions spanning
+from the 60’s, including [1,3,6,13] among many others. This formulation is based on an additional
+approximation to WKB, which seeks to obtain the phase in the form of a quadratic polynomial,
+whose Hessian matrix is evolved along the ray. This approach eliminates ray-bunching at caustics,
+and produces fields which remain bounded.
+However, theoretical convergence as k → ∞ has
+not been established and is believed to be slow. Moreover, the initial beam representation is a
+challenging optimization problem, which leads to errors of a few percent even for propagations
+distances of the order of a small number of wavelengths [13].
+An additional approach, known as Dynamic Surface Extension (DSE, see [11,12]), can success-
+fully propagate wavefronts in a Cartesian discretization. However, the amplitude computations
+present the same limitations as the classical WKB approximation. Finally, the Kinetic Formu-
+lation [4] views each ray tracing equation as describing the motion of a ”particle” (e.g. photon,
+phonon). This method presents severe computational difficulties, as the initial conditions and so-
+lutions are given in terms of Wigner measures, a δ-function that vanishes for incorrect directions p.
+The approach put forth in this paper, on the other hand, is based on the WKB approximation,
+and overcomes the limitations posed by caustics by resorting to a family of curves (or screens)
+on which the total field is decomposed in Fourier modes. Each mode is then propagated for large
+electrical distances (i.e. 20,000λ in the example considered in Figure 6) which are also short enough
+that the presence of caustics is avoided for each Fourier mode.
+2
+The Screened WKB Method
+We consider, as a model problem, the scalar Helmholtz equation
+∆u(r) + k2ε(r)u(r) = 0
+(1)
+u = us + uinc
+(2)
+lim
+r→∞ r
+�∂us
+∂r − ikus
+�
+= 0,
+(3)
+whose character at high frequencies presents challenges often found in diverse fields, such high-
+frequency electromagnetism, acoustics, seismics and quantum mechanics. The proposed screened-
+WKB approach first introduces a family of curves (or screens) Γq, for q = 0, 1, . . . , Ns, as depicted
+in Figure 1. The method proceeds by propagating the solution from one screen to the next on the
+basis of Fourier expansions on the screens Γq and applications of the classical WKB approach for
+each separate Fourier mode.
+For conciseness, we consider planar screens of the form Γq = {(xq, z) : z ∈ (za, zb)}. The initial
+conditions on Γ0 are user-prescribed, and given by:
+u|Γ0 = uinc|Γ0
+(4)
+2
+
+Figure 1: Schematics underlying the proposed S-WKB method. Here and throughout this paper
+flat screens Γq are used, but curved screens could alternatively be utilized, if convenient.
+We then represent the incident field on each screen Γq, arising from propagation of the field from
+Γq−1 (or given by (4) for q = 0) by exploiting certain expressions of the form
+u(x, z) ≈
+N/2−1
+�
+n=−N/2
+Aq
+n(x, z) exp(ikψq
+n(x, z)),
+(5)
+valid between Γq and Γq+1, together with the WKB approximation.
+For simplicity, in our description we consider configurations which may accurately be expressed
+in terms of z-periodic functions, of period [za, zb], which, in particular, can be used to treat cases
+wherein the solution decays rapidly outside a bounded interval in the z variable. (Other arrange-
+ments, including rough top and bottom surfaces and other irregularities, can also be incorporated
+in this framework, but are not considered in this paper at any length.) Under such assumptions,
+for a given screen Γq, a “vertical” DFT can be used by introducing an equi-spaced grid
+{zm : m = −N/2, . . . , N/2 − 1}
+(6)
+in the interval [za, zb] and performing an FFT—which yields
+wq
+j =
+N/2−1
+�
+m=−N/2
+u(xq, zm)e−ijzm.
+(7)
+Then, re-expressing the field u(xq, z) in terms of an inverse DFT, we obtain
+u|Γq ≈ 1
+N
+N/2−1
+�
+m=−N/2
+wq
+jeijzm,
+(8)
+3
+
+2
+I1'T2
+'13Figure 2: Example 1: Ray-tracing leading to a single cusp caustic.
+which may be expressed in terms of (5) by requiring that
+ψq
+n(xq, z) =
+�
+z + zb − za
+2
+�
+2nπ
+k(zb − za).
+(9)
+As a second step, each term in the expansion (5) is obtained up to the next screen Γq+1 by
+means of the classical WKB expansion (see e.g. [7, chapter 3]), which, in particular, requires the
+solution of the Eikonal and Transport equations
+(∇ψ)2 = ε(r)
+(10)
+and
+2∇ψ · ∇A + A∆ψ = 0.
+(11)
+In the present case, the initial conditions for each Fourier mode on Γq are obtained from (9) and
+(10):
+∂zψn(xq, z) =
+2nπ
+k(zb − za)
+(12)
+∂xψn(xq, z) =
+�
+ε(xq, z) −
+�
+2nπ
+k(zb − za)
+�2
+(13)
+This procedure yields a finite number of adequately spaced geometrical-optics rays, and corre-
+sponding values of ψn and An along the rays for the n-th mode. By adequately selecting the spacing
+of the screens Γq it can ensured that all the modes − N
+2 ≤ n ≤ N
+2 − 1 propagate to the next screen
+4
+
+Figure 3: S-WKB solution (top), and physically-exact separation-of-variables solution with super-
+imposed geometrical-optics rays (bottom), with k = 125, along a propagation domain 40 km
+(800, 000 wavelengths) in range.
+5
+
+Figure 4: Error for the “single-caustic” solution displayed in Figure (3): a relative error of the
+order of 10−5 was obtained throughout the propagation domain.
+Γq+1 without incurring caustics. Interpolation can then be used on Γq+1 to obtain approximate
+values of u on the 1D Cartesian grid (xq+1, zm) (− N
+2 ≤ n ≤ N
+2 − 1) on Γq+1:
+u(xq+1, zm) ≈
+N/2−1
+�
+n=−N/2
+Aq
+n(xq+1, zm) exp(ikψq
+n(zm)).
+(14)
+Expanding u(xq+1, z) in a Fourier series along Γq+1 the next iteration of the algorithm can then
+be initiated. Repeating this procedure for all screens the field u over the domain of interest can be
+obtained.
+3
+Numerical Results
+This section presents results of applications of the proposed algorithm to problems of wave propa-
+gation through inhomogeneous media, in two-dimensional configurations, and through wide ranges
+of problem parameters. In order to evaluate the accuracy of the proposed S-WKB method by
+comparisons with solutions obtainable by means of separation of variables, we first consider x-
+invariant permittivities (i.e. permittivities of the form ε(x, z) = ε(z)), as described in what follows,
+for which a simple high-order spectral solver can be used to obtained reference solutions that are
+physically-exact—i.e., which contain no approximations to (1) other than those inherent in the well
+established numerical solver utilized.
+6
+
+Figure 5: Geometrical optics rays for a “ducting” configuration.
+3.1
+High-order reference solutions for x-invariant permittivities
+In order to obtain a valid solution to (1) via separation of variables we seek a solution of the form
+u(x, z) =
+∞
+�
+i=0
+aieiαixφi(z).
+(15)
+Substituting (15) in (1) leads to
+∞
+�
+i=0
+(−α2 + φ′′
+i (z) + k2ε(z))aieiαix = 0.
+(16)
+Using the orthogonality of the complex exponentials we then obtain
+(−α2
+i + φ′′
+i (z) + k2ε(z))ai = 0.
+(17)
+It follows that the non-zero coefficients αi in (15) satisfy the Sturm-Liouville problem:
+φ′′
+i (z) + k2ε(z)φi(z) = α2
+i φi(z)
+(18)
+with radiation boundary conditions:
+lim
+z→±∞
+�
+zφ′
+i − ikφi
+�
+= 0,
+(19)
+Numerically, the radiation boundary conditions can be imposed by considering a sufficiently
+large interval (za, zb) and imposing either Dirichlet or periodic boundary conditions at such points.
+The resulting Sturm-Liouville problem can be discretized with high-order spectral methods. For
+the purposes of the present paper we utilized the spectral eigensolver [10], which is available in the
+ApproxFun.jl Julia package.
+3.2
+Evaluation of the S-WKB accuracy for a Gaussian permittivity model
+In this section we consider the exponential permittivity model
+ε(z) = 1 + ae−bz2
+(20)
+7
+
+Figure 6: “Multiple caustics” test case depicting an idealized “ducting” configuration. S-WKB field
+values (top), and field values with super-imposed geometrical-optics rays (bottom). The geometrical
+optics rays are depicted in Figure 5.
+8
+
+whose physically-exact solution can be obtained by relying on the method described in Section 3.1.
+For our example, an incident field given by a Gaussian beam
+uinc(x, z) =
+� ∞
+−∞
+ei√
+k2−β2x+iβze− β2
+σ2 dβ
+(21)
+is utilized, wherein the integral in the variable β is evaluated via standard numerical integration
+techniques.
+In our first example we consider the permittivity model (20) with parameters a = 10−4 and
+b = 10−4—which, at C-band, results in a configuration that gives rise to a single caustic of cusp
+type for the first 40km (800, 000 wavelengths) in horizontal propagation range. The geometrical
+optics rays are displayed in Fig. 2. The S-WKB solution alongside the Sturm-Liouville solution
+with superimposed ray-tracing are depicted in Fig. 3. As shown in Fig. 4, the relative errors for
+this configuration are of the order of 10−5. Employing N = 400 Fourier modes and a total of 40
+screens, the S-WKB solution in this case was obtained in a computing times of 2 minutes in a
+single-core, whereas the separation-of-variables solution required single-core runs of approximately
+1.5 hours.
+Figure 7: Smooth convex lens simulations produced by the S-WKB method. Note the fine-scale
+fields behind caustics, whose simulation is otherwise quite challenging; cf. e.g. [7, Fig. 6.10].
+For our next example we consider a “ducting” configuration, in which the incident Gaussian
+beam is tilted by an angle of 0.2◦, and where the Gaussian permittivity (20) was used with param-
+eters a = 10−4 and b = 10−3—in such a way that the energy is contained within a bounded interval
+along the z axis. The geometrical-optics rays form a complex system with multiple caustics, as
+9
+
+Figure 8: Permittivity and geometrical-optics rays (left), and rays superimposed on the field de-
+picted in Figure 7.
+depicted in Fig. 5. We consider the propagation of this signal over a range of 200Km (4 million
+wavelengths) in range. A total of n = 800 Fourier modes and 200 of the order of 0.1%.
+4
+Smooth Lens
+We now consider the test case of a smooth convex lens proposed in [4] on the basis of the permittivity
+function given by
+ε(r) =
+�
+1
+d2 > L
+�
+a
+b−cos(πd2)
+�2
+d2 ≤ L
+d2 =
+�x − xc
+xd
+�2
++
+�z − zc
+zd
+�2
+(22)
+For the numerical example depicted in Fig. 7, we have set L = 1, a = 4, b = 3, xc = 0.5, xd =
+0.2, zc = 0, zd = 0.8. The displayed result compares favorably to that presented in [4] on the basis
+of a kinetic formulation, as well as the similar problem demonstrated in [7, Fig. 6.10]. A separation
+of variables solution is not available in this case, and the error could be evaluated by means of
+S-WKB implementation of higher order. In presence of the previous examples, however, we may
+estimate the error in the range of 0.1% to 0.001%.
+References
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+[11] Steven J Ruuth, Barry Merriman, and Stanley Osher. A fixed grid method for capturing the
+motion of self-intersecting wavefronts and related pdes. Journal of Computational Physics,
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+of propagating short acoustic and electromagnetic pulses. Journal of Computational Physics,
+157(2):683–706, 2000.
+[13] Nicolay M Tanushev, Bj¨orn Engquist, and Richard Tsai. Gaussian beam decomposition of
+high frequency wave fields. Journal of Computational Physics, 228(23):8856–8871, 2009.
+[14] Ru-Shan Wu.
+Wide-angle elastic wave one-way propagation in heterogeneous media and
+an elastic wave complex-screen method.
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+Solid Earth,
+99(B1):751–766, 1994.
+11
+

JdE2T4oBgHgl3EQfUgew/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf,len=175
+page_content='WKB Across Caustics: The Screened-WKB Method  Oscar P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Bruno∗  Martin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Maas∗  Abstract  We present a new methodology, based on the WKB approximation and Fast Fourier Trans-  forms, for the evaluation of wave propagation through inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' This method can  accurately resolve fields containing caustics, while still enjoying the computational advantages  of the WKB approximation, namely, the ability to resolve arbitrarily high-frequency problems in  computing times which are orders-of-magnitude shorter than those required by other algorithms  presently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' For example, the proposed approach can simulate with high accuracy (with  errors such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='1%–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='001%) the propagation of 5 cm radar signals across two-dimensional  configurations resembling atmospheric ducting conditions, spanning hundreds of kilometers and  millions of wavelengths in electrical size, in computing times of a few minutes in a single CPU  core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' [Preliminary version]  1  Introduction  Computations of high-frequency wave propagation through inhomogeneous media play a pivotal  roles in diverse fields such as telecommunications, remote sensing, seismics, quantum mechanics,  and optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' A wide range of methodologies have been developed over the last century for the treat-  ment of high-frequency volumetric-propagation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Given that direct numerical simulation  of the configurations of interest, which comprise thousands to millions of wavelengths in acousti-  cal/electrical size, is unfeasible in 2D and even more in 3D, the proposed approaches usually contain  a combination of analytic and numerical approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  The celebrated WKB approximation, also known as the Wentzel-Kramers-Brillouin approxima-  tion [2,8], was the first method to obtain accurate solutions to problems involving propagation over  large distances, and is based on the introduction of a system of ray-coordinates, over which the  amplitude and phase of the solution exhibit slow variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' However, the WKB approximation  can break down in certain situations, particularly when the ray mapping becomes singular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  at caustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Many approaches have been proposed over time to overcome this limitation, most  notably the KMAH-index theory, according to which a correction can be incorporated after the  caustic, of the form (−i)m, where the constant m depends on the number and type of caustics that  the ray has traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' This formulation still breaks down at caustics, is inaccurate near caustics,  and, given its complexity, it is seldom used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  Another notable approach to solve these type of problems is provided by the parabolic approxi-  mation introduced in [9], together with its many subsequent versions and improvements, including  the wide-angle approximation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The method of phase-screens [14], in turn, which assumes a con-  stant refractivity profile along each vertical volumetric screen, is applicable for certain restricted  sets of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' While, unlike the classical WKB approach, these methods are valid at and  around caustics, their limitations arise from its computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' For example, mesh-sizes of  ∗Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='03814v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='comp-ph]  10 Jan 2023  the order of ∆z ≈ λ  4 and ∆x ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='5λ are reported for propagation distances in the order of a  few hundreds of wavelengths (see [7] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' (Here z and x denote the vertical  and range variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=') The parabolic equation methods are most often based on use  of finite-difference approximations, which gives rise to associated dispersion errors, while Fourier  expansions in the vertical axis are only applicable in the lowest-order parabolic approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  The combined effect of large propagation distances, and the presence of dispersion error, and the  requirement of fine spatial discretizations, can lead to extremely large computational cost, under  reasonable error tolerances, for challenging configurations commonly arising in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  Other notable approaches include the Gaussian beams formulation, with contributions spanning  from the 60’s, including [1,3,6,13] among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' This formulation is based on an additional  approximation to WKB, which seeks to obtain the phase in the form of a quadratic polynomial,  whose Hessian matrix is evolved along the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' This approach eliminates ray-bunching at caustics,  and produces fields which remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  However, theoretical convergence as k → ∞ has  not been established and is believed to be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Moreover, the initial beam representation is a  challenging optimization problem, which leads to errors of a few percent even for propagations  distances of the order of a small number of wavelengths [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  An additional approach, known as Dynamic Surface Extension (DSE, see [11,12]), can success-  fully propagate wavefronts in a Cartesian discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' However, the amplitude computations  present the same limitations as the classical WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Finally, the Kinetic Formu-  lation [4] views each ray tracing equation as describing the motion of a ”particle” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' photon,  phonon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' This method presents severe computational difficulties, as the initial conditions and so-  lutions are given in terms of Wigner measures, a δ-function that vanishes for incorrect directions p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  The approach put forth in this paper, on the other hand, is based on the WKB approximation,  and overcomes the limitations posed by caustics by resorting to a family of curves (or screens)  on which the total field is decomposed in Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Each mode is then propagated for large  electrical distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 20,000λ in the example considered in Figure 6) which are also short enough  that the presence of caustics is avoided for each Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  2  The Screened WKB Method  We consider, as a model problem, the scalar Helmholtz equation  ∆u(r) + k2ε(r)u(r) = 0  (1)  u = us + uinc  (2)  lim  r→∞ r  �∂us  ∂r − ikus  �  = 0,  (3)  whose character at high frequencies presents challenges often found in diverse fields, such high-  frequency electromagnetism, acoustics, seismics and quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The proposed screened-  WKB approach first introduces a family of curves (or screens) Γq, for q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' , Ns, as depicted  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The method proceeds by propagating the solution from one screen to the next on the  basis of Fourier expansions on the screens Γq and applications of the classical WKB approach for  each separate Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  For conciseness, we consider planar screens of the form Γq = {(xq, z) : z ∈ (za, zb)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The initial  conditions on Γ0 are user-prescribed, and given by:  u|Γ0 = uinc|Γ0  (4)  2  Figure 1: Schematics underlying the proposed S-WKB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Here and throughout this paper  flat screens Γq are used, but curved screens could alternatively be utilized, if convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  We then represent the incident field on each screen Γq, arising from propagation of the field from  Γq−1 (or given by (4) for q = 0) by exploiting certain expressions of the form  u(x, z) ≈  N/2−1  �  n=−N/2  Aq  n(x, z) exp(ikψq  n(x, z)),  (5)  valid between Γq and Γq+1, together with the WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  For simplicity, in our description we consider configurations which may accurately be expressed  in terms of z-periodic functions, of period [za, zb], which, in particular, can be used to treat cases  wherein the solution decays rapidly outside a bounded interval in the z variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' (Other arrange-  ments, including rough top and bottom surfaces and other irregularities, can also be incorporated  in this framework, but are not considered in this paper at any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=') Under such assumptions,  for a given screen Γq, a “vertical” DFT can be used by introducing an equi-spaced grid  {zm : m = −N/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' , N/2 − 1}  (6)  in the interval [za, zb] and performing an FFT—which yields  wq  j =  N/2−1  �  m=−N/2  u(xq, zm)e−ijzm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content="  (7)  Then, re-expressing the field u(xq, z) in terms of an inverse DFT, we obtain  u|Γq ≈ 1  N  N/2−1  �  m=−N/2  wq  jeijzm,  (8)  3  2  I1'T2  '13Figure 2: Example 1: Ray-tracing leading to a single cusp caustic." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  which may be expressed in terms of (5) by requiring that  ψq  n(xq, z) =  �  z + zb − za  2  �  2nπ  k(zb − za).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (9)  As a second step, each term in the expansion (5) is obtained up to the next screen Γq+1 by  means of the classical WKB expansion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' [7, chapter 3]), which, in particular, requires the  solution of the Eikonal and Transport equations  (∇ψ)2 = ε(r)  (10)  and  2∇ψ · ∇A + A∆ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (11)  In the present case, the initial conditions for each Fourier mode on Γq are obtained from (9) and  (10):  ∂zψn(xq, z) =  2nπ  k(zb − za)  (12)  ∂xψn(xq, z) =  �  ε(xq, z) −  �  2nπ  k(zb − za)  �2  (13)  This procedure yields a finite number of adequately spaced geometrical-optics rays, and corre-  sponding values of ψn and An along the rays for the n-th mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' By adequately selecting the spacing  of the screens Γq it can ensured that all the modes − N  2 ≤ n ≤ N  2 − 1 propagate to the next screen  4  Figure 3: S-WKB solution (top), and physically-exact separation-of-variables solution with super-  imposed geometrical-optics rays (bottom), with k = 125, along a propagation domain 40 km  (800, 000 wavelengths) in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  5  Figure 4: Error for the “single-caustic” solution displayed in Figure (3): a relative error of the  order of 10−5 was obtained throughout the propagation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  Γq+1 without incurring caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Interpolation can then be used on Γq+1 to obtain approximate  values of u on the 1D Cartesian grid (xq+1, zm) (− N  2 ≤ n ≤ N  2 − 1) on Γq+1:  u(xq+1, zm) ≈  N/2−1  �  n=−N/2  Aq  n(xq+1, zm) exp(ikψq  n(zm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (14)  Expanding u(xq+1, z) in a Fourier series along Γq+1 the next iteration of the algorithm can then  be initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Repeating this procedure for all screens the field u over the domain of interest can be  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  3  Numerical Results  This section presents results of applications of the proposed algorithm to problems of wave propa-  gation through inhomogeneous media, in two-dimensional configurations, and through wide ranges  of problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' In order to evaluate the accuracy of the proposed S-WKB method by  comparisons with solutions obtainable by means of separation of variables, we first consider x-  invariant permittivities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' permittivities of the form ε(x, z) = ε(z)), as described in what follows,  for which a simple high-order spectral solver can be used to obtained reference solutions that are  physically-exact—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=', which contain no approximations to (1) other than those inherent in the well  established numerical solver utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  6  Figure 5: Geometrical optics rays for a “ducting” configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='1  High-order reference solutions for x-invariant permittivities  In order to obtain a valid solution to (1) via separation of variables we seek a solution of the form  u(x, z) =  ∞  �  i=0  aieiαixφi(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (15)  Substituting (15) in (1) leads to  ∞  �  i=0  (−α2 + φ′′  i (z) + k2ε(z))aieiαix = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (16)  Using the orthogonality of the complex exponentials we then obtain  (−α2  i + φ′′  i (z) + k2ε(z))ai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  (17)  It follows that the non-zero coefficients αi in (15) satisfy the Sturm-Liouville problem:  φ′′  i (z) + k2ε(z)φi(z) = α2  i φi(z)  (18)  with radiation boundary conditions:  lim  z→±∞  �  zφ′  i − ikφi  �  = 0,  (19)  Numerically, the radiation boundary conditions can be imposed by considering a sufficiently  large interval (za, zb) and imposing either Dirichlet or periodic boundary conditions at such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  The resulting Sturm-Liouville problem can be discretized with high-order spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' For  the purposes of the present paper we utilized the spectral eigensolver [10], which is available in the  ApproxFun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='jl Julia package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='2  Evaluation of the S-WKB accuracy for a Gaussian permittivity model  In this section we consider the exponential permittivity model  ε(z) = 1 + ae−bz2  (20)  7  Figure 6: “Multiple caustics” test case depicting an idealized “ducting” configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' S-WKB field  values (top), and field values with super-imposed geometrical-optics rays (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The geometrical  optics rays are depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  8  whose physically-exact solution can be obtained by relying on the method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  For our example, an incident field given by a Gaussian beam  uinc(x, z) =  � ∞  −∞  ei√  k2−β2x+iβze− β2  σ2 dβ  (21)  is utilized, wherein the integral in the variable β is evaluated via standard numerical integration  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  In our first example we consider the permittivity model (20) with parameters a = 10−4 and  b = 10−4—which, at C-band, results in a configuration that gives rise to a single caustic of cusp  type for the first 40km (800, 000 wavelengths) in horizontal propagation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The geometrical  optics rays are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The S-WKB solution alongside the Sturm-Liouville solution  with superimposed ray-tracing are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 4, the relative errors for  this configuration are of the order of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Employing N = 400 Fourier modes and a total of 40  screens, the S-WKB solution in this case was obtained in a computing times of 2 minutes in a  single-core, whereas the separation-of-variables solution required single-core runs of approximately  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  Figure 7: Smooth convex lens simulations produced by the S-WKB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Note the fine-scale  fields behind caustics, whose simulation is otherwise quite challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' [7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  For our next example we consider a “ducting” configuration, in which the incident Gaussian  beam is tilted by an angle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='2◦, and where the Gaussian permittivity (20) was used with param-  eters a = 10−4 and b = 10−3—in such a way that the energy is contained within a bounded interval  along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The geometrical-optics rays form a complex system with multiple caustics, as  9  Figure 8: Permittivity and geometrical-optics rays (left), and rays superimposed on the field de-  picted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' We consider the propagation of this signal over a range of 200Km (4 million  wavelengths) in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' A total of n = 800 Fourier modes and 200 of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  4  Smooth Lens  We now consider the test case of a smooth convex lens proposed in [4] on the basis of the permittivity  function given by  ε(r) =  �  1  d2 > L  �  a  b−cos(πd2)  �2  d2 ≤ L  d2 =  �x − xc  xd  �2  +  �z − zc  zd  �2  (22)  For the numerical example depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 7, we have set L = 1, a = 4, b = 3, xc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='5, xd =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='2, zc = 0, zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' The displayed result compares favorably to that presented in [4] on the basis  of a kinetic formulation, as well as the similar problem demonstrated in [7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' A separation  of variables solution is not available in this case, and the error could be evaluated by means of  S-WKB implementation of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' In presence of the previous examples, however, we may  estimate the error in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='1% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='001%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  References  [1] Vasilii M Babich and Vladimir Sergeevich Buldyrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  Short-wavelength diffraction theory:  asymptotic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Springer, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  [2] Max Born and Emil Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Principles of optics: electromagnetic theory of propagation, inter-  ference and diffraction of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Elsevier, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  [3] Vlastislav ˇCerven`y, Mikhail M Popov, and Ivan Pˇsenˇc´ık.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Computation of wave fields in inhomo-  geneous media—gaussian beam approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Geophysical Journal International, 70(1):109–128,  1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  [4] Bj¨orn Engquist and Olof Runborg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
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+page_content=' Applications of the split-step fourier method to the numerical solution of nonlinear  and variable coefficient wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' SIAM Review (Chronicles), 15, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content='  [6] Lars H¨ormander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
+page_content=' Linear partial differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
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+page_content='  [7] Finn B Jensen, William A Kuperman, Michael B Porter, Henrik Schmidt, and Alexandra  Tolstoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
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+page_content='  [8] Joseph B Keller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'}
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KdFOT4oBgHgl3EQfzDRI/content/tmp_files/2301.12930v1.pdf.txt

@@ -0,0 +1,3349 @@
+Cause-Effect Inference in Location-Scale Noise Models: Maximum Likelihood
+vs. Independence Testing
+Xiangyu Sun 1 Oliver Schulte 1
+Abstract
+Location-scale noise models (LSNMs) are a class
+of heteroscedastic structural causal models with
+wide applicability, closely related to affine flow
+models.
+Recent likelihood-based methods de-
+signed for LSNMs that infer cause-effect relation-
+ships achieve state-of-the-art accuracy, when their
+assumptions are satisfied concerning the noise dis-
+tributions. However, under misspecification their
+accuracy deteriorates sharply, especially when the
+conditional variance in the anti-causal direction is
+smaller than that in the causal direction. In this pa-
+per, we demonstrate the misspecification problem
+and analyze why and when it occurs. We show
+that residual independence testing is much more
+robust to misspecification than likelihood-based
+cause-effect inference. Our empirical evaluation
+includes 580 synthetic and 99 real-world datasets.
+1. Introduction
+Distinguishing cause and effect is a fundamental problem
+in many disciplines, such as biology, healthcare and fi-
+nance (Zhang & Chan, 2006; Huang, 2021; Mansouri et al.,
+2022). Randomized controlled trials (RCTs) are the gold
+standard for finding causal relationships. However, it may be
+unethical, expensive or even impossible to perform RCTs in
+real-world domains (Peters et al., 2017; Pearl & Mackenzie,
+2018). Causal discovery algorithms aim to find causal rela-
+tionships given observational data alone. Traditional causal
+discovery algorithms can only identify causal relationships
+up to Markov equivalence classes (MECs) (Spirtes & Gly-
+mour, 1991; Kalisch & B¨uhlman, 2007; Colombo et al.,
+2012). To break the symmetry in a MEC, additional as-
+sumptions are needed (Peters et al., 2017; Sch¨olkopf, 2022),
+such as the type of functional dependence of effect on cause.
+Structural causal models (SCMs) specify a functional class
+for the causal relations in the data (Pearl, 2009; Peters et al.,
+2017). In this work, we focus on one particular type of SCM
+1Simon Fraser University, Burnaby, Canada. Correspondence
+to: Xiangyu Sun <xiangyu [email protected]>.
+Preprint.
+called location-scale noise models (LSNMs) (Tagasovska
+et al., 2020; Khemakhem et al., 2021; Xu et al., 2022; Strobl
+& Lasko, 2022; Immer et al., 2022):
+Y := f(X) + g(X) · ZY
+(1)
+where X is the cause, Y is the effect, written X → Y , and
+ZY is a latent noise variable independent of X (i.e., X ⊥
+⊥ ZY ). The functions f and g are twice-differentiable on
+the domain of X, and g is strictly positive on the domain of
+X. LSNMs are closely related to affine flow models, where
+g(X) = exp s(X) (Khemakhem et al., 2021). LSNMs
+generalize the widely studied additive noise models (ANMs,
+where g(x) = 1 for all x) and allow heteroscedasticity
+where the variance of Y conditional on X, (i.e. V[Y |X])
+depends on the value of X.
+Two major approaches for cause-effect inference in SCMs
+are maximum likelihood (ML) and independence testing
+(IT) of residuals vs. the (putative) cause (Mooij et al., 2016).
+Both have been recently evaluated for LSNMs (Khemakhem
+et al., 2021; Immer et al., 2022), with good accuracy, espe-
+cially when the f and g functions are estimated by neural
+networks. Immer et al. note, however, that ML can be less
+robust than IT, in the sense that accuracy deteriorates when
+the noise distribution is not Gaussian.
+In this paper we investigate the robustness of ML vs. IT.
+Our analysis shows that ML cause-effect inference performs
+poorly when two factors coincide: (1) Noise Misspecifica-
+tion: the ML method assumes a different noise distribution
+from the true one. (2) Misleading Conditional Variances
+(CVs): V[Y |X] > V[X|Y ] in the data generated by causal
+direction X → Y . For example, in an experiment on syn-
+thetic datasets shown in Table 1 below, (i) changing the true
+noise distribution from Gaussian to uniform and (ii) manip-
+ulating the CV of effect Y given cause X (i.e. V[Y |X]),
+while keeping other settings equal, can decrease the rate of
+identifying the true causal direction from 100% to 10%. In
+contrast, IT approaches maintain a perfect 100% accuracy.
+Both conditions (1) and (2) often hold in practice. For real-
+world domains, assumptions about the noise distribution
+can be hard to determine or verify. It is also common to
+have misleading CVs in real-world datasets. For example,
+in the T¨ubingen Cause-Effect Pairs Benchmark (Mooij et al.,
+arXiv:2301.12930v1  [cs.LG]  26 Jan 2023
+
+Cause-Effect Inference in Location-Scale Noise Models
+2016), about 40% of the real-world datasets exhibit a mis-
+leading CV (see Appendix Table 4).
+We make the following contributions:
+• Describe experiments and theoretical analysis to show
+that ML methods succeed when the noise distribution
+is known.
+• Demonstrate empirically that ML methods often fail
+when the noise distribution is misspecified and CV is
+misleading.
+• Introduce a new IT method based on an affine flow
+model.
+• Demonstrate empirically that our IT method is robust
+to noise misspecification and misleading CVs.
+The paper is structured as follows. We discuss related works
+and preliminaries in Sections 2 and 3, respectively. Sec-
+tion 4 examines when ML methods succeed and when they
+fail. Section 5 demonstrates the robustness of the IT ap-
+proach. Evaluations on synthetic and real-world datasets are
+given in Section 6. The code and scripts to reproduce all the
+results can be found online 1.
+2. Related Works
+Causal Discovery. Causal discovery methods have been
+well studied in machine learning (Spirtes & Glymour, 1991;
+Kalisch & B¨uhlman, 2007; Colombo et al., 2012). Assum-
+ing causal sufficiency (no latent confounders) and faithful-
+ness, they find causal structure up to a MEC. To identify
+causal relations within a MEC, additional assumptions are
+needed (Peters et al., 2017; Sch¨olkopf, 2022). SCMs exploit
+constraints that result from assumptions about the functional
+dependency of effects on causes. Functional dependencies
+are often studied in the fundamental case of two variables
+X and Y , the simplest MEC (Mooij et al., 2016). We follow
+this line of work and study two-variable LSNMs assuming
+causal sufficiency.
+Structural Causal Models. Assuming a linear non-Gaussian
+acyclic model (LINGAM) (Shimizu et al., 2006), the causal
+direction was proved to be identifiable. The key idea in
+LINGAM is that in the true causal direction X → Y , the
+model residuals are independent of X. We refer to methods
+derived from the LINGAM approach as independence test-
+ing (IT) methods. The more general additive noise model
+(ANM) (Hoyer et al., 2008) allows for nonlinear cause-
+effect relationships and is generally identifiable, except for
+some special cases. There are other identifiable SCMs, such
+as post-nonlinear models (Zhang & Hyvarinen, 2012) and
+Poisson generalized linear models (Park & Park, 2019).
+1https://github.com/xiangyu-sun-789/CAREFL-H
+LSNM identifiability. There are several identifiability re-
+sults for the causal direction in LSNMs. Xu et al. (2022)
+prove LSNMs are identifiable for linear causal dependen-
+cies. Khemakhem et al. (2021) show that nonlinear LSNMs
+are identifiable with Gaussian noise. Strobl & Lasko (2022);
+Immer et al. (2022) prove LSNMs are identifiable except in
+some pathological cases (cf. Section 4.1).
+Cause-Effect Inference in LSNMs. The blueprint is to fit two
+LSNM models for each direction X → Y and X ← Y and
+select the direction with a higher model score. HECI (Xu
+et al., 2022) bins putative cause values and selects a direc-
+tion based on a Bayesian information criterion based score.
+BQCD (Tagasovska et al., 2020) uses nonparamatric quan-
+tile regression to approximate minimum description length
+to select the direction. GRCI (Strobl & Lasko, 2022) finds
+the direction based on mutual information between the pu-
+tative cause and the model residuals. LOCI (Immer et al.,
+2022) models the conditional distribution of effect given
+cause with Gaussian natural parameters. Then, it chooses
+the direction based on either likelihood (LOCI-M) or in-
+dependence (LOCI-H). CAREFL-M (Khemakhem et al.,
+2021) fits affine flow models and scores them by likelihood.
+CAREFL-M is more general than LOCI since LOCI uses a
+fixed Gaussian noise distribution prior, whereas CAREFL-
+M can utilize different prior distributions. We introduce
+CAREFL-H as a new IT method that scores the fitted affine
+flow models based on independence. DECI (Geffner et al.,
+2022) generalizes CAREFL-M to multivariate cases but is
+designed for ANMs. To our knowledge, the problem of mis-
+leading CVs is neither identified nor analyzed in previous
+work.
+3. Preliminaries
+In this section, we define the cause-effect inference problem
+and review the LSNM data likelihood.
+3.1. Problem Definition: Cause-Effect Inference
+Cause-effect inference takes as input a dataset D over two
+random variables X, Y with N observational data pairs
+(X, Y ) = {(x1, y1), (x2, y2), . . . , (xN, yN)} generated by
+a ground-truth LSNM in Equation (1). The binary output
+decision indicates whether X causes Y (X → Y ) or Y
+causes X (X ← Y ). We assume no latent confounders,
+selection bias or feedback cycles.
+3.2. Definition of Maximum Likelihood Approach for
+LSNMs
+An LSNM model for two variables (X, Y ) is a pair (→, PZ).
+The → represents the direction X → Y with parameter
+space f ≡ fθ, g ≡ gψ, PX ≡ PX,ζ. The ← represents
+the direction X ← Y with parameter space h ≡ hθ′, k ≡
+
+Cause-Effect Inference in Location-Scale Noise Models
+kψ′, PY ≡ PY,ζ′. For notational simplicity, we treat the
+model functions directly as model parameters and omit ref-
+erence to their parameterizations. For example, we write
+f ≡ fθ for a function f that is implemented by a neural
+network with weights θ. We refer to PZ as the model prior
+noise distribution.
+A parameterized LSNM model defines a data distribution as
+follows (Immer et al., 2022) (derivation in Appendix A):
+P→,PZ(X, Y ; f, g, PX)
+=PX(X) · PZY (Y − f(X)
+g(X)
+) ·
+1
+g(X)
+P←,PZ(X, Y ; h, k, PY )
+=PY (Y ) · PZX(X − h(Y )
+k(Y )
+) ·
+1
+k(Y )
+(2)
+For conciseness, we use shorter notation P→,PZ(X, Y ) and
+P←,PZ(X, Y ) in later sections. The likelihood of a dataset
+D for a parametrized → model is given by
+P→,PZ(D; f, g, PX) =
+N
+�
+i=1
+P→,PZ(xi, yi; f, g, PX)
+The ML approach estimates the ML parameters and utilizes
+them to score the → model (similarly for the ← model ):
+�f, �g, �PX := arg max
+f,g,PX P→,PZ(D; f, g, PX)
+(3)
+L→,PZ(D) := P→,PZ(D; �f, �g, �
+PX)
+(4)
+4. Analysis of Maximum Likelihood
+Approach
+In this section, we state existing and provide new identi-
+fiability results for LSNMs. We use theoretical analysis
+to understand why misspecified ML methods fail in the
+presence of misleading CVs (cf. Table 1).
+4.1. Identifiability of LSNMs with Correct Noise
+Distribution
+Strobl & Lasko (2022); Immer et al. (2022) prove the iden-
+tifiability of LSNMs, assuming a correctly specified noise
+distribution. That is, given that the data generating distribu-
+tion (X, Y ) follows a LSNM in the direction X → Y , the
+same distribution with equal likelihood cannot be induced
+by a LSNM in the backward direction X ← Y , except in
+some pathological cases. In terms of our notation, direction
+identifiability means that if (→, PZ) is the data generating
+model, then
+P([L→,PZ(D) − L←,PZ(D)] > 0) → 1 as N → ∞
+Immer et al. prove the following identifiability result:
+Theorem 4.1 (Theorem 1 from (Immer et al., 2022)). For
+data (X, Y ) that follows a LSNM in both direction X → Y
+and X ← Y , i.e.,
+Y = f(X) + g(X) · ZY , where X ⊥⊥ ZY
+X = h(Y ) + k(Y ) · ZX, where Y ⊥⊥ ZX
+The following condition must be true:
+(log p(y))′′ +
+g′(x)
+G(x, y) · (log p(y))′
++ ∂2
+∂y2 · νX|Y (x|y) +
+g(x)
+G(x, y) ·
+∂2
+∂y∂x · νX|Y (x|y)
++
+g′(x)
+G(x, y) · ∂
+∂y · νX|Y (x|y) = 0
+(5)
+where G(x, y) = g(x) · f ′(x) + g′(x) · [y − f(x)] ̸= 0 and
+νX|Y (x|y) = log pZX( x−h(y)
+k(y) ) − log k(y).
+They state that Equation (5) will be false except for “patho-
+logical cases”. In addition, Khemakhem et al. (2021) pro-
+vide sufficient conditions for LSNMs with Gaussian noise
+to be identifiable. Our next theorem provides identifiability
+results for some non-Gaussian noise distributions:
+Theorem 4.2. Suppose that the true data-generating dis-
+tribution follows an LSNM model in both X → Y and
+X ← Y directions:
+1. If the noise distribution is Uniform(a, b), then both
+g(X) and k(Y ) are constant functions.
+2. If the noise distribution is ContinuousBernoulli(λ ̸=
+0.5)2 or Exponential(λ), then one of the following
+conditions holds:
+• g(X)−1 and k(Y )−1 are constant functions.
+• g(X)−1 and k(Y )−1 are linear functions with the
+same coefficients on X and Y , respectively.
+The proof is in Appendix B. Essentially, the theorem shows
+that for Uniform, Exponential, and ContinuousBernoulli
+noise distributions, the true LSNM model can be identified
+unless it degenerates to (i) a homoscedastic additive noise
+model or (ii) a heteroscedastic model with the same linear
+scale in both directions.
+4.2. Non-Identifiability of LSNMs with Misspecified
+Noise Distribution
+Existing identifiability results for ML methods require know-
+ing the ground-truth noise distribution. We find that when
+2The case of ContinuousBernoulli(λ = 0.5) is equivalent
+to Uniform(0, 1)).
+
+Cause-Effect Inference in Location-Scale Noise Models
+Table 1: Accuracy over 10 datasets generated by SCM LSNM-sine-tanh (definition in Appendix G) with N = 10, 000
+samples. The task is a binary decision whether X causes Y or Y causes X. Rewrite Equation (1) as Y = f(X)+α·g(X)·ZY ,
+where α is a scale factor to alter the CV. CVs are computed by binning the putative cause. X denotes the ground-truth cause
+and Y denotes the ground-truth effect. We used Gaussian(0, 1) as model noise prior for both CAREFL and LOCI. The
+suffix -M denotes a ML method. The suffix -H denotes the corresponding IT method (more in Section 5).
+True Noise
+Gaussian(0, 1)
+Uniform(−1, 1)
+α
+0.1
+0.5
+1
+5
+10
+0.1
+0.5
+1
+5
+10
+V[Y |X]
+vs.
+V[X|Y ]
+0.166
+vs.
+0.455
+0.615
+vs.
+0.709
+0.834
+vs.
+0.793
+0.990
+vs.
+0.821
+0.997
+vs.
+0.817
+0.044
+vs.
+0.047
+0.404
+vs.
+0.375
+0.673
+vs.
+0.566
+0.975
+vs.
+0.681
+0.994
+vs.
+0.677
+CAREFL-M
+1.0
+1.0
+1.0
+1.0
+1.0
+0.7
+0.6
+0.7
+0.1
+0.1
+LOCI-M
+1.0
+1.0
+1.0
+1.0
+1.0
+0.7
+0.6
+0.7
+0.1
+0.1
+CAREFL-H
+1.0
+1.0
+1.0
+1.0
+1.0
+0.9
+1.0
+1.0
+1.0
+1.0
+LOCI-H
+1.0
+1.0
+1.0
+1.0
+1.0
+0.7
+1.0
+1.0
+1.0
+1.0
+the noise distribution is misspecified for both causal and
+anti-causal models, the anti-causal model may achieve a
+higher data likelihood, even in the sample size limit. In
+other words, ML model selection is not consistent when
+the noise distribution is misspecified. In terms of our nota-
+tion, if (→, PZ) is the data generating model, but the noise
+distribution is misspecified as P ′
+Z, then
+P([L→,P ′
+Z(D) − L←,P ′
+Z(D)] > 0) ̸→ 1 as N → ∞
+We conducted a simple experiment to show that with a
+misspecified noise distribution, ML model selection can
+fail badly. Table 1 shows results for CAREFL-M (Khe-
+makhem et al., 2021) and LOCI-M (Immer et al., 2022) with
+model prior distribution Gaussian(0, 1). For the left half
+of the table, the data was generated with Gaussian(0, 1)
+noise, matching the model specification. In this case, both
+CAREFL-M and LOCI-M work quite well, and increasing
+CV in the causal direction does not affect their accuracy.
+For the right half of the table, the data was generated with
+Uniform(−1, 1) noise, contradicting the model specifica-
+tion. With the misspecified noise, the accuracy of both
+CAREFL-M and LOCI-M decreases to 70%. With both
+misspecified noise and misleading CVs, their accuracy be-
+comes even lower. For example, in the last column when
+V[Y |X] = 0.994 and V[X|Y ] = 0.677, both CAREFL-M
+and LOCI-M give an accuracy of just 10%. Thus they select
+the incorrect anti-causal direction 9 out of 10 times.
+To understand why higher CV corresponds to lower data
+likelihood, note that the model Equation (2) entails the
+following relationships for the X → Y model:
+V [Y |X] = g2(X) · V [ZY ]
+P→,PZ(X, Y ) = PX(X) · PZY (ZY ) ·
+1
+g(X)
+where ZY = Y −f(X)
+g(X) . Therefore we can expect V [Y |X]
+and P→,PZ(X, Y ) to be negatively related:
+1. Increasing V [Y |X] by enlarging g(X) reduces
+1
+g(X),
+which in turn reduces P→,PZ(X, Y ).
+2. High variance typically means small densities. There-
+fore, increasing V [Y |X] by enlarging V [ZY ] often re-
+duces PZY (ZY ), which in turn reduces P→,PZ(X, Y ).
+Figure 1 illustrates these relationships in actual datasets.
+1. CV and likelihood are negatively related (Figures 1a
+and 1b), with correctly or incorrectly specified noise.
+2. When the noise distribution is correctly specified, the
+causal model always has a higher likelihood, even with
+misleading CVs (Figure 1a).
+3. When the noise distribution is misspecified, the anti-
+causal model often has a higher likelihood under mis-
+leading CVs (Figure 1b).
+5. Robustness of Independence Testing
+This section describes IT approaches for cause-effect learn-
+ing in LSNMs, including a new IT method based on affine
+flows. A theoretical analysis explains why IT approaches
+are robust to noise misspecification and misleading CVs.
+5.1. The Independence Testing Approach
+Inspired by the breakthrough LINGAM approach, indepen-
+dence testing has been used in existing methods for various
+SCMs (Hoyer et al., 2008; Shimizu et al., 2011; Peters et al.,
+2014; Strobl & Lasko, 2022; Immer et al., 2022). Like the
+
+Cause-Effect Inference in Location-Scale Noise Models
+(a) Log-Likelihood Difference Under Correct Noise
+Specification: Gaussian(0, 1) noise
+(b) Log-Likelihood Difference Under Noise Misspecifi-
+cation: Uniform(−1, 1) noise
+(c) HSIC Difference Under Correct Noise Specification:
+Gaussian(0, 1) noise
+(d) HSIC Difference Under Noise Misspecification:
+Uniform(−1, 1) noise
+Figure 1: Visualization of Table 1. First row (1a,1b): ML methods. Second row (1c,1d): IT methods. Y-axis < 0.0: A ML
+method returns the incorrect anti-causal direction. Y-axis > 0.0: An IT method returns the incorrect anti-causal direction.
+ML methods may fail under misspecification and misleading CVs (1b). IT methods are more robust (1d).
+ML approach (Equation (4)), IT methods fit the model pa-
+rameters in both directions, typically maximizing the data
+likelihood (Equation (3)). The difference is in the model
+selection step: While ML approaches select the direction
+with the highest likelihood, IT approaches select the direc-
+tion with the highest degree of independence between the
+fitted model residuals and the putative cause. Algorithm 1
+provides pseudo-code.
+As in previous work (e.g., Mooij et al. (2016)), we use
+the Hilbert-Schmidt independence criterion (HSIC) (Gret-
+ton et al., 2005) to measure (in)dependence throughout the
+paper. HSIC measures the squared distance between the
+joint probability of the two variables and the product of
+their marginals embedded in the reproducible kernel Hilbert
+space. We have HSIC(U, V ) = 0 if and only if U ⊥⊥ V .
+To fit the functions in a LSNM, we use the affine flow esti-
+mator T from CAREFL-M (Khemakhem et al., 2021). We
+refer to the resulting IT method as CAREFL-H. This com-
+bination of affine flow with IT appears to be new. Details
+on CAREFL-M and learning the flow transformation T are
+given in Appendix C.
+Another approach to IT methods is to test the independence
+of residuals for both X and Y variables (He et al., 2021).
+We found that this performs similarly to CAREFL-H and
+therefore report experimental results only for the more com-
+mon LINGAM-style approach. More details on IT with
+residuals are in Appendix D.
+5.2. Suitability Theory
+With a consistent HSIC estimator, Mooij et al. (2016) show
+that an IT approach consistently selects the causal direction
+for ANMs if the regression method is suitable. A regres-
+sion method is suitable if the expected mean squared error
+between the predicted residuals �E and the true residuals E
+approaches 0 in the limit of N → ∞:
+lim
+N→∞ ED,D′
+� 1
+N || �E1...N − E1...N||2
+�
+= 0
+(6)
+where D and D′ denote training set and testing set, respec-
+tively. Hence, with enough data a suitable regression method
+reconstructs the ground-truth noise.
+If an HSIC estimator is consistent, the estimated HSIC value
+converges in probability to the population HSIC value.
+�
+HSIC(X, Y )
+P−→ HSIC(X, Y ).
+Mooij et al. (2016) show that even a biased HSIC estimator
+with a fixed bounded kernel is consistent.
+With a consistent HSIC estimator and a suitable regression
+method, the consistency result for ANMs in Mooij et al.
+(2016) extends naturally to LSNMs.
+Proposition 5.1. For identifiable LSNMs with an indepen-
+dent noise term in one causal direction only, if an IT ap-
+
+: Causal - Anti-Causal
+2.00
+1.75
+1.50
+1.25
+1.00
+Log-Likelihood: 
+0.75
+0.50
+0.25
+0.00
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+Mean CV: Causal - Anti-Causal-Causal
+0.6
+Anti.
+0.5
+0.4
+Causal
+0.3
+0.2
+Log-Likelihood:
+0.1
+0.0
+-0.1
+-0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+Mean CV: Causal - Anti-CausalAnti-Causal
+0.000
+-0.005
+一
+-0.010
+: Causal
+-0.015
+HSIC-Value:
+0.020
+-0.025
+0.6-0.4-0.2
+0.0
+0.2
+Mean CV: Causal - Anti-CausalAnti-Causal
+0.0000
+-0.0005
+0.0010
+: Causal
+-0.0015
+HSIC-Value:
+-0.0020
+-0.0025
+-0.0030
+0.0
+0.1
+0.2
+0.3
+0.4
+Mean CV: Causal - Anti-CausalCause-Effect Inference in Location-Scale Noise Models
+Algorithm 1 CAREFL-H
+1: Input: data pairs D := (X, Y ), the flow estimator T
+of CAREFL-M with prior PZ, and an HSIC estimator
+2: Output: estimated causal direction dir
+3: Split D into training set Dtrain := (Xtrain, Ytrain) and
+testing set Dtest := (Xtest, Ytest)
+4: Optimize T�θ, �
+ψ,�ζ(Dtrain; PZ) in X → Y direction via
+ML to estimate �f and �g
+5: Compute the residual �ZY := Ytest− �
+f(Xtest)
+�g(Xtest)
+6: Optimize T �
+θ′,�
+ψ′, �
+ζ′(Dtrain; PZ) in X ← Y direction
+via ML to estimate �h and �k
+7: Compute the residual �ZX := Xtest−�h(Ytest)
+�k(Ytest)
+8: if HSIC(Xtest, �ZY ) < HSIC(Ytest, �ZX) then
+9:
+dir := X → Y
+10: else if HSIC(Xtest, �ZY ) > HSIC(Ytest, �ZX) then
+11:
+dir := X ← Y
+12: else
+13:
+dir := no conclusion
+14: end if
+proach is used with a suitable regression method for LSNMs
+and a consistent HSIC estimator, then the IT approach is
+consistent for inferring causal direction for LSNMs.
+5.3. Suitability: Empirical Results
+The suitability value S is the left-hand side of Equation (6).
+Table 2 shows an empirical evaluation of S for the flow
+estimator T in the causal direction. We generate data from
+3 synthetic LSNMs (definition in Appendix G), and eval-
+uate T under noise misspecification and misleading CVs.
+We find that as the sample size grows, S approaches 0.
+In other words, T is empirically suitable under noise mis-
+specification and misleading CVs. Therefore, according
+to Proposition 5.1, CAREFL-H based on the flow estimator
+T and a consistent HSIC estimator is empirically consis-
+tent for inferring causal direction in LSNMs under noise
+misspecification and misleading CVs.
+Because the T in CAREFL-M (Khemakhem et al., 2021)
+uses neural networks to approximate the observed data dis-
+tribution, it is difficult to provide a theoretical guarantee of
+suitability. Therefore, although our experiments indicate
+that T is often suitable in practice, we do not claim that it
+is suitable for all LSNMs. For example, we have found it
+to be not suitable in the low-noise regime when the LSNMs
+are close to deterministic.
+6. Experiments
+The code and scripts to reproduce all the results are given
+online 3. We show that across hyperparameter choices the IT
+approach (i.e., CAREFL-H) produces much higher accuracy
+than the ML approach (i.e,. CAREFL-M) in the difficult
+settings with noise misspecification and misleading CVs,
+and produces comparable accuracy in the easier settings
+without noise misspecification or misleading CVs. Also the
+IT approach is more robust with real-world data, where the
+ground-truth noise distribution is unknown.
+For all experiments, we start with the same default hyperpa-
+rameter values for both CAREFL-M and CAREFL-H and
+alter one hyperparameter value at a time. The default hyper-
+parameter values are those specified in CAREFL-M (Khe-
+makhem et al., 2021) for the T¨ubingen Cause-Effect Pairs
+Benchmark (Mooij et al., 2016). Please see Appendix F for
+more details on default and alternative hyperparameter val-
+ues. Previous work (Mooij et al., 2016; Immer et al., 2022)
+reported that ML methods perform better with data splitting
+(split data into training set for model fitting and testing set
+for model selection) and IT methods perform better with
+data recycling (the same data is used for both model fitting
+and selection). Therefore, we use both splitting methods:
+(i) CAREFL(0.8): 80% as training and 20% as testing. (ii)
+CAREFL(1.0): training = testing = 100%.
+We use a consistent HSIC estimator with Gaussian ker-
+nels (Pfister et al., 2018). A summary of experimental
+datasets is provided in Appendix Table 4. All the datasets
+are normalized to have mean 0 and variance 1.
+6.1. Synthetic Datasets
+Please see Appendix G for the definition of the 3 ground-
+truth LSNM SCMs and details on how synthetic datasets are
+generated from them. The sample sizes in each synthetic
+dataset are 500 or 5,000. As shown in Appendix Table 4,
+most synthetic datasets generated by such SCMs have mis-
+leading CVs. Based on the analysis of Section 4.2 we formu-
+late the following hypotheses. (i) We expect CAREFL-M
+to be accurate given a correct noise specification with or
+without misleading CVs. (ii) With noise misspecification
+but without misleading CVs, we expect the accuracy of
+CAREFL-M to be reduced. (iii) With both noise misspeci-
+fication and misleading CVs, we expect the accuracy to be
+very low, often below 50%. Overall, the results from the
+experiments confirm our hypotheses.
+6.1.1. NOISE MISSPECIFICATION
+We
+evaluate
+CAREFL-M
+and
+CAREFL-H
+against
+data generated with Uniform(−1, 1), Exponential(1),
+3https://github.com/xiangyu-sun-789/CAREFL-H
+
+Cause-Effect Inference in Location-Scale Noise Models
+Table 2: Suitability of the flow estimator T of CAREFL-M in the causal direction under noise misspecification and
+misleading CVs. T is trained with a Laplace prior. The original dataset with size 2N is split into two: 50% as training set
+and 50% as testing set. V[Y |X] > V[X|Y ] indicates misleading CVs in the dataset.
+(a) LSNM-tanh-exp-cosine and ContinuousBernoulli(0.9) noise. V[Y |X] vs. V[X|Y ]: 0.324 vs. 0.291.
+N=50
+N=500
+N=1000
+N=5000
+[SZ1, SZ2]
+[0.02406, 0.01283]
+[0.00194, 0.00204]
+[0.00094, 0.00092]
+[0.0002, 0.00018]
+(b) LSNM-sine-tanh and Uniform(−1, 1) noise. V[Y |X] vs. V[X|Y ]: 0.422 vs. 0.367.
+N=50
+N=500
+N=1000
+N=5000
+[SZ1, SZ2]
+[0.0081, 0.00285]
+[0.00039, 0.00031]
+[0.0002, 0.00017]
+[0.00003, 0.00003]
+(c) LSNM-sigmoid-sigmoid and Exponential(1) noise. V[Y |X] vs. V[X|Y ]: 0.927 vs. 0.657.
+N=50
+N=500
+N=1000
+N=5000
+[SZ1, SZ2]
+[0.00979, 0.00443]
+[0.00074, 0.00058]
+[0.00045, 0.00026]
+[0.00005, 0.00005]
+Figure 2: Weighted accuracy over 99 datasets from T¨ubingen Cause-Effect Pairs Benchmark.
+ContinuousBernoulli(0.9)
+or
+Beta(0.5, 0.5)
+noise,
+covered by our identifiability Theorem 4.2 (except
+Beta(0.5, 0.5)).
+Khemakhem et al. (2021) claim that
+CAREFL-M is robust to noise misspecification. We show
+that it may fail remarkably.
+We summarize findings from the 336 settings here; the de-
+tailed results are given in Appendix Figures 3 to 14. In 289
+settings (86.01%), both CAREFL-M(0.8) and CAREFL-
+M(1.0) select the correct causal direction with less than 50%
+random accuracy. Furthermore, in 110 settings (32.74%)
+both CAREFL-M(0.8) and CAREFL-M(1.0) fail catastroph-
+ically with an accuracy of 0%. These experiments also
+show that the accuracy of CAREFL-M often decreases as
+N increases. In contrast, CAREFL-H(1.0) achieves better
+accuracy than CAREFL-M in 333 settings (99.11%). The
+accuracy of CAREFL-H(1.0) goes below 50% in only 6 set-
+tings (1.79%). The results demonstrate the robustness of the
+IT approach under noise misspecification and misleading
+CVs, across different hyperparameter choices.
+Appendix Table 4 and Appendix Figure 14 show that with
+misleading CVs, the accuracy of CAREFL-M is close to 0%.
+This is much lower than the corresponding cases without
+misleading CVs in Appendix Figures 6 and 10.
+6.1.2. CORRECT NOISE SPECIFICATION
+These experiments show that CAREFL-H is comparable
+with CAREFL-M under correct noise specification, with
+or without misleading CVs, especially on larger datasets,
+as long as the affine model capacity is sufficient. We eval-
+uate CAREFL-M and CAREFL-H against data generated
+with Gaussian(0, 1) and Laplace(0, 1) noise. The detailed
+results are in Appendix Figures 15 to 20. We find that
+CAREFL-M is more sample efficient than CAREFL-H
+when the model prior matches the data. Consistent with the
+suitability results in Section 5.3, the accuracy of CAREFL-
+H improves with more data. For example, with N = 500,
+there are 49 out of 84 settings (58.33%) where CAREFL-M
+outperforms CAREFL-H(1.0). However, with N = 5, 000,
+CAREFL-H(1.0) achieves similar accuracy as CAREFL-
+M on all datasets (except LSNM-sigmoid-sigmoid with
+Laplace(0, 1) noise.) In addition, CAREFL-H(1.0) may
+underperform CAREFL-M when the number of hidden neu-
+
+0.8
+0.8 ×
+Accuracy
+0.8
+0.6
+0.6
+0.6
+F9'0
+0.6
+0.4
+0.4
+0.4
+0.4
+0.4
+2
+5
+10
+20
+1
+4
+10
+500
+750
+1000
+2000
+0.0
+0.0001 0.001
+0.1
+laplace
+gaussian
+Number of Hidden Neurons
+Number of Sub-Flows
+Number of Epochs
+L2-Penalty
+PriorsCAREFL-M (0.8)
+CAREFL-M (1.0)
+X
+CAREFL-M (0.8)
+CAREFL-M (1.0)
+CAREFL-H (0.8)
+CAREFL-H (1.0)
+6.
+CAREFL-H (0.8)
+CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models
+Table 3: The best accuracy of each method on the SIM and T¨ubingen Cause-Effect Pairs benchmarks. For methods other
+than CAREFL-M and CAREFL-H, we use the results reported in Immer et al. (2022).
+(a) SIM Benchmarks
+LOCI-M
+LOCI-H
+GRCI
+BQCD
+HECI
+CAM
+RESIT
+CAREFL-M
+CAREFL-H
+SIM
+0.53
+0.79
+0.77
+0.62
+0.49
+0.57
+0.77
+0.55
+0.80
+SIM-c
+0.50
+0.83
+0.77
+0.72
+0.55
+0.60
+0.82
+0.58
+0.85
+SIM-ln
+0.79
+0.72
+0.77
+0.80
+0.65
+0.87
+0.87
+0.84
+0.83
+SIM-G
+0.78
+0.82
+0.70
+0.64
+0.56
+0.81
+0.78
+0.82
+0.79
+(b) Weighted Accuracy for T¨ubingen Cause-Effect Pairs Benchmark
+LOCI-M
+LOCI-H
+GRCI
+BQCD
+HECI
+CAM
+RESIT
+CAREFL-M
+CAREFL-H
+T¨ubingen
+Cause-Effect
+Pairs
+0.57
+0.64
+0.82
+0.77
+0.71
+0.58
+0.57
+0.73
+0.82
+rons, sub-flows or training epochs is low. The reason is
+that an IT approach requires more power to fit the LSNM
+functions and produce a good reconstruction of the noise.
+6.2. Synthetic Benchmark
+Similar to Tagasovska et al. (2020); Immer et al. (2022),
+we compare CAREFL-M and CAREFL-H against the SIM
+benchmark suite (Mooij et al., 2016). SIM comprises 4
+sub-benchmarks: default (SIM), with one confounder (SIM-
+c), low noise levels (SIM-ln) and Gaussian noise (SIM-G).
+In this benchmark, most datasets do not have misleading
+CVs (Appendix Table 4), which favors ML methods. Each
+sub-benchmark contains 100 datasets and each dataset has
+N = 1000 data pairs. As shown in Appendix Figure 21,
+CAREFL-M and CAREFL-H(1.0) achieve similar accu-
+racy on SIM-ln and SIM-G across different hyperparame-
+ter choices. For SIM and SIM-c, CAREFL-H, especially
+CAREFL-H(1.0), outperforms CAREFL-M by 20%-30%
+in all settings. The accuracy of CAREFL-M is only about
+random guess (40%-60%) on SIM and SIM-c.
+Table 3a compares CAREFL-H with SOTA methods. For
+each CAREFL method, we report the best accuracy ob-
+tained with the hyperparameter settings considered in Ap-
+pendix Figure 21, without further tuning.
+CAREFL-H
+achieves the best accuracy on SIM and SIM-c, and achieves
+competitive accuracy on SIM-ln and SIM-G.
+6.3. Real-World Benchmark: T¨ubingen Cause-Effect
+We compare CAREFL-M and CAREFL-H against real-
+world datasets from the T¨ubingen Cause-Effect Pairs Bench-
+mark (Mooij et al., 2016). The benchmark is commonly
+used to evaluate cause-effect inference algorithms (Khe-
+makhem et al., 2021; Xu et al., 2022; Immer et al., 2022).
+To be consistent with previous work (Tagasovska et al.,
+2020; Strobl & Lasko, 2022; Immer et al., 2022), we ex-
+clude 6 multivariate and 3 discrete datasets (#47, #52-#55,
+#70, #71, #105, #107) and utilize the remaining 99 bivariate
+datasets. As recommended by Mooij et al. (2016), we re-
+port weighted accuracy. 40% of datasets in the benchmark
+feature misleading CVs. As shown in Figure 2, CAREFL-
+H(1.0) outperforms CAREFL-M in all configurations by
+large margins (7%-30%).
+We also compare CAREFL-H with SOTA methods. Fol-
+lowing Khemakhem et al. (2021), we use a single set of
+hyperparameters for all 99 datasets, found by grid search
+(Appendix H). Table 3b shows that CAREFL-H achieves
+the SOTA accuracy (82%). With the respectively best hyper-
+parameter settings, CAREFL-H is 9% more accurate than
+CAREFL-M (Khemakhem et al., 2021).
+7. Conclusion and Future Work
+We identified a failure of maximum-likelihood (ML) meth-
+ods for cause-effect inference in location-scale noise models.
+Our analysis shows that the failure mode occurs when the
+noise distribution is misspecified and conditional effect vari-
+ances are misleading (i.e., higher in the causal direction).
+Selecting causal models by independence tests (IT) is robust
+even in this difficult setting. Extensive empirical evalu-
+ation compared the ML approach and a new IT method
+based on affine flows, using both synthetic and real-world
+datasets. The IT flow method achieves better accuracy under
+noise misspecification and misleading CVs, with robust per-
+formance across different hyperparameter choices. Future
+directions include improving the sample efficiency of IT
+methods, and improving the robustness of ML methods by
+learning the noise distribution instead of using a fixed prior.
+
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+
+Cause-Effect Inference in Location-Scale Noise Models
+A. Derivation of Equation (2)
+Lemma A.1. For a LSNM model X → Y defined in Equation (1), we have PY |X(Y |X) = PZY ( Y −f(X)
+g(X) ) ·
+1
+g(X), where
+PZY is the noise distribution.
+For the data distribution in the X → Y direction:
+P→,PZ(X, Y ) = PX(X) · PY |X(Y |X) = PX(X) · PZY (Y − f(X)
+g(X)
+) ·
+1
+g(X)
+Similarly, in the X ← Y direction:
+P←,PZ(X, Y ) = PY (Y ) · PX|Y (X|Y ) = PY (Y ) · PZX(X − h(Y )
+k(Y )
+) ·
+1
+k(Y )
+Proof for Lemma A.1. For a LSNM model defined in Equation (1), we have ZY = Y −f(X)
+g(X) . Hence, ∂ZY
+∂Y
+=
+1
+g(X). Since
+g(X) > 0, so | ∂ZY
+∂Y | = |
+1
+g(X)| =
+1
+g(X).
+PY |X(Y |X)
+=PZY (ZY |X) · | det ∂ZY
+∂Y |
+(via change of variables)
+=PZY (ZY ) · | det ∂ZY
+∂Y |
+=PZY (Y − f(X)
+g(X)
+) · |∂ZY
+∂Y |
+=PZY (Y − f(X)
+g(X)
+) ·
+1
+g(X)
+B. Identifiability Proofs
+In this section, we prove Theorem 4.2.
+If the data (x, y) follows a LSNM in the forward (i.e. causal) model:
+y := f(x) + g(x) · zY
+where zX and zY are the noise terms, e ⊥⊥ zX, x ⊥⊥ zY , g(x) > 0 for all x on its domain. We assume f(·) and g(·) are
+twice-differentiable on the domain of x.
+If the data (x, y) follows a LSNM in the backward (i.e. anti-causal) model:
+x := h(y) + k(y) · mX
+where mY and mX are the noise terms, e′ ⊥⊥ mY , y ⊥⊥ mX, k(y) > 0 for all y on its domain. We assume h(·) and k(·)
+are twice-differentiable on the domain of y.
+zX, zY , mY and mX follow one of Uniform(a, b), Exponential(λ) or ContinuousBernoulli(λ) distribution accordingly.
+Proof for Uniform(a, b) Noise. For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x)
+g(x) ) ·
+1
+g(x) = PZY (zY ) ·
+1
+g(x) =
+1
+b−a ·
+1
+g(x) =
+1
+(b−a)·g(x). Similarly, for the backward model, we have PX|Y (x|y) =
+1
+(b−a)·k(y).
+
+Cause-Effect Inference in Location-Scale Noise Models
+The joint likelihood of the observation (x, y) in the causal model is:
+P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) ·
+1
+(b − a) · g(x)
+The joint likelihood of the observation (x, y) in the backward model is:
+P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) ·
+1
+(b − a) · k(y)
+If the data follows both models:
+P→,PZ(x, y) = P←,PZ(x, y)
+PX(x) ·
+1
+(b − a) · g(x) = PY (y) ·
+1
+(b − a) · k(y)
+PX(x) ·
+1
+g(x) = PY (y) ·
+1
+k(y)
+Take the derivative of both sides with respect to x:
+PX(x) · (g(x)−1)′ + (PX(x))′ · g(x)−1 = 0
+PX(x) · −1 · g(x)−2 · g(x)′ + 0 · g(x)−1 = 0
+PX(x) · −1 · g(x)−2 · g(x)′ = 0
+PX(x) > 0
+g(x)−2 =
+1
+g(x)2 > 0
+Therefore, g(x)′ = 0
+Similarly, if we take the derivative of both sides with respect to y instead, we have:
+k(y)′ = 0
+These imply that both g(x) and k(y) are constant functions.
+Proof for Exponential(λ) Noise. For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x)
+g(x) ) ·
+1
+g(x) = λ · e−λ· y−f(x)
+g(x)
+·
+1
+g(x) =
+λ
+g(x) · e−
+λ
+g(x) ·(y−f(x)). Similarly, for the backward model, we have PX|Y (x|y) =
+λ
+k(y) · e−
+λ
+k(y) ·(x−h(y)).
+The joint likelihood of the observation (x, y) in the causal model is:
+P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) ·
+λ
+g(x) · e−
+λ
+g(x) ·(y−f(x))
+log P→,PZ(x, y) = log PX(x) + log λ − log g(x) −
+λ
+g(x) · (y − f(x))
+The joint likelihood of the observation (x, y) in the backward model is:
+P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) ·
+λ
+k(y) · e−
+λ
+k(y) ·(x−h(y))
+log P←,PZ(x, y) = log PY (y) + log λ − log k(y) −
+λ
+k(y) · (x − h(y))
+
+Cause-Effect Inference in Location-Scale Noise Models
+If the data follows both models:
+log P→,PZ(x, y) = log P←,PZ(x, y)
+log PX(x) + log λ − log g(x) −
+λ
+g(x) · (y − f(x))
+= log PY (y) + log λ − log k(y) −
+λ
+k(y) · (x − h(y))
+log PX(x) − log g(x) − λ · g(x)−1 · y + λ · g(x)−1 · f(x)
+= log PY (y) − log k(y) − λ · k(y)−1 · x + λ · k(y)−1 · h(y)
+Take the derivative of both sides with respect to x:
+(log PX(x))′ − (log g(x))′ − λ · y · (g(x)−1)′ + λ · (g(x)−1 · f(x))′
+= −λ · k(y)−1
+Take the derivative of both sides with respect to y:
+−λ · (g(x)−1)′ = −λ · (k(y)−1)′
+∂g(x)−1
+∂x
+= ∂k(y)−1
+∂y
+They can be equal only if both sides are constants. Therefore, g(x)−1 and k(y)−1 are both constants or both linear functions
+with the same coefficient on x and y, respectively.
+Proof for ContinuousBernoulli(λ ̸= 0.5) Noise. Please refer to Uniform for ContinuousBernoulli(λ = 0.5), which
+equals to Uniform(0, 1). For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x)
+g(x) ) ·
+1
+g(x) =
+Cλ·λ
+y−f(x)
+g(x) ·(1−λ)1− y−f(x)
+g(x) ·
+1
+g(x), where Cλ is the normalizing constant of the continuous Bernoulli distribution. Similarly,
+for the backward model, we have PX|Y (x|y) = Cλ · λ
+x−h(y)
+k(y)
+· (1 − λ)1− x−h(y)
+k(y)
+·
+1
+k(y).
+The joint likelihood of the observation (x, y) in the causal model is:
+P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) ·
+1
+g(x) · Cλ · λ
+y−f(x)
+g(x)
+·(1 − λ)1− y−f(x)
+g(x)
+log P→,PZ(x, y) = log PX(x) − log g(x) + log Cλ + log λ
+y−f(x)
+g(x)
++ log(1 − λ)1− y−f(x)
+g(x)
+The joint likelihood of the observation (x, y) in the backward model is:
+P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) ·
+1
+k(y) · Cλ · λ
+x−h(y)
+k(y)
+·(1 − λ)1− x−h(y)
+k(y)
+log P←,PZ(x, y) = log PY (y) − log k(y) + log Cλ + log λ
+x−h(y)
+k(y)
++ log(1 − λ)1− x−h(y)
+k(y)
+
+Cause-Effect Inference in Location-Scale Noise Models
+If the data follows both models:
+log P→,PZ(x, y) = log P←,PZ(x, y)
+log PX(x) − log g(x) + log Cλ + log λ
+y−f(x)
+g(x)
++ log(1 − λ)1− y−f(x)
+g(x)
+= log PY (y) − log k(y) + log Cλ + log λ
+x−h(y)
+k(y)
++ log(1 − λ)1− x−h(y)
+k(y)
+log PX(x) − log g(x) + log Cλ + y − f(x)
+g(x)
+· log λ + (1 − y − f(x)
+g(x)
+)
+· log(1 − λ)
+= log PY (y) − log k(y) + log Cλ + x − h(y)
+k(y)
+· log λ + (1 − x − h(y)
+k(y)
+)
+· log(1 − λ)
+log PX(x) − log g(x) + log Cλ + g(x)−1 · log λ · y − g(x)−1·
+log λ · f(x) + log(1 − λ) − log(1 − λ) · g(x)−1 · y + log(1 − λ) · g(x)−1 · f(x)
+= log PY (y) − log k(y) + log Cλ + k(y)−1 · log λ · x − k(y)−1·
+log λ · h(y) + log(1 − λ) − log(1 − λ) · k(y)−1 · x + log(1 − λ) · k(y)−1 · h(y)
+Take the derivative of both sides with respect to x:
+(log PX(x))′ − g(x)−1 · g(x)′ − 1 · g(x)−2 · g(x)′ · log λ · y
+− (g(x)−1 · log λ · f(x))′ − log(1 − λ) · −1 · g(x)−2 · g(x)′ · y
++ (log(1 − λ) · g(x)−1 · f(x))′ = k(y)−1 · log λ − log(1 − λ) · k(y)−1
+Take the derivative of both sides with respect to y:
+− g(x)−2 · g(x)′ · log λ − log(1 − λ) · −1 · g(x)−2 · g(x)′
+= −1 · k(y)−2 · k(y)′ · log λ − log(1 − λ) · −1 · k(y)−2 · k(y)′
+g(x)−2 · g(x)′ · log λ − log(1 − λ) · g(x)−2 · g(x)′
+= k(y)−2 · k(y)′ · log λ − log(1 − λ) · k(y)−2 · k(y)′
+g(x)−2 · g(x)′ · (log λ − log(1 − λ)) = k(y)−2 · k(y)′ · (log λ − log(1 − λ))
+Since λ ̸= 0.5, therefore log λ − log(1 − λ) ̸= 0
+g(x)−2 · g(x)′ = k(y)−2 · k(y)′
+∂g(x)−1
+∂x
+= ∂k(y)−1
+∂y
+They can be equal only if both sides are constants. Therefore, g(x)−1 and k(y)−1 are both constants or both linear functions
+with the same coefficient on x and y, respectively.
+C. CAREFL-M
+CAREFL-M (Khemakhem et al., 2021) models a LSNM in Equation (1) via affine flows T. Each sub-flow Tk ∈ T is
+defined as the following:
+X = t1 + es1 · ZX
+Y = t2(X) + es2(X) · ZY
+(7)
+where X is the putative cause and Y is the putative effect in X → Y direction. t1 and s1 are constants. t2 and s2 are
+functions parameterized using neural networks. Without loss of generality, X is assumed to be a function of latent noise
+
+Cause-Effect Inference in Location-Scale Noise Models
+Algorithm 2 CAREFL-M
+1: Input: data pairs D := (X, Y ), and the flow estimator T with prior PZ
+2: Output: estimated causal direction dir
+3: Split D into training set Dtrain := (Xtrain, Ytrain) and testing set Dtest := (Xtest, Ytest)
+4: Optimize T�t1,�s1,�t2,�s2(Dtrain; PZ) in X → Y direction via ML
+5: Compute the likelihood �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) in X → Y direction
+6: Optimize T�t′1,�s′1,�t′2,�s′2(Dtrain; PZ) in X ← Y direction via ML
+7: Compute the likelihood �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) in X ← Y direction
+8: if �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) > �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) then
+9:
+dir := X → Y
+10: else if �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) < �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) then
+11:
+dir := X ← Y
+12: else
+13:
+dir := no conclusion
+14: end if
+variable ZX. If t1 = 0 and s1 = 0, then X = ZX. The exponential function e ensures the multipliers to Z are positive
+without expression loss. Similarly, for the backward direction X ← Y :
+Y = t′
+1 + es′
+1 · ZY
+X = t′
+2(Y ) + es′
+2(Y ) · ZX
+(8)
+where Y is the putative cause and X is the putative effect in X ← Y direction. t′
+1 and s′
+1 are constants. t′
+2 and s′
+2 are
+functions parameterized using neural networks.
+Given Equation (7), the joint log-likelihood of (x, y) in X → Y direction is:
+log P→,PZ(x, y) = log PZX
+�
+e−s1 · (x − t1)
+�
++ log PZY
+�
+e−s2(x) · (y − t2(x))
+�
+− s1 − s2(x)
+(9)
+Similarly for the X ← Y direction. Note that the priors PZ = {PZX, PZY } in Equation (9) may mismatch the unknown
+ground-truth noise distribution P ∗
+Z = {P ∗
+ZX, P ∗
+ZY }. Both CAREFL-M and CAREFL-H optimizes Equation (9) for each
+direction over the training set. For CAREFL-M, it chooses the direction with ML Score L over the testing set as the
+estimated causal direction. Detailed procedure for CAREFL-M is given in Algorithm 2. To map the parameters θ, ψ, ζ, θ′,
+ψ′ and ζ′ in LSNM (Equation (2)) to the flow estimator T of CAREFL (Equations (7) and (8)), we have fθ ≡ t2, gψ ≡ es2,
+PX,ζ ≡ {t1, es1}, fθ′ ≡ t′
+2, gψ′ ≡ es′
+2 and PY,ζ′ ≡ {t′
+1, es′
+1}.
+D. CAREFL-H Alternative Independence Testing
+In Algorithm 1, CAREFL-H tests independence between the putative cause and the residual of the putative effect in each
+direction, i.e., between X and �ZY in X → Y direction, and between Y and �ZX in X ← Y direction. An alternative way of
+testing independence is to test between the residual of the putative cause and the residual of the putative effect (He et al.,
+2021), i.e., between �ZX and �ZY in both directions. Please see Algorithm 3 for complete steps.
+Theorem D.1. By optimizing the log-likelihood in the causal direction, may or may not under noise misspecification,
+CAREFL-H gives the reconstructed residual of the ground-truth cause (i.e. �ZX) identical to the ground-truth cause (i.e. X)
+and to the latent noise variable of the cause (i.e. ZX) up to shifting and scaling.
+Proof is in Appendix E. Although we prove only for the cause, empirically we find that in the causal direction the
+reconstructed residual of the ground-truth effect (i.e., �ZY ) is also close to the latent noise variable of the effect (i.e.,
+ZY ). According to Theorem D.1, testing HSIC(X, �ZY ) in Algorithm 1 and HSIC( �ZX, �ZY ) in Algorithm 3 for the
+causal direction are equivalent. The difference comes from testing HSIC(Y, �ZX) in Algorithm 1 and HSIC( �ZX, �ZY )
+in Algorithm 3 for the anti-causal direction. Although in our experiments the two algorithms often produce the same
+estimation of causal direction, we prefer Algorithm 1, since it relies on few estimations of the residuals.
+
+Cause-Effect Inference in Location-Scale Noise Models
+Algorithm 3 CAREFL-H (Between Residuals)
+1: Input: data pairs D := (X, Y ), the flow estimator T of CAREFL-M with prior PZ, and an HSIC estimator
+2: Output: estimated causal direction dir
+3: Split D into training set Dtrain := (Xtrain, Ytrain) and testing set Dtest := (Xtest, Ytest)
+4: Optimize T�t1,�s1,�t2,�s2(Dtrain; PZ) in X → Y direction via ML to estimate �t1, �s1, �t2 and �s2
+5: Compute the residuals �ZX,→ := Xtest−�t1
+e�s1
+and �ZY,→ := Ytest−�t2(X)
+e�s2(X)
+6: Optimize T�t′1,�s′1,�t′2,�s′2(Dtrain; PZ) in X ← Y direction via ML to estimate �t′1, �s′1, �t′2 and �s′2
+7: Compute the residuals �ZY,← := Ytest−�t′1
+e
+�
+s′1
+and �ZX,← := Xtest−�t′2(Y )
+e
+�
+s′2(Y )
+8: if HSIC( �ZX,→, �ZY,→) < HSIC( �ZX,←, �ZY,←) then
+9:
+dir := X → Y
+10: else if HSIC( �ZX,→, �ZY,→) > HSIC( �ZX,←, �ZY,←) then
+11:
+dir := X ← Y
+12: else
+13:
+dir := no conclusion
+14: end if
+E. Proof for Theorem D.1
+In this section, we prove Theorem D.1.
+The flow estimator T in CAREFL-M (Khemakhem et al., 2021) models a LSNM in Equation (1) as the following:
+X = t1 + es1 · ZX
+Y = t2(X) + es2(X) · ZY
+(10)
+where X and Y are putative cause and effect, respectively. ZX and ZY are assumed to follow a prior distribution, e.g.
+Gaussian(0, 1) or Laplace(0, 1). t1 and s1 are constants. It also means ZX is identical to X up to shifting and scaling.
+If t1 = 0 and s1 = 0, then X = ZX. t2 and s2 are functions parameterized using neural networks. Let (xn, yn), where
+n ∈ {1, . . . , N}, be the n-th data pair.
+Fact 1, invert Equation (10):
+zn
+X = e−s1 · (xn − t1)
+zn
+Y = e−s2(xn) · (yn − t2(xn))
+Fact 2, how �zn
+X and �zn
+Y are computed in the flow estimator T:
+�zn
+X = e−�s1 · (xn − �t1)
+�zn
+Y = e−�s2(xn) · (yn − �t2(xn))
+
+Cause-Effect Inference in Location-Scale Noise Models
+Fact 3, how the flow estimator T is trained via ML (may under noise misspecification):
+pX(xn, yn) = pZX(�zn
+X) · pZY (�zn
+Y ) · | det
+� ∂�zn
+X
+∂xn
+∂�zn
+X
+∂yn
+∂�zn
+Y
+∂xn
+∂�zn
+Y
+∂yn
+�
+|
+= pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn)))
+· | det
+�e−�s1
+0
+∂�zn
+Y
+∂xn
+e−�s2(xn)
+�
+|
+= pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn)))
+· |e−�s1 · e−�s2(xn) − 0|
+= pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn))) · e−�s1 · e−�s2(xn)
+= pZX(e−�s1 · (es1 · zn
+X + t1 − �t1))
+· pZY (e−�s2(xn) · (es2(xn) · zn
+Y + t2(xn) − �t2(xn)))
+· e−�s1 · e−�s2(xn)
+= pZX(e−�s1 · (es1 · zn
+X + t1 − �t1)) · e−�s1
+· pZY (e−�s2(xn) · (es2(xn) · zn
+Y + t2(xn) − �t2(xn))) · e−�s2(xn)
+= pZX(es1−�s1 · zn
+X + e−�s1 · (t1 − �t1)) · e−�s1
+· pZY (es2(xn)−�s2(xn) · zn
+Y + e−�s2(xn) · (t2(xn) − �t2(xn))) · e−�s2(xn)
+Assume the priors pZX and pZY are standard Gaussian N(0, 1) for math convenience. The proof is analogous with other
+prior distributions, e.g., standard Laplace. We do not make assumptions on the ground-truth noise distribution, which allows
+noise misspecification.
+pX(xn, yn) = N(es1−�s1 · zn
+X + e−�s1 · (t1 − �t1); µ = 0, σ2 = 1) · e−�s1
+· N(es2(xn)−�s2(xn) · zn
+Y + e−�s2(xn) · (t2(xn) − �t2(xn)); µ = 0, σ2 = 1)
+· e−�s2(xn)
+(the PDF of N(x; 0, 1) is
+1
+√
+2π · e− 1
+2 ·(x)2)
+=
+1
+√
+2π · e− 1
+2 ·(es1−�s1·zn
+X+e−�s1·(t1−�t1))2 · e−�s1
+·
+1
+√
+2π · e− 1
+2 ·(es2(xn)−�s2(xn)·zn
+Y +e−�s2(xn)·(t2(xn)−�t2(xn)))2 · e−�s2(xn)
+ln pX(xn, yn) = ln
+1
+√
+2π − 1
+2 · (es1−�s1 · zn
+X + e−�s1 · (t1 − �t1))2 − �s1
++ ln
+1
+√
+2π − 1
+2 · (es2(xn)−�s2(xn) · zn
+Y + e−�s2(xn) · (t2(xn) − �t2(xn)))2
+− �s2(xn)
+Lemma E.1. To maximize the log-likelihood, may or may not under noise misspecification, �t1 = E[X], a constant.
+Lemma E.2. To maximize the log-likelihood, may or may not under noise misspecification, �s1 = E[ln |�t1 − X|], a constant.
+�zn
+X = e−�s1 · (xn − �t1)
+�zn
+X = e−E[ln |�t1−X|] · (xn − �t1)
+�zn
+X = e−E[ln |E[X]−X|] · (xn − E[X])
+�zn
+X = C1 · (xn − C2)
+
+Cause-Effect Inference in Location-Scale Noise Models
+Therefore, �ZX and X are identically distributed up to shifting and scaling.
+Since xn := t1 + es1 · zn
+X, we have:
+�zn
+X = C1 · (xn − C2)
+= C1 · (es1 · zn
+X + t1 − C2)
+= C1 · es1 · zn
+X + C1 · t1 − C1 · C2
+= C3 · zn
+X + C4 − C5
+= C3 · zn
+X + C6
+Therefore, �ZX and ZX are also identically distributed up to shifting and scaling.
+Proof for Lemma E.1.
+∂ ln pX(xn, yn)
+∂�t1
+= e−�s1 · (zn
+X · es1−�s1 + e−�s1 · (t1 − �t1))
+Let e−�s1 · (zn
+X · es1−�s1 + e−�s1 · (t1 − �t1)) = 0
+�t1 = t1 + zn
+X · es1 = xn
+Similarly, for ∂ ln pX(xn=1,yn=1)
+∂�t1
+, �t1 = xn=1; for ∂ ln pX(xn=2,yn=2)
+∂�t1
+, �t1 = xn=2. Therefore,
+�t1 = 1
+n ·
+N
+�
+n=1
+xn = E[X]
+Since ∂2 ln pX(xn,yn)
+∂2�t1
+is negative, �t1 maximizes the log-likelihood.
+Proof for Lemma E.2.
+∂ ln pX(xn, yn)
+∂�s1
+= (−zn
+X · es1−�s1 − (t1 − �t1) · e−�s1)2 − 1
+Let (−zn
+X · es1−�s1 − (t1 − �t1) · e−�s1)2 − 1 = 0
+± e�s1 = �t1 − xn
+e�s1 = ±(�t1 − xn)
+e· must be positive.
+e�s1 = |�t1 − xn|
+�s1 = ln |�t1 − xn|
+Similarly, for ∂ ln pX(xn=1,yn=1)
+∂�s1
+, �s1 = ln |�t1 − xn=1|; for ∂ ln pX(xn=2,yn=2)
+∂�s1
+, �s1 = ln |�t1 − xn=2|. Therefore,
+�s1 = 1
+n ·
+N
+�
+n=1
+ln |�t1 − xn| = E[ln |�t1 − X|]
+Since ∂2 ln pX(xn,yn)
+∂2�s1
+is negative, �s1 maximizes the log-likelihood.
+
+Cause-Effect Inference in Location-Scale Noise Models
+F. Default and Alternative Hyperparameter Values Used in Section 6
+We use the reported hyperparameter values in CAREFL-M (Khemakhem et al., 2021) for the T¨ubingen Cause-Effect Pairs
+Benchmark (Mooij et al., 2016) as the default hyperparameter values in all our experiments:
+• The flow estimator T is parameterized with 4 sub-flows (alternatively: 1, 7 and 10).
+• For each sub-flow, f, g, h and k are modelled as four-layer MLPs with 5 hidden neurons in each layer (alternatively: 2,
+10 and 20).
+• Prior distribution is Laplace (alternatively: Gaussian prior).
+• Adam optimizer (Kingma & Ba, 2014) is used to train each model for 750 epochs (alternatively: 500, 1000 and 2000).
+• L2-penalty strength is 0 by default (alternatively: 0.0001, 0.001, 0.1).
+Although we also observe that LOCI-H is more robust than LOCI-M under noise misspecification and misleading CVs, we
+omit their results because: (1) CAREFL often outperforms LOCI (see Tables 1 and 3). (2) LOCI is fixed as a Gaussian
+distribution, whereas CAREFL can specify different prior distributions. (3) For conciseness. LOCI uses different set of
+hyperparameters than CAREFL. So their results cannot be merged into the same figures.
+G. Synthetic SCMs Used in Section 6.1
+We use the following SCMs to generate synthetic datasets:
+• LSNM-tanh-exp-cosine: Y := tanh(X · θ1) · θ2 + ecos(X·ψ1)·ψ2 · ZY
+• LSNM-sine-tanh: Y := sin(X · θ1) · θ2 + (tanh(X · ψ) + φ) · ZY
+• LSNM-sigmoid-sigmoid: Y := σ(X · θ1) · θ2 + σ(X · ψ1) · ψ2 · ZY
+where ZY is the ground-truth noise sampled from one of the following distributions: ContinuousBernoulli(0.9),
+Uniform(−1, 1), Exponential(1), Beta(0.5, 0.5), Gaussian(0, 1) and Laplace(0, 1), and σ is the sigmoid function. Al-
+though we did not prove identifiability with Beta(0.5, 0.5) noise in Theorem 4.2, empirically we find it is identifiable.
+Following (Zheng et al., 2018; 2020), each θ and ψ are sampled uniformly from range [−2, −0.5] ∪ [0.5, 2]. φ is sampled
+uniformly from range [1, 2] to make the tanh function positive. The number of data pairs in each synthetic dataset is
+N ∈ {500, 5000}.
+H. Hyperparameter Values of CAREFL-H For Table 3b
+To acquire the result of CAREFL-H in Table 3b, the hyperparameter values used are as follows:
+• Number of hidden neurons in each layer of the MLPs: 2
+• Number of sub-flows: 10
+• Training dataset = testing dataset = 100%.
+The rest of the hyperparameter values are identical to the default ones.
+
+Cause-Effect Inference in Location-Scale Noise Models
+Table 4: (Best viewed in color) Summary of datasets. Settings in Section 6.1 are color-coded. (1) Blue: with noise
+misspecification; (2) Red: with more than 50% of datasets having misleading CVs; (3) Brown: both (1) and (2).
+Type
+Name
+Noise
+α
+Number of
+Datasets
+Percentage of Datasets
+With Misleading CVs
+(%)
+Synthetic
+(Table 1)
+LSNM-sine-tanh
+Gaussian(0, 1)
+0.1
+10
+30
+0.5
+10
+50
+1
+10
+70
+5
+10
+100
+10
+10
+100
+LSNM-sine-tanh
+Uniform(−1, 1)
+0.1
+10
+30
+0.5
+10
+90
+1
+10
+100
+5
+10
+100
+10
+10
+100
+Synthetic
+(Section 6.1)
+LSNM-tanh-exp-cosine
+Uniform(−1, 1)
+N/A
+10
+60
+Beta(0.5, 0.5)
+10
+80
+ContinuousBernoulli(0.9)
+10
+70
+Exponential(1)
+10
+20
+Gaussian(0, 1)
+10
+40
+Laplace(0, 1)
+10
+40
+LSNM-sine-tanh
+Uniform(−1, 1)
+N/A
+10
+100
+Beta(0.5, 0.5)
+10
+100
+ContinuousBernoulli(0.9)
+10
+60
+Exponential(1)
+10
+20
+Gaussian(0, 1)
+10
+60
+Laplace(0, 1)
+10
+80
+LSNM-sigmoid-sigmoid
+Uniform(−1, 1)
+N/A
+10
+90
+Beta(0.5, 0.5)
+10
+100
+ContinuousBernoulli(0.9)
+10
+80
+Exponential(1)
+10
+70
+Gaussian(0, 1)
+10
+90
+Laplace(0, 1)
+10
+100
+Synthetic
+(Section 6.2)
+SIM
+N/A
+N/A
+100
+40
+SIM-c
+100
+46
+SIM-ln
+100
+23
+SIM-G
+100
+29
+Real-World
+(Section 6.3)
+T¨ubingen Cause-Effect Pairs
+N/A
+N/A
+99
+40
+
+Cause-Effect Inference in Location-Scale Noise Models
+(a) N = 500
+(b) N = 5000
+Figure 3: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Uniform(−1, 1) noise.
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+Figure 4: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Beta(0.5, 0.5) noise.
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+Figure 6: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Exponential(1) noise.
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+Figure 8: Accuracy over 10 datasets: LSNM-sine-tanh and Beta(0.5, 0.5) noise.
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+Figure 12: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Beta(0.5, 0.5) noise.
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+Figure 14: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Exponential(1) noise.
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+CAREFL-H (0.8)
+CAREFL-H (1.0)
+6.
+CAREFL-H (0.8)
+CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models
+(a) N = 500
+(b) N = 5000
+Figure 19: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Gaussian(0, 1) noise.
+(a) N = 500
+(b) N = 5000
+Figure 20: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Laplace(0, 1) noise.
+
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+6.
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+CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models
+(a) Accuracy over 100 datasets from SIM sub-benchmark.
+(b) Accuracy over 100 datasets from SIM-c sub-benchmark.
+(c) Accuracy over 100 datasets from SIM-ln sub-benchmark.
+(d) Accuracy over 100 datasets from SIM-G sub-benchmark.
+Figure 21: Results with SIM benchmarks.
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+X
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+CAREFL-H (0.8)
+CAREFL-H (1.0)
+6.
+CAREFL-H (0.8)
+CAREFL-H (1.0)

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