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+Crucial role of Fe in determining the hard magnetic properties of Nd2Fe14B
+Juba Bouaziz∗
+Department of Physics, University of Warwick, Coventry CV4 7AL, UK and
+Peter Gr¨unberg Institut and Institute for Advanced Simulation,
+Forschungszentrum J¨ulich & JARA, D-52425 J¨ulich, Germany
+Christopher E. Patrick
+Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK
+Julie B. Staunton†
+Department of Physics, University of Warwick, Coventry CV4 7AL, UK
+(Dated: January 10, 2023)
+Nd2Fe14B’s unsurpassed, hard magnetic properties for a wide range of temperatures result from
+a combination of a large volume magnetization from Fe and a strong single-ion anisotropy from
+Nd. Here, using finite temperature first-principles calculations, we focus on the other crucial roles
+played by the Fe atoms in maintaining the magnetic order on the Nd sublattices, and hence the large
+magnetic anisotropy, and directly generating significant uniaxial anisotropy at high temperatures.
+We identify effective spins for atomistic modelling from the material’s interacting electrons and
+quantify pairwise and higher order, non-pairwise magnetic interactions among them. We find the
+Nd spins couple most strongly to spins on sites belonging to two specific Fe sublattices, 8j1, 8j2.
+Moreover the Fe 8j1 sublattice also provides the electronic origin of the unusual, nonmonotonic
+temperature dependence of the anisotropy of Y2Fe14B. Our work provides atomic-level resolution
+of the properties of this fascinating magnetic material.
+The elemental lanthanides show remarkable magnetic
+properties deriving from their partially-filled shells of
+atomic-like 4f electrons.
+However, direct exploitation
+of these properties is hindered by low magnetic order-
+ing temperatures.
+No elemental lanthanide retains its
+magnetism at room temperature, with the highest Curie
+temperature Tc being 292 K for Gd [1].
+Combining
+the lanthanides with other elements can strengthen the
+magnetic interactions and allow ordering to persist to
+higher temperatures.
+The most successful example of
+this paradigm is the rare-earth/transition-metal (RE-
+TM) family of permanent magnets [2]. Specifically, Nd-
+Fe-B demonstrates exceptional magnetic strength over a
+wide range of temperatures. Having revolutionized com-
+puter hard disk technology in the last century, Nd-Fe-B is
+again under intense investigation owing to its use in elec-
+tric vehicle motors and renewable energy turbines [3].
+The RE-TM magnetic interactions are most simply de-
+scribed in terms of the exchange field Bexch.
+In this
+picture, the TM-3d electrons produce an effective mag-
+netic field which couples to the spin magnetic moments
+of the RE ions. A minimal model to describe the RE ions
+and the high magnetic anisotropy they generate combines
+this exchange field with the interaction with an external
+field Bext and the crystal field ˆVCF, which describes the
+(predominantly) electrostatic interaction of the 4f charge
+cloud with its environment [4–7]:
+HRE = 2µB ˆS · Bexch + µB (ˆL + 2ˆS) · Bext + ˆVCF. (1)
+ˆS and ˆL are the total spin and orbital angular mo-
+mentum operators.
+Values of the exchange field can
+be extracted by fitting Eq. 1 to experimental data ob-
+tained in inelastic neutron scattering (INS) or magneti-
+zation measurements. Experimental estimates of Bexch
+are far stronger than fields achievable in the laboratory
+(µBBexch/kB >∼ 300 K [8], i.e. Bexch >∼ 450 T) as required
+to maintain magnetic order above room temperature.
+Going beyond a phenomenological understanding of
+RE ordering requires an atomistic picture of the mag-
+netic interactions among effective spins. Nd2Fe14B has a
+tetragonal crystal structure with 68 atoms per unit cell
+([8] [9]). The RE atoms occupy two crystallographically
+distinct sites (RE4f and RE4g), which (together with Fe4c
+and B4g atoms) form planes encapsulating the remain-
+ing 5 Fe sublattices (4e, 8j1, 8j2, 16k1, 16k2). For the
+Nd sites the spins come from the localized f-electrons
+but for the TM sites the local effective spins, or local
+moments, emerge from the material’s itinerant electron
+fluid [10].
+Spin-polarized regions at atomic sites form
+from co-operative behavior of the valence electrons and
+at finite temperatures their orientations fluctuate on rel-
+atively long time scales compared to the remaining elec-
+tronic degrees of freedom. These local magnetic moments
+are the pertinent, effective spins for the TM aspect of the
+atomistic modelling.
+A conceptually simple model assumes interactions only
+between pairs of spins (ij) according to the classical
+Heisenberg model, −Jij ˆSi · ˆ
+Sj where ˆSi represents an
+effective spin. Previous works [12, 13] calculate such Jij
+parameters from first principles within density-functional
+theory (DFT) [12, 13], and use them directly in atomistic
+spin dynamics simulations. With the TM magnetocrys-
+talline anisotropy (MCA) modelled as a sum of single
+arXiv:2301.02868v1  [cond-mat.mtrl-sci]  7 Jan 2023
+
+2
+c)
+(c)
+(b)
+Nd spin 
+Nd+Fe spin 
+Nd orb
+Fe orb
+Exp
+(a)
+FIG. 1. (a) Nd2Fe14B’s magnetization versus (T/Tc) from DLM-DFT calculations compared to experiment [11]. (b) Sub-lattice
+resolved magnetic order parameters and (c) Weiss fields. The dots indicate the full DLM-DFT results, dashed lines from a
+pair-wise interaction model and continuous lines from a fit of the DLM-DFT results to the model discussed in the text Eq. (3).
+ion-like terms, assumed to be substantial, and RE crystal
+field coefficients taken from experiment, the simulations
+can reproduce the magnetization behavior of Nd2Fe14B,
+including the spin reorientation transition at low tem-
+perature, and represent the current state-of-the-art in
+modelling these magnets [14]. Although such a pair-wise
+Heisenberg model is computationally straightforward to
+implement, it is nonetheless a clear presumption for a
+magnetic metal like Nd-Fe-B. Despite the huge technical
+importance of the material, the role of “beyond Heisen-
+berg” itinerant electron spin features has yet to be elu-
+cidated for Nd-Fe-B. Moreover the MCA from the spin-
+orbit coupling of the itinerant d-electrons is also not guar-
+anteed to be single-ion like [15]. In this letter we quantify
+the significance of both these aspects and propose ways
+to improve atomistic spin modelling.
+The disordered local moment (DLM) picture imple-
+mented within DFT provides an appropriate ab initio
+framework [10, 15, 16]. The approach combines statisti-
+cal mechanics of the effective spins (local moments, {ei})
+and DFT, to describe the complementary evolution of
+electronic and magnetic structure as a material’s tem-
+perature is varied. Strongly correlated 4f-electron effects
+are treated with a parameter free, self interaction correc-
+tion (SIC) approach [17, 18] which incorporates Hund’s
+rules naturally [19].
+As such DLM-DFT can describe
+temperature-dependent magnetic properties of perma-
+nent magnets as shown recently for the RECo5 fam-
+ily [7]. The crucial RE contribution to the anisotropy is
+accounted by crystal field theory, calculating the CF coef-
+ficients within DFT using a robust numerical method [16]
+so that the modelling is independent of any prior fit of
+phenomenological parameters.
+Here we investigate the nature of magnetic order in
+Nd2Fe14B, sublattice-resolved, and describe the mag-
+netic interactions among the effective spins associated
+with both RE and TM sites. We show that the interac-
+tions among the TM spins are influenced by the global
+magnetic order and its impact and link with the spin-
+polarised electrons of the system. This is in essence a
+multi-spin coupling effect. We find significant diversity
+in the behavior of the Fe local moments depending on
+their location in the unit cell. While most TM spins are
+ferromagnetically-coupled, some interact antiferromag-
+netically with each other. This leads to some frustration
+and a peculiar strong suppression of magnetic order on
+the 8j1 sites which are located roughly midway between
+the Nd-containing layers. We also find that the Nd spins
+couple most strongly to spins on sites belonging to this
+Fe sublattice along with those on another (8j2). Further-
+more we discover a link between this 8j1 sublattice and
+the unusual non-monotonic temperature dependence of
+the non-RE MCA of the isostructural material Y2Fe14B,
+resolving a longstanding a puzzle [11, 20]. Finally our
+calculation of the anisotropy field of Nd2Fe14B across a
+range of temperatures agrees well with experiment and
+confirms the vital role played by the Fe spins for the
+functionality of this champion magnet.
+Apart from the local moments themselves, the cen-
+tral quantities in DLM-DFT theory are Weiss fields {hi}
+which, analogously to the exchange field of Eq.1, drive
+the ordering of the local moments.
+However, unlike
+Bexch, the Weiss fields are not phenomenological, but
+instead are rigorously defined by thermal averages over
+the local moment orientational configurations {ei} of the
+magnetic energy Ω{ei} [10, 15], i.e.
+hi =
+�
+3
+4π ⟨Ω⟩ei;T dei .
+(2)
+where ⟨X⟩ei denotes the average of X with the restric-
+tion that the orientation of the moment on site i is fixed
+as ei and the order parameters of the local moments,
+{mi} are the averages {⟨ei⟩} [10, 15]. ⟨Ω⟩�ei;T is calcu-
+lated from DFT [10]. Crucially no prior prescription is
+assumed for the form of the magnetic interactions inher-
+ent in the first-principles Ω. For a pairwise Heisenberg
+model, the magnetic energy Ω{ei} = −1/2 �
+ij Jijei · ej
+with Weiss fields linear in the {mi}, hi = �
+j Jijmj.
+
+3
+16k1
+16k2
+8j1
+8j2
+f g 4e 4c
+FIG. 2. The relative strengths of interactions between sites in
+the unit cell (boron sites not included), Jij, (Eq. 3) highlight-
+ing the RE-TM ones (sites 48–51 and 52–55 correspond to 4f
+and 4g respectively). Numerical values in meV are given in [9]
+along with specific site coordinates. Red/blue color indicates
+FM/AF interactions.
+Consequently beyond-pairwise terms are clearly identi-
+fied in DLM-DFT theory from the non-linear dependence
+of the Weiss fields on the {mi} [21–25]. The {mi} order
+parameters, describing an equilibrium state at a temper-
+ature T, are given by the self-consistent solution of Eq.2
+and mi = (−1/βhi + coth βhi), the Langevin function,
+(β = 1/kBT).
+Figure 1(a) shows the magnetization as a function of
+T compared to experiment and resolved into the RE and
+TM spin and orbital components.
+The magnetization
+is directed along θ = 45◦ in the (xz)-plane.
+Full cal-
+culational details are given in the Supplemental Mate-
+rial [9] and references [16, 26–28]. The contribution from
+a particular site i is found by multiplying its local mo-
+ment magnitude, µi, by the order parameter mi(T). The
+Fe and Nd spin moments interact antiferromagnetically
+(AF) and order in an anti-parallel alignment in a fer-
+rimagnetic state, but the large orbital moment of Nd,
+pointing opposite to its spin, leads to overall ferromag-
+netic (FM) order.
+The Fe orbital moments are small
+(∼ 0.05µB/atom). The calculated Tc is 1058 K, which,
+although an overestimate of 473 K in comparison to the
+experiment [11], is reasonable for a first-principles theory
+which uses a mean field approximation for the statistical
+mechanics of the effective spins [28].
+On each of the six Fe and two Nd sublattices ([8, 9]
+the magnetic order varies from complete, {mi = 1},
+at T = 0K to zero above Tc, {mi = 0}.
+Figure 1(b)
+shows how the temperature dependence of magnetic or-
+der varies across the sublattices. The Nd sublattices dis-
+order more quickly than all the Fe sublattices except
+the 8j1 one.
+Complementary information in Fig. 1(c)
+shows that Weiss fields, {hi}, promote strong ordering
+when large and have considerable sublattice variation,
+notably the factor ∼4 difference between the 8j1 and
+8j2 sites. Analysis of {hi}, Eq.2, reveals the presence
+and importance of interactions that fall outside those of
+a Heisenberg-like model. For such a pairwise model the
+Jij interactions (Fig. 2), directly obtained from the Weiss
+fields for small values of the {mi}, are used to construct
+the model’s Weiss fields and {mi} at all T (dashed lines
+in Fig. 1(c)). There are large discrepancies from the full
+ab initio DLM-DFT data away from Tc, leading us to
+propose a more realistic representation of the interac-
+tions which is straightforward to incorporate into atom-
+istic spin modelling of the magnet’s properties. It leads
+to a magnetic energy per unit cell
+¯Ω = −1
+2
+�
+ij
+Jijmi · mj − 1
+4
+�
+i
+BI(mi · M)2,
+(3)
+where i, j run over the sites in the unit cell, I denotes one
+of the 8 sub-lattices to which the site i belongs and M
+is the total magnetization per unit cell, M = �
+i µimi
+where the order parameters on the RE sites are anti-
+parallel to the TM sites for the ferrimagnetic state. The
+second, higher order term captures the effect of the over-
+all spin-polarization of the electronic structure on the
+effective interactions between the local moments. Com-
+puting Weiss fields from this expression fits the DLM-
+DFT calculations very well as shown by the full curves in
+Fig. 1(c) and ¯Ω closely approximates ⟨Ω⟩T . Table I lists
+the BI parameters that measure the sublattice-dependent
+size of these higher order, multi-spin terms.
+System
+4c
+4e
+8j1
+8j2 16k1 16k2
+Rf
+Rg
+Nd2Fe14B -15.42 14.31 -5.06 -1.38 3.82 4.44 -2.53 -1.41
+Y2Fe14B
+-13.68 9.91 -4.07 1.27 4.25 4.07
+0.0
+0.0
+TABLE I. Effective, multi-spin interaction constants (in µeV),
+BI, for Nd2Fe14B and Y2Fe14B.
+Fig. 2 shows the relative strengths of the Jij interac-
+tions between pairs of sites. They are represented on a
+64 × 64 grid (56 Fe sites and 8 RE sites and arranged
+according to sublattice). Numerical values are given as
+Supplemental Material [9]. Assuming a range less than
+roughly 5˚A, they can be directly used in atomistic spin
+simulations together with the terms from Table I. The
+figure illustrates the vital importance on the RE mag-
+netic ordering of the hexagonal nets of Fe atoms [8, 9]
+from the k1, k2 and notably sites on the j1 and j2 sublat-
+tices. Indeed the largest contributions to the Weiss fields
+at the RE sites originate from the j1 and j2 sublattices.
+The TM-TM interactions are particularly varied rang-
+ing from FM (red) for the majority to AF (blue). The
+j1 sites have AF interactions with e, c and RE sites and
+strong FM ones with j2 sites.
+This frustration drives
+this sublattice’s aversion to magnetic order. The diver-
+sity of the interactions stems from the profound effect
+
+60
+40
+30
+40
+20
+Site
+10
+20
+0
+-10
+20
+40
+60
+Site i4
+that atom coordination and spacing of Fe atoms in a
+metallic material has on its magnetism. The archetype
+for this quintessentially itinerant electron effect is fcc
+Fe where squeezing the lattice turns ferromagnetic order
+anti-ferromagnetic and then destroys it [29, 30].
+The diverse nature of the magnetic order on the six
+Fe sub-lattices also has an impact on the intrinsic MCA
+generated by the system’s itinerant valence electrons. As
+found for other TM metal magnets [15, 26], a simple fit in
+terms of a single ion model is unsatisfactory and, as found
+for other itinerant electron magnets, two-ion type terms
+should be included in the model [15, 31, 32]. Further-
+more, on general grounds, modelling the MCA as a sum
+of single ion anisotropy terms must be done extremely
+carefully. The various Fe sites have different crystallo-
+graphic point symmetries, and their unique symmetry
+axes do not necessarily point along the c direction [33].
+There are important implications for atomistic spin dy-
+namics simulations [14, 31, 34] where it is not correct
+to assign a single ion anisotropy to each Fe atom with
+the same angular dependence and same symmetry axes.
+Rather a simpler and more rigorous alternative would
+be to compute an anisotropy energy based on the vector
+sum of all moments in the same sublattice so that a single
+symmetry axis is recovered [5].
+While significantly smaller than from the f-electrons of
+the Nd atoms, the primarily TM component of the MCA
+grows in importance with rising T.
+As the RE MCA
+drops swiftly along with the magnetic order
+[35, 36],
+the TM contribution can actually increase as shown ex-
+plicitly in measurements on Y2Fe14B [11, 20].
+Such
+non-monotonic temperature variation is puzzling, and
+has been attributed to a magnetostructural effect from
+an anisotropic expansion of the crystal lattice, compet-
+ing single-ion-like contributions [20] or competing single
+and two-ion MCA using atomistic spin dynamics simula-
+tions [31, 32] Since fully relativistic effects such as spin-
+orbit coupling are included in our DLM-DFT theory we
+investigate the MCA temperature dependence directly
+and show our results in Fig. 3 for Y2Fe14B.
+Using the highly accurate, full potential (FP) KKR
+code [37, 38], we first calculate the MCA at T = 0K to
+be ≈ 0.9 meV/formula unit (FU) which agrees well with
+experimental values [11]. The same rapid loss of magnetic
+order with increasing temperature which we find for the
+Fe 8j1 sites in Nd2Fe14B (Fig. 1(b)) is also evident in
+Y2Fe14B [9] and this points to a significant role for this
+sublattice in the anomalous MCA T-dependence.
+We
+therefore carry out further FP MCA calculations where
+now the Fe 8j1 sites are constrained to be magnetically
+disordered (m8j1 = 0) via an equal weighting of local
+moments on each of these sites along the ±x and ±z
+directions. The effect on the computed MCA is striking
+- it increases greatly to ≈ 1.7 meV/FU - and we infer
+that the much faster decrease of 8j1 magnetic order with
+temperature relative to that on the other Fe sublattices
+is the key driver for the MCA T-dependence.
+To test this proposition, we calibrate DLM-DFT MCA
+values against our T = 0K FP MCA calculations, given
+the current implementation with an atomic sphere ap-
+proximation (ASA). Although the ASA values are smaller
+than the FP ones, the same large increase of the value
+when the Fe 8j1 sites are magnetically disordered is
+found. In Fig. 3 we show the DLM-DFT temperature de-
+pendent MCA both using the ASA (red curve) and also
+scaled by the ratio between the FP and our ASA T = 0K
+values (green). The increase with temperature is evident,
+peaking at T/Tc = 50% in line with experiment [11] con-
+firming our proposition. Since the calculations are for a
+fixed lattice structure, we can exclude thermal expansion
+as a cause of the non-monotonic behavior. We also show
+the effect on the MCA of forcing the Fe8j1 sublattice to
+remain magnetically disordered at all T, i.e. m8j1 = 0.
+The resulting unscaled MCA, shown in black in Fig. 3, is
+dramatically altered - the peak has gone and the MCA
+decays linearly with temperature and the T = 0K value
+is enhanced significantly. Clearly, establishment of mag-
+netic order on the Fe8j1 sublattice correlates with a sub-
+stantial drop in the (uniaxial) MCA.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T/TC
+0.0
+0.5
+1.0
+1.5
+2.0
+K1 (meV/FU)
+Exp
+Th
+Th-8j1
+Th-scl
+FIG. 3. The T-dependence of the leading anisotropy constant,
+K1, of Y2Fe14B from DLM-DFT theory (red curve) and ex-
+periment (blue) [11]. The green curve shows the theory values,
+scaled to account for the difference between FP [37] and ASA
+([9]) at T = 0K. The black curve shows K1 (unscaled) if the
+Fe 8j1 sublattice is constrained to be disordered magnetically.
+Our ultimate goal is to describe Nd2Fe14B’s large mag-
+netic anisotropy and its temperature variation. So to the
+TM MCA we add the dominant RE components. These
+are calculated [6, 39] from the solution of Eq. 1 where the
+crystal field coefficients [9] are determined from first prin-
+ciples [16], and exchange field, Bexch provided directly by
+the DLM-DFT Weiss field for each Nd site (Fig. 1) di-
+vided by the computed Nd spin moment of 3.66 µB.
+Our calculated exchange fields of 699 and 725 T for
+the RE f and g sites respectively are somewhat larger
+than those used in fits of experimental magnetization
+
+5
+0.4
+0.6
+0.8
+1.0
+T/Tc
+0
+5
+10
+µ0HA(T)
+Computed
+Computed-scl
+Exp (Hirosawa 1986)
+Exp (Grossinger 1986)
+FIG. 4. Evolution of the anisotropy field, HA, versus T/Tc
+from theory compared to experimental measurements (from
+Refs. [42], black curve,
+[11], blue curve). The agreement is
+good above Tc/2. The red and green curves use the non-RE
+MCA taken from the red and green curves of Fig. 3.
+data (450–520 T [40]), but as pointed out in Ref. [8],
+the large number of parameters in Eq. 1 introduce sig-
+nificant uncertainties. In principle, INS data would pro-
+vide a direct measure of the exchange fields but are not
+available for Nd-Fe-B. Our proposed values are, however,
+supported by the good agreement between INS experi-
+ments [41] and our DLM-DFT calculations for the re-
+lated Gd2Fe14B magnet (324 T vs 307/319 T). The Gd
+exchange fields are substantially smaller than those cal-
+culated for Nd. The relative difference (∼2) mirrors that
+of the spin moments (7.46 vs. 3.66 µB) and reflects the
+similar Weiss fields we calculate for the two materials.
+Using the method of Ref. [39] we calculate effective
+anisotropy constants K1(T), K2(T) and anisotropy field,
+µ0HA = 2K1/M ab initio which is shown in Figure 4.
+The red/green curves show µ0HA which includes the non-
+RE contribution to the MCA of the red/green plots in
+Fig. 3. Fig. 4 also shows the experimental measurements
+from Refs. [11, 42]. Below T/Tc ∼ 0.5 there is some dis-
+crepancy between the two sets of experimental data, but
+above there is consistency between both the experiments
+and our calculations.
+The calculations show the clear
+importance of the Fe-dominated MCA to the anisotropy
+field at high temperatures - the red curve is over 1 T less
+than the green one over a range of temperatures despite
+the contributions from the non-RE MCAs differing by
+less than 30 µeV per Fe atom.
+Nd2Fe14B’s spin reorientation transition (SRT) at
+135K [8, 14, 43] is not described by our calculations owing
+to an underestimate of the high order crystal field coeffi-
+cients [14, 44, 45]. This shortcoming exemplifies a more
+general challenge for theory modelling of low T strongly
+correlated f-electron effects to construct a robust way to
+significantly enhance the value of these coefficients [46].
+Around room temperature and above, however, the ef-
+fects on the MCA from these high order terms are small.
+This is also the temperature regime where the tenets of
+our DLM-DFT theory are valid.
+Nd2Fe14B and the RE-TM permanent magnet family
+to which it belongs have a compelling set of attributes.
+Their technological value is enormous and growing and
+their magnetic properties, at a fundamental level, come
+from a rich and subtle combination of RE, localized, and
+TM, itinerant electron, effects.
+To enhance magnetic
+functionality and extract pointers for the development of
+even better materials, multiple interrelated aspects have
+to be accounted for. Our ab initio DLM-DFT modelling
+has shown the importance of describing accurately the
+rich and complex itinerant electron magnetism associ-
+ated with the Fe sites and valence electrons generally for
+the production of the robust exchange field acting on the
+RE atoms, the higher order effective spin interactions
+and the nature of the non-f electron MCA. The modifi-
+cations proposed here should be incorporated into future
+atomistic, effective spin and micromagnetic modelling to
+correctly describe these phenomena.
+The work was supported by EPSRC (UK) Grant No.
+EP/M028941/1 (J.B. and J.B.S.) and Royal Society Re-
+search Grant RGS\R1\201151 (C.E.P.).
+∗ [email protected]
+† [email protected]
+[1] R. J. Elliott, in Magnetic Properties of Rare Earth Metals
+(Plenum Press, London and New York, 1972) p. 2.
+[2] J. M. D. Coey, Hard Magnetic Materials: A Perspective,
+IEEE Trans. Magn. 47, 4671 (2011).
+[3] S. Hirosawa, Preface to the viewpoint set on: Permanent
+magnets and underlining material science, Scripta Mat.
+154, 245 (2018).
+[4] M. Richter, Band structure theory of magnetism in 3d-4f
+compounds, Journal of Physics D: Applied Physics 31,
+1017 (1998).
+[5] M. D. Kuz’min and A. M. Tishin, Chapter three theory
+of crystal-field effects in 3d-4f intermetallic compounds,
+Handbook of Magnetic Materials 17, 149 (2007).
+[6] C. E. Patrick, M. Matsumoto, and J. B. Staunton,
+First-principles calculations of the magnetocrystalline
+anisotropy of the prototype 2:17 cell boundary phase
+Y(Co1−x−yFexCuy)5, Journal of Magnetism and Mag-
+netic Materials 477, 147 (2019).
+[7] C.
+E.
+Patrick
+and
+J.
+B.
+Staunton,
+Temperature-
+dependent
+magnetocrystalline
+anisotropy
+of
+rare
+earth/transition metal permanent magnets from first
+principles:
+The light RCo5(R
+=
+Y, La-Gd) inter-
+metallics, Phys. Rev. Materials 3, 101401 (2019).
+[8] J. F. Herbst, R2Fe14B materials: Intrinsic properties and
+technological aspects, Rev. Mod. Phys. 63, 819 (1991).
+[9] See Supplemental Material for a picture of the crys-
+tal structure, site-resolved local, electronic spin-polarized
+densities of states, further information about multi-
+spin interactions, Y2Fe14B magnetic properties, the 4f-
+atomic Hamiltonian and numerical values of magnetic in-
+
+6
+teractions between pairs of sites (see, also, references [47–
+53] therein).
+[10] B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks,
+and H. Winter, A first-principles theory of ferromagnetic
+phase transitions in metals, Journal of Physics F: Metal
+Physics 15, 1337 (1985).
+[11] S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura,
+M. Sagawa, and H. Yamauchi, Magnetization and mag-
+netic anisotropy of R2Fe14B measured on single crystals,
+Journal of applied physics 59, 873 (1986).
+[12] A. Liechtenstein, M. Katsnelson, and V. Gubanov, Ex-
+change interactions and spin-wave stiffness in ferromag-
+netic metals, Journal of Physics F: Metal Physics 14,
+L125 (1984).
+[13] A. I. Liechtenstein, M. Katsnelson, V. Antropov, and
+V. Gubanov, Local spin density functional approach to
+the theory of exchange interactions in ferromagnetic met-
+als and alloys, Journal of Magnetism and Magnetic Ma-
+terials 67, 65 (1987).
+[14] Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi,
+T. Miyake, and A. Sakuma, Monte carlo analysis for
+finite-temperature magnetism of Nd2Fe14B permanent
+magnet, Phys. Rev. B 94, 174433 (2016).
+[15] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy,
+S. Ostanin, and L. Udvardi, Temperature dependence of
+magnetic anisotropy: An ab initio approach, Phys. Rev.
+B 74, 144411 (2006).
+[16] C. E. Patrick and J. B. Staunton, Crystal field coef-
+ficients for yttrium analogues of rare-earth/transition-
+metal magnets using density-functional theory in the
+projector-augmented wave formalism, Journal of Physics:
+Condensed Matter 31, 305901 (2019).
+[17] J. P. Perdew and A. Zunger, Self-interaction correction
+to density-functional approximations for many-electron
+systems, Phys. Rev. B 23, 5048 (1981).
+[18] M. L¨uders, A. Ernst, M. D¨ane, Z. Szotek, A. Svane,
+D. K¨odderitzsch, W. Hergert, B. L. Gy¨orffy, and W. M.
+Temmerman, Self-interaction correction in multiple scat-
+tering theory, Phys. Rev. B 71, 205109 (2005).
+[19] C. E. Patrick and J. B. Staunton, Rare-earth/transition-
+metal magnets at finite temperature:
+Self-interaction-
+corrected relativistic density functional theory in the dis-
+ordered local moment picture, Phys. Rev. B 97, 224415
+(2018).
+[20] J. Cadogan and H.-S. Li, Analysis of the unusual tem-
+perature dependence of the anisotropy constant K1 of
+Y2Fe14B, Journal of magnetism and magnetic materials
+110, L15 (1992).
+[21] J. B. Staunton, R. Banerjee, M. d. S. Dias, A. Deak, and
+L. Szunyogh, Fluctuating local moments, itinerant elec-
+trons, and the magnetocaloric effect: Compositional hy-
+persensitivity of FeRh, Phys. Rev. B 89, 054427 (2014).
+[22] E. Mendive-Tapia and J. B. Staunton, Ab initio theory
+of the gibbs free energy and a hierarchy of local moment
+correlation functions in itinerant electron systems: The
+magnetism of the mn3a materials class, Phys. Rev. B 99,
+144424 (2019).
+[23] D. Boldrin, E. Mendive-Tapia, J. Zemen, J. B. Staunton,
+T.
+Hansen,
+A.
+Aznar,
+J.-L.
+Tamarit,
+M.
+Barrio,
+P. Lloveras, J. Kim, X. Moya, and L. F. Cohen, Multi-
+site exchange-enhanced barocaloric response in Mn3NiN,
+Phys. Rev. X 8, 041035 (2018).
+[24] E. Mendive-Tapia and J. B. Staunton, Theory of mag-
+netic ordering in the heavy rare earths: Ab initio elec-
+tronic origin of pair- and four-spin interactions, Phys.
+Rev. Lett. 118, 197202 (2017).
+[25] E. Mendive-Tapia, D. Paudyal, L. Petit, and J. B.
+Staunton, First-order ferromagnetic transitions of lan-
+thanide local moments in divalent compounds: An itin-
+erant electron positive feedback mechanism and fermi
+surface topological change, Phys. Rev. B 101, 174437
+(2020).
+[26] M. Matsumoto, R. Banerjee, and J. B. Staunton, Im-
+provement of magnetic hardness at finite temperatures:
+Ab initio disordered local-moment approach for YCo5,
+Phys. Rev. B 90, 054421 (2014).
+[27] C. E. Patrick, S. Kumar, G. Balakrishnan, R. S. Ed-
+wards, M. R. Lees, E. Mendive-Tapia, L. Petit, and J. B.
+Staunton, Rare-earth/transition-metal magnetic interac-
+tions in pristine and (Ni,Fe)-doped YCo5 and GdCo5,
+Phys. Rev. Materials 1, 024411 (2017).
+[28] C. E. Patrick and J. B. Staunton, MARMOT: mag-
+netism, anisotropy, and more, using the relativistic disor-
+dered local moment picture at finite temperature, Elec-
+tronic Structure (2022).
+[29] J. Kuebler, magnetic moments of ferromagnetic and an-
+tiferromagnetic bcc and fcc iron, Physics Letters A 81,
+81 (1981).
+[30] F. J. Pinski, J. Staunton, B. Gyorffy, D. D. John-
+son, and G. M. Stocks, Ferromagnetism versus anti-
+ferromagnetism in face-centered cubic iron, Phys. Rev.
+B 56, 2096 (1986).
+[31] R. Cuadrado, R. F. Evans, T. Shoji, M. Yano, A. Kato,
+M. Ito, G. Hrkac, T. Schrefl, and R. W. Chantrell,
+First principles and atomistic calculation of the magnetic
+anisotropy of Y2Fe14B, Journal of Applied Physics 130,
+023901 (2021).
+[32] R. F. Evans, L. R´ozsa, S. Jenkins, and U. Atxitia, Tem-
+perature scaling of two-ion anisotropy in pure and mixed
+anisotropy systems, Physical Review B 102, 020412
+(2020).
+[33] Y. Miura, H. Tsuchiura, and T. Yoshioka, Magnetocrys-
+talline anisotropy of the fe-sublattice in Y2Fe14B sys-
+tems, Journal of Applied Physics 115, 17A765 (2014).
+[34] Q. Gong, M. Yi, R. F. L. Evans, B.-X. Xu, and O. Gut-
+fleisch, Calculating temperature-dependent properties of
+Nd2Fe14B permanent magnets by atomistic spin model
+simulations, Phys. Rev. B 99, 214409 (2019).
+[35] H. B. Callen and E. Callen, The present status of the tem-
+perature dependence of magnetocrystalline anisotropy,
+and the l(l+1)/2 power law, Journal of Physics and
+Chemistry of Solids 27, 1271 (1966).
+[36] C. E. Patrick, G. A. Marchant, and J. B. Staunton, Spin
+orientation and magnetostriction of Tb1−xDyxFe2 from
+first principles, Phys. Rev. Applied 14, 014091 (2020).
+[37] The jukkr developers, The J¨ulich KKR Codes (2020).
+[38] N. Papanikolaou, R. Zeller, and P. H. Dederichs, Con-
+ceptual improvements of the KKR method, Journal of
+Physics: Condensed Matter 14, 2799 (2002).
+[39] C. E. Patrick, S. Kumar, G. Balakrishnan, R. S. Ed-
+wards, M. R. Lees, L. Petit, and J. B. Staunton, Calcu-
+lating the magnetic anisotropy of rare-earth–transition-
+metal ferrimagnets, Phys. Rev. Lett. 120, 097202 (2018).
+[40] M. Loewenhaupt, I. Sosnowska, and B. Frick, Ground-
+state multiplet of rare-earth 3+ ions in R2Fe14B investi-
+gated by inelastic neutron scattering, Phys. Rev. B 42,
+3866 (1990).
+[41] M. Loewenhaupt and I. Sosnowska, Exchange and crystal
+
+7
+fields in R2Fe14B studied by inelastic neutron scattering
+(invited), Journal of Applied Physics 70, 5967 (1991).
+[42] R. Gr¨ossinger, R. Krewenka, X. Sun, R. Eibler, H. Kirch-
+mayr, and K. Buschow, Magnetic phase transitions
+and magnetic anisotropy in Nd2Fe14−xCoxB compounds,
+Journal of the Less Common Metals 124, 165 (1986).
+[43] M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa,
+Crystal-field analysis of the magnetization process in a
+series of Nd2Fe14B-type compounds, Phys. Rev. B 38,
+620 (1988).
+[44] K. Hummler and M. F¨ahnle, Full-potential linear-muffin-
+tin-orbital calculations of the magnetic properties of
+rare-earth–transition-metal intermetallics. ii. Nd2Fe14B,
+Phys. Rev. B 53, 3290 (1996).
+[45] Y. Tatetsu, Y. Harashima, T. Miyake, and Y. Gohda,
+Role of typical elements in Nd2Fe14X (X = B, C, N, O,
+F), Phys. Rev. Materials 2, 074410 (2018).
+[46] L. V. Pourovskii, J. Boust, R. Ballou, G. G. Eslava, and
+D. Givord, Higher-order crystal field and rare-earth mag-
+netism in rare-earth–Co5 intermetallics, Phys. Rev. B
+101, 214433 (2020).
+[47] O. Isnard, W. B. Yelon, S. Miraglia, and D. Fruchart,
+Neutron-diffraction study of the insertion scheme of hy-
+drogen in Nd2Fe14B, Journal of Applied Physics 78, 1892
+(1995).
+[48] Y.-K. Huang, C. Wu, Y. Chuang, F.-M. Yang, and
+F. De Boer, First-order magnetic transition in (nd, pr)
+2fe14b, Journal of the Less Common Metals 132, 317
+(1987).
+[49] F. Bolzoni, F. Leccabue, O. Moze, L. Pareti, M. Solzi,
+and A. Deriu, 3 d and 4 f magnetism in nd2fe14- x co
+x b and y2fe14- x co x b compounds, Journal of applied
+physics 61, 5369 (1987).
+[50] K. Stevens, Matrix elements and operator equivalents
+connected with the magnetic properties of rare earth ions,
+Proceedings of the Physical Society. Section A 65, 209
+(1952).
+[51] J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen,
+M. Du�lak,
+L. Ferrighi,
+J. Gavnholt,
+C. Glinsvad,
+V. Haikola, H. Hansen, et al., Electronic structure cal-
+culations with gpaw: a real-space implementation of the
+projector augmented-wave method, Journal of physics:
+Condensed matter 22, 253202 (2010).
+[52] D. S. G. Bauer, Development of a relativistic full-
+potential
+first-principles
+multiple
+scattering
+Green
+function method applied to complex magnetic textures
+of nanostructures at surfaces, Forschungszentrum J¨ulich
+http://publications.rwth-aachen.de/record/229375
+(2014).
+[53] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-
+dependent electron liquid correlation energies for local
+spin density calculations: a critical analysis, Canadian
+Journal of Physics 58, 1200 (1980).
+

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+Lifting-wing Quadcopter Modeling and Unified Control
+Quan Quan∗, Shuai Wang, and Wenhan Gao
+Hybrid unmanned aerial vehicles (UAVs) integrate the efficient forward flight of fixed-
+wing and vertical takeoff and landing (VTOL) capabilities of multicopter UAVs. This paper
+presents the modeling, control and simulation of a new type of hybrid micro-small UAVs,
+coined as lifting-wing quadcopters.
+The airframe orientation of the lifting wing needs to
+tilt a specific angle often within 45 degrees, neither nearly 90 nor approximately 0 degrees.
+Compared with some convertiplane and tail-sitter UAVs, the lifting-wing quadcopter has a
+highly reliable structure, robust wind resistance, low cruise speed and reliable transition flight,
+making it potential to work fully-autonomous outdoor or some confined airspace indoor. In the
+modeling part, forces and moments generated by both lifting wing and rotors are considered.
+Based on the established model, a unified controller for the full flight phase is designed. The
+controller has the capability of uniformly treating the hovering and forward flight, and enables
+a continuous transition between two modes, depending on the velocity command. What is more,
+by taking rotor thrust and aerodynamic force under consideration simultaneously, a control
+allocation based on optimization is utilized to realize cooperative control for energy saving.
+Finally, comprehensive Hardware-In-the-Loop (HIL) simulations are performed to verify the
+advantages of the designed aircraft and the proposed controller.
+Nomenclature
+𝑜e𝑥e𝑦e𝑧e
+=
+Earth-Fixed Coordinate Frame(eF )
+𝑜b𝑥b𝑦b𝑧b
+=
+Quadcopter-Body Coordinate Frame(bF )
+𝑜l𝑥l𝑦l𝑧l
+=
+Lifting-Wing Coordinate Frame(lF )
+𝑜w𝑥w𝑦w𝑧w
+=
+Wind Coordinate Frame(wF )
+ep
+=
+Position in eF
+ev
+=
+Velocity in eF
+bva, lva
+=
+Airspeed vector in bF and lF , respectively
+evw
+=
+Wind velocity in eF
+𝑉a
+=
+Airspeed
+∗Corresponding Author, is with the School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China.
+Email: [email protected].
+arXiv:2301.00730v1  [cs.RO]  2 Jan 2023
+
+𝜙, 𝜃, 𝜓
+=
+Euler angles in bF
+𝜔𝑥b, 𝜔𝑦b, 𝜔𝑧b
+=
+Angular velocity in bF
+𝛼
+=
+Angle of attack in lF
+𝛽
+=
+Sideslip angle in lF
+𝐶𝐿
+=
+Aerodynamic lift coefficient
+𝐶𝐷
+=
+Aerodynamic drag coefficient
+𝐶𝑚
+=
+Aerodynamic pitch moment coefficient
+𝐶𝑌
+=
+Aerodynamic lateral force coefficient
+𝐶𝑙
+=
+Aerodynamic roll moment coefficient
+𝐶𝑛
+=
+Aerodynamic yaw moment coefficient
+𝜅
+=
+Installation angle of the lifting wing
+𝜂
+=
+Installation angle of the motor
+𝑐
+=
+Mean chord of the lifting wing
+𝑏
+=
+Wingspan of the lifting wing
+𝑆
+=
+Area of the lifting wing
+I. Introduction
+A. Why Lifting-wing Quadcopter
+Unmanned aerial vehicles (UAVs) have attracted lots of recent attention due to their outstanding performances in
+many fields, such as aerial photography, precision farming, and unmanned cargo. According to [1], UAV platforms are
+currently dominated by three types: fixed-wing UAV, rotorcraft UAV, and their hybrid that integrates the advantages of
+the first two. The hybrid UAVs have the capability of Vertical Take-off and Landing (VTOL), which enables more
+accessible grounding or holding by hovering. This might be mandated by authorities in high traffic areas such as lower
+altitudes in the urban airspace. Furthermore, hybrid UAVs are categorized into two types: convertiplane and tail-sitter.
+A convertiplane maintains its airframe orientation in all flight modes, but a tail-sitter is an aircraft that takes off and
+lands vertically on its tail, and the entire airframe needs to tilt nearly 90◦ to accomplish forward flight [1, 2].
+In March 2015, Google announced that the tail-sitter UAV for a packet delivery service was scrapped, because it is
+still too difficult to control in a reliable and robust manner according to the conclusion came by the project leader [3].
+Some studies try to remedy this by newly designed controllers [4]. However, unlike this way, we will study a new type
+of hybrid UAV, coined as the lifting-wing quadcopter [5, 6], to overcome the difficulty Google’s Project Wing faced. A
+lifting-wing quadcopter is a quadcopter [7, 8] with a lifting wing installed at a specific mounting angle. During the
+flight, the quadcopter will provide thrust upward and forward simultaneously; and the lifting wing also contributes a
+2
+
+a) VertiKUL 2
+b)  Vespertilio
+c) RflyLW2
+d) Prime Air
+Fig. 1
+Prototypes of lifting-wing quadcopters.
+lifting force partially.
+As shown in Fig. 1, as far as we know, some prototypes of lifting-wing quadcopters in public can be found, such as
+VertiKUL2 by the University of Leuven (Fig. 1 (a), Sept 2015)[9], Vespertilio by the VOLITATION company(Fig. 1
+(b))[10], the latest version of the Prime Air delivery drone unveiled by Amazon(Fig. 1(d), Jun 2019)[11] and, RflyLW2
+by us(Fig. 1 (c)) [5, 6].
+The lifting-wing quadcopter is a new type of hybrid UAV because convertiplane and tail-sitter UAVs in their cruise
+phase work as fixed-wing UAVs, so the airframe orientation is defined as the head of the fixed-wing. But, the lifting-wing
+quadcopter is more like a quadcopter. The airframe orientation of the lifting wing needs to tilt a specific angle often
+within 45◦, neither nearly 90◦ (corresponding to tail-sitter UAVs) nor approximately 0◦(corresponding to convertiplane
+UAVs). Fig. 2 shows the full flight phase of the three VTOL UAVs. The design and performance evaluation of
+lifting-wing quadcopters have been studied extensively in [5, 6]. In order to make this paper self-contained, we briefly
+introduce the advantages of the lifting-wing quadcopter compared with some convertiplane and tail-sitter UAVs.
+• Highly reliable structure. It does not require extra transition actuators. This is a reliable structure by eliminating
+the need for complicated control.
+• Robust wind resistance. It has a shorter lifting wing compared with the corresponding fixed wing of convertiplanes
+and tail-sitter UAVs, because rotors can share the lift. Moreover, as shown in Fig. 4, it does not have a vertical
+rudder. Instead, this function is replaced by the yaw control of the quadcopter component. In order to improve the
+3
+
+融天
+福阳融天
+MOITATIJOVAsmirgc9iocobret①   
+②   
+③
+②   
+②   
+②   
+③
+④
+④
+①   
+①   
+②   
+③
+④
+②   
+(c) Lifting-wing quadcopter
+(b) Convertiplane UAV
+(a) Tail-sitter UAV
+① Vertical Take-off
+② Transition
+③ Level Flight
+④ Vertical Landing
+a) Tail-sitter UAV
+b) Convertiplane UAV
+c) Lifting-wing quadcopter
+①   
+②   
+③
+②   
+④
+②   
+③
+②   
+①   
+②   
+③
+②   
+④
+① Vertical Take-off
+② Transition
+③ Forward Flight
+④ Vertical Landing
+④
+①   
+Fig. 2
+Different flight modes of some VTOL UAVs.
+yaw control ability, the axes of rotors do not point only upward anymore as shown in Fig. 4(a). This implies that
+the thrust component by rotors can change the yaw directly rather than merely counting on the reaction torque of
+rotors. From the above, the wind interference is significantly reduced on the one hand; on the other hand, the yaw
+control ability is improved. As a result, it has better maneuverability and hover control to resist the disturbance of
+wind than those by tail-sitter and convertiplane UAVs.
+• Low cruise speed. It can make a cruise at a lower speed than that by convertiplanes and tail-sitter UAVs,
+meanwhile saving energy compared with corresponding quadcopters. This is very useful when a UAV flies in
+confined airspace such as a tunnel, where the high speed is very dangerous. Although current hybrid UAVs can
+have a big or long wing for low cruise speed, they cannot work in many confined airspace due to their long
+wingspan. However, the lifting-wing quadcopter can be small.
+• Reliable transition flight. When a tail-sitter UAV performs transition flight, the velocity and angle of attack will
+change dramatically, leading to complicated aerodynamics even stall. Unlike tail-sitter UAVs, the lifting-wing
+quadcopter only has to tilt a specific angle often smaller than 45◦ rather than 90◦. The airflow on the lifting wing
+is stable, and lift and drag are changed linearly with the angle of attack. These will avoid great difficulty (or say
+danger) in controlling within the full flight envelope.
+With the four features above, it is potential for the lifting-wing quadcopter to work fully-autonomously outdoors or
+in some confined airspace indoors replacing with corresponding quadcopters. The further comparisons with multicopter,
+4
+
+noitiansTno-seT IeoinsV
+gnibasI IeoinsV
+A ELTable 1
+Comparison of different VTOL UAVs.
+Endurance
+Reliability
+Wind Resistance at Hover
+Flight Range
+Multicopter tilt-rotor/wing convertiplane
+4
+2
+2
+4
+Multicopter dual-system convertiplane
+3
+4
+2
+3
+Multicopter tail-sitter
+4
+4
+1
+4
+Lifting-wing multicopter
+2
+4
+4
+2
+Multicopter
+1
+5
+5
+1
+Note: bigger number implies better.
+Endurance
+Flight Range
+Wind Resistance at Hover
+Reliability
+Multicopter tilt-rotor/wing 
+convertiplane
+Multicopter dual-system 
+convertiplane
+Multicopter tail-sitter
+Lifting-wing Multicopter 
+Multicopter 
+Fig. 3
+Comparison of different VTOL UAVs.
+tilt-rotor/wing convertiplane, multicopter dual-system convertiplane and multicopter tail-sitter [2] are summarized in
+Tab. 1 and Fig. 3. As shown, the lifting-wing quadcopter possesses the feature between current hybrid UAVs and
+quadcopters, and it is more like a quadcopter.
+B. Control of Current Hybrid UAVs
+The control of the lifting-wing quadcopter has the following two distinguishing features.
+• Unified control for full flight phases. Hybrid UAVs often have three different flight modes, including the hover,
+the transition flight, and the forward flight. By taking the multicopter tilt-rotor/wing convertiplane, multicopter
+dual-system convertiplane and multicopter-tail sitter, for example, their take-off and landing are controlled only by
+the quadcopter component, while the forward flight is controlled like a fixed-wing aircraft. The two control ways
+are very different, so the transition flight is challenging due to the nonlinearities and uncertainties. However, a full
+flight phase of the lifting-wing quadcopter always involves thrust by the quadcopter and aerodynamic force by the
+5
+
+19i2list1oto1itluM--
+9li-1oitlMliniwo-lititl
+19voH1690n6t2i291bniW
+豪0
+....p!!
+2
+Euqnlgucelifting wing. Therefore, the lifting-wing quadcopter can be considered under only the transition flight mode in the
+full flight phase (hover control here also will take the aerodynamic force into consideration due to wind on the
+lifting wing). As a result, a unified control is needed. Fortunately, the lifting-wing quadcopter only needs to tilt a
+specific angle often smaller than 45◦, rather than 90◦ like tail sitter UAVs. This reduces the possibility of having a
+stall.
+• Cooperative control for energy saving. The transition flight for current hybrid UAVs is very short, so not
+too much attention needs to pay to energy consumption in practice. However, it should be considered for the
+lifting-wing quadcopter as it is under the transition flight mode in the full flight phase. Cooperative control for
+energy saving is feasible. For example, roll control can be performed by both the quadcopter component and the
+ailerons by the lifting wing. Obviously, the aileron control is more energy-saving.
+Among the control phase of a hybrid UAV, the transition control is the most challenging issue [12], especially for
+tail-sitter UAVs. Since the actuators of tail-sitter UAVs are like those of lifting-wing quadcopters, the existing transition
+control of tail-sitter UAVs can be used for reference.
+(i) Trajectory open-loop control for transition flight.
+The open-loop trajectory tracking control is very straightforward. The principle is to make the UAV enter into
+another mode’s condition by focusing on controlling some variables like altitude other than the trajectory,
+then switch to the controller of the next mode. For example, increasing thrust and reducing its pitch angle at
+the same time can make a tail-sitter UAV enter into forwarding flight [13, 14]. The aim is to keep the altitude
+the same [14]. Because the transition time for tail-sitter UAVs is short, the trajectory will not change too
+much. Obviously, this method is inapplicable to lifting-wing quadcopters.
+(ii) Trajectory closed-loop control for transition flight.
+• Linearization method based on optimization. According to the trajectory and the model, the reference
+state and feedforward are derived by optimization in advance [15, 16]. Based on them, the linearization
+can be performed. With the resulting linear model, existing controllers against uncertainties and
+disturbance are designed [17, 18]. As for the lifting-wing quadcopter, this method is applicable when
+the model and transition trajectory are known prior. Furthermore, cooperative control of the quadcopter
+component or the ailerons of the lifting wing can be performed by taking energy-saving into optimization.
+However, in practice, the model is often uncertain as the payload is often changed, such as parcel
+delivery. Also, this method is not very flexible due to that the trajectory has to be known a priori.
+• Nonlinear control method. One way is to take all aerodynamic forces as disturbances, and only the
+quadcopter component works for the flight transition [19, 20]. This requires that the quadcopter has
+a strong control ability to reject the aerodynamic force. Another way takes the aerodynamic force
+into consideration explicitly to generate a proper attitude [21, 22]. How to cooperatively control the
+6
+
+quadcopter component and the actuators of fixed-wing is not found so far. This is because, we guess,
+the transition flight is often short, and more attention is paid to making the UAV stable by reducing the
+possibility of stall rather than optimization.
+As shown above, the linearization method based on optimization is somewhat not flexible, but cooperative control for
+energy saving can be performed in an open-loop manner. The nonlinear control method is flexible but not considering
+how to control cooperatively for energy saving.
+C. Our Work and Contributions
+In this paper, we will consider designing a unified controller for the full flight phase of a lifting-wing quadcopter.
+What is more, the quadcopter component and the ailerons of the lifting wing work cooperatively to save energy. First,
+we build the model of the lifting-wing quadcopter. Unlike the tail-sitter UAV, it does not have a rudder, and its tilted
+rotors will generate force components on the XY-plane in the quadcopter-body coordinate frame (traditional quadcopters
+do not have the force component on the XY-plane). Because of this, the translational dynamic involves five control
+variables, namely three-dimensional force in the quadcopter-body coordinate frame and two Euler angles (pitch and roll
+angles), further by considering the aerodynamic force determined by Euler angles. However, it is difficult and a bit
+too early to determine the five control variables according to the three-dimensional desired acceleration, because it is
+hard to obtain the bounds of these control variables. An improper choice may not be realized by actuators. To this
+end, we only choose the 𝑜b𝑧b force (the main force component) in the quadcopter-body coordinate frame and two Euler
+angles (pitch and roll angles) to determine the desired acceleration uniquely, leaving the other two force components as
+a lumped disturbance. This adopts the controlling idea of quadcopters [7, 23], but the computation method is different
+due to the existence of the aerodynamic force. With the determined Euler angles, moments are further determined in the
+lifting wing coordinate frame. So far, the unified control for the full flight phase is accomplished. Finally, we will utilize
+the control allocation to realize cooperative control for energy saving. The 𝑜b𝑧b force and three-dimensional moments
+will be realized by four rotors and two ailerons. This is why we have the freedom to optimize the allocation for saving
+energy. The principle behind this is to make the aerodynamic force (two ailerons) undertake the control task as much as
+possible because aileron control is more energy-saving than rotor control. As a result, cooperative control for energy
+saving is accomplished.
+The contributions of this paper are: (i) establish the model of a lifting-wing quadcopter for the first time; (ii) a
+unified controller design for the full flight phase of the lifting-wing quadcopter; (iii) control allocation for energy-saving
+performance. Comprehensive HIL simulation experiments are performed to show (i) the proposed lifting-wing
+quadcopter is more energy-saving with aileron; (ii) synthesizing the angular rate command from the coordinated turn in
+high-speed flight can reduce sideslip; (iii) the transition phase of the proposed lifting-wing quadcopter is significantly
+better than the tail-sitter and UAVs.
+7
+
+ex
+ey
+bx
+by
+
+c
+b
+x
+d
+y
+d
+1
+2
+3
+4
+a) Front view
+b) Left view
+c) Top view
+d) 3D view
+ey
+ez
+bz
+by
+
+
+CoG
+ex
+lx
+lz
+bz
+
+
+bx
+ar
+
+al
+
+CoG
+CoG
+
+al
+
+ar
+
+Fig. 4
+Coordinate frames and nomenclatures.
+II. Coordinate Frame
+A lifting-wing quadcopter is divided into two components, the lifting-wing component and the quadcopter component
+as shown in Fig. 4. According to these, the following coordinate frames are defined.
+A. Earth-Fixed Coordinate Frame (eF )
+The earth-fixed coordinate frame 𝑜e𝑥e𝑦e𝑧e is an inertial frame. The 𝑜e𝑧e axis points perpendicularly to the ground,
+and the 𝑜e𝑥e axis points to a certain direction in the horizontal plane. Then, the 𝑜e𝑦e axis is determined according to the
+right-hand rule. This frame is fixed, the initial position of the lifting-wing quadcopter or the center of the Earth is often
+set as the coordinate origin 𝑜e.
+B. Quadcopter-Body Coordinate Frame (bF )
+The quadcopter-body coordinate frame 𝑜b𝑥b𝑦b𝑧b is fixed to the quadcopter component of a lifting-wing quadcopter.
+The Center of Gravity (CoG) of the lifting-wing quadcopter is chosen as the origin 𝑜b of bF . The 𝑜b𝑥b axis points to
+the nose direction in the symmetric plane of the quadcopter. The 𝑜b𝑧b axis is in the symmetric plane of the quadcopter,
+pointing downward, perpendicular to the 𝑜b𝑥b axis. The 𝑜b𝑦b axis is determined according to the right-hand rule.
+8
+
+C. Lifting-Wing Coordinate Frame (lF )
+The lifting-wing coordinate frame 𝑜l𝑥l𝑦l𝑧l is fixed to the lifting-wing component. The origin 𝑜l of lF is also set at
+the CoG of the lifting-wing quadcopter. The 𝑜l𝑥l axis is in the symmetric plane pointing to the nose of the lifting wing.
+The 𝑜l𝑧l axis is in the symmetric plane of the lifting wing, pointing downward, perpendicular to the 𝑜l𝑥l axis, and the
+𝑜l𝑦l axis is determined according to the right-hand rule. The installation angle of the lifting wing, that is the angle
+between the 𝑜l𝑥l axis and the 𝑜l𝑥l𝑦l plane, is denoted by 𝜅 ∈ R as shown in Fig. 4(b).
+D. Wind Coordinate Frame (wF )
+The origin 𝑜w of the wind coordinate frame 𝑜w𝑥w𝑦w𝑧w is also at the CoG of the lifting-wing quadcopter. The 𝑜w𝑥w
+axis is aligned with the airspeed vector. The 𝑜w𝑧w axis is in the symmetric plane of the lifting wing, pointing downward,
+perpendicular to the 𝑜w𝑥w axis, and the 𝑜w𝑦w axis is determined according to the right-hand rule. The angle of attack
+(AoA), denoted by 𝛼 ∈ R, is defined as the angle between the projection of the airspeed vector on the 𝑜l𝑥l𝑧l plane and
+the 𝑜l𝑥l as shown in Fig. 4(b). The sideslip angle, denoted by 𝛽 ∈ R, is defined as the angle between the airspeed vector
+and the 𝑜l𝑥l𝑧l plane as shown in Fig. 4(c).
+To convert the aerodynamic forces and moments acting on frame wF and lF to bF respectively, two rotation
+matrices are defined as followed:
+Rb
+w(𝜆) =
+��������
+cos 𝜆 cos 𝛽
+− cos 𝜆 sin 𝛽
+sin 𝜆
+sin 𝛽
+cos 𝛽
+0
+− sin 𝜆 cos 𝛽
+sin 𝜆 sin 𝛽
+cos 𝜆
+��������
+, Rb
+l =
+��������
+cos 𝜅
+0
+sin 𝜅
+0
+1
+0
+− sin 𝜅
+0
+cos 𝜅
+��������
+,
+where 𝜆 = 𝜅 − 𝛼. And the rotation matrix Re
+b maps a vector from frame bF to eF , defined by
+Re
+b =
+��������
+cos 𝜃 cos 𝜓 − sin 𝜃 sin 𝜙 sin 𝜓
+− sin 𝜓 cos 𝜙
+cos 𝜓 sin 𝜃 + cos 𝜃 sin 𝜙 cos 𝜓
+sin 𝜃 sin 𝜙 cos 𝜓 + cos 𝜃 sin 𝜓
+cos 𝜙 cos 𝜓
+sin 𝜓 sin 𝜃 − cos 𝜙 cos 𝜃 sin 𝜙
+− cos 𝜙 sin 𝜃
+sin 𝜙
+cos 𝜙 cos 𝜃
+��������
+.
+III. MODELING
+A. Assumptions
+For the sake of model simplicity, the following assumptions are made:
+Assumption 1. The body structure is rigid and symmetric about the 𝑜l𝑥l𝑦l plane.
+Assumption 2. The mass and the moments of inertia are constant.
+Assumption 3. The geometric center of the lifting-wing quadcopter is the same as the CoG.
+Assumption 4. The aircraft is only subjected to gravity, aerodynamic forces, and the forces generated by rotors.
+9
+
+B. Flight Control Rigid Model
+By Assumptions 1-2, the Newton’s equation of motion is applied to get the translational motion as follows
+e �p = ev
+e�v = Re
+b
+bf
+𝑚
+(1)
+where ep =
+�
+𝑝𝑥e 𝑝𝑦e 𝑝𝑧e
+�T and ev =
+�
+𝑣𝑥e 𝑣𝑦e 𝑣𝑧e
+�T are the position and velocity expressed in frame eF respectively;
+𝑚 is the mass, bf is the total force acting on the airframe expressed in frame bF .
+To facilitate attitude control and combine the control characteristics of the rotor and lifting wing, the rotational
+dynamics is carried out in frame lF . It is given by Euler’s equation of motion as
+�Re
+l = Re
+l
+�l𝝎
+�
+×
+J · l �𝝎 = lm − l𝝎 × (J·l𝝎)
+(2)
+where the rotational matrix is derived by Re
+l = Re
+b Rb
+l , Rb
+l being a constant matrix; lm is the total moment acting
+on the airframe expressed in frame lF , l𝝎 =
+�
+𝜔𝑥l 𝜔𝑦l 𝜔𝑧l
+�T is the angular velocity in frame lF ,
+�l𝝎
+�
+× denotes the
+skew-symmetric matric
+�l𝝎
+�
+× =
+��������
+0
+−𝜔𝑧l
+𝜔𝑦l
+𝜔𝑧l
+0
+−𝜔𝑥l
+−𝜔𝑦l
+𝜔𝑥l
+0
+��������
+,
+and J ∈ R3×3 is the inertia matrix given by
+J =
+��������
+𝐽𝑥
+0
+−𝐽𝑥𝑧
+0
+𝐽𝑦
+0
+−𝐽𝑥𝑧
+0
+𝐽𝑧
+��������
+.
+C. Forces and Moments
+By Assumptions 3-4, the total forces and moments acting on the UAV are decomposed into three parts: the
+aerodynamic forces and moments acting on the airframe (fa and ma), the forces and moments generated by rotors (fr and
+mr), and the gravitational forces f𝑔, where f𝑔 = [0 0 𝑔]T, 𝑔 is the gravitational acceleration. The front two types of
+forces and moments will be described detailly in the following two subsections.
+10
+
+Lf
+Df
+Yf
+a
+v
+x
+av
+z
+av
+y
+av
+1T
+2T
+3T
+4T
+
+lo
+ly
+lx
+lz
+
+
+Fig. 5
+Forces act on a lifting-wing quadcopter.
+1. Forces and Moments in the Quadcopter Component
+In the quadcopter part, the thrust and torque produced by one rotor are given by
+𝑇𝑖 = 𝐾 𝑓 𝜛𝑖2, 𝑀𝑖 = 𝐾𝑚𝜛𝑖2 = 𝐾𝑚
+𝐾 𝑓
+𝑇𝑖
+(3)
+where 𝐾 𝑓 > 0 is the lift force coefficient, 𝐾𝑚 > 0 is the drag torque coefficient, and 𝜛𝑖 is the angular rate of the ith
+rotor, 𝑖 = 1, 2, 3, 4. In order to improve the controllability during performing yaw, an installation angle 𝜂 is set as shown
+in Fig. 4(a). The left motors tilt to the positive left and right motors tilt to the positive right.
+Because of the installation angle 𝜂, the forces and moments produced by rotors are expressed by the thrust on each
+propeller 𝑇𝑖 as
+�������������
+𝑓𝑟𝑦
+𝑓𝑟𝑧
+𝑚𝑟𝑥
+𝑚𝑟𝑦
+𝑚𝑟𝑧
+�������������
+=
+�������������
+sin 𝜂
+− sin 𝜂
+− sin 𝜂
+sin 𝜂
+− cos 𝜂
+− cos 𝜂
+− cos 𝜂
+− cos 𝜂
+−𝑑𝑦 cos 𝜂
+𝑑𝑦 cos 𝜂
+𝑑𝑦 cos 𝜂
+−𝑑𝑦 cos 𝜂
+𝑑𝑥 cos 𝜂
+−𝑑𝑥 cos 𝜂
+𝑑𝑥 cos 𝜂
+−𝑑𝑥 cos 𝜂
+𝐾1
+𝐾1
+−𝐾1
+−𝐾1
+�������������
+����������
+𝑇1
+𝑇2
+𝑇3
+𝑇4
+����������
+,
+(4)
+where 𝐾1 = 𝐾𝑚
+�𝐾 𝑓 + 𝑑𝑥 sin 𝜂, 𝑑𝑥 and 𝑑𝑦 are the components of the distance from the center of the lifting-wing
+quadcopter to a propeller on the 𝑜b𝑥b𝑦b plane, as shown in Fig. 4(c).
+2. Forces and Moments in the Lifting-wing Component
+The aerodynamic forces and moments acting on the lifting wing are mainly generated by the lifting wing itself and
+the ailerons at the trailing edge, as shown in Fig. 5.
+11
+
+Let evw be the wind velocity in eF . Then
+bva = bv − (Re
+b)T · evw.
+(5)
+Thus, the airspeed vector lva =
+�
+𝑣a𝑥 𝑣a𝑦 𝑣a𝑧
+�T and airspeed 𝑉𝑎 are defined as
+lva = (Rb
+l )T · bva,
+(6)
+𝑉a =
+√︃
+𝑣2a𝑥 + 𝑣2a𝑦 + 𝑣2a𝑧.
+(7)
+The aerodynamic angles 𝛼 and 𝛽 are defined as
+𝛼 = tan−1( 𝑣a𝑧
+𝑣a𝑥
+), 𝛽 = sin−1(
+𝑣a𝑦
+𝑉a
+).
+(8)
+In the longitudinal plane, lift, drag, and pitching moment acting on the lifting-wing body are given by
+𝑓𝐿 = 𝑄𝑆(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒)
+𝑓𝐷 = 𝑄𝑆(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒)
+𝑚 = 𝑄𝑆𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒).
+(9)
+The lateral force and the roll and yaw moments acting on the lifting-wing body are given by
+𝑓𝑌 = 𝑄𝑆(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎)
+𝑙 = 𝑄𝑆𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎)
+𝑛 = 𝑄𝑆𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎)
+(10)
+where 𝑄 = 1
+2 𝜌𝑉2
+𝑎; 𝐶𝐿, 𝐶𝐷, 𝐶𝑚,𝐶𝑌 , 𝐶𝑙 and 𝐶𝑛 are nondimensional aerodynamic coefficients, 𝐶𝐿 𝛿𝑒, 𝐶𝑚𝛿𝑒, 𝐶𝐷 𝛿𝑒,𝐶𝑌 𝛿𝑎,
+𝐶𝑛𝛿𝑎 and 𝐶𝑙𝛿𝑎 are control derivative; 𝑆 is the area of the lifting wing, 𝑐 is the mean chord of the lifting wing, 𝑏 is the
+wingspan of the lifting-wing aircraft, 𝛿𝑒 and 𝛿𝑎 are calculated using the right and the left aileron (𝛿𝑎𝑟 and 𝛿𝑎𝑙, as shown
+in Fig. 4(d)) as
+�
+𝛿𝑒
+𝛿𝑎
+�
+=
+�
+1
+1
+−1
+1
+� �
+𝛿𝑎𝑟
+𝛿𝑎𝑙
+�
+.
+(11)
+The external forces and moments are summarized as
+12
+
+Table 2
+Lifting-wing quadcopter structure parameters, lift force and drag torque coefficients
+𝑚
+Aircraft mass
+1.92 kg
+𝜅
+Installation angle of lifting wing
+34 deg
+𝜂
+Installation angle of motor
+10 deg
+𝑑𝑥
+The distance from 𝑜𝑏𝑥𝑏 to a propeller
+0.25 m
+𝑑𝑦
+The distance from 𝑜𝑏𝑦𝑏 to a propeller
+0.2125 m
+[𝐽𝑥𝑥 𝐽𝑦𝑦 𝐽𝑧𝑧]
+Moment of inertia
+[5.12 5.54 7.6] × 10−2 kg · m2
+𝑏
+Wingspan of the lifting-wing aircraft
+0.94 m
+𝑐
+Mean chord of the lifting wing
+0.17 m
+𝐾𝑚
+Drag moment coefficient
+5.875e-07 kg · m2
+𝐾 𝑓
+Lift force coefficient
+2.824e-05 kg · m2
+bf =
+��������
+0
+𝑓𝑟𝑦
+𝑓𝑟𝑧
+��������
++ 𝑄𝑆Rb
+w
+��������
+−(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒)
+(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎)
+−(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒)
+��������
++ 𝑚Rb
+ef𝑔
+(12)
+lm = Rl
+b
+��������
+𝑚𝑟𝑥
+𝑚𝑟𝑦
+𝑚𝑟𝑧
+��������
++ 𝑄𝑆
+��������
+𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎)
+𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒)
+𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎)
+��������
+.
+(13)
+The structure parameters, lift force and drag torque coefficients of the lifting-wing quadcopter are given in Tab.2.
+IV. CONTROLLER DESIGN
+The successive loop closure is a common control architecture for UAVs [24], which consists of an outer-loop
+controlling the position and an inner-loop for attitude, as illustrated in Fig. 6. The basic idea behind successive
+loop closure is to close several simple feedback loops in succession around the open-loop plant dynamics rather than
+designing a single control system. The position controller receives the desired position and then computes the desired
+acceleration. Then the desired acceleration is mapped to the collective thrust and attitude. The attitude command is sent
+to the inner-loop, while the thrust command skips directly to the control allocation. To facilitate the control experiment
+step by step, the attitude can also be commanded by the pilot. Furthermore, the attitude controller receives the desired
+attitude and generates the desired moment. Finally, the control allocation algorithm distributes the moment command
+from the inner-loop and the direct force command from the outer-loop to corresponding ailerons and rotors.
+13
+
+Position 
+Controller
+Attitude 
+Controller
+Control 
+Allocation
+d
+m
+Lifting-wing 
+Quadcopter 
+Dynamics
+,
+T δ
+d
+p
+Mannual
+Inputs
+Signal
+Distribution
+d
+Θ
+df
+d
+Θ
+df
+df
+d
+Θ
+a
+,
+p v
+av
+q
+Fig. 6
+Control structure
+A. Controller Design Model
+Since 𝛼, 𝛽 are not easy to obtain, we consider that 𝛼 ≈ 𝜅 + 𝜃 and 𝛽 ≈ 0. The translational dynamic involves five
+control variables, namely three-dimensional forces in frame bF and two Euler angles (pitch and roll angles). However, it
+is a bit too early to determine the five control variables according to the three-dimensional desired acceleration because
+it is hard to obtain the bounds of these control variables. An improper choice may not be realized by ailerons. To
+this end, we only choose 𝑓𝑧 (the main force component) in frame bF and two Euler angles (pitch and roll angles) to
+determine the desired acceleration uniquely, and the desired yaw angle is directly specified, leaving 𝑓𝑦 as a disturbance.
+This adopts the controlling idea of quadcopters[7, 23], but the computation method is different due to the existence of
+the aerodynamic force. According to the idea above, we rewrite the system Eqs. (1) and (2) in the form as
+e �p = ev
+e�v = u + g + d1
+�Re
+l = Re
+l
+�l𝝎
+�
+×
+l �𝝎 = J−1 · lm + d2
+.
+(14)
+Here
+u =
+Re
+b
+𝑚
+����
+�
+��������
+0
+0
+− 𝑓𝑧
+��������
++ 𝑄𝑆Rb
+w
+��������
+−𝐶𝐷
+0
+−𝐶𝐿
+��������
+����
+�
+,
+and d1, d2 are disturbances, where
+d1 =
+Re
+b
+𝑚
+����
+�
+��������
+0
+𝑓𝑦
+0
+��������
++ 𝑄𝑆Rb
+w
+��������
+−𝐶𝐷 𝛿𝑒𝛿𝑒
+(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎)
+−𝐶𝐿 𝛿𝑒𝛿𝑒
+��������
+����
+�
+d2 = −J−1 · l𝝎 × (J · l𝝎).
+14
+
+B. Position Control
+Given a twice differentiable trajectory pd(𝑡), in order to satisfy lim
+𝑡→∞ ∥ep(𝑡) − pd(𝑡)∥ = 0, the desired ud for Eq. (14)
+can be designed as a PID controller in the form
+ud = −g + �pd − KPd (ev − �pd) − KPp (ep − pd) − KPi
+∫
+(ep − pd) d𝑠
+(15)
+where KPp, KPi, KPd ∈ R3×3 are diagonal matrices acting as control gains. The left work is to determine desired thrust
+by rotors 𝑓d ∈ Ω 𝑓 and 𝜃d, 𝜙d ∈ Ω𝑎 such that
+( 𝑓d, 𝜃d, 𝜙d) =
+arg min
+𝑓𝑧 ∈Ω 𝑓 ,𝜃,𝜙∈Ω𝑎
+∥u ( 𝑓𝑧, 𝜃, 𝜙) − ud∥
+(16)
+where Ω 𝑓 is a set to confine the force, and Ω𝑎 is a set to confine the pitch and roll. In order to reduce drag, the vehicle’s
+nose should be consistent with the current direction of the vehicle velocity, that is
+𝜓d = tan−1
+� 𝑣𝑦e
+𝑣𝑥e
+�
+.
+(17)
+The attitude command can also be given by the pilot, in case the position controller fails with GPS denied, as shown
+in Fig. 6. Finally, the desired attitude is given as 𝚯d = [𝜙d 𝜃d 𝜓d]T.
+C. Attitude Control
+The attitude controller generates the desired moment from the output of the position controller or the attitude given
+by the pilot, as shown in Fig. 6. As far as we know, studies about the hybrid UAV hardly consider the lateral control,
+such as turning right or left. As for the considered UAV, the control on yaw is quite different between the multicopter
+mode and the fixed-wing mode. To establish a unified control, the attitude control is performed on the lifting-wing
+frame lF , so l𝚯d = [𝜙d 𝜃d + 𝜅 𝜓d]T.
+1. Basic Attitude Control
+The attitude error is presented in the form of quaternion based on which the corresponding controller is designed.
+This can guarantee a uniform and good convergence rate for all initial attitude errors [25]
+qe = q∗
+d ⊗ qe
+l =
+�
+𝑞e0 𝑞e1 𝑞e2 𝑞e3
+�T,
+(18)
+15
+
+where qd is transformed from l𝚯d with ‘ZXY’ rotational sequence, (·)∗ is the conjugate of a quaternion, and ⊗ is the
+quaternion product. Then qe is transformed into the axis-angle form qe =
+�
+cos 𝜗
+2 𝝃T
+e sin 𝜗
+2
+�T by
+𝜗 = wrap𝜋
+�2acos �𝑞e0
+��
+𝝃e =
+�
+[0 0 0]T,
+𝜃 = 0
+sign(𝑞e0)
+𝜗
+sin 𝜗/2
+�
+𝑞e1 𝑞e2 𝑞e3
+�T,
+𝜃 ≠ 0.
+(19)
+The function wrap𝜋(𝜗) constrains the 𝜗 in [−𝜋 𝜋] to ensure the shortest rotation path. To eliminate the attitude error,
+the attitude control is designed as
+l𝝎ac = sat �K𝚯p𝝃e, 𝝎min, 𝝎max
+�
+(20)
+where K𝚯p ∈ R3×3 is the diagonal matrix acting as the control gain, 𝝎min and 𝝎max ∈ R3 are the minimum and maximum
+angular control rates, the function sat (x, xmin, xmax) is defined as
+sat (x, xmin, xmax) ≜
+��������
+sat �𝑥1, 𝑥1,min, 𝑥1,max
+�
+...
+sat �𝑥𝑛, 𝑥𝑛,min, 𝑥𝑛,max
+�
+��������
+, sat �𝑥𝑘, 𝑥𝑘,min, 𝑥𝑘,max
+� ≜
+�����
+�����
+𝑥𝑘,min,
+𝑥𝑘 < 𝑥𝑘,min
+𝑥𝑘,max,
+𝑥𝑘 > 𝑥𝑘,max
+𝑥𝑘,
+else
+.
+(21)
+2. Lateral Compensation
+When the UAV is in high-speed flight, a roll command given to track a specified trajectory will cause a lateral skid.
+In order to reduce the sideslip angle when the UAV turns at a high speed, the coordinated turn should be considered. In
+lF frame, if it is assumed that there is no wind, the coordinated turn equation is expressed as [24]
+�𝜓d = 𝑔 tan 𝜙
+𝑉𝑎
+.
+(22)
+It should be noted that the Euler angles are the attitude presentation between lF . So the desired yaw rate generated by
+coordinated turn is
+l𝜔ct = �𝜓d cos 𝜃 cos 𝜙.
+(23)
+3. Angular Rate Command Synthesis
+In the lifting-wing coordinated turn, 𝑉𝑎 being zero makes no sense. In addition, considering that coordinated turn
+should not be used when airspeed is small, a weight coefficient related to airspeed is added. So the desired angular rates
+are rewritten as
+l𝜔d =
+�
+𝜔dac,𝑥 𝜔dac,𝑦
+�
+𝜔dac,𝑧 + 𝑤·l𝜔ct
+��
+(24)
+16
+
+where 𝑤 = sat
+�
+𝑉𝑎−𝑉min
+𝑉max−𝑉min , 0, 1
+�
+. When the airspeed is slower than 𝑉min, the desired yaw rate is completely decided by
+the basic attitude controller. In contrast, when l𝜔d𝑧 = 0 and the airspeed reaches the specified value 𝑉max, the desired
+yaw rate is completely decided by the coordinated turn.
+4. Attitude Rate Control
+To eliminate the attitude rate error, the controller is designed as
+lmd = sat
+�
+J(−K𝝎p(l𝝎 − l𝝎d) − md,I − K𝝎d(l �𝝎 − l �𝝎d)), −md,max, md,max
+�
+(25)
+where md,I = sat
+�
+K𝝎i
+∫
+(l𝝎 − l𝝎d)d𝑠, − md,Imax, md,Imax
+�
+, K𝜔p, K𝜔i, K𝝎d ∈ R3×3 are diagonal matrices acting as
+control gains, md,Imax is maximum amplitude of integral action, md,max is the maximum moment generated by actuators.
+So far, we have obtained 𝑓d and lmd, which will be further realized by 𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙.
+D. Control Allocation
+The lifting-wing quadcopter is an over-actuated aircraft, which provides six independent control inputs, namely
+𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙 to meet a specific thrust 𝑓d ∈ R and moment demand lmd ∈ R3. A method of control allocation
+based on optimization is proposed. Recalling the control in translational dynamic, if we determined the three-dimensional
+force in bF and two Euler angles (pitch and roll angles) by an optimization before, then the six control variables (plus
+desired yaw angle) will be determined by six actuators uniquely. If so, however, the optimization in the control of the
+translational dynamic is not related to energy-saving directly. This is why we only choose the 𝑜b𝑧b force (the main force
+component) and two Euler angles (pitch and roll angles) to determine the desired acceleration as in Eq.(16).
+First, Eqs. (12) and (13) are rearranged as
+����������
+𝑓𝑧
+l𝑚𝑥 − 𝑄𝑆𝑏𝐶𝑙
+l𝑚𝑦 − 𝑄𝑆𝑐𝐶𝑚
+l𝑚𝑧 − 𝑄𝑆𝑏𝐶𝑛
+����������
+����������������������������������������
+uv
+=
+����������
+− cos 𝜂
+− cos 𝜂
+− cos 𝜂
+− cos 𝜂
+0
+0
+−𝐾2
+𝐾3
+𝐾2
+−𝐾3
+−𝑄𝑆𝑏𝐶𝑙 𝛿𝑎
+𝑄𝑆𝑏𝐶𝑙 𝛿𝑎
+𝑑𝑥 cos 𝜂
+−𝑑𝑥 cos 𝜂
+𝑑𝑥 cos 𝜂
+−𝑑𝑥 cos 𝜂
+𝑄𝑆𝑐𝐶𝑚𝛿𝑒
+𝑄𝑆𝑐𝐶𝑚𝛿𝑒
+−𝐾5
+𝐾4
+𝐾5
+−𝐾4
+−𝑄𝑆𝑏𝐶𝑛𝛿𝑎
+𝑄𝑆𝑏𝐶𝑛𝛿𝑎
+����������
+������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
+B
+����������������
+𝑇1
+𝑇2
+𝑇3
+𝑇4
+𝛿𝑎𝑟
+𝛿𝑎𝑙
+����������������
+����������
+𝜹
+(26)
+where 𝐾2 = 𝑑𝑦 cos 𝜂 cos 𝜅 +𝐾1 sin 𝜅, 𝐾3 = 𝑑𝑦 cos 𝜂 cos 𝜅 −𝐾1 sin 𝜅, 𝐾4 = 𝑑𝑦 cos 𝜂 sin 𝜅 +𝐾1 cos 𝜅, 𝐾5 = 𝑑𝑦 cos 𝜂 sin 𝜅 −
+𝐾1 cos 𝜅. The value 𝜹 is the control input of the actuator , uv is the virtual control, B is the control efficiency matrix.
+As shown in the Eq.(26), rank (B) = 4, the dimension of 𝜹 is 6, which is higher than that of uv, so Eq.(26) has a
+minimum norm solution. To take the control priority of the uv components, actuator priority of 𝜹 and actuator saturation
+17
+
+under consideration, the control allocation is formulated to be an optimization problem as
+min
+��Wu
+�B𝜹 − uv,d
+���2 + 𝛾
+��W𝜹
+�𝜹 − 𝜹p
+���2
+s.t.
+𝜹− ⩽ 𝜹 ⩽ ¯𝜹
+(27)
+where uv,d =
+�
+𝑓d l𝑚d𝑥 − 𝑄𝑆𝑏𝐶𝑙 l𝑚d𝑦 − 𝑄𝑆𝑐𝐶𝑚 l𝑚d𝑧 − 𝑄𝑆𝑏𝐶𝑛
+�T is the desired virtual control, 𝜹p is the preferred
+control vector which will be specified later, Wu ∈ R6×6 is a positive definite weighting matrix that prioritizes the
+commands in case the desired virtual input uv,d cannot be achieved, W𝜹 ∈ R6×6 is a positive definite weighting matrix
+that prioritizes the different actuators, 𝜹− = max(𝜹min, 𝜹l − Δ𝜹) , ¯𝜹 = min(𝜹max, 𝜹l + Δ𝜹) are lower and upper bounds at
+each sampling instant of actuators, 𝜹min and 𝜹max ∈ R6 are actuator position limits, Δ𝜹 ∈ R6 is the rate limit, and 𝜹l is
+the last value of 𝜹.
+There are two optimization objectives in Eq.(27), namely
+��Wu
+�B𝜹 − uv,d
+���2 and
+��W𝜹
+�𝜹 − 𝜹p
+���2. The first one is
+the primary objective of minimizing the slack variables weighted by Wu, so the weighting factor 𝛾 is often chosen to be
+a small value. In many cases, the preferred control vector is set to the last value of 𝜹, 𝜹p = 𝜹l. But, in order to save
+energy, we prefer to use the aerodynamic force because the change of rotor force implies the motor’s rotational speed
+change, which is power-hungry. According to the consideration above, we can give more weight to 𝛿𝑎𝑟 and 𝛿𝑎𝑙 and set
+first four elements of 𝜹p to 𝑇1+𝑇2+𝑇3+𝑇4
+4
+, and last two elements to that of 𝜹l.
+The hardware resources of the flight control are very limited, so it is very important to ensure the real-time operation
+of the proposed algorithm. Several different algorithms, like redistributed pseudoinverse, interior-point methods, and
+active set methods have been proposed to solve the constrained quadratic programming problem. Among them, active
+set methods[26] perform well in the considered control allocation problems, because they have the advantage that their
+initialization can take advantage of the solution from the previous sample (known as the warm start), which is often a
+good guess for the optimal solution at the current sample. This can reduce the number of iterations needed to find the
+optimal solution in many cases. The study [27] shows that the computational complexity of the active set methods is
+similar to the redistributed pseudoinverse method and the fixed-point algorithm, but the active set methods produce
+solutions with better accuracy.
+V. Simulation Experiments
+In order to verify the three main advantages of the designed aircraft: control performance, energy saving and reliable
+transition flight, several experiments are conducted. The verification of these performances is mainly determined by two
+factors, namely, with and without ailerons, and with and without coordinated turn.
+Therefore, experiments are primarily composed of two parts. One is the comparison of control performance, energy
+saving and transition flight with and without aileron. And the other is the comparison of control performance with and
+18
+
+without coordinated turn, but both with aileron. In addition, the transition flight of three different VTOL vehicles, as
+shown in Fig. 2, is analyzed in the HIL simulation environment.
+A. Simulation Platform
+The HIL simulation is carried out in the RflySim platform [28–30], which provides CopterSim simulator and
+supports Pixhawk/PX4 autopilot.When perform the HIL simulation, CopterSim sends sensor data such as accelerometer,
+barometer, magnetometer, which is generated by a mathematical model, to the Pixhawk system via a USB serial port.
+The Pixhawk/PX4 autopilot will receive the sensors data for state estimation by the EKF2 filter and send the estimated
+states to the controller through the internal uORB message bus as the feedback. Then the controller sends the control
+signal of each motor as output back to CopterSim. Thereby a close-loop is established in the HIL simulation. Compared
+with the numerical simulation, the control algorithm is deployed and run in a real embedded system as the real flight
+does. After HIL simulation, the controller will be directly deployed to a real vehicle and further verification experiments
+will be performed.
+PixHawk Autopilot Hardware System
+HIL Real-Time Simulation System
+Control 
+Signals
+Sensor 
+Data
+Multicopter Simulation + 3D Display + Fault 
+Injection
+Fig. 7
+HIL simulation of RflyLW2.
+The lifting-wing quadcopter model is built in MATLAB/Simulink according to Eqs. (1),(2) (12), (13). Besides, the
+dynamics of the motor and servo are modeled as first-order transfers as follows:
+𝜛 =
+1
+𝑇m𝑠 + 1 (𝐶m𝜎𝑖 + 𝜛b), 𝛿a =
+1
+𝑇a𝑠 + 1 (𝐶a𝜎𝑗 + 𝛿b), 𝑖 = 1, 2, 3, 4, 𝑗 = 5, 6
+(28)
+where 𝜎𝑖 ∈ [0, 1] is 𝑖th throttle command, 𝑇m and 𝑇a are the time constant parameter of motor and servo response, 𝐶m,
+𝐶a, 𝛿b and 𝜛b are constant parameters. The sensors used in HIL simulation include IMU, magnetometer, barometer,
+GPS and airspeed meter, are modeled reference to [7].
+Due to the vehicle’s VTOL capabilities, the aerodynamic parameters must not only be modeled up to the stall AoA,
+19
+
+L
+inrgps中
+9 0
+0
+1  [0
+1 4= [0
+X0
+EX+ BI (CCIT)
+3
+ x2* 14
+[e000 rry
+[30
+(+9
+30
+3
+ xg sa
+玉净大量
+(去+)的内
+To
+#4
+F¥4
+[880 xten,
+0*   -[o 
+50
+SB
+30u
+%
+20
+O
+COWBOr eBOFb
+BRITYBIE AFTEBI
+一VMWIE0.00
+000
+00.0
+0:00
+0:00
+000OMES
+128
+MAd
+LETEW
+LETEWS
+2MUCH
+CAIAIR32
+26K11D2V
+NE
+LMEbut also in the post-stall region, in order to cover the entire flight envelope. The full angle aerodynamic parameters
+are obtained by combining the small angle aerodynamic parameters obtained by CFD simulation and the empirical
+aerodynamic model [31]. The aerodynamic characteristics at low and large AoA are approximated as
+�����
+�����
+𝐶𝐿𝑆 (𝛼) =
+0.5𝑐2
+2
+(𝑐2 − 𝑐3)cos2(𝛼) + 𝑐3
+sin(2𝛼)
+𝐶𝐷𝑆 (𝛼) = 𝑐0 +
+𝑐2𝑐3
+(𝑐2 − 𝑐3) cos2 (𝛼) + 𝑐3
+sin2 (𝛼)
+,
+����
+����
+𝐶𝐿𝐿 (𝛼) = 𝑐1 sin (2𝛼)
+𝐶𝐷𝐿 (𝛼) = 𝑐0 + 2𝑐1sin2 (𝛼)
+,
+and a pseudo-sigmoid function
+𝜎 (𝛼0, 𝑘, 𝛼) =
+1 + tanh �𝑘𝛼2
+0 − 𝑘𝛼2�
+1 + tanh
+�
+𝑘𝛼2
+0
+�
+, 𝛼 ∈ [−𝜋, 𝜋)
+is used to blend low and large AoA regions together
+����
+����
+𝐶𝐿 (𝛼) = 𝐶𝐿𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐿, 𝛼) + 𝐶𝐿𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐿, 𝛼)]
+𝐶𝐷 (𝛼) = 𝐶𝐷𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐷, 𝛼) + 𝐶𝐷𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐷, 𝛼)]
+.
+(29)
+The low and large AOA parameters 𝑐0, 𝑐1, 𝑐2, 𝑐3, and blending parameters 𝛼0, 𝑘𝐿, 𝑘𝐷 are turned according to CFD results,
+and their values are set to 𝑐0 = 0.055, 𝑐1 = 0.9, 𝑐2 = 13.0, 𝑐3 = 3.3, 𝛼0 = 3deg, 𝑘𝐿 = 38, 𝑘𝐷 = 48. Corresponding
+Aerodynamic curves are shown in Fig. 8.
+(deg)
+
+(deg)
+
+L
+C
+D
+C
+and
+L
+C
+D
+C
+L
+D
+C
+C
+Fig. 8
+Aerodynamic parameters obtained by CFD
+B. Simulation Experiments
+1. Verifying the Effectiveness of Aileron
+In order to show the effectiveness of ailerons, two groups of comparative experiments are carried out. The attitude
+control and position control are the same, but the control allocation of the first group uses the aileron, while the second
+20
+
+021-
+-J00
+20
+0
+J00
+J20
+500
+2.0
+2.0
+2.0500
+021-
+-J00
+-20
+0
+J00
+J20
+500
+-J2
+-JO
+2-
+0
+2
+JO
+J2Time(s)
+Throttle×1000
+Power(W)
+Fitted Power
+Measured Power
+Fig. 9
+Throttle command and identification result of the motor
+group does not. The second group only depends on the quadcopter control allocation. In this experiment, the aircraft
+tracks the specified trajectory, as shown in Fig. 10(a), which is a straight line plus a circle with a radius of 200m.
+As shown in Fig. 10(b), after adding the aileron, the control amplitude of the four motors is almost the same trend,
+especially when the aircraft transits between straight and circular trajectories, indicating that the attitude control is
+mainly realized by the aileron, and motors are more like a thruster on a fixed-wing in this case. However, when the
+attitude changes sharply, as shown in Fig. 10(b) during 𝑡 = 22 ∼ 23s, the motor and aileron will participate in the
+control at the same time to improve the control performance. When the state of the UAV is in the slow adjustment, the
+aileron undertakes the control as much as possible to save energy.
+In order to quantify the influence of ailerons, the relationship between throttle command and energy consumption of
+the motor and actuator is identified. A sinusoidal signal with the constant amplitude and linear increasing frequency
+is designed, and the energy consumption test experiments are carried out on the servo and motor at the same time.
+Measurement and identification results of the motor are shown in the Fig.9, in which the amplitude of the throttle
+decreases as the frequency increasing, because a low-pass filter in Eq.(28) is applied to it. We find that the motor power
+has a quadratic function relationship with a fixed throttle, and the power is different when the motor accelerates and
+decelerates. When the change frequency of the throttle command becomes faster, the power is increased. Therefore, the
+following formula is established to fit the relationship between throttle command and motor power
+𝑃𝑇 =
+4
+∑︁
+𝑖=1
+𝑝1𝜎𝑖2 + 𝑝2𝜎𝑖 + 𝑝3 �𝜎 𝑝4
+𝑢𝑝,𝑖 + 𝑝5 �𝜎 𝑝6
+𝑑𝑜𝑤𝑛,𝑖 + 𝑝7
+(30)
+21
+
+180
+500
+SSO
+J40
+seo
+580
+300
+400
+420
+eoo
+Q20180
+500
+SSO
+540
+seo
+580
+300
+20
+100
+J20
+50OWith the given lifting-wing quadcopter platform, 𝑝1 = 563.7, 𝑝2 = −147.4, 𝑝3 = 15, 𝑝4 = 1.05, 𝑝5 = 4, 𝑝6 = 1, 𝑝7 =
+0.05538. With respect to the servo, its power is negligible, because the power of servo is only 0.2W, far less than the
+motor’s, even if the 100g weight is suspended on the two ailerons.
+The trajectory tracking experiment is performed three times, with the forward flight speed of 5m/s, 10m/s and 20m/s,
+respectively. The powers in three flight speeds are shown in Fig. 10(c). It can be observed that when the speed is 5m/s,
+the power is the same, because the aileron is not used at slow flight speed. When the speed increases to 10m/s, the
+power with the aileron is slightly smaller. Further when the speed increases to 20m/s, the power with the aileron is
+greatly smaller because the aerodynamic force is stronger at high speed.
+Time(s), without aileron
+T1~T4(N)
+Time(s), with aileron
+b) Control output with and without aileron when the forward flight speed is 20m/s
+c) Control indexs in different forward flight speeds
+a) Flight path
+Time(s), 5m/s
+x(m)
+y(m)
+-z(m)
+Time(s), 10m/s
+Time(s), 20m/s
+Power(W)
+without aileron
+with aileron
+ar
+al
+ and 
+(deg)
+
+
+Fig. 10
+The attitude control performance and mixer output with and without aileron
+2. Verifying the Effectiveness of Coordinated Turn
+As for the lateral control, experiments are carried out in hover and high-speed forward flight, and the results are
+shown in Fig. 11. In the high-speed forward flight, as shown in Fig. 11 yellow region, when a turn command is given
+for the first time during 30.8 ∼ 42.2s, the lifting-wing quadcopter with coordinated turn has no sideslip angle and has a
+better lateral tracking performance, as shown in Fig.11(a.1 vs. b.1, and a.5 vs. b.5). In this state, the control of the
+lifting-wing quadcopter is similar to the fixed-wing, and the desired yaw rate is generated by Eq.(23) as the feedforward.
+When the turn command is given for the second time during 60.8 ∼ 72.2s, the lifting-wing quadcopter is in low-speed
+flight, and the control of the lifting-wing quadcopter is similar to the quadcopter, so a big sideslip angle appears both in
+experiments with and without coordinated turn.
+22
+
+0
+0
+J00
+J20
+S00
+5O
+0
+5O0
+0
+J00
+J20
+S00
+0
+2
+10
+J2400
+e00
+0
+500
+Q00
+400
+5OO
+0-
+-05
+40-
+eo-
+80-50
+JI
+53
+54
+52
+e'812
+5O
+52
+30
+32
+40
+42
+20
+J20
+500
+520
+30012
+50
+52
+30
+32
+40
+42
+300
+400
+200J2
+50
+52
+30
+32
+40
+42
+20
+J20F
+500
+520
+3000
+20
+J00
+J20
+500
+-
+0
+I0
+0
+J00
+J20
+S00
+2
+10
+J2Time(s)
+b) Vehicle states with coordinated turn
+Time(s)
+a) Vehicle states without coordinated turn
+Desired
+Response
+>10m/s
+6~10m/s
+6~10m/s
+>10m/s
+(deg)
+
+(deg)
+
+30.8-42.2
+60.8-72.2
+(a.1)
+(a.2)
+(a.3)
+(a.4)
+(a.5)
+(b.1)
+(b.2)
+(b.3)
+(b.4)
+(b.5)
+Desired Yaw Rate
+Yaw Response
+(rad)
+
+a(m s)
+V
+Fig. 11
+The control response with and without coordinated turn
+3. Transition Flight
+The transition flight is often defined as the aircraft changing from hover to level flight. To quantify the phase, the
+transition process is defined as the time it takes for the aircraft to start a transitional flight to the airspeed greater than
+18m/s. The quality of the transition is mainly reflected in the transition time and control accuracy of altitude. In the
+simulation, the aircraft will take off to a certain altitude, and then a step signal of −30◦ is given to the pitch channel.
+This is because the wind CFD test results show that the maximum energy efficiency is obtained when 𝛼 = 4◦. As a
+comparison, the same experiment is performed on a tail-sitter quadcopter. The model of the tail-sitter quadcopter is built
+on the lifting-wing quadcopter with the installation angle of the lifting-wing from 34◦ to 90◦, as shown in Fig. 2(a).
+Fig.12 shows the response curves of the pitch angle, altitude and airspeed of the lifting-wing quadcopter and
+tail-sitter UAV in the transition phrase in the HIL simulation. It can be observed that during the transition phase of
+23
+
+0
+O
+40
+eo
+80
+J00
+-50
+0
+5O5O
+40
+eo
+80
+J00
+-50
+0
+5O5O
+40
+e
+80
+J00
+-30F
+50
+-J0
+050
+40
+eo
+80
+J00
+-40
+50
+00
+O
+40
+eo
+80
+0
+JO
+SO5O
+40
+e
+80
+J00
+JO
+5O0
+5O
+40
+eo
+80
+30
+-50
+-JO
+00
+5O
+40
+eo
+80
+0
+5O
+40
+eo0
+SO
+40
+eo
+80
+2.I-
+-J
+2.0-
+050
+40
+eo
+80
+J00
+-J
+0Time(s)
+Va(m/s)
+RflyLW2
+Tail-sitter-60°
+Tail-sitter-70°
+Take 
+off
+Hovering Transition 
+Flight
+Time(s)
+Time(s)
+Take 
+off
+Hovering Transition 
+Flight
+Take 
+off
+Hovering Transition 
+Flight
+RflyLW2
+Tail-sitter-60°
+Tail-sitter-70°
+RflyLW2
+Tail-sitter-60°
+Tail-sitter-70°
+Pitch(rad)
+Altitude(m)
+a) Airspeed response 
+b) Pitch response 
+c) Altitude response 
+Fig. 12
+Transition flight of lifting-wing quadcopter and tail-sitter UAV.
+the lifting-wing quadcopter, the time for the pitch angle to reach the desired value is 1.1s, as shown in Fig.12(b), and
+the time it takes for the airspeed to reach the set 18m/s is 4.7s as shown in Fig.12(a). So the transition time of the
+lifting-wing quadcopter is 4.7s. Furthermore, the altitude decreases as soon as the transition starts, with a maximum
+altitude error of 0.09 m at 𝑡 = 21.68 s. As for tail-sitter UAV, the transition time is 7.1s, when the desired pitch angle is
+−70◦. But after the transition, the altitude drops sharply, as shown in Fig.12(c). When the desired pitch angle is −60◦,
+the altitude is stable, but the transition time is 20.48s much longer than that of the RflyLW2. Thus, the transition phase of
+the lifting-wing quadcopter is significantly better than the tail-sitter UAV, and does not require an additional controller.
+VI. Conclusion
+The modeling, controller design, and HIL simulation verification of the lifting-wing quadcopter—a novel type
+of hybrid aircraft—are presented in this paper. The modeling portion takes into account the forces and moments
+produced by the lifting wing and rotors. A unified controller for the entire flight phase is created based on the existing
+model. Depending on the velocity command, the controller can regard hovering and forward flight equally and enable
+a seamless transition between the two modes. The experimental results show that the proposed aircraft outperforms
+the tail-sitter UAV in terms of tracking performance during the transition phase. In addition, the controller combines
+the characteristics of the quadcopter and fixed-wing control law, allowing it to retain yaw during the hover phase and
+decrease sideslip angle during the cruise phase. What is more, a control allocation based on optimization is utilized
+to realize cooperative control for energy saving, by taking rotor thrust and aerodynamic force under consideration
+simultaneously. Through identification, we find that compared with the motor, the aileron can reduce the energy
+consumption when implementing high-frequency control inputs.
+References
+[1] Saeed, A. S., Younes, A. B., Cai, C., and Cai, G., “A survey of hybrid unmanned aerial vehicles,” Progress in Aerospace
+Sciences, Vol. 98, 2018, pp. 91 – 105.
+24
+
+0
+10
+S0
+30
+40
+08-
+-eo
+-40
+-50
+00
+10
+S0
+30
+40
+0
+10
+300
+S0
+30
+40
+0
+SO[2] Raj, N., Banavar, R., Abhishek, and Kothari, M., “Attitude control of a novel tailsitter: swiveling biplane-quadrotor,” Journal of
+Guidance, Control, and Dynamics, Vol. 43, No. 3, 2020, pp. 599–607.
+[3] wsj.com, “Google tail-sitter UAV,” https://www.wsj.com/articles/BL-DGB-40935, 2015. Accessed January 1, 2023.
+[4] Ritz, R., and D’Andrea, R., “A global strategy for tailsitter hover control,” Robotics Research, Springer, 2018, pp. 21–37.
+[5] Xiao, K., Meng, Y., Dai, X., Zhang, H., and Quan, Q., “A lifting wing fixed on multirotor UAVs for long flight ranges,” 2021
+International Conference on Unmanned Aircraft Systems (ICUAS), IEEE, 2021, pp. 1605–1610.
+[6] Zhang, H., Tan, S., Song, Z., and Quan, Q., “Performance evaluation and design method of lifting-wing multicopters,”
+IEEE/ASME Transactions on Mechatronics, Vol. 27, No. 3, 2021, pp. 1606–1616.
+[7] Quan, Q., Introduction to multicopter design and control, Springer, 2017.
+[8] Emran, B. J., and Najjaran, H., “A review of quadrotor: An underactuated mechanical system,” Annual Reviews in Control,
+Vol. 46, 2018, pp. 165–180.
+[9] Theys, B., De Vos, G., and De Schutter, J., “A control approach for transitioning VTOL UAVs with continuously varying
+transition angle and controlled by differential thrust,” 2016 international conference on unmanned aircraft systems (ICUAS),
+IEEE, 2016, pp. 118–125.
+[10] tx tech.cn, “Volitation vespertilio,” http://www.tx-tech.cn/en/col.jsp?id=133, 2019. Accessed January 1, 2023.
+[11] theverge.com,
+“Amazon
+Prime
+Air,”
+https://www.theverge.com/2019/6/5/18654044/amazon-prime-air-
+delivery-drone-new-design-safety-transforming-flight-video, 2019. Accessed January 1, 2023.
+[12] Hassanalian, M., and Abdelkefi, A., “Classifications, applications, and design challenges of drones: A review,” Progress in
+Aerospace Sciences, Vol. 91, 2017, pp. 99–131.
+[13] Argyle, M. E., Beard, R. W., and Morris, S., “The vertical bat tail-sitter: dynamic model and control architecture,” 2013
+American Control Conference, IEEE, 2013, pp. 806–811.
+[14] Escareno, J., Stone, R., Sanchez, A., and Lozano, R., “Modeling and control strategy for the transition of a convertible tail-sitter
+UAV,” 2007 European Control Conference (ECC), IEEE, 2007, pp. 3385–3390.
+[15] Oosedo, A., Abiko, S., Konno, A., and Uchiyama, M., “Optimal transition from hovering to level-flight of a quadrotor tail-sitter
+UAV,” Autonomous Robots, Vol. 41, No. 5, 2017, pp. 1143–1159.
+[16] Li, B., Sun, J., Zhou, W., Wen, C.-Y., Low, K. H., and Chen, C.-K., “Transition optimization for a VTOL tail-sitter UAV,”
+IEEE/ASME Transactions on Mechatronics, 2020.
+[17] Li, B., Sun, J., Zhou, W., Wen, C.-Y., Low, K. H., and Chen, C.-K., “Nonlinear robust flight mode transition control for tail-sitter
+aircraft,” IEEE/ASME transactions on mechatronics, Vol. 25, No. 5, 2020, pp. 2534–2545.
+25
+
+[18] Smeur, E. J., Bronz, M., and de Croon, G. C., “Incremental control and guidance of hybrid aircraft applied to a tailsitter
+unmanned air vehicle,” Journal of Guidance, Control, and Dynamics, Vol. 43, No. 2, 2020, pp. 274–287.
+[19] Lyu, X., Gu, H., Zhou, J., Li, Z., Shen, S., and Zhang, F., “Simulation and flight experiments of a quadrotor tail-sitter vertical
+take-off and landing unmanned aerial vehicle with wide flight envelope,” International Journal of Micro Air Vehicles, Vol. 10,
+No. 4, 2018, pp. 303–317.
+[20] Liu, H., Peng, F., Lewis, F. L., and Wan, Y., “Robust tracking control for tail-sitters in flight mode transitions,” IEEE Transactions
+on Aerospace and Electronic Systems, Vol. 55, No. 4, 2018, pp. 2023–2035.
+[21] Zhou, J., Lyu, X., Li, Z., Shen, S., and Zhang, F., “A unified control method for quadrotor tail-sitter uavs in all flight modes:
+Hover, transition, and level flight,” 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE,
+2017, pp. 4835–4841.
+[22] Flores, A., de Oca, A. M., and Flores, G., “A simple controller for the transition maneuver of a tail-sitter drone,” 2018 IEEE
+Conference on Decision and Control (CDC), IEEE, 2018, pp. 4277–4281.
+[23] Nascimento, T. P., and Saska, M., “Position and attitude control of multi-rotor aerial vehicles: A survey,” Annual Reviews in
+Control, Vol. 48, 2019, pp. 129–146.
+[24] Beard, R. W., and McLain, T. W., Small unmanned aircraft: Theory and practice, Princeton university press, 2012.
+[25] Lyu, X., Gu, H., Zhou, J., Li, Z., Shen, S., and Zhang, F., “A hierarchical control approach for a quadrotor tail-sitter VTOL UAV
+and experimental verification,” 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE,
+Vancouver, BC, 2017, pp. 5135–5141.
+[26] Johansen, T. A., and Fossen, T. I., “Control allocation—A survey,” Automatica, Vol. 49, No. 5, 2013, pp. 1087–1103.
+[27] Harkegard, O., “Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation,”
+Proceedings of the 41st IEEE Conference on Decision and Control, 2002., Vol. 2, IEEE, 2002, pp. 1295–1300.
+[28] Wang, S., Dai, X., Ke, C., and Quan, Q., “RflySim: A rapid multicopter development platform for education and research
+based on Pixhawk and MATLAB,” 2021 International Conference on Unmanned Aircraft Systems (ICUAS), IEEE, 2021, pp.
+1587–1594.
+[29] Dai, X., Ke, C., Quan, Q., and Cai, K.-Y., “RFlySim: Automatic test platform for UAV autopilot systems with FPGA-based
+hardware-in-the-loop simulations,” Aerospace Science and Technology, Vol. 114, 2021, p. 106727.
+[30] rflysim.com, “rflysim,” https://rflysim.com/en/index.html, 2020. Accessed January 1, 2023.
+[31] Pucci, D., Hamel, T., Morin, P., and Samson, C., “Nonlinear control of PVTOL vehicles subjected to drag and lift,” 2011 50th
+IEEE Conference on Decision and Control and European Control Conference, IEEE, 2011, pp. 6177–6183.
+26
+

0tFQT4oBgHgl3EQf0TbP/content/tmp_files/2301.13416v1.pdf.txt

@@ -0,0 +1,1018 @@
+Structure Flow-Guided Network for Real Depth Super-Resolution
+Jiayi Yuan*, Haobo Jiang*, Xiang Li, Jianjun Qian, Jun Li†, Jian Yang†
+PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education
+Jiangsu Key Lab of Image and Video Understanding for Social Security
+School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China
+{jiayiyuan, jiang.hao.bo, xiang.li.implus, csjqian, junli, csjyang}@njust.edu.cn
+Real LR
+Groundtruth
+RGB image
+FDSR
+Ours
+(c)
+(e)
+(d)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(a)
+(b)
+Synthetic LR
+Figure 1: In this paper, we propose a novel structure flow-guided method for real-world DSR. Our method obtains better depth
+edge recovery (g-h), compared to (e) and (f) using the SOTA method, FDSR (He et al. 2021). (a-b) Synthetic LR depth maps;
+(c) Real LR depth map with the structural distortion; (d) Real LR depth map with the edge noise (e.g., holes); (i-j) Ground-truth
+HR depth maps; (k-l) RGB image guidance.
+Abstract
+Real depth super-resolution (DSR), unlike synthetic settings,
+is a challenging task due to the structural distortion and the
+edge noise caused by the natural degradation in real-world
+low-resolution (LR) depth maps. These defeats result in sig-
+nificant structure inconsistency between the depth map and
+the RGB guidance, which potentially confuses the RGB-
+structure guidance and thereby degrades the DSR quality. In
+this paper, we propose a novel structure flow-guided DSR
+framework, where a cross-modality flow map is learned to
+guide the RGB-structure information transferring for pre-
+cise depth upsampling. Specifically, our framework consists
+of a cross-modality flow-guided upsampling network (CFU-
+Net) and a flow-enhanced pyramid edge attention network
+(PEANet). CFUNet contains a trilateral self-attention module
+combining both the geometric and semantic correlations for
+reliable cross-modality flow learning. Then, the learned flow
+maps are combined with the grid-sampling mechanism for
+coarse high-resolution (HR) depth prediction. PEANet targets
+*These authors contributed equally.
+†corresponding authors
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+at integrating the learned flow map as the edge attention into
+a pyramid network to hierarchically learn the edge-focused
+guidance feature for depth edge refinement. Extensive exper-
+iments on real and synthetic DSR datasets verify that our ap-
+proach achieves excellent performance compared to state-of-
+the-art methods.
+Introduction
+With the fast development of cheap RGB-D sensors, depth
+maps have played a much more important role in a variety
+of computer vision applications, such as object recognition
+(Blum et al. 2012; Eitel et al. 2015), 3D reconstruction (Hou,
+Dai, and Nießner 2019; Newcombe et al. 2011), and virtual
+reality (Meuleman et al. 2020)). However, the defects (e.g.,
+low resolution and structural distortion) lying in the cheap
+RGB-D sensors (e.g., Microsoft Kinect and HuaweiP30Pro),
+still hinder their more extensive applications in real world.
+Also, although the popular DSR methods (Song et al. 2020;
+Kim, Ponce, and Ham 2021; Sun et al. 2021) have achieved
+excellent DSR accuracy on synthetic LR depth maps, the
+significant domain gap between the real and the synthetic
+data largely degrades their DSR precision on the real data.
+arXiv:2301.13416v1  [cs.CV]  31 Jan 2023
+
+This domain gap is mainly caused by different genera-
+tion mechanisms of the LR depth map. The synthetic LR
+depth map is usually generated via artificial degradation
+(e.g., down-sampling operation), while the real one is from
+natural degradation (e.g., noise, blur, and distortion). Differ-
+ent from the synthetic data, there are two challenges of the
+real-data DSR as below. The first one is the severe structural
+distortion (see Fig. 1 (c)), especially for the low-reflection
+glass surface or the infrared-absorbing surface. The second
+one is the edge noise even the holes (see Fig. 1 (d)), caused
+by the physical limitations or the low processing power of
+the depth sensors. Both of the challenges above present
+a significant difference between the real and the synthetic
+data, which inherently degrades the generalization precision
+of the synthetic DSR methods to the real data.
+In this paper, we develop a novel structure flow-guided
+DSR framework to handle the above challenges. For the
+structural distortion, we propose a cross-modality flow-
+guided upsampling network (CFUNet) that learns a struc-
+tured flow between the depth map and the RGB image to
+guide their structure alignment for the recovery of the dis-
+torted depth structure. It includes two key components: a
+trilateral self-attention module and a cross-modality cross-
+attention module. In detail, the former leverages the geomet-
+ric and semantic correlations (i.e., coordinate distance, pixel
+difference, feature difference) to guide the relevant depth-
+feature aggregation into each depth feature to supplement
+the missing depth-structure information. The latter utilizes
+the enhanced depth feature and the RGB feature as the in-
+put for their sufficient message passing and flow-map gen-
+eration. Finally, we combine the flow map with the grid-
+sampling mechanism for the coarse HR depth prediction.
+For the edge noise, we present a flow-enhanced pyramid
+edge attention network (PEANet) that integrates the learned
+structure flow map as the edge attention into a pyramid net-
+work to learn the edge-focused guidance feature for the edge
+refinement of the coarse HR depth map predicted above.
+Considering the structure clue (i.e., edge region tends to
+own significant flow-value fluctuations) lying in the learned
+flow map, we combine the flow map with the RGB fea-
+ture to form the flow-enhanced RGB feature for highlighting
+the RGB-structure region. Then, we feed the flow-enhanced
+RGB feature into an iterative pyramid network for its edge-
+focused guidance feature learning. The low-level guidance
+features effectively filter the RGB-texture noise (guided by
+the flow map), while the high-level guidance features exploit
+the rich context information for more precise edge-feature
+capture. Finally, we pass the learned guidance feature and
+the depth feature into a decoder network to predict the edge-
+refined HR depth map. Extensive experiments on challeng-
+ing real-world datasets verify the effectiveness of our pro-
+posed method (see examples in Fig. 1(g-h)). In summary,
+our contributions are as follows:
+• We propose an effective cross-modality flow-guided up-
+sampling network (CFUNet), where a structure flow map
+is learned to guide the structure alignment between the
+depth map and the RGB image for the recovery of the
+distorted depth edge.
+• We present a flow-enhanced pyramid edge attention net-
+work (PEANet) that integrates the flow map as edge at-
+tention into a pyramid network to hierarchically learn the
+edge-focused guidance feature for edge refinement.
+• Extensive experiments on the real and synthetic datasets
+verify the effectiveness of the proposed framework, and
+we achieve state-of-the-art restoration performance on
+multiple DSR dataset benchmarks.
+Related Work
+Synthetic Depth Super-Resolution
+The synthetic depth super-resolution (DSR) architectures
+can be divided into the pre-upsampling methods and the
+progressive upsampling methods (Wang et al. 2020). The
+pre-upsampling DSR methods first upsample the input depth
+with interpolation algorithms (e.g., bicubic) from LR to HR,
+and then feed it into depth recovery network layers. (Li
+et al. 2016) introduce the first pre-upsampling network ar-
+chitecture. As this method handles arbitrary scaling factor
+depth, more and more similar approaches have been pre-
+sented to further facilitate DSR task (Li et al. 2019; Lutio
+et al. 2019; Zhu et al. 2018; Chen and Jung 2018; Hao et al.
+2019; Su et al. 2019). However, upsampling in one step is
+not suitable for large scaling factors simply because it usu-
+ally leads to losing much detailed information. To tackle
+these issues, a progressive upsampling structure is designed
+in MSG-net(Tak-Wai, Loy, and Tang 2016), which gradually
+upsamples the LR depth map by transposed convolution lay-
+ers at different scale levels. Since then, various progressive
+upsample-based methods have been proposed that greatly
+promote the development of this domain(Tak-Wai, Loy, and
+Tang 2016; Guo et al. 2019; He et al. 2021; Zuo et al. 2019).
+Recently, the joint-task learning framework achieves im-
+pressive performance, such as DSR & completion (Yan et al.
+2022), depth estimation & enhancement (Wang et al. 2021)
+and DSR & depth estimation (Tang et al. 2021; Sun et al.
+2021). Inspired by these joint-task methods, we combine the
+alignment task with the super-resolution task to distill the
+cross-modality knowledge for robust depth upsampling.
+Real-world Depth Super-Resolution
+In recent years, the super-resolution for real-world images
+has been under the spotlight, which involves upsampling,
+denoising, and hole-filling. Early traditional depth enhance-
+ment methods (Yang et al. 2014; Liu et al. 2016, 2018) are
+based on complex and time-consuming optimization. For
+fast CNN-based DSR, AIR (Song et al. 2020) simulates
+the real LR depth map by combining the interval degrada-
+tion and the bicubic degradation, and proposes a channel at-
+tention based network for real DSR. PAC (Su et al. 2019)
+and DKN (Kim, Ponce, and Ham 2021) utilize the adap-
+tive kernels calculated by the neighborhood pixels in RGB
+image for robust DSR. FDSR(He et al. 2021) proposes the
+octave convolution for frequency domain separation, which
+achieves outstanding performance in real datasets. Although
+these methods handle the large modality gap between the
+guidance image and depth map, the structure misalignment
+between the depth map and the RGB image still leads them
+
+Cross
+Attention
+Trilateral
+Self-
+attention
+Grid 
+Sample
+�������������������������������������
+������������������������������������
+������������������������������������������������������������������������������������
+Encoder
+������������������������������������
+Cross-modality Flow-guided Upsampling       
+  Flow-enhanced Pyramid Edge Attention
+Edge 
+Decoder
+������������������������������������������������������������������������������������
+Flow-enhanced 
+Pyramid Attention 
+Add
+Multi-scale
+features 
+Flow 
+Decoder
+Upsample
+Decoder
+Flow maps 
+Figure 2: The pipeline of our structure flow-guided DSR framework. Given the LR depth map and the RGB image, the left
+block (blue) first generates the flow maps through a trilateral self-attention module and a cross-attention module, and predicts
+the coarse depth map Dcoarse with the flow-based grid-sampling. Then, the right block (yellow) integrates the RGB/depth
+features and the flow map (as edge attention) to learn the edge-focused guidance feature for edge refinement (Drefine).
+to suffer from serious errors around the edge regions. Dif-
+ferent from the general paradigms, we introduce a novel
+structure flow-guided framework, which exploits the cross-
+modality flow map to guide the RGB-structure information
+transferring for real DSR.
+Approach
+In the following, we introduce our structure flow-guided
+DSR framework for robust real-world DSR. As shown in
+Fig. 2, our framework consists of two modules: a cross-
+modality flow-guided upsampling network (CFUNet) and a
+flow-enhanced pyramid edge attention network (PEANet).
+Given an LR depth map DLR ∈ RH0×W0 and its cor-
+responding HR RGB image I ∈ RH×W ×3 (H/H0 =
+W/W0 = s and s is the scale factor), CFUNet first learns
+the cross-modality flow to guide the structure alignment be-
+tween depth the RGB for coarse HR depth prediction. Then,
+PEANet exploits the structure flow as edge attention to learn
+the edge-focused guidance feature for edge refinement.
+Cross-modality Flow-guided Upsampling Network
+As demonstrated in Fig. 1 (c), the structural distortion of the
+real LR depth map leads to the significant structure misalign-
+ment between the RGB image and the depth map, which po-
+tentially damages the structure guidance of RGB images for
+depth edge recovery. To handle it, our solution is to learn an
+effective cross-modality flow map between the depth and the
+RGB to identify their structure relationship. Then, guided by
+the learned flow map, we align the structure of the depth map
+to the RGB image for the recovery of the distorted depth
+edge. Next, we will describe our network in terms of the
+feature extraction, the trilateral attention-based flow genera-
+tion, and the flow-guided depth upsampling.
+Feature extraction. To achieve the consistent input size,
+we first upsample the LR depth map DLR to a resolution
+map DBic ∈ RH×W with the bicubic interpolation.
+Then, we feed the upsampled depth map and the RGB
+image into an encoder for their feature extraction: {Fl ∈
+RH×W ×D}L
+l=1 and {Gl ∈ RH×W ×D}L
+l=1 where the sub-
+script l denotes the feature output in l-th layer of the encoder.
+Trilateral attention-based flow generation. The key to
+generating a reliable cross-modality flow map is to model
+a robust relationship between the RGB and the depth map.
+Nevertheless, the serious structural distortion caused by the
+natural degradation potentially increases the modality gap
+between the depth and the RGB. Thereby, it’s difficult to
+directly exploit a general attention mechanism to model
+such a relationship. To mitigate it, we target at enhanc-
+ing the depth feature through a proposed trilateral self-
+attention block so that the distorted depth-structure informa-
+tion can be largely complemented for relationship modeling.
+As shown in Fig. 3, our trilateral self-attention block fuses
+the geometric-level correlation and the semantic-level cor-
+relation to jointly guide the depth feature enhancement. It’s
+noted that we just enhance the depth feature FL in the last
+layer (L-th layer):
+¯F(i)
+L =
+�
+j
+αi,j · (βlow
+i,j + βhigh
+i,j
+) · γi,j · F(j)
+L + F(j)
+L ,
+(1)
+where F(j)
+L (1 ≤ j ≤ H × W) denotes the j-th depth-pixel
+feature and ¯F(i)
+L
+denotes the i-th enhanced depth feature
+(1 ≤ i ≤ H × W). The geometric-level correlation con-
+tains a spatial kernel α ∈ R(H×W )×(H×W ) and a low-level
+color kernel βlow ∈ R(H×W )×(H×W ), while the semantic-
+level correlation contains a high-level color semantic ker-
+nel βhigh ∈ R(H×W )×(H×W ) and a depth semantic kernel
+γ ∈ R(H×W )×(H×W ). In detail, we formulate the spatial
+kernel as a coordinate distance-aware Gaussian kernel:
+αi,j = Gaussian(∥ Coor(i) − Coor(j)∥2, σs),
+(2)
+where Gaussian(x, σ) =
+1
+σ
+√
+2π exp (− x2
+2σ2 ) is the Gaussian
+function. Coor(i) ∈ R2 denotes the row-column coordi-
+nates of pixel i at the depth map and σs is the kernel vari-
+ance. The low-level and high-level color kernels are defined
+
+K
+V
+Q
+Cross-attention Module
+Depth Transformation
+Self-
+Attention
+Add & 
+Norm
+Self-
+Attention
+Add & 
+Norm
+Color Transformation
+V
+K
+Q
+�������������������������
+������������������������
+������������������������′
+C
+������������������������+������������
+C Concatenation
+P Position Embedding
+Element-wise Production
+P
+Trilateral Self-attention Module
+{������������������������}������������>������������
+������������
+������������������������������������������������
+������������������������������������������������������������
+������������
+Geometric-level
+Semantic-level
+������������������������
+������������������������
+Figure 3: The architecture of the trilateral self-attention module and the cross-attention module.
+by the Gaussian kernels with the low-level and the semantic-
+level RGB feature similarity, whose kernel sum is:
+βlow
+i,j + βhigh
+i,j
+=
+L
+�
+l=0
+Gaussian(∥G(i)
+l
+− G(j)
+l ∥2, σc).
+(3)
+The depth semantic kernel is designed based on the depth
+feature similarity in the L-th layer:
+γi,j = Gaussian(∥F(i)
+L − F(j)
+L ∥2, σd).
+(4)
+Guided by the geometric and semantic kernels above, the
+correlated depth information can be effectively aggregated
+into each depth feature through Eq.1 for depth feature com-
+pletion and enhancement.
+Then, we feed the enhanced depth feature ¯FL and the
+RGB feature GL into the cross-attention block for their ef-
+ficient cross-modality feature intersection:
+˜F(i)
+L = ¯F(i)
+L + MLP(
+�
+j
+softmaxj(φq(¯F(i)
+L )⊤φk(G(j)
+L ))φv(G(j)
+L )),
+˜G(i)
+L = G(i)
+L + MLP(
+�
+j
+softmaxj(φq(G(i)
+L )⊤φk(¯F(j)
+L )φv(¯F(j)
+L )),
+(5)
+where φq, φk and φv are the projection functions of the
+query, the key and the value in our nonlocal-style cross-
+attention module. With the query-key similarity, the value
+can be retrieved for feature enhancement. Then, we concate-
+nate the enhanced depth feature ˜FL and RGB feature ˜GL
+and pass them into a multi-layer convolutional network to
+obtain their correlated feature at each layer {Gl}L′
+l=L+1. Fi-
+nally, following (Dosovitskiy et al. 2015), based on the pre-
+viously extracted features {Gl}L
+l=1 and the correlated fea-
+tures {Gl}L′
+l=L+1, we exploit a decoder network to generate
+the multi-layer flow maps {∆l}L′
+l=1, where the flow genera-
+tion in layer l can be formulated as:
+Gflow
+l+1 , ∆l+1 = deconv(Cat[Gflow
+l
+, ∆l, GL′−l−1]), (6)
+where Gflow
+l
+denotes the intermediate flow feature and
+deconv consisting of a deconvolution operation and a con-
+volutional block (Gflow
+1
+, ∆1 = deconv(GL′)).
+Flow-guided depth upsampling module. With the
+learned flow map ∆L′ in the last layer, we combine it with
+the grid-sampling strategy for the HR depth map predic-
+tion. In detail, the value of the HR depth map is the bilinear
+interpolation of the neighborhood pixels in LR depth map
+DLR, where the neighborhoods are defined according to the
+learned flow field, which can be formulated as:
+Dcoarse = Grid-Sample(DLR, ∆L′),
+(7)
+where Grid-Sample denotes the upsampling operation com-
+puting the output using pixel values from neighborhood pix-
+els and pixel locations from the grid (Li et al. 2020).
+Flow-enhanced Pyramid Edge Attention Network
+In order to further improve our DSR precision in the case of
+the edge noise problem, we propose a flow-enhanced pyra-
+mid network, where the learned structure flow is served as
+the edge attention to hierarchically mine edge-focused guid-
+ance feature from the RGB image for the edge refinement
+of Dcoarse. Specifically, we first feed the previously pre-
+dicted HR depth map Dcoarse and the RGB image into an
+encoder network to extract their features: {Fcoarse
+t
+}T +1
+t=1 and
+{Gt}T
+t=1, where subscript t indicates the extracted feature at
+the t-th layer. Then, we propose the flow-enhanced pyramid
+attention module and the edge decoder module as follows
+for refined HR depth prediction.
+Flow-enhanced pyramid attention module. In this mod-
+ule, we target at combining the RGB feature and the flow
+map to learn the edge-focused guidance feature {Gguide
+t
+}
+at each layer. In detail, for the t-th layer, with the RGB fea-
+ture Gt and its corresponding flow map ∆L′−t, we first fuse
+the flow information into the RGB feature to form the flow-
+enhanced RGB feature,
+Gflow
+t
+= ∆L′−t · Gt + Gt,
+(8)
+
+Decoded Search Feature
+Decoder
+Add&Ins.Norm
+formator
+MaskT
+Encoded
+Add&Ins.Norm
+Mul & Iins, Norm
+Template Feature
+Cross-Attention
+Cross-Attention
+Encoder
+?
+Add&ins.Nom
+Add&Ins.Norm
+Self-Attention
+Self-Attention
+eight Sharing
+TemplateFeature
+SearchFeature
+Element-wiseProduction
+@ Template Feature Mask
+Figure 4. An overview of the proposed transformer architectureC
+Scale Unify & Concat
+CONV
+CONV
+CONV
+CONV
+CONV
+CONV
+∆������������′−������������
+Flow-enhanced Pyramid Attention Module
+×K
+������������������������
+������������������������
+������������������������������������������������������������������������
+������������������������
+������������������������������������������������������������
+Figure 4: The architecture of the pyramid attention module.
+The subscript t denotes the feature output in the t-th layer of
+the encoder (1 ≤ t ≤ T). ‘×K’ indicates the iteration times
+of the guidance feature updating.
+where ∆L′−t ·Gt is expected to exploit the significant flow-
+value fluctuations at the edge region in ∆L′−t to better
+highlight the structure region of the RGB feature. To fur-
+ther smooth the texture feature in Gflow
+t
+, we concatenate
+it with the texture-less depth feature Fcoarse
+t
+to obtain the
+texture-degraded RGB feature ˜Gflow
+t
+. Then, we feed ˜Gflow
+t
+into a pyramid network to extract its edge-focused guid-
+ance features { ˜Gflow
+t,k
+}K
+k=1 at different scales. The low-level
+guidance feature is to filter the texture noise (guided by the
+flow map) while the high-level is to exploit the rich context
+information for edge-feature capture. After that, we unify
+the scales of the hierarchical feature { ˜Gflow
+t,k
+}K
+k=1 using the
+bicubic interpolation and pass the concatenated feature into
+a convolutional block to generate the flow-enhanced RGB
+guidance feature Gguide
+t
+at the t-th layer. Notably, we de-
+sign an iterative architecture to progressively refine the RGB
+guidance feature as illustrated in Fig. 4.
+Edge decoder. Guided by the flow-based guidance fea-
+tures {Gguide
+t
+}T
+t=1 learned at each layer, we progressively
+decode the depth feature in an iterative manner:
+Fedge
+t+1 = FU(Cat(Fedge
+t
+, Gguide
+T −t+1, Fcoarse
+T −t+1)),
+(9)
+where FU function indicates the fusion and upsampling op-
+eration following (Guo et al. 2020) and the initial feature
+Fedge
+1
+is obtained by the convolutional operation on Fcoarse
+T +1 .
+Finally, we pass Fedge
+T +1 into a convolutional block to obtain
+the edge-refined HR depth map Drefine.
+Loss Function
+We train our model by minimizing the smooth-L1 loss be-
+tween the ground-truth depth map Dgt and the network out-
+put of each sub-network, including the coarse depth predic-
+tion Dcoarse and the refined one Drefine:
+Ldsr =
+H×W
+�
+i=1
+ℓ
+�
+Dcoarse
+i
+− Dgt
+i
+�
++ ℓ
+�
+Drefine
+i
+− Dgt
+i
+�
+, (10)
+where the subscript i denote the pixel index and the smooth-
+L1 loss function is defined as:
+ℓ(u) =
+�0.5u2,
+if |u| ≤ 1
+(|u| − 0.5) ,
+otherwise.
+(11)
+Experiments
+Experimental Setting
+To evaluate the performance of our method, we perform ex-
+tensive experiments on real-world RGB-D-D dataset (He
+et al. 2021), ToFMark dataset (Ferstl et al. 2013) and syn-
+thetic NYU-v2 dataset (Silberman et al. 2012). We imple-
+ment our model with PyTorch and conduct all experiments
+on a server containing an Intel i5 2.2 GHz CPU and a TITAN
+RTX GPU with almost 24 GB. During training, we randomly
+crop patches of resolution 256 × 256 as groundtruth and the
+training and testing data are normalized to the range [0, 1].
+In order to balance the training time and network perfor-
+mance, the parameters L, L′, K, T are set to 3, 6, 3, 2 in this
+paper. We quantitatively and visually compare our method
+with 13 state-of-the-art (SOTA) methods: TGV (Ferstl et al.
+2013), FBS (Barron and Poole 2016), MSG (Tak-Wai, Loy,
+and Tang 2016), DJF (Li et al. 2016), DJFR (Li et al. 2019),
+GbFT (AlBahar and Huang 2019), PAC (Su et al. 2019),
+CUNet (Deng and Dragotti 2020), FDKN (Kim, Ponce, and
+Ham 2021), DKN (Kim, Ponce, and Ham 2021), FDSR (He
+et al. 2021), CTKT (Sun et al. 2021) and DCTNet (Zhao
+et al. 2022). For simplicity, we name our Structure Flow-
+Guided method as SFG.
+Experiments on Real Datasets
+Depth maps captured by cheap depth sensors usually suf-
+fer from structural distortion and edge noise. To verify the
+efficiency and robustness of our proposed method, we em-
+ploy our method on two challenging benchmarks: RGB-D-D
+dataset and ToFMark dataset.
+Evaluation on hand-filled RGB-D-D. To evaluate the per-
+formance of our method on real LR depth maps, we conduct
+experiments on RGB-D-D datasets captured by two RGB-
+D sensors: Huawei P30 Pro (captures RGB images and LR
+depth maps) and Helios TOF camera (captures HR depth
+maps). The LR inputs are shown in Fig. 5, which suffer from
+the low resolution (LR size is 192 × 144 and target size is
+512 × 384) and random structural missing in the edge re-
+gion. Following FDSR (He et al. 2021), we first use 2215
+hand-filled RGB/D pairs for training and 405 RGB/D pairs
+for testing. As listed in the first row of Table 1, the proposed
+model outperforms SOTA methods by a significant margin.
+The first two rows in Fig. 5 show the visual DSR com-
+parisons on hand-filled RGB-D-D dataset. We can see that
+edges in the results of DKN (Kim, Ponce, and Ham 2021)
+and DCTNet (Zhao et al. 2022) are over-smoothed and the
+artifacts are visible in the FDSR results. In contrast, our re-
+sults show more accurate structures without texture copying.
+Evaluation on incomplete RGB-D-D. To further verify the
+DSR performance of our method in the case of edge noise
+(e.g., edge holes), instead of the hole completion above, we
+directly test SFG on unfilled RGB-D dataset and achieve the
+
+(a) LR depth
+(f) Groundtruth
+(d) DCTNet
+(c) FDSR
+(e) SFG (ours)
+(b) DKN
+Hand-filled
+Incomplete
+Figure 5: Visual comparison on RGB-D-D dataset. The first (last) two rows show DSR results of hand-filled (incomplete) LR.
+RMSE
+Bicubic
+MSG
+DJF
+DJFR
+CUNet
+DKN
+FDKN
+FDSR
+DCTNet
+SFG (ours)
+Hand-filled
+7.17
+5.50
+5.54
+5.52
+5.84
+5.08
+5.37
+5.34
+5.28
+3.88
+Incomplete
+-
+7.90
+5.70
+5.52
+6.54
+5.43
+5.87
+5.59
+5.49
+4.79
+Noisy
+11.57
+10.36
+5.62
+5.71
+6.13
+5.16
+5.54
+5.63
+5.16
+4.45
+Table 1: Quantitative comparison on RGB-D-D dataset. Best and second best results are in bold and underline, respectively.
+(b) Bicubic
+(d) DCTNet
+(c) DKN
+(e) SFG (ours)
+(a) Groundtruth
+Figure 6: Visual comparison on ToFMark dataset.
+DJFR
+DKN
+FDKN
+FDSR
+DCTNet
+SFG (ours)
+RMSE
+0.27
+0.26
+0.28
+0.28
+0.27
+0.25
+Table 2: Quantitative comparison on ToFMark dataset.
+lowest RMSE as shown in the second row of Table 1. More-
+over, as shown in the last two rows in Fig. 5, the edges recov-
+ered by our method are sharper with fewer artifacts and vi-
+sually closest to the ground-truth map. It’s mainly attributed
+to the edge-focused guidance feature learning with our flow-
+enhanced pyramid edge attention network.
+Evaluation on noisy RGB-D-D and ToFMark. We evalu-
+ate the denoising and generalization ability of our method on
+ToFMark dataset consisting of three RGB-D pairs. The LR
+inputs have irregular noise and limited resolution (LR depth
+is 120 × 160 and target size is 610 × 810). To simulate the
+similar degradation for training, we add the Gaussian noise
+(mean 0 and standard deviation 0.07) and the Gaussian blur
+(kernel size 5) on the 2215 RGB-D pairs from RGB-D-D
+dataset to generate the noisy training dataset. Testing dataset
+consists of 405 RGB-D pairs from noisy RGB-D-D dataset
+and 3 RGB-D pairs from ToFMark dataset. As shown in the
+last row of Table 1 and Table 2, our method achieves the low-
+est RMSE in noisy RGB-D-D dataset and the lowest RMSE
+in ToFMark dataset, which proves its ability for noise re-
+moving. As shown in Fig. 6, it is observed that DKN (Kim,
+Ponce, and Ham 2021) and DCTNet (Zhao et al. 2022) intro-
+duce some texture artifacts and noise in the low-frequency
+region, while SFG recovers clean surface owing to PEA with
+effective texture removing.
+Experiments on Synthetic Datasets
+Since most popular methods are designed for synthetic
+datasets, we further evaluate our method on NYU-v2
+datasets for a more comprehensive comparison. Following
+the widely used data splitting criterion, we sample 1000
+
+T(a) LR depth
+(b) DJFR
+(c) DKN
+(d) FDSR
+(e) CFUNet
+(f) SFG (ours)
+(g) Groundtruth
+× 8
+× 16 
+Figure 7: Visual comparison of ×8 and ×16 DSR results on NYU-v2 dataset.
+RMSE
+TGV
+FBS
+DJFR
+GbFT
+PAC
+CUNet
+FDKN
+DKN
+FDSR
+DCTNet
+CTKT
+SFG (ours)
+×4
+4.98
+4.29
+2.38
+3.35
+2.39
+1.89
+1.86
+1.62
+1.61
+1.59
+1.49
+1.45
+×8
+11.23
+8.94
+4.94
+5.73
+4.59
+3.58
+3.33
+3.26
+3.18
+3.16
+2.73
+2.84
+×16
+28.13
+14.59
+9.18
+9.01
+8.09
+6.96
+6.78
+6.51
+5.86
+5.84
+5.11
+5.56
+Table 3: Quantitative comparison on NYU-v2 dataset in terms of average RMSE (cm).
+Model
+RMSE
+CFUNet
+4.22
+CFUNet w/o TriSA
+4.34
+CFUNet w/o cross-attention
+4.57
+Table 4: Ablation study of CFUNet on RGB-D-D dataset.
+Datasets
+SFG
+SFG w/o PEANet
+RGB-D-D
+3.88
+4.22
+NYU-v2 (×4)
+1.45
+1.82
+NYU-v2 (×8)
+2.84
+3.76
+NYU-v2 (×16)
+5.55
+5.90
+Table 5: Ablation study (in RMSE) of PEANet.
+RGB-D pairs for training and the rest 449 RGB-D pairs
+for testing. As shown in the Table 3, the proposed method
+still achieves comparable results with the SOTA methods
+on all upsampling cases (×4, ×8, ×16). In addition, Fig. 7
+presents that our ×8 and ×16 upsampled depth maps own
+higher accuracy and more convincing results. It verifies that
+our method not only performs DSR well in low-quality maps
+with noise and missing structure, but also achieves high-
+quality precision in the case of large-scale upsampling.
+Ablation Analysis
+Ablation study on CFUNet. As shown in the first row of the
+Table 4, we still achieve the lowest RMSE criterion just with
+the single CFUNet (SFG w/o PEANet) on RGB-D-D dataset
+when compare with SOTA methods. It proves the effective-
+ness of the learned structure flow map for real DSR. The
+Table 4 also shows that removing the trilateral self-attention
+(TriSA) and cross-attention module in CFUNet causes per-
+formance degradation on RGB-D-D datasets, which verifies
+the necessary of the depth feature enhancement for reliable
+flow map generation.
+K=0 (w/o FPA)
+K=1
+K=2
+K=3
+Figure 8: Visual comparison of guidance features using FPA
+with different iteration times K, i.e., from 0 (w/o FPA) to 3.
+Ablation study on PEANet. To analyze the effectiveness of
+PEANet, we train the network with and without PEANet on
+the synthetic dataset (NYU-v2) and the real-world dataset
+(RGB-D-D). As shown in the Table 5, PEANet consistently
+brings the RMSE gain under both real and synthetic dataset
+settings. It’s mainly due to our edge-focused guidance fea-
+ture learning for robust edge refinement. In addition, Fig. 8
+shows the guidance features under varying iteration times
+in FPA (Flow-enhanced Pyramid Attention) module from
+0 (w/o FPA) to 3. Visually, as the number of iterations in-
+creases, the edge regions tend to receive more attention.
+Conclusion
+In this paper, we proposed a novel structure flow-guided
+DSR framework for real-world depth super-resolution,
+which deals with issues of structural distortion and edge
+noise. For the structural distortion, a cross-modality flow-
+guided upsampling network was presented to learn a reli-
+able cross-modality flow between depth and the correspond-
+ing RGB guidance for the reconstruction of the distorted
+depth edge, where a trilateral self-attention combines the ge-
+ometric and semantic correlations for structure flow learn-
+ing. For the edge noise, a flow-enhanced pyramid edge at-
+tention network was introduced to produce edge attention
+based on the learned flow map and learn the edge-focused
+guidance feature for depth edge refinement with a pyramid
+network. Extensive experiments on both real-world and syn-
+thetic datasets demonstrated the superiority of our method.
+
+Acknowledgement
+This work was supported by the National Science Fund of
+China under Grant Nos. U1713208 and 62072242.
+References
+AlBahar, B.; and Huang, J.-B. 2019. Guided image-to-image
+translation with bi-directional feature transformation.
+In
+ICCV, 9016–9025.
+Barron, J. T.; and Poole, B. 2016. The fast bilateral solver.
+In ECCV, 617–632.
+Blum, M.; Springenberg, J. T.; W¨ulfing, J.; and Riedmiller,
+M. 2012. A learned feature descriptor for object recognition
+in RGB-D data. In ICRA, 1298–1303.
+Chen, B.; and Jung, C. 2018.
+Single depth image super-
+resolution using convolutional neural networks. In ICASSP,
+1473–1477.
+Deng, X.; and Dragotti, P. L. 2020. Deep convolutional neu-
+ral network for multi-modal image restoration and fusion.
+IEEE transactions on pattern analysis and machine intelli-
+gence, PP(99): 1–1.
+Dosovitskiy, A.; Fischer, P.; Ilg, E.; Hausser, P.; Hazirbas,
+C.; Golkov, V.; van der Smagt, P.; Cremers, D.; and Brox, T.
+2015. FlowNet: Learning Optical Flow With Convolutional
+Networks. In ICCV, 2758–2766.
+Eitel, A.; Springenberg, J. T.; Spinello, L.; Riedmiller, M.;
+and Burgard, W. 2015. Multimodal deep learning for robust
+RGB-D object recognition. In IROS, 681–687.
+Ferstl, D.; Reinbacher, C.; Ranftl, R.; R¨uther, M.; and
+Bischof, H. 2013. Image guided depth upsampling using
+anisotropic total generalized variation. In ICCV, 993–1000.
+Guo, C.; Li, C.; Guo, J.; Cong, R.; Fu, H.; and Han, P. 2019.
+Hierarchical Features Driven Residual Learning for Depth
+Map Super-Resolution. IEEE Transactions on Image Pro-
+cessing, 2545–2557.
+Guo, Y.; Chen, J.; Wang, J.; Chen, Q.; Cao, J.; Deng, Z.; Xu,
+Y.; and Tan, M. 2020. Closed-loop matters: Dual regression
+networks for single image super-resolution. In CVPR, 5407–
+5416.
+Hao, X.; Lu, T.; Zhang, Y.; Wang, Z.; and Chen, H. 2019.
+Multi-Source Deep Residual Fusion Network for Depth Im-
+age Super-resolution. In RSVT, 62–67.
+He, L.; Zhu, H.; Li, F.; Bai, H.; Cong, R.; Zhang, C.; Lin,
+C.; Liu, M.; and Zhao, Y. 2021. Towards Fast and Accurate
+Real-World Depth Super-Resolution: Benchmark Dataset
+Baseline and. In CVPR, 9229–9238.
+Hou, J.; Dai, A.; and Nießner, M. 2019. 3d-sis: 3d semantic
+instance segmentation of rgb-d scans. In CVPR, 4421–4430.
+Kim, B.; Ponce, J.; and Ham, B. 2021. Deformable kernel
+networks for joint image filtering. International Journal of
+Computer Vision, 129(2): 579–600.
+Li, X.; You, A.; Zhu, Z.; Zhao, H.; Yang, M.; Yang, K.; Tan,
+S.; and Tong, Y. 2020. Semantic flow for fast and accurate
+scene parsing. In ECCV, 775–793. Springer.
+Li, Y.; Huang, J.-B.; Ahuja, N.; and Yang, M.-H. 2016. Deep
+joint image filtering. In ECCV, 154–169.
+Li, Y.; Huang, J.-B.; Ahuja, N.; and Yang, M.-H. 2019. Joint
+image filtering with deep convolutional networks.
+IEEE
+transactions on pattern analysis and machine intelligence,
+41(8): 1909–1923.
+Liu, W.; Chen, X.; Yang, J.; and Wu, Q. 2016. Robust color
+guided depth map restoration. IEEE Transactions on Image
+Processing, 26(1): 315–327.
+Liu, X.; Zhai, D.; Chen, R.; Ji, X.; Zhao, D.; and Gao, W.
+2018. Depth restoration from RGB-D data via joint adaptive
+regularization and thresholding on manifolds. IEEE Trans-
+actions on Image Processing, 28(3): 1068–1079.
+Lutio, R. d.; D’aronco, S.; Wegner, J. D.; and Schindler, K.
+2019. Guided super-resolution as pixel-to-pixel transforma-
+tion. In ICCV, 8829–8837.
+Meuleman, A.; Baek, S.-H.; Heide, F.; and Kim, M. H. 2020.
+Single-shot monocular rgb-d imaging using uneven double
+refraction. In CVPR, 2465–2474.
+Newcombe, R. A.; Izadi, S.; Hilliges, O.; Molyneaux, D.;
+Kim, D.; Davison, A. J.; Kohi, P.; Shotton, J.; Hodges, S.;
+and Fitzgibbon, A. 2011. Kinectfusion: Real-time dense sur-
+face mapping and tracking. In ISMAR, 127–136. Ieee.
+Silberman, N.; Hoiem, D.; Kohli, P.; and Fergus, R. 2012.
+Indoor segmentation and support inference from rgbd im-
+ages. In ECCV, 746–760.
+Song, X.; Dai, Y.; Zhou, D.; Liu, L.; Li, W.; Li, H.; and Yang,
+R. 2020. Channel attention based iterative residual learning
+for depth map super-resolution. In CVPR, 5631–5640.
+Su, H.; Jampani, V.; Sun, D.; Gallo, O.; Learned-Miller, E.;
+and Kautz, J. 2019. Pixel-adaptive convolutional neural net-
+works. In CVPR, 11166–11175.
+Sun, B.; Ye, X.; Li, B.; Li, H.; Wang, Z.; and Xu, R. 2021.
+Learning Scene Structure Guidance via Cross-Task Knowl-
+edge Transfer for Single Depth Super-Resolution. In CVPR,
+7792–7801.
+Tak-Wai; Loy, C. C.; and Tang, X. 2016. Depth Map Super-
+Resolution by Deep Multi-Scale Guidance. In ECCV, 353–
+369.
+Tang, Q.; Cong, R.; Sheng, R.; He, L.; Zhang, D.; Zhao, Y.;
+and Kwong, S. 2021. BridgeNet: A Joint Learning Network
+of Depth Map Super-Resolution and Monocular Depth Esti-
+mation. In ACMMM, 2148–2157.
+Wang, K.; Zhang, Z.; Yan, Z.; Li, X.; Xu, B.; Li, J.; and
+Yang, J. 2021. Regularizing Nighttime Weirdness: Efficient
+Self-Supervised Monocular Depth Estimation in the Dark.
+In ICCV, 16055–16064.
+Wang, Z.; Ye, X.; Sun, B.; Yang, J.; Xu, R.; and Li, H. 2020.
+Depth upsampling based on deep edge-aware learning. Pat-
+tern Recognition, 103: 107274.
+Yan, Z.; Wang, K.; Li, X.; Zhang, Z.; Li, G.; Li, J.; and Yang,
+J. 2022. Learning Complementary Correlations for Depth
+Super-Resolution With Incomplete Data in Real World.
+IEEE Transactions on Neural Networks and Learning Sys-
+tems.
+Yang, J.; Ye, X.; Li, K.; Hou, C.; and Wang, Y. 2014. Color-
+guided depth recovery from RGB-D data using an adaptive
+
+autoregressive model. IEEE transactions on image process-
+ing, 23(8): 3443–3458.
+Zhao, Z.; Zhang, J.; Xu, S.; Lin, Z.; and Pfister, H. 2022.
+Discrete cosine transform network for guided depth map
+super-resolution. In CVPR, 5697–5707.
+Zhu, J.; Zhai, W.; Cao, Y.; and Zha, Z.-J. 2018. Co-occurrent
+structural edge detection for color-guided depth map super-
+resolution. In MMM, 93–105.
+Zuo, Y.; Wu, Q.; Fang, Y.; An, P.; Huang, L.; and Chen,
+Z. 2019. Multi-scale frequency reconstruction for guided
+depth map super-resolution via deep residual network. IEEE
+Transactions on Circuits and Systems for Video Technology,
+30(2): 297–306.
+

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+P3DC-Shot: Prior-Driven Discrete Data Calibration for
+Nearest-Neighbor Few-Shot Classification
+Shuangmei Wanga,∗, Rui Maa,b,∗, Tieru Wua,b,∗∗, Yang Caoa,∗∗
+aJilin University, No. 2699 Qianjin Street, Changchun, 130012, China
+bEngineering Research Center of Knowledge-Driven Human-Machine Intelligence, MOE, No. 2699 Qianjin Street, Changchun, 130012, China
+Abstract
+Nearest-Neighbor (NN) classification has been proven as a simple and effective approach for few-shot learning. The
+query data can be classified efficiently by finding the nearest support class based on features extracted by pretrained
+deep models. However, NN-based methods are sensitive to the data distribution and may produce false prediction if
+the samples in the support set happen to lie around the distribution boundary of different classes. To solve this issue,
+we present P3DC-Shot, an improved nearest-neighbor based few-shot classification method empowered by prior-
+driven data calibration. Inspired by the distribution calibration technique which utilizes the distribution or statistics of
+the base classes to calibrate the data for few-shot tasks, we propose a novel discrete data calibration operation which is
+more suitable for NN-based few-shot classification. Specifically, we treat the prototypes representing each base class
+as priors and calibrate each support data based on its similarity to different base prototypes. Then, we perform NN
+classification using these discretely calibrated support data. Results from extensive experiments on various datasets
+show our efficient non-learning based method can outperform or at least comparable to SOTA methods which need
+additional learning steps.
+Keywords:
+Few-Shot Learning, Image Classification, Prototype, Calibration
+1. Introduction
+Deep learning has triggered significant breakthroughs
+in many computer vision tasks, such as image classifi-
+cation [1, 2, 3], object detection [4, 5, 6], and seman-
+tic segmentation [7, 8, 9] etc. One key factor for the
+success of deep learning is the emergence of large-scale
+datasets, e.g., ImageNet [2], MSCOCO [10], Cityscapes
+[11], just to name a few. However, it is difficult and
+expensive to collect and annotate sufficient data sam-
+ples to train a deep model with numerous weights. The
+data limitation has become a main bottleneck for more
+broader application of deep leaning, especially for the
+tasks involving rarely seen samples. On the other hand,
+human can learn to recognize novel visual concepts
+from only a few samples. There is still a notable gap
+• This work is supported in part by the National Key Research
+and Development Program of China (Grant No. 2020YFA0714103)
+and the National Natural Science Foundation of China (Grant No.
+61872162 and 62202199).
+∗Co-first authors.
+∗∗Corresponding authors.
+between human intelligence and the deep learning based
+artificial intelligence. Few-shot learning (FSL) aims to
+learn neural models for novel classes with only a few
+samples. Due to its ability for generalization, FSL has
+attracted extensive interests in recent years [12, 13, 14].
+Few-shot classification is the most widely studied
+FSL task which attempts to recognize new classes or
+classify data in an unseen query set. Usually, few-shot
+classification is formulated in a meta-learning frame-
+work [15, 16, 17, 18, 19, 20, 21, 22, 23].
+In the
+meta-training stage, the N-way K-shot episodic training
+paradigm is often employed to learn generalizable clas-
+sifiers or feature extractors for data of the base classes.
+Then, in the meta-testing stage, the meta-learned clas-
+sifiers can quickly adapt to a few annotated but unseen
+data in a support set and attain the ability to classify the
+novel query data. Although meta-learning has shown
+the effectiveness for few-shot classification, it is unclear
+how to set the optimal class number (N) and per-class
+sample number (K) when learning the classifiers. Also,
+the learned classifier may not perform well when the
+sample number K used in meta-testing does not match
+Preprint submitted to Elsevier
+January 3, 2023
+arXiv:2301.00740v1  [cs.CV]  2 Jan 2023
+
+the one used in the meta-training [24].
+On the other hand, nearest-neighbor (NN) based clas-
+sification has been proven as a simple and effective ap-
+proach for FSL. Based on features obtained from the
+meta-learned feature extractor [15, 16] or the pretrained
+deep image models [25], the query data can be effi-
+ciently classified by finding the nearest support class.
+Specifically, the prediction is determined by measuring
+the similarity or distance between the query feature and
+the prototypes (i.e., average or centroid) of the support
+features. From the geometric view, NN-based classi-
+fication can be solved using a Voronoi Diagram (VD)
+which is a partition of the space formed by the support
+features [26, 27]. Given a query feature, its class can
+be predicted by computing the closest Voronoi cell that
+corresponds to a certain support class. With proper VD
+construction and feature distance metrics, the state-of-
+the-art performance can be achieved for few-shot clas-
+sification [28].
+However, due to the limited number
+of support samples, NN-based few-shot classification is
+sensitive to the distribution of the sampled data and may
+produce false prediction if the samples in the support set
+happen to lie around the distribution boundary of differ-
+ent classes (see Figure 1 left).
+To solve above issues, various efforts have been paid
+to more effectively utilize the knowledge or priors from
+the base classes for few-shot classification. One natural
+way is to learn pretrained classifiers or image encoders
+with the abundant labeled samples of base classes and
+then adapt them the novel classes via transfer learning
+[29, 30, 31, 23]. Meanwhile, it has been shown that
+variations in selecting the base classes can lead to dif-
+ferent performance on the novel classes [32, 33, 34] and
+how to select the base classes for better feature repre-
+sentation learning still needs more investigation.
+On
+the other hand, a series of works [35, 36, 37, 38] per-
+form data calibration to the novel classes so that the re-
+sults are less affected by the limited number of support
+samples. One representative is Distribution Calibration
+(DC) [38] which assumes the features of the data fol-
+low the Gaussian distribution and transfers the statis-
+tics from the similar base classes to the novel classes.
+Then, DC trains a simple logistic regression classifier
+to classify the query features using features sampled
+from the calibrated distributions of the novel classes.
+Although DC has achieved superior performance than
+previous meta-learning [19, 21, 22] or transfer-learning
+[29, 30, 31, 23] based methods, it relies on the strong as-
+sumption for Gaussian-like data distribution and it can-
+not be directly used for NN-based few-shot classifica-
+tion.
+In this paper, we propose P3DC-Shot, an improved
+Support sample                               Query sample                       Calibrated support sample
+Figure 1: When samples in the support set lie around the distribution
+boundary of different classes, the NN classifier may produce false pre-
+diction. By performing discrete calibration for each support sample
+using priors from the base classes, the calibrated support data is trans-
+formed closer to the actual class centroid and can lead to less-biased
+NN classification. The colored regions represent the underlying data
+distribution of different classes. The gray lines are the predicted deci-
+sion boundaries by the NN classifier.
+NN-based few-shot classification method that employs
+prior information from base classes to discretely cali-
+brate or adjust the support samples so that the calibrated
+data is more representative for the underlying data dis-
+tribution (Figure 1 right). Our main insight is even the
+novel classes have not been seen before, they still share
+similar features to some base classes, and the prior in-
+formation from the base classes can serve as the context
+data for the novel classes. When only a few support
+samples are available for the novel classes, performing
+prior-driven calibration can alleviate the possible bias
+introduced by the few-shot support samples. With the
+calibrated support samples, the query data can be more
+accurately classified by a NN-based classifier.
+Specifically, for the prior information, we compute
+the prototype, i.e., the average of features, for each base
+class. Then, we propose three different schemes for se-
+lecting the similar prototypes to calibrate the support
+data. Firstly, we propose the sample-level calibration
+which selects the top M most similar base prototypes for
+each support sample and then apply weighted averaging
+between each support sample and selected prototypes to
+obtain the calibrated support sample. Secondly, to uti-
+lize more context from the base classes, we propose the
+task-level calibration which combines the most similar
+base prototypes for each support sample into a union
+and performs the calibration for the support samples us-
+ing each prototype in the union. In addition, we pro-
+pose a unified calibration scheme that combines the two
+above schemes so that the calibration can exploit dif-
+ferent levels of prior information from the base classes.
+To utilize the calibrated support samples for the NN-
+based classification, we further obtain the prototypes of
+2
+
+the support class using an attention-weighted averaging,
+while the attention weights are computed between the
+query sample and each calibrated support sample. Fi-
+nally, the classification of a query sample is simply de-
+termined by finding its nearest support prototype mea-
+sured by the cosine similarity.
+Comparing to DC, our P3DC-Shot adopts the simi-
+lar idea of transferring the information or statistics from
+the base classes to the novel classes. The key differ-
+ence is our data calibration is performed on each indi-
+vidual support sample rather than the distribution pa-
+rameters and we employ the NN-based classification in-
+stead of the learned classifier as in DC. Comparing to
+other NN-based few-shot classification methods such as
+SimpleShot [25], since our support data is calibrated,
+the NN classification is less affected by the sampling
+bias for the support data, e.g, the calibrated data is more
+likely to be close to the center of the corresponding
+novel class. We conduct extensive comparisons with re-
+cent state-of-the-art few-shot classificaiton methods on
+miniImageNet [2], tiredImageNet [39] and CUB [40]
+and the results demonstrate the superiority and general-
+izability of our P3DC-Shot. Ablation studies on differ-
+ent calibration schemes, i.e., different weights between
+the sample-level and task-level calibration also show the
+necessity of combining two schemes for better results.
+In summary, our contributions are as follows:
+1. We
+propose
+P3DC-Shot,
+a
+prior-driven
+dis-
+crete data calibration strategy for nearest-neighbor
+based few-shot classification to enhance the
+model’s robustness to the distribution of the sup-
+port samples.
+2. Without additional training and expensive compu-
+tation, the proposed method can efficiently cali-
+brate each support sample using information from
+the prototypes of the similar base classes.
+3. We conduct extensive evaluations on three discrete
+calibration schemes on various datasets and the re-
+sults show our efficient non-learning based method
+can outperform or at least comparable to SOTA
+few-shot classification methods.
+2. Related Work
+In this section, we first review the representative
+meta-learning and transfer learning based few-shot clas-
+sification techniques. Then, we summarize the nearest-
+neighbor and data calibration based approaches which
+are most relevant to our P3DC-Shot.
+Meta-learning based few-shot classification. Meta-
+learning [41] has been widely adopted for few-shot clas-
+sification.
+The core idea is to leverage the episodic
+training paradigm to learn generalizable classifiers or
+feature extractors using the data from the base classes
+in an optimization-based framework [18, 19, 20, 21,
+22], as well as learn a distance function to measure
+the similarity between the support and query samples
+through metric-learning [42, 15, 17, 43, 44, 37]. For
+example, MAML [19] is one of the most representa-
+tive optimization-based meta-learning method for few-
+shot classification and its goal is to learn good net-
+work initialization parameters so that the model can
+quickly adapt to new tasks with only a small amount
+of new training data from the novel classes. For metric-
+learning based methods such as the Matching Networks
+[15], Prototypical Networks [16] and Relation Net-
+works [17], the network is trained to either learn an
+embedding function with a given distance function or
+learn both the embedding and the distance function in
+a meta-learning architecture. Unlike the optimization
+and metric-learning based methods which require so-
+phisticated meta-learning steps, our method can directly
+utilize the features extracted by the pretrained models
+and perform the prior-driven calibration to obtain less-
+biased support features for classification.
+Transfer learning based few-shot classification.
+Transfer learning [45, 46, 47] is a classic machine learn-
+ing or deep learning technique that aims to improve
+the the learning of a new task through the transfer of
+knowledge from one or more related tasks that have al-
+ready been learned. Pretraining a deep network on the
+base dataset and transferring knowledge to the novel
+classes via fine-tuning [31, 48, 30] has been shown as
+the strong baseline for the few-shot classification. To
+learn better feature representations which can lead to
+improved few-shot fine-tuning performance, Mangla et
+al. [29] propose S2M2, the Self-Supervised Manifold
+Mixup, to apply regularization over the feature mani-
+fold enriched via the self-supervised tasks. In addition
+to training new linear classifiers based on the pretrained
+weights learned from the base classes, Meta-Baseline
+[23] performs meta-learning to further optimize the pre-
+trained weights for few-shot classification. On the other
+hand, it has been shown the results of the transfer learn-
+ing based methods depend on different selections of the
+base classes for pretraining [32, 33], while how to se-
+lect the base classes to achieve better performance is
+still challenging [34]. In comparison, our P3DC-shot
+does not need the additional cost for feature represen-
+tation learning and can more effectively utilize the base
+classes in a NN-based classification framework.
+Nearest neighbor based few-shot classification.
+NN-based classification has also been investigated for
+few-shot classification. The main idea is to compute the
+3
+
+prototypes of the support samples, i.e., the mean or cen-
+troid of the support features, and classify the query sam-
+ple using metrics such as L2 distance, cosine similarity
+or a learned distance function. In SimpleShot [25], it
+shows nearest neighbor classification with features sim-
+ply normalized by L2 norm and measured by Euclidean
+distance can achieve competitive few-shot classification
+results. Instead of performing nearest neighbor classifi-
+cation on the image-level features, Li et al. [49] intro-
+duces a Deep Nearest Neighbor Neural Network which
+performs nearest neighbor search over the deep local
+descriptors and defines an image-to-class measure for
+few-shot classification. From a geometric view, Ma et
+al. [50] utilize the Cluster-induced Voronoi Diagram
+(CIVD) to incorporate cluster-to-point and cluster-to-
+cluster relationships to the nearest neighbor based clas-
+sification.
+Similar to above methods, our method is
+based on the nearest prototype classification, while
+we perform the prior-driven data calibration to obtain
+less-biased support data for the prototype computation.
+Meanwhile, computing the attentive or reweighted pro-
+totypes [51, 52, 53] that are guided by the base classes
+or query samples has also been investigated recently.
+We follow the similar idea and compute the attention-
+weighted prototypes for NN-based classification.
+Data calibration for few-shot classification. Due to
+the limited number of samples, the prototypes or cen-
+troids computed from the few-shot support data may be
+biased and cannot represent the underlying data distri-
+bution. Simply performing NN-based classification on
+these biased prototypes will lead to inaccurate classi-
+fication. Several methods have been proposed to cali-
+brate or rectify the data to obtain better samples or pro-
+totypes of the support class [35, 36, 37, 54, 38]. Using
+the images in the base classes, RestoreNet [35] learns
+a class agnostic transformation on the feature of each
+image to move it closer to the class center in the fea-
+ture space. To reduce the bias caused by the scarcity
+of the support data, Liu et al., [36] employ the pseudo-
+labeling to add unlabelled samples with high prediction
+confidence into the support set for prototype rectifica-
+tion. In [37], Guo et al. propose a Pair-wise Similar-
+ity Module to generate calibrated class centers that are
+adapted to the query sample. Instead of calibrating in-
+dividual support samples, Distribution Calibration (DC)
+[38] aims to calibrate the underlying distribution of the
+support classes by transferring the Gaussian statistics
+from the base classes. With sufficient new support data
+sampled from the calibrated distribution, an additional
+classifier is trained in [38] to classify the query sam-
+ple. In contrast to these methods, we do not require
+additional training or assumption of the underlying dis-
+tribution. Instead, we directly use the prototypes of the
+base classes to calibrate each support sample individ-
+ually and we adopt the NN-based classification which
+makes the whole pipeline discrete and efficient. One
+recent work that is similar to ours is Xu et al.
+[54]
+which proposes the Task Centroid Projection Removing
+(TCPR) module and transforms all support and query
+features in a given task to alleviate the sample selection
+bias problem. Comparing to [54], we only calibrate the
+support samples using the priors from the base classes
+and keep the query samples unchanged.
+3. Method
+To effectively utilize the prior knowledge from the
+base classes, we first propose two independent calibra-
+tion strategies, i.e., sample-level calibration and task-
+level calibration, which exploit different levels of infor-
+mation from the base classes. Then, we combine the
+sample-level and task-level calibration together to ob-
+tain the final calibrated support samples which will be
+used for the nearest neighbor classification.
+Figure 2 shows an illustration of the P3DC-Shot
+pipeline. Given a pretrained feature extractor F and a
+set of prototypes of base classes, we perform the prior-
+driven discrete calibration to the normalized features of
+the support data. Initially, the query sample in green
+is closer to the support sample in yellow.
+After the
+proposed calibration using the related base class proto-
+types, the query sample becomes closer to the calibrated
+support sample in blue. In the following, we provide
+technical details of the P3DC-Shot for few-shot classi-
+fication.
+3.1. Problem Statement
+In this paper, we focus on the few-shot image clas-
+sification which aims to classify the new image sam-
+ples from the novel classes with just a few labeled im-
+age samples. Normally, the new data sample is called
+a query sample and the labelled samples are called sup-
+port samples. With the aid of a set of base classes rep-
+resented by their prototypes Pb = {pb
+i }nb
+i=1, our goal is to
+calibrate the support samples from novel-class so that
+they can be better matched with the query samples by
+a nearest neighbor classifier. Here, all data samples are
+represented by the features computed from a pretrained
+feature extractor F(·) : X → Rd, while X is the domain
+of the image space and d is the dimension of the feature
+space; pb
+i is the prototype of a base class, which is com-
+puted as the average feature of the samples within the
+4
+
+L2 norm
+Calibration
+������������
+Feature
+extraction
+������������1
+������������2
+������������
+Support data
+Query data
+Final calibrated support features
+Endpoint of sample-level calibration
+Endpoint of task-level calibration
+All base class prototypes
+or
+Selected prototypes for a sample
++
+Selected prototypes for a task
+̅������������1
+̅������������1
+̅������������2
+̅������������2
+������������1
+������������
+������������2
+������������
+̅������������1
+�������������
+̅������������2
+Figure 2: An illustration of the P3DC-Shot pipeline for the 2-way 1-shot scenario. Note that the direct interpolation of the three triangle vertices
+return a feature on the triangle plane. After normalization, the final calibrated features ¯xu
+1 and ¯xu
+2 are on the hypersphere of the normalized space.
+class; nb is the number of all base classes. For simplic-
+ity, we directly use xi to represent the feature F(xi) of
+an image xi.
+We follow the conventional few-shot learning setting,
+i.e., build a series of N-way K-shot tasks where N is the
+number of novel classes and K is the number of sup-
+port samples in each task.
+Formally, each task con-
+sists of a support set S = {(xi, yi)}N×K
+i=1
+and a query set
+Q = {qi}N×K+N×Q
+i=N×K+1 . Here, yi is the label of the corre-
+sponding sample, which is known for the support set
+and unknown for the query set; Q is the number of query
+sample for each novel class in the current task. Given a
+support feature xi, we perform our prior-driven calibra-
+tion to obtain the calibrated support feature xc
+i = C(xi),
+where C(·) : Rd → Rd conducts feature transformation
+based on the information from the base classes. Then,
+we predict the label of a query feature by performing
+nearest neighbor classification w.r.t the novel class pro-
+totypes computed from the calibrated support feature(s).
+3.2. Prior-Driven Discrete Data Calibration
+Before we perform calibration to the support data, we
+first apply L2 normalization to the support and query
+features.
+It is shown in SimpleShot [25] that using
+L2-normalized feature with a NN-based classifier can
+lead to competitive results for few-shot classification.
+Hence, we obtain ¯xi for a support feature xi by:
+¯xi = normalize(xi) =
+xi
+∥xi∥2
+.
+(1)
+Similarly, the normalization of the query features are
+also computed: ¯qi = normalize(qi). By working with
+the normalized features, we can obviate the absolute
+scales of the features and focus on the similarities and
+differences on their directions. Note that, the normal-
+ized features are used in the feature combination step
+(Eq. 7, 10 and 11) for obtaining the interpolation be-
+tween the normalized features and in the NN-based clas-
+sification step (Eq. 12) for performance improvement.
+Next, we propose the sample-level and task-level cal-
+ibration, and their combination to utilize the priors from
+the base classes for obtaining the less-biased support
+features.
+3.2.1. Sample-Level Calibration
+According to previous works [55, 38] which also use
+the information from base classes for classifying the
+new classes, the base classes with higher similarities
+to the query classes are more important than other base
+classes. Hence, we first propose to perform calibration
+based on the top similar base classes for each support
+sample. Moreover, following DC [38], we apply the
+Tukeys’s Ladder of Powers transformation [56] to the
+features of the support samples before the calibration:
+˜xi =
+� xλ
+i
+if λ � 0
+log(xi)
+if λ = 0
+(2)
+Here, λ is a hyperparameter which controls the distri-
+bution of the transformed feature, with a smaller λ can
+lead to a less skewed feature distribution. We set λ = 0.5
+and obtain the transformed support feature ˜xi from the
+original feature xi.
+Then, we select the top M base classes with higher
+similarities to a transformed support feature ˜xi:
+ΛM
+i = {pb
+j| j ∈ topM(Si)},
+(3)
+where Si = {< ˜xi, pb
+j > | j ∈ {1, . . . nb}}.
+(4)
+Here, ΛM
+i stores the M nearest base prototypes with re-
+spect to a transformed support feature vector ˜xi; topM(·)
+is an operator that returns the index of top M elements
+from Si, the similarity set of ˜xi, while the similarity be-
+tween ˜xi and a base prototype pb
+j is computed by the
+5
+
+inner product < ·, · >. In DC [38], the distributions
+of the base and novel classes are assumed as Gaussian
+distribution and the statistics (mean and co-variance) of
+the base classes are used to calibrate the distribution of
+the novel classes. In contrast, we directly use the sim-
+ilar base prototypes to calibrate each support feature.
+Specifically, the calibration for ˜xi driven by base proto-
+types pb
+j ∈ ΛM
+i is computed as:
+si = ˜xi +
+�
+j∈ΛM
+i
+wijpb
+j,
+(5)
+where the weights of the M nearest base classes proto-
+types in ΛM
+i
+are obtained by applying Softmax to the
+similarities between ˜xi and these prototypes:
+wij =
+e<˜xi,pb
+j>
+�
+k∈ΛM
+i e<˜xi,pb
+k> , j ∈ ΛM
+i .
+(6)
+It should be noted that, in Eq. 5, the support feature ˜xi
+is a transformed feature, while the base prototypes are
+in the original feature space. This setting is the same
+as DC does for calibrating the distribution of the novel
+classes and it can be understood as follows: 1) the trans-
+formation can initially reduce the skewness of the few-
+shot-sampled support features; 2) the term wijpb
+j can be
+regarded as the projection of ˜xi w.r.t prototype pb
+j; 3)
+˜xi is calibrated based on its projects to all of its similar
+base prototypes in ΛM
+i .
+Finally, the sample-level calibration for a normalized
+support sample ¯xi is defined as:
+¯xs
+i = normalize((1 − α)¯xi + α¯si),
+(7)
+where α ∈ [0, 1] is a parameter to linearly combine
+the normalized support feature ¯xi and normalized base-
+prototypes-driven calibration ¯si = norm(si). As shown
+in Figure 2, ¯xi and ¯si form a line in the normalized fea-
+ture space and ¯xs
+i is the normalization of a in-between
+point on this line. In general, the sample-level calibra-
+tion can rectify each support sample based on its own
+top M most similar base classes.
+3.2.2. Task-Level Calibration
+By performing the sample-level calibration, the bias
+induced by the few-shot support samples can be reduced
+to a certain degree. However, when the sampling bias
+is too large, e.g., the support sample is lying near the
+boundary of a class, the set of similar base classes ΛM
+i
+obtained by Eq. 3 may also be biased. To alleviate such
+bias, we propose the task-level calibration which utilizes
+the base prototypes related to all support samples when
+calibrating each individual support feature. Concretely,
+for a support set S = {(xi, yi)}N×K
+i=1 w.r.t a task T , we col-
+lect the top M similar base prototypes for each support
+sample and form a union of related base prototypes for
+T :
+ΛT =
+N×K
+�
+i=1
+ΛM
+i .
+(8)
+Then, for a transformed support sample ˜xi obtained
+by Eq. 2, the calibration using all of the task-related
+base prototypes is computed by:
+ti = ˜xi +
+�
+j∈ΛT
+wi jpb
+j,
+(9)
+where wi j is calculated in the similar way as Eq. 6, but
+the similarities are computed using the prototypes from
+ΛT instead of ΛM
+i . By involving more prototypes to cal-
+ibrate the support samples, the bias caused by only using
+nearby prototypes for a near-boundary support sample
+can be reduced.
+Then, we define the task-level calibration for a nor-
+malized support sample ¯xi as:
+¯xt
+i = normalize((1 − β)¯xi + β¯ti),
+(10)
+where ¯ti is the normalization of ti. Similar to the sample-
+level calibration, ¯xi and ¯ti also form a line in the normal-
+ized feature space, while the calibration for each support
+sample is based on the union of all related base proto-
+types ΛT .
+3.2.3. Unified Model
+The sample-level and task-level calibration utilize
+different levels of information from the base classes to
+rectify the support samples in a discrete manner. To fur-
+ther attain the merits of both calibration schemes, we
+propose a unified model which linearly combines the
+sample-level and task-level calibration:
+xc
+i = ¯xu
+i = normalize((1 − α − β)¯xi + α¯si + β¯ti).
+(11)
+Here, ¯xu
+i which is also denoted as xc
+i , is the final calibra-
+tion for a normalized support sample ¯xi . Geometrically,
+xc
+i can be understood as the normalization of an interpo-
+lated feature point xu
+i locating in the triangle formulated
+by the three vertices ¯xi, ¯si and ¯ti, while 1 − α − β, α and
+β are the barycentric coordinates of xu
+i . Different α and
+β values can lead to different calibration effects. When
+β = 0, the unified model degenerates to the sample-
+level calibration, while when α = 0, the model becomes
+to the task-level calibration. We quantitatively evaluate
+the effects of different α and β values in Section 4.4.
+6
+
+3.3. Nearest Prototype Classifier
+With the calibrated support set Sc = {(xc
+i , yi)}N×K
+i=1 , we
+compute the prototypes {pn}N
+n=1 for the novel classes and
+perform cosine similarity based nearest classification
+for a query feature q. To simplify the notation, we fur-
+ther represent Sc = {Sc
+n}N
+n=1, while Sc
+n = {(xc
+k, yk = n)}K
+k=1
+is the support set for a novel class CLS n.
+For the 1-shot case, each calibrated support sample
+becomes one prototype and the class of the query fea-
+ture is predicted by the nearest prototype classifier:
+y∗ = max
+pn cos(¯q, pn),
+(12)
+where pn = xc
+n is the calibrated prototype for novel class
+CLS n and ¯q is the normalization of query q.
+For the multi-shot case, one way to obtain the pro-
+totype for a novel class is simply to compute the av-
+erage of all support features for the given class as in
+Prototypical Networks [16]. However, merely using the
+unweighted average of the support features as prototype
+does not consider the importance of the support samples
+w.r.t the query. Therefore, we adopt the idea of attentive
+prototype which is proposed in recent works [51, 53] for
+query-guided prototype computation. In our implemen-
+tation, we define the attention-weighted prototype as:
+pq
+n =
+�
+xc
+k∈Scn
+akxc
+k,
+(13)
+where ak =
+e<q,xc
+k>
+�
+xcm∈Scn e<q,xcm> .
+(14)
+Here, xc
+k and xc
+m are the calibrated support samples be-
+longing to the CLS n’s support set Sc
+n and ak is the atten-
+tion weight computed by applying Softmax to the sim-
+ilarities between query q and these calibrated support
+samples; pq
+n is the CLS n’s prototype guided by query
+q. Similar to Eq. 12, the prediction for a query q is
+obtained by finding the novel class with the nearest pro-
+totype pq
+n.
+4. Experiments
+In this section, we perform quantitative compar-
+isons between our P3DC-Shot and state-of-the-art
+few-shot classification methods on three represen-
+tative datasets.
+We also conduct ablation studies
+on evaluating different hyperparameters and design
+choices for our methods.
+Our code is available at:
+https://github.com/breakaway7/P3DC-Shot.
+4.1. Datasets
+We evaluate our prior-driven data calibration strate-
+gies on three popular datasets for benchmarking few
+shot classificaiton: miniImageNet [2], tieredImageNet
+[39] and CUB [40]. miniImageNet and tieredImageNet
+contain a broad range of classes including various an-
+imals and objects, while CUB is a more fine-grained
+dataset that focuses on various species of birds.
+Specifically, the miniImageNet [2] is derived from
+the ILSVRC-2012 [58] and it contains a subset of 100
+classes, each of which consisting of 600 images. We
+follow the split used in [18] and obtain 64 base, 16 val-
+idation and 20 novel classes for miniImageNet. Comar-
+ing to miniImageNet, the tieredImageNet [39] is a larger
+subset of [58] which contains 608 classes and therefore
+more challenging. We follow [39] and split the tiered-
+ImageNet into 351, 97, and 160 classes for base, vali-
+dation, and novel classes, respectively. For CUB [40], it
+is the short name for Caltech-UCSD Birds 200 dataset,
+which contains a total of 11,788 images covering 200
+categories of different bird species. We split the CUB
+dataset into 100 base, 50 validation and 50 novel classes
+following [31]. Note that the set formed by the base
+classes can also be regarded as the train set and the novel
+classes correspond to the test set.
+4.2. Implementation Details
+For each image in the dataset, we represent it as a
+640-dimensional feature vector which is extracted us-
+ing the WideResNet [59] pretrained by the S2M2 [29]
+work. Our calibration pipeline can efficiently proceed
+in four steps: 1) find the M = 5 nearby base prototypes
+for each support sample xi; 2) compute the endpoint
+of the sample-level calibration for xi, i.e., si; 3) col-
+lect all nearby base prototypes for all support samples
+in the task and compute the endpoint of the task-level
+calibration for xi, i.e., ti; 4) combine the sample-level
+and task-level calibration and obtain the final calibrated
+support sample xc
+i . The parameter α and β for weighting
+the sample-level and task-level calibration are selected
+based on the best results obtained on the validation set
+for each dataset. All experiments are conducted on a
+PC with a 2.70GHz CPU and 16G memory. No GPU
+is needed during the calibration. On average, for a 5-
+way 5-shot task, it takes 0.027 seconds to calibrate the
+support samples and 0.002 seconds for performing the
+nearest prototype classification.
+4.3. Comparison and Evaluation
+To evaluate the performance of our P3DC-Shot, we
+first conduct quantitative comparisons with some rep-
+resentative and state-of-the-art few-short classification
+7
+
+Table 1:
+Quantitative comparison on the test set of miniImageNet, tieredImageNet and CUB. The 5-way 1-shot and 5-way 5-shot classification
+accuracy (%) with 95% confidence intervals are measured. Best results are highlighted in bold and second best are in italic. The last line shows the
+α and β selected based on the valiation set for each dataset. * 8 and 20 are the number of ensembles in DeepVoro and DeepVoro++. † The results
+of [54] on tieredImageNet are obtained using its released code.
+Methods
+miniImageNet
+tieredImageNet
+CUB
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+Meta-learning (metric-learning)
+MatchingNet [15] (2016)
+64.03 ± 0.20
+76.32 ± 0.16
+68.50 ± 0.92
+80.60 ± 0.71
+73.49 ± 0.89
+84.45 ± 0.58
+ProtoNet [16] (2017)
+54.16 ± 0.82
+73.68 ± 0.65
+65.65 ± 0.92
+83.40 ± 0.65
+72.99 ± 0.88
+86.64 ± 0.51
+RelationNet [17] (2018)
+52.19 ± 0.83
+70.20 ± 0.66
+54.48 ± 0.93
+71.32 ± 0.78
+68.65 ± 0.91
+81.12 ± 0.63
+Meta-learning (optimization)
+MAML [19] (2017)
+48.70 ± 1.84
+63.10 ± 0.92
+51.67 ± 1.81
+70.30 ± 0.08
+50.45 ± 0.97
+59.60 ± 0.84
+LEO [21] (2019)
+61.76 ± 0.08
+77.59 ± 0.12
+66.33 ± 0.15
+81.44 ± 0.09
+68.22 ± 0.22
+78.27 ± 0.16
+DCO [22] (2019)
+62.64 ± 0.61
+78.63 ± 0.46
+65.99 ± 0.72
+81.56 ± 0.53
+-
+-
+Transfer learning
+Baseline++ [31] (2019)
+57.53 ± 0.10
+72.99 ± 0.43
+60.98 ± 0.21
+75.93 ± 0.17
+70.40 ± 0.81
+82.92 ± 0.78
+Negative-Cosine [57] (2020)
+62.33 ± 0.82
+80.94 ± 0.59
+-
+-
+72.66 ± 0.85
+89.40 ± 0.43
+S2M2R [29] (2020)
+64.65 ± 0.45
+83.20 ± 0.30
+68.12 ± 0.52
+86.71 ± 0.34
+80.14 ± 0.45
+90.99 ± 0.23
+Nearest neighbor
+SimpleShot [25] (2019)
+64.29 ± 0.20
+81.50 ± 0.14
+71.32 ± 0.22
+86.66 ± 0.15
+-
+-
+DeepVoro(8)∗ [50] (2022)
+66.45 ± 0.44
+84.55 ± 0.29
+74.02 ± 0.49
+88.90 ± 0.29
+80.98 ± 0.44
+91.47 ± 0.22
+DeepVoro++(20)∗ [50] (2022)
+68.38 ± 0.46
+83.27 ± 0.31
+74.48 ± 0.50
+-
+80.70 ± 0.45
+-
+Data calibration
+RestoreNet [35] (2020)
+59.28 ± 0.20
+-
+-
+-
+74.32 ± 0.91
+-
+DC [38] (2021)
+67.79 ± 0.45
+83.69 ± 0.31
+74.24 ± 0.50
+88.38 ± 0.31
+79.93 ± 0.46
+90.77 ± 0.24
+MCL-Katz+PSM [37] (2022)
+67.03
+84.03
+69.90
+85.08
+85.89
+93.08
+S2M2+TCPR† [54] (2022)
+68.05 ± 0.41
+84.51 ± 0.27
+72.67 ± 0.48
+87.96 ± 0.31
+-
+-
+P3DC-Shot (α = 0, β = 0)
+65.93 ± 0.45
+84.06 ± 0.30
+73.56 ± 0.49
+88.50 ± 0.32
+81.61 ± 0.43
+91.36 ± 0.22
+P3DC-Shot (α = 1, β = 0)
+68.41 ± 0.44
+83.06 ± 0.32
+74.84 ± 0.49
+88.01 ± 0.33
+81.51 ± 0.44
+90.83 ± 0.24
+P3DC-Shot (α = 0, β = 1)
+68.67 ± 0.44
+83.64 ± 0.31
+75.20 ± 0.48
+88.29 ± 0.33
+81.58 ± 0.44
+91.02 ± 0.23
+P3DC-Shot (α = 1
+3, β = 1
+3)
+68.33 ± 0.44
+84.19 ± 0.30
+74.91 ± 0.49
+88.54 ± 0.32
+81.75 ± 0.43
+91.21 ± 0.23
+P3DC-Shot (selected α, β)
+68.68 ± 0.44
+84.37 ± 0.30
+75.20 ± 0.48
+88.67 ± 0.32
+81.86 ± 0.43
+91.36 ± 0.23
+(0.0, 0.9)
+(0.0, 0.4)
+(0.0, 1.0)
+(0.0, 0.3)
+(0.2, 0.4)
+(0.0, 0.4)
+methods. Then, we compare with different data trans-
+formation or calibration schemes and provide qualita-
+tive visualization for showing the difference of our cali-
+bration results w.r.t existing works. In addition, we eval-
+uate the generalizability of our method by performing
+classification tasks with different difficulties.
+Quantitative comparisons. As there are numerous
+efforts have been paid to the few-shot classification,
+we mainly compare our P3DC-Shot with representative
+and SOTA works which cover different types of few-
+shot learning schemes. The compared methods include
+the metric-learning based meta-learning [15, 16, 17],
+optimization-based meta-learning [19, 21, 22], transfer
+learning [31, 57, 29], nearest neighbor [25, 50] and cal-
+ibration [35, 38, 37, 54] based methods. For certain
+methods such as [29, 28], we only compare with their
+basic versions and do not consider their model trained
+with data augmentation. Note that as not every method
+has conducted experiments on all three datasets, we
+mainly compare with their reported results. One excep-
+tion is for [54], we compare with its results generated
+using its released code.
+For our method, we report the results of our model
+with different hyperparameters α and β. In particular,
+we consider the case when α and β are both zero, which
+makes our method a simple NN-based method with no
+data calibration and only shows the effect for using the
+query-guided prototype computation (Eq. 13). We also
+compare with the results of α or β is 1, or both of them
+are equal to 1
+3, which correspond to the cases that the
+endpoint of the sample-level or task-level calibration or
+the barycenter of the calibration triangle (Figure 2). In
+the end, we provide our best results with the α or β se-
+lected based on the validation set.
+For each dataset, we evaluate on the 5-way 1-shot
+and 5-way 5-shot classification setting. For each set-
+ting, 2,000 testing tasks, each of which contains 5 × K
+(K = 1 or 5) samples for the support set and 5 × 15
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+(a) 
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+Figure 3: T-SNE visualization of the calibration on example support samples from the test set of miniImageNet (a), tieredImageNet (b), and CUB
+(c). The colored dots are data from the same underlying classes as the selected sample and the star is the center of each class. Given a support
+sample (represented in square), the upside down triangle is our calibration result and the lozenge is the calibration result of DC [38].
+samples for the query set, are randomly generated from
+the test split of the corresponding dataset. Table 1 shows
+the quantitative comparison results on three datasets. It
+can be seen that our best results outperform most meth-
+ods in the 5-way 1-shot setting and are comparable to
+the SOTA methods [28, 38] for the 5-way 5-shot set-
+ting. Note that although [37] achieves best results on the
+CUB dataset, it is inferior on miniImageNet and tiered-
+ImageNet. Moreover, since [37] follows a metric-based
+few-shot learning pipeline, it still requires to train the
+feature extractor and the metric module for each dataset.
+For [28], it performs generally well on all three datasets,
+but as an ensemble-based method, its computation time
+is much longer than our method, especially when the
+ensemble number is large. In contrast, our method does
+not require any training and only needs to perform an
+efficient calibration step for each testing task.
+Also, from results of our method with different α and
+β values in Table 1, it can be found when α and β is
+zero, the query-guided prototype computation can lead
+to better performance than the simple NN-based Sim-
+pleShot [25]. When either the sample-level or task-level
+calibration is applied, i.e., α or β is not zero, the results
+are better than the non-calibrated version, showing the
+calibration can indeed reduce the bias for the support
+samples. Meanwhile, which calibration type is more
+suitable is depending on the underlying data distribu-
+tion of the dataset. By selecting the α and β based on
+the validation set of each dataset, the results are further
+improved. In the ablation study, we perform more ex-
+periments and analysis of different α and β values.
+Comparison with different data transformation or
+calibration schemes. To further verify the effectiveness
+Table 2: Comparison with different data transformation or calibration
+schemes. Accuracy (%) for 5-way 1-shot task on the test set of mini-
+ImageNet are measured.
+Model
+miniImageNet
+CUB
+5-way 1-shot
+5-way 1-shot
+NN
+47.50
+76.40
+L2N+NN
+65.93
+81.61
+CL2N+NN
+65.96
+81.54
+DC+L2N+NN
+66.23
+79.49
+P3DC-Shot
+68.68
+81.86
+(selected α, β)
+(0.0,0.9)
+(0.2,0.4)
+of our prior-driven data calibration, we compare with
+several NN-based baseline methods which perform dif-
+ferent data transformation or calibration schemes and
+the results are shown in Table 2. In this experiment, all
+methods are based on the pretrained WideResNet fea-
+tures.
+Also, only the 5-way 1-shot classification ac-
+curacy is measured so that the comparison is focused
+on feature transformation instead of the prototype com-
+putation schemes. The first baseline is NN, which is
+a naive inner product based nearest neighbor classifier.
+Then, L2N and CL2N represent L2 normalization and
+centered L2 normalization which have been shown as
+effective in SimpleShot [25]. In addition, another base-
+line that follows the data calibration scheme in DC [38]
+is compared. Comparing to the original DC, this base-
+line directly takes the calibrated and then normalized
+features and employs NN for classification instead of
+training new classifiers using the sampled data. From
+Table 2, it can be observed the data normalization or cal-
+ibration can significantly improve the NN-based classi-
+9
+
+★Table 3: Generalizability test on different N in N-way 1-shot tasks. Accuracy (%) on the test set of miniImageNet are measured. For our P3DC-Shot,
+the same α = 0 and β = 0.9 selected based on the validation set for the 5-way 1-shot case are used for all experiments.
+Models
+5-way
+7-way
+9-way
+11-way
+13-way
+15-way
+20-way
+RestroreNet [35]
+59.56
+50.55
+44.54
+39.98
+36.34
+33.52
+28.48
+L2N+NN
+65.93
+57.86
+52.45
+48.25
+44.80
+42.12
+37.06
+CL2N+NN
+65.96
+57.69
+52.23
+47.93
+44.36
+41.85
+36.65
+P3DC-Shot
+68.68
+60.58
+55.03
+50.75
+47.21
+44.43
+39.33
+fication. In addition, our data calibration achieves the
+best results comparing to other baselines. The main rea-
+son is the L2N and CL2N only perform transformation
+rather than calibration using the base priors, while the
+modified DC does not consider the attentive similarity
+between the support samples and the base classes when
+performing the calibration.
+Visualization of the calibration.
+To qualitatively
+verify the effectiveness of our calibration, we show the
+T-SNE [60] visualization of the calibration results for
+some example support samples in Figure 3. The results
+of calibrating the same sample using DC [38] are also
+compared. It can be seen from Figure 3 that our calibra-
+tion can more effectively transform the support samples
+closer to the center of the underlying classes. For DC,
+the calibration may be minor or even be far away from
+the center. The reason is still due to it treats the nearby
+base classes with the same weights. In contrast, our cal-
+ibration pays more attention to the similar base classes
+when determining the weights for combining the base
+prototypes (Eq. 5 and 9).
+Generalizability test on different N in N-way clas-
+sification. Following [35], we conduct a series of N-
+way 1-shot experiments on miniImageNet to test the
+generalizability of the proposed calibration for differ-
+ent classification tasks. Table 3 shows the results of the
+baseline methods [35], L2N and CL2N and ours. Note
+that with the N increases, there are more data samples in
+a test task and the classification becomes more difficult.
+It can be observed that our P3DC-Shot achieves con-
+sistent best results comparing to the baseline methods,
+verifying our method is generalizable to classification
+tasks with different difficulties.
+4.4. Ablation Study
+In this section, we perform ablation studies to ver-
+ify the effectiveness of different modules and design
+choices of our method. First, we conduct experiments
+on different hyperparameter α and β to see how the
+sample-level and task-level calibration can affect the fi-
+nal results. Then, we perform the study on the effec-
+tiveness of using the query-guided attentive prototypes
+in the NN classification step.
+Effect on different hyperparameter α, β. Differ-
+ent α and β values correspond to different degrees of
+sample-level and task-level calibration applied to the in-
+put data. Geometrically, α, β and 1 − α − β can also be
+understood as the coordinates of the calibration result
+w.r.t to the triangle formed by the three points ¯xi, si, ti.
+To quantitatively reveal how these two hyperparameters
+can affect the results, we enumerate different α and β
+values on both the validation and test sets of different
+datasets. From the results in Figure 4, it can be found
+the accuracy near the origin of the figures are smaller,
+which means performing calibration can improve upon
+using the original features for classification, i.e., α and
+β is zero. Also, different datasets prefer different α and
+β combinations for achieving higher performance. For
+example, miniImageNet shows better results when α+β
+is around 0.9 and CUB prefers a relatively smaller cal-
+ibration, i.e., α + β is around 0.6. For tieredImageNet,
+better results are obtained around the topper left of the
+figure, showing the task-level calibration is more help-
+ful than the sample-level. Overall, the trend on the test
+set is consistent with the validation set. From above ex-
+periments, it shows the sample-level and task-level cali-
+bration are consistently effective, while how to selecting
+the good α and β values are dataset dependent. There-
+fore, for our best results, we use the α and β selected
+based on the validation set and report their performance
+on the test set.
+Effect on using attentive prototypes in NN classifi-
+cation. To improve the conventional prototype based
+NN classificaiton, we propose to compute the query-
+guided attentive prototypes to represent the support
+class. To verify the effectiveness of this scheme, we per-
+form ablation study for 5-way 5-shot tasks on different
+tasks using different prototype computation schemes.
+Specifically, we take the calibrated support features
+and compute the prototypes for the support classes by
+performing the conventional average operation or our
+query-guided attentive averaging (Eq. 13). The results
+10
+
+Figure 4: The effect of different α and β on the validation (top) and test (bottom) set of different datasets. Accuracy (%) for 5-way 1-shot task on
+miniImageNet, tieredImageNet and CUB are measured. The warmer color corresponds to higher accuracy.
+Table 4: Ablation study on using the query-guided attentive proto-
+types in NN classification. Accuray (%) on the test set of miniIma-
+geNet, tieredImageNet and CUB are measured.
+Model
+miniImageNet tieredImageNet
+CUB
+5-way 5-shot
+5-way 5-shot
+5-way 5-shot
+Average
+84.11
+88.54
+91.27
+Attentive
+84.37
+88.67
+91.36
+in Table 4 show that the attentive prototypes can lead to
+better performance. Hence, we adopt the attentive pro-
+totypes in our NN-based classification.
+5. Conclusion
+In this paper, we propose a simple yet effective frame-
+work, named P3DC-Shot, for few-shot classification.
+Without any retraining and expensive computation, our
+prior-driven discrete data calibration method can effi-
+ciently calibrate the support samples based on prior-
+information from the base classes to obtain the less-
+biased support data for NN-based classification. Exten-
+sive experiments show that our method can outperform
+or at least comparable to SOTA methods which need ad-
+ditional learning steps or more computation. One lim-
+itation of our method is we rely on the whole valida-
+tion set to select the good hyperparameters α and β to
+determine which degree of the sample-level and task-
+level calibration is more suitable for the given dataset.
+Investigating a more general scheme to combine the
+sample-level and task-level calibration is an interesting
+future work. Moreover, when exploring the combina-
+tion schemes, we only focus on exploring the inner area
+of the calibration triangle. It is worthy to extend the
+parameter search to a larger area, i.e., by extrapolation
+of the calibration triangle, to find whether better results
+can be obtained.
+References
+[1] K. Simonyan, A. Zisserman,
+Very deep convolutional net-
+works for large-scale image recognition,
+arXiv preprint
+arXiv:1409.1556 (2014).
+[2] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma,
+Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al., Ima-
+genet large scale visual recognition challenge, Int. J. Comput.
+Vis. 115 (2015) 211–252.
+11
+
+[3] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for
+image recognition, in: IEEE Conf. Comput. Vis. Pattern Recog.,
+2016, pp. 770–778.
+[4] R. Girshick, Fast r-cnn, in: Int. Conf. Comput. Vis., 2015, pp.
+1440–1448.
+[5] S. Ren, K. He, R. Girshick, J. Sun, Faster r-cnn: Towards real-
+time object detection with region proposal networks, Adv. Neu-
+ral Inform. Process. Syst. 28 (2015).
+[6] J. Redmon, S. Divvala, R. Girshick, A. Farhadi, You only look
+once: Unified, real-time object detection, in: IEEE Conf. Com-
+put. Vis. Pattern Recog., 2016, pp. 779–788.
+[7] J. Long, E. Shelhamer, T. Darrell, Fully convolutional networks
+for semantic segmentation, in: IEEE Conf. Comput. Vis. Pattern
+Recog., 2015, pp. 3431–3440.
+[8] K. He, G. Gkioxari, P. Doll´ar, R. Girshick, Mask r-cnn, in: Int.
+Conf. Comput. Vis., 2017, pp. 2961–2969.
+[9] L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, A. L.
+Yuille, Deeplab: Semantic image segmentation with deep con-
+volutional nets, atrous convolution, and fully connected crfs,
+IEEE Trans. Pattern Anal. Mach. Intell. 40 (2017) 834–848.
+[10] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ra-
+manan, P. Doll´ar, C. L. Zitnick, Microsoft coco: Common ob-
+jects in context, in: Eur. Conf. Comput. Vis., 2014, pp. 740–755.
+[11] M. Cordts, M. Omran, S. Ramos, T. Rehfeld, M. Enzweiler,
+R. Benenson, U. Franke, S. Roth, B. Schiele, The cityscapes
+dataset for semantic urban scene understanding, in: IEEE Conf.
+Comput. Vis. Pattern Recog., 2016, pp. 3213–3223.
+[12] Y. Wang, Q. Yao, J. T. Kwok, L. M. Ni, Generalizing from a few
+examples: A survey on few-shot learning, ACM Comput Surv
+53 (2020) 1–34.
+[13] J. Lu, P. Gong, J. Ye, C. Zhang, Learning from very few sam-
+ples: A survey, arXiv preprint arXiv:2009.02653 (2020).
+[14] G. Huang, I. Laradji, D. V´azquez, S. Lacoste-Julien, P. Ro-
+driguez, A survey of self-supervised and few-shot object de-
+tection, IEEE Trans. Pattern Anal. Mach. Intell. (2022).
+[15] O. Vinyals, C. Blundell, T. Lillicrap, D. Wierstra, et al., Match-
+ing networks for one shot learning, Adv. Neural Inform. Pro-
+cess. Syst. 29 (2016).
+[16] J. Snell, K. Swersky, R. Zemel, Prototypical networks for few-
+shot learning, Adv. Neural Inform. Process. Syst. 30 (2017).
+[17] F. Sung, Y. Yang, L. Zhang, T. Xiang, P. H. Torr, T. M.
+Hospedales, Learning to compare: Relation network for few-
+shot learning (2018) 1199–1208.
+[18] S. Ravi, H. Larochelle, Optimization as a model for few-shot
+learning (2016).
+[19] C. Finn, P. Abbeel, S. Levine, Model-agnostic meta-learning for
+fast adaptation of deep networks, in: Int. Conf. Mach. Learn.,
+PMLR, 2017, pp. 1126–1135.
+[20] M. A. Jamal, G.-J. Qi, Task agnostic meta-learning for few-shot
+learning, in: IEEE Conf. Comput. Vis. Pattern Recog., 2019.
+[21] A. A. Rusu, D. Rao, J. Sygnowski, O. Vinyals, R. Pascanu,
+S. Osindero, R. Hadsell, Meta-learning with latent embedding
+optimization, in: Int. Conf. Learn. Represent., 2019.
+[22] K. Lee, S. Maji, A. Ravichandran, S. Soatto, Meta-learning with
+differentiable convex optimization, in: IEEE Conf. Comput. Vis.
+Pattern Recog., 2019, pp. 10657–10665.
+[23] Y. Chen, X. Wang, Z. Liu, H. Xu, T. Darrell,
+A new meta-
+baseline for few-shot learning, arXiv preprint arXiv:2003.11539
+(2020).
+[24] T. Cao, M. T. Law, S. Fidler, A theoretical analysis of the num-
+ber of shots in few-shot learning, in: Int. Conf. Learn. Repre-
+sent., 2019.
+[25] Y. Wang, W.-L. Chao, K. Q. Weinberger, L. van der Maaten,
+Simpleshot: Revisiting nearest-neighbor classification for few-
+shot learning, arXiv preprint arXiv:1911.04623 (2019).
+[26] F. Aurenhammer, Voronoi diagrams—a survey of a fundamental
+geometric data structure, ACM Comput Surv 23 (1991) 345–
+405.
+[27] D. Z. Chen, Z. Huang, Y. Liu, J. Xu, On clustering induced
+voronoi diagrams,
+SIAM Journal on Computing 46 (2017)
+1679–1711.
+[28] C. Ma, Z. Huang, M. Gao, J. Xu, Few-shot learning as cluster-
+induced voronoi diagrams: A geometric approach, in: Int. Conf.
+Learn. Represent., 2022.
+[29] P. Mangla, N. Kumari, A. Sinha, M. Singh, B. Krishnamurthy,
+V. N. Balasubramanian, Charting the right manifold: Manifold
+mixup for few-shot learning, in: WACV, 2020, pp. 2218–2227.
+[30] Y. Tian, Y. Wang, D. Krishnan, J. B. Tenenbaum, P. Isola, Re-
+thinking few-shot image classification: a good embedding is all
+you need?, in: Eur. Conf. Comput. Vis., Springer, 2020, pp.
+266–282.
+[31] W.-Y. Chen, Y.-C. Liu, Z. Kira, Y.-C. F. Wang, J.-B. Huang,
+A closer look at few-shot classification, in: Int. Conf. Learn.
+Represent., 2019.
+[32] W. Ge, Y. Yu,
+Borrowing treasures from the wealthy: Deep
+transfer learning through selective joint fine-tuning, in: IEEE
+Conf. Comput. Vis. Pattern Recog., 2017.
+[33] O. Sbai, C. Couprie, M. Aubry, Impact of base dataset design on
+few-shot image classification, Eur. Conf. Comput. Vis. (2020).
+[34] L. Zhou, P. Cui, X. Jia, S. Yang, Q. Tian, Learning to select base
+classes for few-shot classification, in: IEEE Conf. Comput. Vis.
+Pattern Recog., 2020, pp. 4624–4633.
+[35] W. Xue, W. Wang, One-shot image classification by learning to
+restore prototypes, in: AAAI, volume 34, 2020, pp. 6558–6565.
+[36] J. Liu, L. Song, Y. Qin,
+Prototype rectification for few-shot
+learning, in: Eur. Conf. Comput. Vis., Springer, 2020, pp. 741–
+756.
+[37] Y. Guo, R. Du, X. Li, J. Xie, Z. Ma, Y. Dong, Learning cali-
+brated class centers for few-shot classification by pair-wise sim-
+ilarity, IEEE Trans. Image Process. 31 (2022) 4543–4555.
+[38] S. Yang, L. Liu, M. Xu, Free lunch for few-shot learning: Dis-
+tribution calibration, in: Int. Conf. Learn. Represent., 2021.
+[39] M. Ren, E. Triantafillou, S. Ravi, J. Snell, K. Swersky, J. B.
+Tenenbaum, H. Larochelle, R. S. Zemel,
+Meta-learning for
+semi-supervised few-shot classification, in: Int. Conf. Learn.
+Represent., 2018.
+[40] C. Wah, S. Branson, P. Welinder, P. Perona, S. Belongie, The
+caltech-ucsd birds-200-2011 dataset (2011).
+[41] T. Hospedales, A. Antoniou, P. Micaelli, A. Storkey,
+Meta-
+learning in neural networks: A survey 44 (2021) 5149–5169.
+[42] G. Koch, R. Zemel, R. Salakhutdinov, et al., Siamese neural net-
+works for one-shot image recognition, in: ICML deep learning
+workshop, 2015.
+[43] W. Xu, Y. Xu, H. Wang, Z. Tu, Attentional constellation nets
+for few-shot learning, in: Int. Conf. Learn. Represent., 2021.
+[44] Y. Liu, T. Zheng, J. Song, D. Cai, X. He, Dmn4: Few-shot learn-
+ing via discriminative mutual nearest neighbor neural network,
+in: AAAI, volume 36, 2022, pp. 1828–1836.
+[45] L. Torrey, J. Shavlik, Transfer learning, in: Handbook of re-
+search on machine learning applications and trends: algorithms,
+methods, and techniques, IGI global, 2010, pp. 242–264.
+[46] C. Tan, F. Sun, T. Kong, W. Zhang, C. Yang, C. Liu, A survey on
+deep transfer learning, in: International conference on artificial
+neural networks, Springer, 2018, pp. 270–279.
+[47] F. Zhuang, Z. Qi, K. Duan, D. Xi, Y. Zhu, H. Zhu, H. Xiong,
+Q. He, A comprehensive survey on transfer learning, Proceed-
+ings of the IEEE 109 (2020) 43–76.
+[48] G. S. Dhillon, P. Chaudhari, A. Ravichandran, S. Soatto,
+A
+baseline for few-shot image classification, in: Int. Conf. Learn.
+Represent., 2020.
+12
+
+[49] W. Li, L. Wang, J. Xu, J. Huo, Y. Gao, J. Luo, Revisiting local
+descriptor based image-to-class measure for few-shot learning,
+in: IEEE Conf. Comput. Vis. Pattern Recog., 2019, pp. 7260–
+7268.
+[50] C. Ma, Z. Huang, M. Gao, J. Xu, Few-shot learning as cluster-
+induced voronoi diagrams: A geometric approach (2022).
+[51] F. Wu, J. S. Smith, W. Lu, C. Pang, B. Zhang, Attentive proto-
+type few-shot learning with capsule network-based embedding,
+in: Eur. Conf. Comput. Vis., 2020, pp. 237–253.
+[52] Z. Ji, X. Chai, Y. Yu, Z. Zhang, Reweighting and information-
+guidance networks for few-shot learning, Neurocomputing 423
+(2021) 13–23.
+[53] X. Wang, J. Meng, B. Wen, F. Xue,
+Racp: A network with
+attention corrected prototype for few-shot speaker recognition
+using indefinite distance metric, Neurocomputing 490 (2022)
+283–294.
+[54] J. Xu, X. Luo, X. Pan, W. Pei, Y. Li, Z. Xu, Alleviating the sam-
+ple selection bias in few-shot learning by removing projection
+to the centroid, in: Adv. Neural Inform. Process. Syst., 2022.
+[55] L. Zhou, P. Cui, S. Yang, W. Zhu, Q. Tian, Learning to learn
+image classifiers with visual analogy, in: IEEE Conf. Comput.
+Vis. Pattern Recog., 2019, pp. 11497–11506.
+[56] J. W. Tukey, et al., Exploratory data analysis, volume 2, Read-
+ing, MA, 1977.
+[57] B. Liu, Y. Cao, Y. Lin, Q. Li, Z. Zhang, M. Long, H. Hu, Nega-
+tive margin matters: Understanding margin in few-shot classifi-
+cation, in: Eur. Conf. Comput. Vis., Springer, 2020.
+[58] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, L. Fei-Fei, Im-
+agenet: A large-scale hierarchical image database,
+in: IEEE
+Conf. Comput. Vis. Pattern Recog., Ieee, 2009, pp. 248–255.
+[59] S. Zagoruyko, N. Komodakis, Wide residual networks, arXiv
+preprint arXiv:1605.07146 (2016).
+[60] L. van der Maaten, G. E. Hinton, Visualizing data using t-sne,
+Journal of Machine Learning Research 9 (2008) 2579–2605.
+13
+

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+arXiv:2301.02607v1  [eess.SP]  6 Jan 2023
+EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US
+1
+A Data-Driven Gaussian Process Filter for
+Electrocardiogram Denoising
+Mircea Dumitru, Qiao Li, Erick Andres Perez Alday, Ali Bahrami Rad, Gari D. Clifford, Reza Sameni*
+Abstract—Objective: Gaussian Processes (GP)-based filters,
+which have been effectively used for various applications includ-
+ing electrocardiogram (ECG) filtering can be computationally
+demanding and the choice of their hyperparameters is typically
+ad hoc. Methods: We develop a data-driven GP filter to address
+both issues, using the notion of the ECG phase domain — a time-
+warped representation of the ECG beats onto a fixed number
+of samples and aligned R-peaks, which is assumed to follow a
+Gaussian distribution. Under this assumption, the computation of
+the sample mean and covariance matrix is simplified, enabling an
+efficient implementation of the GP filter in a data-driven manner,
+with no ad hoc hyperparameters. The proposed filter is evaluated
+and compared with a state-of-the-art wavelet-based filter, on
+the PhysioNet QT Database. The performance is evaluated by
+measuring the signal-to-noise ratio (SNR) improvement of the
+filter at SNR levels ranging from –5 to 30 dB, in 5 dB steps,
+using additive noise. For a clinical evaluation, the error between
+the estimated QT-intervals of the original and filtered signals is
+measured and compared with the benchmark filter. Results: It is
+shown that the proposed GP filter outperforms the benchmark
+filter for all the tested noise levels. It also outperforms the state-
+of-the-art filter in terms of QT-interval estimation error bias
+and variance. Conclusion: The proposed GP filter is a versatile
+technique for preprocessing the ECG in clinical and research
+applications, is applicable to ECG of arbitrary lengths and
+sampling frequencies, and provides confidence intervals for its
+performance.
+Index Terms—ECG Bayesian filter, Gaussian processes, ECG
+denoising, ECG wavelet denoising, QT-interval estimation
+I. INTRODUCTION
+Electrocardiogram (ECG) denoising is a recurrent problem
+in traditional and wearable cardiac monitors. The problem
+has been addressed by various approaches, including model-
+based and non-model-based filters. A powerful non-parametric
+framework for ECG filtering is via Gaussian process (GP)
+models [1], [2], which considers the ECG beats as GPs with
+common parameters. The choice of the beats GP hyperparam-
+eters, namely the mean and kernel functions is non-evident and
+ad hoc. For GP models with no beat assumptions, beside the
+ambiguity in parameter selection, the GP filter implementation
+involves the inversion of large covariance matrices, which
+precludes the use of this framework for long ECG records.
+In this paper, ECG filtering is addressed via a data-driven
+non-parametric GP model. The novelty of the proposed filter
+is that it requires no ad hoc GP model hyperparameters and it
+is computationally efficient, making it suitable for any length
+ECG records; it is based on the assumption that each phase
+The authors are with the Department of Biomedical Informatics, School
+of Medicine, Emory University. G. D. Clifford is also with the Biomedical
+Engineering Department, Georgia Institute of Technology. Corresponding
+author: R. Sameni (email: [email protected]).
+domain beat — a time-warped (stretched or squeezed) repre-
+sentation of the ECG beats onto a fixed number of samples
+and aligned R-peaks — is an ensemble of an underlying
+GP. The mean and the kernel function are set via the phase
+domain sample mean and covariance matrix, computed via
+the available ensembles, which are transformed back to the
+time-domain and used to derive the posterior mean using the
+Bayesian formalism.
+This proposed filter is data-driven, does not presume any
+parametric model for the underlying GP, and is computation-
+ally efficient.
+The filter is evaluated in terms of signal-to-noise ratio (SNR)
+improvement, using as benchmark a wavelet-based ECG de-
+noiser that was demonstrated in [3] to outperform adaptive
+filters [4], Tikhonov regularization and Extended Kalman
+filters [5], in terms of SNR improvement. The proposed filter’s
+clinical performance is evaluated by measuring the QT-interval
+error between the clean ECG and its corresponding filtered
+version.
+II. GAUSSIAN PROCESS-BASED ECG FILTERING
+A. The mathematical model
+The ECG measurement x(t) is assumed to be an additive
+mixture of a clean ECG s(t), assumed to be a GP contami-
+nated by additive white noise:
+x(t) = s(t) + n(t),
+t ∈ {t1 . . . tN}
+∆= TN,
+(1)
+where n(t) ∼ N(0, vn), vn denotes the noise variance and
+N denotes the number of measurements. The signal x(t) is
+assumed to be baseline-wander (BW) and powerline noise
+removed, which are relatively straightforward, with classical
+filtering pipelines (cf. Section III-A). Therefore, the filter
+design objective is focused on in-band ECG noise removal.
+For the beat i, Ti =
+�
+ti1 . . . tiRi . . . tiNi
+�
+denotes the set of
+time samples, ti1 representing the first sample, tiRi the sample
+corresponding to the R-peak and tiNi the last sample. We
+further define xi = [x(t)]i∈Ti, si = [s(t)]i∈Ti, ni = [n(t)]i∈Ti
+as vectorial representations of the measurement, clean ECG
+and noise, respectively. Therefore, xi = si + ni.
+Next, we define matrices Θi ∈ RT ×Ni to map the time
+domain beats xi, si and ni to the phase domain beats
+ξi = Θixi, ςi = Θisi, ηi = Θini,
+(2)
+with aligned R-peaks and the same number of samples T
+(Fig. 1). The Θi matrices are defined by considering T knots
+
+2
+EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US
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+150
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+200
+−0.4
+−0.2
+0.0
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+0.4
+0.6
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+1.0
+amplitude [mv]
+x[1]
+x[2]
+x[3]
+x[4]
+x[5]
+x[6]
+0
+50
+100
+150
+200
+time [samples]
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+amplitude [mV]
+ξ[1]
+ξ[2]
+ξ[3]
+ξ[4]
+ξ[5]
+ξ[6]
+Fig. 1. Time-domain measurements beats (top) and the corresponding phase
+domain ECG beats (bottom), with the same number T of samples for the
+first 6 beats of sel100 record from QTDB [6] with 0 dB Gaussian additive
+noise. Transformation matrices Θi are defined via (3).
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
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+10
+15
+20
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+0
+5
+10
+15
+20
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Fig. 2.
+Corner detail example of transformation matrix Θi (left), ΘT
+i
+(middle) and the corresponding (diagonal) Gramian Gi = ΘT
+i Θi (right).
+equidistantly distributed in the interval [1, Ni] and assigning
+Θi(j, k) =
+�
+1, if j − 1 ≤ (k − 1) Ni−1
+T −1 < j
+0, otherwise
+,
+(3)
+with j = 1, . . . , Ni − 1, k = 1, . . . , T and T ≥ maxi {Ni}.
+With this choice, the corresponding Gramian matrices Gi, are
+diagonal matrices (Fig. 2),
+Gi = ΘT
+i Θi = diag [gi] and diag
+�
+ΘiΘT
+i
+�
+= 1T ,
+(4)
+with gi ∈ RNi and 1T ∈ RT . Therefore, Gi is invertible and
+the back transformation from the phase to the time domain is
+given by Ψi = G−1
+i ΘT
+i .
+From (1) and (2), the ECG beats satisfy ξi = ςi + ηi.
+As shown in Fig. 1, in the phase domain the beats have
+been normalized in lengths and the R-peaks are aligned.
+Therefore, the phase-domain sample variations are only due to
+the stochastic inter-beat variations of the ECG beats and noise.
+As our working model, we assume that the phase domain beats
+ξi to be ensembles of an underlying GP
+ξi ∼ N (µξ, Kξ) .
+(5)
+Moreover, from the time domain noise assumption and (2),
+the phase domain noise beats also have a zero-mean normal
+distribution ηi ∼ N
+�
+0, vnΘiΘT
+i
+�
+. Therefore, the phase do-
+main ECG beats follow ςi ∼ N
+�
+µξ, Kξ − vnΘiΘT
+i
+�
+, where
+the model parameters µξ and Kξ can be estimated by the
+sample mean ¯µξ := B−1 �B
+i=1 ξi and the sample covariance
+¯
+Kξ := B−1 �B
+i=1(ξi− ¯µξ)(ξi− ¯µξ)T , where B is the number
+of beats. Therefore, the time domain (clean) ECG beats follow
+a Normal distribution si ∼ N (µsi, Ksi) with parameters
+µsi = Ψi ¯µξ, Ksi = Ψi
+� ¯
+Kξ − ˆvnΘiΘT
+i
+�
+ΨT
+i ,
+(6)
+where ˆvn represents the noise variance estimate and the
+covariance matrix corresponding to time domains beats xi is
+given by
+Kxi = Ψi ¯
+KξΨi.
+(7)
+Finally, the filtered beats are defined as the time domain
+posterior mean, using (6) and (7):
+ˆsi = µsi + KsiK−1
+xi (xi − µsi) ,
+(8)
+In the sequel, we refer to µsi and ˆsi as prior-based and
+posterior-based GP filter results.
+B. The GP filter with diagonal covariance matrix
+The direct implementation of the filter in (8) requires the
+inversion of covariance matrices that typically have huge
+condition numbers. The matrix inversion can be avoided if
+we consider the diagonal case of ¯
+Kξ:
+¯kξ = diag
+� ¯
+Kξ
+�
+, kηi
+(4)= ˆvn1T
+(9)
+In this case, the corresponding time domain matrices are also
+diagonal and can be computed via
+kxi =
+�
+ΘT
+i ¯kξ
+�
+⊘ g2
+i , ksi =
+�
+ΘT
+i
+�¯kξ − kηi
+��
+⊘ g2
+i
+(10)
+with ◦ and ⊘ denoting the Hadamard product and division,
+respectively (element-wise product and division), g2
+i := gi◦gi,
+the time domain (prior) mean computed via
+µsi =
+�
+ΘT
+i ¯µξ
+�
+⊘ gi,
+(11)
+and the corresponding filter given by
+ˆsi = µsi + ksi ⊘ kxi ◦ (xi − µsi) .
+(12)
+The overall algorithm for GP ECG filtering is summarized in
+Algorithm 1 and is available online in our Git repository [7].
+Algorithm 1 GP ECG filtering
+1: {tiRi}i = RPeakDetector(x)
+[Section III-B]
+2: ˆvn = NoiseVartianceEstimator(x) [Section II-D]
+Input: x, {tiRi}i
+Output: {�si}i
+3: function GPDIAG(x, {tiRi}i, ˆvn)
+⊲ GP diagonal filter
+4:
+for all beats do
+⊲ phase domain computations
+5:
+compute transformation matrices Θi via (3)
+6:
+compute the vectors gi = diag
+�
+ΘT
+i Θi
+�
+7:
+compute kηi via (9)
+8:
+compute the phase beats ξi via (2)
+9:
+end for
+10:
+compute phase domain sample mean ¯
+µξ
+11:
+compute phase domain sample variance vector ¯kξ
+12:
+for all beats do
+⊲ time domain computations
+13:
+compute ECG prior mean µsi via (11)
+14:
+compute ECG variance ksi via (10)
+15:
+compute measurements variance kxi via (10)
+16:
+compute the filtered ECG ˆsi via (12)
+17:
+end for
+18: end function
+
+A DATA-DRIVEN ECG GAUSSIAN PROCESS FILTER, M. DUMITRU ET AL.
+3
+C. Computational cost and model selection
+The direct implementation of a GP filter (without the hereby
+proposed phase-domain model) would be as follows [1], [2]:
+�s = µs + KsK−1
+x (x − µs) ,
+(13)
+with the computational complexity O(N 3), dominated by the
+inversion of the measurement covariance matrix Kx. In this
+approach the model’s hyperparameters are the mean µs, the
+covariance matrix Ks) and the noise variance vn (or more
+generally the noise covariance matrix) and optimizing them
+via classical methods (e.g. maximum evidence, leave-one-
+out cross validation, [8, Ch. 5]) adds to the computational
+complexity. For long ECGs, the application of this model
+is not possible. Previous research considered the GP beat-
+wise formulation and adopted a model-based approach to
+confine the structure of the covariance matrices [1], [2], but the
+choice of the particular model-based mean and kernel function
+families remains ad-hoc and difficult to justify.
+The proposed model infers the GP mean and covariance
+matrix in a data-driven way, based on the sample mean and
+covariance matrix from the phase domain (6) and (7), and
+in the diagonal case, Algorithm 1, does not require any
+inversion. The fundamental assumption allowing the data-
+driven computation is the assumption that the phase domain
+beats ξi are ensembles from the same underlying GP, (5).
+D. Hyperparameter selection
+The number of phase domain beat samples T is chosen
+greater than the longest beat in the time domain; this allows the
+choice of the transformation and back transformation matrices
+such that the time-phase-time transition can be done without
+(transformation) errors. The noise variance ˆvn can be com-
+puted via maximum evidence or practically from the baseline
+segment of the ECG beats, where the heart is electrically silent
+and only the noise is exhibited in the ECG.
+III. RESULTS
+A. Baseline wander removal
+The BW is removed via two successively zero-phase first
+order forward-backward lowpass filters (filtfilt in MAT-
+LAB/Python SciPy) with cut-off frequencies set at fc =
+5.0 Hz and fc = 80.0 Hz, respectively. While the resulting
+passband frequency range is rather narrow and eliminates some
+ECG-related components, it enables us to assess the filtering
+performance for the dominant ECG frequency band.
+B. R-peak detection and heartbeat segmentation
+The proposed filter requires the ECG R-peaks. The beats
+are defined relative to the R-peaks, segmenting the mea-
+surements at the midpoints between successive R-peaks. The
+R-peak estimation is done using a modified version of the
+Pan–Tompkins algorithm [9]. Specifically, the version used in
+this paper estimates the R-peaks by successively applying a
+band pass filter, an outlier saturation filter via the hyperbolic
+tangent function, a square root moving average filter and a
+thresholding.
+C. Evaluation
+The PhysioNet QT Database (QTDB) [6] is used to evaluate
+the developed filter. QTDB consists of 15 minutes 2-lead
+ECGs sampled at fs = 250 Hz. The baseline wander was
+removed as detailed in Section III-B. The required software
+for preprocessing and R-peak detection were adopted from the
+Open-Source Electrophysiological Toolbox (OSET) [10].
+The benchmark filter is a wavelet denoiser with a Symlet–5
+mother wavelet, soft thresholding, Stein’s unbiased risk es-
+timate (SURE) shrinkage rule, rescaling using a single-level
+noise level estimation and four levels of decomposition. In a
+previous study, this combination was proved to outperform
+other ECG filtering schemes [3]. The filter evaluation is
+measured in terms of SNR improvement and QT-interval
+estimation error.
+D. SNR improvement performance
+The ECG records were contaminated by additive white
+Gaussian noise at SNR levels ranging from –5 to 30 dB, in
+5 dB steps. An example of the noisy and filtered ECG are
+shown in Fig. 3. The average and standard deviation of the
+SNR improvement is reported for each noise level, for the
+proposed and benchmark methods in Fig. 4. Accordingly, the
+proposed posterior-based filter improves the SNR for every
+level of noise tested and outperforms the prior-based and the
+benchmark filter for all tested levels of noise.
+E. Clinical parameters preservation
+The accuracy of QT-interval estimation is considered to test
+the quality of the proposed methods for clinical ECG parame-
+ters. For this, the QT-interval estimation error (∆QT) between
+the QT-interval estimated from the filtered ECG and the QT-
+interval estimated from the noiseless ECG is measured and
+compared between the benchmark and the proposed method at
+variable input noise levels. The QT-interval estimation method
+used is adopted from [11]. Fig. 5 shows the median and the
+interquartile range (IQR) of ∆QT for the benchmark wavelet
+and the proposed filter, measured over QTDB. Accordingly,
+compared with the benchmark method, the GP posterior filter
+is reducing the median error for all levels of input noise.
+IV. DISCUSSION AND CONCLUSION
+In this work we addressed the problem of ECG denoising
+via a data-driven based GP model, with beat-wise computa-
+tions. Compared with the existing non-parametric ECG filters,
+the proposed filter makes no ad hoc assumptions about the GP
+model and can be used for ECG records of arbitrary length,
+since the computational cost has been significantly reduced
+as compared with conventional GP filters. The proposed filter
+is efficient in terms of SNR improvement, outperforming the
+benchmark performances for all tested noise levels (Fig. 4) and
+also clinically, with an improved QT-interval estimation error
+compared with the benchmark wavelet denoiser, for all tested
+levels of noise (Fig. 5). Another advantage of the proposed
+filter is its Bayesian formulation, which allows us to quantify
+the filter’s uncertainty (via the estimated variances). It also
+
+4
+EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US
+−0.6
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+0.9
+amplitude [mV]
+x
+GP Prior [ΔSNR=12.0]
+−0.6
+−0.3
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+0.3
+0.6
+0.9
+amplitude [mV]
+GP Po terior [ΔSNR=13.3]
+0.0
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+(a) input SNR = 0 dB
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+0.64
+0.90
+amplitude [mV]
+x
+GP P io  [ΔSNR=7.3]
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+−0.14
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+0.90
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+GP Poste io  [ΔSNR=10.6]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+time [s]
+−0.40
+−0.14
+0.12
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+0.64
+0.90
+amplitude [mV]
+Wavelet [ΔSNR=6.1]
+(b) input SNR = 5 dB
+−0.40
+−0.14
+0.12
+0.38
+0.64
+0.90
+amplitude [mV]
+x
+GP Prior [ΔSNR=2.4]
+−0.40
+−0.14
+0.12
+0.38
+0.64
+0.90
+amplitude [mV]
+GP Po terior [ΔSNR=8.1]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+time [ ]
+−0.40
+−0.14
+0.12
+0.38
+0.64
+0.90
+amplitude [mV]
+Wavelet [ΔSNR=5.1]
+(c) input SNR = 10 dB
+Fig. 3. The sel100 recording from the PhysioNet QTDB [6]. From top to bottom the measurements x vs. the prior estimate (11), the posterior estimate
+(12), and the wavelet denoiser (Section III-C), at different input SNR levels. The post-filtering SNR improvement is noted in each case.
+−5
+0
+5
+10
+15
+20
+25
+30
+Input SNR [dB]
+−25
+−20
+−15
+−10
+−5
+0
+5
+10
+15
+20
+SNR improvemen  [dB]
+Wavele  [benchmark]
+GP Prior 
+GP Pos erior
+Fig. 4. Mean and standard deviation SNR improvement using the proposed
+GP filter and the benchmark wavelet denoiser [3] across all samples of the
+PhysioNet QTDB [6], in leads I and II, with 5 repetitions using different noise
+instances per record.
+-5
+0
+5
+10
+15
+20
+25
+30
+Input SNR [dB]
+−100
+−50
+0
+50
+100
+150
+200
+QT es ima ion error [ms]
+Wavele 
+GP Fil er
+Fig. 5.
+The median and the interquartile range for ∆QT estimations
+corresponding to the proposed and benchmark filters across all samples of
+the PhysioNet QTDB [6].
+provides a framework that allows for synthetic ECG generation
+via data-driven learned parameters, which can be used in
+generative models for producing synthetic ECG records for
+data greedy machine learning and deep learning applications.
+In future studies, the fundamental assumption of the model,
+namely the same underlying Gaussian distribution for all the
+beats in the phase domain can be relaxed, by clustering the
+beats and assuming different underlying distributions for the
+beats in each cluster. Also, comparison with expert annotated
+QT-interval (and other clinical parameters) is required and
+statistical hypothesis testing should be performed to investigate
+if the differences are statistically insignificant. The proposed
+filter requires the R-peaks for aligning the ECG beats in the
+phase-domain, which requires investigating to what extend the
+filtering performance is susceptible to mis-detection of the
+R-peaks and morphological variations due to ectopic beats.
+The Python codes corresponding to the Algorithm 1 and the
+reported results are available in [7].
+V. ACKNOWLEDGEMENTS
+The authors acknowledge support from the National Insti-
+tute of Biomedical Imaging and Bioengineering under the NIH
+grant R01EB030362, and the National Center for Advancing
+Translational Sciences under the NIH Award UL1TR002378.
+REFERENCES
+[1] B. Rivet, M. Niknazar, and C. Jutten, “Non parametric modelling of
+ECG: Applications to denoising and single sensor fetal ECG extraction,”
+in LVA/ICA 2012 - 10th International Conference on Latent Variable
+Analysis and Signal Separation, vol. LNCS 7191.
+Tel-Aviv, Israel:
+Springer, Mar 2012, pp. 470–477.
+[2] M. Niknazar, B. Rivet, and C. Jutten, “Fetal ECG extraction from a
+single sensor by a non-parametric modeling,” in 2012 Proceedings of
+the 20th European Signal Processing Conference, 2012, pp. 949–953.
+[3] R. Sameni, “Online filtering using piecewise
+smoothness priors:
+Application to normal and abnormal electrocardiogram denoising,”
+Signal Processing, vol. 133, pp. 52–63, Apr. 2017. [Online]. Available:
+https://doi.org/10.1016/j.sigpro.2016.10.019
+[4] P. Laguna, R. Jane, O. Meste, P. Poon, P. Caminal, H. Rix, and
+N. Thakor, “Adaptive filter for event-related bioelectric signals using an
+impulse correlated reference input: comparison with signal averaging
+techniques,” IEEE Transactions on Biomedical Engineering, vol. 39,
+no. 10, pp. 1032–1044, 1992.
+[5] R. Sameni, M. B. Shamsollahi, C. Jutten, and G. D. Clifford, “A
+nonlinear bayesian filtering framework for ECG denoising,” Biomedical
+Engineering, IEEE Transactions on, vol. 54, no. 12, pp. 2172–2185,
+December 2007. [Online]. Available: https://doi.org/10.1109/TBME.
+2007.897817
+[6] P. Laguna, R. Mark, A. Goldberg, and G. Moody, “A database for
+evaluation of algorithms for measurement of QT and other waveform
+intervals in the ECG,” in Computers in Cardiology 1997.
+IEEE, 1997.
+[Online]. Available: https://doi.org/10.1109/cic.1997.648140
+[7] M.
+Dumitru,
+Data
+Driven
+Gaussian
+Process
+filter
+for
+ECG,
+2022. [Online]. Available: https://github.com/alphanumericslab/OSET/
+tree/master/UnderDevelopment/DataDrivenGPFilter
+[8] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine
+Learning, ser. Adaptive computation and machine learning.
+The MIT
+Press, 2005.
+[9] J. Pan and W. J. Tompkins, “A Real-Time QRS Detection Algorithm,”
+Biomedical Engineering, IEEE Transactions on, vol. BME-32, no. 3, pp.
+230–236, 1985.
+[10] R. Sameni, The Open-Source Electrophysiological Toolbox (OSET),
+version
+3.14,
+2018.
+[Online].
+Available:
+https://github.com/
+alphanumericslab/OSET
+[11] Q. Li, M. Dumitru, and E.A. Perez Alday, et al., “QT-Interval Estimation
+Improved with Fusion of Multiple Automated Algorithms,” in Interna-
+tional Society for Computerized Electrocardiology (ISCE), April 6-10,
+2022, Las Vegas, NV, 2022.
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='02607v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='SP]  6 Jan 2023  EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US  1  A Data-Driven Gaussian Process Filter for  Electrocardiogram Denoising  Mircea Dumitru, Qiao Li, Erick Andres Perez Alday, Ali Bahrami Rad, Gari D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Clifford, Reza Sameni*  Abstract—Objective: Gaussian Processes (GP)-based filters,  which have been effectively used for various applications includ-  ing electrocardiogram (ECG) filtering can be computationally  demanding and the choice of their hyperparameters is typically  ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Methods: We develop a data-driven GP filter to address  both issues, using the notion of the ECG phase domain — a time-  warped representation of the ECG beats onto a fixed number  of samples and aligned R-peaks, which is assumed to follow a  Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Under this assumption, the computation of  the sample mean and covariance matrix is simplified, enabling an  efficient implementation of the GP filter in a data-driven manner,  with no ad hoc hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The proposed filter is evaluated  and compared with a state-of-the-art wavelet-based filter, on  the PhysioNet QT Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The performance is evaluated by  measuring the signal-to-noise ratio (SNR) improvement of the  filter at SNR levels ranging from –5 to 30 dB, in 5 dB steps,  using additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' For a clinical evaluation, the error between  the estimated QT-intervals of the original and filtered signals is  measured and compared with the benchmark filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Results: It is  shown that the proposed GP filter outperforms the benchmark  filter for all the tested noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' It also outperforms the state-  of-the-art filter in terms of QT-interval estimation error bias  and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Conclusion: The proposed GP filter is a versatile  technique for preprocessing the ECG in clinical and research  applications, is applicable to ECG of arbitrary lengths and  sampling frequencies, and provides confidence intervals for its  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Index Terms—ECG Bayesian filter, Gaussian processes, ECG  denoising, ECG wavelet denoising, QT-interval estimation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' INTRODUCTION  Electrocardiogram (ECG) denoising is a recurrent problem  in traditional and wearable cardiac monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The problem  has been addressed by various approaches, including model-  based and non-model-based filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' A powerful non-parametric  framework for ECG filtering is via Gaussian process (GP)  models [1], [2], which considers the ECG beats as GPs with  common parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The choice of the beats GP hyperparam-  eters, namely the mean and kernel functions is non-evident and  ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' For GP models with no beat assumptions, beside the  ambiguity in parameter selection, the GP filter implementation  involves the inversion of large covariance matrices, which  precludes the use of this framework for long ECG records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  In this paper, ECG filtering is addressed via a data-driven  non-parametric GP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The novelty of the proposed filter  is that it requires no ad hoc GP model hyperparameters and it  is computationally efficient, making it suitable for any length  ECG records;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' it is based on the assumption that each phase  The authors are with the Department of Biomedical Informatics, School  of Medicine, Emory University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Clifford is also with the Biomedical  Engineering Department, Georgia Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Corresponding  author: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Sameni (email: rsameni@dbmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  domain beat — a time-warped (stretched or squeezed) repre-  sentation of the ECG beats onto a fixed number of samples  and aligned R-peaks — is an ensemble of an underlying  GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The mean and the kernel function are set via the phase  domain sample mean and covariance matrix, computed via  the available ensembles, which are transformed back to the  time-domain and used to derive the posterior mean using the  Bayesian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  This proposed filter is data-driven, does not presume any  parametric model for the underlying GP, and is computation-  ally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The filter is evaluated in terms of signal-to-noise ratio (SNR)  improvement, using as benchmark a wavelet-based ECG de-  noiser that was demonstrated in [3] to outperform adaptive  filters [4], Tikhonov regularization and Extended Kalman  filters [5], in terms of SNR improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The proposed filter’s  clinical performance is evaluated by measuring the QT-interval  error between the clean ECG and its corresponding filtered  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' GAUSSIAN PROCESS-BASED ECG FILTERING  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The mathematical model  The ECG measurement x(t) is assumed to be an additive  mixture of a clean ECG s(t), assumed to be a GP contami-  nated by additive white noise:  x(t) = s(t) + n(t),  t ∈ {t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' tN}  ∆= TN,  (1)  where n(t) ∼ N(0, vn), vn denotes the noise variance and  N denotes the number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The signal x(t) is  assumed to be baseline-wander (BW) and powerline noise  removed, which are relatively straightforward, with classical  filtering pipelines (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Therefore, the filter  design objective is focused on in-band ECG noise removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  For the beat i, Ti =  �  ti1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' tiRi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' tiNi  �  denotes the set of  time samples, ti1 representing the first sample, tiRi the sample  corresponding to the R-peak and tiNi the last sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' We  further define xi = [x(t)]i∈Ti, si = [s(t)]i∈Ti, ni = [n(t)]i∈Ti  as vectorial representations of the measurement, clean ECG  and noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Therefore, xi = si + ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Next, we define matrices Θi ∈ RT ×Ni to map the time  domain beats xi, si and ni to the phase domain beats  ξi = Θixi, ςi = Θisi, ηi = Θini,  (2)  with aligned R-peaks and the same number of samples T  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The Θi matrices are defined by considering T knots  2  EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US  0  25  50  75  100  125  150  175  200  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
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+page_content='0  amplitude [mv]  x[1]  x[2]  x[3]  x[4]  x[5]  x[6]  0  50  100  150  200  time [samples]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
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+page_content='0  amplitude [mV]  ξ[1]  ξ[2]  ξ[3]  ξ[4]  ξ[5]  ξ[6]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Time-domain measurements beats (top) and the corresponding phase  domain ECG beats (bottom), with the same number T of samples for the  first 6 beats of sel100 record from QTDB [6] with 0 dB Gaussian additive  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Transformation matrices Θi are defined via (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  0  5  10  15  20  0  5  10  15  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
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+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Corner detail example of transformation matrix Θi (left), ΘT  i  (middle) and the corresponding (diagonal) Gramian Gi = ΘT  i Θi (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  equidistantly distributed in the interval [1, Ni] and assigning  Θi(j, k) =  �  1, if j − 1 ≤ (k − 1) Ni−1  T −1 < j  0, otherwise  ,  (3)  with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' , Ni − 1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' , T and T ≥ maxi {Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  With this choice, the corresponding Gramian matrices Gi, are  diagonal matrices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 2),  Gi = ΘT  i Θi = diag [gi] and diag  �  ΘiΘT  i  �  = 1T ,  (4)  with gi ∈ RNi and 1T ∈ RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Therefore, Gi is invertible and  the back transformation from the phase to the time domain is  given by Ψi = G−1  i ΘT  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  From (1) and (2), the ECG beats satisfy ξi = ςi + ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 1, in the phase domain the beats have  been normalized in lengths and the R-peaks are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Therefore, the phase-domain sample variations are only due to  the stochastic inter-beat variations of the ECG beats and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  As our working model, we assume that the phase domain beats  ξi to be ensembles of an underlying GP  ξi ∼ N (µξ, Kξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  (5)  Moreover, from the time domain noise assumption and (2),  the phase domain noise beats also have a zero-mean normal  distribution ηi ∼ N  �  0, vnΘiΘT  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Therefore, the phase do-  main ECG beats follow ςi ∼ N  �  µξ, Kξ − vnΘiΘT  i  �  , where  the model parameters µξ and Kξ can be estimated by the  sample mean ¯µξ := B−1 �B  i=1 ξi and the sample covariance  ¯  Kξ := B−1 �B  i=1(ξi− ¯µξ)(ξi− ¯µξ)T , where B is the number  of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Therefore, the time domain (clean) ECG beats follow  a Normal distribution si ∼ N (µsi, Ksi) with parameters  µsi = Ψi ¯µξ, Ksi = Ψi  � ¯  Kξ − ˆvnΘiΘT  i  �  ΨT  i ,  (6)  where ˆvn represents the noise variance estimate and the  covariance matrix corresponding to time domains beats xi is  given by  Kxi = Ψi ¯  KξΨi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  (7)  Finally, the filtered beats are defined as the time domain  posterior mean, using (6) and (7):  ˆsi = µsi + KsiK−1  xi (xi − µsi) ,  (8)  In the sequel, we refer to µsi and ˆsi as prior-based and  posterior-based GP filter results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The GP filter with diagonal covariance matrix  The direct implementation of the filter in (8) requires the  inversion of covariance matrices that typically have huge  condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The matrix inversion can be avoided if  we consider the diagonal case of ¯  Kξ:  ¯kξ = diag  � ¯  Kξ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' kηi  (4)= ˆvn1T  (9)  In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' the corresponding time domain matrices are also  diagonal and can be computed via  kxi =  �  ΘT  i ¯kξ  �  ⊘ g2  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' ksi =  �  ΘT  i  �¯kξ − kηi  ��  ⊘ g2  i  (10)  with ◦ and ⊘ denoting the Hadamard product and division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  respectively (element-wise product and division),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' g2  i := gi◦gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  the time domain (prior) mean computed via  µsi =  �  ΘT  i ¯µξ  �  ⊘ gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  (11)  and the corresponding filter given by  ˆsi = µsi + ksi ⊘ kxi ◦ (xi − µsi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  (12)  The overall algorithm for GP ECG filtering is summarized in  Algorithm 1 and is available online in our Git repository [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Algorithm 1 GP ECG filtering  1: {tiRi}i = RPeakDetector(x)  [Section III-B]  2: ˆvn = NoiseVartianceEstimator(x) [Section II-D]  Input: x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' {tiRi}i  Output: {�si}i  3: function GPDIAG(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' {tiRi}i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' ˆvn)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='⊲ GP diagonal filter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='for all beats do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='⊲ phase domain computations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute transformation matrices Θi via (3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute the vectors gi = diag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='ΘT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='i Θi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute kηi via (9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='8:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute the phase beats ξi via (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute phase domain sample mean ¯  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='µξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='11:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute phase domain sample variance vector ¯kξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='for all beats do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='⊲ time domain computations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='13:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute ECG prior mean µsi via (11)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute ECG variance ksi via (10)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute measurements variance kxi via (10)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='compute the filtered ECG ˆsi via (12)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='18: end function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='A DATA-DRIVEN ECG GAUSSIAN PROCESS FILTER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' DUMITRU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  3  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Computational cost and model selection  The direct implementation of a GP filter (without the hereby  proposed phase-domain model) would be as follows [1], [2]:  �s = µs + KsK−1  x (x − µs) ,  (13)  with the computational complexity O(N 3), dominated by the  inversion of the measurement covariance matrix Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' In this  approach the model’s hyperparameters are the mean µs, the  covariance matrix Ks) and the noise variance vn (or more  generally the noise covariance matrix) and optimizing them  via classical methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' maximum evidence, leave-one-  out cross validation, [8, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 5]) adds to the computational  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' For long ECGs, the application of this model  is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Previous research considered the GP beat-  wise formulation and adopted a model-based approach to  confine the structure of the covariance matrices [1], [2], but the  choice of the particular model-based mean and kernel function  families remains ad-hoc and difficult to justify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The proposed model infers the GP mean and covariance  matrix in a data-driven way, based on the sample mean and  covariance matrix from the phase domain (6) and (7), and  in the diagonal case, Algorithm 1, does not require any  inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The fundamental assumption allowing the data-  driven computation is the assumption that the phase domain  beats ξi are ensembles from the same underlying GP, (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Hyperparameter selection  The number of phase domain beat samples T is chosen  greater than the longest beat in the time domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' this allows the  choice of the transformation and back transformation matrices  such that the time-phase-time transition can be done without  (transformation) errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The noise variance ˆvn can be com-  puted via maximum evidence or practically from the baseline  segment of the ECG beats, where the heart is electrically silent  and only the noise is exhibited in the ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Baseline wander removal  The BW is removed via two successively zero-phase first  order forward-backward lowpass filters (filtfilt in MAT-  LAB/Python SciPy) with cut-off frequencies set at fc =  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='0 Hz and fc = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='0 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' While the resulting  passband frequency range is rather narrow and eliminates some  ECG-related components, it enables us to assess the filtering  performance for the dominant ECG frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' R-peak detection and heartbeat segmentation  The proposed filter requires the ECG R-peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The beats  are defined relative to the R-peaks, segmenting the mea-  surements at the midpoints between successive R-peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The  R-peak estimation is done using a modified version of the  Pan–Tompkins algorithm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Specifically, the version used in  this paper estimates the R-peaks by successively applying a  band pass filter, an outlier saturation filter via the hyperbolic  tangent function, a square root moving average filter and a  thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Evaluation  The PhysioNet QT Database (QTDB) [6] is used to evaluate  the developed filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' QTDB consists of 15 minutes 2-lead  ECGs sampled at fs = 250 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The baseline wander was  removed as detailed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The required software  for preprocessing and R-peak detection were adopted from the  Open-Source Electrophysiological Toolbox (OSET) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The benchmark filter is a wavelet denoiser with a Symlet–5  mother wavelet, soft thresholding, Stein’s unbiased risk es-  timate (SURE) shrinkage rule, rescaling using a single-level  noise level estimation and four levels of decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' In a  previous study, this combination was proved to outperform  other ECG filtering schemes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The filter evaluation is  measured in terms of SNR improvement and QT-interval  estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' SNR improvement performance  The ECG records were contaminated by additive white  Gaussian noise at SNR levels ranging from –5 to 30 dB, in  5 dB steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' An example of the noisy and filtered ECG are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The average and standard deviation of the  SNR improvement is reported for each noise level, for the  proposed and benchmark methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Accordingly, the  proposed posterior-based filter improves the SNR for every  level of noise tested and outperforms the prior-based and the  benchmark filter for all tested levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Clinical parameters preservation  The accuracy of QT-interval estimation is considered to test  the quality of the proposed methods for clinical ECG parame-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' For this, the QT-interval estimation error (∆QT) between  the QT-interval estimated from the filtered ECG and the QT-  interval estimated from the noiseless ECG is measured and  compared between the benchmark and the proposed method at  variable input noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The QT-interval estimation method  used is adopted from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 5 shows the median and the  interquartile range (IQR) of ∆QT for the benchmark wavelet  and the proposed filter, measured over QTDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Accordingly,  compared with the benchmark method, the GP posterior filter  is reducing the median error for all levels of input noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  In this work we addressed the problem of ECG denoising  via a data-driven based GP model, with beat-wise computa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Compared with the existing non-parametric ECG filters,  the proposed filter makes no ad hoc assumptions about the GP  model and can be used for ECG records of arbitrary length,  since the computational cost has been significantly reduced  as compared with conventional GP filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The proposed filter  is efficient in terms of SNR improvement, outperforming the  benchmark performances for all tested noise levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 4) and  also clinically, with an improved QT-interval estimation error  compared with the benchmark wavelet denoiser, for all tested  levels of noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Another advantage of the proposed  filter is its Bayesian formulation, which allows us to quantify  the filter’s uncertainty (via the estimated variances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' It also  4  EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
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+page_content='1]  (c) input SNR = 10 dB  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The sel100 recording from the PhysioNet QTDB [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' From top to bottom the measurements x vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' the prior estimate (11), the posterior estimate  (12), and the wavelet denoiser (Section III-C), at different input SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The post-filtering SNR improvement is noted in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  −5  0  5  10  15  20  25  30  Input SNR [dB]  −25  −20  −15  −10  −5  0  5  10  15  20  SNR improvemen  [dB]  Wavele  [benchmark]  GP Prior  GP Pos erior  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Mean and standard deviation SNR improvement using the proposed  GP filter and the benchmark wavelet denoiser [3] across all samples of the  PhysioNet QTDB [6], in leads I and II, with 5 repetitions using different noise  instances per record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  5  0  5  10  15  20  25  30  Input SNR [dB]  −100  −50  0  50  100  150  200  QT es ima ion error [ms]  Wavele  GP Fil er  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The median and the interquartile range for ∆QT estimations  corresponding to the proposed and benchmark filters across all samples of  the PhysioNet QTDB [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  provides a framework that allows for synthetic ECG generation  via data-driven learned parameters, which can be used in  generative models for producing synthetic ECG records for  data greedy machine learning and deep learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  In future studies, the fundamental assumption of the model,  namely the same underlying Gaussian distribution for all the  beats in the phase domain can be relaxed, by clustering the  beats and assuming different underlying distributions for the  beats in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Also, comparison with expert annotated  QT-interval (and other clinical parameters) is required and  statistical hypothesis testing should be performed to investigate  if the differences are statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' The proposed  filter requires the R-peaks for aligning the ECG beats in the  phase-domain, which requires investigating to what extend the  filtering performance is susceptible to mis-detection of the  R-peaks and morphological variations due to ectopic beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The Python codes corresponding to the Algorithm 1 and the  reported results are available in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  The authors acknowledge support from the National Insti-  tute of Biomedical Imaging and Bioengineering under the NIH  grant R01EB030362, and the National Center for Advancing  Translational Sciences under the NIH Award UL1TR002378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  REFERENCES  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Rivet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Niknazar, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Jutten, “Non parametric modelling of  ECG: Applications to denoising and single sensor fetal ECG extraction,”  in LVA/ICA 2012 - 10th International Conference on Latent Variable  Analysis and Signal Separation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' LNCS 7191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Tel-Aviv, Israel:  Springer, Mar 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 470–477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Niknazar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Rivet, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Jutten, “Fetal ECG extraction from a  single sensor by a non-parametric modeling,” in 2012 Proceedings of  the 20th European Signal Processing Conference, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 949–953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Sameni, “Online filtering using piecewise  smoothness priors:  Application to normal and abnormal electrocardiogram denoising,”  Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 133, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 52–63, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Available:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='sigpro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='019  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Laguna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Jane, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Meste, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Poon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Caminal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Rix, and  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Thakor, “Adaptive filter for event-related bioelectric signals using an  impulse correlated reference input: comparison with signal averaging  techniques,” IEEE Transactions on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 39,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 1032–1044, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Sameni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Shamsollahi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Jutten, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Clifford, “A  nonlinear bayesian filtering framework for ECG denoising,” Biomedical  Engineering, IEEE Transactions on, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 2172–2185,  December 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='1109/TBME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='897817  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Laguna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Mark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Goldberg, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Moody, “A database for  evaluation of algorithms for measurement of QT and other waveform  intervals in the ECG,” in Computers in Cardiology 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  IEEE, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='1109/cic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='648140  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Dumitru,  Data  Driven  Gaussian  Process  filter  for  ECG,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Available: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='com/alphanumericslab/OSET/  tree/master/UnderDevelopment/DataDrivenGPFilter  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Rasmussen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Williams, Gaussian Processes for Machine  Learning, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Adaptive computation and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  The MIT  Press, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Pan and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Tompkins, “A Real-Time QRS Detection Algorithm,”  Biomedical Engineering, IEEE Transactions on, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' BME-32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  230–236, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Sameni, The Open-Source Electrophysiological Toolbox (OSET),  version  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='14,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='  Available:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='com/  alphanumericslab/OSET  [11] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Dumitru, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=' Perez Alday, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}
+page_content=', “QT-Interval Estimation  Improved with Fusion of Multiple Automated Algorithms,” in Interna-  tional Society for Computerized Electrocardiology (ISCE), April 6-10,  2022, Las Vegas, NV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'}

39A0T4oBgHgl3EQfNf8t/vector_store/index.pkl

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+Combining Dynamic Mode Decomposition with Ensemble Kalman
+Filtering for Tracking and Forecasting
+Stephen A Falconer1, David J.B. Lloyd1, and Naratip Santitissadeekorn1
+1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
+January 16, 2023
+Abstract
+Data assimilation techniques, such as ensemble Kalman filtering, have been shown to be a
+highly effective and efficient way to combine noisy data with a mathematical model to track
+and forecast dynamical systems. However, when dealing with high-dimensional data, in many
+situations one does not have a model, so data assimilation techniques cannot be applied. In
+this paper, we use dynamic mode decomposition to generate a low-dimensional, linear model
+of a dynamical system directly from high-dimensional data, which is defined by temporal and
+spatial modes, that we can then use with data assimilation techniques such as the ensemble
+Kalman filter.
+We show how the dynamic mode decomposition can be combined with the
+ensemble Kalman filter (which we call the DMDEnKF) to iteratively update the current state
+and temporal modes as new data becomes available. We demonstrate that this approach is able
+to track time varying dynamical systems in synthetic examples, and experiment with the use
+of time-delay embeddings. We then apply the DMDEnKF to real world seasonal influenza-like
+illness data from the USA Centers for Disease Control and Prevention, and find that for short
+term forecasting, the DMDEnKF is comparable to the best mechanistic models in the ILINet
+competition.
+Keywords
+Dynamic mode decomposition; Ensemble Kalman filter; Data-driven modelling; Data assimilation;
+Dynamical systems
+1
+Introduction
+Data assimilation refers to the collection of methods that integrate vast data sets with sophisticated
+mathematical models, to track and forecast systems that may evolve or change [38]. The majority
+of its applications lie in the earth sciences [51], however due to the generality of its techniques they
+have also been successfully applied in a wide range of areas from medicine [18] to ecology [40].
+The Kalman filter [36] is one such data assimilation technique widely used throughout industry [3]
+that optimally combines predictions from a linear model with Gaussian data. Whilst traditionally
+applied to a model’s state, the parameters of the model can simultaneously be filtered, leading to
+1
+arXiv:2301.05504v1  [math.DS]  13 Jan 2023
+
+what is known as the joint state-parameter estimation problem [33]. If the system being filtered is
+nonlinear, alternative versions of the Kalman filter can be utilized such as the extended Kalman filter
+[53], unscented Kalman filter [59] or ensemble Kalman filter (EnKF) [23]. The EnKF represents
+the distribution of a system’s state with an ensemble of random samples, that can then be used to
+estimate useful statistics like the state’s covariance via the sample covariance or a point estimate
+of the state via the sample mean [23] and is well-suited for high-dimensional problems. All of these
+methods require a model of the system, however if no model exists then one must be generated and
+the most generalizable way to do this is via data-driven modelling.
+Dynamic mode decomposition (DMD) is a data-driven modelling technique for identifying low di-
+mensional, spatial and temporal patterns within a dynamical system directly from high-dimensional
+data [54]. It does this by postulating the state vector is evolved via a linear system and looking for
+a low-dimensional approximation of the eigenvalues (temporal modes) and corresponding eigenvec-
+tors (spatial modes). Spatial modes can be thought of as modes that decompose state variables
+into separate components that evolve together linearly in time. The corresponding temporal modes
+describe whether a spatial mode is growing, decaying or stationary in time. DMD has been used
+to approximate dynamical systems from measurement data in a multitude of fields, ranging from
+epidemiology [49], finance [43] and neuroscience [12]. Due to its popularity, it has been extended
+to systems that are nonlinear in their recorded measurement functions via Extended/Kernel DMD
+[61]/[44], with one such extension Hankel-DMD [2] employing time-delay embeddings of the original
+observables. In the presence of measurement noise, the standard DMD has been shown to induce a
+systematic bias by asymmetrically attributing all noise to the model’s target output measurements
+and none to its inputs during training [15]. This systematic bias, prompted the creation of noise
+handling variants of DMD that directly account for the noise term [15], the Forward Backward
+DMD [15] that performs DMD forwards and backward in time and combines the results, and To-
+tal DMD (TDMD) [31] that minimizes the total least squares error as opposed to minimizing the
+ordinary least squares error.
+The aim of this paper is to develop an algorithm that iteratively improves the temporal modes
+(eigenvalues) and state estimates produced by DMD with the EnKF as new data becomes available.
+This would be highly useful for dynamical systems that make a change from growing or decaying
+behaviour over time. While estimating just the state of the system using the DMD modes can be
+done using a standard Kalman filter, without also filtering the model’s temporal mode, estimates
+are likely to suffer if the system changes over time. Methods already exist that combine DMD with
+the Kalman filter [45] or extended Kalman filter [46], which apply filtering to estimate the entire
+system dynamics matrix. The filtering in our work is instead focused on efficiently tracking the
+system’s temporal modes, and forecasting the system’s future states. DMD produces a linear model
+which makes it a natural fit for the Kalman filter, however when a system’s state and temporal
+modes are estimated simultaneously the filtering process becomes nonlinear. Hence, we need to use
+a filter designed for a nonlinear model, and we chose the EnKF due to its versatility, scalability to
+large dynamical systems, and ease of implementation [52]. While any DMD variant that produces
+temporal modes would be compatible with the DMDEnKF framework, we use TDMD to remain
+consistent with the EnKF’s assumption that noise is present in the data. In tandem, we apply
+the DMDEnKF using a total least squares version of Hankel-DMD, henceforth referred to as the
+Hankel-DMDEnKF, to investigate the effect time-delay embeddings have on our framework.
+2
+
+To demonstrate the DMDEnKF method, we first test it on synthetically generated datasets. Ini-
+tially, on a simple noisy oscillating system with a decreasing period of oscillation, we use the DM-
+DEnKF to track the system’s temporal modes and compare results with the Hankel-DMDEnKF,
+other iterative DMD variants, and “gold standard” filtering methods. Next, we simulate a pan-
+demic and evaluate the DMDEnKF’s ability to track the system’s temporal modes and generate
+multistep ahead forecasts.
+Figure 1: ILI consultations as a percentage of total weekly GP consultations in the US from 2003 to end of
+2018. The data shows the annual peaks in ILI consultations that vary in size, timing and shape, which
+would make them difficult to model with a simple SIR-type model.
+Finally, we apply the DMDEnKF and Hankel-DMDEnKF to real seasonal influenza-like illness
+(ILI) data in the United States from the Centers for Disease Control and Prevention (CDC) ILINet
+[25] shown in Figure 1, with the aim of investigating their forecasting skills for ILI consultation
+rates. ILI is defined as a fever with a cough or sore throat that has no known cause other than
+influenza [25] and infects between 9 and 35 million people in the US each year [24]. Due to its
+prevalence, a multitude of methods have already been developed to model the spread of ILI [14, 47]
+and the approaches these models take can broadly be classified as either mechanistic or statistical
+[37]. Mechanistic methods [5, 48] make explicit hypotheses about what is driving the spread of an
+infectious disease, before then fitting parameters in the proposed models to the data. They have
+the advantage of being highly interpretable making them useful when trying to understand how
+one can control the spread of a disease [39], however can make assumptions that are oversimplified
+[4]. For example, a simple SIR-type model would struggle to describe specific behaviours like the
+drop in ILI consultations around Christmastime seen in Figure 1.
+Statistical methods [11, 60]
+are generally more versatile as they require fewer domain-specific assumptions, but both methods
+achieve a similar predictive skill in real time on the ILINet dataset [50]. The DMDEnKF attempts
+to find a middle ground between the two methods, remaining versatile by virtue of being purely
+data-driven but also providing some level of interpretability via the associated DMD modes.
+The remainder of this paper will be structured as follows. First, a brief summary of DMD, Hankel-
+3
+
+8
+7
+6
+4
+三
+%3
+2
+1
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+DateDMD and EnKF algorithms for completeness.
+After which, the DMDEnKF algorithm will be
+described in full. We will then apply the DMDEnKF and Hankel-DMDEnKF to synthetic data
+and compare their performance against other pre-existing, iterative DMD variants. Finally, we will
+use the DMDEnKF and Hankel-DMDEnKF on ILINet data to forecast the rate of ILI consultations
+up to 4 weeks into the future and examine their performance.
+2
+DMDEnKF
+2.1
+Dynamic Mode Decomposition (DMD)
+Consider an n dimensional state xk ∈ Rn measured at regular time intervals k = 1, ..., m. Assuming
+this time-series data was generated by a linear dynamical system, the consecutive states xk and
+xk+1 are connected via
+xk+1 = Axk
+(2.1)
+for some unknown matrix A ∈ Rn×n. By denoting
+X =
+�
+��
+|
+|
+|
+x1
+x2
+...
+xm−1
+|
+|
+|
+�
+�� ,
+X′ =
+�
+��
+|
+|
+|
+x2
+x3
+...
+xm
+|
+|
+|
+�
+�� ,
+(2.2)
+equation (2.1) can be written succinctly over all consecutive data pairs as
+X′ = AX.
+(2.3)
+To minimize the mean squared error term �m−1
+k=1 ∥xk+1 − Axk∥2
+2, the standard DMD defines
+A = X′X+,
+(2.4)
+where X+ is the Moore-Penrose pseudoinverse [6] of X. Efficiently solving for the eigendecom-
+position of A is the primary purpose of DMD, as these eigenvalues/eigenvectors correspond to
+spatio-temporal patterns in the data.
+The DMD method starts by applying the Singular Value Decomposition (SVD) to the data matrix
+X, representing it as the matrix multiplication of 2 real-valued, orthonormal matrices (complex and
+unitary if X ∈ Cn×m) U ∈ Rn×n, V ∈ Rm×m and a rectangular diagonal matrix with decreasing
+non-negative real values (Σ ∈ Rn×m) in the form
+X = UΣV∗.
+(2.5)
+The best rank r approximation of a matrix according to the Eckart-Young Theorem [22] is obtained
+by truncating its SVD, hence by truncating equation (2.5) to a suitable rank r [26] we can compress
+the data matrix with minimal loss of information, which we write as
+X ≈ UrΣrV∗
+r.
+(2.6)
+By performing this compression, we are implicitly assuming that there exists a low dimensional
+(≤ r), linear structure within the high-dimensional data.
+4
+
+The Moore-Penrose pseudoinverse can be found directly from the SVD computed in equation (2.5)
+as VΣ−1U∗. We use the rank r truncated matrices from equation (2.6) for reasons of efficiency,
+setting
+A = X′X+,
+≈ X′VrΣ−1
+r U∗
+r.
+(2.7)
+This approximation of A now acts only on an r dimensional subspace defined by Col(Ur). Hence,
+we can restrict A onto this r dimensional subspace (representing the largest r POD modes of X)
+and denote the restricted A ∈ Rr×r as
+˜A = U∗
+rAUr,
+≈ U∗
+rX′VrΣ−1
+r .
+(2.8)
+We calculate the eigenvalues (λi), and corresponding eigenvectors (vi) of ˜A, and define
+Λ =
+�
+���
+λ1
+0
+0
+0
+...
+0
+0
+0
+λr
+�
+��� ,
+W =
+�
+��
+|
+|
+|
+v1
+v2
+...
+vr
+|
+|
+|
+�
+�� .
+(2.9)
+Reconstructing the eigenvalues/eigenvectors of the original operator A will provide insights into
+the structure of the system [1] and allow us to propagate it forward in time. The eigenvalues of ˜A
+(Λ) can be shown to be equal to the eigenvalues of the original operator A [34], however recovering
+the original eigenvectors is more involved and can be done using either projected or exact DMD.
+We use the exact DMD method introduced by Tu et al. [34] as it finds the exact DMD modes (Φ)
+for all eigenvectors with non-zero λi, where Φ is defined as
+Φ = X′VrΣ−1
+r W.
+(2.10)
+DMD modes with zero eigenvalues have no effect on the system’s dynamics, so this restriction of
+exact DMD is of little consequence. This method finds A such that AX = X′ exactly provided
+r ≥ rank(X) and X and X′ are linearly consistent [34].
+With Λ and Φ in hand, we can construct a r dimensional approximation of A, however still need
+to find the initial phase and amplitude of each mode. The standard method [54] for computing this
+vector (b) is to rewrite the initial state x1 in a basis of the DMD modes via
+b = Φ+x1.
+(2.11)
+It is worth noting that there exist alternative methods for example [13, 35] that focus on optimizing
+b over all data points with additional conditions.
+To summarise, the final solution to the discrete system can be written as
+xk = ΦΛkb.
+(2.12)
+In the remainder of the paper, we call Λ the temporal modes and Φ the spatial modes.
+5
+
+2.1.1
+Hankel-DMD
+Hankel-DMD first augments the original, measured state xk ∈ Rn, by appending to it measurements
+of the state at the previous d − 1 time steps
+h(xk) =
+�
+xkT
+xk−1T
+. . .
+xk−(d−1)T �T
+,
+(2.13)
+to form a new state h(xk) ∈ Rdn. This is known as a time-delay embedding, and we refer to d
+as the delay-embedding dimension. Taking time-delay embeddings, h(xk), to be our new states,
+matrices X and X′ from equation (2.2) now become
+X =
+�
+���
+xd
+. . .
+xm−1
+...
+...
+...
+x1
+. . .
+xm−d
+�
+��� ,
+X′ =
+�
+���
+xd+1
+. . .
+xm
+...
+...
+...
+x2
+. . .
+xm−(d−1)
+�
+��� .
+(2.14)
+With X and X′ defined above, Hankel-DMD proceeds exactly as the standard DMD algorithm,
+generating eigenvalues Λ, DMD modes Φ and their initial states b as described above. The original
+system can be reconstructed/forecast for all time steps from d onwards, by applying equation (2.12)
+and restricting the result to the first n rows.
+2.1.2
+Iterative DMD Variants
+There exists other variants of DMD that are designed to be applied iteratively, and in this paper we
+will compare these with the DMDEnKF in their ability to track a system’s eigenvalues and make
+future state predictions. Streaming DMD [32] is an adaption of the standard DMD algorithm to
+efficiently process new data as it becomes available, and the noise aware variant Streaming TDMD
+[30] is the first variant we wish to compare against. The second method we will use for comparison
+is Windowed DMD [62], where the standard DMD described above is applied over a sliding window
+of the w most recent data snapshots only. The final method we will be comparing against is Online
+DMD [62], specifically the variant of this algorithm that places an exponentially decaying weight ρ
+on the importance of past measurements.
+2.2
+Ensemble Kalman Filter (EnKF)
+Consider a discrete-time, nonlinear dynamical system with a stochastic perturbation
+xk = F(xk−1) + wk,
+wk ∼ N(0, Qk),
+(2.15)
+where F is a nonlinear function F : Rn → Rn, xk ∈ Rn is the system’s state, wk ∈ Rn is a
+stochastic perturbation and N is the normal distribution with mean 0 and covariance matrix Qk.
+A measurement equation that relates what we observe to the true state of the system is given by
+yk = H(xk) + vk,
+vk ∼ N(0, Rk),
+(2.16)
+where H : Rn → Rl is the system’s observation operator, yk ∈ Rl is an observation of the system,
+vk ∈ Rl is the noise in the observation and N is the normal distribution with mean 0 and covariance
+matrix Rk. We focus on the instance relevant to our use case where H is linear, so can be represented
+by a matrix H ∈ Rl×n.
+6
+
+In general, filtering methods aim to combine information from the state-transition model (2.15) and
+observation model (2.16) to compute the conditional density p(xk|Yk), where Yk = (y1, ..., yk).
+The Kalman filter is the optimal filter if F and H are both linear and the stochastic perturbations
+are normal [36]. The EnKF was developed to deal with the filtering problem where either the linear
+or normal assumption (or both) is violated [23]. It exploits the Kalman formulation to propagate
+an ensemble of the state into a region of high probability in such a way that the ensemble spread
+would be consistent with the linear and normal model.
+To begin the EnKF algorithm, an initial ensemble of N state estimates ˆx(1)
+0 ,..., ˆx(N)
+0
+is required. If
+an ensemble is not available, one can be generated from initial state estimates ˆx0 and covariance
+matrix P0 by taking N independent draws from N(ˆx0, P0).
+Algorithm
+The EnKF then acts as follows [23]:
+Step 1: Propagate forward in time each ensemble member using equation (2.15) for i = 1, ..., N
+via
+ˆx(i)
+k|k−1 = F(ˆx(i)
+k−1|k−1) + w(i)
+k .
+(2.17)
+The notation ˆx(i)
+k|k−1 denotes the state estimate at time k of the ith ensemble member ˆx(i)
+k
+using
+only information up to time k − 1, and ˆx(i)
+k−1|k−1 represents the same ensemble member at time
+k − 1 using information up to time k − 1. Each w(i)
+k
+is independently drawn from N(0, Qk). The
+current covariance matrix can also now be estimated via the sample covariance of the ensemble,
+which we denote as ˆPk|k−1. This can then be used to estimate the Kalman Gain matrix ˆKk as
+ˆKk = ˆPk|k−1HT (HˆPk|k−1HT + Rk)−1.
+(2.18)
+Step 2: Calculate the measurement innovation utilizing equation (2.16).
+From measurement yk, we again use i = 1, ..., N and generate simulated measurements
+y(i)
+k
+= yk + v(i)
+k
+(2.19)
+where each v(i)
+k
+is an independent draw from N(0, Rk). These simulated measurements y(i)
+k
+are
+combined with the ensemble members ˆx(i)
+k|k−1 from equation (2.17) to define N measurement inno-
+vations
+e(i)
+k = y(i)
+k − Hˆx(i)
+k|k−1.
+(2.20)
+The e(i)
+k
+represent samples from the distribution of the distance of the model’s prediction from the
+measured value.
+Step 3: Combine the model estimates in equation (2.17) and measurement innovation of equation
+(2.20) via the estimated Kalman gain from (2.18) to update each ensemble member’s state estimate
+ˆx(i)
+k|k = ˆx(i)
+k|k−1 + ˆKke(i)
+k .
+(2.21)
+We can generate a point estimate for the state ˆxk using the mean of the N updated ensemble
+members. This process then repeats every time a new state measurement becomes available, with
+the updated ensemble from the previous data point becoming the initial ensemble for the new one.
+We combine these 2 previously described techniques to form the DMDEnKF. This new, hybrid
+method uses DMD to generate a low dimensional model of a dynamical system that is then itera-
+tively improved by the EnKF as new data emerges.
+7
+
+2.3
+DMDEnKF
+We now describe how we carry out filtering of the temporal modes and state of the system, while
+keeping the spatial modes found by one’s chosen version of DMD on the “spin-up” fixed. We note
+that once we allow the temporal modes to vary with the spatial modes being fixed, these are no
+longer eigenvalues/eigenvectors, and we then call them temporal modes. Consider an n dimensional
+state xk ∈ Rn measured at regular time intervals k = 1, ..., m and then measured iteratively at times
+k = m + 1, ....
+Algorithm
+Step 1: Perform the chosen version of DMD on the dataset x1, ..., xm, defining X, X′ as before in
+equation (2.2) to obtain the expression
+xk = ΦΛkb,
+(2.22)
+where
+Λ =
+�
+���
+λ1
+0
+0
+0
+...
+0
+0
+0
+λr
+�
+��� ,
+Φ =
+�
+��
+|
+|
+|
+d1
+d2
+...
+dr
+|
+|
+|
+�
+�� ,
+b =
+�
+���
+b1
+...
+br
+�
+��� ,
+(2.23)
+and defining λi, di, bi as the ith temporal mode, DMD mode, initial condition triplet of the r
+retained modes. This acts as a spin-up process to generate a model we can then filter using the
+EnKF.
+Step 2: Define the matrices required for the EnKF’s ensemble initialisation, propagation via
+equation (2.15), and measurement using equation (2.16).
+First, rewrite each of the r temporal modes in polar coordinates as
+λi = τieθii,
+(2.24)
+where τi ≥ 0, 0 ≤ θi < 2π and i2 = −1. As xk ∈ Rn, the temporal modes in the DMD model’s
+spectrum will either be real or in a complex conjugate pair. When filtering, we view the temporal
+modes as a time varying parameter.
+However, we must enforce that the real temporal modes
+remain real and complex conjugate pairs remain intact, as this ensures the state output by the
+model will still be real. We do this by defining the filterable model parameters µi as new variables
+for i = 1, ..., r
+µi =
+�
+τi,
+if θi = 0, or ∄ j for j < i such that λ∗
+j = λi,
+θi,
+otherwise.
+(2.25)
+Written in this way, these µi’s represent all the possible degrees of freedom in the model’s temporal
+modes under the additional constraint of producing a real state estimate. By maintaining a note
+of the positional indexes of each complex conjugate pair produced in the initial DMD, it is possible
+to recover the λi representation from the µi’s. While this transformation technically requires the
+full list of µi’s, we informally write Λ(µi) = λi to symbolize the reversion from µi’s back to λi’s.
+Ensemble initialisation: We can now define an augmented joint parameter state z0 ∈ Rn+r to
+be used as the initial state for the EnKF
+z0 =
+�
+−
+xm
+−
+µ1
+. . .
+µr
+�T
+.
+(2.26)
+8
+
+We denote the joint parameter state at time m + k as zk ∈ Rn+r. To generate an initial ensemble
+from this state, we first define sample covariance C = (1/m)(X′ − ΦΛΦ+X)(X′ − ΦΛΦ+X)T ,
+which represents the current state uncertainty based on prediction errors in the spin-up DMD. We
+then form the initial covariance matrix
+P0 =
+�
+C
+0
+0
+α2Ir
+�
+,
+(2.27)
+where α2 > 0, Ir is the r-dimensional identity matrix, and the α2Ir term determines the initial
+uncertainty in the spin-up DMD’s temporal modes. Take independent draws from N(z0, P0) until
+the ensemble is sufficiently large. The optimal ensemble size will vary from problem to problem,
+adding ensemble members will increase accuracy but at the cost of computational efficiency.
+Propagation: Using the notation zi
+k to signify the ith element of zk, we define the matrix Λzk ∈
+Rr×r for state zk as
+Λzk =
+�
+���
+Λ(zn+1
+k
+)
+0
+0
+0
+...
+0
+0
+0
+Λ(zn+r
+k
+)
+�
+��� .
+(2.28)
+The EnKF’s propagation equation can be written as
+zk+1 =
+�
+ΦΛzkΦ+
+0
+0
+Ir
+�
+zk + wk.
+(2.29)
+For convenience, we introduce notation zi:j
+k ∈ Rj−i+1 to denote the ith through to the jth element
+of zk where i ≤ j
+zi:j
+k =
+�
+zi
+k
+. . .
+zj
+k
+�T
+.
+(2.30)
+Equation (2.29) propagates z1:n
+k
+representing the state in the DMD framework xm+k forward in
+time using the standard DMD equation with the updated temporal modes from Λzk. The vector
+zn+1:n+r
+k
+represents the current estimate of the temporal modes in their µi representation and is
+unchanged other than the addition of noise by the propagation equation, for although we assume
+the temporal modes vary in time no direction of drift in the parameters is explicitly foreknown.
+The vector wk ∈ Rn+r is a normally distributed variable wk ∼ N(0, Qk), and this represents the
+uncertainty within the model of the system. We construct Qk as follows,
+Qk =
+�
+α1In
+0
+0
+α2Ir
+�
+,
+(2.31)
+where α1 and α2 are constants determined by the user such that α2 ≪ α1. This construction
+with Qk a diagonal matrix assumes model errors for each element of zk are uncorrelated with
+one another.
+The condition α2 ≪ α1 ensures that the state of the DMD system z1:n
+k
+changes
+significantly faster than its temporal modes zn+1:n+r
+k
+, as parameters by definition should vary
+slowly in time. Furthermore, for the temporal mode’s moduli being filtered, it prevents the strictly
+positive modulus dropping below 0.
+Measurement: We write the EnKF’s measurement equation as
+yk =
+�
+In
+0
+0
+0
+�
+zk + vk,
+(2.32)
+9
+
+where yk ∈ Rn are observations of the DMD state xm+k, and vk ∈ Rn is a normally distributed
+variable ∼ N(0, Rk) representing the noise in the measurements. We assume new measurements
+yk to be available for the full DMD state z1:n
+k
+but not its temporal modes zn+1:n+r
+k
+, as this is
+consistent with the format of the data used to generate the spin-up DMD model. We also assume
+uncorrelated measurement noise on each dimension of the state, so choose a diagonal matrix Rk.
+Step 3 State measurements xm+k at times k = 1, ... are being iteratively generated. By setting
+yk = xm+k as each new measurement arrives, we can iteratively apply the EnKF to produce a
+hybrid estimate for zk that combines model predictions from zk−1 and noisy measurement yk.
+A brief summary of how the EnKF does this is provided in Section 2.2, and a more expansive
+description can be found at [42].
+Step 4: The state of the original system xm+k can be reconstructed from zk by simply taking it’s
+first n elements z1:n
+k . Predictions p steps ahead at time m + k can also be forecast from zk via
+xm+k+p = ΦΛp
+zkΦ+z1:n
+k .
+(2.33)
+The Hankel-DMDEnKF is defined algorithmically in exactly the same way, with the only difference
+being that Hankel-DMD is applied over the “spin-up” period as opposed to standard DMD.
+3
+Synthetic Applications
+3.1
+Comparison against other iterative DMD variants
+To test the DMDEnKF, we first apply it to data generated from a synthetic system with time
+varying eigenvalues, which we aim to track. The dynamics of this system are governed by the 2
+dimensional rotation matrix, where the angle of rotation θk increases linearly from π/64 to π/8
+over the course of 500 time steps. The evolution of the state xk of the system can hence be written
+as
+xk+1 =
+�
+cos(θk)
+− sin(θk)
+sin(θk)
+cos(θk)
+�
+xk,
+x1 =
+�
+1
+0
+�
+,
+(3.1)
+where θk = π/64 + (k−1)(7π/64)
+499
+and k = (1, ..., 500).
+We assume noisy measurement values yk to be available for the state at each time step, such that
+yk = xk + vk,
+vk ∼ N(0, σ2I2),
+(3.2)
+where each experiment σ = 0.05 or 0.5 to simulate a low or high level of measurement noise
+respectively.
+The 500 values of yk (shown in Figure 2) are used to train the DMDEnKF and
+Hankel-DMDEnKF, with the first 100 time steps being used for the spin-up process described in
+Step 1 of the DMDEnKF algorithm to produce the output described in equation (2.23).
+We will also train the iterative variants of DMD described at the end of Section 2.1 (Streaming
+TDMD1, Windowed DMD and Online DMD) on this dataset to compare their ability to track the
+1As the synthetic dataset is small, it is computationally tractable to apply batch methods to the data. Hence,
+instead of applying the true Streaming TDMD algorithm, we use batch TDMD over all data up to the current
+time step as a proxy for Streaming TDMD utilizing code from the PyDMD library [19].
+As Streaming TDMD
+approximates the results of TDMD with the only differences occurring due to additional data compression steps in
+Streaming TDMD’s algorithm, we believe this to be an acceptable substitution.
+10
+
+Figure 2: Time series for a synthetic system with a linearly increasing eigenvalue argument, showing the
+state’s first dimension with no, low (σ = 0.05) and high (σ = 0.5) measurement noise.
+system’s time varying eigenvalues against that of the DMDEnKF. Within the Windowed DMD
+algorithm, we replace DMD with TDMD to allow for this method to effectively handle the noise
+in the data, henceforth referring to this amalgamation of the two methods as Windowed TDMD.
+To implement Online DMD, we use code made available by its creators here [29]. Computational
+parameters were set as follows; window size w = 10 for Windowed TDMD, exponential decay
+rate ρ = 0.9 for Online DMD, delay-embedding dimension d = 50 for the Hankel-DMDEnKF and
+spin-up time steps m = 100 for the DMDEnKF as previously stated.
+At each time step k, the system’s true eigenvalues can be written in modulus-argument form as
+λk = 1e±θki,
+(3.3)
+and for each time step where the models are defined their estimates of the system’s eigenvalues can
+also be written as
+ˆλk = ˆτke±ˆθki.
+(3.4)
+We start by comparing the errors in each method’s estimate of the constant eigenvalue modulus
+(ˆτk−1). A thousand runs of the synthetic data were generated for each value of σ, and the difference
+of each method’s eigenvalue modulus and argument from their true values at every time step after
+the spin-up period were collected. When any of the methods failed to identify the eigenvalues as a
+complex conjugate pair at a given time step in a run, the dominant eigenvalue’s modulus was used
+for modulus error calculations. The average errors in the eigenvalue modulus estimates are shown
+in Table 1.
+For all levels of measurement noise, Streaming TDMD estimated the eigenvalue modulus the most
+accurately. This is due to the method’s assumption of a stationary system, hence assigning an
+equal weight to the importance of each data point, which works well in the case of estimating a
+constant parameter. At low levels of measurement noise as seen in the first column of Table 1,
+Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly
+11
+
+2
+dimension
+0
+1
+XkTrue State
+yk for  = 0.05
+-2
+yk for o = 0.5
+0
+100
+200
+300
+400
+500
+TimestepsIterative DMD
+Mean Eigenvalue Modulus Error
+Variant
+σ = 0.05
+σ = 0.5
+Windowed TDMD
+9.82 × 10−3
+1.39
+Online DMD
+6.04 × 10−3
+3.06 × 10−1
+Streaming TDMD
+2.31 × 10−4
+2.50 × 10−3
+DMDEnKF
+8.07 × 10−3
+1.89 × 10−2
+Hankel-DMDEnKF
+9.49 × 10−3
+1.38 × 10−2
+Table 1: Mean absolute errors in the synthetic system’s eigenvalue modulus estimates produced by each
+iterative DMD variant over all time steps over the course of all 1000 runs. Measurement noise is set to
+either low levels with σ = 0.05 (left) or high levels with σ = 0.5 (right). Streaming TDMD scored
+significantly lower errors than all other methods, and as noise levels increased, errors in Windowed TDMD
+and Online DMD grew significantly larger than those produced by the DMDEnKF and Hankel-DMDEnKF.
+well with mean eigenvalue modulus errors below 0.01. As errors in the eigenvalue modulus grow
+exponentially when forecasting future states, these 4 methods could produce acceptable short term
+forecasts but would quickly diverge from the true state as the forecast horizon was extended. At
+high levels of noise shown in the second column of Table 1, Windowed TDMD and Online DMD’s
+eigenvalue modulus estimates degrade significantly, making them unsuitable for forecasting in this
+scenario. The errors in the DMDEnKF and Hankel-DMDEnKF remain fairly small, however are
+still an order of magnitude greater than those produced by Streaming TDMD.
+A typical trajectory of the eigenvalue argument estimates (ˆθk) for each method over the course
+of one run from the end of the spin-up period onwards can be seen in Figures 3a and 3c. The
+error distributions for each method’s eigenvalue argument estimates (ˆθk − θk) over all 1000 runs
+are plotted in Figures 3b and 3d.
+At low levels of noise as seen in Figures 3a and 3b, all 5 methods on average underestimated the
+eigenvalue argument of the system. This is to be expected as the eigenvalue argument is increasing
+with time, meaning that all but the last data pair available to each method would have been
+generated using an argument smaller than its current value. Streaming TDMD exhibited the worst
+performance, again due to its equal weighting of every data point, however in this instance being a
+negative quality as it hampers the model’s ability to adapt to fresh data that reflects the changing
+parameter. Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed
+similarly. Online DMD produced a tighter error distribution, but with a slightly larger bias than
+Windowed TDMD. This suggests that Online DMD’s soft thresholding reduces the model volatility
+caused by measurement noise compared to the hard cut-off employed by Windowed TDMD. For
+this same reason however, Online DMD is slower to adapt to new measurements than Windowed
+TDMD, leading to a larger bias below the system’s true eigenvalue argument. The DMDEnKF
+and Hankel-DMDEnKF performed very similar to Windowed TDMD at this noise level, however
+tweaks to the magnitude of the DMDEnKF’s system uncertainty matrix can be made to balance
+the speed of model innovation with its volatility and produce distributions closer to that of Online
+DMD if required.
+At higher noise levels shown in Figures 3c and 3d, the performance of Windowed TDMD and Online
+DMD significantly degrades. Placing a larger weight on more recent samples allowed these methods
+12
+
+(a) Eigenvalue argument trajectory for σ = 0.05.
+(b) Error distribution for σ = 0.05.
+(c) Eigenvalue argument trajectory for σ = 0.5.
+(d) Error distribution for σ = 0.5.
+Figure 3:
+Estimates of the synthetic system’s eigenvalue argument produced by each iterative DMD
+variant. Presented are typical trajectories of the eigenvalue argument at each time step over the course of 1
+experiment’s run (left) and error distributions of the difference between the true system’s eigenvalue
+argument and the estimated eigenvalue argument over all time steps over the course of all 10 runs (right).
+Measurement noise is set to either low levels with σ = 0.05 (top) or high levels with σ = 0.5 (bottom). The
+DMDEnKF and Hankel-DMDEnKF experience similar errors to Online DMD and Windowed TDMD at
+low measurement noise, but track the eigenvalue argument much more accurately than them for high
+measurement noise.
+to quickly adapt to changes in the system’s parameters, however as the noise increases this induces
+an extreme volatility in their respective models.
+The performance of Streaming TDMD is not
+largely changed from the low noise case, still lagging behind the true system values but somewhat
+insulated from the noise by its symmetric treatment of all data points. Here the benefit of explicit
+inclusion of measurement noise in the DMDEnKF framework becomes apparent, as at this noise
+level the DMDEnKF and Hankel-DMDEnKF are the only techniques tested capable of producing
+an accurate eigenvalue argument estimate.
+Furthermore, here we see the first significant difference in the performance of the DMDEnKF and
+Hankel-DMDEnKF, as the DMDEnKF’s error distribution has a thin tail extending down to −π/8
+which is not present in the error distribution of the Hankel-DMDEnKF. These additional errors are
+caused by the spin-up of the DMD for the DMDEnKF method occasionally failing to identify the
+system’s eigenvalues as a complex conjugate pair (empirically, this happens ∼ 3% of the time), due
+to the increased noise in the data. When this happens, the DMDEnKF catastrophically fails for
+the EnKF is unable to generate complex eigenvalues from real ones regardless of how many future
+time steps are filtered due to its formulation in equation (2.25). This failure of the DMDEnKF can
+be mitigated in the following way. If the errors produced by the DMDEnKF during the filtering
+stage are deemed too large (e.g. exceed a given threshold) for a prolonged period of time, then the
+spin-up DMD process can be rerun on an extended dataset consisting of the original spin up data,
+13
+
+0.5
+True Eigenvalue
+Streaming TDMD
+igenvalue Argument
+0.4
+Windowed TDMD
+Online DMD
+0.3
+Hankel-DMDEnKF
+DMDEnKF
+0.2
+0.1
+0.0
+100
+200
+300
+400
+500
+TimestepsStreaming TDMD
+70
+WindowedTDMD
+60
+Online DMD
+Hankel-DMDEnKE
+50
+DMDEnKF
+Density
+40
+30
+20
+10
+%.20
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+Distance from True Eigenvalues Argument0.5
+TrueEigenvalue
+Streaming TDMD
+0.4
+Windowed TDMD
+OnlineDMD
+0.3
+Hankel-DMDEnKF
+DMDEnKF
+0.2
+0.1
+0.0
+100
+200
+300
+400
+500
+TimestepsStreaming TDMD
+70
+Windowed TDMD
+60
+Online DMD
+Hankel-DMDEnKF
+50
+DMDEnKF
+Density
+40
+30
+20
+10
+-%.20
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+Distance from True Eigenvalues Argumentplus the newly available data used so far in the filtering step. By including more data in the spin-up
+process, the spin-up DMD model is more likely to successfully capture the signal component in the
+data as a pose to measurement noise, and hence produce eigenvalues with the same structure as
+those of the true system. Time-delay embeddings make the SVD step in the DMD algorithm more
+robust to measurement noise [16]. Hence, while the Hankel-DMDEnKF is similarly restricted by
+the eigenvalues it can produce at the filtering stage, in all 1000 runs of the synthetic data the spin
+up Hankel-DMD was able to identify the system’s eigenvalues to be a complex conjugate pair, so
+this was not an issue for the Hankel-DMDEnKF.
+3.2
+Comparing against DMD with a particle filter
+Having compared the performance of the DMDEnKF against other iterative DMD variants, we
+now focus on evaluating the filtering component of the algorithm. Since the linear DMD model
+acts nonlinearly in the filter when applied to both the model’s state and eigenvalues, we compare
+the EnKF filter with a particle filter. Particle filters [27] have been show to converge to the optimal
+filter as the number of particles tends to infinity for general nonlinear models with non-Gaussian
+noise [17]. However, particle filters are restricted to low dimensional systems only, as the number of
+particles required scales approximately exponentially with the dimension of the state [57]. Hence,
+we compare the DMDEnKF and Hankel-DMDEnKF with a DMD plus particle filter which we will
+take to be the “gold standard” estimation to assess how well the EnKF does with the nonlinear
+filtering problem.
+We use the same synthetic system (3.1) with a linearly increasing eigenvalue argument as in the
+previous subsection to generate data with high levels of measurement noise (σ = 0.5); a trajectory
+of which can be seen in Figure 2. Again, the time-delay embedding dimension d = 50 for the
+Hankel-DMDEnKF, and the first 100 time steps are used to train a spin-up DMD model, with the
+next 400 used to filter the state and spin-up model’s eigenvalues.
+The DMDEnKF’s filter state thus has dimension 4 (2 state dimensions and 2 temporal modes),
+while the Hankel-DMDEnKF’s filter state is of dimension 102 (100 state dimensions and 2 temporal
+modes). To generate a “gold standard” solution, at the filtering step we use a particle filter with
+10,000 particles, applying multinomial importance resampling [20] every time the effective sample
+size falls below half the number of particles to avoid sample degeneracy [21]. For the DMDEnKF
+and Hankel-DMDEnKF at the filtering step, we run the EnKF with varying numbers of ensemble
+members (N), to see if as N increases their estimates mean and covariance will tend to that of
+the particle filter ensemble. We generated 1000 runs of the synthetic data to apply the particle
+filter/EnKF with each value of N to and collected the errors in the eigenvalue argument estimates
+for each method at every time step.
+As can be seen in Figure 4a, the DMD particle filter with 10,000 particles produces an extremely
+tight error distribution that is slightly biased to produce estimates below that of the true eigen-
+value’s argument. This is to be expected, as mentioned in the previous subsection, due to the
+system’s eigenvalue argument constantly increasing. There is also a thin tail in the error distribu-
+tion that extends down to −π/8. This is again a result of the spin up DMD sometimes failing to
+identify a complex conjugate eigenvalue pair, trapping the particle filter in the faulty assumption
+that the eigenvalues are real.
+14
+
+(a) Error distributions.
+(b) Mean squared errors.
+Figure 4: Error distributions (left) and mean squared errors (right) for estimates of the synthetic system’s
+eigenvalue arguments produced by the DMDEnKF and Hankel-DMDEnKF with varying numbers of
+ensemble members (N) against those produced by a particle filter with 10,000 particles. Increasing N
+quickly leads to error levels in the DMDEnKF and Hankel-DMDEnKF that are similar to those produced by
+their respective “gold standards”.
+For low numbers of ensemble members (N = 5), the DMDEnKF and Hankel-DMDEnKF are centred
+at a similar value to the “gold standard”. However, they produce a far larger spread with long tails
+in both directions that imply a lack of robustness with this few ensemble members. With only a
+small increase to N = 10, both methods become more stable, as although they still have a larger
+variance than the particle filter, the long positive tails from N = 5 have been eliminated. A similar
+pattern occurs as we move to N = 20, with more ensemble members resulting in a tighter error
+distribution. At this point, the Hankel-DMDEnKF’s distribution can be distinguished from that
+of the DMDEnKF and DMD particle filter by its aforementioned lack of a persistent thin negative
+tail. By N = 40, the main peaks of the DMDEnKF, Hankel-DMDEnKF and “gold standard” are
+almost indistinguishable on the graphical scale, with the DMDEnKF and DMD particle filter both
+sharing a thin negative tail.
+Figure 4b shows how the mean squared error for the eigenvalue arguments predicted by the DM-
+DEnKF and Hankel-DMDEnKF are affected by varying the number of ensemble members. For
+the DMDEnKF, errors initially sharply decline as N is increased, however on this small synthetic
+system returns diminish quickly after N = 20. By N = 50, we achieve a mean squared error with
+the DMDEnKF only ∼ 3% larger than that of the “gold standard”, despite using 200 times fewer
+particles. When comparing the Hankel-DMDEnKF to the “gold standard”, the errors in the DMD
+particle filter’s eigenvalue estimates are skewed by the runs in which the spin up DMD was unable
+to identify a complex conjugate eigenvalue pair, as Hankel-DMD did not encounter this problem
+on these synthetic examples. To attempt to fairly compare the filtering methods, we remove all
+runs in which the spin up DMD failed in this way, before again calculating the mean squared error
+for the DMD particle filter and recording it in Figure 4b. A similar pattern of reducing errors with
+diminishing returns can be seen for the Hankel-DMDEnKF as ensemble size is increased, and by
+N = 50 its mean squared error is within 5% of the newly calculated DMD particle filter’s score.
+Our results show that in this simple, synthetic case at least, the EnKF is an efficient and effective
+solution to the nonlinear filtering problem that arise within the DMDEnKF framework.
+15
+
+No. of Ensemble Members = 5
+No. of Ensemble Members = 10
+DMDParticleFilter
+DMDParticle Filter
+20
+with10,000particles
+with10,000particles
+15
+Hankel-DMDEnKF
+Hankel-DMDEnKF
+DMDEnKF
+DMDEnKF
+10
+5
+Densit
+No. of Ensemble Members = 20
+No. of Ensemble Members = 40
+DMDParticleFilter
+DMDParticleFilter
+20
+with10,000 particles
+with10,000 particles
+15
+Hankel-DMDEnKF
+Hankel-DMDEnKF
+DMDEnKF
+DMDEnKF
+10
+5
+0
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.20.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Distance from TrueEigenvalues ArgumentHankel-DMDEnKF
+10-1
+DMDEnKF
+Mean Squared Error
+10-2
+DMD Particle Filter
+with 10,000 particles
+10-3
+DMD Particle Filter
+with 10,000 particles
+and failed runs removed
+10
+20
+30
+40
+50
+No. of Ensemble Members3.3
+Tracking a synthetically generated pandemic
+Lastly, we test the DMDEnKF’s performance on synthetic data designed to simulate a simple
+pandemic with a state xk representing the level of infection in 3 different population classes. The
+system’s dynamics are governed by a matrix A ∈ R3×3 that we randomly generate with non-
+negative elements each being drawn from the Uniform distribution U[0, 1). The (i, j)th element of
+A represents how the level of infection in class j at time k will affect the level of infection in class i
+at time k + 1. To control whether the synthetic pandemic is spreading or dying off, we then define
+a new matrix ˆA = A
+λ1 where λ1 is the largest eigenvalue of A, thus ensuring the spectral radius
+ρ(ˆA) = 1. By introducing a constant γ, we can replace A with γ ˆA causing the state to grow if
+γ > 1 or decay for γ < 1. To simulate a pandemic, we linearly decrease γ from 1.01 to 0.99 over the
+course of the experiment’s 1000 time steps. The initial state used is a vector of ones. The system
+that generates the synthetic data can be written as
+xk+1 = γk ˆAxk,
+x1 =
+�
+1
+1
+1
+�T
+,
+(3.5)
+where the state xk ∈ R3, γk = 1.01 − 0.02(k−1)
+999
+and k = (1, ..., 1000). We assume not to have access
+to the true state of the system xk but instead noisy measurements yk defined by
+yk = xk + vk,
+(3.6)
+The constant σ that governs the level of measurement noise is set to σ = 0.05 to represent low
+noise and σ = 0.5 for high noise as in (3.2). Figure 5 shows the values of the system’s three state
+dimensions and the respective available measurements over the course of one run.
+Figure 5:
+Time series for a synthetic system that simulates a pandemic, showing all 3 dimensions of the
+state with no, low (σ = 0.05) and high (σ = 0.5) measurement noise.
+All five DMD variants tested had their computational parameters set to the same values as those
+used in the synthetic experiments in Section 3.1. The only small difference was that Streaming
+TDMD, Windowed TDMD, the DMDEnKF and Hankel-DMDEnKF truncated the data by remov-
+ing the smallest singular value to reduce model instability caused by what was often a very low
+16
+
+16
+Xk True State
+14
+ykforg=0.05
+12
+yk for g= 0.5
+Values
+10
+8
+State
+6
+4
+0
+0
+200
+400
+600
+800
+1000
+Timestepssignal-to-noise ratio in this direction. Online DMD did not apply any truncation to the data as the
+method was not designed to do so, however it did not appear to suffer from any stability issues as
+a consequence.
+The first 100 measurements (y1, ..., y100) were used to initialize the models, and as each new data
+point (y100, ..., y1000) was successively fed into the models, they produced 50 step ahead forecasts
+(ˆx150, ..., ˆx1050).
+We generate 1000 data points, however a standard flu season lasts around 20
+weeks. For this reason, we chose to forecast 50 steps ahead to mimic forecasting 1 week ahead in
+a more realistic timescale. The relative prediction errors ˆek = ∥xk−ˆxk∥
+∥xk∥
+could then be calculated
+for k = (150, ..., 1000) and the mean of these errors was the main metric we used to evaluate the
+forecasting skill of each method over the course of one run.
+A thousand runs were performed
+for both low and high levels of noise and the empirical cumulative distributions of 50 step ahead
+forecast mean run relative errors for low noise (σ = 0.05) can be seen in Figure 6.
+Figure 6:
+Cumulative error distributions of the mean run relative errors for the 50 step ahead forecasts of
+each iterative DMD variant. Mean relative errors were calculated over all time steps for each run of the
+experiment, with the results from 1000 runs under low levels of measurement noise (σ = 0.05) displayed.
+Forecast errors had a wide range for some methods, due to exponentially compounding errors caused by
+forecasting 50 steps ahead. The DMDEnKF, Hankel-DMDEnKF and Online DMD produced errors orders
+of magnitude smaller than those of Streaming TDMD and Windowed TDMD.
+The first noteworthy feature of the cumulative error distributions is the wide range in some method’s
+forecast errors. This is a result of the 50 step ahead forecasts being produced by training each
+model to forecast 1 step ahead, then applying the trained model to the data 50 times. As such,
+forecast errors compound exponentially and small errors over a 1-step forecast horizon can become
+vast after 50 iterations. Inspecting the individual methods, we see Windowed TDMD to be the
+worst performing method. This is due to its aforementioned instability under measurement noise
+caused by considering only a small subset of the data at a time. This instability could be reduced
+by increasing the window size (w) computational parameter, however as w increases the model’s
+ability to track a system that changes with time diminishes. Streaming TDMD had the second-
+largest errors, caused by the method’s assumption of a stationary system hindering its ability to
+17
+
+1.0
+Streaming TDMD
+Windowed TDMD
+Online DMD
+DMDEnKF
+0.8
+Hankel-DMDEnKF
+ Proportion of runs
+0.6
+0.4 
+0.2 
+0.0
+1010
+1029
+1048
+1067
+1086
+10105
+10124
+10143
+50 step ahead forecast mean run relative errorcorrectly adapt to the system’s changing eigenvalues as new data became available. In the majority
+of cases, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly well. All
+three methods exhibited cumulative error distributions tightly concentrated around a low error
+value, however in a few runs, the DMDEnKF became unstable and produced large errors. It is
+clear even at low levels of noise that the forecasting performance of Online DMD, the DMDEnKF
+and Hankel-DMDEnKF are far superior on this type of system to those of Windowed TDMD and
+Streaming TDMD. Hence, we now focus exclusively on these top three performing methods to allow
+for a thorough comparison of them on an appropriate error scale.
+(a) σ = 0.05
+(b) σ = 0.5
+Figure 7: Error distributions of the mean run relative errors for the 50 step ahead forecasts of Online
+DMD, the DMDEnKF and Hankel-DMDEnKF, attained over 1000 runs under low σ = 0.05 (left) and high
+σ = 0.5 (right) levels of measurement noise. Similar errors were found at both noise levels, with Online
+DMD performing better at low measurement noise and the DMDEnKF/Hankel-DMDEnKF performing
+better at high measurement noise.
+In Figure 7 for Online DMD, the DMDEnKF and Hankel-DMDEnKF, we plot the distributions of
+50 step ahead forecast mean run relative errors at both low and high levels of measurement noise.
+At low levels of noise, Online DMD’s errors peak at a lower level than those of the DMDEnKF
+and Hankel-DMDEnKF, however as noise levels increase we see the peaks switch sides, and the
+DMDEnKF/Hankel-DMDEnKF become the better performing methods. At both noise levels, the
+peak in the DMDEnKF’s error distribution is centred at the same value as the Hankel-DMDEnKF’s
+peak, however it is less dense due to the additional probability mass stored in the long tail of the
+DMDEnKF’s error distribution, which is not present in that of the Hankel-DMDEnKF.
+These disproportionately large errors in the DMDEnKF distribution’s tail occur when the spin-up
+DMD process fails to produce a model similar enough to the system’s true dynamics. As briefly
+touched upon in the first synthetic example, if the spin-up DMD model is sufficiently inaccurate
+then it can stop the EnKF from effectively assimilating new data, leading to the catastrophic
+failure of the DMDEnKF. In this example, as the signal-to-noise ratio in the direction of the
+second-largest singular value was often low, an unfortunate random draw of the system dynamics
+(A) and measurement noise (vk) in the spin-up period could produce large errors in DMD’s second
+eigenvalue. Empirically, using the interquartile range method to detect outlying forecast errors,
+this DMDEnKF failure occurred 5.5% of the time for σ = 0.05. The errors would persist in the
+filtering step as new data was processed, whereas other methods were able to recover from poor
+initial model estimates more effectively. The quality of the model produced by the initial DMD is
+dependent on the quality and volume of the spin-up data, hence this problem was exacerbated and
+occurred much more regularly at higher noise levels (21.9% of the time for σ = 0.5). It could be
+18
+
+35
+Online DMD
+Hankel-DMDEnKF
+30
+DMDEnKF
+25
+10
+5
+0
+3 × 10-2
+4 × 10-2
+6 × 10-2
+10-1
+1052
+1056
+5o step ahead forecast mean run relative errorOnline DMD
+16
+Hankel-DMDEnKF
+14
+DMDEnKF
+12
+8
+6
+4
+2
+0
+10-1
+2 × 10-1
+3 × 10-14× 10-1
+6 × 10-1
+1043
+1047
+5o step ahead forecast mean run relative errormitigated somewhat by increasing the number of time steps in the spin-up stage as described at
+the end of Section 3.1, however similarly to Windowed TDMD as the system is assumed to be time
+varying there likely exists a point of negative returns once the spin-up period becomes too long due
+to the stationarity assumption of batch DMD becoming progressively more violated.
+Unlike the DMDEnKF, the Hankel-DMDEnKF and Online DMD do not suffer from a long tail in
+their error distributions, and perform consistently well over all 1000 runs. At both noise levels,
+their error distributions have a similar variance, with the Hankel-DMDEnKF’s errors being slightly
+more tightly grouped than those of Online DMD. Hence, the average error is the main factor when
+differentiating between the method’s performance in this example, meaning Online DMD is the
+preferred method at low noise and the Hankel-DMDEnKF (or DMDEnKF provided the spin-up
+DMD does not catastrophically fail) is more accurate at high noise. As both methods posses useful
+yet differing attributes, we generate a typical data trajectory (one for which the DMDEnKF does
+not fail) for both low and high measurement noise. We then investigate how each model’s 50 step
+ahead forecasts and dominant eigenvalue estimates change over the course of each run, as shown
+in Figure 8.
+(a) 50 step ahead forecasts for σ = 0.05.
+(b) Dominant eigenvalue’s modulus for σ = 0.05.
+(c) 50 step ahead forecasts for σ = 0.5.
+(d) Dominant eigenvalue’s modulus for σ = 0.5.
+Figure 8: Typical trajectories of the 50 step ahead forecasts for the value of the state’s first dimension (left)
+and estimates of the dominant eigenvalue’s current modulus (right) under low (σ = 0.05) and high
+(σ = 0.5) levels of measurement noise for Online DMD, the DMDEnKF and Hankel-DMDEnKF over the
+course of 1 run. Online DMD forecasts 50 steps ahead more accurately at low noise, and the
+DMDEnKF/Hankel-DMDEnKF more accurately at high noises, however when signal-to-noise ratio is low
+(at the start and end of the experiment) Online DMD’s eigenvalue estimates become unstable.
+First, observing the low noise forecasts in Figure 8a, it is clear Online DMD produces forecasts
+that are more robust and closer to the true state’s value than those of the DMDEnKF and Hankel-
+DMDEnKF. This was to be expected, by virtue of Online DMD’s lower average errors in the error
+distributions of Figure 7a at this noise level. As noise is increased, the forecasts in Figure 8c show
+the DMDEnKF and Hankel-DMDEnKF becoming the more accurate methods, however Online
+19
+
+14
+Xk True State
+Online DMD
+12
+Hankel-DMDEnKF
+State in dimension 
+DMDEnKF
+10
+8
+6
+4
+2
+200
+400
+600
+800
+1000
+Timesteps1.010
+True Eigenvalue Mod
+OnlineDMD
+1.005
+Hankel-DMDEnKE
+DMDEnKF
+1.000
+0.995
+0.990
+0.985
+200
+400
+600
+800
+1000
+Timesteps18
+XkTrueState
+16
+OnlineDMD
+14
+Hankel-DMDEnKE
+dimension
+DMDEnKF
+12
+10
+8
+in
+State i
+6
+4
+0
+200
+400
+600
+800
+1000
+TimestepsModulus
+.02
+1.01
+1.00
+PT
+0.99
+TrueEigenvalueMod
+0.98
+OnlineDMD
+Hankel-DMDEnKF
+0.97
+DMDEnKF
+200
+400
+600
+800
+1000
+TimestepsDMD’s forecasts remains fairly stable, and still appear to be a viable forecasting option.
+Analysing the eigenvalue estimates in Figures 8b and 8d, we see that over the middle section of data
+where k = (250, ..., 750), Online DMD is able to track the dominant eigenvalue effectively. However,
+at the beginning and end of the dataset when states and hence the signal component of each new
+data point is small relative to the measurement noise, Online DMD’s eigenvalue estimates become
+progressively more unstable. In the low noise case this is not a problem, as Online DMD’s estimates
+are significantly more accurate than those of the DMDEnKF/Hankel-DMDEnKF, so even in the
+poorly performing sections of the data it’s estimates still better/match those of the DMDEnKF.
+For higher noise however, Online DMD provides significantly less robust estimates of the dominant
+eigenvalue at the start and end of the datasets than those generated by the DMDEnKF and Hankel-
+DMDEnKF. In the epidemiological context of an infectious disease outbreak, which this synthetic
+example attempts to mimic, scientists will often try to calculate the basic reproduction number
+(R0) [58] using noisy data from the small number of initial infections. If R0 > 1 the number of
+infections will grow exponentially if left unchecked, and if R0 < 1 the number of infections will
+decay naturally to 0.
+Within this example, using the DMDEnKF/Hankel-DMDEnKF one can
+quickly determine that initially R0 > 1 and take any required action thanks to the stability of it’s
+early eigenvalue estimates, whereas it takes significantly longer and a higher level of infection for
+Online DMD to consistently determine if R0 is above or below the growth/decay threshold.
+4
+Seasonal Influenza-like Illness Application
+4.1
+Problem setup
+DMD based methods have previously been applied to infectious disease data [49]. In this case,
+DMD modes can be viewed as stationary, spatial modes used to create a reduced order model in
+which only the amplitudes and frequencies are time varying [9]. Hence, modelling influenza-like
+illness (ILI) data is a prime potential application for the DMDEnKF/Hankel-DMDEnKF.
+The CDC’s ILINet data [25] we will be using records the number of ILI General Practitioner (GP)
+consultations in the US each week, alongside the number of total GP consultations which can be
+used to normalize the ILI data. We use a subset of the data from the start of 2003, the first year
+when data is available all year round, to the end of 2018 as seen in Figure 1. We then split each
+week’s data into demographics, consisting of 4 age groups (0-4, 5-24, 25-24, 65+) and 10 Health
+and Human Services (HHS) regions. Each region consists of the following locations:
+• Region 1 - Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont.
+• Region 2 - New Jersey, New York, Puerto Rico, and the U.S. Virgin Islands.
+• Region 3 - Delaware, District of Columbia, Maryland, Pennsylvania, Virginia, and West
+Virginia.
+• Region 4 - Alabama, Florida, Georgia, Kentucky, Mississippi, North Carolina, South Carolina,
+and Tennessee.
+• Region 5 - Illinois, Indiana, Michigan, Minnesota, Ohio, and Wisconsin.
+20
+
+• Region 6 - Arkansas, Louisiana, New Mexico, Oklahoma, and Texas.
+• Region 7 - Iowa, Kansas, Missouri, and Nebraska.
+• Region 8 - Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming.
+• Region 9 - Arizona, California, Hawaii, and Nevada.
+• Region 10 - Alaska, Idaho, Oregon, and Washington.
+Whilst ILI consultation data is available over all 40 of these strata, total GP consultation data
+is only provided by region. To generate an age breakdown for a region’s total consultations we
+linearly interpolate using census data to approximate the US population’s age demographics for
+a given week. We then allocate the region’s total consultations to each age group based on the
+proportion of the total population they represent. This method assumes that all age groups have
+a similar likelihood of attending the GP’s, which may be flawed but we believe it to be sufficient
+for the purpose of demonstrating the DMDEnKF on real-world data.
+4.2
+Building the spin-up DMD model
+The format of the ILI data used in the DMDEnKF is thus a 40 dimensional vector for each week,
+recording ILI consultations as a percentage of total GP consultations over every demographic. This
+data exists in R40
++ however DMD computes modes in R40. Hence, to ensure the model’s estimates
+are non-negative, we first transform the data by adding a small constant (c = 1) then taking
+the natural logarithm of each element. For the Hankel-DMDEnKF, this transformed data is then
+delay-embedded with the previous 99 time steps (d = 100) to form a state in R4000. We use data
+up to the end of 2012 as training data for the spin-up DMD processes of the DMDEnKF/Hankel-
+DMDEnKF detailed in Step 1 of the DMDEnKF algorithm, and then filter the remaining years
+from 2013-2018. The transformed, centred data with the split where the spin-up process ends and
+the filtering begins marked is shown in Figure 9.
+We initially choose to truncate to 8 DMD modes for the DMDEnKF to demonstrate the method.
+We discuss the effect of changing the truncation on the DMDEnKF method below, but at 8 DMD
+modes approximately the amount of additional variance in the data that is retained by keeping more
+modes diminishes significantly. This is evidenced by the “elbow” seen in the cumulative variance
+plot of Figure 14a, at the point where the graph transitions from rapidly increasing in variance
+with r to a more gradual ascent. We also truncate the Hankel-DMDEnKF to 8 DMD modes, to
+allow for a more direct comparison between the two variants.
+The spectrum and dominant DMD/Hankel-DMD modes associated with each frequency identified
+by the spin-up DMD processes can be seen in Figure 10. All eigenvalues shown in Figures 10a
+and 10c had a modulus of ∼ 1, meaning in both cases each mode was expected to persist in the
+data without growing exponentially. The major difference between the two methods spectra is that
+Hankel-DMD identifies the most dominant mode to have a period of one year, whereas DMD does
+not detect any modes with this period. Annual peaks in ILI consultations occurring at a relatively
+similar time each year indicates that the data contains a strong mode of period one year, and this
+is supported by Fourier analysis [10] which also identifies one year as the dominant period. Hence,
+DMD is missing the yearly mode present in the data which Hankel-DMD is able to detect, and
+21
+
+Figure 9: ILI consultations as a percentage of total weekly GP consultations across 4 age brackets and the
+10 HHS regions in the US, log transformed and centred. The peaks of varying size, timing and shape in
+Figure 1 are visible here as vertical red areas of varying width and intensity that encompass most
+demographics.
+this is likely due to Hankel-DMD’s aforementioned enhanced robustness to measurement noise.
+There are two clear patterns in the structure of the dominant DMD and Hankel-DMD modes seen
+in Figures 10b and 10d. Firstly, their strata generally move together. This is shown by the vast
+majority of entries for each DMD mode, and entries within the same delay-embedded week (denoted
+by a vertical slice through the mode) for Hankel-DMD modes, sharing the same sign. This implies
+that the percentage of ILI consultations increases and decreases at a similar time across all ages
+and regions. Secondly, the variance is higher for the younger age groups. This is demonstrated
+by the absolute value of elements in the top two rows of each region generally being larger than
+those in the bottom two. From Figure 10b, this is visible trivially for the DMD modes. In Figure
+10d, age groups are arranged in ascending order for each region, so this effect is evidenced in the
+Hankel-DMD modes by the presence of more intensely coloured horizontal lines of width 2, followed
+by less intensely coloured lines of width 2 repeating over each of the 10 regions. This indicates that
+there are sharper peaks and deeper troughs in the percentage of ILI consultations for the young,
+while the rates for those 25 and over remain more stable.
+4.3
+Applying the filter
+The filtering steps of the DMDEnKF/Hankel-DMDEnKF are then applied over the remaining data
+using the spatial and temporal modes from the spin-up DMD and spin-up Hankel-DMD respectively.
+Producing a 4-week ahead ILI forecast for the ILINet data that consistently outperforms a simple
+historical baseline prediction is difficult even for state-of-the-art models [50]. As such, to test the
+DMDEnKF/Hankel-DMDEnKF we use a forecast horizon of 4 weeks when making predictions.
+In Figure 11, the DMDEnKF and Hankel-DMDEnKF’s forecasting of total ILI consultations as
+22
+
+0-4
+2
+consultations
+2
+3
+65 +
+centred % ILI 
+Demographic
+25.64
+0-4
+65+
+Re
+25.64
+0
+65 ±
+Log transformed,
+Region
+2
+0-4
+6
+25.64
+-2
+65 +
+2003
+2005
+2007
+2009
+2011
+Split
+2015
+2017
+2019
+201
+Date(a) DMD Eigenvalue Spectrum.
+(b) Dominant DMD Modes.
+(c) Hankel-DMD Eigenvalue Spectrum.
+(d) Dominant Hankel-DMD Modes.
+Figure 10: Eigenvalue Spectrum (left) and DMD modes in descending order of dominance (right) generated
+by the DMD (top)/Hankel-DMD (bottom) applied to the data in Figure 9 up to the spin-up date. In both
+cases, all eigenvalues lie approximately on the unit circle, and dominant modes feature the same sign for
+most demographics with a magnitude that varies with age. The DMD modes are more interpretable, but
+Hankel-DMD identifies the mode with period 1 year, which DMD does not.
+23
+
+1
+0.5
+Imaginary
+0
+-0.5
+-1 
+-1
+-0.5
+0
+0.5
+1
+RealMode l:
+Eigenvalue Period = 1.9 Years
+0-4
+5-24
+25-64
+65 +
+Mode 2:
+EigenvaluePeriod=0.6Years
+0-4
+0.3
+5-24
+25-64
+0.2
+65 +
+es
+Mode 3:
+Eigenvalue Period = 4.4 Years
+0.1
+0-4
+5-24
+0.0
+25-64
+-0.1
+65 +
+Mode 4:
+Eigenvalue Period = 22.5'Years
+-0.2
+0-4
+5-24
+25-64
+65 +
+1
+2
+3
+4
+5
+6
+8
+9
+ion
+Region
+Region
+Region
+ion
+ion
+Region 
+Region
+Region
+Regi
+Regi
+Regi
+Regions1
+0.5
+Imaginary
+0
+-0.5
+-1 
+-1
+-0.5
+0.5
+1
+RealMode l:
+Eigenvalue Period = l.0 Years
+21
+28
+35
+Mode 2:
+Eigenvalue Period = 13.5 Years
+07
+0.04
+14
+Age/Regions
+21
+28
+0.02
+35
+Mode 3:
+Eigenvalue Period = 1.6 Years
+07
+0.00
+14
+-0.02
+21
+28
+35
+-0.04
+Mode 4:
+Eigenvalue Period = 0.6 Years
+07:
+14
+21
+28
+35
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Weeks Delay-Embededa percentage of total weekly GP consultations can be seen in full. Up until 2012, we generate
+4-week ahead reconstructions using the spin-up DMD models only, estimating each state by taking
+the state measurement from 4 weeks prior, then iteratively applying the model to it 4 times. The
+4-week ahead DMD reconstruction in Figure 11a captures more fluctuations in the data than that
+of Hankel-DMD, however these high frequency fluctuations can also indicate the effect of noise in
+the measurements. The Hankel-DMD reconstruction shown in Figure 11b is much less sensitive
+to noise, although fails to identify the sharper peaks in the data, which suggest it may be over-
+smoothing. From 2012 onwards the filtering begins, and forecasts are generated as described in the
+DMDEnKF algorithm using equation (2.33). The DMDEnKF forecasts become significantly more
+stable, while the Hankel-DMDEnKF forecasts improve in capturing the true shape of the data,
+however both suffered from some degree of lag in their predictions.
+During this second section of the data, the models are producing actual forecasts, as the DM-
+DEnKFs only have access to data up to 4 weeks prior to the prediction target’s date. Hence, it
+is in this section of the data we compare the models’ performance against that of the historical
+baseline. The historical baseline prediction was created in a similar manner to that used in [8],
+taking ILI consultation rates from the same week of every previous year in the data (excluding the
+pandemic year of 2009) and then producing a probability distribution for the current week’s con-
+sultations via Gaussian kernel density estimation (KDE) [55]. KDE Bandwidths were determined
+using Silverman’s rule of thumb [56], and when point estimates were required they were taken as
+the median of the distribution. The results of the comparisons can be seen in Figure 12 and Table
+2. Here it is worth noting that although we use data dated 4 weeks prior to the prediction date,
+in reality this data is often subject to revisions so the ILINet data as it currently stands would not
+necessarily be available in real time [50].
+4.4
+Evaluating the DMDEnKF’s performance
+Figure 12 demonstrates graphically the 4-step ahead DMDEnKF, Hankel-DMDEnKF and historical
+baseline forecasts. The DMDEnKFs more successfully capture the shape and height of each flu sea-
+son’s peak, however tend to predict the peaks late, whilst the historical baseline consistently under-
+predicts the peak rates but is fairly accurate on the timings. The Hankel-DMDEnKF’s forecasts are
+smoother than those of the DMDEnKF, however do not capture smaller details within the shape of
+the peaks. We also plot the 95% confidence intervals for the DMDEnKF and Hankel-DMDEnKF’s
+forecasts in Figure 12, generated using the ensemble that is maintained and propagated in the
+EnKF framework. At all times, the real data lies within the DMDEnKF’s confidence interval,
+which is not true for the Hankel-DMDEnKF. The DMDEnKF’s confidence interval is significantly
+wider than that of the Hankel-DMDEnKF, and this is due to Hankel-DMD’s robustness to noise,
+meaning that when the ensemble is propagated through the model, a large amount of probability
+mass is concentrated in a small area of the state space. This then leads to the Hankel-DMDEnKF
+underestimating the uncertainty in the system, and hence some real data values falling outside the
+boundaries of it’s 95% confidence interval.
+To numerically compare performance, we used metrics designed for the Forecast the Influenza
+Season Collaborative Challenge (FluSight), in which multiple teams would submit predictions about
+the weekly ILINet consultation rates for the upcoming flu season at a national and HHS regional
+level [7], [8]. The FluSight challenge evaluated models abilities to generate 1-4-week ahead forecasts
+24
+
+(a) DMDEnKF 4-week ahead forecast.
+(b) Hankel-DMDEnKF 4-week ahead forecast.
+Figure 11: ILI consultations as a percentage of total weekly GP consultations forecast 4 weeks ahead using
+the DMDEnKF (top) and Hankel-DMDEnKF (bottom). The DMD reconstruction captures the shape of the
+data well but is unstable, whereas the Hankel-DMD reconstruction is less sensitive to noise but
+over-smooths. The DMDEnKF and Hankel-DMDEnKF forecasts help reduce these issues present in their
+respective reconstructions, but both suffer from some degree of lag in their predictions.
+25
+
+10
+Spin-up DMD
+Real Data
+DMDEnKE
+8
+Iconsultations
+4
+%
+2
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+Date10
+Real Data
+Spin-up Hankel-DMD
+Hankel-DMDEnKF
+8
+Iconsultations
+三
+%
+2
+0
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+DateFigure 12: ILI consultations as a percentage of total weekly GP consultations, forecast 4 weeks ahead using
+the DMDEnKF (top) and Hankel-DMDEnKF (bottom). A 95% confidence interval for each forecast, and
+historical baseline predictions are also shown. The Hankel-DMDEnKF forecasts are smoother than those of
+the DMDEnKF, but both forecasts contain some lag. The real data always lies within the DMDEnKF’s
+confidence interval but not the Hankel-DMDEnKF’s, however this is likely due to the DMDEnKF’s
+confidence interval being significantly wider than that of the Hankel-DMDEnKF.
+26
+
+18
+Real Data
+16
+Historical Baseline
+DMDEnKF
+95% CI
+14
+consultations
+12
+10
+8
+%
+6
+4
+2
+0
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+Date18
+Real Data
+Historical Baseline
+16
+Hankel-DMDEnKF
+95% CI
+14
+consultations
+12
+10
+8
+三
+%
+6
+4
+2
+0
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+DateForecast
+Log Score
+Mean Squared Error
+Historical Baseline
+0.28
+1.24
+1-week ahead DMDEnKF
+0.49
+0.33
+2-week ahead DMDEnKF
+0.38
+0.61
+3-week ahead DMDEnKF
+0.32
+0.87
+4-week ahead DMDEnKF
+0.27
+1.16
+1-week ahead Hankel-DMDEnKF
+0.41
+0.49
+2-week ahead Hankel-DMDEnKF
+0.33
+0.70
+3-week ahead Hankel-DMDEnKF
+0.29
+0.97
+4-week ahead Hankel-DMDEnKF
+0.23
+1.26
+Table 2: The log scores and mean squared errors for the DMDEnKF and Hankel-DMDEnKF with differing
+forecast horizons, and the historical baseline prediction. The DMDEnKF achieves a higher log score and
+mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where it attains a
+similar level of forecast skill. The Hankel-DMDEnKF consistently underperforms against the DMDEnKF
+in both metrics over these short forecast horizons. Scores are calculated over the 6 flu seasons from
+2012/13 to 2017/18.
+known as short-term targets over the course of a flu season, as well as other longer term targets,
+known as seasonal targets, before the season had begun. The DMDEnKF is primarily intended to
+be a tool for tracking and short-term forecasting, hence we focus on forecasting these short-term
+targets only. For this purpose we used two different metrics, the log probability measure (log score)
+slightly adjusted from the FluSight challenge as used in [50] and the mean squared error due to its
+popular use in regression problems. The log score represents the geometric average probability of
+each model’s prediction being accurate, with accuracy deemed as a forecast within +/ − 0.5 of the
+true ILI consultation rate. The higher the log score, the better the forecast. Metrics are calculated
+from week 40 to week 20 of the following year to prioritize evaluation of forecasts during the flu
+season, and we use the 6 full seasons from 2012/13 to 2017/18.
+The results for the historical baseline prediction and DMDEnKF/Hankel-DMDEnKF’s forecasts at
+a national level can be seen in Table 2. As one would expect, the accuracy of both DMDEnKFs
+degrade as they make predictions further into the future. The DMDEnKF achieves a higher log score
+and mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where
+it attains a similar level of forecast skill. For forecasts of 5 or more weeks ahead, the DMDEnKF is
+unable to outperform the historical baseline in either metric. The Hankel-DMDEnKF consistently
+underperforms against the DMDEnKF in both metrics over these short forecast horizons. The top
+3 statistical models and top 3 mechanistic models in the FluSight challenge achieved log scores of
+0.32 and 0.3 respectively for their 4-week ahead forecasts, hence the DMDEnKF has lower (but
+comparable) forecasting skill than current state of the art ILI models. As the forecast horizon is
+extended up to 12 weeks ahead, the DMDEnKF’s forecast scores continue to decrease monotonically,
+whereas the Hankel-DMDEnKF’s log scores for 9-12 weeks ahead are no worse than those for 5-8
+weeks ahead. As such, the DMDEnKF is preferred for short-term forecasting, while the Hankel-
+DMDEnKF is considered superior when forecasting over longer timescales.
+Figure 13 shows the log scores for the 4-week ahead DMDEnKF forecast, and how these compare to
+27
+
+(a) DMDEnKF 4-week ahead forecast.
+(b) DMDEnKF 4-week ahead forecast - historical
+baseline.
+Figure 13: Log scores over all ages and regions for the DMDEnKF’s 4-week ahead forecast (left), followed
+by those same scores with the log scores of the historical baseline prediction subtracted (right). The
+Hankel-DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so we do not include its
+breakdown to avoid redundancy. In the top figure, the generally increasing intensity of red as one moves
+down the age groups shows the DMDEnKF performing more accurately for older age groups. The bottom
+figure’s varying areas of red/blue shows the DMDEnKF/historical baseline vary in superiority of forecasting
+skill depending on the age and region being forecast, with the historical baseline scoring more highly for
+most regions.
+the scores attained by the historical baseline prediction at an age and regional level. The Hankel-
+DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so its breakdown is
+rather similar to that of the DMDEnKF, and we do not include it to avoid redundancy. In the
+DMDEnKF’s log scores, we see a major pattern in the older age groups scoring higher and hence
+being better predicted than the younger demographics. This pattern does not persist when the
+historical baseline’s scores are removed, indicating it is a more general trait of the data as opposed
+to a specific quality of the DMDEnKF’s modelling technique. There is also a significant difference in
+the predictability from region to region. For example, region 1 was the most predictable region for
+both the DMDEnKF and historical baseline, which is consistent with the findings in [50]. However,
+the DMDEnKF improved on the historical baseline’s forecast for only two of the four age groups
+in this region. In [50] it was found that the most overall improvement gained by forecasting for a
+region using a model as opposed to the historical baseline prediction also occurred in region 1, so
+one would expect to see improvements by the DMDEnKF over the historical baseline in all four age
+groups. As log score is heavily influenced by the amount of uncertainty in a forecast, it is possible
+that the covariance matrices used in the DMDEnKF were too large for this region. Hence, setting
+the magnitude of the DMDEnKF’s variance on a region by region basis could lead to better results
+and more accurate forecasts. Region 6 was the worst forecast region by the DMDEnKF, and the
+historical baseline predictions were slightly more accurate. Again, this is consistent with [50] where
+region 6 was the lowest scoring region for the models. In that work however, region 6 experienced
+the second most improvement by using a model over the historical baseline prediction. Hence, for
+this region a larger variance within the DMDEnKF may have been more appropriate to account
+for its extra unpredictability, further supporting the idea of varying the variance by region in the
+future.
+28
+
+0-4
+0.40
+0.35
+5-24
+0.30
+score
+Ages
+0.25
+s
+25-64
+0.20
+Log
+0.15
+65 +
+0.10
+0.05
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Region
+Region
+Region
+Region
+egion
+egion
+Region
+Region
+Region
+Region
+Re
+e
+R
+R
+Regions-0.05
+0-4
+0.00
+5-24
+-0.05
+core
+Ages 
+-0.10S
+25-64
+Log
+-0.15
+65 +
+-0.20
+Region 1
+Region 2.
+3
+Region 4
+5
+Region 6.
+Region 7.
+8
+9
+Region 10
+Region 3
+Region 
+Region 8
+Region 9
+Regions4.5
+Varying the truncation rank
+Having analysed the DMDEnKF and Hankel-DMDEnKF’s ILI forecasting with 8 DMD modes,
+we now investigate the effect different truncation ranks (r) have on their performance in Figure
+14. From Figure 14a, the subjective process of identifying an “elbow” in the data could lead an
+observer to determine a suitable rank for truncation as low as 4 or as high as 12. Application of
+the algorithm of Gavish and Donoho for identifying the optimal truncation threshold [26] also finds
+the truncation rank to be 12, hence we will focus on investigating values of r in the interval from
+4 to 12.
+(a) % of total variance in the data.
+(b) DMDEnKF log score/mean squared errors for
+r = 4, ..., 12.
+(c) % of total variance in the delay-embedded data.
+(d) Hankel-DMDEnKF log score/mean squared errors
+for r = 4, ..., 12.
+Figure 14: On the left, the % of the total variance in the data (top) and delay-embedded data (bottom),
+dependent on the number of singular values that are retained (r). An “elbow” in the data occurs around
+r = 8 where we choose to truncate, however determining the exact position of the “elbow” is subjective and
+could be considered anywhere from r = 4 to r = 12. On the right, the log score and mean squared errors for
+4-step ahead forecasts generated using the DMDEnKF (top) and Hankel-DMDEnKF (bottom) with differing
+values of r. In both cases, log score is maximised and mean squared error minimised for r = 8.
+Figures 14b and 14d show how the metrics we use to measure the DMDEnKF and Hankel-
+DMDEnKF’s forecasting skill vary with r. An ideal forecast will have a high log score reflecting
+a relatively tight and accurate probability distribution, with a low mean squared error indicating
+a point estimate close to the true percentage of ILI consultations. For both methods, log score
+is maximised and mean squared error minimised by r = 8, indicating this is the optimal rank to
+truncate at for our models. For r = 4, we have the simplest model tested, hence it has a low degree
+of uncertainty resulting in a relatively high log score, however is too simple to properly model the
+system so receives a high mean squared error. By increasing r, we allow the DMDEnKFs more
+freedom to capture complexity within the system, resulting in a more accurate representation of
+the true dynamics and hence a generally lower mean squared error. When the number of eigen-
+29
+
+lained
+100%
+95%
+ex
+90%
+iance 
+85%
+vari
+80%
+75%
+ total
+70%
+Rank cut off
+of
+65%
+at r=8.
+%
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Singular values retained (r)1.6
+.7
+rror
+1.5
+.5
+1.4
+Mean Squared
+1.3
+.4
+1.2
+.6
+1.1
+.10
+.12
+.11
+1.0
+.9
+.8
+0.9
+0.16
+0.18
+0.20
+0.22
+0.24
+0.26
+0.28
+Log Score<plained
+100%
+95%
+ex
+90%
+iance 
+85%
+vari
+80%
+75%
+ total
+70%
+Rank cut off
+of
+65%
+at r=8.
+%
+0
+50
+100
+150
+200
+250
+300
+350
+Singular values retained (r)1.7
+Error
+1.6
+ Squared
+1.5
+1.4
+4
+.10
+5
+Mean
+.12
+1.3
+11
+1.2
+.8
+1.1
+0.19
+0.20
+0.21
+0.22
+0.23
+0.24
+Log Scorevalues is increased too far however, it begins modelling elements of the noise in the system which
+negatively impacts future predictions, as seen in the increase in mean squared errors for r > 8.
+The additional freedom afforded to the DMDEnKF by increasing r also means the model contains
+more parameters, each of which have an associated degree of uncertainty. This causes the overall
+forecast’s probability distribution to become more spread out, and when no longer offset by the
+increased model accuracy up to r = 8, reduces the forecasts log score.
+5
+Conclusion
+To conclude, we have defined two new algorithms, the DMDEnKF and Hankel-DMDEnKF, that
+combine dynamic mode decomposition and Hankel dynamic mode decomposition respectively with
+ensemble Kalman filtering, to update state and temporal mode estimates of a dynamical system
+as new data becomes available. When applied to simple, synthetic systems with a time varying
+parameter and low measurement noise, the DMDEnKFs performed similarly to other iterative DMD
+variants tested in tracking the system’s time varying parameter and forecasting future states. As
+measurement noise was increased, the DMDEnKFs outperformed the other methods tested in both
+metrics, and the Hankel-DMDEnKF produced more stable forecasts than those of the DMDEnKF.
+Both DMDEnKFs achieved similar performance levels to their equivalent DMD Particle Filters
+(an alteration to the DMDEnKF algorithms where the ensemble Kalman filters were switched for
+Particle Filters), while requiring significantly fewer ensemble members. When forecasting influenza-
+like illness across age groups and HHS regions in the US using data from the CDC, the DMDEnKF
+produced more accurate forecasts than a historical baseline prediction up to 3 weeks ahead, and
+forecasts approximately as accurate 4 weeks ahead. The Hankel-DMDEnKF produced less accurate
+forecasts for these short-term targets than the DMDEnKF, however in general it’s forecasts were
+more stable. Also, the Hankel-DMDEnKF was able to identify the presence of a mode with period 1
+year, which is strongly visible in the data, yet not identified by the DMDEnKF. Both DMDEnKFs
+exhibited lower forecasting skill than current state of the art influenza-like illness models.
+A natural extension of the DMDEnKF would be to apply extended/kernel DMD in the spin-up
+DMD phase, allowing the algorithm to be used more effectively on dynamical systems that act
+nonlinearly in their measured states. Instead of taking the observed values alone as the system’s
+state xk, these variants use for the state a collection of functions on the observables g(xk), which
+often increases the state’s dimension n. The EnKF is well suited to this pairing, as it scales more
+computationally efficiently in the state dimension than other Kalman filtering methods [23]. The
+best choice of the collection of functions g(xk) as an embedding for nonlinear systems so that DMD
+may be effectively utilized is an interesting area of future work. Many methods have been developed
+that propose ways of generating g(xk), for example using deep learning [41] or reservoir computing
+[28], and this remains a promising avenue for future work.
+Code availability
+Codes used to produce the results in this paper are available at https://github.com/falconical/DMDEnKF.
+30
+
+Data availability statement
+All data used to produce the results in this paper will be made available upon reasonable request.
+Acknowledgements
+This work was supported by the UKRI, whose Doctoral Training Partnership Studentship helped
+fund Stephen Falconers PhD. He would also like to thank for their valuable discussions, Nadia
+Smith and Spencer Thomas from the National Physics Laboratory.
+References
+[1] H. Abdi, The eigen-decomposition: Eigenvalues and eigenvectors, Encyclopedia of measure-
+ment and statistics, (2007).
+[2] H. Arbabi and I. Mezi´c, Ergodic theory, dynamic mode decomposition, and computation of
+spectral properties of the koopman operator, SIAM Journal on Applied Dynamical Systems, 16
+(2017), pp. 2096–2126.
+[3] F. Auger, M. Hilairet, J. Guerrero, E. Monmasson, T. Orlowska-Kowalska,
+and S. Katsura, Industrial applications of the kalman filter: A review, IEEE Transactions
+on Industrial Electronics, 60 (2013), p. 5458.
+[4] R. Baker, J.-M. Pe˜na, J. Jayamohan, and A. J´erusalem, Mechanistic models versus
+machine learning, a fight worth fighting for the biological community?, Biology Letters, 14
+(2018), p. 20170660.
+[5] D. Balcan, V. Colizza, B. Gonc¸alves, H. Hu, J. J. Ramasco, and A. Vespignani,
+Multiscale mobility networks and the spatial spreading of infectious diseases, Proceedings of
+the National Academy of Sciences, 106 (2009), pp. 21484–21489.
+[6] J. C. A. Barata and M. S. Hussein, The moore–penrose pseudoinverse: A tutorial review
+of the theory, Brazilian Journal of Physics, 42 (2011), p. 146–165.
+[7] M. Biggerstaff, D. Alper, M. Dredze, S. Fox, I. C.-H. Fung, K. S. Hickmann,
+B. Lewis, R. Rosenfeld, J. Shaman, M.-H. Tsou, et al., Results from the centers
+for disease control and prevention’s predict the 2013–2014 influenza season challenge, BMC
+infectious diseases, 16 (2016), pp. 1–10.
+[8] M. Biggerstaff, M. Johansson, D. Alper, L. C. Brooks, P. Chakraborty, D. C.
+Farrow, S. Hyun, S. Kandula, C. McGowan, N. Ramakrishnan, R. Rosenfeld,
+J. Shaman, R. Tibshirani, R. J. Tibshirani, A. Vespignani, W. Yang, Q. Zhang,
+and C. Reed, Results from the second year of a collaborative effort to forecast influenza
+seasons in the united states, Epidemics, 24 (2018), pp. 26–33.
+[9] D. A. Bistrian, G. Dimitriu, and I. M. Navon, Processing epidemiological data using
+dynamic mode decomposition method, AIP Conference Proceedings, 2164 (2019), p. 080002.
+31
+
+[10] R. N. Bracewell and R. N. Bracewell, The Fourier transform and its applications,
+vol. 31999, McGraw-Hill New York, 1986.
+[11] L. Brooks, D. Farrow, S. Hyun, R. Tibshirani, and R. Rosenfeld, Flexible modeling
+of epidemics with an empirical bayes framework, PLOS Computational Biology, 11 (2014).
+[12] B. W. Brunton, L. A. Johnson, J. G. Ojemann, and J. N. Kutz, Extracting spa-
+tial–temporal coherent patterns in large-scale neural recordings using dynamic mode decompo-
+sition, Journal of Neuroscience Methods, 258 (2016), p. 1–15.
+[13] K. Chen, J. Tu, and C. Rowley, Variants of dynamic mode decomposition: Boundary
+condition, koopman, and fourier analyses, Journal of Nonlinear Science, 22 (2012).
+[14] J.-P. Chretien, D. George, J. Shaman, R. Chitale, and F. McKenzie, Influenza
+forecasting in human populations: A scoping review, PloS one, 9 (2014), p. e94130.
+[15] S. T. M. Dawson, M. S. Hemati, M. O. Williams, and C. W. Rowley, Characterizing
+and correcting for the effect of sensor noise in the dynamic mode decomposition, Experiments
+in Fluids, 57 (2016).
+[16] A. de Cheveign´e and J. Z. Simon, Denoising based on time-shift pca, Journal of Neuro-
+science Methods, 165 (2007), pp. 297–305.
+[17] P. Del Moral, Nonlinear filtering:
+Interacting particle resolution, Comptes Rendus de
+l’Acad´emie des Sciences - Series I - Mathematics, 325 (1997), pp. 653–658.
+[18] M. D’Elia, L. Mirabella, T. Passerini, M. Perego, M. Piccinelli, C. Vergara, and
+A. Veneziani, Applications of variational data assimilation in computational hemodynamics,
+Modeling, Simulation and Applications, 5 (2012).
+[19] N. Demo, M. Tezzele, and G. Rozza, PyDMD: Python Dynamic Mode Decomposition,
+The Journal of Open Source Software, 3 (2018), p. 530.
+[20] R. Douc and O. Capp´e, Comparison of resampling schemes for particle filtering, in ISPA
+2005. Proceedings of the 4th International Symposium on Image and Signal Processing and
+Analysis, 2005., IEEE, 2005, pp. 64–69.
+[21] A. Doucet and A. Johansen, A tutorial on particle filtering and smoothing: Fifteen years
+later, Handbook of Nonlinear Filtering, 12 (2009).
+[22] C. Eckart and G. Young, The approximation of one matrix by another of lower rank,
+Psychometrika, 1 (1936), pp. 211–218.
+[23] G. Evensen, The ensemble kalman filter: Theoretical formulation and practical implementa-
+tion, Ocean dynamics, 53 (2003), pp. 343–367.
+[24] S. I. Flu, Estimated influenza illnesses, medical visits, hospitalizations, and deaths averted by
+vaccination in the united states, Prevent, (2007), pp. 2006–2007.
+[25] C. for Disease Control and Prevention, Cdc u.s. influenza surveillance system: Pur-
+pose and methods. https://www.cdc.gov/flu/weekly/overview.htm, 2021. Accessed: 2021-
+04-27.
+32
+
+[26] M. Gavish and D. L. Donoho, The optimal hard threshold for singular values is 4/sqrt(3),
+2014.
+[27] N. Gordon, D. Salmond, and A. F. M. Smith, Novel approach to nonlinear/non-gaussian
+bayesian state estimation, IEE Proceedings F (Radar and Signal Processing), 140 (1993),
+pp. 107–113(6).
+[28] M. Gulina and A. Mauroy, Two methods to approximate the koopman operator with a reser-
+voir computer, Chaos: An Interdisciplinary Journal of Nonlinear Science, 31 (2021), p. 023116.
+[29] C. W. R. Hao Zhang, Online dmd github repository. https://github.com/haozhg/odmd,
+2020. Accessed: 2021-01-19.
+[30] M. Hemati, E. Deem, M. Williams, C. W. Rowley, and L. N. Cattafesta, Improving
+separation control with noise-robust variants of dynamic mode decomposition, in 54th AIAA
+Aerospace Sciences Meeting, 2016, p. 1103.
+[31] M. S. Hemati, C. W. Rowley, E. A. Deem, and L. N. Cattafesta, De-biasing the dy-
+namic mode decomposition for applied koopman spectral analysis of noisy datasets, Theoretical
+and Computational Fluid Dynamics, 31 (2017), p. 349–368.
+[32] M. S. Hemati, M. O. Williams, and C. W. Rowley, Dynamic mode decomposition for
+large and streaming datasets, Physics of Fluids, 26 (2014), p. 111701.
+[33] R. Isermann and M. M¨unchhof, State and Parameter Estimation by Kalman Filtering,
+Springer Berlin Heidelberg, Berlin, Heidelberg, 2011, pp. 539–551.
+[34] T. Jonathan H., R. Clarence W., L. Dirk M., B. Steven L., and K. J. Nathan, On
+dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics,
+1 (2014), p. 391.
+[35] M. R. Jovanovi´c, P. J. Schmid, and J. W. Nichols, Sparsity-promoting dynamic mode
+decomposition, Physics of Fluids, 26 (2014), p. 024103.
+[36] R. E. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of
+Basic Engineering, 82 (1960), pp. 35–45.
+[37] S. Kandula, T. Yamana, S. Pei, W. Yang, H. Morita, and J. Shaman, Evaluation of
+mechanistic and statistical methods in forecasting influenza-like illness, Journal of The Royal
+Society Interface, 15 (2018), p. 20180174.
+[38] K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, Data assimilation: A mathematical
+introduction, 2015.
+[39] J. Lessler and D. Cummings, Mechanistic models of infectious disease and their impact on
+public health, American Journal of Epidemiology, 183 (2016), p. kww021.
+[40] Y. Luo, K. Ogle, C. Tucker, S. Fei, C. Gao, S. LaDeau, J. S. Clark, and D. S.
+Schimel, Ecological forecasting and data assimilation in a data-rich era, Ecological Applica-
+tions, 21 (2011), pp. 1429–1442.
+33
+
+[41] B. Lusch, J. N. Kutz, and S. L. Brunton, Deep learning for universal linear embeddings
+of nonlinear dynamics, Nature Communications, 9 (2018).
+[42] J. Mandel, A brief tutorial on the ensemble kalman filter, 2009.
+[43] J. Mann and J. N. Kutz, Dynamic mode decomposition for financial trading strategies, 2015.
+[44] I. G. K. ”Matthew O. Williams”, ”Clarence W. Rowley”, A kernel-based method for
+data-driven koopman spectral analysis, Journal of Computational Dynamics, 2 (2015), p. 247.
+[45] T. Nonomura, H. Shibata, and R. Takaki, Dynamic mode decomposition using a kalman
+filter for parameter estimation, AIP Advances, 8 (2018), p. 105106.
+[46] T. Nonomura, H. Shibata, and R. Takaki, Extended-kalman-filter-based dynamic mode
+decomposition for simultaneous system identification and denoising, PloS one, 14 (2019),
+p. e0209836.
+[47] E. O. Nsoesie, J. S. Brownstein, N. Ramakrishnan, and M. V. Marathe, A system-
+atic review of studies on forecasting the dynamics of influenza outbreaks, Influenza and Other
+Respiratory Viruses, 8 (2014), pp. 309–316.
+[48] D. Osthus, K. S. Hickmann, P. C. Caragea, D. Higdon, and S. Y. D. Valle, Fore-
+casting seasonal influenza with a state-space SIR model, The Annals of Applied Statistics, 11
+(2017), pp. 202 – 224.
+[49] J. Proctor and P. Welkhoff, Discovering dynamic patterns from infectious disease data
+using dynamic mode decomposition, International health, 7 (2015), pp. 139–45.
+[50] N. G. Reich, L. C. Brooks, S. J. Fox, S. Kandula, C. J. McGowan, E. Moore,
+D. Osthus, E. L. Ray, A. Tushar, T. K. Yamana, M. Biggerstaff, M. A. Johansson,
+R. Rosenfeld, and J. Shaman, A collaborative multiyear, multimodel assessment of seasonal
+influenza forecasting in the united states, Proceedings of the National Academy of Sciences,
+116 (2019), pp. 3146–3154.
+[51] R. H. Reichle, Data assimilation methods in the earth sciences, Advances in water resources,
+31 (2008), pp. 1411–1418.
+[52] R. H. Reichle, J. P. Walker, R. D. Koster, and P. R. Houser, Extended versus
+ensemble kalman filtering for land data assimilation, Journal of Hydrometeorology, 3 (01 Dec.
+2002), pp. 728 – 740.
+[53] M. Ribeiro and I. Ribeiro, Kalman and extended kalman filters: Concept, derivation and
+properties, 04 2004.
+[54] P. Schmid and J. Sesterhenn, Dynamic Mode Decomposition of numerical and experi-
+mental data, in APS Division of Fluid Dynamics Meeting Abstracts, vol. 61 of APS Meeting
+Abstracts, Nov. 2008, p. MR.007.
+[55] S. J. Sheather, Density estimation, Statistical Science, 19 (2004), pp. 588–597.
+[56] B. W. Silverman, Density estimation for statistics and data analysis, Routledge, 2018.
+34
+
+[57] C. Snyder, T. Bengtsson, P. Bickel, and J. L. Anderson, Obstacles to high-
+dimensional particle filtering, Monthly Weather Review, 136 (2008), pp. 4629–4640.
+[58] P. Van den Driessche, Reproduction numbers of infectious disease models, Infectious Disease
+Modelling, 2 (2017), pp. 288–303.
+[59] E. A. Wan and R. Van Der Merwe, The unscented kalman filter for nonlinear estimation,
+in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications,
+and Control Symposium (Cat. No.00EX373), 2000, pp. 153–158.
+[60] Z. Wang, P. Chakraborty, S. R. Mekaru, J. S. Brownstein, J. Ye, and N. Ra-
+makrishnan, Dynamic poisson autoregression for influenza-like-illness case count prediction,
+in Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery
+and Data Mining, KDD ’15, New York, NY, USA, 2015, Association for Computing Machinery,
+p. 1285–1294.
+[61] M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data–driven approximation of
+the koopman operator: Extending dynamic mode decomposition, Journal of Nonlinear Science,
+25 (2015), p. 1307–1346.
+[62] H. Zhang, C. W. Rowley, E. A. Deem, and L. N. Cattafesta, Online dynamic mode
+decomposition for time-varying systems, 2017.
+35
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6dE3T4oBgHgl3EQfpwoB/content/tmp_files/2301.04644v1.pdf.txt

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+DOES PROGRESS ON IMAGENET TRANSFER
+TO REAL-WORLD DATASETS?
+Alex Fang
+University of Washington
+[email protected]
+Simon Kornblith∗
+Google Research, Brain Team
+[email protected]
+Ludwig Schmidt∗
+University of Washington, Allen Institute for AI
+[email protected]
+ABSTRACT
+Does progress on ImageNet transfer to real-world datasets? We investigate this
+question by evaluating ImageNet pre-trained models with varying accuracy (57% -
+83%) on six practical image classification datasets. In particular, we study datasets
+collected with the goal of solving real-world tasks (e.g., classifying images from
+camera traps or satellites), as opposed to web-scraped benchmarks collected for
+comparing models. On multiple datasets, models with higher ImageNet accuracy
+do not consistently yield performance improvements. For certain tasks, interven-
+tions such as data augmentation improve performance even when architectures
+do not. We hope that future benchmarks will include more diverse datasets to
+encourage a more comprehensive approach to improving learning algorithms.
+1
+INTRODUCTION
+ImageNet is one of the most widely used datasets in machine learning. Initially, the ImageNet com-
+petition played a key role in re-popularizing neural networks with the success of AlexNet in 2012.
+Ten years later, the ImageNet dataset is still one of the main benchmarks for state-of-the-art com-
+puter vision models (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016; Liu
+et al., 2018; Howard et al., 2019; Touvron et al., 2021; Radford et al., 2021). As a result of Ima-
+geNet’s prominence, the machine learning community has invested tremendous effort into develop-
+ing model architectures, training algorithms, and other methodological innovations with the goal of
+increasing performance on ImageNet. Comparing methods on a common task has important benefits
+because it ensures controlled experimental conditions and results in rigorous evaluations. But the
+singular focus on ImageNet also raises the question whether the community is over-optimizing for
+this specific dataset.
+As a first approximation, ImageNet has clearly encouraged effective methodological innovation be-
+yond ImageNet itself. For instance, the key finding from the early years of ImageNet was that
+large convolution neural networks (CNNs) can succeed on contemporary computer vision datasets
+by leveraging GPUs for training. This paradigm has led to large improvements in other computer
+vision tasks, and CNNs are now omnipresent in the field. Nevertheless, this clear example of trans-
+fer to other tasks early in the ImageNet evolution does not necessarily justify the continued focus
+ImageNet still receives. For instance, it is possible that early methodological innovations transferred
+more broadly to other tasks, but later innovations have become less generalizable. The goal of our
+paper is to investigate this possibility specifically for neural network architecture and their transfer
+to real-world data not commonly found on the Internet.
+When discussing the transfer of techniques developed for ImageNet to other datasets, a key ques-
+tion is what other datasets to consider. Currently there is no comprehensive characterization of the
+many machine learning datasets and transfer between them. Hence we restrict our attention to a
+limited but well-motivated family of datasets. In particular, we consider classification tasks derived
+from image data that were specifically collected with the goal of classification in mind. This is in
+∗Equal contribution
+1
+arXiv:2301.04644v1  [cs.CV]  11 Jan 2023
+
+contrast to many standard computer vision datasets – including ImageNet – where the constituent
+images were originally collected for a different purpose, posted to the web, and later re-purposed for
+benchmarking computer vision methods. Concretely, we study six datasets ranging from leaf dis-
+ease classification over melanoma detection to categorizing animals in camera trap images. Since
+these datasets represent real-world applications, transfer of methods from ImageNet is particularly
+relevant.
+We find that on four out of our six real-world datasets, ImageNet-motivated architecture improve-
+ments after VGG resulted in little to no progress (see Figure 1). Specifically, when we fit a line to
+downstream model accuracies as a function of ImageNet accuracy, the resulting slope is less than
+0.05. The two exceptions where post-VGG architectures yield larger gains are the Caltech Camera
+Traps-20 (CCT-20) (Beery et al., 2018) dataset (slope 0.11) and the Human Protein Atlas Image
+Classification (Ouyang et al., 2019) dataset (slope 0.29). On multiple other datasets, we find that
+task-specific improvements such as data augmentations or extra training data lead to larger gains
+than using a more recent ImageNet architecture. We evaluate on a representative testbed of 19 Im-
+ageNet models, ranging from the seminal AlexNet (Krizhevsky et al., 2012) over VGG (Simonyan
+& Zisserman, 2015) and ResNets (He et al., 2016) to the more recent and higher-performing Effi-
+cientNets (Tan & Le, 2019) and ConvNexts (Liu et al., 2022) (ImageNet top-1 accuracies 56.5% to
+83.4%). Our testbed includes three Vision Transformer models to cover non-CNN architectures.
+Interestingly, our findings stand in contrast to earlier work that investigated the aforementioned
+image classification benchmarks such as CIFAR-10 (Krizhevsky & Hinton, 2009), PASCAL VOC
+2007 (Everingham et al., 2010), and Caltech-101 (Fei-Fei et al., 2004) that were scraped from the
+Internet. On these datasets, Kornblith et al. (2019) found consistent gains in downstream task ac-
+curacy for a similar range of architectures as we study in our work. Taken together, these findings
+indicate that ImageNet accuracy may be a good predictor for other web-scraped datasets, but less
+informative for real-world image classification datasets that are not sourced through the web. On the
+other hand, the CCT-20 data point shows that even very recent ImageNet models do help on some
+downstream tasks that do not rely on images from the web. Overall, our results highlight the need
+for a more comprehensive understanding of machine learning datasets to build and evaluate broadly
+useful data representations.
+2
+RELATED WORK
+Transferability of ImageNet architectures. Although there is extensive previous work investigat-
+ing the effect of architecture upon the transferability of ImageNet-pretrained models to different
+datasets, most of this work focuses on performance on datasets collected for the purpose of bench-
+marking. Kornblith et al. (2019) previously showed that ImageNet accuracy of different models is
+strongly correlated with downstream accuracy on a wide variety of web-scraped object-centric com-
+puter vision benchmark tasks. Later studies have investigated the relationship between ImageNet
+and transfer accuracy for self-supervised networks (Ericsson et al., 2021; Kotar et al., 2021; Nayman
+et al., 2022), adversarially trained networks (Salman et al., 2020), or networks trained with different
+loss functions (Kornblith et al., 2021), but still evaluate primarily on web-scraped benchmark tasks.
+The Visual Task Adaptation Benchmark (VTAB) (Zhai et al., 2019) comprises a more diverse set of
+tasks, including natural and non-natural classification tasks as well as non-classification tasks, but
+nearly all consist of web-scraped or synthetic images. In the medical imaging domain, models have
+been extensively evaluated on real-world data, with limited gains from newer models that perform
+better on ImageNet (Raghu et al., 2019; Bressem et al., 2020; Ke et al., 2021).
+Most closely related to our work, Tuggener et al. (2021) investigate performance of 500 CNN archi-
+tectures on yet another set of datasets, several of which are not web-scraped, and find that accuracy
+correlates poorly with ImageNet accuracy when training from scratch, but correlations are higher
+when fine-tuning ImageNet-pretrained models. Our work differs from theirs in our focus solely on
+real-world datasets (e.g., from Kaggle competitions) and in that we perform extensive tuning in order
+to approach the best single-model performance obtainable on these datasets whereas Tuggener et al.
+(2021) instead devote their compute budget to increasing the breadth of architectures investigated.
+Transferability of networks trained on other datasets. Other work has evaluated transferability
+of representations of networks trained on datasets beyond ImageNet. Most notably, Abnar et al.
+(2022) explore the relationship between upstream and downstream accuracy for models pretrained
+on JFT and ImageNet-21K and find that, on many tasks, downstream accuracy saturates with up-
+2
+
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+64
+66
+68
+70
+72
+74
+76
+78
+Accuracy
+Caltech Camera Traps 20
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+0.895
+0.900
+0.905
+0.910
+0.915
+0.920
+0.925
+0.930
+Quadratic weighted kappa
+APTOS 2019 Blindness
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Macro F1 score
+Human Protein Atlas
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+83
+84
+85
+86
+87
+88
+89
+Accuracy
+Cassava Leaf Disease
+60
+65
+70
+75
+80
+ImageNet top-1 accuracy
+98.0
+98.2
+98.4
+98.6
+98.8
+99.0
+99.2
+99.4
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+ViT-B/16
+Figure 1: Overview of transfer performance across models from ImageNet to each of the datasets we study.
+Although there seems to be a strong linear trends between ImageNet accuracy and the target metrics (green),
+these trends become less certain when we restrict the models to those above 70% ImageNet accuracy (blue).
+Versions with error bars and spline interpolation can be found in Appendix B.
+stream accuracy. However, they evaluate representational quality using linear transfer rather than
+end-to-end fine-tuning. Other studies have investigated the impact of relationships between pre-
+training and fine-tuning tasks (Zamir et al., 2018; Mensink et al., 2021) or the impact of scaling the
+model and dataset (Goyal et al., 2019; Kolesnikov et al., 2020).
+Another direction of related work relates to the effect of pretraining data on transfer learning. Huh
+et al. (2016) look into the factors that make ImageNet good for transfer learning. They find that
+fine-grained classes are not needed for good transfer performance, and that reducing the dataset size
+and number of classes only results in slight drops in transfer learning performance. Though there is
+a common goal of exploring what makes transfer learning work well, our work differs from this line
+of work by focusing on the fine-tuning aspect of transfer learning.
+Other studies of external validity of benchmarks. Our study fits into a broader literature inves-
+tigating the external validity of image classification benchmarks. Early work in this area identified
+lack of diversity as a key shortcoming of the benchmarks of the time (Ponce et al., 2006; Torralba
+& Efros, 2011), a problem that was largely resolved with the introduction of the much more di-
+verse ImageNet benchmark (Deng et al., 2009; Russakovsky et al., 2015). More recent studies have
+investigated the extent to which ImageNet classification accuracy correlates with accuracy on out-
+of-distribution (OOD) data (Recht et al., 2019; Taori et al., 2020) or accuracy as measured using
+higher-quality human labels (Shankar et al., 2020; Tsipras et al., 2020; Beyer et al., 2020).
+As in previous studies of OOD generalization, transfer learning involves generalization to test sets
+that differ in distribution from the (pre-)training data. However, there are also key differences be-
+tween transfer learning and OOD generalization. First, in transfer learning, additional training data
+from the target task is used to adapt the model, while OOD evaluations usually apply trained models
+to a new distribution without any adaptation. Second, OOD evaluations usually focus on settings
+with a shared class space so that evaluations without adaptation are possible. In contrast, transfer
+learning evaluation generally involves downstream tasks with classes different from those in the pre-
+training dataset. These differences between transfer learning and OOD generalization are not only
+conceptual but also lead to different empirical phenomena. Miller et al. (2021) has shown that in-
+3
+
+distribution accuracy improvements often directly yield out-of-distribution accuracy improvements
+as well. This is the opposite of our main experimental finding that ImageNet improvements do not
+directly yield performance improvements on many real-world downstream tasks. Hence our work
+demonstrates an important difference between OOD generalization and transfer learning.
+3
+DATASETS
+As mentioned in the introduction, a key choice in any transfer study is the set of target tasks on which
+to evaluate model performance. Before we introduce our suite of target tasks, we first describe three
+criteria that guided our dataset selection: (i) diverse data sources, (ii) relevance to an application,
+and (iii) availability of well-tuned baseline models for comparison.
+3.1
+SELECTION CRITERIA
+Prior work has already investigated transfer of ImageNet architectures to many downstream
+datasets (Donahue et al., 2014; Sharif Razavian et al., 2014; Chatfield et al., 2014; Simonyan &
+Zisserman, 2015). The 12 datasets used by Kornblith et al. (2019) often serve as a standard evalu-
+ation suite (e.g., in (Salman et al., 2020; Ericsson et al., 2021; Radford et al., 2021)). While these
+datasets are an informative starting point, they are all object-centric natural image datasets, and do
+not represent the entire range of image classification problems. There are many applications of com-
+puter vision; the Kaggle website alone lists more than 1,500 datasets as of May 2022. To understand
+transfer from ImageNet more broadly, we selected six datasets guided by the following criteria.
+Diverse data sources. Since collecting data is an expensive process, machine learning researchers
+often rely on web scraping to gather data when assembling a new benchmark. This practice has led to
+several image classification datasets with different label spaces such as food dishes, bird species, car
+models, or other everyday objects. However, the data sources underlying these seemingly different
+tasks are actually often similar. Specifically, we surveyed the 12 datasets from Kornblith et al.
+(2019) and found that all of these datasets were harvested from the web, often via keyword searches
+in Flickr, Google image search, or other search engines (see Appendix K). This narrow range of
+data sources limits the external validity of existing transfer learning experiments. To get a broader
+understanding of transfer from ImageNet, we focus on scientific, commercial, and medical image
+classification datasets that were not originally scraped from the web.
+Application relevance. In addition to the data source, the classification task posed on a given set of
+images also affects how relevant the resulting problem is for real-world applications. For instance,
+it would be possible to start with real-world satellite imagery that shows multiple building types
+per image, but only label one of the building types for the purpose of benchmarking (e.g., to avoid
+high annotation costs). The resulting task may then be of limited value for an actual application
+involving the satellite images that requires all buildings to be annotated. We aim to avoid such
+pitfalls by limiting our attention to classification tasks that were assembled by domain experts with
+a specific application in mind.
+Availability of baselines. If methodological progress does not transfer from ImageNet to a given
+target task, we should expect that, as models perform better on ImageNet, accuracy on the target
+task saturates. However, observing such a trend in an experiment is not sufficient to reach a conclu-
+sion regarding transfer because there is an alternative explanation for this empirical phenomenon.
+Besides a lack of transfer, the target task could also simply be easier than the source task so that
+models with sub-optimal source task accuracy already approach the Bayes error rate. As an illus-
+trative example, consider MNIST as a target task for ImageNet transfer. A model with mediocre
+ImageNet accuracy is already sufficient to get 99% accuracy on MNIST, but this finding does not
+mean that better ImageNet models are insufficient to improve MNIST accuracy — the models have
+already hit the MNIST performance ceiling.
+More interesting failures of transfer occur when ImageNet architectures plateau on the target task,
+but it is still possible to improve accuracy beyond what the best ImageNet architecture can achieve
+without target task-specific modifications. In order to make such comparisons, well-tuned base-
+lines for the target task are essential. If improving ImageNet accuracy alone is insufficient to reach
+these well-tuned baselines, we can indeed conclude that architecture transfer to this target task is
+limited. In our experiments, we use multiple datasets from Kaggle competitions since the resulting
+leaderboards offer well-tuned baselines arising from a competitive process.
+4
+
+3.2
+DATASETS STUDIED
+Table 1: We examine a variety of real-world datasets that cover different types of tasks.
+Dataset
+# of classes
+Train size
+Eval size
+Eval metric
+Kaggle
+Caltech Camera Traps
+15
+14,071
+15,215
+Accuracy
+APTOS 2019 Blindness
+5
+2,930
+732
+Quadratic
+
+weighted kappa
+Human Protein Atlas
+28
+22,582
+5,664
+Macro F1 score
+
+SIIM-ISIC Melanoma
+2
+46,372
+11,592
+Area under ROC
+
+Cassava Leaf Disease
+5
+17,118
+4,279
+Accuracy
+
+EuroSAT
+10
+21,600
+5,400
+Accuracy
+Caltech Camera
+Traps-20
+APTOS 2019
+Blindness Detection
+Human Protein Atlas
+Image Classification
+SIIM-ISIC Melanoma
+Classification
+Cassava Leaf Disease
+Classification
+EuroSAT
+Figure 2: Sample images from each of the datasets.
+The datasets studied in this work are practical and cover a variety of applications. We choose four
+of the most popular image classification competitions on Kaggle, as measured by number of com-
+petitors, teams, and submissions. Each of these competitions is funded by an organization with the
+goal of advancing performance on that real-world task. Additionally, we supplement these datasets
+with Caltech Camera Traps (Beery et al., 2018) and EuroSAT (Helber et al., 2019) to broaden the
+types of applications studied. Details for each dataset can be found in Table 1 1.
+4
+MAIN EXPERIMENTS
+We run our experiments across 19 model architectures, including both CNNs and Vision Transform-
+ers (ViT and DeiT). They range from 57% to 83% ImageNet top-1 accuracy, allowing us to observe
+the relationship between ImageNet performance and target dataset performance. In order to get the
+best performance out of each architecture, we do extensive hyperparameter tuning over learning
+rate, weight decay, optimizer, and learning schedule. Details about our experiment setup can be
+found in Appendix C. We now present our results for each of the datasets we investigated. Figure 1
+summarizes our results across all datasets, with additional statistics in Table 2. Appendix A contains
+complete results for all datasets across the hyperparameter grids.
+4.1
+CALTECH CAMERA TRAPS
+Table 2: We summarize the blue regression lines from
+Figure 1, calculated on models above 70% ImageNet
+accuracy, with their correlation and slope. Slope is cal-
+culated so that all metrics have a range from 0 to 100.
+Dataset
+Correlation
+Slope
+Caltech Camera Traps
+0.17
+0.11
+APTOS 2019 Blindness
+0.06
+0.01
+Human Protein Atlas
+0.26
+0.29
+SIIM-ISIC Melanoma
+0.44
+0.05
+Cassava Leaf Disease
+0.12
+0.02
+EuroSAT
+0.05
+0.00
+Beery et al. (2018) created Caltech Camera
+Traps-20 (CCT-20) using images taken from
+camera traps deployed to monitor animal pop-
+ulations. The images contain 15 different ani-
+mal classes, as well as an empty class that we
+remove for our experiments 2. The dataset con-
+tains two sets of validation and test sets which
+differ by whether they come from locations that
+are the same as or different from the training set
+locations. While one of the goals of the dataset
+is to study generalization to new environments,
+here we only study the sets from the same locations. Although CCT-20 is not a Kaggle competition,
+it is a subset of the iWildCam Challenge 2018, whose yearly editions have been hosted on Kaggle.
+We see in Figure 1 (top-left) an overall positive trend between ImageNet performance and CCT-
+20 performance. The overall trend is unsurprising, given the number of animal classes present in
+ImageNet. But despite the drastic reduction in the number of classes when compared to ImageNet,
+1Dataset download links and PyTorch datasets and splits can be found at https://github.com/
+mlfoundations/imagenet-applications-transfer.
+2Empty class is removed for the classification experiments in Table 1 of Beery et al. (2018)
+5
+
+店CCT-20 has its own set of challenges. Animals are often pictured at difficult angles, and sometimes
+are not even visible in the image because a sequence of frames triggered by activity all have the
+same label. Despite these challenges, an even higher performing model still does better on this task
+- we train a CLIP ViT L/14-336px model (85.4% ImageNet top-1) with additional augmentation to
+achieve 83.4% accuracy on CCT-20.
+4.2
+APTOS 2019 BLINDNESS DETECTION
+This dataset was created for a Kaggle competition run by the Asia Pacific Tele-Ophthalmology
+Society (APTOS) with the goal of advancing medical screening for diabetic retinopathy in rural
+areas (Asia Pacific Tele-Ophthalmology Society, 2019). Images are taken using fundus photography
+and vary in terms of clinic source, camera used, and time taken. Images are labeled by clinicians on
+a scale of 0 to 4 for the severity of diabetic retinopathy. Given the scaled nature of the labels, the
+competition uses quadratic weighted kappa (QWK) as the evaluation metric. We create a local 80%
+to 20% random class-balanced train/validation split, as the competition test labels are hidden.
+We find that models after VGG do not show significant improvement. Similar to in CCT-20, DeiT
+and EfficientNets performs slightly worse, while deeper models from the same architecture slightly
+help performance. We also find that accuracy has a similar trend as QWK, despite it being an inferior
+metric in the context of this dataset.
+When performance stagnates, one might ask whether we have reached a performance limit for our
+class of models on the dataset. To answer this question, we compare with the Kaggle leaderboard’s
+top submissions. The top Kaggle submission achieves 0.936 QWK on the private leaderboard (85%
+of the test set) (Xu, 2019). They do this by using additional augmentation, using external data,
+training on L1-loss, replacing the final pooling layer with generalized mean pooling, and ensembling
+a variety of models trained with different input sizes. The external data consists of 88,702 images
+from the 2015 Diabetic Retinopathy Detection Kaggle competition.
+Even though performance saturates with architecture, we find that additional data augmentation and
+other interventions still improve accuracy. We submitted our ResNet-50 and ResNet-152 models
+with additional interventions, along with an Inception-ResNet v2 (Szegedy et al., 2017b) model with
+hyperparameter tuning. We find that increasing color and affine augmentation by itself can account
+for a 0.03 QWK point improvement. Once we train on 512 input size, additional augmentation, and
+additional data, our ResNet-50 and Inception-ResNet v2 both achieve 0.896 QWK on the private
+leaderboard, while ResNet-152 achieves 0.890 QWK, once again suggesting that better ImageNet
+architectures by themselves do not lead to increased performance on this task.
+As a comparison, the ensemble from the top leaderboard entry included a single model Inception-
+ResNet v2 trained with additional interventions that achieves 0.927 QWK. We submitted the original
+models we trained to Kaggle as well, finding that the new models trained with additional interven-
+tions do at least 0.03 QWK points better. See Appendix F for additional experimental details. Both
+this result and the gap between our models and the top leaderboard models show that there exist
+interventions that do improve task performance.
+4.3
+HUMAN PROTEIN ATLAS IMAGE CLASSIFICATION
+The Human Protein Atlas runs the Human Protein Atlas Image Classification competition on Kaggle
+to build an automated tool for identifying and locating proteins from high-throughput microscopy
+images (Ouyang et al., 2019). Images can contain multiple of the 28 different proteins, so the
+competition uses the macro F1 score. Given the multi-label nature of the problem, this requires
+thresholding for prediction. We use a 73% / 18% / 9% train / validation / test-validation split created
+by a previous competitor (Park, 2019). We report results on the validation split, as we find that the
+thresholds selected for the larger validation split generalize well to the smaller test-validation split.
+We find a slightly positive trend between task performance and ImageNet performance, even when
+ignoring AlexNet and MobileNet. This is surprising because ImageNet is quite visually distinct from
+human protein slides. These results suggest that models with more parameters help with downstream
+performance, especially for tasks that have a lot of room for improvement.
+Specific challenges for this dataset are extreme class imbalance, multi-label thresholding, and gen-
+eralization from the training data to the test set. Competitors were able to improve performance
+beyond the baselines we found by using external data as well as techniques such as data cleaning,
+6
+
+additional training augmentation, test time augmentation, ensembling, and oversampling (Dai, 2019;
+Park, 2019; Shugaev, 2019). Additionally, some competitors modified commonly-used architectures
+by substituting pooling layers or incorporating attention (Park, 2019; Zheng, 2019). Uniquely, the
+first place solution used metric learning on top of a single DenseNet121 (Dai, 2019). These tech-
+niques may be useful when applied to other datasets, but are rarely used in a typical workflow.
+4.4
+SIIM-ISIC MELANOMA CLASSIFICATION
+The Society for Imaging Informatics in Medicine (SIIM) and the International Skin Imaging Collab-
+oration (ISIC) jointly ran this Kaggle competition for identifying Melanoma (SIIM & ISIC, 2020),
+a serious type of skin cancer. Competitors use images of skin lesions to predict the probability that
+each observed image is malignant. Images come from the ISIC Archive, which is publicly available
+and contains images from a variety of countries. The competition provided 33,126 training images,
+plus an additional 25,331 images from previous competitions. We split the combined data into an
+80% to 20% class-balanced and year-balanced train/validation split. Given the imbalanced nature of
+the data (8.8% positive), the competition uses area under ROC curve as the evaluation metric.
+We find only a weak positive correlation (0.44) between ImageNet performance and task perfor-
+mance, with a regression line with a normalized slope of close to zero (0.05). But if we instead look
+at classification accuracy, Appendix H shows that there is a stronger trend for transfer than that of
+area under ROC curve, as model task accuracy more closely follows the same order as ImageNet
+performance. This difference shows that characterizing the relationship between better ImageNet
+models and better transfer performance is reliant on the evaluation metric as well. We use a rela-
+tively simple setup to measure the impact of ImageNet models on task performance, but we know we
+can achieve better results with additional strategies. The top two Kaggle solutions used models with
+different input size, ensembling, cross-validation and a significant variety of training augmentation
+to create a stable model that generalized to the hidden test set (Ha et al., 2020; Pan, 2020).
+4.5
+CASSAVA LEAF DISEASE CLASSIFICATION
+The Makerere Artificial Intelligence Lab is an academic research group focused on applications
+that benefit the developing world. Their goal in creating the Cassava Leaf Disease Classification
+Kaggle competition (Makerere University AI Lab, 2021) was to give farmers access to methods
+for diagnosing plant diseases, which could allow farmers to prevent these diseases from spreading,
+increasing crop yield. Images were taken with an inexpensive camera and labeled by agricultural
+experts. Each image was classified as healthy or as one of four different diseases. We report results
+using a 80%/20% random class-balanced train/validation split of the provided training data.
+Once we ignore models below 70% ImageNet accuracy, the relationship between the performance on
+the two datasets has both a weak positive correlation (0.12) and a near-zero normalized slope (0.02).
+While these are natural images similar to portions of ImageNet, it is notable that ImageNet contains
+very few plant classes (e.g., buckeye, hip, rapeseed). Yet based on a dataset’s perceived similarity to
+ImageNet, it is surprising that leaf disease classification is not positively correlated with ImageNet,
+while the microscopy image based Human Protein Atlas competition is. Our results are supported
+by Kaggle competitors: the first place solution found that on the private leaderboard, EfficientNet
+B4 (Tan & Le, 2019), MobileNet, and ViT (Dosovitskiy et al., 2021b) achieve 89.5%, 89.4%, and
+88.8% respectively (Hanke, 2021). Their ensemble achieves 91.3% on the private leaderboard.
+4.6
+EUROSAT
+Helber et al. (2019) created EuroSAT from Sentinel-2 satellite images to classify land use and land
+cover. Past work has improved performance on the dataset through additional training time tech-
+niques (Naushad et al., 2021) and using 13 spectral bands (Yassine et al., 2021). We use RGB
+images and keep our experimental setup consistent to compare across a range of models. Since
+there is no set train/test split, we create a 80%/20% class-balanced split.
+All models over 60% ImageNet accuracy achieve over 98.5% EuroSAT accuracy, and the majority
+of our models achieve over 99.0% EuroSAT accuracy. There are certain tasks where using better
+ImageNet models does not improve performance, and this would be the extreme case where perfor-
+mance saturation is close to being achieved. While it is outside the scope of this study, a next step
+would be to investigate the remaining errors and find other methods to reduce this last bit of error.
+7
+
+5
+ADDITIONAL STUDIES
+5.1
+AUGMENTATION ABLATIONS
+In our main experiments, we keep augmentation simple to minimize confounding factors when com-
+paring models. However, it is possible pre-training and fine-tuning with different combinations of
+augmentations may have different results. This is an important point because different architectures
+may have different inductive biases and often use different augmentation strategies at pre-training
+time. To investigate these effects, we run additional experiments on CCT-20 and APTOS to explore
+the effect of data augmentation on transfer. Specifically, we take ResNet-50 models pre-trained with
+standard crop and flip augmentation, AugMix (Hendrycks et al., 2020), and RandAugment (Cubuk
+et al., 2020), and then fine-tune on our default augmentation, AugMix, and RandAugment. We also
+study DeiT-tiny and Deit-small models by fine-tuning on the same three augmentations mentioned
+above. We choose to examine DeiT models because they are pre-trained using RandAugment and
+RandErasing (Zhong et al., 2020). We increase the number of epochs we fine-tune on from 30 to 50
+to account for augmentation. Our experimental results are found in Appendix G.
+In our ResNet-50 experiments, both AugMix and RandAugment improve performance on ImageNet,
+but while pre-training with RandAugment improves performance on downstream tasks, pre-training
+with AugMix does not. Furthermore, fine-tuning with RandAugment usually yields additional per-
+formance gains when compared to our default fine-tuning augmentation, no matter which pre-trained
+model is used. For DeiT models, we found that additional augmentation did not significantly in-
+crease performance on the downstream tasks. Thus, as with architectures, augmentation strategies
+that improve accuracy on ImageNet do not always improve accuracy on real-world tasks.
+5.2
+CLIP MODELS
+A natural follow-up to our experiments is to change the source of pre-training data. We exam-
+ine CLIP models from Radford et al. (2021), which use diverse pre-training data and achieve high
+performance on a variety of downstream datasets. We fine-tune CLIP models on each of our down-
+stream datasets by linear probing then fine-tuning (LP-FT) (Kumar et al., 2022).3 Our results are
+visualized by the purple stars in Appendix I Figure 8. We see that by using a model that takes larger
+images we can do better than all previous models, and even without the larger images, ViT-L/14
+does better on four out of the six datasets. While across all CLIP models the change in pre-training
+data increases performance for CCT-20, the effect on the other datasets is more complicated. When
+controlling for architecture changes by only looking at ResNet-50 and ViT/B16, we see that the ad-
+ditional pre-training data helps for CCT-20, HPA, and Cassava, the former two corresponding to the
+datasets that empirically benefit most from using better ImageNet models. Additional results can be
+found in Appendix I, while additional fine-tuning details can be found in Appendix J.
+6
+DISCUSSION
+Alternative explanations for saturation. Whereas Kornblith et al. (2019) reported a high degree
+of correlation between ImageNet and transfer accuracy, we find that better ImageNet models do not
+consistently transfer better on our real-world tasks. We believe these differences are related to the
+tasks themselves. Here, we rule out alternative hypotheses for our findings.
+Comparison of datasets statistics suggests that the number of classes and dataset size also do not
+explain the differences from Kornblith et al. (2019). The datasets we study range from two to 28
+classes. Although most of the datasets studied in Kornblith et al. (2019) have more classes, CIFAR-
+10 has 10. In Appendix E, we replicate CIFAR-10 results from Kornblith et al. (2019) using our
+experimental setup, finding a strong correlation between ImageNet accuracy and transfer accuracy.
+Thus, the number of classes is likely not the determining factor. Training set sizes are similar
+between our study and that of Kornblith et al. (2019) and thus also do not seem to play a major role.
+A third hypothesis is that it is parameter count, rather than ImageNet accuracy, that drives trends.
+We see that VGG BN models appear to outperform their ImageNet accuracy on multiple datasets,
+and they are among the largest models by parameter count. However, in Appendix L, we find that
+model size is also not a good indicator of improved transfer performance on real world datasets.
+3We use LP-FT because, in past experiments, we have found that LP-FT makes hyperparameter tuning
+easier for CLIP models, but does not significantly alter performance when using optimal hyperparameters.
+8
+
+Differences between web-scraped datasets and real-world images We conjecture that it is possi-
+ble to perform well on most, if not all, web-scraped target datasets simply by collecting a very large
+amount of data from the Internet and training a very large model on it. Web-scraped target datasets
+are by definition within the distribution of data collected from the web, and a sufficiently large model
+can learn that distribution. In support of this conjecture, recent models such as CLIP (Radford et al.,
+2021), ALIGN (Jia et al., 2021), ViT-G (Zhai et al., 2022), BASIC (Pham et al., 2021), and CoCa (Yu
+et al., 2022) are trained on very large web-scraped datasets and achieve high accuracy on a variety of
+web-scraped benchmarks. However, this strategy may not be effective for non-web-scraped datasets,
+where there is no guarantee that we will train on data that is close in distribution to the target data,
+even if we train on the entire web. Thus, it makes sense to distinguish these two types of datasets.
+There are clear differences in image distribution between the non-web-scraped datasets we consider
+and web-scraped datasets considered by previous work. In Figure 3 and Appendix M, we compute
+Figure 3: FID scores vs ImageNet for the datasets
+we study in this work (red), and the web-scraped
+datasets studied by Kornblith et al. (2019) (blue).
+Fr´echet inception distance (FID) (Heusel et al.,
+2017) between ImageNet and each of the datasets
+we study in this work as well as the ones found in
+Kornblith et al. (2019). The real-world datasets are
+further away from ImageNet than those found in Ko-
+rnblith et al. (2019), implying that there is a large
+amount of distribution shift between web-scraped
+datasets and real-world datasets. However, FID is
+only a proxy measure and may not capture all fac-
+tors that lead to differences in transferability.
+Whereas web-scraped data is cheap to acquire, real-
+world data can be more expensive. Ideally, progress
+in computer vision architectures should improve per-
+formance not just on web-scraped data, but also on
+real-world tasks. Our results suggest that the latter has not happened. Gains in ImageNet accuracy
+over the last decade have primarily come from improving and scaling architectures, and past work
+has shown that these gains generally transfer to other web-scraped datasets, regardless of size (Sun
+et al., 2017; Kornblith et al., 2019; Mahajan et al., 2018; Xie et al., 2020; Kolesnikov et al., 2020).
+However, we find that improvements arising from architecture generally do not transfer to non-web-
+scraped tasks. Nonetheless, data augmentation and other tweaks can provide further gains on these
+tasks.
+Recommendations towards better benchmarking. While it is unclear whether researchers have
+over-optimized for ImageNet, our work suggests that researchers should explicitly search for meth-
+ods that improve accuracy on real-world non-web-scraped datasets, rather than assuming that meth-
+ods that improve accuracy on ImageNet will provide meaningful improvements on real-world
+datasets as well. Just as there are methods that improve accuracy on ImageNet but not on the tasks
+we investigate, there may be methods that improve accuracy on our tasks but not ImageNet. The
+Kaggle community provides some evidence for the existence of such methods; Kaggle submissions
+often explore architectural improvements that are less common in traditional ImageNet pre-trained
+models. To measure such improvements on real-world problems, we suggest simply using the aver-
+age accuracy across our tasks as a benchmark for future representation learning research.
+Further analysis of our results shows consistencies in the accuracies of different models across the
+non-web-scraped datasets, suggesting that accuracy improvements on these datasets may translate
+to other datasets. For each dataset, we use linear regression to predict model accuracies on the target
+dataset as a linear combination of ImageNet accuracy and accuracy averaged across the other real-
+world datasets. We perform an F-test to determine whether the average accuracy on other real-world
+datasets explains significant variance beyond that explained by ImageNet accuracy. We find that this
+F-test is significant on all datasets except EuroSAT, where accuracy may be very close to ceiling
+(see further analysis in Appendix N.1). Additionally, in Appendix N.2 we compare the Spearman
+rank correlation (i.e., the Pearson correlation between ranks) between each dataset and the accuracy
+averaged across the other real-world datasets to the Spearman correlation between each dataset and
+ImageNet. We find that the correlation with the average over real-world datasets is higher than
+the correlation with ImageNet and statistically significant for CCT-20, APTOS, HPA, and Cassava.
+Thus, there is some signal in the average accuracy across the datasets that we investigate that is not
+captured by ImageNet top-1 accuracy.
+9
+
+Web (count)
+Non-web (count)
+3
+山
+Number of datasets
+2
+0 H
+00
+00
+00
+00
+00
+00
+100
+50
+125
+FIDWhere do our findings leave ImageNet? We suspect that most of the methodological innovations
+that help on ImageNet are useful for some real-world tasks, and in that sense it has been a successful
+benchmark. However, the innovations that improve performance on industrial web-scraped datasets
+such as JFT (Sun et al., 2017) or IG-3.5B-17k (Mahajan et al., 2018) (e.g., model scaling) may be
+almost entirely disjoint from the innovations that help with the non-web-scraped real-world tasks
+studied here (e.g., data augmentation strategies). We hope that future benchmarks will include more
+diverse datasets to encourage a more comprehensive approach to improving learning algorithms.
+7
+ACKNOWLEDGEMENTS
+We would like to thank Samuel Ainsworth, Sara Beery, Gabriel Ilharco, Pieter-Jan Kindermans,
+Sarah Pratt, Matthew Wallingford, Ross Wightman, and Mitchell Wortsman for valuable conversa-
+tions while working on this project. We would especially like to thank Sarah Pratt for help with
+early experimentation and brainstorming.
+We would also like to thank Hyak computing cluster at the University of Washington and the Google
+TPU Research Cloud program for access to compute resources that allowed us to run our experi-
+ments.
+This work is in part supported by the NSF AI Institute for Foundations of Machine Learning (IFML),
+Open Philanthropy, Google, and the Allen Institute for AI.
+REFERENCES
+Samira Abnar, Mostafa Dehghani, Behnam Neyshabur, and Hanie Sedghi. Exploring the limits of
+large scale pre-training. In International Conference on Learning Representations, 2022. URL
+https://openreview.net/forum?id=V3C8p78sDa.
+Asia Pacific Tele-Ophthalmology Society. Aptos 2019 blindness detection, 2019. URL https:
+//www.kaggle.com/competitions/aptos2019-blindness-detection/
+overview.
+Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Vittorio Ferrari,
+Martial Hebert, Cristian Sminchisescu, and Yair Weiss (eds.), Computer Vision - ECCV 2018 -
+15th European Conference, Munich, Germany, September 8-14, 2018, Proceedings, Part XVI,
+volume 11220 of Lecture Notes in Computer Science, pp. 472–489. Springer, 2018. doi: 10.1007/
+978-3-030-01270-0\ 28.
+URL https://doi.org/10.1007/978-3-030-01270-0_
+28.
+Lucas Beyer, Olivier J H´enaff, Alexander Kolesnikov, Xiaohua Zhai, and A¨aron van den Oord. Are
+we done with imagenet? arXiv preprint arXiv:2006.07159, 2020.
+Keno K Bressem, Lisa C Adams, Christoph Erxleben, Bernd Hamm, Stefan M Niehues, and Janis L
+Vahldiek. Comparing different deep learning architectures for classification of chest radiographs.
+Scientific reports, 10(1):1–16, 2020.
+Ken Chatfield, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Return of the devil in the
+details: Delving deep into convolutional nets. arXiv preprint arXiv:1405.3531, 2014.
+Ekin
+Dogus
+Cubuk,
+Barret
+Zoph,
+Jonathon
+Shlens,
+and
+Quoc
+Le.
+Randaugment:
+Practical
+automated
+data
+augmentation
+with
+a
+reduced
+search
+space.
+In
+Hugo
+Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien
+Lin (eds.), Advances in Neural Information Processing Systems 33:
+Annual Conference
+on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020,
+virtual,
+2020.
+URL https://proceedings.neurips.cc/paper/2020/hash/
+d85b63ef0ccb114d0a3bb7b7d808028f-Abstract.html.
+Shubin
+Dai.
+A
+cnn
+classifier
+and
+a
+metric
+learning
+model,
+1st
+place
+so-
+lution,
+2019.
+URL
+https://www.kaggle.com/competitions/
+human-protein-atlas-image-classification/discussion/78109.
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hi-
+erarchical image database. In 2009 IEEE conference on computer vision and pattern recognition,
+pp. 248–255. Ieee, 2009.
+10
+
+Jeff Donahue, Yangqing Jia, Oriol Vinyals, Judy Hoffman, Ning Zhang, Eric Tzeng, and Trevor
+Darrell. Decaf: A deep convolutional activation feature for generic visual recognition. In Inter-
+national conference on machine learning, pp. 647–655. PMLR, 2014.
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas
+Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszko-
+reit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition
+at scale.
+In 9th International Conference on Learning Representations, ICLR 2021, Virtual
+Event, Austria, May 3-7, 2021. OpenReview.net, 2021a. URL https://openreview.net/
+forum?id=YicbFdNTTy.
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas
+Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszko-
+reit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition
+at scale.
+In 9th International Conference on Learning Representations, ICLR 2021, Virtual
+Event, Austria, May 3-7, 2021. OpenReview.net, 2021b. URL https://openreview.net/
+forum?id=YicbFdNTTy.
+Linus Ericsson, Henry Gouk, and Timothy M Hospedales. How well do self-supervised models
+transfer? In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recogni-
+tion, pp. 5414–5423, 2021.
+Mark Everingham, Luc Van Gool, Christopher K. I. Williams, John M. Winn, and Andrew Zis-
+serman.
+The pascal visual object classes (VOC) challenge.
+Int. J. Comput. Vis., 88(2):
+303–338, 2010. doi: 10.1007/s11263-009-0275-4. URL https://doi.org/10.1007/
+s11263-009-0275-4.
+Li Fei-Fei, Rob Fergus, and Pietro Perona. Learning generative visual models from few training
+examples: An incremental bayesian approach tested on 101 object categories. In IEEE Conference
+on Computer Vision and Pattern Recognition Workshops, CVPR Workshops 2004, Washington,
+DC, USA, June 27 - July 2, 2004, pp. 178. IEEE Computer Society, 2004. doi: 10.1109/CVPR.
+2004.383. URL https://doi.org/10.1109/CVPR.2004.383.
+Priya Goyal, Dhruv Mahajan, Abhinav Gupta, and Ishan Misra. Scaling and benchmarking self-
+supervised visual representation learning. In Proceedings of the ieee/cvf International Conference
+on computer vision, pp. 6391–6400, 2019.
+Qishen Ha, Bo Liu, and Fuxu Liu. Identifying melanoma images using efficientnet ensemble: Win-
+ning solution to the SIIM-ISIC melanoma classification challenge. CoRR, abs/2010.05351, 2020.
+URL https://arxiv.org/abs/2010.05351.
+Jannis Hanke. 1st place solution, 2021. URL https://www.kaggle.com/competitions/
+cassava-leaf-disease-classification/discussion/221957.
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image
+recognition.
+In 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR
+2016, Las Vegas, NV, USA, June 27-30, 2016, pp. 770–778. IEEE Computer Society, 2016. doi:
+10.1109/CVPR.2016.90. URL https://doi.org/10.1109/CVPR.2016.90.
+Patrick Helber, Benjamin Bischke, Andreas Dengel, and Damian Borth. Eurosat: A novel dataset
+and deep learning benchmark for land use and land cover classification. IEEE J. Sel. Top. Appl.
+Earth Obs. Remote. Sens., 12(7):2217–2226, 2019. doi: 10.1109/JSTARS.2019.2918242. URL
+https://doi.org/10.1109/JSTARS.2019.2918242.
+Dan Hendrycks, Norman Mu, Ekin Dogus Cubuk, Barret Zoph, Justin Gilmer, and Balaji Lakshmi-
+narayanan. Augmix: A simple data processing method to improve robustness and uncertainty. In
+8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia,
+April 26-30, 2020. OpenReview.net, 2020. URL https://openreview.net/forum?id=
+S1gmrxHFvB.
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter.
+Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Isabelle
+11
+
+Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vish-
+wanathan, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 30:
+Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long
+Beach, CA, USA, pp. 6626–6637, 2017.
+URL https://proceedings.neurips.cc/
+paper/2017/hash/8a1d694707eb0fefe65871369074926d-Abstract.html.
+Andrew Howard, Ruoming Pang, Hartwig Adam, Quoc V. Le, Mark Sandler, Bo Chen, Weijun
+Wang, Liang-Chieh Chen, Mingxing Tan, Grace Chu, Vijay Vasudevan, and Yukun Zhu. Search-
+ing for mobilenetv3. In 2019 IEEE/CVF International Conference on Computer Vision, ICCV
+2019, Seoul, Korea (South), October 27 - November 2, 2019, pp. 1314–1324. IEEE, 2019. doi:
+10.1109/ICCV.2019.00140. URL https://doi.org/10.1109/ICCV.2019.00140.
+Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected
+convolutional networks. In 2017 IEEE Conference on Computer Vision and Pattern Recognition,
+CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 2261–2269. IEEE Computer Society,
+2017. doi: 10.1109/CVPR.2017.243. URL https://doi.org/10.1109/CVPR.2017.
+243.
+Minyoung Huh, Pulkit Agrawal, and Alexei A. Efros. What makes imagenet good for transfer
+learning? CoRR, abs/1608.08614, 2016. URL http://arxiv.org/abs/1608.08614.
+Forrest N. Iandola, Matthew W. Moskewicz, Khalid Ashraf, Song Han, William J. Dally, and Kurt
+Keutzer. Squeezenet: Alexnet-level accuracy with 50x fewer parameters and <1mb model size.
+CoRR, abs/1602.07360, 2016. URL http://arxiv.org/abs/1602.07360.
+Chao Jia, Yinfei Yang, Ye Xia, Yi-Ting Chen, Zarana Parekh, Hieu Pham, Quoc Le, Yun-Hsuan
+Sung, Zhen Li, and Tom Duerig. Scaling up visual and vision-language representation learning
+with noisy text supervision. In International Conference on Machine Learning, pp. 4904–4916.
+PMLR, 2021.
+Alexander Ke, William Ellsworth, Oishi Banerjee, Andrew Y Ng, and Pranav Rajpurkar. Chextrans-
+fer: performance and parameter efficiency of imagenet models for chest x-ray interpretation. In
+Proceedings of the Conference on Health, Inference, and Learning, pp. 116–124, 2021.
+Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Joan Puigcerver, Jessica Yung, Sylvain Gelly,
+and Neil Houlsby. Big transfer (bit): General visual representation learning. In European confer-
+ence on computer vision, pp. 491–507. Springer, 2020.
+Simon Kornblith, Jonathon Shlens, and Quoc V Le. Do better imagenet models transfer better? In
+Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 2661–
+2671, 2019.
+Simon Kornblith, Ting Chen, Honglak Lee, and Mohammad Norouzi. Why do better loss functions
+lead to less transferable features? Advances in Neural Information Processing Systems, 34, 2021.
+Klemen Kotar, Gabriel Ilharco, Ludwig Schmidt, Kiana Ehsani, and Roozbeh Mottaghi. Contrasting
+contrastive self-supervised representation learning pipelines. In Proceedings of the IEEE/CVF
+International Conference on Computer Vision, pp. 9949–9959, 2021.
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Tech-
+nical report, University of Toronto, 2009.
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep con-
+volutional neural networks.
+In Peter L. Bartlett, Fernando C. N. Pereira, Christopher J. C.
+Burges, L´eon Bottou, and Kilian Q. Weinberger (eds.), Advances in Neural Information Pro-
+cessing Systems 25: 26th Annual Conference on Neural Information Processing Systems 2012.
+Proceedings of a meeting held December 3-6, 2012, Lake Tahoe, Nevada, United States, pp.
+1106–1114, 2012. URL https://proceedings.neurips.cc/paper/2012/hash/
+c399862d3b9d6b76c8436e924a68c45b-Abstract.html.
+Ananya Kumar, Aditi Raghunathan, Robbie Matthew Jones, Tengyu Ma, and Percy Liang. Fine-
+tuning can distort pretrained features and underperform out-of-distribution. In International Con-
+ference on Learning Representations, 2022.
+URL https://openreview.net/forum?
+id=UYneFzXSJWh.
+12
+
+Chenxi Liu, Barret Zoph, Maxim Neumann, Jonathon Shlens, Wei Hua, Li-Jia Li, Li Fei-Fei, Alan L.
+Yuille, Jonathan Huang, and Kevin Murphy. Progressive neural architecture search. In Vittorio
+Ferrari, Martial Hebert, Cristian Sminchisescu, and Yair Weiss (eds.), Computer Vision - ECCV
+2018 - 15th European Conference, Munich, Germany, September 8-14, 2018, Proceedings, Part
+I, volume 11205 of Lecture Notes in Computer Science, pp. 19–35. Springer, 2018. doi: 10.1007/
+978-3-030-01246-5\ 2. URL https://doi.org/10.1007/978-3-030-01246-5_2.
+Zhuang Liu, Hanzi Mao, Chao-Yuan Wu, Christoph Feichtenhofer, Trevor Darrell, and Saining Xie.
+A convnet for the 2020s. CoRR, abs/2201.03545, 2022. URL https://arxiv.org/abs/
+2201.03545.
+Ningning Ma, Xiangyu Zhang, Hai-Tao Zheng, and Jian Sun. Shufflenet V2: practical guidelines
+for efficient CNN architecture design. In Vittorio Ferrari, Martial Hebert, Cristian Sminchis-
+escu, and Yair Weiss (eds.), Computer Vision - ECCV 2018 - 15th European Conference, Mu-
+nich, Germany, September 8-14, 2018, Proceedings, Part XIV, volume 11218 of Lecture Notes
+in Computer Science, pp. 122–138. Springer, 2018. doi: 10.1007/978-3-030-01264-9\ 8. URL
+https://doi.org/10.1007/978-3-030-01264-9_8.
+Dhruv Mahajan, Ross Girshick, Vignesh Ramanathan, Kaiming He, Manohar Paluri, Yixuan Li,
+Ashwin Bharambe, and Laurens Van Der Maaten. Exploring the limits of weakly supervised
+pretraining. In Proceedings of the European conference on computer vision (ECCV), pp. 181–
+196, 2018.
+Makerere University AI Lab.
+Cassava leaf disease classification, 2021.
+URL https://
+www.kaggle.com/competitions/cassava-leaf-disease-classification/
+overview.
+Thomas Mensink, Jasper Uijlings, Alina Kuznetsova, Michael Gygli, and Vittorio Ferrari. Factors
+of influence for transfer learning across diverse appearance domains and task types. IEEE Trans-
+actions on Pattern Analysis and Machine Intelligence, pp. 1–1, 2021. doi: 10.1109/TPAMI.2021.
+3129870.
+John Miller, Rohan Taori, Aditi Raghunathan, Shiori Sagawa, Pang Wei Koh, Vaishaal Shankar,
+Percy Liang, Yair Carmon, and Ludwig Schmidt. Accuracy on the line: on the strong correla-
+tion between out-of-distribution and in-distribution generalization. In Marina Meila and Tong
+Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, ICML
+2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Re-
+search, pp. 7721–7735. PMLR, 2021. URL http://proceedings.mlr.press/v139/
+miller21b.html.
+Raoof Naushad, Tarunpreet Kaur, and Ebrahim Ghaderpour. Deep transfer learning for land use
+and land cover classification: A comparative study. Sensors, 21(23):8083, 2021. doi: 10.3390/
+s21238083. URL https://doi.org/10.3390/s21238083.
+Niv Nayman, Avram Golbert, Asaf Noy, Tan Ping, and Lihi Zelnik-Manor. Diverse imagenet models
+transfer better. arXiv preprint arXiv:2204.09134, 2022.
+Wei Ouyang, Casper F. Winsnes, Martin Hjelmare, Anthony J. Cesnik, Lovisa ˚Akesson, Hao Xu,
+Devin P. Sullivan, Shubin Dai, Jun Lan, Park Jinmo, Shaikat M. Galib, Christof Henkel, Kevin
+Hwang, Dmytro Poplavskiy, Bojan Tunguz, Russel D. Wolfinger, Yinzheng Gu, Chuanpeng Li,
+Jinbin Xie, Dmitry Buslov, Sergei Fironov, Alexander Kiselev, Dmytro Panchenko, Xuan Cao,
+Runmin Wei, Yuanhao Wu, Xun Zhu, Kuan-Lun Tseng, Zhifeng Gao, Cheng Ju, Xiaohan Yi,
+Hongdong Zheng, Constantin Kappel, and Emma Lundberg. Analysis of the human protein at-
+las image classification competition. Nature Methods, 16(12):1254–1261, 2019. doi: 10.1038/
+s41592-019-0658-6. URL https://doi.org/10.1038/s41592-019-0658-6.
+Ian Pan.
+[2nd place] solution overview, 2020.
+URL https://www.kaggle.com/
+competitions/siim-isic-melanoma-classification/discussion/
+175324.
+Jinmo Park.
+3rd place solution with code., 2019.
+URL https://www.kaggle.com/c/
+human-protein-atlas-image-classification/discussion/77320.
+13
+
+Hieu Pham, Zihang Dai, Golnaz Ghiasi, Hanxiao Liu, Adams Wei Yu, Minh-Thang Luong, Mingx-
+ing Tan, and Quoc V Le.
+Combined scaling for zero-shot transfer learning.
+arXiv preprint
+arXiv:2111.10050, 2021.
+Jean Ponce, Tamara L Berg, Mark Everingham, David A Forsyth, Martial Hebert, Svetlana Lazeb-
+nik, Marcin Marszalek, Cordelia Schmid, Bryan C Russell, Antonio Torralba, et al. Dataset issues
+in object recognition. In Toward category-level object recognition, pp. 29–48. Springer, 2006.
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal,
+Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual
+models from natural language supervision. In International Conference on Machine Learning,
+pp. 8748–8763. PMLR, 2021.
+Maithra Raghu, Chiyuan Zhang, Jon Kleinberg, and Samy Bengio. Transfusion: Understanding
+transfer learning for medical imaging. Advances in neural information processing systems, 32,
+2019.
+Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal Shankar. Do imagenet classifiers
+generalize to imagenet?
+In International Conference on Machine Learning, pp. 5389–5400.
+PMLR, 2019.
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng
+Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual
+recognition challenge. International journal of computer vision, 115(3):211–252, 2015.
+Hadi Salman, Andrew Ilyas, Logan Engstrom, Ashish Kapoor, and Aleksander Madry. Do adver-
+sarially robust imagenet models transfer better?
+Advances in Neural Information Processing
+Systems, 33:3533–3545, 2020.
+Vaishaal Shankar, Rebecca Roelofs, Horia Mania, Alex Fang, Benjamin Recht, and Ludwig
+Schmidt. Evaluating machine accuracy on imagenet. In International Conference on Machine
+Learning, pp. 8634–8644. PMLR, 2020.
+Ali Sharif Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson. Cnn features off-
+the-shelf: an astounding baseline for recognition. In Proceedings of the IEEE conference on
+computer vision and pattern recognition workshops, pp. 806–813, 2014.
+Maxim
+Shugaev.
+Pretrained
+resnet34
+with
+rgby
+(0.460
+public
+lb),
+2019.
+URL
+https://www.kaggle.com/code/iafoss/
+pretrained-resnet34-with-rgby-0-460-public-lb/notebook.
+SIIM and ISIC. Siim-isic melanoma classification, 2020. URL https://www.kaggle.com/
+competitions/siim-isic-melanoma-classification/overview.
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image
+recognition. In Yoshua Bengio and Yann LeCun (eds.), 3rd International Conference on Learning
+Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceed-
+ings, 2015. URL http://arxiv.org/abs/1409.1556.
+Andreas Steiner, Alexander Kolesnikov, Xiaohua Zhai, Ross Wightman, Jakob Uszkoreit, and Lucas
+Beyer. How to train your vit? data, augmentation, and regularization in vision transformers.
+CoRR, abs/2106.10270, 2021. URL https://arxiv.org/abs/2106.10270.
+Chen Sun, Abhinav Shrivastava, Saurabh Singh, and Abhinav Gupta. Revisiting unreasonable ef-
+fectiveness of data in deep learning era. In Proceedings of the IEEE international conference on
+computer vision, pp. 843–852, 2017.
+Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A. Alemi.
+Inception-v4,
+inception-resnet and the impact of residual connections on learning. In Satinder Singh and Shaul
+Markovitch (eds.), Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence,
+February 4-9, 2017, San Francisco, California, USA, pp. 4278–4284. AAAI Press, 2017a. URL
+http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14806.
+14
+
+Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A. Alemi.
+Inception-v4,
+inception-resnet and the impact of residual connections on learning. In Satinder Singh and Shaul
+Markovitch (eds.), Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence,
+February 4-9, 2017, San Francisco, California, USA, pp. 4278–4284. AAAI Press, 2017b. URL
+http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14806.
+Mingxing Tan and Quoc V. Le. Efficientnet: Rethinking model scaling for convolutional neural
+networks. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th
+International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, Cali-
+fornia, USA, volume 97 of Proceedings of Machine Learning Research, pp. 6105–6114. PMLR,
+2019. URL http://proceedings.mlr.press/v97/tan19a.html.
+Rohan Taori, Achal Dave, Vaishaal Shankar, Nicholas Carlini, Benjamin Recht, and Ludwig
+Schmidt. Measuring robustness to natural distribution shifts in image classification. Advances
+in Neural Information Processing Systems, 33:18583–18599, 2020.
+Antonio Torralba and Alexei A Efros. Unbiased look at dataset bias. In CVPR 2011, pp. 1521–1528.
+IEEE, 2011.
+Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and
+Herv´e J´egou. Training data-efficient image transformers & distillation through attention. In Ma-
+rina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine
+Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine
+Learning Research, pp. 10347–10357. PMLR, 2021. URL http://proceedings.mlr.
+press/v139/touvron21a.html.
+Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Andrew Ilyas, and Aleksander Madry. From
+imagenet to image classification: Contextualizing progress on benchmarks. In International Con-
+ference on Machine Learning, pp. 9625–9635. PMLR, 2020.
+Lukas Tuggener, J¨urgen Schmidhuber, and Thilo Stadelmann. Is it enough to optimize cnn architec-
+tures on imagenet? arXiv preprint arXiv:2103.09108, 2021.
+Ross Wightman, Hugo Touvron, and Herv´e J´egou.
+Resnet strikes back: An improved training
+procedure in timm. CoRR, abs/2110.00476, 2021. URL https://arxiv.org/abs/2110.
+00476.
+Qizhe Xie, Minh-Thang Luong, Eduard Hovy, and Quoc V Le. Self-training with noisy student
+improves imagenet classification. In Proceedings of the IEEE/CVF conference on computer vision
+and pattern recognition, pp. 10687–10698, 2020.
+Saining Xie, Ross B. Girshick, Piotr Doll´ar, Zhuowen Tu, and Kaiming He. Aggregated resid-
+ual transformations for deep neural networks. In 2017 IEEE Conference on Computer Vision
+and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 5987–5995.
+IEEE Computer Society, 2017. doi: 10.1109/CVPR.2017.634. URL https://doi.org/10.
+1109/CVPR.2017.634.
+Guanshuo Xu.
+1st place solution summary, 2019.
+URL https://www.kaggle.com/
+competitions/aptos2019-blindness-detection/discussion/108065.
+H. Yassine, K. Tout, and M. Jaber.
+Improving Lulc Classification from Satellite Imagery Us-
+ing Deep Learning - Eurosat Dataset.
+ISPRS - International Archives of the Photogram-
+metry, Remote Sensing and Spatial Information Sciences, 43B3:369–376, June 2021.
+doi:
+10.5194/isprs-archives-XLIII-B3-2021-369-2021.
+Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui
+Wu.
+Coca:
+Contrastive captioners are image-text foundation models.
+arXiv preprint
+arXiv:2205.01917, 2022.
+Amir R Zamir, Alexander Sax, William Shen, Leonidas J Guibas, Jitendra Malik, and Silvio
+Savarese. Taskonomy: Disentangling task transfer learning. In Proceedings of the IEEE con-
+ference on computer vision and pattern recognition, pp. 3712–3722, 2018.
+15
+
+Xiaohua Zhai, Joan Puigcerver, Alexander Kolesnikov, Pierre Ruyssen, Carlos Riquelme, Mario
+Lucic, Josip Djolonga, Andre Susano Pinto, Maxim Neumann, Alexey Dosovitskiy, et al. The
+visual task adaptation benchmark. 2019.
+Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, and Lucas Beyer. Scaling vision transformers.
+In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.
+12104–12113, 2022.
+Kevin
+Zheng.
+39th
+solution-attention
+gated
+resnet18
+(single
+model
+with-
+out
+cv),
+2019.
+URL
+https://www.kaggle.com/competitions/
+human-protein-atlas-image-classification/discussion/77637.
+Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data aug-
+mentation. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The
+Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The
+Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New
+York, NY, USA, February 7-12, 2020, pp. 13001–13008. AAAI Press, 2020.
+URL https:
+//ojs.aaai.org/index.php/AAAI/article/view/7000.
+16
+
+Appendix
+A
+DETAILED EXPERIMENT RESULTS
+Table 3: For each ImageNet pre-trained model, we provide the best performing model when fine-tuned on each
+dataset across our hyperparameter grid
+Model
+ImageNet top-1
+CCT20
+APTOS
+HPA
+Melanoma
+Cassava
+EuroSAT
+AlexNet
+56.5
+63.59
+0.8835
+0.3846
+0.9283
+82.58
+97.93
+SqueezeNet 1.1
+58.2
+66.36
+0.9021
+0.3972
+0.9073
+85.15
+98.07
+ShuffleNetV2x0.5
+60.6
+66.37
+0.9227
+0.5867
+0.9289
+85.64
+98.56
+MobileNet V3 small
+67.7
+66.01
+0.9230
+0.6108
+0.9455
+85.81
+99.15
+ShuffleNetV2x1.0
+69.4
+69.27
+0.9202
+0.6202
+0.9418
+87.33
+98.91
+VGG-13 BN
+71.6
+75.06
+0.9268
+0.6794
+0.9529
+88.99
+98.85
+DeiT-tiny
+72.2
+68.77
+0.9130
+0.5777
+0.9510
+86.25
+99.11
+VGG-16 BN
+73.4
+75.93
+0.9287
+0.6791
+0.9531
+88.45
+98.93
+DenseNet-121
+74.4
+74.66
+0.9287
+0.7019
+0.9514
+87.80
+99.06
+ResNet-50
+76.1
+73.96
+0.9215
+0.6718
+0.9524
+87.75
+99.19
+ResNeXt-50-32x4d
+77.6
+73.73
+0.9212
+0.6906
+0.9588
+88.15
+99.24
+EfficientNet B0
+77.7
+71.02
+0.9195
+0.6942
+0.9456
+87.63
+98.80
+ResNet-152
+78.3
+74.05
+0.9228
+0.6732
+0.9562
+87.75
+99.15
+ViT-B/16
+78.7
+72.07
+0.9262
+0.5852
+0.9600
+86.63
+99.28
+DeiT-small
+79.9
+71.41
+0.9205
+0.6148
+0.9583
+87.19
+99.20
+Inception-ResNet v2
+80.4
+70.68
+0.9168
+0.6882
+0.9483
+87.84
+98.93
+ConvNext-tiny
+82.5
+78.51
+0.9297
+0.6992
+0.9628
+88.89
+99.11
+PNASNet-5 large
+82.9
+75.21
+0.9271
+0.6941
+0.9584
+87.77
+99.17
+EfficientNet B4
+83.4
+73.49
+0.9211
+0.6954
+0.9552
+88.36
+98.70
+See the following link for experiment results across hyperparameters: https://docs.google.
+com/spreadsheets/d/1aDeuTH0V1Kid_JMRUt3sF1N76LUCAMDQ007Ykjo3Z4U/
+edit?usp=sharing.
+17
+
+B
+MAIN FIGURE VARIATIONS
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+64
+66
+68
+70
+72
+74
+76
+78
+Accuracy
+Caltech Camera Traps 20
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.88
+0.90
+0.92
+0.94
+Quadratic weighted kappa
+APTOS 2019 Blindness
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Macro F1 score
+Human Protein Atlas
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+82
+84
+86
+88
+90
+Accuracy
+Cassava Leaf Disease
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+97.50
+97.75
+98.00
+98.25
+98.50
+98.75
+99.00
+99.25
+99.50
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+ViT-B/16
+Figure 4: Figure 1 with error bars. Green is linear trend of all models, while blue is linear trend for models
+above 70% ImageNet accuracy. We use 95% confidence intervals computed with Clopper-Pearson for accuracy
+metrics and bootstrap with 10,000 trials for other metrics.
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+62.5
+65.0
+67.5
+70.0
+72.5
+75.0
+77.5
+80.0
+Accuracy
+Caltech Camera Traps 20
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.88
+0.90
+0.92
+0.94
+Quadratic weighted kappa
+APTOS 2019 Blindness
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Macro F1 score
+Human Protein Atlas
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.89
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+82
+84
+86
+88
+90
+Accuracy
+Cassava Leaf Disease
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+97.50
+97.75
+98.00
+98.25
+98.50
+98.75
+99.00
+99.25
+99.50
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+ViT-B/16
+Figure 5: Figure 4 with spline interpolation fits instead of linear fits.
+18
+
+C
+EXPERIMENT SETUP
+C.1
+MODELS
+Table 4: We examine the effectiveness of transfer learning from a number of models pretrained on ImageNet,
+including both CNNs and Vision Transformers.
+Model
+ImageNet top-1
+# params
+Year Released
+AlexNet (Krizhevsky et al., 2012)
+56.5
+61M
+2012
+SqueezeNet 1.1 (Iandola et al., 2016)
+58.2
+1.2M
+2016
+ShuffleNetV2x0.5 (Ma et al., 2018)
+60.6
+1.4M
+2018
+MobileNet V3 small (Howard et al., 2019)
+67.7
+2.5M
+2019
+ShuffleNetV2x1.0 (Ma et al., 2018)
+69.4
+2.3M
+2018
+VGG-13 BN (Simonyan & Zisserman, 2015)
+71.6
+133M
+2014/2015
+DeiT-tiny (Touvron et al., 2021)
+72.2
+5.7M
+2020
+VGG-16 BN (Simonyan & Zisserman, 2015)
+73.4
+138M
+2014/2015
+DenseNet-121 (Huang et al., 2017)
+74.4
+8.0M
+2016
+ResNet-50 (He et al., 2016)
+76.1
+26M
+2015
+ResNeXt-50-32x4d (Xie et al., 2017)
+77.6
+25M
+2016
+EfficientNet B0 (Tan & Le, 2019)
+77.7
+5.3M
+2019
+ResNet-152 (He et al., 2016)
+78.3
+60M
+2015
+ViT-B/16 (Dosovitskiy et al., 2021a; Steiner et al., 2021)
+78.7
+304M
+2020
+DeiT-small (Touvron et al., 2021)
+79.9
+22M
+2020
+Inception-ResNet v2 (Szegedy et al., 2017a)
+80.4
+56M
+2016
+ConvNext-tiny (Liu et al., 2022)
+82.5
+29M
+2022
+PNASNet-5 large (Liu et al., 2018)
+82.9
+86M
+2017
+EfficientNet B4 (Tan & Le, 2019)
+83.4
+19M
+2019
+We examine 19 model architectures in this work that cover a diverse range of accuracies on ImageNet
+in order to observe the relationship between ImageNet performance and target dataset performance.
+In addition to the commonly used CNNs, we also include data-efficient image transformers (DeiT)
+due to the recent increase in usage of Vision Transformers. Additional model details are in Table 4.
+C.2
+HYPERPARAMETER GRID
+Hyperparameter tuning is a key part of neural network training, as using suboptimal hyperparameters
+can lead to suboptimal performance. Furthermore, the correct hyperparameters vary across both
+models and training data. To get the best performance out of each model, we train each model
+on AdamW with a cosine decay learning rate schedule, SGD with a cosine decay learning rate
+schedule, and SGD with a multi-step decay learning rate schedule. We also grid search for optimal
+initial learning rate and weight decay combinations, searching logarithmically between 10−1 to
+10−4 for SGD learning rate, 10−2 to 10−5 for AdamW learning rate, and 10−3 to 10−6 as well as
+0 for weight decay. All models are pretrained on ImageNet and then fine-tuned on the downstream
+task. Additional training details for each dataset can be found in Appendix D. We also run our
+hyperparameter grid on CIFAR-10 in Appendix E to verify that we find a strong relationship between
+ImageNet and CIFAR-10 accuracy as previously reported by Kornblith et al. (2019).
+D
+TRAINING DETAILS BY DATASET (IMAGENET MODELS)
+Experiments on Cassava Leaf Disease, SIIM-ISIC Melanoma, and EuroSAT datasets were ran on
+TPU v2-8s, while all other datasets were ran on NVIDIA A40s.
+All experiments were ran with mini-batch size of 128.
+For SGD experiments, we use Nesterov momentum, set momentum to 0.9, and try learning rates of
+1e-1, 1e-2, 1e-3, and 1e-4. For AdamW experiments, we try learning rates of 1e-2, 1e-3, 1e-4, 1e-5.
+For all experiments, we try weight decays of 1e-3, 1e-4, 1e-5, 1e-6, and 0.
+For all experiments, we use weights that are pretrained on ImageNet. AlexNet, DenseNet, Mo-
+bileNet, ResNet, ResNext, ShuffleNet, SqueezeNet and VGG models are from torchvision, while
+ConvNext, DeiT, EfficientNet, InceptionResNet, and PNASNet models are from timm. Addition-
+ally, we normalize images to ImageNet’s mean and standard deviation.
+For EuroSAT we random resize crop to 224 with area at least 0.65.
+For all other datasets, we random resize crop with area at least 0.65 to 224 for DeiT models, and 256
+for all other models. Additionally, we use horizontal flips. For Human Protein Atlas, Cassava Leaf
+Disease, and SIIM-ISIC Melanoma, we also use vertical flips.
+19
+
+For SIIM-ISIC Melanoma, we train for 10 epochs, and for the step scheduler decay with factor 0.1
+at 5 epochs.
+For all other datasets, we train for 30 epochs, and for the step scheduler decay with factor 0.1 at 15,
+20, and 25 epochs.
+E
+CIFAR-10 ON HYPERPARAMETER GRID
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+93
+94
+95
+96
+97
+98
+99
+Accuracy
+CIFAR-10
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+Figure 6: Transfer performance across models from ImageNet to CIFAR-10. Green linear trend is computed
+across all models, while blue linear trend is restricted to models above 70% ImageNet accuracy. We use 95%
+confidence intervals computed with Clopper-Pearson.
+F
+APTOS 2019 BLINDNESS DETECTION ABLATIONS
+Scores presented are submissions to the Kaggle leaderboard. All scores are evaluated with quadratic
+weighted kappa. Within each entry, we first present the private leaderboard score, then the pub-
+lic leaderboard score. The private leaderboard represents 85% of the test data, while the public
+leaderboard is the remaining 15%.
+Models used here are trained using AdamW with a cosine scheduler. We random resize crop to 512,
+use random rotations, and use color jitter (brightness=0.2, contrast=0.2, saturation=0.2, hue=0.1).
+We train on all the available training data, no longer using the local train/validation split mentioned
+in the main text. This includes both the training data in the 2019 competition, as well as data from a
+prior 2015 diabetic retinopathy competition.
+Table 5: Comparing various models with additional interventions by evaluating on the Kaggle leaderboard.
+lr \wd
+1.00E-04
+1.00E-05
+1.00E-06
+ResNet-50
+1.00E-03
+0.8610 / 0.6317
+0.8570 / 0.6180
+0.8548 / 0.6646
+1.00E-04
+0.8952 / 0.7531
+0.8918 / 0.7204
+0.8961 / 0.7547
+ResNet-152
+1.00E-03
+0.8658 / 0.6812
+0.8686 / 0.6612
+0.8640 / 0.6554
+1.00E-04
+0.8898 / 0.7164
+0.8836 / 0.6946
+0.8859 / 0.6947
+Inception-Resnet-v2
+1.00E-03
+0.8933 / 0.7748
+0.8905 / 0.7565
+0.8960 / 0.7585
+1.00E-04
+0.8897 / 0.7210
+0.8929 / 0.7420
+0.8944 / 0.7439
+20
+
+Table 6: Comparing the effect of augmentation on Kaggle leaderboard scores. More augmentation is as de-
+scribed earlier in this section. Less augmentation only uses random resize crop with at least 0.65 area and
+horizontal flips.
+lr \wd
+1.00E-04
+1.00E-05
+1.00E-06
+ResNet-50
+less aug
+1.00E-03
+0.8669 / 0.6405
+0.8520 / 0.6013
+0.8613 / 0.6269
+1.00E-04
+0.8525 / 0.6115
+0.8570 / 0.6431
+0.8483 / 0.6147
+1.00E-05
+0.8186 / 0.5071
+0.8287 / 0.5647
+0.8288 / 0.5328
+ResNet-50
+more aug
+1.00E-03
+0.8440 / 0.6432
+0.8547 / 0.6856
+0.8524 / 0.7125
+1.00E-04
+0.8948 / 0.7490
+0.8972 / 0.7693
+0.8999 / 0.7758
+1.00E-05
+0.8724 / 0.7370
+0.8685 / 0.7567
+0.8623 / 0.7376
+G
+AUGMENTATION ABLATION DETAILS
+Table 7:
+We examine the effect of pre-training augmentation and fine-tuning augmentation on downstream
+transfer performance. The model specifies the architecture and pre-training augmentation, while each column
+specifies the downstream task and fine-tuning augmentation. We find that augmentation strategies that improve
+ImageNet accuracy do not always improve accuracy on downstream tasks. Pre-trained augmentation models
+are from Wightman et al. (2021).
+Model
+ImageNet
+CCT-20
+CCT-20
+CCT-20
+APTOS
+APTOS
+APTOS
+Acc
+Base Aug
+AugMix
+RandAug
+Base Aug
+AugMix
+RandAug
+ResNet-50
+76.1
+72.02
+72.24
+73.57
+0.9210
+0.9212
+0.9250
+ResNet-50
+77.5
+71.63
+71.53
+72.39
+0.9239
+0.9152
+0.9222
+w/ AugMix
+ResNet-50
+78.8
+72.94
+73.54
+73.76
+0.9190
+0.9204
+0.9302
+w/ RandAug
+Deit-tiny
+72.2
+66.57
+66.47
+66.95
+0.9153
+0.9197
+0.9172
+Deit-small
+79.9
+70.65
+69.72
+70.07
+0.9293
+0.9212
+0.9277
+H
+MELANOMA METRIC COMPARISON
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma ROC
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+93.0
+93.5
+94.0
+94.5
+95.0
+95.5
+96.0
+96.5
+Accuracy
+SIIM-ISIC Melanoma Acc
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+Figure 7: Comparing transfer performance from ImageNet to Melanoma when using different metrics. Green
+linear trend is computed across all models, while blue linear trend is restricted to models above 70% ImageNet
+accuracy. Using accuracy implies that better ImageNet models transfer better; however, ROC is a better metric
+for this task.
+21
+
+I
+CLIP EXPERIMENT DETAILS
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+65
+70
+75
+80
+Accuracy
+Caltech Camera Traps 20
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.88
+0.90
+0.92
+0.94
+Quadratic weighted kappa
+APTOS 2019 Blindness
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Macro F1 score
+Human Protein Atlas
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+0.98
+Area under ROC
+SIIM-ISIC Melanoma
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+82
+84
+86
+88
+90
+Accuracy
+Cassava Leaf Disease
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+97.5
+98.0
+98.5
+99.0
+99.5
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+ViT-B/16
+CLIP-RN50
+CLIP-RN101
+CLIP-B32
+CLIP-B16
+CLIP-L14
+CLIP-L14@336
+Figure 8:
+Figure 4 with CLIP models overlaid (purple stars). The best CLIP models do better than all the
+ImageNet models, but when looking across all CLIP models, the patterns are more complicated.
+Table 8: For each CLIP pre-trained model, we provide the best performing model when fine-tuned on each
+dataset across our LP-FT hyperparameter grid
+Model
+ImageNet top-1
+CCT20
+APTOS
+HPA
+Melanoma
+Cassava
+EuroSAT
+CLIP-RN50
+73.3
+74.45
+0.9135
+0.7053
+0.9350
+87.89
+98.80
+CLIP-RN101
+75.7
+75.19
+0.9235
+0.6909
+0.9378
+87.68
+99.11
+CLIP-B32
+76.1
+70.57
+0.9137
+0.5338
+0.9546
+86.28
+99.26
+CLIP-B16
+80.2
+77.81
+0.9213
+0.6365
+0.9619
+87.82
+99.24
+CLIP-L14
+83.9
+79.99
+0.9330
+0.6687
+0.9717
+88.82
+99.33
+CLIP-L14@336
+85.4
+83.17
+0.9337
+0.7131
+0.9738
+89.24
+99.48
+Table 9: We directly compare models pre-trained on ImageNet with models pre-trained on OpenAI’s CLIP
+data. Specifically, we look at ResNet 50 and ViT B/16.
+Model
+ImageNet top-1
+CCT20
+APTOS
+HPA
+Melanoma
+Cassava
+EuroSAT
+IN-ResNet-50
+76.1
+73.96
+0.9215
+0.6718
+0.9524
+87.75
+99.19
+CLIP-RN50
+73.3
+74.45
+0.9135
+0.7053
+0.9350
+87.89
+98.80
+IN-ViT-B/16
+78.7
+72.07
+0.9262
+0.5852
+0.9600
+86.63
+99.28
+CLIP-B16
+80.2
+77.81
+0.9213
+0.6365
+0.9619
+87.82
+99.24
+J
+CLIP FINE-TUNING DETAILS
+We fine-tune by running a linear probe, followed by end-to-end fine-tuning on the best model from
+the first part. We keep total epochs consistent with the previous models, with a third of the epochs
+going toward linear probing. We use AdamW with a cosine decay schedule. During the linear probe,
+we search over 10−1, 10−2, and 10−3 learning rates, and during fine-tuning, we search over 10−4,
+10−5, and 10−6 learning rates. For both parts, we search over 10−3 to 10−6 and 0 for weight decay.
+22
+
+K
+CREATION INFORMATION FOR DATASETS STUDIED IN KORNBLITH ET AL.
+(2019)
+Table 10: We find that the 12 datasets studied in Kornblith et al. (2019) come from web scraping.
+Dataset
+Origin
+Additional information
+Food-101
+foodspotting.com
+Users upload an image of their food and anno-
+tate the type of food; categories chosen by pop-
+ularity
+CIFAR-10
+TinyImages
+Web crawl
+CIFAR-100
+TinyImages
+Web crawl
+Birdsnap
+Flickr
+Also used MTurk
+SUN397
+Web search engines
+Also used WordNet
+Stanford Cars
+Flickr,
+Google,
+Bing
+Also used MTurk
+FGVC Aircraft
+airliners.net
+Images taken by 10 photographers
+Pascal VOC 2007 Cls.
+Flickr
+N/A
+Describable Textures
+Google and Flickr
+Also used MTurk
+Oxford-IIT Pets
+Flickr, Google,
+Catster, Dogster
+Catster and Dogster are social websites for col-
+lecting and discussing pet images
+Caltech-101
+Google
+97 categories chosen from Webster Collegiate
+Dictionary categories associated with a drawing
+Oxford 102 Flowers
+Mostly collected
+from web
+A small number of images acquired by the pa-
+per authors taking the pictures
+L
+RELATIONSHIP BETWEEN MODEL SIZE AND TRANSFER PERFORMANCE
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+62.5
+65.0
+67.5
+70.0
+72.5
+75.0
+77.5
+80.0
+Accuracy
+Caltech Camera Traps 20
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+0.88
+0.90
+0.92
+0.94
+Quadratic weighted kappa
+APTOS 2019 Blindness
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+0.4
+0.5
+0.6
+0.7
+0.8
+Macro F1 score
+Human Protein Atlas
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+82
+84
+86
+88
+90
+Accuracy
+Cassava Leaf Disease
+0
+20
+40
+60
+80
+100
+120
+140
+# of parameters (in millions)
+97.50
+97.75
+98.00
+98.25
+98.50
+98.75
+99.00
+99.25
+99.50
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+Figure 9: We compare model size with downstream transfer performance. Again we use separate trend lines
+for all models (green) and only those above 70% ImageNet accuracy (blue). We use 95% confidence intervals
+computed with Clopper-Pearson for accuracy metrics and bootstrap with 10,000 trials for other metrics.
+23
+
+M
+FID SCORE DETAILS
+Table 11: We calculate FID scores between the ImageNet validation set and each of the datasets we study, as
+well as between the ImageNet validation set and each of the datasets in Kornblith et al. (2019). We found that
+dataset size affects FID score, so we take a 3,662 subset of each downstream dataset. Note that 3,662 is the size
+of APTOS, which is the smallest dataset.
+Dataset
+FID
+CCT-20
+162.69
+APTOS
+196.24
+HPA
+230.70
+Cassava
+179.24
+Melanoma
+186.34
+EuroSAT
+151.85
+Food-101
+108.35
+CIFAR-10
+132.53
+CIFAR-100
+120.72
+Birdsnap
+94.08
+SUN397
+62.95
+Stanford Cars
+143.35
+FGVC Aircraft
+183.35
+Pascal VOC 2007 Cls.
+39.84
+Describable Textures
+89.13
+Oxford-IIT Pets
+77.27
+Caltech-101
+50.77
+Oxford 102 Flowers
+140.21
+N
+PREDICTIVE POWER OF ACCURACY ON NON-WEB-SCRAPED DATASETS ON
+NOVEL DATASETS
+We observe that, on many non-web-scraped datasets, accuracy correlates only weakly with Ima-
+geNet accuracy. It is thus worth asking whether other predictors might correlate better. In this
+section, we examine the extent to which accuracy on a given non-web-scraped target dataset can be
+predicted from the accuracy on the other non-web-scraped target datasets.
+N.1
+F-TEST
+We can further measure the extent to which the averages of the five other datasets beyond the pre-
+dictive power provided by ImageNet by using F-tests. For each target task, we fit a linear regression
+model that predicts accuracy as either ImageNet accuracy or the average accuracy on the other five
+non-web-scraped datasets, and a second linear regression model that predicts accuracy as a func-
+tion of both ImageNet accuracy and the average accuracy on the other five datasets. Since the first
+model is nested within the second, the second model must explain at least as much variance as the
+first. The F-test measures whether the increase in explained variance is significant. For these ex-
+periments, we logit-transform accuracy values and standardize them to zero mean and unit variance
+before computing the averages, as in the middle column of Table 13.
+Results are shown in Table 12. The average accuracy across the other five datasets explains variance
+beyond that explained by ImageNet accuracy alone on five of the six datasets. The only exception
+is EuroSAT, where the range of accuracies is low (most models get ∼99%) and a significant fraction
+of the variance among models may correspond to noise. By contrast, ImageNet accuracy explains
+variance beyond the average accuracy only on two datasets (APTOS and Melanoma). These results
+indicate that there are patterns in how well different models transfer to non-web-scraped data that
+are not captured by ImageNet accuracy alone, but are captured by the accuracy on other non-web-
+scraped datasets.
+24
+
+Table 12: Results of the F-test described in Section N.1. “+Avg. across datasets” tests whether a model that
+includes both ImageNet accuracy and the average accuracy across the 5 other datasets explains more variance
+than a model that includes only ImageNet accuracy. “+ImageNet” tests whether a model that includes both
+predictors explains more variance than a model that includes only the average accuracy across the 5 other
+datasets. In addition to F and p values, we report adjusted R2 for all models. p-values < 0.05 are bold-faced.
++Avg. across datasets
++ImageNet
+Dataset
+F (1, 16)
+p-value
+F (1, 16)
+p-value
+Adj. R2
+(ImageNet-only)
+Adj. R2
+(Average-only)
+Adj. R2
+(Both predictors)
+CCT-20
+8.2
+0.01
+0.69
+0.42
+0.56
+0.70
+0.69
+APTOS
+31.0
+0.00004
+4.6
+0.047
+0.34
+0.71
+0.76
+HPA
+11.8
+0.003
+0.84
+0.37
+0.60
+0.76
+0.76
+Melanoma
+5.8
+0.03
+7.8
+0.01
+0.74
+0.71
+0.79
+Cassava
+13.2
+0.002
+0.14
+0.71
+0.55
+0.75
+0.74
+EuroSAT
+2.9
+0.11
+0.72
+0.41
+0.43
+0.52
+0.49
+N.2
+SPEARMAN CORRELATION
+Table 13: We measure the Spearman correlation between each dataset with either the average of the 5 other
+datasets we study, or with ImageNet. Normalization is done by logit transforming accuracies, and then stan-
+dardizing to zero mean and unit variance. The results suggest that using additional datasets is more predictive
+of model performance than just using ImageNet.
+Avg of 5 others
+(unnormalized)
+Avg of 5 others
+(normalized)
+ImageNet
+Dataset
+ρ
+p-value
+ρ
+p-value
+ρ
+p-value
+CCT-20
+0.8684
+0.0000
+0.9263
+0.0000
+0.5825
+0.0089
+APTOS
+0.7205
+0.0005
+0.6950
+0.0010
+0.3010
+0.2105
+HPA
+0.7351
+0.0003
+0.6825
+0.0013
+0.6491
+0.0026
+Melanoma
+0.6561
+0.0023
+0.7807
+0.0000
+0.7667
+0.0001
+Cassava
+0.8872
+0.0000
+0.7442
+0.0003
+0.5222
+0.0218
+EuroSAT
+0.3030
+0.2073
+0.3821
+0.1065
+0.4734
+0.0406
+25
+
+O
+PRE-TRAINING AUGMENTATION DETAILS
+Table 14: For each ImageNet pre-trained model, we provide the augmentation strategy used during pre-training
+time.
+Model
+Augmentation
+AlexNet
+Resize + Crop + Flip
+SqueezeNet 1.1
+Resize + Crop + Flip
+ShuffleNetV2x0.5
+AutoAugment (TrivialAugmentWide) + RandErasing + MixUp + CutMix
+MobileNet V3 small
+AutoAugment (ImageNet/Default)+ RandErasing
+ShuffleNetV2x1.0
+AutoAugment (TrivialAugmentWide) + RandErasing + MixUp + CutMix
+VGG-13 BN
+Resize + Crop + Flip
+DeiT-tiny
+RandAugment + RandErasing
+VGG-16 BN
+Resize + Crop + Flip
+DenseNet-121
+Resize + Crop + Flip
+ResNet-50
+Resize + Crop + Flip
+ResNeXt-50-32x4d
+Resize + Crop + Flip
+EfficientNet B0
+RandAugment
+ResNet-152
+Resize + Crop + Flip
+ViT-B/16
+RandAugment + MixUp
+DeiT-small
+RandAugment + RandErasing
+Inception-ResNet v2
+Inception Preprocessing (Color Distort + Resize + Crop + Flip)
+ConvNext-tiny
+AutoAugment (TrivialAugmentWide) + RandErasing + MixUp + CutMix
+PNASNet-5 large
+Whiten + Resize + Crop + Flip
+EfficientNet B4
+RandAugment
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+64
+66
+68
+70
+72
+74
+76
+78
+Accuracy
+Caltech Camera Traps 20
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.88
+0.90
+0.92
+0.94
+Quadratic weighted kappa
+APTOS 2019 Blindness
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Macro F1 score
+Human Protein Atlas
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+0.90
+0.91
+0.92
+0.93
+0.94
+0.95
+0.96
+0.97
+Area under ROC
+SIIM-ISIC Melanoma
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+82
+84
+86
+88
+90
+Accuracy
+Cassava Leaf Disease
+55
+60
+65
+70
+75
+80
+85
+ImageNet top-1 accuracy
+97.50
+97.75
+98.00
+98.25
+98.50
+98.75
+99.00
+99.25
+99.50
+Accuracy
+EuroSAT
+AlexNet
+MobileNetV3-small
+VGG-13 BN
+DeiT-tiny
+ResNet-50
+ResNet-152
+DeiT-small
+PNASNet-5
+Inception-ResNet v2
+VGG-16 BN
+EfficientNet B0
+EfficientNet B4
+DenseNet-121
+ResNeXt-50-32x4d
+ShuffleNetV2x1.0
+ConvNext-tiny
+ShuffleNetV2x0.5
+SqueezeNet 1.1
+ViT-B/16
+Figure 10: Figure 1 with points colored by general pre-training augmentation strategy. Cyan points use simple
+augmentation (resize, crops, flips, etc.), and red points use automatic augmentation (RandAugment, AutoAug-
+ment, TrivialAugmentWide).
+26
+

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+arXiv:2301.01005v1  [math.AG]  3 Jan 2023
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+ANANYO DAN AND INDER KAUR
+Abstract. In this article we study the (cohomological) Hodge conjecture for singular varieties.
+We prove the conjecture for simple normal crossing varieties that can be embedded in a family
+where the Mumford-Tate group remains constant.
+We show how to produce such families.
+Furthermore, we show for varieties with worse singularities the conjecture can be expressed
+solely in terms of the algebraic classes.
+1. Introduction
+The underlying field will always be C.
+Recall, the classical Hodge conjecture claims that
+given a smooth projective variety X, every (rational) Hodge class in X is the cohomology class
+of an algebraic cycle in X. The conjecture is known in some cases (see [20, 32] for a survey
+of known results and [6, 30] for related results), but is open in general. A typical strategy has
+been to consider smooth, projective low dimensional varieties that are birational to already
+known cases. This is primarily because the exceptional divisors arising from the resolution of
+the indeterminacy locus satisfy the Hodge conjecture. However, this strategy fails in higher
+dimension. Another approach is to consider families of varieties (e.g. in the case of abelian
+varieties) and then use a Noether-Lefschetz-type argument to conclude that the Hodge classes
+in a very general fiber in the family are powers of the first Chern class of a line bundle. This
+implies the Hodge conjecture for a very general fiber. In this article, we combine ideas from
+both these approaches.
+It is well-known that any smooth projective variety X is birational to a hypersurface Xhyp in
+a projective space. This hypersurface Xhyp is almost always singular. Note that there is homo-
+logical version of the Hodge conjecture for singular varieties given by Jannsen [13, Conjecture
+7.2] (see also [18]). He proved that the classical Hodge conjecture is equivalent to the singular
+version (see [13, Theorem 7.9], see also [19]). Therefore, proving the singular Hodge conjecture
+for Xhyp would imply the Hodge conjecture for X.
+In the present article, we give a cohomological formulation of the Hodge conjecture for singular
+varieties. There are obvious reasons why this interpretation has so far been unexplored. Firstly
+for X singular, the classical Chow group is not compatible with pull-back morphisms. In [9,
+Chapter 17] (see also [10, Proposition 4]), Fulton and MacPherson developed the operational
+Chow group, denoted Ap(X) which is compatible with pull-back morphisms and for smooth
+varieties coincides with the classical Chow group. However, even for the operational Chow group,
+we know by [29] that in general, there is no map Ap(X) → H2p(X, Q) with good properties.
+Date: January 4, 2023.
+2010 Mathematics Subject Classification. 14C15, 14C30, 32S35, 32G20, 14D07, 14C05.
+Key words and phrases. Hodge conjecture, Limit mixed Hodge structures, Operational Chow group, Cycle
+class map, flag Hilbert schemes, singular varieties.
+A.D. is funded by EPSRC grant number EP/T019379/1. I. K. was funded by the DFG, TRR 326 Geometry
+and Arithmetic of Uniformized Structures, project number 444845124 and is currently funded by EPSRC grant
+number EP/W026554/1.
+1
+
+2
+A. DAN AND I. KAUR
+Nevertheless, by the work of Bloch-Gillet-Soul´e (see [2]) there is a (functorial) cycle class map:
+clp : Ap(X) ⊗ Q → GrW
+2pH2p(X, Q).
+Using this we formulate the cohomological singular Hodge conjecture as follows:
+Singular Hodge conjecture. Let X be a projective variety such that the dimension of the
+singular locus is at most p − 1. Then, the image of the cycle class map clp coincides with
+H2p
+Hdg(X) := GrW
+2pH2p(X, Q) ∩ F pGrW
+2pH2p(X, C).
+If X is of dimension n and the above conjecture holds for X, then we say that X satisfies
+SHC(p, n). Of course, if X is non-singular then the singular Hodge conjecture is the same as
+the classical Hodge conjecture. In this case, we say that X satisfies HC(p, n). The Lefschetz
+(1, 1)-theorem implies HC(1, n) holds true, for any n.
+Recall, a very general hypersurface of any dimension satisfies the Hodge conjecture (as the
+cohomology ring is generated by the class of the hyperplane section). Therefore we can always
+embed Xhyp in a one parameter family of hypersurfaces such that a general fibre satisfies the
+Hodge conjecture. One then expects that the Hodge classes on Xhyp “spread out” to Hodge
+classes in the family. Since a general member of the family satisfies the Hodge conjecture, we
+know that the Hodge class away from the centre is the cohomology class of an algebraic cycle.
+By the simple operation of taking closure, one can then extend the algebraic cycles on the
+general fiber to the central fiber. One needs to check that the cohomology class of this “new”
+algebraic cycle on the central fiber coincides with the Hodge class we started with. However,
+there are several technical problems. Heuristically, the specialization map is not injective and
+hence Hodge classes need not “spread out”. Even if a Hodge class does spread out, it might
+not restrict to a Hodge class on the general fibre! In this article we study these problems and
+give several examples of families of varieties where these problems can be circumvented. Let us
+make this precise.
+Let X be a singular, projective variety of dimension n and π : X → ∆ be a flat family of
+projective varieties, smooth over ∆∗ with the central fiber X. Fix an integer p. Denote by h
+the universal cover for ∆∗ and by X∞ the pull-back of X to h. By Ehresmann’s theorem, for
+every u ∈ h there is an isomorphism of cohomology groups H2p(X∞, Q) and H2p(Xu, Q). The
+natural Hodge filtration on H2p(Xu, Q) induces a filtration F p
+u on H2p(X∞, Q). The limit Hodge
+filtration on H2p(X∞, Q) arises as the limit of this filtration as the imaginary part of u tends to
+∞ (see §2.3 for details). However, there may be rational points H2p(X∞, Q) ∩ F pH2p(X∞, C)
+of the limit Hodge filtration that do not come from the rational points of the filtration F p
+u.
+The Noether-Lefschetz locus gives examples of this phenomena even for smooth families (see
+Example 3.3). As a result, H2p(X∞, Q) may contain more Hodge classes than that on a general
+fiber! This means that although a Hodge class on X0 maps to a Hodge class on X∞ via the
+specialization map, it need not extend to a Hodge class on the family.
+The jump in the rank of the Hodge lattice is captured by Mumford-Tate groups (see §3.1
+for the definition). We call π a Mumford-Tate family if the rank of the Mumford-Tate group
+remains “constant in the limit” (see §3.2 for precise definitions). Moreover, we call a singular,
+projective variety MT-smoothable if it can be embedded as a central fiber of a Mumford-Tate
+family where the general fiber satisfies the Hodge conjecture. We prove the following:
+Theorem 1.1. Let X be a projective variety of dimension 4 with strict normal crossings sin-
+gularities. If X is MT-smoothable, then X satisfies SHC(p, 4) for every p.
+In Theorem 5.2 below, we prove Theorem 1.1 for any dimension. Clearly Theorem 1.1 leads
+to the following questions:
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+3
+• Question 1: How to find Mumford-Tate families?
+• Question 2: Can we generalize Theorem 1.1 to varieties with worse singularities?
+For an exhaustive answer of Question 1 one would need a complete description of the Noether-
+Lefschetz locus for families of hypersurfaces in all dimensions greater than 3. This problem
+is largely open.
+However in §6, we give a general method to obtain Mumford-Tate families
+from known ones using the theory of correspondences. Recall, that given a coherent sheaf E
+on a product of two smooth, projective varieties X × Y , the i-th Chern class of E induces a
+morphism of pure Hodge structures from H2m−k(X) to H2i−k(Y ) for all integers i and k, where
+m = dim(X) (see §6.2). Let us denote such a morphism by Φ(i,k)
+E
+. We say Y is cohomologically
+generated by (X, E) if the cohomology ring H∗(Y ) is generated (as a ring) by the images of
+morphisms of the form Φ(i,k)
+E
+as i and k varies over all integers (see Definition 6.3).
+Note
+that several examples of cohomologically generated varieties appear in existing literature. For
+example, in [23] Mumford and Newstead proved that the moduli space of stable rank 2 bundles
+with odd degree determinant over a curve C is cohomologically generated by the pair (C, U),
+where U is the universal bundle associated to the moduli space. In [21,22] Markmann showed a
+similar result for moduli spaces of sheaves over certain surfaces. In §6 we show how this notion
+of cohomologically generated leads to producing more Mumford-Tate families.
+Theorem 1.2. Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families. Assume
+that there exists a coherent sheaf U over X ∗ ×∆∗ Y∗ such that it is flat over ∆∗. Suppose that for
+general t ∈ ∆∗, Yt is cohomologically generated by (Xt, Ut), where Ut := U|Xt×Yt. If the family
+π1 is (strictly) Mumford-Tate family, then so is the family π2.
+See Theorem 6.5 for the precise formulation. An obvious choice for π1 is a family of smooth
+curves degenerating to a singular curve (with arbitrary singularities). See Proposition 6.1 for a
+proof in the case when the singular curve is nodal.
+Let us turn to Question 2. Suppose X is a singular projective variety of dimension n and p be
+an integer such that dim(Xsing) ≤ p − 1. Suppose φ : �
+X → X is any resolution of singularities
+and E is the exceptional divisor. By [25, Corollary-Definition 5.37], we have an exact sequence
+on cohomology
+H2p(X) → H2p( �
+X) → H2p(E).
+We conjecture that taking algebraic cohomology groups preserves the exactness of the sequence:
+Conjecture A. The following sequence is exact:
+H2p
+A (X) → H2p
+A ( �
+X) → H2p
+A (E).
+Note that, this conjecture does not involve Hodge classes.
+Surprisingly, we prove that if
+X is MT-smoothable, then this conjecture is equivalent to the singular Hodge conjecture. In
+particular,
+Theorem 1.3. Let X be as above. If X satisfies SHC(p, n), then X satisfies Conjecture A.
+Conversely, if HC(p−1, n−1) holds true, X is MT-smoothable and satisfies Conjecture A, then
+X satisfies SHC(p, n).
+See Theorem 5.5 for the precise statement.
+Outline: The paper is organised as follows: in §2 we briefly recall the necessary preliminaries
+on limit mixed Hodge structures and flag Hilbert schemes. In §3 we recall the definition of a
+Mumford-Tate group and introduce Mumford-Tate families. We give both examples and non-
+examples of such families. In §4, we define limit algebraic cohomology groups and limit Hodge
+classes. We recall the preliminaries on Operational Chow group and the Bloch-Gillet-Soul´e cycle
+
+4
+A. DAN AND I. KAUR
+class map. We give the singular Hodge conjecture and prove some of the preliminary results
+which we use later. In §5, we prove the main results of this article. Finally, in §6 we give a
+method to produce Mumford-Tate families.
+2. Preliminaries
+In this section we briefly recall some of the basics on limit mixed Hodge structures and flag
+Hilbert schemes. Limit mixed Hodge structures play an important role throughout this article.
+See [25, §11] for a detailed treatment of the topic.
+2.1. Setup. Consider a flat family of projective varieties,
+π : X → ∆,
+smooth over ∆∗ of relative dimension n. Suppose the central fiber X0 := π−1(0) is a reduced,
+simple normal crossings divisor. Denote by π′ : X∆∗ → ∆∗ the restriction of π to the punctured
+disc ∆∗. Denote by X1, ..., Xr the irreducible components of the central fiber X0. For m ≥ 2,
+denote by X(m) the disjoint union of the intersections of m number of irreducible components
+of X0 i.e.,
+X(m) :=
+�
+|I|=m
+I=(1≤i1<i2<...<im≤r)
+� m
+�
+k=1
+Xik
+�
+.
+Let e : h → ∆∗ be the exponential map from the upper half plane h to the punctured disc
+∆∗. Denote by X∞ := X∆∗ ×∆∗ h the base change of X∆∗ to h via the exponential map e.
+2.2. Monodromy operator. Since h is simply connected, the natural inclusion
+is : Xe(s) ֒→ X∞
+for any s ∈ h, induces an isomorphism of cohomology groups:
+i∗
+s : H2p(X∞, Z) ∼
+−→ H2p(Xe(s), Z).
+Note that, the morphism i∗
+s changes even if e(s) does not. In particular, we have the monodromy
+operator associate to the family π given by the composition:
+T : H2p(X∞, Z)
+i∗
+s+1
+−−→
+∼
+H2p(Xe(s), Z)
+(i∗
+s)−1
+−−−−→
+∼
+H2p(X∞, Z).
+See [16, p. 67, (2.4.13)] for further details. Denote by N := −(1/2πi) log(T). Using this operator
+N we will recall the limit Hodge filtration.
+2.3. Limit Hodge filtration. Denote by
+F •
+s H2p(X∞, C) := (i∗
+s)−1(F •H2p(Xe(s), C))
+the preimage of the Hodge filtration on H2p(Xe(s), C). The dimension of F k
+s H2p(X∞, C), denoted
+mk, does not depend on the choice of s ∈ h. Consider the Grassmann variety parameterizing
+mk-dimensional subspaces of H2p(X∞, C), denoted Grass(mk, H2p(X∞, C)). There is a natural
+map:
+h → Grass(mk, H2p(X∞, C)) sending s ∈ h to exp(2πisN)F k
+s H2p(X∞, C).
+This map is invariant under the translation s �→ s + 1 and tends to a limit F kH2p(X∞, C) as
+the imaginary part of s tends to ∞ i.e.,
+F kH2p(X∞, C) :=
+lim
+Im(s)→∞ exp(2πisN)F k
+s H2p(X∞, C).
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+5
+See [16, §I.2.6] or [26, p. 254, 255] for further details. Clearly,
+lim
+Im(s)→∞ exp(2πisN)(F p
+s H2p(X∞, C) ∩ H2p(X∞, Q)) ⊂ F pH2p(X∞, C) ∩ H2p(X∞, Q).
+(2.1)
+This inclusion will play an important role in the definition of the Mumford-Tate family in §3.
+2.4. Limit weight filtration. One can observe that the decreasing filtration
+F 0H2p(X∞, C) ⊇ F 1H2p(X∞, C) ⊇ ... ⊇ F 2pH2p(X∞, C) ⊇ 0
+need not be a Hodge filtration i.e., F k ∩ F
+2p+1−k need not be 0. It was observed by Schmid
+that H2p(X∞, Q) can be equipped with an increasing limit weight filtration W•, arising from
+the monodromy action by T, such that the two filtrations F • and W• together define a mixed
+Hodge structure on H2p(X∞, Q) (see [26, Theorem 6.16]). Steenbrink in [28] retrieved the limit
+weight filtration using a spectral sequence. We recall the E1-terms of the spectral sequence:
+Theorem 2.1 ( [25, Corollary 11.23]). The spectral sequence
+∞Ep,q
+1
+:=
+�
+k≥max{0,p}
+Hq+2p−2k(X(2k − p + 1), Q)(p − k)
+with the differential map d : ∞Ep−1,q
+1
+→ ∞Ep,q
+1
+being a combination of the restriction morphism
+and the Gysin morphism, degenerates at E2. Moreover, ∞Ep,q
+1
+⇒ Hp+q(X∞, Q) with the weight
+filtration given by ∞Ep,q
+2
+= GrW
+q Hp+q(X∞, Q).
+2.5. Specialization map. By the identification between H2p(X∞, Z) and H2p(Xs, Z) men-
+tioned above, we get a specialization morphism (see [1, §2]) which is a morphism of mixed
+Hodge structures:
+sp : H2p(X0, Z) → H2p(X∞, Z),
+where H2p(X∞, Q) is equipped with the limit mixed Hodge structure. Using the Mayer-Vietoris
+sequence observe that the weight filtration on H2p(X0, Q) arises from the spectral sequence with
+E1-terms:
+Ep,q
+1
+= Hq(X(p + 1), Q) ⇒ Hp+q(X0, Q)
+where the differential d : Ep−1,q
+1
+→ Ep,q
+1
+is the restriction morphism (see [28, Example 3.5]).
+Note that, the spectral sequence degenerates at E2.
+Remark 2.2. By the definition of Ej,q
+1
+and ∞Ej,q
+1
+given above, we have a natural morphism
+from Ej,q
+1
+to ∞Ej,q
+1 , which commutes with the respective differential maps d. As a result, this
+induces a morphism of spectral sequences:
+φ : Ep,q
+2
+→ ∞Ep,q
+2 .
+(2.2)
+We now compute the kernel over the weight graded pieces of the specialization morphism:
+Proposition 2.3. For p ≥ 0, we have an exact sequence of the form:
+Hq−2(X(p + 2), Q) → Ep,q
+2
+φ−→
+∞Ep,q
+2
+where the first morphism is induced by the Gysin morphism
+Hq−2(X(p + 2), Q) → Hq(X(p + 1), Q) = Ep,q
+1
+and φ is as in (2.2).
+
+6
+A. DAN AND I. KAUR
+Proof. Note that the composed morphism
+Hq−2(X(p + 2), Q) → Hq(X(p + 1), Q) → Hq(X(p + 2), Q) is the zero map,
+where the first morphism is simply the Gysin morphism and the second morphism is the restric-
+tion map. Therefore, there is a natural map from Hq−2(X(p + 2), Q) to Ep,q
+2 . The difference
+between the spectral sequences Ep,q
+1
+and ∞Ep,q
+1
+is that the differential map in the latter case also
+allows Gysin morphism. Therefore, the kernel of the morphism φ is isomorphic to the image of
+the Gysin map. This proves the proposition.
+□
+2.6. Flag Hilbert schemes. We refer the reader to [27, §4.5] for a detailed study of flag Hilbert
+schemes. Let
+π : X∆∗ → ∆∗
+be a smooth, projective morphism over the punctured disc ∆∗. Fix a relative polarization L on
+X∆∗ inducing a closed immersion of X∆∗ into a relative projective space PN
+∆∗ for some integer
+N. By the constancy of Hilbert polynomials in flat, projective families, every fiber of π has the
+same Hilbert polynomial (with respect to the polarization L), say Q (see [12, Theorem III.9.9]).
+Recall, given a Hilbert polynomial P, there exists a projective scheme, denoted HilbP,Q, called
+a flag Hilbert scheme parameterizing pairs of the form (Y ⊂ X ⊂ PN), where Y (resp. X) is of
+Hilbert polynomial P (resp. Q).
+The flag Hilbert scheme HilbP,Q is equipped with an universal family Y ⊂ Xuniv with Y, Xuniv
+flat over HilbP,Q and for every s ∈ HilbP,Q, the corresponding fiber Ys (resp. Xs) has Hilbert
+polynomial P (resp. Q) satisfying the universal property: if there exists a closed subscheme
+Z ⊂ X∆∗, flat over ∆∗ with fibers having Hilbert polynomial P, then there exists an unique
+morphism f : ∆∗ → HilbP,Q such that the pull-back of the universal family Y ⊂ Xuniv to ∆∗ is
+isomorphic to Z ⊂ X∆∗ (see [27, Theorem 4.5.1]).
+Lemma 2.4. For every 0 < ǫ ∈ R small enough, there exists sǫ ∈ ∆∗ of distance less than
+ǫ from the origin, such that every closed subvariety Zsǫ of codimension p in Xsǫ extends to a
+∆∗-flat closed subscheme Z ⊂ X∆∗ such that the fiber Z ∩ Xsǫ over sǫ is isomorphic to Zsǫ.
+Proof. Since the Hilbert polynomial of the fibers of π is Q, by the universal property of Hilbert
+schemes there is a natural morphism
+f : ∆∗ → HilbQ
+such that the pull-back of the universal family on HilbQ to ∆∗ is isomorphic to X∆∗. Let S be
+the set of Hilbert polynomials P of degree n − p such that the image of the natural projection
+morphism from HilbP,Q to HilbQ does not contain the image of f i.e., intersects properly the
+image of f. Clearly, S is a countable set. Note that the union of countably many proper closed
+subsets in ∆∗ does not contain any open subsets. Hence, for every 0 < ǫ ∈ R small enough, there
+exists sǫ ∈ ∆∗ of distance less than ǫ from the origin, such that f(sǫ) does not lie in the image
+of the projection from HilbP,Q to HilbQ, as P varies in the set S. In other words, every closed
+subscheme in Xsǫ extends to to a ∆∗-flat closed subscheme of X∆∗. This proves the lemma.
+□
+3. Mumford-Tate families
+In this section we introduce the concept of Mumford-Tate families. These are smooth families
+of projective varieties such that the associated limit mixed Hodge structure has “as many” Hodge
+classes as a general fiber in the family. The motivation behind the name is that Mumford-Tate
+groups are determined uniquely by the set of Hodge classes in the associated tensor algebra. Let
+us first recall the definition of the Mumford-Tate group.
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+7
+3.1. Mumford-Tate groups. Denote by S the Weil restriction of scalars for the field extension
+C/R. Let V be a Q-vector space. A pure Hodge structure of weight n on V is given by a non-
+constant homomorphism of R-algebraic groups
+φ : C∗ = S(R) → GL(V )(R)
+such that φ(r) = rnId for all r ∈ R∗ ⊂ S(R) = C∗.
+Let VC := V ⊗Q C.
+To this group
+homomorphism one associates the Hodge decomposition:
+VC =
+�
+p+q=n
+V p,q where V p,q := {v ∈ VC| φ(z)v = zpzqv for all z ∈ C∗}.
+The Mumford-Tate group associated to the pure Hodge structure (V, φ), denoted MT(V, φ),
+is the smallest Q-algebraic subgroup of GL(V ) whose set of real points contain the image of φ.
+Denote by
+T m,n(V ) := V ⊗m ⊗ Hom(V, Q)⊗n.
+Note that, the Hodge structure on V induces a pure Hodge structure on T m,n(V ). Elements of
+F 0(T m,n(VC)) ∩ T m,n(V )
+are called Hodge tensors. The Mumford-Tate group as the largest subgroup of GL(VQ) which
+fixes the Hodge tensors (see [11, §I.B]).
+Example 3.1. We now recall some well-known examples of Mumford-Tate groups.
+(1) Let X be an abelian variety and V = H1(X, Q). The Mumford-Tate group associated
+to the pure Hodge structure on V will be denoted by MT(X). The polarization on X
+corresponds to a non-degenerate alternating form φ : V ⊗ V → Q. Denote by GSp(V, φ)
+the group of symplectic simplitudes with respect to the symplectic form φ:
+GSp(V, φ) := {g ∈ GL(V ) | ∃ λ ∈ C∗ such that φ(gv, gw) = λφ(v, w) ∀ v, w ∈ V }.
+Recall, for any abelian variety X, the Mumford-Tate group of X is contained in the
+group of symplectic simplitudes i.e. MT(X) ⊆ GSp(V, φ). An abelian variety is called
+simple if it does not contain an abelian subvariety other than 0 and X. If X is simple
+and dim(X) = p, where p is a prime number, then MT(X) = GSp(V, φ).
+(2) Let c be a positive integer. Let X be a general complete intersection subvariety contained
+in P2m+c of codimension c, for some m ≥ 1. Assume that the degree of X is at least 5.
+Denote by V := Hn(X, Q)prim and φ : V ⊗ V → Q the polarization on V . Let GO(V, φ)
+be the group of orthogonal simplitudes with respect φ:
+GO(V, φ) := {g ∈ GL(V ) | ∃ λ ∈ C∗ such that φ(gv, gw) = λφ(v, w) ∀ v, w ∈ V }.
+Then the Mumford-Tate group of X, MT(X) = GO(V, φ).
+3.2. Mumford-Tate families. Keep setup as in §2.1. Given any s ∈ h, recall the exponential
+map e from h to ∆∗ and the natural inclusion is from Xe(s) into X∞. Recall,
+π : X∆∗ → ∆∗
+the family of smooth, projective varieties. For any s ∈ h, H2p(Xe(s), Q) is equipped with a natural
+pure Hodge structure. Denote by MTp(Xe(s)) the Mumford-Tate group associated to this pure
+Hodge structure on H2p(Xe(s), Q). We say that π is a Mumford-Tate family of weight p if for any
+class γ ∈ F pH2p(X∞, C) ∩ H2p(X∞, Q) satisfying Nγ = 0, the pullback i∗
+s(γ) ∈ H2p(Xe(s), Q) is
+fixed by MTp(Xe(s)) for a general s ∈ h. We say that π is Mumford-Tate if it is Mumford-Tate
+of all weights.
+Example 3.2. We now give some examples of Mumford-Tate families:
+
+8
+A. DAN AND I. KAUR
+(1) By Lefschetz hyperplane section theorem, for any smooth hypersurface X in P2m for
+m ≥ 2, we have H2p(X, Q) ∼= Q for any 0 ≤ p ≤ 2m − 1. This implies if π parametrizes
+smooth, hypersurfaces in P2m, then π is Mumford-Tate.
+(2) Let π : X → ∆ be a smooth family of prime dimensional abelian varieties such that the
+central fiber π−1(0) is simple. Then π is a Mumford-Tate family. Indeed, since π is a
+smooth family, the local system Vp := R2pπ∗Q has no monodromy over the punctured
+disc. Hence, H2p(X∞, Q) ∼= H2p(X0, Q) as pure Hodge structures, for all p and the local
+system Vp is trivial. By the same argument, R1π∗Q is a trivial local system. A choice
+of the trivialization fixes an identification:
+ψt : V0
+∼
+−→ Vt, where Vt := H1(Xt, Q) for any t ∈ ∆.
+Note that the natural polarizations on V0 and Vt commutes with the identification ψt.
+This induces an isomorphism:
+GSp(Vt, φt) ∼
+−→ GSp(V0, φ0) sending
+�
+Vt
+g−→
+∼ Vt
+�
+to
+�
+V0
+ψt
+−→
+∼ Vt
+g−→
+∼ Vt
+ψ−1
+t
+−−→
+∼
+V0
+�
+.
+(3.1)
+Now, γ0 ∈ H2p(X∞, Q) = H2p(X0, Q) is a Hodge class if and only if it is fixed by the
+Mumford-Tate group MT(X0). Since X0 is simple, MT(X0) = GSp(V0, φ0). Using the
+identification (3.1), since the Hodge class γ0 is fixed by GSp(V0, φ0), i∗
+s(γ) = φs(γ) is
+fixed by GSp(Vs, φs) for any s ∈ ∆∗. Since MT(Xs) is contained in GSp(Vs, φs), φs(γ)
+is fixed by MT(Xs). Hence, φs(γ) is a Hodge class in H2p(Xs, Q). This proves the claim
+that π is a Mumford-Tate family.
+(3) Let π : X → ∆ be a smooth family of complex intersection subvarieties of codimension
+c and let π−1(0) = X0. Suppose that MT(X0) = GO(Hn(X0, Q)prim, φ). Then π is a
+Mumford-Tate family. The proof for this is the same as that of (2) above with GSp
+replaced by GO.
+Example 3.3. (Examples of non Mumford-Tate families) Recall for d ≥ 4, the Noether-
+Lefschetz theorem states that a very general smooth, degree d surface in P3 has Picard number
+1. The Noether-Lefschetz locus parametrizes smooth degree d surfaces in P3 with Picard number
+at least 2. See [3–5] for some its geometric properties. This means that there are smooth families
+π : X → ∆ of hypersurfaces in P3 such that 0 ∈ ∆ lies on the Noether-Lefschetz locus and ∆���
+does not intersect the Noether-Lefschetz locus. Since π is a smooth family, the local system
+R2π∗Q does not have any monodromy over the punctured disc. Then, H2(X∞, Q) ∼= H2(X0.Q)
+as pure Hodge structures. In particular, by the condition on the central fiber X0, the rank of
+the Hodge lattice in H2(X∞, Q) is at least 2. But the rank of the Hodge lattice in H2(Xs, Q) is
+1 for any s ∈ ∆∗. Since the pullback morphism i∗
+s is an isomorphism, this implies that there is
+a Hodge class on H2(X∞, Q) that does not pullback to a Hodge class on H2(Xs, Q). Hence, π
+cannot be a Mumford-Tate family.
+4. A cohomological version of the Hodge conjecture for singular varieties
+In this section we define limit algebraic cohomology classes and limit Hodge classes. We show
+that the limit algebraic cohomology classes are contained in the monodromy invariant limit
+Hodge classes and the converse holds for Mumford-Tate families. In subsection 4.3 and 4.4 we
+recall the necessary preliminaries for the Operational Chow group and the Bloch-Gillet-Soul´e
+cycle class map. In 4.5 we state the Singular Hodge conjecture and in 4.6 we show that the
+cohomology classes of algebraic cycles on a simple normal crossings variety are contained in the
+Hodge classes.
+We begin by recalling the classical Hodge conjecture.
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+9
+4.1. The classical Hodge conjecture. Let X be a smooth, projective variety.
+Given an
+integer p > 0, denote by Zp(X) the free abelian group generated by codimension p algebraic
+subvarieties. There is a natural cycle class map:
+clp : Zp(X) → H2p(X, Z)
+which associates to an algebraic subvariety W ⊂ X of codimension p, the fundamental class
+[W] ∈ H2p(X, Z) (see [31, §11.1.2] for further details) and extend linearly. Furthermore, by [31,
+Proposition 11.20], the image of the cycle class map clp lies in Hp,p(X, C) ∩ H2p(X, Z) i.e., the
+cohomology class of an algebraic variety is a Hodge class. Tensoring the cycle class map by
+rationals gives:
+clp : Zp(X) ⊗Z Q → H2p(X, Q) ∩ Hp,p(X, C).
+We denote by H2p
+Hdg(X) := H2p(X, Q) ∩ Hp,p(X, C) the space of Hodge classes and the space of
+algebraic classes H2p
+A (X) ⊂ H2p(X, Q) is the image of the (rational) cycle class map clp. The
+(rational) Hodge conjecture claims that the (rational) cycle class map clp is surjective for all p
+i.e., the natural inclusion H2p
+A (X) ⊂ H2p
+Hdg(X) is an equality for all p.
+Definition 4.1. Let X be a smooth, projective variety of dimension n. We say that X satisfies
+HC(p, n) if the natural inclusion H2p
+A (X) ⊂ H2p
+Hdg(X) is an equality. We say that X satisfies
+the Hodge conjecture if it satisfies HC(p, n) for every p ≥ 0. We say that HC(p, n) holds true to
+mean that every smooth, projective variety of dimension n satisfies HC(p, n).
+4.2. Relative cycle class. Let
+π : X∆∗ → ∆∗
+be a smooth, projective morphism of relative dimension n. Let Z ⊂ X∆∗ be a closed subscheme
+of X∆∗, flat over ∆∗ and of relative dimension n − p. The fundamental class of Z defines a
+global section γZ of the local system H2p := R2pπ∗Z such that for every t ∈ ∆∗, the value
+γZ(t) ∈ H2p(Xt, Z) of γZ at the point t is simply the fundamental class of Zt := Z ∩ Xt in Xt
+(see [9, §19.2] and [25, §B.2.9] for details). The pull-back of the local system H2p under the
+exponential map e : h → ∆∗ is a trivial local system with fiber H2p(X∞, Z). The global section
+γZ defines an element of H2p(X∞, Z), which we again denote by γZ, such that for every s ∈ h,
+the image i∗
+s(γZ) is the fundamental class of Z ∩ Xe(s) in Xe(s), where is is the natural inclusion
+of Xe(s) into X∞.
+Definition 4.2. Denote by H2p
+A (X∞) the sub-vector space of H2p(X∞, Q) generated by all such
+elements of the form γZ arising from a ∆∗-flat closed subscheme of relative dimension n − p
+in X∆∗. We call H2p
+A (X∞) the limit algebraic cohomology group. We define the limit Hodge
+cohomology group
+H2p
+Hdg(X∞) := F pH2p(X∞, C) ∩ W2pH2p(X∞, Q).
+Note that, H2p
+Hdg(X∞) need not be monodromy invariant. Recall, N is a morphism of mixed
+Hodge structures from H2p(X∞, Q) to H2p(X∞, Q)(−1). We denote by H2p
+Hdg(X∞)inv the mon-
+odromy invariant part of H2p
+Hdg(X∞) i.e.,
+H2p
+Hdg(X∞)inv := ker
+�
+H2p
+Hdg(X∞) ֒→ H2p(X∞, Q) N
+−→ H2p
+Hdg(X∞, Q)
+�
+.
+We now prove that the limit algebraic cohomology group lies in the limit Hodge cohomology
+group. This is the asymptotic version of a classical result in Hodge theory.
+Proposition 4.3. The limit algebraic cohomology group is contained in the monodromy invari-
+ant part of the limit Hodge cohomology group i.e., the natural inclusion H2p
+A (X∞) ⊂ H2p(X∞, Q)
+factors through H2p
+Hdg(X∞)inv.
+
+10
+A. DAN AND I. KAUR
+Proof. Take γ ∈ H2p
+A (X∞). By construction, there exist ∆∗-flat closed subschemes Z1, ..., Zr of
+relative dimension n − p in X∆∗ such that γ = � aiγZi for ai ∈ Q and γZi ∈ H2p
+A (X∞) is as
+defined above, arising from the fundamental class of Zi. By construction, each γZi arises from
+a global section of the local system H2p. Hence, γZi is monodromy invariant i.e., T(γZi) = γZi
+for 1 ≤ i ≤ r. This implies NγZi = 0 for 1 ≤ i ≤ r.
+As the cohomology class of Zi ∩ Xe(s) lies in F pH2p(Xe(s), Q), we have γZi ∈ F p
+s H2p(X∞, Q)
+for all s ∈ h (notations as in §2.3). This implies γZi lies in exp(2πisN)F p
+s H2p(X∞, Q) for every
+s ∈ h. Recall from §2.3 that F pH2p(X∞, Q) contains the limit of exp(2πisN)F p
+s H2p(X∞, C)
+as Im(s) approaches ∞. Hence, γZi ∈ F pH2p(X∞, Q). As γZi is monodromy invariant and a
+rational class, it must lie in W2pH2p(X∞, Q) (use the invariant cycle theorem along with the
+fact that the degree 2p cohomology of the central fiber is of weight at most 2p). Therefore,
+γ ∈ H2p
+Hdg(X∞)inv. This proves the first part of the proposition.
+□
+We now ask when is H2p
+A (X∞) isomorphic to H2p
+Hdg(X∞)inv? One can naively guess that if the
+general fibers in the family π satisfy the Hodge conjecture then this happens. However, this is
+not enough (see Example 3.3 above). In particular, one needs to additionally assume that the
+family π is Mumford-Tate. We prove:
+Proposition 4.4. Suppose that π is a Mumford-Tate family of weight p. If a general fiber of π
+satisfies HC(p, n), then the inclusion from H2p
+A (X∞) to H2p
+Hdg(X∞)inv is an isomorphism.
+Note that, by general in the statement of the proposition, we mean the complement of finitely
+many proper, closed subvarieties of the punctured disc ∆∗.
+Proof. We need to show that every element in H2p
+Hdg(X∞)inv lies in H2p
+A (X∞).
+Since π is a
+Mumford-Tate family, we have
+H2p
+Hdg(X∞)inv =
+lim
+Im(s)→∞(F p
+s H2p(X∞, Q) ∩ H2p(X∞, Q)inv).
+(4.1)
+It therefore suffices to show that
+lim
+Im(s)→∞(F p
+s H2p(X∞, Q) ∩ H2p(X∞, Q)inv)
+is contained in H2p
+A (X∞).
+By Lemma 2.4 for every 0 < ǫ ∈ R small enough, there exists sǫ ∈ ∆∗ of distance less than ǫ
+from the origin, such that Xsǫ satisfies HC(p, n) and every closed subvariety Zsǫ of codimension
+p in Xsǫ extends to a ∆∗-flat closed subscheme Z ⊂ X∆∗ such that the fiber Z ∩ Xsǫ over sǫ
+is isomorphic to Zsǫ.
+As observed before Definition 4.2, the fundamental class of Z defines
+a section γZ ∈ H2p
+A (X∞) and is monodromy invariant. Since F pH2p(Xsǫ, Q) is isomorphic to
+H2p
+A (Xsǫ), this implies
+H2p(X∞, Q)inv ∩ F p
+sǫH2p(X∞, Q) = (i∗
+sǫ)−1(H2p
+A (Xsǫ)) ⊆ H2p
+A (X∞),
+where isǫ is the natural inclusion of Xe(sǫ) into X∞. Therefore, the limit as Im(s) tends to ∞,
+of H2p(X∞, Q)inv ∩ F p
+s H2p(X∞, Q) is contained in H2p
+A (X∞). This proves the proposition.
+□
+4.3. Operational Chow group. Let Y be a quasi-projective variety (possibly singular), of
+dimension say n. Consider a non-singular hyperenvelope of a compactification of Y (see [10,
+§1.4.1] for the definition and basic properties of hyperenvelopes). The hyperenvelope gives rise
+to a cochain complex of motives (see [10, §2.1]). For any positive integer p, one can then obtain
+an abelian group R0CHp(Y ) arising as the cohomology group after applying the functor CHp(−)
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+11
+to the cochain complex of motives (see [10, §3.1.4]). Observe that R0CHp(Y ) does not depend
+on the choice of the compactification or the hyperenvelope. Note that,
+Theorem 4.5. Fix a positive integer p. Then, the following holds true for R0CHp(Y ):
+(1) if Y is projective, then R0CHp(Y ) is the operational Chow group Ap(Y ) defined by
+Fulton and MacPherson (see [9, Chapter 17]),
+(2) if Y is non-singular (but not necessarily projective), then Ap(Y ) is the free abelian group
+generated by the codimension p subvarieties in Y , upto rational equivalence,
+(3) if Y is non-singular and Y is a compactification of Y with boundary Z := Y \Y , we then
+have the exact sequence:
+0 → R0CHp(Y ) → R0CHp(Y ) → R0CHp(Z)
+(4.2)
+(4) if Y is the union of two proper closed subvarieties Y1 and Y2, then we have the exact
+sequence:
+0 → R0CHp(Y ) → R0CHp(Y1) ⊕ R0CHp(Y2) → R0CHp(Y1 ∩ Y2).
+(4.3)
+Proof.
+(1) This is [10, Proposition 4].
+(2) This is [9, Proposition 17.3.1 and Corollary 17.4].
+(3) This is [10, Theorem 2(iii) and §3.1.1].
+(4) This is [10, Theorem 2(iv) and §3.1.1].
+□
+Notation 4.6. If Y is quasi-projective but not projective, we denote by Ap
+c(Y ) := R0CHp(Y ),
+the compactly supported operational Chow cohomology.
+Given any compactification Y of Y ,
+Theorem 4.5 implies that we have the following exact sequence
+0 → Ap
+c(Y ) → Ap(Y ) → Ap(Y \Y )
+(4.4)
+For Y a projective variety, there are natural functorial cycle class maps (see [2] or [17, §2]):
+clp : Ap(Y ) → GrW
+2pH2p(Y, Q) and clc
+p : Ap
+c(Ysm) → GrW
+2pH2p
+c (Ysm, Q)
+which agree with the usual cycle class map (see [31, §11.1.2]) if Y is non-singular (here Ysm
+denotes the smooth locus of Y ). For Y projective, define the algebraic cohomology group denoted
+by H2p
+A (Y ) ⊂ GrW
+2pH2p(Y, Q) to be the image of the cycle class map clp.
+4.4. Bloch-Gille-Soul´e Cycle class map. Let Y be a scheme and φ : U → Y , γ : V → U×Y U
+be envelopes. Let pi : V → U denote the compositions of γ with the projections U ×Y U → U.
+Theorem 4.7. ( [2, Theorem A.3]) There is a left-exact sequence of Chow cohomology groups
+0 → CH∗(Y )
+φ∗
+−→ CH∗(U)
+p∗
+1−p∗
+2
+−−−−→ CH∗(V ).
+Using the cycle map over smooth, quasi-projective varieties U and V , Bloch-Gillet-Soul´e uses
+the above theorem to conclude:
+Corollary 4.8. ( [2, Corollary A.4]) On the category of varieties over C, there is a “cycle class”
+natural transformation of contravariant functors to the category of commutative, graded rings:
+�
+p
+clp :
+�
+p
+CHp(−) →
+�
+p
+GrW
+0 H2p(− , Q(p)).
+
+12
+A. DAN AND I. KAUR
+4.5. Singular Hodge conjecture. We are now ready to give a formulation of the Hodge
+conjecture for singular varieties. Let Y be a projective variety of dimension n. Fix a positive
+integer p ≤ n. We say that Y satisfies SHC(p, n) if the singular locus of Y is of dimension at
+most p − 1 and the algebraic cohomology group H2p
+A (Y ) coincides with
+H2p
+Hdg(Y ) := GrW
+2pH2p(Y, Q) ∩ F 2pGrW
+2pH2p(Y, C).
+In the case when Y is non-singular and projective, this simply is the classical Hodge conjecture
+(in weight p), which we already denote by HC(p, n).
+4.6. Algebraic cycles on simple normal crossings divisors. We now prove that the coho-
+mology classes of algebraic cycles on a simple normal crossings variety are Hodge classes. This
+is a generalization to the singular case of a classical result in Hodge theory. Recall, X0 is called a
+simple normal crossings variety if X0 is connected, X0 = X1 ∪ ... ∪ Xr with Xi irreducible, non-
+singular for all i and the intersection of any p of the irreducible components of X0 is non-singular
+of codimension p, for any p ≥ 1.
+Lemma 4.9. Let X0 be a simple normal crossings variety. Then, the cycle class map clp from
+Ap(X0) to GrW
+2pH2p(X0, Q) factors through
+H2p
+Hdg(X0) := F pGrW
+2pH2p(X0, C) ∩ GrW
+2pH2p(X0, Q).
+Proof. We use recursion on the components of X0. Let X0, ..., Xr be the irreducible components
+of X0. Denote by Zi := X0\(X1 ∪ ... ∪ Xi), the complement of the components X1, ..., Xi for
+i ≥ 1. Let Z0 := X0. Since Xi, Xj and Xi ∩ Xj are non-singular for all i, j, they have pure
+Hodge structures. Moreover by [25, Theorem 5.39], H2p−1(Xi ∩Zi, Q) is of weight at most 2p−1
+i.e., GrW
+2pH2p−1(Xi ∩ Zi, Q) = 0 for all 1 ≤ i ≤ r − 1. Therefore for all 1 ≤ i ≤ r − 1, we have
+the following exact sequence of pure Hodge structures:
+0 → GrW
+2pH2p(Zi−1, Q) → H2p(Xi, Q) ⊕ GrW
+2pH2p(Zi, Q) → GrW
+2pH2p(Xi ∩ Zi, Q)
+(4.5)
+Moreover, by Theorem 4.5, we have the exact sequence:
+0 → Ap(Zi−1) → Ap(Xi) ⊕ Ap(Zi) → Ap(Xi ∩ Zi)
+(4.6)
+By the functoriality of the cycle class maps clp, we have the following diagram
+0
+✲ Ap(Zi−1)
+✲ Ap(Xi) ⊕ Ap(Zi)
+✲ Ap(Xi ∩ Zi)
+0
+✲ GrW
+2pH2p(Zi−1, Q)
+clp
+❄
+✲ H2p(Xi, Q) ⊕ GrW
+2pH2p(Zi, Q)
+clp
+❄
+✲ GrW
+2pH2p(Xi ∩ Zi, Q)
+clp
+❄
+For the base case, consider i = r−1. Note that, Zr−1 = Xr. Since Xr is non-singular, Ap(Zr−1)
+is the usual Chow group. Therefore, clp(Ap(Zr−1)) ⊂ H2p
+Hdg(Zr−1).
+Now for the recursion step. Assume that clp(Ap(Zi)) ⊂ H2p
+Hdg(Zi). Since the exact sequence
+(4.5) is a morphism of pure Hodge structures, the commutativity of the left hand square implies
+that clp(Ap(Zi−1)) ⊂ H2p
+Hdg(Zi−1). This proves the lemma.
+□
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+13
+5. Main results
+In this section we introduce the concept of MT-smoothable varieties. Consider a simple normal
+crossings variety X (in the sense of §4.6). Denote by X(2) the disjoint union of intersection of
+any 2 irreducible components of X. We prove that if X is MT-smoothable and X(2) satisfies
+HC(p − 1, n − 1) then X satisfies SHC(p, n) (see Theorem 5.2).
+This is a generalization of
+Theorem 1.1 in the introduction. Moreover, if there is an irreducible component Xi of X such
+that the restriction morphism on cohomology is surjective, then Xi satisfies the classical Hodge
+conjecture (see Corollary 5.3). Finally, if the variety has worse singularities than simple normal
+crossings, then we reduce the singular Hodge conjecture to a question solely on the algebraic
+classes (see Theorem 5.5).
+Definition 5.1. Let X be a singular projective variety of dimension n and p be an integer such
+that dim(Xsing) ≤ p − 1. We say that X is MT-smoothable of weight p if there exists a flat,
+projective, Mumford-Tate family
+π0 : Y → ∆
+smooth over ∆∗, containing X as a central fiber and a general fiber satisfying HC(p, n). We call
+π0 a MT-smoothing of weight p of X.
+Given a normal crossings variety X, We prove:
+Theorem 5.2. Let X be a simple normal crossings variety of dimension n. Assume that every
+irreducible component of X(2) satisfies HC(p − 1, n − 1). If X is MT-smoothable of weight p,
+then X satisfies SHC(p, n) i.e.,
+H2p
+A (X, Q) ∼= H2p
+Hdg(X, Q).
+Moreover, for every irreducible component Xi of X, the image of the restriction morphism from
+H2p
+Hdg(X, Q) to H2p
+Hdg(Xi, Q) are cohomology classes of algebraic cycles i.e., the image
+Im(H2p
+Hdg(X, Q) → H2p
+Hdg(Xi, Q))
+is contained in H2p
+A (Xi, Q).
+Proof. Since X is MT-smoothable of weight p, there exists a Mumford-Tate family of weight p
+π : X → ∆
+with central fiber X and general fibers satisfying HC(p, n). By Proposition 4.4 and Lemma 4.9,
+we have a morphism spA from H2p
+A (X) to H2p
+A (X∞) given by the composition:
+spA : H2p
+A (X) ֒→ H2p
+Hdg(X)
+sp
+−→ H2p
+Hdg(X∞)inv ∼= H2p
+A (X∞).
+We claim that spA is surjective. Recall from Definition 4.2, H2p
+A (X∞) is generated as a Q-vector
+space by classes γZ where Z ⊂ X∆∗ is a ∆∗-flat closed subscheme of relative dimension n − p.
+Denote by Z the closure of Z in X. By [9, §6.1], the intersection product Z.Xi of Z with Xi
+is of codimension p in Xi . Denote by γi ∈ H2p(Xi, Q) the cohomology class of the intersection
+product Z.Xi for 1 ≤ i ≤ r. By the associativity of intersection product (see [9, Proposition
+8.1.1 or Proposition 8.3]), for any pair of integers 1 ≤ i < j ≤ r, the image of γi (resp. γj) under
+the restriction morphisms from H2p(Xi, Q) (resp. H2p(Xj, Q)) to H2p(Xi ∩ Xj, Q) coincides.
+Using (4.5) one can observe that there exists an algebraic cohomology class γ ∈ H2p
+A (X) such
+that the image of γ under the restriction morphism from H2p
+A (X) to H2p
+A (Xi) is γi for 1 ≤ i ≤ r.
+In other words, the cohomology class of Z in H2p(X, Q) (see [25, §B.2.9]) pulls back to γ in
+H2p(X, Q) and to the cohomology class [Z ∩ Xt] ∈ H2p(Xt, Q) over Xt, for any t ∈ ∆∗. This
+
+14
+A. DAN AND I. KAUR
+means that under the specialization morphism sp from H2p(X, Q) to H2p(X∞, Q), γ maps to
+γZ. This proves our claim.
+By Proposition 2.3, the kernel of the specialization morphism
+GrW
+2pH2p(X, Q) = E0,2p
+2
+sp
+−→ ∞E0,2p
+2
+= GrW
+2pH2p(X∞, Q)
+is isomorphic to the image of the Gysin morphism from H2p−2(X(2), Q) to H2p(X, Q) (as X(2)
+is non-singular, H2p−2(X(2), Q) has a pure Hodge structure of weight 2p − 2). By assumption,
+every irreducible component of X(2) satisfies HC(p − 1, n − 1).
+Then, we get the following
+commutative diagram of exact sequences:
+H2p
+A (X(2))
+✲ H2p
+A (X)
+spA✲ H2p
+A (X∞)
+✲ 0
+⟲
+⟲
+H2p
+Hdg(X(2))
+∼=
+❄
+✲ H2p
+Hdg(X)
+❄
+∩
+sp✲ H2p
+Hdg(X∞)inv
+∼=
+❄
+By diagram chase (or using four lemma for the diagram of exact sequences), we conclude that the
+middle morphism from H2p
+A (X) to H2p
+Hdg(X) is surjective, hence an isomorphism. This proves
+the first part of the theorem. The second part of the theorem follows immediately from the
+following commutative diagram, which arises from the Mayer-Vietoris sequence:
+H2p
+A (X) ⊂
+✲ H2p
+A (Xi) ⊕ H2p
+A (X\Xi)
+⟲
+H2p
+Hdg(X)
+∼=
+❄
+⊂✲ H2p
+Hdg(Xi) ⊕ H2p
+Hdg(X\Xi)
+❄
+∩
+This proves the theorem.
+□
+Corollary 5.3. Notations and hypothesis as in Theorem 5.2. Let X1 be an irreducible com-
+ponent in X such that the complement Xc
+1 := X\X1 (the closure of X\X1 in X) satisfies:
+Im(H2p
+Hdg(X1) → H2p
+Hdg(Xc
+1 ∩ X1)) ⊂ Im(H2p
+Hdg(Xc
+1) → H2p
+Hdg(Xc
+1 ∩ X1)).
+(5.1)
+Then, X1 satisfies HC(p, n).
+Proof. Using the Mayer-Vietoris sequence we have the following commutative diagram:
+0
+✲ H2p
+A (X)
+✲ H2p
+A (X1) ⊕ H2p
+A (Xc
+1)
+⟲
+0
+✲ H2p
+Hdg(X)
+∼=
+❄
+⊂✲ H2p
+Hdg(X1) ⊕ H2p
+Hdg(Xc
+1)
+❄
+∩
+✲ H2p
+Hdg(Xc
+1 ∩ X1)
+where the isomorphism of the first vertical arrow follows from Theorem 5.2 and the bottom row
+is exact. If (5.1) is satisfied then for any γ ∈ H2p
+Hdg(X1), there exists γ′ ∈ H2p
+Hdg(Xc
+1) such that
+their restrictions to X1 ∩ Xc
+1 agree. In other words, γ ⊕ γ′ maps to zero in H2p
+Hdg(Xc
+1 ∩ X1).
+By diagram chase, one observes that there exists γA ∈ H2p
+A (X1) which maps to γ. This proves
+H2p
+A (X1) ∼= H2p
+Hdg(X1). In other words, X1 satisfies HC(p, n). This proves the corollary.
+□
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+15
+One immediately asks whether there are examples where (5.1) is satisfied?
+Example 5.4. Let X be a projective variety of dimension n with only ordinary double point
+singularities. Suppose also that X is smoothable. Then, there exists a flat, projective family
+π0 : Y → ∆
+smooth over ∆∗, X as the central fiber and Y is a regular variety. Moreover, there exists a
+semi-stable reduction of π0:
+π : X → ∆
+such that the central fiber X0 := �
+X ∪ E, where E is a disjoint union of quadric hypersurfaces in
+Pn+1 and E ∩ �
+X0 is the intersection of E by hyperplanes in copies of Pn+1. If n = 2p for some
+p, then the n-th rational cohomology of a quadric hypersurface in Pn is isomorphic to Q. This
+implies the natural restriction morphism from H2p(E) to H2p(E ∩ �
+X) is surjective. In this case,
+taking X1 := �
+X, (5.1) is satisfied.
+A natural conjecture arises from our observations:
+Conjecture A. Let X be a singular projective variety, φ :
+�
+X → X be any resolution of
+singularities and E be the exceptional divisor. Let p be an integer such that dim(Xsing) ≤ p−1.
+We then have an exact sequence on cohomology (see [25, Corollary-Definition 5.37]):
+H2p(X) → H2p( �
+X) → H2p(E)
+We conjecture that taking algebraic cohomology groups preserves the exactness of the sequence
+i.e., the following sequence is exact:
+H2p
+A (X) → H2p
+A ( �
+X) → H2p
+A (E).
+We now observe that this conjecture is closely related to the singular Hodge conjecture (which
+is equivalent to the Hodge conjecture).
+Theorem 5.5. Let X be a singular projective variety of dimension n and p be an integer such
+that dim(Xsing) ≤ p − 1. If X satisfies SHC(p, n), then X satisfies Conjecture A. Conversely, if
+HC(p − 1, n − 1) holds true, X is MT-smoothable of weight p and satisfies Conjecture A, then
+X satisfies SHC(p, n).
+Proof. If X satisfies the SHC(p, n), then H2p
+A (X) ∼= H2p
+Hdg(X). Let
+φ : �
+X → X
+be a resolution of X and E be the exceptional divisor. We then have the following commutative
+diagram:
+H2p
+A (X)
+✲ H2p
+A ( �
+X)
+✲ H2p
+A (E)
+⟲
+⟲
+H2p
+Hdg(X)
+∼=
+❄
+⊂✲ H2p
+Hdg( �
+X)
+❄
+∩
+✲ H2p
+Hdg(E)
+❄
+∩
+(5.2)
+where the bottom row is exact, injective on the left and the top row is a complex. To prove
+Conjecture A, we need to show that the top row is exact in the middle. For this, take γ ∈ H2p
+A ( �
+X)
+which maps to zero in H2p
+A (E). By diagram chase it is easy to check that there exists γ′ ∈ H2p
+A (X)
+which maps to γ. In other words, the top row of (5.2) is exact in the middle. This proves the
+first part of the theorem.
+
+16
+A. DAN AND I. KAUR
+We now assume that X satisfies Conjecture A. Let π0 : Y → ∆ be a MT-smoothing of weight
+p of X. By the semi-stable reduction theorem (see [15, Chapter II]) there exists a flat, projective
+family π : X → ∆ which has the same fiber over ∆∗ as π0, X is regular, the central fiber X0 is
+a reduced simple normal crossings divisor with one of the irreducible components, say �
+X being
+proper birational to X. Furthermore, the complement �
+Xc := X0\ �
+X satisfies:
+X0\ �
+Xc ∼= �
+X\( �
+Xc ∩ �
+X) ∼= X\Xsing
+i.e., X is isomorphic to Y away from Xsing. Using the Mayer-Vietoris sequence and Conjecture
+A we have the following commutative diagram of exact sequences:
+H2p
+A (X)
+✲ H2p
+A ( �
+X)
+✲ H2p
+A ( �
+X ∩ �
+Xc)
+⟲
+⟲
+H2p
+A (X0)
+❄
+∩
+✲ H2p
+A ( �
+X) ⊕ H2p
+A ( �
+Xc)
+❄
+∩
+✲ H2p
+A ( �
+X ∩ �
+Xc)
+∼=
+❄
+(5.3)
+where the first vertical morphism is induced by the pullback from X to X0 and the second one
+is the natural inclusion. By snake lemma, this gives rise to the exact sequence:
+0 → H2p
+A (X) → H2p
+A (X0) → H2p
+A ( �
+Xc)
+(5.4)
+Since Xsing is of dimension at most p − 1, Hi(Xsing) = 0 for i ≥ 2p − 1. Then, the long exact
+sequences in cohomology associated to the pairs (X, Xsing) and (X0, �
+Xc) (see [25, Proposition 5.46
+and Corollary B.14])) implies GrW
+2pH2p
+c (U) ∼= GrW
+2pH2p(X) where U := X\Xsing. Furthermore,
+0 → GrW
+2pH2p
+c (U, Q) → GrW
+2pH2p(X0, Q) → GrW
+2pH2p( �
+Xc, Q)
+is an exact sequence of pure Hodge structures. This gives rise to the exact sequence:
+0 → H2p
+Hdg(X) → H2p
+Hdg(X0) → H2p
+Hdg( �
+Xc)
+(5.5)
+of Q-vector spaces. Then, there is a natural morphism of exact sequences from (5.4) to (5.5):
+0
+✲ H2p
+A (X)
+✲ H2p
+A (X0)
+✲ H2p
+A ( �
+Xc)
+⟲
+⟲
+0
+✲ H2p
+Hdg(X)
+❄
+∩
+✲ H2p
+Hdg(X0)
+∼=
+❄
+✲ H2p
+Hdg( �
+Xc)
+❄
+∩
+where the isomorphism of the middle vertical arrow follows from Theorem 5.2. Applying snake
+lemma once again we conclude that the first vertical morphism is surjective. In other words, X
+satisfies SHC(p, n). This proves the converse and hence the theorem.
+□
+6. Examples of Mumford-Tate families
+In §3 we introduced Mumford-Tate families.
+For such families, the central fiber displays
+interesting properties. For example, if the central fiber is smooth, then it is easy to check that it
+satisfies the Hodge conjecture if a general fiber satisfies the Hodge conjecture. More generally,
+if the central fiber is a reduced, simple normal crossings divisor, then it satisfies the singular
+Hodge conjecture if the general fiber satisfies the Hodge conjecture (see Theorem 5.2). In this
+section we use correspondences to give a general method to produce Mumford-Tate families (see
+Theorem 6.5). We give examples in Corollary 6.6.
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+17
+6.1. Strict Mumford-Tate families. Let π1 : X ∗ → ∆∗ be a smooth, projective morphism
+over the punctured disc ∆∗. Recall that π1 is called a Mumford-Tate family if the pullback
+of every monodromy invariant Hodge class on H2p(X∞, Q) to a general fiber is fixed by the
+associated Mumford-Tate group, for every p. Here we generalize this condition to the tensor
+algebra of the cohomology ring H∗(X∞, Q). This is a slightly stronger notion. In particular, it
+is possible that wedge product of two elements from odd degree cohomology groups become a
+Hodge class, although they are individually not Hodge classes. This is a common phenomena
+appearing in the cohomology of abelian varieties, for example. This will play a crucial role below
+to produce new examples of Mumford-Tate families.
+In order to study the tensor algebras more effectively, we separate the odd cohomology groups
+from the even ones. We take exterior algebra of the odd cohomology groups and the symmetric
+algebra of the even ones. This is done to preserve compatibility with cup-products. Given two
+r-tuple of positive integers m := (m1, ..., mr) and k := (k1, ..., kr), denote by
+Tk
+m :=
+k1
+�
+Hm1(X∞, Q) ⊗ ... ⊗
+kr
+�
+Hmr(X∞, Q), if each mi is odd,
+Tk
+m := Symk1Hm1(X∞, Q) ⊗ ... ⊗ SymkrHmr(X∞, Q), if each mi is even.
+Given an r-tuple of even positive integers m := (m1, ..., mr), an l-tuple of odd positive integers
+n := (n1, ..., nl) and an r (resp. l) tuple of arbitrary positive integers k := (k1, ..., kr) (resp.
+k′ := (k′
+1, ..., k′
+l)), denote by
+T(k,k′)
+(m,n) the pure part of Tk
+m ⊗ Tk′
+n i.e., T(k,k′)
+(m,n) := GrW
+a Tk
+m ⊗ Tk′
+n ,
+where a := �r
+i=1 miki + �l
+j=1 njk′
+j. Denote by
+T(m,n) :=
+�
+(k,k′)
+T(k,k′)
+(m,n),
+(6.1)
+where k and k′ ranges over all k-tuple and l-tuple of positive integers, respectively. Denote by
+Ts
+(m,n) the same as T(m,n) with X∞ replaced by Xs for any s ∈ ∆∗.
+Note that, the Hodge structure on Hm(Xs, Q) is pure for all m, so the “pure part” condition
+is redundant in this case. Let MTs
+m be the Mumford-Tate group associated to the pure Hodge
+structure Hm(Xs, Q). Then, the product of the Mumford-Tate groups
+MTs
+(m,n) := MTs
+m1 × MTs
+m2 × ... × MTs
+mr × MTs
+n1 × MTs
+n2 × ... × MTs
+nl
+acts on Ts
+(m,n). The family π is called strictly Mumford-Tate with respect to (m, n) if for any
+Hodge class γ ∈ T(m,n) and s ∈ h general, j∗
+s(γ) is fixed by MTs
+(m,n), where
+j∗
+s : T(m,n) → Ts
+(m,n)
+is induced by the pullback of the natural inclusion of Xs inside X∞.
+Proposition 6.1. Let π1 : X → ∆ be a flat, projective family of genus g curves for g ≥ 2. We
+assume that π1 is smooth over ∆∗ and the central fiber is a very general irreducible nodal curve
+(in the sense of [7]). Then, π1 is strictly Mumford-Tate with respect to ((0, 2), (1)).
+Proof. Consider the family of Jacobians associated to the family of curves π1,
+π2 : J → ∆∗ i.e., for all t ∈ ∆∗, π−1
+2 (t) = Jac(Xt).
+By the definition of cohomology of abelian varieties, there is a natural isomorphism of mixed
+Hodge structures between H1(X∞, Q) and H1(J∞, Q). This induces an isomorphism of mixed
+
+18
+A. DAN AND I. KAUR
+Hodge structures,
+∗�
+H1(X∞, Q) ∼
+−→ H∗(J∞, Q).
+By [7, Theorem 4.3], we have
+H∗
+Hdg(J∞, Q) ∼= Q[θ]/(θg+1), where g = genus(Xt), t ∈ ∆∗.
+Note that, Sym∗H0(X∞, Q) ∼= Q[T0] and Sym∗H2(X∞, Q) ∼= Q[T1] where T0 and T1 are Hodge
+classes. Consider the direct sum of vector spaces T(0,2),(1) as in (6.1) associated to the family π1.
+Then, the space of Hodge classes THdg in T(0,2),(1) is isomorphic to Q[T0, T1, θ]/(θg+1). Similarly,
+the set of Hodge class Ts
+Hdg in Ts
+(0,2),(1) contains Q[T s
+0 , T s
+1 , θs]/((θs)g+1), where (−)s := j∗
+s(−).
+Hence, T s
+0 , T s
+1 and θs are fixed by the Mumford-Tate group MTs
+(0,2),(1). Therefore, π1 is strictly
+Mumford-Tate with respect to ((0, 2), (1)). This proves the proposition.
+□
+6.2. Cohomologies generated by Chern classes. Let X, Y be smooth, projective varieties
+of dimension m and n, respectively. Combining K¨unneth decomposition with Poincare duality,
+we have for every i, k ≥ 0,
+H2i−k(X × Y ) ≃
+�
+k
+H2n−k(X) ⊗ H2i−k(Y )
+∨ ≃
+�
+k
+Hom(H2m−k(X), H2i−k(Y )).
+(6.2)
+Let E be a coherent sheaf on the fibre product X ×Y and ci(E) be the i-th Chern class of E. De-
+note by Φ(i,k)
+E
+the projection of ci(E) in H2i−k(Y ) to the component Hom(H2m−k(X), H2i−k(Y )).
+By [31, Lemma 11.41], the induced morphism
+Φ(i,k)
+E
+: H2m−k(X) → H2i−k(Y ) is a morphism of pure Hodge structures.
+(6.3)
+Theorem 6.2. Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families of
+relative dimensions m and n, respectively. Assume that there exists a coherent sheaf U over
+X ∗ ×∆∗ Y∗ such that it is flat over ∆∗. Then the morphism
+Φ(i,k)
+Ut
+: H2m−k(Xt) → H2i−k(Yt)
+induces a morphism of (limit) mixed Hodge structures:
+Φ(i,k)
+U,∞ : H2m−k(X∞) → H2i−k(Y∞).
+Furthermore, the morphisms Φ(i,k)
+U,∞ and Φ(i,k)
+Ut
+commute with pullback to closed fibers i.e., for
+any u ∈ h with e(u) = t (where e is the exponential map) we have the following commutative
+diagram:
+H2m−k(X∞)
+Φ(i,k)
+U,∞
+✲ H2i−k(Y∞)
+⟲
+H2m−k(Xt)
+(ju)∗ ∼=
+❄
+Φ(i,k)
+Ut ✲ H2i−k(Yt)
+(j′
+u)∗ ∼=
+❄
+(6.4)
+where ju : Yt ֒→ Y∞ and j′
+u : Xt ֒→ X∞ are natural inclusions.
+Proof. Consider the natural projective morphisms:
+π : X ∗ ×∆∗ Y∗ → ∆∗, π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗.
+Consider the local system H2i := R2iπ∗Z over ∆∗. We denote by
+Hi
+X ∗ := Riπ1∗Z and Hi
+Y∗ := Riπ2∗Z.
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+19
+By K¨unneth decomposition in families (see [14, Ex. II.18]), we have
+H2i ≃
+�
+k
+(Hk
+X ∗ ⊗ H2i−k
+Y∗ )
+Applying Poincare duality to the local system Hk
+X ∗ (see [16, §I.2.6]), we get:
+H2i ≃
+�
+k
+(H2m−k
+X ∗
+)∨ ⊗ H2i−k
+Y∗
+≃
+�
+k
+Hom(H2m−k
+X ∗
+, H2i−k
+Y∗ ).
+For any i, the i-th Chern class ci(U) defines a global section of H2i. Consider the projection
+φ of ci(U) to Hom(H2m−k
+X ∗
+, H2i−k
+Y∗ ). Pulling back the morphism φ of local systems on ∆∗ to the
+upper half plane h and taking global sections, we get the morphism
+Φ(i,k)
+U,∞ : H2m−k(X∞) → H2i−k(Y∞).
+Restricting the morphism to the fiber over u ∈ h gives us the morphism Φ(i,k)
+Ut
+, where t := e(u).
+In particular, we have commutative diagram (6.4).
+It remains to check that Φ(i,k)
+U,∞ is a morphism of limit mixed Hodge structures. By (6.3), Φ(i,k)
+Ut
+is a morphism of pure Hodge structures. Since the limit Hodge filtrations on X∞ and Y∞ arise
+simply as a limit of these Hodge filtrations, we conclude that Φ(i,k)
+U,∞ preserves the limit Hodge
+filtrations. It remains to check that Φ(i,k)
+U,∞ preserves the limit weight filtration. Equivalently,
+using the diagram (6.4) we need to prove that Φ(i,k)
+Ut
+preserves the weight filtration where the
+weight filtration on Xt and Yt is induced by X∞ and Y∞, respectively (via the isomorphisms j∗
+u
+and j′
+u
+∗, respectively). Recall, the weight filtration on Xt and Yt is induced by the log of the
+monodromy operators (see [25, Lemma-Definition 11.9]):
+NX := log(TX ) and NY := log(TY).
+So, it suffices to check that for all γ ∈ H2m−k(Xt), we have Φ(i,k)
+Ut
+(NX (γ)) = NYΦ(i,k)
+Ut
+(γ). Since
+ci(U) is a global section of the local system, it is monodromy invariant. This means the induced
+morphism φ from H2m−k
+X ∗
+to H2i−k
+Y∗
+commutes with the monodromy operators i.e., for every
+t ∈ ∆∗, we have following commutative diagram:
+H2m−k(Xt)
+Φ(i,k)
+Ut✲ H2i−k(Yt)
+⟲
+H2m−k(Xt)
+TX
+❄
+Φ(i,k)
+Ut✲ H2i−k(Yt)
+TY
+❄
+(6.5)
+where TX and TY are the monodromy operators and Φ(i,k)
+Ut
+is as in (6.3). This implies for all
+γ ∈ H2m−k(Xt), we have Φ(i,k)
+Ut (TX (γ)) = TYΦ(i,k)
+Ut
+(γ). Hence,
+Φ(i,k)
+Ut
+(TX − Id)(γ) = Φ(i,k)
+Ut
+(TX (γ)) − Φ(i,k)
+Ut
+(γ) = TY(Φ(i,k)
+Ut
+(γ)) − Φ(i,k)
+Ut
+(γ) = (TY − id)Φ(i,k)
+Ut
+(γ).
+More generally, this implies for all m ≥ 1,
+Φ(i,k)
+Ut
+(TX − Id)m(γ) = Φ(i,k)
+Ut (TX − Id)(TX − Id)m−1(γ) = (TY − Id)Φ(i,k)
+Ut
+(TX − Id)m−1(γ)
+Therefore, by recursion we have Φ(i,k)
+Ut
+(TX −Id)m(γ) = (TY −Id)mΦ(i,k)
+Ut
+(γ). Using the logarithmic
+expansion of NX and NY we conclude:
+Φ(i,k)
+Ut
+(NX (γ)) = NYΦ(i,k)
+Ut
+(γ), for all γ ∈ H2m−k(Xt).
+
+20
+A. DAN AND I. KAUR
+This implies that Φ(i,k)
+Ut
+preserves the limit weight filtration. This proves the theorem.
+□
+Definition 6.3. Let X,Y be smooth, projective varieties of dimensions m and n, respectively.
+Denote by E a coherent sheaf on X ×k Y . The variety Y is said to be cohomologically generated
+by (X, E) if there is a collection SY (X, E) of pairs of integers (k, i) such that H∗(Y ) is generated
+as a cohomology ring by the direct sum of the images of
+Φ(i,k)
+E
+: H2m−k(X) → H2i−k(Y )
+as the pair (k, i) varies over all the elements in SY (X, E). Note that pr1(SY (X, E)) need not
+contain all integers from 0 to 2m. We call SY (X, E) an associated indexing set.
+Notations and Conventions 6.4. We fix the following notations:
+Seven := {(k, i) ∈ SY (X, E) | k even} and Sodd := {(k, i) ∈ SY (X, E) | k odd}
+p(Seven) := {2m − k|(k, i) ∈ Seven} and p(Sodd) := {2m − k|(k, i) ∈ Sodd}
+q(Seven) := {2i − k|(k, i) ∈ Seven} and q(Sodd) := {2i − k|(k, i) ∈ Sodd}
+Theorem 6.5. Let π1 : X ∗ → ∆∗ and π2 : Y∗ → ∆∗ be two smooth, projective families of
+relative dimensions m and n, respectively. Assume that there exists a coherent sheaf U over
+X ∗ ×∆∗ Y∗ such that it is flat over ∆∗ and for general t ∈ ∆∗, Yt is cohomologically generated
+by (Xt, Ut) by an indexing set SYt(Xt, Ut) such that π1 is strictly Mumford-Tate with respect to
+(p(Seven), p(Sodd)). Then, the family π2 is Mumford-Tate.
+Proof. Let t ∈ ∆∗ be such that Yt is cohomologically generated by (Xt, Ut) with indexing set
+SYt(Xt, Ut) such that π1 is strictly Mumford-Tate with respect to (p(Seven), p(Sodd)).
+Using
+Ehresmann’s theorem one can check that for any s ∈ ∆∗, Ys is cohomologically generated by
+(Xs, Us) and we have an equality of indexing sets SYt(Xt, Ut) = SYs(Xs, Us). Denote by
+TX := T(p(Seven),p(Sodd)) and TY := T(q(Seven),q(Sodd)) with X∞ replaced by Y∞.
+Recall, for any (k, i) ∈ SYt(Xt, Ut) we have the morphism Φ(i,k)
+U,∞ of mixed Hodge structures from
+H2m−k(X∞) to H2i−k(Y∞). This induces a morphism of mixed Hodge structures:
+φ : TX → TY.
+Recall, the cup-product morphism is a morphism of mixed Hodge structures [8, Lemma 6.16].
+So, the composition of the cup-product morphism with φ:
+Φ : TX
+φ−→ TY
+�
+−→ H∗(Y∞, Q)
+is a morphism of mixed Hodge structures. Given s ∈ ∆∗, denote by (see §6.1)
+TXs := Ts
+(p(Seven),p(Sodd)) and TYs := Ts
+(q(Seven),q(Sodd)) with Xs replaced by Ys.
+As before, we have the following composed morphism of Hodge structures:
+Φs : TXs → TYs
+�
+−→ H∗(Ys, Q),
+where the first morphism arises from Φ(i,k)
+Us
+as (k, i) ranges over entries in SYs(Xs, Us).
+By
+Theorem 6.2 we then have the following commutative diagram:
+TX
+Φ✲ H∗(Y∞, Q)
+⟲
+TXs
+j∗
+s
+❄
+Φs✲ H∗(Ys, Q)
+(j′
+s)∗
+❄
+
+MUMFORD TATE GROUPS AND THE HODGE CONJECTURE
+21
+where js (resp. j′
+s) is the natural inclusion of Xs (resp. Ys) into X∞ (resp. Y∞).
+Take γ ∈ F pH2p(Y∞, Q) i.e., γ is a Hodge class. We need to prove that j′
+s
+∗(γ) is a Hodge class
+in H2p(Ys, Q). Since Ys is cohomologically generated by (Xs, Us) and Φ is a morphism of mixed
+Hodge structures, there exists a Hodge class γ′ ∈ TX such that Φ(γ′) = γ. As π1 is strictly
+Mumford-Tate with respect to (p(Seven), p(Sodd)), we have j∗
+s(γ′) is fixed by MTs
+(p(Seven),p(Sodd)).
+Hence, j∗
+s(γ′) is a Hodge class in TXs. Since Φs is a morphism of Hodge structures, this means
+(j′
+s)∗(γ) = Φs ◦ j∗
+s(γ′) is a Hodge class.
+Therefore, π2 is a Mumford-Tate family. This proves the theorem.
+□
+We now use the above theorem to get an explicit example.
+Corollary 6.6. Let π1 : X → ∆ be a flat, projective family of curves satisfying the hypothesis
+in Proposition 6.1. Fix an invertible sheaf L on X ∗ := π−1
+1 (∆∗) of (relative) odd degree over the
+punctured disc ∆∗. Let
+π2 : M(2, L) → ∆∗
+be a relative moduli space of rank 2 semi-stable sheaves with fixed determinant L over X ∗.
+Then, π2 is a Mumford-Tate family.
+Proof. Consider the universal bundle U over X ∗ ×∆∗ M(2, L). It is well-known that for each
+t ∈ ∆∗, the fiber M(2, L)t := π−1
+2 (t) is cohomologically generated by (Xt, Ut) with the associated
+indexing set (see [24, Theorem 1]):
+{(0, 1), (0, 2), (1, 2), (2, 2)}
+By Proposition 6.1, π1 is strictly Mumford-Tate.
+Then, Theorem 6.5 implies that π2 is a
+Mumford-Tate family. This proves the corollary.
+□
+Remark 6.7. In fact, the relative moduli space M(2, L) mentioned in Corollary 6.6 degenerates
+to a singular variety. A desingularization of this variety satisfies the classical Hodge conjecture.
+See [7, Theorem 5.2] for details.
+Acknowledgements
+This article was motivated by some questions asked by Prof. C. Simpson, after the second
+author gave a talk on the article [7] at the workshop ‘Moduli of bundles and related structures’
+held at ICTS, Bengaluru, India. We thank Prof. Simpson for his interest and the organisers
+for organising the workshop. We also thank Prof. R. Laterveer for his comments on an earlier
+draft.
+References
+[1] S. Basu, A. Dan, and I. Kaur. Degeneration of intermediate Jacobians and the Torelli theorem. Documenta
+Mathematica, 24:1739–1767, 2019.
+[2] S. Bloch, H. Gillet, and C. Soul´e. Non-archimedean Arakelov theory. Journal of Algebraic Geometry, 4(4):427–
+486, 1995.
+[3] A. Dan. On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci. Communications
+in Contemporary Mathematics, 17(5):1550002, 2015.
+[4] A. Dan. On generically non-reduced components of Hilbert schemes of smooth curves. Mathematische
+Nachrichten, 290(17-18):2800–2814, 2017.
+[5] A. Dan. On a conjecture of Harris. Communications in Contemporary Mathematics, 23(07):2050028, 2021.
+[6] A. Dan and I. Kaur. Semi-regular varieties and variational Hodge conjecture. Comptes Rendus Mathematique,
+354(3):297–300, 2016.
+[7] A. Dan and I. Kaur. Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve. Annali
+di Matematica Pura ed Applicata (1923-), pages 1–20, 2022.
+
+22
+A. DAN AND I. KAUR
+[8] T. Fujisawa. Polarizations on limiting mixed Hodge structures. Journal of Singularities, 8:146–193, 2014.
+[9] W. Fulton. Intersection theory, volume 2. Springer Science & Business Media, 2013.
+[10] H Gillet and C Soul´e. Descent, motives and K-theory. Journal f¨ur die reine und angewandte Mathematik,
+478:127–176, 1996.
+[11] M. Green, P. A. Griffiths, and M. Kerr. Mumford-Tate Groups and Domains, Their Geometry and Arithmetic,
+volume 183 of Annals of Mathematics Studies. Princeton University Press, 2012.
+[12] R. Hartshorne. Algebraic Geometry. Graduate text in Mathematics-52. Springer-Verlag, 1977.
+[13] U. Jannsen. Mixed motives and algebraic K-theory, volume 1400. Springer, 2006.
+[14] M. Kashiwara and P. Schapira. Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, 292.
+[15] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat. Toroidal embeddings 1, volume 339. Springer,
+2006.
+[16] V. S. Kulikov. Mixed Hodge structures and singularities, volume 132. Cambridge University Press, 1998.
+[17] R. Laterveer. Surjectivity of cycle maps for singular varieties. Geometriae Dedicata, 179(1):265–278, 2015.
+[18] J. D. Lewis. A generalization of Mumford’s theorem, II. Illinois Journal of Mathematics, 39(2):288–304, 1995.
+[19] J. D. Lewis. The Hodge conjecture for a certain class of singular varieties. Mathematische Zeitschrift,
+224(1):25–31, 1997.
+[20] J. D. Lewis and B. B. Gordon. A survey of the Hodge conjecture, volume 10. American Mathematical Soc.,
+2016.
+[21] E. Markman. Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. Journal
+fur die reine und angewandte Mathematik, 544, 2002.
+[22] E. Markman. Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces.
+Advances in Mathematics, 208(2):622–646, 2007.
+[23] D. Mumford and P. Newstead. Periods of a moduli space of bundles on curves. American Journal of Mathe-
+matics, 90(4):1200–1208, 1968.
+[24] P. E. Newstead. Characteristic classes of stable bundles of rank 2 over an algebraic curve. Transactions of
+the American Mathematical Society, 169:337–345, 1972.
+[25] C. Peters and J. H. M. Steenbrink. Mixed Hodge structures, volume 52. Springer Science & Business Media,
+2008.
+[26] W. Schmid. Variation of Hodge structure: the singularities of the period mapping. Inventiones mathematicae,
+22(3-4):211–319, 1973.
+[27] E. Sernesi. Deformaions of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften-334.
+Springer-Verlag, 2006.
+[28] J. Steenbrink. Limits of Hodge structures. Inventiones mathematicae, 31:229–257, 1976.
+[29] B. Totaro. Chow groups, Chow cohomology, and linear varieties. In Forum of Mathematics, Sigma, volume 2.
+Cambridge University Press, 2014.
+[30] C. Voisin. A counterexample to the Hodge conjecture extended to K¨ahler varieties. International Mathematics
+Research Notices, 2002(20):1057–1075, 2002.
+[31] C. Voisin. Hodge Theory and Complex Algebraic Geometry-I. Cambridge studies in advanced mathematics-76.
+Cambridge University press, 2002.
+[32] C. Voisin. Some aspects of the Hodge conjecture. Japanese Journal of Mathematics, 2(2):261–296, 2007.
+School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road,
+S3 7RH, UK
+Email address: [email protected]
+Department of Mathematical Sciences, Loughborough University, LE11 3TU, U.K
+Email address: [email protected]
+

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+Even if Explanations: 
+Prior Work, Desiderata & Benchmarks for Semi-Factual XAI 
+Saugat Aryal1,2 , Mark T. Keane1,2  
+1School of Computer Science, University College Dublin, Dublin, Ireland 
+2 Insight Centre for Data Analytics, Dublin, Ireland  
+ 
+[email protected], [email protected] 
+ 
+Abstract 
+Recently, eXplainable AI (XAI) research has 
+focused on counterfactual explanations as post-
+hoc justifications for AI-system decisions (e.g., a 
+customer refused a loan might be told “if you 
+asked for a loan with a shorter term, it would have 
+been approved”). Counterfactuals explain what 
+changes to the input-features of an AI system 
+change the output-decision. However, there is a 
+sub-type of counterfactual, semi-factuals, that 
+have received less attention in AI (though the 
+Cognitive 
+Sciences 
+have 
+studied 
+them 
+extensively). This paper surveys these literatures 
+to summarise historical and recent breakthroughs 
+in this area. It defines key desiderata for semi-
+factual XAI and reports benchmark tests of 
+historical algorithms (along with a novel, na¨ıve 
+method) to provide a solid basis for future 
+algorithmic developments. 
+1 Introduction 
+With the emergence of deep learning there have been rising 
+concern about the opacity of Artifical Intelligence (AI) 
+systems and their impact on public and private life [Adadi 
+and Berrada, 2018; Guidotti et al., 2018]. Currently, 
+governments are taking steps to protect people’s rights in 
+these areas, to regulate the AI industry and ensure that 
+these technologies are not abused (e.g., the EU’s GDPR 
+[Goodman and Flaxman, 2017]). Research on eXplainable AI 
+(XAI) tries to address these issues using automated 
+explanations to improve the transparency of black-box 
+models, to facilitate the auditing of datasets and to ensure 
+fairness, accountability and trustworthiness [Gunning and 
+Aha, 2019; Sokol and Flach, 2019; Birhane et al., 2022]. 
+Recently, significant research effort have been expended 
+on counterfactual explanations for XAI [Byrne, 2019; Miller, 
+2019; Keane et al., 2021; Karimi et al., 2022]; for instance, a 
+recent survey paper reports 350 papers on the topic [Verma 
+et al., 2022]. In this paper, we survey a less-researched 
+special case of the counterfactual using semi-factual 
+explanations. In this review, we profile the literature on 
+semi-factuals, we define desiderata for this explanation 
+method, identify key evaluation metrics and implement 
+baselines to provide a solid base for future work. 
+Counterfactuals aim to explain algorithmic decisions in a 
+post-hoc fashion, as an after-the-fact justification, by 
+showing end-users what features could change an 
+automated decision (e.g., a customer refused a loan might 
+be told “if you asked for a loan with a shorter term, it would 
+have been approved”). In XAI, counterfactuals are typically 
+used to explain what changes to the input-features of an AI 
+system will change the output-decision (e.g., a class change, 
+loan-refused to the loan-approved; see also Fig. 1). 
+Technically, they could be called “outcome-counterfactuals” 
+as they capture changes to the world that change the 
+outcome (here, to be consistent with the literature, we will 
+mostly call them “counterfactuals”). 
+Semi-factuals are a special-case of the counterfactual; 
+they differ from outcome-counterfactuals in that they show 
+endusers the feature changes that do not change a 
+Figure 1: A and B are two semi-factuals (in blue) for the 
+query Q (in green) all in the same class (i.e. the negative 
+one), whereas the counterfactual C (in red) is over the 
+decision boundary in the positive class. A is considered 
+to be a better semi-factual than B, because A is further 
+from Q and closer to the decision boundary. 
+ 
+
+manifold
+data
+AO
+B
+C
+?
+-
+-decisionoutcome (e.g., “Even if you asked for a lower car-
+loan, you would still have been refused the loan” or “Even if 
+you doubled your income, you would still be refused”). They 
+are “counterfactual” in that they convey possibilities that 
+“counter” what actually occurred, even though the outcome 
+does not change (see Fig.1). Philosophers have argued over 
+whether 
+semi-factuals 
+really 
+differ 
+from 
+outcomecounterfactuals (see [Bennett, 2003; Goodman, 
+1947]), but they have been shown to differ in their 
+psychological impacts [McCloy and Byrne, 2002; Parkinson 
+and Byrne, 2017]. 
+We believe that the benefits accruing to counterfactuals also 
+accrue to semi-factuals in XAI; namely, that they have many 
+legal [Wachter et al., 2017], psychological [Byrne, 2019] and 
+technical benefits [Keane et al., 2021]. For example, in 
+medicine it is often important to know what changes (e.g., 
+inflammation or cell changes) occur just before an illness 
+emerges (e.g., an ulcer or cancer). Similarly, semi-factuals 
+can reveal errors in causal models (e.g., a farmer might be 
+told “Even if you doubled fertiliser use, your yield would not 
+increase” because of soil factors). However, as we shall see, 
+semi-factuals also differ significantly in several respects from 
+counterfactuals (see desiderata, section 4). 
+Outline of Paper & Contributions: In this paper, we 
+systematically 
+review 
+prior 
+work 
+on 
+semi-factuals 
+(henceforth, SFs) in the Cognitive Sciences and AI, beginning 
+with a discussion of key examples from the early literature in 
+Philosophy and Psychology (see section 2). From this work 
+we define desiderata for SFs (section 3). In section 4, we 
+report the results of a systematic survey before sketching 
+the brief history of semi-factual algorithms for explanation 
+(section 5). We then report a benchmarking study 
+implementing key historical algorithms along with a newly-
+proposed na¨ıve benchmark (see section 6), before closing 
+with some conclusions (see section 7). As such, the paper 
+makes several novel contributions to this emerging area of 
+XAI, providing: 
+• A comprehensive survey of the relevant literature. 
+• A first statement of desiderata for semi-factual XAI. 
+• A na¨ıve benchmark algorithm, based on the new idea 
+of Most Distant Neighbors (MDNs). 
+• Novel comparative tests of historical benchmarks, 
+toidentify the best for future use. 
+• A publically-available repository of metrics, data, re-
+sults, benchmarks and an annotated bibliography (see 
+https://github.com/itsaugat/sf survey). 
+2 Philosophy & Psychology of Semi-Factuals 
+Semi-factuals have been studied under different guises in 
+Philosophy and Psychology for several decades. In 
+Philosophy, counterfactuals (if only...) and semi-factuals 
+(even if...) are often compared to conditionals (if...then) with 
+ 
+1 Because Philip is allergic to the ice-cream in both desserts. 
+a view to analysing their logic, truth conditions and role in 
+causation [Chisholm, 1946; Goodman, 1947; Bennett, 1982; 
+Barker, 1991; Bennett, 2003]. For example, [Bennett, 1982] 
+and [Barker, 1991] argue about how the words “even” and 
+“still” affect the interpretation of examples, such as: 
+(1) Even if the United States has used nuclear weapons in 
+Vietnam, it would still have lost the war. 
+where the semi-factual asserts that even if the military-force 
+expended by United States significantly increased, the 
+Vietnam War would still have been lost. In AI terms, the 
+semifactual says increasing the feature-value of military-
+force would not change the outcome. So, [Iten, 2002] 
+proposes “scalar” analyses of even and even if; “Even Neville 
+passed the exam” puts Neville low on an academic-ability 
+scale). 
+In Psychology, as in AI, semi-factual research has grown 
+out of counterfactual studies, specifically, from proposals on 
+counterfactual thinking in human cognition [Kahneman and 
+Tversky, 1982; Byrne, 2007; Handley and Feeney, 2007; 
+Epstude and Roese, 2008]. Byrne [2007] proposed a mental 
+models theory of semi-factuals that has been tested in 
+several psychological studies (see e.g., [McCloy and Byrne, 
+2002; Parkinson and Byrne, 2017]). McCloy & Byrne’s [2002] 
+seminal work explicitly compared people’s reasoning using 
+matched scenarios for counterfactuals and semi-factuals, 
+akin to the case of Philip who has an allergic reaction to an 
+icecream sundae: 
+(2) If only Philip had not chosen the ice-cream sundae, he 
+wouldn’t have had an allergic reaction. 
+(Counterfactual) 
+(3) Even if Philip had chosen the banana split, he would 
+still have had an allergic reaction1. (Semi-factual) 
+McCloy & Byrne found that counterfactuals lead people to 
+judge the antecedent event (i.e., the choice of dessert) to be 
+more causally-related to the outcome, but semi-factuals had 
+the opposite effect, leading people to judge the antecedent 
+event to be less causally-related to the outcome. So, semi-
+factuals weaken the causal link between the inputs and 
+outcome, convincing people that outcome would have 
+occurred anyway (people also differ in their emotional 
+reactions to these events). In another experiment, they also 
+found that counterfactuals lead people to focus on 
+alternative antecedents that undo the outcome (e.g., “If only 
+Philip had chosen the cheese cake they would not have had 
+a reaction”), whereas semi-factuals lead people to focus on 
+alternative antecedents that do not undo the outcome (e.g., 
+“Even if Philip had chosen the baked-alaska he would still 
+have had a reaction”). Subsequent studies have tested other 
+psychological aspects of semi-factuals [Parkinson and Byrne, 
+2017; Moreno-Rios et al., 2008; Espino et al., 2022]. 
+
+Taken together these psychological findings show that 
+semi-factuals have very different psychological effects than 
+counterfactuals. 
+Unlike 
+counterfactuals, 
+semi-factuals 
+convince people of the status quo, they dissuade them from 
+questioning outcomes [Green, 2008], and weaken the causal 
+link between features and outcomes. 
+3 Desiderata for Semi-Factuals 
+Several desiderata are suggested by these analyses of 
+semifactuals. These desiderata cover computational (i.e., 
+“what needs to be computed”) and psychological 
+requirements (i.e., the response to be elicited in users) and 
+are defined as follows. 
+Assume (i) a query instance, Q, that has a vector, x, and an 
+outcome, y, that occurs when x holds and (ii) a semi-factual 
+instance, SF, that has a vector, x′, and an outcome, y′, that 
+occurs when x′ holds. SF will be a good explanation of Q if: 
+a) Q is factually the case and SF counters some of Q’s facts 
+but not Q’s outcome; so the vectors x and x′ differ, 
+diff(x, x′), with no outcome change, y = y′ 
+b) Ideally, SF relies on sparse changes to a key-feature(s), 
+f, of Q, with other features being equal1; ideally, one 
+feature change (i.e., diff(x,x′)=1) 
+c) The 
+key-feature(s) 
+changed 
+should 
+be 
+plausible/mutable/actionable; that is, the SF produced 
+by the change should be within the data-manifold. 
+d) People should find the SF convincing even though it 
+may seem to be unexpected/surprising/counter-
+intuitive; for instance, they may expect the key-feature 
+change to change the outcome, where y ̸= y′ 
+e) If people accept SF, it will change their perception of 
+the causal role of the key-feature(s), f, in the domain. 
+So, their causal model of the domain will change (e.g., 
+causes may be updated/deleted/refined). 
+f) For fairness and ethical reasons, the asserted 
+differencesbetween Q and SF, should not be 
+misleading. For instance, (i) the key-feature should not 
+be a proxy variable, (ii) the change should not just 
+address a small local region in the decision space, (iii) 
+though the change may be unexpected it should not 
+violate the domain’s causality, (iii) the change assumes 
+ceteris paribus (i.e., “other things being equal”), 
+verifiably so (i.e., the unchangedoutcome shown 
+should not depend on subtle interactions with other 
+variables). 
+These desiderata present a high bar for semi-factual 
+explanation methods; indeed, it is unclear whether any 
+current method meets all of them. Furthermore, some of 
+them may require further computational specification (e.g., 
+how 
+keyfeatures 
+are 
+selected) 
+and 
+psychological 
+ 
+2 Equal may not mean the features have identical values, they may just 
+be within some threshold difference. 
+specification in operational definitions for user studies (e.g., 
+for the notions of plausibility, convincingness and surprise). 
+4 Systematic Survey: Even if Explanations 
+A systematic search of the AI, Philosophy and Psychology 
+literatures on semi-factuals was conducted using a 
+bottomup citation-search and top-down keyword-searches 
+(see Table 1). Ten searches were carried out between 
+October 12th, 2022 and December 19th, 2022, consisting of 
+(i) a bottomup search checking GoogleScholar citations to 
+three key papers (i.e., [Cummins and Bridge, 2006; Nugent 
+et al., 2009; Kenny and Keane, 2021], (ii) nine top-down 
+searches using keywords in GoogleScholar (see Table 1). The 
+papers found (N=1,150) were title-and-abstact screened to 
+check whether they were just citing semi-factuals or 
+substantially researching them as a topic. Subsequent 
+selections then identified the core papers of relevance (see 
+here for PRISMA diagram). 
+Table 1: Ten searches used in the systematic survey of 
+GoogleScholar (12-10-2022 to 19-12-2022) with the number 
+of papers found and unique papers reviewed further (n.b., 
+“sf”, “ai” and “xp” are short for “semi-factual”, “artificial 
+intelligence” and “explanation”,respectively). 
+ 
+4.1 Survey Results 
+Of the 1,150 original results checked, 92 potentially-relevant 
+papers were selected to be read in depth from which 62 core 
+papers were identified (41 cited here; note, 145 duplicates 
+Search Terms 
+# 
+Papers 
+Found 
+Unique 
+Papers 
+*no search terms* 
+(citation search of key papers) 
+1 
+108 
+17 
+“sf”, “nearest-neighbor” 
+2 
+20 
+3 
+“sf”, “ai” 
+3 
+95 
+12 
+“sf”, “ai”, “xp” 
+4 
+86 
+12 
+“sf”, “xai” 
+5 
+44 
+0 
+“ai”, “xp”, (“near-hit” OR 
+“nearest-hit”) 
+6 
+230 
+20 
+“ai”, “xp”,“nearest-
+likeneighbors” 
+7 
+12 
+0 
+“sf”, “xp”, “philosophy” 
+8 
+203 
+11 
+“sf”, “xp”, “psychology” 
+9 
+228 
+3 
+“xp”, “even if conditionals”, 
+“linguistic”, “philosophy” 
+10 
+124 
+14 
+Totals 
+ 
+1,150 
+92 
+
+were removed). As we shall see in the next section on history 
+(section 5), from a low base semi-factual research in AI has 
+expanded considerably in the last two years. Note, many 
+semi-factual papers in Philosophy, Psychology and 
+Linguistics were checked but few are specifically relevant to 
+explanation (e.g., in Philosophy the focus tends to be on the 
+truth conditions of counterfactual statements and the 
+linguistic functions of “even” and “still”). Finally, it should  
+also be said that many excluded papers were from closely-
+related areas that do not cover semi-factuals per se, but 
+which could provide insights for future work; areas that 
+include research on (i) case difference learning (e.g., 
+[Hanney and Keane, 1996; Ye et al., 2021]), (ii) feature 
+selection using near misses (e.g., [Kira et al., 1992; 
+Herchenbach et al., 2022]), (iii) counterfactual explanation 
+(e.g., [Keane et al., 2021; Verma et al., 2022]), (iv) flip-points 
+in learning (e.g., [Yousefzadeh and O’Leary, 2019]) and (v) 
+dynamic critiquing in recommenders (e.g., [Reilly et al., 
+2004]). These papers are recorded in a publically-available 
+annotated biblography (see https://github.com/itsaugat/sf 
+survey). 
+5 A Brief History of Semi-Factual XAI 
+Thought absent in AI, there are long-standing literatures on 
+semi-factuals in Philosophy and Psychology [Bennett, 2003; 
+Byrne, 2007]. Much of the initial work emerged from 
+CaseBased 
+Reasoning 
+(CBR) 
+research 
+on 
+post-hoc, 
+examplebased explanations [Sørmo et al., 2005; Keane and 
+Kenny, 2019]. In this AI research, semi-factual explanations 
+have been variously cast as a fortori arguments [Nugent et 
+al., 2005; Nugent et al., 2009] and precedent-based 
+explanations [Cummins and Bridge, 2006; Bridge and 
+Cummins, 2005]. More recently, Kenny & Keane [2021] re-
+connected this work to the Cognitive Science literatures by 
+calling them “semifactuals”. Arguably, there are four distinct 
+phases in the development of semi-factual explanation ideas 
+in AI: (i) initial utility-based proposals, (ii) proximity-based 
+methods, (iii) local-region methods and (iv) the more recent 
+“modern-era” of counterfactually-inspired proposals. In the 
+following subsections, we describe each in turn and the 
+intuitions behind them. We end this section by defining a 
+new benchmarkmethod based on the notion of Most Distant 
+Neighbors (MDNs). 
+ 
+5.1 Semi-Factuals Based on Feature-Utility 
+Doyle et al. [2004] appears as the first AI paper in our 
+searches to propose using semi-factuals to explain 
+automated decisions, under the rubric of a fortori reasoning. 
+An a fortori argument is defined as one that uses a stronger 
+version of an already-convincing proposition (i.e., “EU 
+countries cannot afford standing armies, sure even the US 
+can hardly afford its standing army”). Doyle et al. [2004] 
+noted that nearestneighbor, example-based explanations 
+can often be less convincing than neighbors that have more 
+extreme feature-values within the same class. For example, 
+if patient-x with a moderate temperature is judged to be 
+dischargeable then a semifactual past case, patient-y with a 
+much higher temperature who was discharged is more 
+convincing than pointing to another patient with the same 
+moderate temperature being discharged [Doyle et al., 2006]. 
+So, this semi-factual method computes a set of nearest 
+neighbours as explanatory cases and then re-ranks them 
+using utility functions on selected features to find a more 
+convincing a fortori case, as follows: 
+where q is the query, x is an instance, c is a class label and 
+ξ( ) measures the contribution to explanation utility of the 
+feature f. The ξ() function uses relative-differences in 
+feature-values to assign utilities. For example, for the 
+temperature feature, the measure might assign higher utility 
+to a 10◦C difference than to a 5◦C difference between a 
+query and semi-factual case. This method priorities 
+explanatory instances with more convincing feature-values, 
+and 
+may 
+compute 
+these 
+over 
+multiple 
+features. 
+Furthermore, these utilities are seen as being class-specific 
+and, even, user-specific, depending on what a given user 
+may find convincing. Furthermore, these utility values often 
+decrease as instances approach the decision boundary, 
+rather than just being linearly increasing functions. 
+However, this method was knowledge-intensive, the 
+utility values for each feature had to be hand-coded for each 
+class (and, presumably, for each end-user). Indeed, in one of 
+their user tests, the utility measures had to be re-defined 
+half-way through the study to better reflect end-users’ 
+assessments [Doyle et al., 2006]. This is a major drawback 
+for the technique, as it begs the critical question about what 
+featuredifferences will actually be more convincing. 
+Accordingly, this utility method is not a plausible benchmark, 
+though we do use their intuition about feature-differences 
+to define a new, useful benchmark method (see section 5.4). 
+5.2 NUN-Related Semi-Factuals 
+Cummins & Bridge’s [2006] “Knowledge-Light based 
+Explanation-Oriented 
+Retrieval” 
+(KLEOR) 
+approach 
+proposed three methods based on similarity to Nearest 
+Unlike Neighbors (NUNs). These KLEOR variants use the NUN 
+to find the best semi-factual for a given query (n.b., they 
+called the NUN, a Nearest Miss). In modern parlance, the 
+NUN is the closest counterfactual in the dataset to the query 
+(see [Keane and Smyth, 2020]). 
+The first variant, Sim-Miss, selects an instance to be the 
+semi-factual which is most similar to the NUN but in the 
+same class as the query q: 
+
+Utility(q, ,c) = wfs (qf,&f,c)
+(1)
+fEF
+SFUtility (q, , c) = argmax Utility(q, &, c)
+(2)SFsim-Miss (q, nun, G) = arg max Sim(r, nun)
+(3)where q is the query, x is the instance, G represents the set 
+of all instances in the same class as the query, and nun is the 
+Nearest Unlike Neighbor, with Sim being Euclidean Distance 
+or Cosine Similarity. This variant is the most naieve as it 
+assumes an simple decision boundary. The second variant, 
+Global-Sim method, is more sophisticated in that it requires 
+the semi-factual be closer to q than to the nun (to avoid SFs 
+far from the query but close to the NUN): 
+using the global similarity between instances. Finally, the 
+third variant, Attr-Sim, computes more fine-grained 
+similarities for each feature-attribute, ensuring that the 
+semi-factual lies between the q and nun across the majority 
+of features: 
+ 
+where F is the feature-dimension set and a is a 
+featureattribute. These methods rely on the interesting 
+intuition that a known counterfactual can guide finding a 
+good semi-factual explanation. Furthermore, Cummins & 
+Bridge also showed, using computational and user 
+evaluations, that SFSim-Miss and SFAttr-Sim can do as well as 
+SFUtility, without the knowledge engineering overheads of 
+the latter, albeit on a single dataset. Accordingly, this 
+method is used in the present benchmarking study (see 
+section 6). 
+5.3 Semi-Factuals Near Local-Region Boundaries 
+Nugent et al. [2009] proposed another a fortori method, by 
+finding marginal instances in the local region around the 
+query. Here, a surrogate model, specifically, logistic 
+regression was used to capture the local neighborhood 
+around the query, built using perturbations of it (akin to 
+LIME [Ribeiro et al., 2016]) . Then, candidate nearest 
+neighbors are tested using this local model to give a 
+probability, with the marginalprobability instance closest to 
+the decision boundary, being chosen as the semi-factual 
+explanation, as follows:  
+where, C is the set of candidate neighbors and LR() is the 
+local logistic regression model providing the probability 
+score. 
+The intuition here is that good semi-factuals should be 
+close to the query’s local decision boundary, while being as 
+far as possible from it in this local space (see Fig. 1). So, a 
+convincing semi-factual explanation should be locally close 
+to the query but as distant from it as possible within this local 
+region. Unfortunately, Nugent et al. [2009] did not evaluate 
+this method beyond providing indicative outputs, that seem 
+to be informative semi-factuals. Accordingly, it is also used 
+in the present benchmarking study (see section 6). 
+ 
+5.4 A New Benchmark: Most Distant Neighbors 
+Analogies between counterfactual XAI and semi-factuals 
+suggest another na¨ıve benchmark that has not been 
+proposed before in the literature. Early counterfactual 
+methods often used Nearest Unlike Neighbors (NUNs), the 
+nearest classdifferent instance in the dataset to the query, 
+as counterfactual explanations [Cunningham et al., 2003; 
+Wexler et al., 2019]. NUNs are reasonable first-pass at 
+counterfactuals that are guaranteed to be within-domain 
+(though they have other weaknesses). An analogous 
+solution for semi-factual explanations relies on the notion of 
+Most Distant Neighbors (MDNs); namely, the most distant 
+same-class instance in the dataset to the query on some key-
+feature. MDNs should be good semi-factuals because they 
+reflect many of the desiderata and are, by definition, within 
+domain. 
+To compute MDNs, for a given feature of q, its neighbours 
+on the dimension are partitioned into instance-sets that 
+have higher values (i.e., HighSet) or lower values (i.e., 
+LowSet) than the query. Each of these sets are ranked-
+ordered separately using the “Semi-Factual Score” (sfs) 
+function, a distance messure that prioritises instances that 
+are sparse (few feature differences) while also having the 
+highest valuedifferences on a key-feature, as follows: 
+ 
+ 
+where S is HigherSet or LowerSet and x ∈ S, same() counts 
+the features that are equal between q and x, F is the total 
+number of features, diff() gives the difference-value of 
+keyfeature, f, and diffmax() is the maximum difference-value 
+for that key-feature in the HighSet/LowSet. Basically, the 
+instance with the highest overall sfs value from the 
+HighSet/LowSet is the best candidate for that feature. This 
+computation is done for each feature of q, independently, 
+
+SFAttr-Sim(q, nun, G) = arg max Sim(α, nun)
+EC
++ maxcount[Sim(qa, aa) > Sim(qa, nuna)]
+aeF
+(5)Algorithm1MDN Semi-factual
+Input: query q
+Output:Semi-factual(q)
+1:InitializeI=0,F=0
+2: for feature f = fi, f2, fs, ..., fn do
+3:
+S=:or≤]
+High/Low Set
+4:
+foraESdo
+5:
+I ← I Usfs(x)
+Equation 7
+6:
+end for
+7:
+F←FUmax(I)
+8:
+end for
+9: SF(q) ←max(F)
+10: return SF(q)same(q, a)
+diff(qf, cf)
+sfs(q, S, F) =
+(7)
+F
+diffmar(qf, Sf)SFGlobal-Sim(q, nun,G) = arg max Sim(c, nun)
+CEG
+(4)
++ Sim(q,r) > Sim(q,nunSFLocal-Region(q, C) = arg min LR(α)
+(6)with the best of the best instances (i.e., with the highest sfs 
+value across all features) being chosen as the overall semi-
+factual for the query (see Algorithm 1). 
+The intuition behind MDNs is that if one can find a instance 
+that has some features in common with the query but is as 
+far from it on a key-feature, then it will make a good 
+semifactual (see Desiderata). This new method was also 
+added to benchmarking study to compare it to the historical 
+methods. 
+5.5 The Modern Era: Post-2020 Methods 
+Kenny & Keane [2021] instigated, what could be called, the 
+modern-era of semi-factual AI research when they proposed 
+a GAN-based counterfactual method for images, called 
+PIECE, that also computed semi-factuals. PIECE finds 
+“exceptional” and “normal” features for a given class and 
+then modifies the query’s “exceptional” features to create 
+instances that have the “normal” features of the 
+counterfactual 
+class, 
+using 
+the 
+GAN 
+to 
+generate 
+visualisations. As successive exceptionalfeatures are 
+changed the generated instances move away from the query 
+towards the counterfactual class, with the instance 
+generated just before the decision boundary being identified 
+as the semi-factual. Kenny & Keane showed that these 
+generated semi-factuals were more distant from the query 
+than those produced by other perturbation techniques (see 
+their Expt.2). In one sense, this solution re-imagines the 
+CumminsBridge intuition that good semi-factuals can be 
+found somewhere between the query and a counterfactual, 
+close to the decision boundary. 
+PIECE kicked off a renewed interest in semi-factual XAI as 
+researchers have looked to improve on it and to apply semi-
+factuals in different application contexts. So, Zhao et al. 
+[2022] have proposed a class-to-class variational encoder 
+(C2C-VAR) which is less computationally expensive than 
+PIECE that can generate semi-factuals (and counterfactuals). 
+Vats et al. [2022] have used StyleGAN2 [Karras et al., 2020] 
+to find semi-factual explanations for classifications of 
+medical images of ulcers. While these works try to explain 
+model capabilities, others have proposed using semifactuals 
+to explain model limits. Artelt & Hammer [2022] use semi-
+factuals to explain the “reject option”; that is, the option 
+where an AI system rejects inputs because “a prediction with 
+an unacceptable lower certainty” can only be made. Their 
+perturbation-based optimisation method uses a loss 
+function that promotes diverse semi-factuals that are (i) in 
+the same class as the query (they are also rejected), (ii) 
+sparse (they aim for 1-feature-difference), (iii) “sufficiently 
+distant” from the query, and (iv) of higher certainty than the 
+query (to make them more convincing). Notably, here, the 
+key-feature being varied is the certainty of the instance’s 
+prediction. In a similar vein, Lu et al. [2022] argue that semi-
+factuals may be used to explain spurious patterns using 
+human-in-the-loop ML. Finally, Mertes et al. [2022] propose 
+what appears to be a wholly new type of counterfactual, 
+called “alterfactuals”, to explore the“irrelevant feature” 
+space of the model; they describe these as semi-factuals that 
+“move parallel to the decision boundary, indicating which 
+features would not modify the model’s decision”. Other 
+proposals have also been made that suffer from a poor 
+knowledge of the literature (see e.g., [Fernandez et al., 2022; 
+Herchenbach et al., 2022]). 
+Finally, from the user perspective, Mueller et al. [2021] 
+include a semi-factual module in their cognitive tutorial for 
+training users about “cognitively-challenging aspects of an AI 
+system” and [Salimi, 2022] reports user-tests for 
+trustworthiness after using semi-factuals. 
+These recent papers reflect a rapidly-expanding interest in 
+semi-factual XAI. In time, these modern-era methods will 
+need to be comparatively evaluated relative to the 
+benchmarks and metrics proposed here, to determine which 
+fare best in explaining predictions to end-users. 
+6 Benchmarking Study 
+To provide a firm empirical basis for future work on 
+semifactual XAI, we ran a benchmark study of five methods, 
+the four historical methods [i.e., the three KLEOR methods 
+(SimMiss, Global-Sim, Attr-sim) and the Local-Region one) 
+and the newly-proposed MDN method. Standard evaluation 
+metrics from prior XAI work were used to compare these 
+methods, using the five measures detailed below. 
+Query-to-SF Distance: The L2-norm from the Query to the 
+SF, where higher scores are better, as the semi-factual 
+should be far from from the query 
+Query-to-SF kNN (%): This is a measure of the percentage 
+of instances (within the whole dataset) in the k-NN set 
+surrounding the Query that occur before the SF is included 
+(i.e., as k is successively increased upto the appearance of 
+the SF); it is an alternative measure for how far the SF is from 
+the Query in the dataset, so higher values are better. 
+SF-to-Query-Class Distance: A within-distribution measure 
+for the closeness of the SF to the distribution of the Query-
+Class using Mahalanobis distance, where lower values 
+indicate that the SF is closer to the query-class distribution. 
+MDN Distance: The sfs function, a semi-factual-oriented 
+distance for comparing Queries and a candidate-SFs, can 
+also be used to determine how far the SFs selected by 
+historical methods are from the Query; this metric allows us 
+to assess whether historical methods find “better” MDNs 
+than the MDN-method itself, where higher sfs values 
+indicate the SF is a better MDN for the Query 
+Sparsity (%): The L0-norm counting the number of feature-
+differences between the Query and SF, divided into three 
+levels (i.e., 1-diff, 2-diff and >3-diff) where the percent of SFs 
+selected by the method at each level is recorded; obviously, 
+methods with higher percentages at lower difference levels 
+are better (ideally, high-percentages at the 1-diff level). 
+
+SFMDN(q, S) = arg max sfs()
+(8)
+ES6.1 Method 
+We performed leave-one-out cross-validation for each of the 
+five methods on seven datasets to find a semi-factual for 
+every instance in the dataset, treating each as a query. We 
+used 3-NN model to implement the KLEOR variants. For the 
+Local Region method, we consider a minimum of 200 
+instances from each class to build the local model for a 
+query. In the MDN method, a “20% of the standard 
+deviation” threshold was used to determine whether values 
+for a given feature were essentially “the same”. The seven 
+datasets were benchmark, publically-available, tabular 
+datasets commonly used in the counterfactual literature, 
+which were binary-classed: AdultIncome (N=26,540, 12 
+features), Blood Alcohol (N=2,000, 5 features), Default 
+Credit Card (N=30,000, 23 features), Pima Diabetes (N=392, 
+8 features), German Credit (N=1,000, 20 features), HELOC 
+(8,291 instances, 20 features), Lending Club (N=39,239, 8 
+features). All the experiments were carried out in Python 3.9 
+on Ubuntu 16.04 machine with 40 core Intel Xeon(R) 
+processor with an approximate run-time of 40 hours. All 
+programs, 
+data 
+and 
+results 
+are 
+available 
+at 
+https://github.com/itsaugat/sf survey. 
+ 
+6.2 Results & Discussion 
+Figures 2 summarises the overall results for the five methods 
+(as mean ranks over datasets) on the five benchmark 
+measures (Figures 3 and 4 show results by-dataset). The 
+summary shows that MDN does best on three of the five 
+measures (i.e., Query-to-SF Distance, Query-to-SF kNN, 
+MDN Distance), with the Local Region method being a close 
+second; performance on the two other metrics (SF-to-
+Query-Class 
+Distance, 
+Sparcity) 
+require 
+further 
+interpretation. 
+On the Query-to-SF Distance metric (Figure 3a) it can be 
+seen that MDN produces the highest Query-to-SF distances 
+for 4 of the 7 datasets, showing that it tends to find the 
+furthest SF-instances from the query. On the Query-SF kNN 
+metric (Figure 3b) MDN again scores the highest in 3 of 7 
+datasets with overall percentages that stand out; so, MDN 
+finds SFs separated from the Query by many instances. On 
+the SF-to-Q-Class Distance measure (Figure 3c) MDN scores 
+less well (overall it is ranked 4th); though all these SFs are 
+by-definition within distribution (as valid datapoints), MDN 
+probably scores lower as it is finding more instances at the 
+edges of the distribution. On the MDN-Distance metric 
+(Figure 3d) the four historical methods mainly produce lower 
+scores across datasets (except for the HELOC dataset) 
+showing that the MDN method is finding the furthest SFs 
+from the Query in the dataset. 
+The one wrinkle in MDN’s performance is on the sparsity 
+measure. As a rough reckoning, in Figure 4, the higher the 
+blue-portion of the bars [i.e., the % of 1-diff SFs] for a given 
+method-dataset pair, the better the performance. In Figure 
+4, we can see that MDN does the worst of all the methods in 
+three datasets where 100% of its SFs have >3-
+featuredifferences (though in three others it fares better). 
+This performance could probably be improved by fine-tuning 
+the sfs function [see formula (7)]. Recall, that this function 
+has two equally-weighted components, that compute (i) 
+samefeatures and (ii) relative-differences in the key-feature. 
+If a higher weight was given to the same-features 
+component, then the method should select sparser SFs 
+(perhaps also aided by a scoring threshold). For the present 
+work, we felt it was better to provide a vanilla sfs function to 
+get a clear sense of how a baseline-MDN method might 
+work. 
+Overall, in conclusion, though it seems that the MDN and 
+the Local Region methods provide the best candidates for 
+semi-factual baselines. The Local Region method provides 
+reasonable, solid results with decent sparsity, whereas the 
+MDN method shows the furthest point in the dataset than 
+an SF can be from the Query (as type of upper limit to beat). 
+ 
+ 
+Figure 2: Mean Ranks of Success of the Five Benchmark 
+Methods on Five Different Measures, for the Tested 
+Datasets. 
+ 
+ 
+
+1
+m
+4
+6
+8
+9
+Query-to-SFDistance
+Query-to-SFkNN
+SFtoQuery-ClassDistance
+MDNDistance
+Sparsity
+Sim-MissGlobal-SimAttr-SimLocal-RegionMDN7 Conclusion 
+In recent years, counterfactual explanations has been 
+heavily researched as a significant explanation strategy in 
+XAI. Yet, very little attention has been given to an, arguably, 
+equally useful method that relies on semi-factuals (where 
+changes to input features do not lead to output changes). In 
+this paper, from a systematic survey, we aim to remedy this 
+deficit and place this topic area on a firm footing with 
+defined desiderata, benchmarked methods and suitable 
+metrics. In conclusion, several limitations and caveats are to 
+be noted. 
+With respect to limitations, it is to be noted that in the 
+current benchmark study we have concentrated on tabular 
+data, largely to respect the focus of historical methods. 
+However, the desiderata and evaluation metrics should 
+equally apply to image dataset (and possibly time-series 
+data), albeit relying more on latent features (as has been 
+demonstrated in [Kenny and Keane, 2021]). The paucity of 
+user studies is another severe limitation; until some 
+carefully-controlled studies are carried out, we do not really 
+know how users will respond to these explanations in the AI 
+context. 
+With respect to caveats, we believe that it is important to 
+reiterate the ethical point about the use of semi-factuals (a 
+point that also applies to counterfactuals [Asher et al., 
+2022]). These explanatory methods have significant 
+cognitive impacts on people’s understanding of AI systems 
+and domains, they convince and dissuade people. But, they 
+could be misused if certain assumptions are violated (e.g., if 
+the SF is not representative of the data). So, future 
+implementations of these methods will need to provide 
+metrics to audit these assumptions, to ensure they are being 
+properly and fairly applied in advice to end-users. 
+ 
+Figure 4: Sparsity Results Showing Precentages of 1-diff, 2-
+diff and >3-diff SFs for each Method across Different 
+Datasets. 
+ 
+Figure 3: Benchmark Results: Performance of Five Semi-Factual Methods on Seven Tabular Datasets for Four Key Evaluation 
+Measures, the (a) Query-to-SF Distance, (b) Query-to-SF kNN (%), (c) SF-to-Q-Class Distance, (d) MDN Distance Measures. 
+ 
+
+100
+90
+80
+OL
+Sparsity (%)
+60
+50
+40
+20
+10
+Adult-Income
+Blood Alcohol
+Default Credit Card
+Diabetes
+German Credit
+HELOC
+Lending Club
+Adult-Income
+BloodAlcohol
+Default Credit Card
+Diabetes
+German Credit
+HELOC
+Lending Club
+Adult-Income
+Blood Alcohol
+Default Credit Card
+Diabetes
+German Credit
+HELOC
+Lending Club
+Adult-Income
+Blood Alcohol
+Default Credit Card
+Diabetes
+German Credit
+HELOC
+Lending Club
+Adult-Income
+Blood Alcohol
+Default Credit Card
+Diabetes
+German Credit
+HELOC
+LendingClub
+Sim-Miss
+Global-Sim
+Attr-Sim
+Local-Region
+MDN
+1-diff
+2-diff
+>3-diff(a)Query-to-SFDistance
+(b)Query-to-SFkNN(%)
+1
+70
+0.9
+60
+0.8
+40
+0.5
+30
+00.3
+0.2
+10
+0.1
+0
+Sim-Miss
+Global-Sim
+Attr-Sim
+Local-Region
+MDN
+Sim-Miss
+Global-Sim
+Attr-Sim
+Local-Region
+MDN
+(c)SF-to-Query-ClassDistance
+(d)MDNDistance
+5
+2
+4.5
+1.9
+1.8
+4
+1.7
+3.5
+1.6
+3
+1.5
+1.4
+1.3
+2
+1.2
+1.5
+1.1
+Sim-Miss
+Global-Sim
+Attr-Sim
+Local-Region
+MDN
+Sim-Miss
+Global-Sim
+Attr-Sim
+Local-Region
+MDN
+■Adulit-Income
+BloodAlcohol
+DefaultCreditCard
+Diabetes
+GermanCredit
+HELOC
+LendingClubAnnotated Bibliography  
+  
+*     means cited in original shorter paper  
+SF_AI    means core papers related to SFs in AI  
+SF_PSY    means articles related to SFs in Psychology  
+SF_PHL   means papers related to SFs in Philosophy  
+CF   means related to Counterfactual XAI  
+SURV   means survey/review article related to XAI  
+REL   means areas closely related to SF  
+  
+  
+*SURV [Adadi and Berrada, 2018] Amina Adadi and 
+Mohammed Berrada. Peeking inside the black-box: A 
+survey on explainable artificial intelligence (xai). IEEE 
+Access, 6:52138–52160, 2018.  
+SF_AI [Armengol and Plaza, 2006] Eva Armengol and Enric 
+Plaza. Symbolic explanation of similarities in case-based 
+reasoning. Computing and informatics, 25(2-3):153–171, 
+2006.  
+*SF_AI [Artelt and Hammer, 2022] Andre´ Artelt and 
+Barbara Hammer. ” even if...”–diverse semifactual 
+explanations of reject. arXiv preprint arXiv:2207.01898, 
+2022.  
+*CF [Asher et al., 2022] Nicholas Asher, Lucas De Lara, 
+Soumya Paul, and Chris Russell. Counterfactual models 
+for fair and adequate explanations. Machine Learning and 
+Knowledge Extraction, 4(2):316–349, 2022.  
+*SF_PHL [Barker, 1991] Stephen Barker. ” even, still” and 
+counterfactuals. Linguistics and Philosophy, pages 1–38, 
+1991.  
+SF_PHL 
+[Barker, 
+1994] 
+Stephen 
+J 
+Barker. 
+The 
+consequententailment problem foreven if. Linguistics and 
+Philosophy, 17(3):249–260, 1994.  
+*SF_PHL [Bennett, 1982] Jonathan Bennett. Even if. 
+Linguistics and Philosophy, 5(3):403–418, 1982.  
+*SF_PHL [Bennett, 2003] Jonathan Bennett. A philosophical 
+guide to conditionals. Clarendon Press, 2003.  
+*REL [Birhane et al., 2022] Abeba Birhane, Vinay Uday 
+Prabhu, and John Whaley. Auditing saliency cropping 
+algorithms. In Proceedings of the IEEE/CVF Winter 
+Conference on Applications of Computer Vision, pages 
+4051–4059, 2022.  
+REL [Bolon-Canedo and Remeseiro, 2020] Veronica 
+BolonCanedo and Beatriz Remeseiro. Feature selection in 
+image analysis: a survey. Artificial Intelligence Review, 
+53(4):2905–2931, 2020.  
+REL [Booth et al., 2021] Serena Booth, Yilun Zhou, Ankit 
+Shah, and Julie Shah. Bayes-trex: a bayesian sampling 
+approach to model transparency by example. In 
+Proceedings of the AAAI Conference on Artificial 
+Intelligence, volume 35, pages 11423–11432, 2021.  
+SF_PHL [Booth, 2014] Charles Booth. Boundary work in 
+theory and practice: Past, present and future. PhD thesis, 
+University of the West of England, 2014.  
+SF_PSY [Branscombe et al., 1996] Nyla R Branscombe, 
+Susan Owen, Teri A Garstka, and Jason Coleman. Rape 
+and accident counterfactuals: Who might have done 
+otherwise and would it have changed the outcome? 1. 
+Journal of Applied Social Psychology, 26(12):1042–
+1067, 1996.  
+SF_PSY [Branscombe et al., 1997] Nyla R Branscombe, 
+Ahogni N’gbala, Diane Kobrynowicz, and Daniel L 
+Wann. Self and group protection concerns influence 
+attributions 
+but 
+they 
+are 
+not 
+determinants 
+of 
+counterfactual mutation focus. British Journal of Social 
+Psychology, 36(4):387–404, 1997.  
+*SF_AI [Bridge and Cummins, 2005] Derek G Bridge and 
+Lisa Cummins. Knowledge lite explanation oriented 
+retrieval. In ExaCt, pages 35–42, 2005.  
+SF_PHL [Butcher, 1983] David Butcher. An incompatible   
+pair 
+of 
+subjunctive 
+conditional 
+modal 
+axioms.  
+Philosophical Studies: An International Journal for  
+Philosophy in the Analytic Tradition, 44(1):71–110, 
+1983.  
+SF_PSY [Byrne, ] Ruth MJ Byrne. Counterfactuals, causes 
+and exceptions.  
+SF_PSY [Byrne, 2007a] Ruth MJ Byrne. Precis of the 
+rational imagination: How people create alternatives to 
+reality. Behavioral and Brain Sciences, 30(5-6):439– 453, 
+2007.  
+*SF_PSY [Byrne, 2007b] Ruth MJ Byrne. The rational 
+imagination: How people create alternatives to reality. 
+MIT press, 2007.   
+*CF [Byrne, 2019] Ruth MJ Byrne. Counterfactuals in 
+explainable artificial intelligence (xai): evidence from 
+human reasoning. In Proceedings of the Twenty-Eighth 
+International Joint Conference on Artificial Intelligence, 
+IJCAI- 19, pages 6276–6282, 2019.  
+REL [Carvalho, 2022] Maria Manuel Domingos Carvalho. 
+Towards biometrically-morphed medical case-based 
+explanations. 2022.  
+*SF_PHL [Chisholm, 1946] Roderick M Chisholm. The 
+contrary-to-fact conditional. Mind, 55(220):289–307, 
+1946.  
+CF [Cho and Shin, 2023] Soo Hyun Cho and Kyung-shik 
+Shin. Feature-weighted counterfactual-based explanation 
+for 
+bankruptcy 
+prediction. 
+Expert 
+Systems 
+with 
+Applications, 216:119390, 2023.  
+REL [Craw et al., 2006] Susan Craw, Nirmalie Wiratunga, 
+and Ray C Rowe. Learning adaptation knowledge to 
+improve case-based reasoning. Artificial intelligence, 
+170(16- 17):1175–1192, 2006.  
+*SF_AI [Cummins and Bridge, 2006] Lisa Cummins and 
+Derek Bridge. Kleor: A knowledge lite approach to 
+explanation 
+oriented 
+retrieval. 
+Computing 
+and 
+Informatics, 25(2- 3):173–193, 2006.  
+*SF_AI [Cunningham et al., 2003] Pa´draig Cunningham, 
+Do´nal Doyle, and John Loughrey. An evaluation of the 
+usefulness of case-based explanation. In International 
+conference on case-based reasoning, pages 122–130. 
+Springer, 2003.  
+CF [Dandl et al., 2020] Susanne Dandl, Christoph Molnar, 
+Martin Binder, and Bernd Bischl. Multi-objective 
+counterfactual explanations. In International Conference 
+on Parallel Problem Solving from Nature, pages 448– 
+469. Springer, 2020.  
+REL [Dash and Liu, 1997] Manoranjan Dash and Huan Liu. 
+Feature selection for classification. Intelligent data 
+analysis, 1(1-4):131–156, 1997.  
+SF_PHL [Declerck and Reed, 2001] Renaat Declerck and 
+Susan Reed. Some truths and nontruths about even if. 
+Linguistics, 39:203–255, 01 2001.  
+CF [Dhurandhar et al., 2018] Amit Dhurandhar, Pin-Yu 
+Chen, Ronny Luss, Chun-Chen Tu, Paishun Ting, 
+Karthikeyan Shanmugam, and Payel Das. Explanations 
+based on the missing: Towards contrastive explanations 
+with pertinent negatives. Advances in neural information 
+processing systems, 31, 2018.  
+*SF_AI [Doyle et al., 2004] Do´nal Doyle, Pa´draig 
+Cunningham, Derek Bridge, and Yusof Rahman.  
+Explanation oriented retrieval. In European Conference 
+on Case-Based Reasoning, pages 157-168. Springer, 
+2004.  
+*SF_AI [Doyle et al., 2006] Do´nal Doyle, Pa´draig 
+Cunningham, and Paul Walsh. An evaluation of the 
+usefulness of explanation in a case-based reasoning system 
+for 
+decision 
+support 
+in 
+bronchiolitis 
+treatment. 
+Computational Intelligence, 22(3- 4):269–281, 2006.  
+REL [d’Aquin et al., 2022] Mathieu d’Aquin, Emmanuel 
+Nauer, and Jean Lieber. A factorial study of neural network 
+
+learning from differences for regression. In CaseBased 
+Reasoning Research and Development: 30th International 
+Conference, ICCBR 2022, Nancy, France, September 12– 
+15, 2022, Proceedings, pages 289–303. Springer, 2022.  
+*SF_PSY [Epstude and Roese, 2008] Kai Epstude and Neal J 
+Roese. The functional theory of counterfactual thinking. 
+Personality and Social Psychology Review, 12(2):168– 
+192, 5 2008.  
+*SF_PSY [Espino et al., 2022] Orlando Espino, Isabel 
+Orenes, and Sergio Moreno-R´ıos. Inferences from the 
+negation of counterfactual and semifactual conditionals. 
+Memory & Cognition, 50(5):1090–1102, 2022.  
+SF_PSY [Feeney et al., 2011] Aidan Feeney, Simon J 
+Handley, et al. Suppositions, conditionals, and causal 
+claims. Under- standing counterfactuals and causation: 
+Issues in philosophy and psychology, pages 242–262, 
+2011.  
+CF [Feiman, 2008] Roman Feiman. Possible worlds and 
+counterfactuals: 
+Critique 
+and 
+commentary 
+on 
+complicating causation. Episteme, 19(1):4, 2008.  
+*SF_AI [Ferna´ndez et al., 2022] Rube´n R Ferna´ndez, Isaac  
+Mart´ın de Diego, Javier M Moguerza, and Francisco 
+Herrera. Explanation sets: A general framework for 
+machine learning explainability. Information Sciences, 
+617:464–481, 2022.  
+REL [Freitas et al., 2008] Alex A Freitas, Daniela C Wieser, 
+and Rolf Apweiler.  On the importance of comprehensible 
+classification models for protein function prediction. 
+IEEE/ACM Transactions on Computational Biology and 
+Bioinformatics, 7(1):172–182, 2008.  
+SURV [Gates and Leake, 2021] Lawrence Gates and David 
+Leake. Evaluating cbr explanation capabilities: Survey and 
+next steps. In ICCBR Workshops, pages 40–51, 2021.  
+SF_PHL [Gomes, 2020] Gilberto Gomes. Concessive 
+conditionals 
+without 
+even 
+if 
+and 
+nonconcessive 
+conditionals with even if. Acta Analytica, 35(1):1–21, 
+2020.  
+REL [Goodman and Flaxman, 2017] Bryce Goodman and 
+Seth Flaxman. European union regulations on algorithmic 
+decision-making and a “right to explanation”. AI 
+magazine, 38(3):50–57, 2017.  
+*SF_PHL [Goodman, 1947] Nelson Goodman. The problem 
+of counterfactual conditionals. The Journal of Philosophy, 
+44(5):113–128, 1947.  
+*SF_PSY [Green, 2008] David W Green. Persuasion and the 
+contexts of dissuasion: Causal models and informal 
+arguments. Thinking & reasoning, 14(1):28–59, 2008.  
+*SURV [Guidotti et al., 2018] Riccardo Guidotti, Anna 
+Monreale, Salvatore Ruggieri, Franco Turini, Fosca 
+Giannotti, and Dino Pedreschi. A survey of methods for 
+explaining black box models. ACM computing surveys 
+(CSUR), 51(5):1–42, 2018.  
+SF_PHL [Gu¨ngo¨r, ] Hu¨seyin Gu¨ngo¨r. Truthmaking even 
+ifs.   
+*SURV [Gunning and Aha, 2019] David Gunning and David 
+W Aha. Darpa’s explainable artificial intelligence 
+program. AI Magazine, 40(2):44-58, 2019.   
+SF_AI [Hagos et al., 2022] Misgina Tsighe Hagos, Kathleen 
+M Curran, and Brian Mac Namee. Identifying spurious 
+correlations 
+and 
+correcting 
+them 
+with 
+an 
+explanationbased 
+learning. 
+arXiv 
+preprint 
+arXiv:2211.08285, 2022.  
+SF_PSY [Handley and Feeney, 2004] Simon J Handley and 
+Aidan Feeney. Reasoning and pragmatics: The case of 
+even-if. In Experimental pragmatics, pages 228–253. 
+Springer, 2004.  
+*SF_PSY [Handley and Feeney, 2007] Simon J Handley and 
+Aidan Feeney. Semifactual: Byrne’s account of even-if. 
+Behavioral and Brain Sciences, 30(5-6):458–459, 2007.  
+*REL [Hanney and Keane, 1996] Kathleen Hanney and 
+Mark T Keane. Learning adaptation rules from a 
+casebase. In European Workshop on Advances in Case-
+Based Reasoning, pages 179–192. Springer, 1996.  
+*SF_AI [Herchenbach et al., 2022] Marvin Herchenbach, 
+Dennis Mu¨ller, Stephan Scheele, and Ute Schmid. 
+Explaining image classifications with near misses, near 
+hits and prototypes. In International Conference on 
+Pattern Recognition and Artificial Intelligence, pages 
+419–430. Springer, 2022.  
+CF [Ho¨ltgen et al., 2021] Benedikt Ho¨ltgen, Lisa Schut, Jan 
+M Brauner, and Yarin Gal. Deduce: Generating 
+counterfactual explanations efficiently. arXiv preprint 
+arXiv:2111.15639, 2021.  
+*SF_PHL [Iten, 2002] Corinne Iten. Even if and even: The 
+case for an inferential scalar account. UCL Working 
+Papers in Linguistics, 14:119, 2002.  
+REL [Jalali et al., 2017] Vahid Jalali, David Leake, and 
+Najmeh Forouzandehmehr. Learning and applying case 
+adaptation rules for classification: An ensemble approach. 
+In IJCAI, pages 4874–4878, 2017.  
+*SF_PSY [Kahneman and Tversky, 1982] Daniel Kahneman 
+and Amos Tversky. The Simulation Heuristic. In Daniel 
+Kahneman, Paul Slovic, and Amos Tversky, editors, 
+Judgment Under Uncertainty: Heuristics and Biases, 
+pages 201–8. Cambridge University Press, New York, 
+1982.  
+*SURV [Karimi et al., 2022] Amir-Hossein Karimi, Gilles 
+Barthe, Bernhard Scho¨lkopf, and Isabel Valera. A survey 
+of algorithmic recourse: contrastive explanations and 
+consequential recommendations. ACM Computing  
+Surveys, 55(5):1– 29, 2022.  
+*REL [Karras et al., 2020] Tero Karras, Samuli Laine, Miika 
+Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. 
+Analyzing and improving the image quality of stylegan. In 
+Proceedings of the IEEE/CVF conference on computer 
+vision and pattern recognition, pages 8110–8119, 2020.  
+*SURV [Keane and Kenny, 2019] Mark T Keane and Eoin M 
+Kenny. How case-based reasoning explains neural 
+networks: A theoretical analysis of xai using post-hoc 
+explanation-by-example from a survey of ann-cbr 
+twinsystems. In International Conference on Case-Based 
+Reasoning, pages 155–171. Springer, 2019.  
+*CF [Keane and Smyth, 2020] Mark T Keane and Barry 
+Smyth. Good counterfactuals and where to find them: A 
+case- based technique for generating counterfactuals for 
+explainable ai (xai). In Proceedings of the 28th 
+International Conference on Case-Based Reasoning 
+(ICCBR-20), pages 163–178. Springer, 2020.  
+*CF [Keane et al., 2021] Mark T Keane, Eoin M Kenny, Eoin 
+Delaney, and Barry Smyth. If only we had better counter- 
+factual explanations. In Proceedings of the 30th 
+International Joint Conference on Artificial Intelligence 
+(IJCAI-21), 2021.  
+*SF_AI [Kenny and Keane, 2021] Eoin M. Kenny and Mark 
+T. Keane. On generating plausible counterfactual and semi-
+factual explanations for deep learning. In Proceedings of 
+the 35th AAAI Conference on Artificial Intelligence (AAAI-
+21), pages 11575–11585, 2021.  
+*REL [Kira et al., 1992] Kenji Kira, Larry A Rendell, et al. 
+The feature selection problem: Traditional methods and a 
+new algorithm. In Aaai, volume 2, pages 129–134, 1992.  
+CF [Kuhl et al., 2022] Ulrike Kuhl, Andre´ Artelt, and Barbara 
+Hammer. Keep your friends close and your counterfactuals 
+closer: Improved learning from closest rather than plausible 
+counterfactual explanations in an abstract setting. arXiv 
+preprint arXiv:2205.05515, 2022.  
+REL [Kukar et al., 1996] Matjazˇ Kukar, Igor Kononenko, and 
+Toma Silvester. Machine learning in prognosis of the 
+femoral neck fracture recovery. Artificial intelligence in 
+medicine, 8(5):431-451, 1996.  
+
+REL [Liao et al., 2018] Chieh-Kang Liao, Alan Liu, and Yu- 
+Sheng Chao. A machine learning approach to case 
+adaptation. In 2018 IEEE First International Conference 
+on Artificial Intelligence and Knowledge Engineering 
+(AIKE), pages 106–109. IEEE, 2018.  
+REL [Lin and Shaw, 1997] Fu-Ren Lin and Michael J Shaw.  
+Active training of backpropagation neural networks using 
+the learning by experimentation methodology. Annals of 
+Operations Research, 75:105–122, 1997.  
+REL [Lou and Obradovic, 2012] Qiang Lou and Zoran 
+Obradovic. Margin-based feature selection in incomplete 
+data. In Proceedings of the AAAI Conference on Artificial 
+Intelligence, volume 26, pages 1040–1046, 2012.  
+*SF_AI [Lu et al., 2022] Jinghui Lu, Linyi Yang, Brian Mac 
+Namee, and Yue Zhang. A rationale-centric framework 
+for human-in-the-loop machine learning. arXiv preprint 
+arXiv:2203.12918, 2022.  
+REL [Luss et al., 2021] Ronny Luss, Pin-Yu Chen, Amit 
+Dhurandhar, 
+Prasanna 
+Sattigeri, 
+Yunfeng 
+Zhang, 
+Karthikeyan Shanmugam, and Chun-Chen Tu.  
+Leveraging latent features for local explanations. In 
+Proceedings of the 27th ACM SIGKDD Conference on 
+Knowledge Discovery & Data Mining, pages 1139–1149, 
+2021.  
+SF_PHL [Lycan, 1991] William G Lycan. Even” and” even 
+if. Linguistics and Philosophy, pages 115–150, 1991.  
+REL [McCarthy et al., 2005] Kevin McCarthy, James Reilly, 
+Lorraine McGinty, and Barry Smyth. Experiments in 
+dynamic critiquing. In Proceedings of the 10th 
+international conference on Intelligent user interfaces, 
+pages 175–182, 2005.  
+*SF_PSY [McCloy and Byrne, 2002] Rachel McCloy and 
+Ruth MJ Byrne. Semifactual “even if” thinking. Thinking 
+& Reasoning, 8(1):41–67, 2002.  
+SF_AI [McSherry, 2004] David McSherry. Explaining the 
+pros and cons of conclusions in cbr. In European 
+Conference on Case-Based Reasoning, pages 317–330. 
+Springer, 2004.  
+*SF_AI [Mertes et al., 2022] Silvan Mertes, Christina Karle, 
+Tobias Huber, Katharina Weitz, Ruben Schlagowski, and 
+Elisabeth 
+Andre´. 
+Alterfactual 
+explanations–the 
+relevance of irrelevance for explaining ai systems. arXiv 
+preprint arXiv:2207.09374, 2022.  
+SF_PHL [Kvart, 2001] Igal Kvart. The counterfactual analysis 
+of cause. Synthese, 127(3):389–427, 2001.  
+CF [Laugel et al., 2019] Thibault Laugel, Marie-Jeanne 
+Lesot, Christophe Marsala, Xavier Renard, and Marcin 
+Detyniecki. The dangers of post-hoc interpretability: 
+Unjustified counterfactual explanations. arXiv preprint 
+arXiv:1907.09294, 2019.  
+REL [Leake et al., 2022] David Leake, Zachary Wilkerson, 
+and David Crandall. Extracting case indices from 
+convolutional neural networks: A comparative study. In 
+Case- Based Reasoning Research and Development: 30th 
+International Conference, ICCBR 2022, Nancy, France, 
+September 12–15, 2022, Proceedings, pages 81–95. 
+Springer, 2022.  
+*SURV [Miller, 2019] Tim Miller. Explanation in artificial 
+intelligence: Insights from the social sciences. Artificial 
+Intelligence, 267:1–38, 2019.  
+REL [Montenegro et al., 2021] Helena Montenegro, Wilson 
+Silva, and Jaime S Cardoso. Privacy-preserving 
+generative 
+adversarial 
+network 
+for 
+case-based 
+explainability in medical image analysis. IEEE Access, 
+9:148037–148047, 2021.  
+*SF_PSY [Moreno-Rios et al., 2008] Sergio Moreno-Rios, 
+Juan A Garcia-Madruga, and Ruth MJ Byrne. Inferences 
+from 
+semifactual 
+‘even 
+if’conditionals. 
+Acta 
+Psychologica, 128(2):197–209, 2008.  
+CF [Mothilal et al., 2020] Ramaravind K Mothilal, Amit 
+Sharma, and Chenhao Tan. Explaining machine learning 
+classifiers through diverse counterfactual explanations. In 
+Proceedings of the 2020 conference on fairness, 
+accountability, and transparency, pages 607-617, 2020.   
+*SF_AI [Mueller et al., 2021] Shane Mueller, Yin-Yin Tan, 
+Anne Linja, Gary Klein, and Robert Hoffman. Authoring 
+guide for cognitive tutorials for artificial intelligence: 
+Purposes and methods. 2021.  
+REL [Nimmy et al., 2023] Sonia Farhana Nimmy, Omar K 
+Hussain, Ripon K Chakrabortty, Farookh Khadeer Hussain, 
+and Morteza Saberi. Interpreting the antecedents of a 
+predicted output by capturing the interdependencies among 
+the system features and their evolution over time. 
+Engineering Applications of Artificial Intelligence, 
+117:105596, 2023.  
+*SF_AI [Nugent et al., 2005] Conor Nugent, Pa´draig 
+Cunningham, and Do´nal Doyle. The best way to instil 
+confidence is by being right. In International Conference 
+on Case-Based Reasoning, pages 368–381. Springer, 2005.  
+*SF_AI [Nugent et al., 2009] Conor Nugent, Do´nal Doyle, 
+and Pa´draig Cunningham. Gaining insight through 
+casebased explanation. Journal of Intelligent Information 
+Systems, 32(3):267–295, 2009.  
+SF_AI [Olsson et al., 2014] Tomas Olsson, Daniel Gillblad, 
+Peter Funk, and Ning Xiong. Explaining probabilistic fault 
+diagnosis and classification using case-based reasoning. In 
+International Conference on Case-Based Reasoning, pages 
+360–374. Springer, 2014.  
+REL [Olsson et al., 2014] Tomas Olsson, Daniel Gillblad, 
+Peter Funk, and Ning Xiong. Case-based reasoning for 
+explaining probabilistic machine learning. International 
+Journal of Computer Science and Information  
+Technology, 6:87–101, 2014.  
+*SF_PSY [Parkinson and Byrne, 2017] Mary Parkinson and 
+Ruth MJ Byrne. Counterfactual and semi-factual thoughts 
+in moral judgements about failed attempts to harm. 
+Thinking & Reasoning, 23(4):409–448, 2017.  
+SF_PHL [Paul, 2009] Laurie Ann Paul. Counterfactual 
+theories. The Oxford handbook of causation, pages 158– 
+184, 2009.  
+CF [Pedapati et al., 2020] Tejaswini Pedapati, Avinash 
+Balakrishnan, 
+Karthikeyan 
+Shanmugam, 
+and 
+Amit 
+Dhurandhar. Learning global transparent models consistent 
+with local contrastive explanations. Advances in neural 
+information processing systems, 33:3592–3602, 2020.  
+SF_AI [Rabold et al., 2022] Johannes Rabold, Michael 
+Siebers, and Ute Schmid. Generating contrastive 
+explanations for inductive logic programming based on a 
+near miss approach. Machine Learning, 111(5):1799– 
+1820, 2022.  
+REL [Raman and Ioerger, 2003] Baranidharan Raman and 
+Thomas R Ioerger. Enhancing learning using feature and 
+example selection. Journal of Machine Learning Research 
+(submitted for publication), 2003.  
+*REL [Reilly et al., 2004] James Reilly, Kevin McCarthy, 
+Lorraine McGinty, and Barry Smyth. Dynamic critiquing. 
+In European Conference on Case-Based Reasoning, pages 
+763– 777. Springer, 2004.  
+*REL [Ribeiro et al., 2016] Marco Tulio Ribeiro, Sameer 
+Singh, and Carlos Guestrin. ” why should i trust you?” 
+explaining the predictions of any classifier. In Proceedings 
+of the 22nd ACM SIGKDD international conference on 
+knowledge discovery and data mining, pages 1135–1144, 
+2016.  
+REL [Ribeiro et al., 2018] Marco Tulio Ribeiro, Sameer 
+Singh, and Carlos Guestrin. Anchors: High-precision 
+model- agnostic explanations. In Proceedings of the AAAI 
+conference on artificial intelligence, volume 32, 2018.  
+REL [Rissland and Skalak, 1989a] Edwina L Rissland and 
+David B Skalak. Combining case-based and rule-based 
+reasoning: A heuristic approach. In IJCAI, pages 524– 
+530, 1989.  
+REL [Rissland and Skalak, 1989b] Edwina L Rissland and 
+David B Skalak. Interpreting statutory predicates. In 
+
+Proceedings of the 2nd international conference on 
+Artificial intelligence and law, pages 46–53, 1989.  
+SURV [Rissland, 2006] Edwina L Rissland. Ai and 
+similarity. IEEE Intelligent Systems, 21(03):39–49, 2006.  
+SURV [Rissland, 2009] Edwina L Rissland. Black swans, 
+gray cygnets and other rare birds. In International 
+Conference on Case-Based Reasoning, pages 6–13. 
+Springer, 2009.  
+*SF_AI [Salimi, 2022] Pedram Salimi. Addressing trust and 
+mutability issues in xai utilising case based reasoning. 
+ICCBR Doctoral Consortium 2022, 1613:0073, 2022.  
+SF_PSY[Santamar´ıa et al., 2005] Carlos Santamar´ıa, 
+Orlando Espino, and Ruth MJ Byrne. Counterfactual and 
+semifactual conditionals prime alternative possibilities. 
+Journal of Experimental Psychology: Learning, Memory, 
+and Cognition, 31(5):1149, 2005.  
+SF_PHL [Sarasvathy, 2021] Saras D Sarasvathy. Even-if: 
+Sufficient, yet unnecessary conditions for worldmaking. 
+Organization Theory, 2(2):26317877211005785, 2021.  
+CF [Schleich et al., 2021] Maximilian Schleich, Zixuan 
+Geng, Yihong Zhang, and Dan Suciu. Geco: quality 
+counterfactual explanations in real time. arXiv preprint 
+arXiv:2101.01292, 2021.  
+*CF [Sokol and Flach, 2019] Kacper Sokol and Peter A 
+Flach. Counterfactual explanations of machine learning 
+predictions: opportunities and challenges for ai safety. 
+SafeAI@ AAAI, 2019.  
+*SURV [Sørmo et al., 2005] Frode Sørmo, Jo¨rg Cassens, 
+and Agnar Aamodt. Explanation in case-based reasoning– 
+perspectives and goals. Artificial Intelligence Review, 
+24(2):109–143, 2005.  
+REL [Sun, 2007] Yijun Sun. Iterative relief for feature 
+weighting: algorithms, theories, and applications. IEEE 
+transactions 
+on 
+pattern 
+analysis 
+and 
+machine 
+intelligence, 29(6):1035– 1051, 2007.  
+SF_PHL [Tellings, 2017] Jos Tellings. Still as an additive 
+particle in conditionals. In Semantics and Linguistic 
+Theory, vol- ume 27, pages 1–21, 2017.  
+REL [Urbanowicz et al., 2018] Ryan J Urbanowicz, Melissa 
+Meeker, William La Cava, Randal S Olson, and Jason H 
+Moore. Relief-based feature selection: Introduction and 
+review. Journal of biomedical informatics, 85:189–203, 
+2018.  
+*SF_AI [Vats et al., 2022] Anuja Vats, Ahmed Mohammed, 
+Marius Pedersen, and Nirmalie Wiratunga. This changes 
+to that: Combining causal and non-causal explanations to 
+generate disease progression in capsule endoscopy. arXiv 
+preprint arXiv:2212.02506, 2022.  
+*SURV [Verma et al., 2022] Sahil Verma, Varich 
+Boonsanong, Minh Hoang, Keegan E. Hines, John P. 
+Dickerson, and Chirag Shah. Counterfactual explanations 
+and algorithmic recourses for machine learning: A review, 
+2022.  
+SF_PSY [Vidal and Baratgin, 2017] Mathieu Vidal and Jean 
+Baratgin. A psychological study of unconnected 
+conditionals. Journal of Cognitive Psychology, 29(6):769–
+781, 2017.  
+SF_PHL [Vidal, 2017] Mathieu Vidal. A compositional 
+semantics for ‘even if’conditionals. Logic and Logical 
+Philosophy, 26(2):237–276, 2017.  
+*CF [Wachter et al., 2017] Sandra Wachter, Brent 
+Mittelstadt, 
+and 
+Chris 
+Russell. 
+Counterfactual 
+explanations without opening the black box. Harv. JL & 
+Tech., 31:841, 2017.  
+*CF [Wexler et al., 2019] James Wexler, Mahima Pushkarna, 
+Tolga Bolukbasi, Martin Wattenberg, Fernanda Vie´gas, 
+and Jimbo Wilson. The what-if tool. IEEE transactions on 
+visualization and computer graphics, 26(1):56–65, 2019.  
+CF [Wijekoon et al., 2022] Anjana Wijekoon, Nirmalie 
+Wiratunga, Ikechukwu Nkisi-Orji, Chamath  
+Palihawadana, David Corsar, and Kyle Martin. How close 
+is too close? the role of feature attributions in discovering 
+counterfactual explanations. In Case-Based Reasoning 
+Research 
+and 
+Development: 
+30th 
+International 
+Conference, ICCBR 2022, Nancy, France, September 12– 
+15, 2022, Proceedings, pages 33–47. Springer, 2022.  
+SF_AI [Winston, 1970] Patrick H Winston. Learning 
+structural descriptions from examples. 1970.  
+CF [Wiratunga et al., 2021] Nirmalie Wiratunga, Anjana  
+Wi- jekoon, Ikechukwu Nkisi-Orji, Kyle Martin, 
+Chamath Pal- ihawadana, and David Corsar. Discern: 
+Discovering counterfactual explanations using relevance 
+features from neighbourhoods. In 2021 IEEE 33rd 
+International Conference on Tools with Artificial 
+Intelligence (ICTAI), pages 1466–1473. IEEE, 2021.  
+SF_PSY [Yang, 2022] Yujing Yang. On the study of chinese 
+double-false counterfactual conditionals. In 
+2021 
+International Conference on Social Development and 
+Media Commu- nication (SDMC 2021), pages 1553–
+1563. Atlantis Press, 2022.  
+SF_AI [Ye et al., 2020] Xiaomeng Ye, David Leake, 
+William Huibregtse, and Mehmet Dalkilic. Applying 
+classto-class siamese networks to explain classifications 
+with supportive and contrastive cases. In International 
+Conference on Case-Based Reasoning, pages 245–260. 
+Springer, 2020.  
+*REL [Ye et al., 2021] Xiaomeng Ye, Ziwei Zhao, David 
+Leake, Xizi Wang, and David Crandall. Applying the 
+case difference heuristic to learn adaptations from deep 
+network features. arXiv preprint arXiv:2107.07095, 
+2021.  
+*REL  
+[Yousefzadeh  and  O’Leary,  2019]  Roozbeh  
+Yousefzadeh and Dianne P O’Leary. Interpreting neural 
+networks 
+using 
+flip 
+points. 
+arXiv 
+preprint 
+arXiv:1903.08789, 2019.  
+REL [Zheng et al., 2019] Jianyang Zheng, Hexing Zhu, 
+Fangfang Chang, and Yunlong Liu. An improved relief 
+feature selection algorithm based on monte-carlo tree 
+search. Systems Science & Control Engineering, 
+7(1):304–310, 2019.  
+*SF_AI  [Zhao et al., 2022] Ziwei Zhao, David Leake, 
+Xiaomeng Ye, and David Crandall. Generating 
+counterfactual images: Towards a c2c-vae approach. 
+2022. 
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+   
+   
+  
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+arXiv:2301.02862v1  [math.MG]  7 Jan 2023
+AN INTEGER PARALLELOTOPE WITH SMALL SURFACE AREA
+ASSAF NAOR AND ODED REGEV
+ABSTRACT. We prove that for any n ∈ N there is a convex body K ⊆ Rn whose surface area is at most n
+1
+2 +o(1),
+yet the translates of K by the integer lattice Zn tile Rn.
+1. INTRODUCTION
+Given n ∈ N and a lattice Λ ⊆ Rn, a convex body K ⊆ Rn is called a Λ-parallelotope (e.g., [12]) if the
+translates of K by elements of Λ tile Rn, i.e., Rn = Λ+K = �
+x∈Λ(x +K ), and the interior of (x +K )∩(y +K )
+is empty for every distinct x, y ∈ Λ. One calls K a parallelotope (parallelogon if n = 2 and parallelohedron
+if n = 3; some of the literature calls a parallelotope in Rn and n-dimensional parallellohedron; e.g., [1, 11])
+if it is a Λ-parallelotope for some lattice Λ ⊆ Rn. We call a Zn-parallelotope an integer parallelotope.
+The hypercube [− 1
+2, 1
+2]n is an integer parallelotope whose surface area equals 2n. By [16, Corollary A.2],
+for every n ∈ N there exists an integer parallelotope K ⊆ Rn whose surface area is smaller than 2n by a
+universal constant factor. Specifically, the surface area of the integer parallelotope K that was considered
+in [16] satisfies voln−1(∂K ) ⩽ σ(n +O(n2/3)), where σ = 2�∞
+s=1(s/e)s/(s3/2s!) ⩽ 1.23721. To the best of our
+knowledge, this is the previously best known upper bound on the smallest possible surface area of an
+integer parallelotope. The main result of the present work is the following theorem:
+Theorem 1. For every n ∈ N there exists an integer parallelotope whose surface area is n
+1
+2 +o(1).
+Because the covolume of Zn is 1, the volume of any integer parallelotope K ⊆ Rn satisfies voln(K ) = 1.
+Consequently, by the isoperimetric inequality we have1
+voln−1(∂K ) ⩾ voln−1(Sn−1)
+voln(Bn)
+n−1
+n
+≍
+�
+n,
+(1)
+where Bn def
+= {(x1,...,xn) ∈ Rn : x2
+1 +···+ x2
+n ⩽ 1} denotes the Euclidean ball and Sn−1 def
+= ∂Bn.
+Thanks to (1), Theorem 1 is optimal up to the implicit lower order factor. It remains open to determine
+whether this lower-order factor could be removed altogether, namely to answer the following question:
+Question 2. For every n ∈ N, does there exist an integer parallelotope K ⊆ Rn with voln−1(∂K ) ≍ �n?
+Question 2 goes back to [24], though such early investigations were (naturally, from the perspective of
+crystallography) focused on n = 3 and asked for the exact value of the smallest possible surface area of
+a parallelohedron; see Conjecture 7.5 in [5] and the historical discussion in the paragraph that precedes
+it. The corresponding question about precisely determining the minimum perimeter when n = 2 was
+answered in [7] (its solution for general parallelogons rather than integer parallelogons is due to [17]; see
+also [22], which treats tiles that need not be convex). Finding the exact minimum when n = 3 remains
+A.N. was supported by NSF grant DMS-2054875, BSF grant 201822, and a Simons Investigator award. O.R. was supported by
+NSF grant CCF-1320188 and a Simons Investigator award.
+1We use the following conventions for asymptotic notation, in addition to the usual O(·),o(·),Ω(·),Θ(·) notation. For a,b > 0,
+by writing a ≲ b or b ≳ a we mean that a ⩽ Cb for a universal constant C > 0, and a ≍ b stands for (a ≲ b)∧(b ≲ a). If we need
+to allow for dependence on parameters, we indicate it by subscripts. For example, in the presence of an auxiliary parameter ε,
+the notation a ≲ε b means that a ⩽ C(ε)b, where C(ε) > 0 may depend only on ε, and analogously for a ≳ε b and a ≍ε b.
+1
+
+open; we will not review the substantial literature on this topic, referring instead to the monograph [4]
+(see also [28] for an exact solution of a different isoperimetric-type question for parallelohedra).
+The higher dimensional asymptotic nature of Question 2 differs from the search for exact minimizers
+in lower dimensions on which the literature has focused, but it is a natural outgrowth of it and it stands
+to reason that it was considered by researchers who worked on this topic over the past centuries. Never-
+theless, we do not know of a published source that mentions Question 2 prior to the more recent interest
+in this topic that arose due to its connection to theoretical computer science that was found in [16] and
+were pursued in [33, 25, 3, 26, 6]; specifically, Question 2 appears in [6, Section 6].
+In [25] it was proved that Question 2 has a positive answer if one drops the requirement that the tiling
+set is convex, i.e., by [25, Theorem 1.1] for every n ∈ N there is a compact set Ω ⊆ Rn such that Rn = Zn+Ω,
+the interior of (x + Ω) ∩ (y + Ω) is empty for every distinct x, y ∈ Zn, and voln−1(∂Ω) ≲ �n; see also the
+proof of this result that was found in [3]. The lack of convexity of Ω is irrelevant for the applications to
+computational complexity that were found in [16]. The proofs in [25, 3] produce a set Ω that is decidedly
+non-convex. Our proof of Theorem 1 proceeds via an entirely different route and provides a paralletotope
+whose surface area comes close to the guarantee of [25] (prior to [25], the best known upper bound on
+the smallest possible surface area of a compact Zn-tiling set was the aforementioned 1.23721n of [16]).
+While it could be tempting to view the existence of the aforementioned compact set Ω as evidence
+for the availability of an integer parallelotope with comparable surface area, this is a tenuous hope be-
+cause the convexity requirement from a parallelotope imposes severe restrictions. In particular, by [30]
+for every n ∈ N there are only finitely many combinatorial types of parallelotopes in Rn.2 In fact, by com-
+bining [10, Section 6] with [30, 36] we see that K ⊆ Rn is a parallelotope if and only if K is a centrally
+symmetric polytope, all of the (n − 1)-dimensional faces of K are centrally symmetric, and the orthog-
+onal projection of K along any of its (n − 2)-dimensional faces is either a parallelogram or a centrally
+symmetric hexagon.
+Of course, Theorem 1 must produce such a constrained polytope. To understand how this is achieved,
+it is first important to stress that this becomes a straightforward task if one only asks for a parallelotope
+with small surface area rather than for an integer parallelotope with small surface area. Namely, it follows
+easily from the literature that for every n ∈ N there exist a rank n lattice Λ ⊆ Rn whose covolume is 1 and
+a Λ-parallelotope K ⊆ Rn that satisfies voln−1(∂K ) ≲ �n. Indeed, by [34] there is a rank n lattice Λ ⊆ Rn
+of covolume 1 whose packing radius is at least c�n, where c > 0 is a universal constant. Let K be the
+Voronoi cell of Λ, namely K consists of the points in Rn whose (Euclidean) distance to any point of Λ is
+not less than their distance to the origin. Then, K is a Λ-parallelotope, voln(K ) = 1 since the covolume of
+Λ is 1, and K ⊇ c�nBn since the packing radius of Λ is at least c�n. Consequently, the surface area of K is
+at most c−1�n by the following simple lemma that we will use multiple times in the proof of Theorem 1:
+Lemma 3. Fix n ∈ N and R > 0. Suppose that a convex body K ⊆ Rn satisfies K ⊇ RBn. Then,
+voln−1(∂K )
+voln(K )
+⩽ n
+R .
+Lemma 3 is known (e.g., [19, Lemma 2.1]); for completeness we will present its short proof in Section 2.
+Even though the packing radius of Zn is small, the above observation drives our inductive proof of
+Theorem 1, which proceeds along the following lines. Fix m ∈ {1,...,n−1} and let V be an m-dimensional
+subspace of Rn. If the lattice V ⊥ ∩Zn has rank n −m and its packing radius is large, then Lemma 3 yields
+a meaningful upper bound on the (n −m −1)-dimensional volume of the boundary of the Voronoi cell of
+V ⊥ ∩Zn. We could then consider the lattice Λ ⊆ V which is the orthogonal projection of Zn onto V , and
+inductively obtain a Λ-parallelotope (residing within V ) for which the (m −1)-dimensional volume of its
+boundary is small. By considering the product (with respect to the identification of Rn with V ⊥ ×V ) of
+the two convex bodies thus obtained, we could hope to get the desired integer parallelotope.
+2Thus, just for the sake concreteness (not important for the present purposes): Since antiquity it was known that there are 2
+types of parallelogons; by [13] there are 5 types of parallelohedra; by [8, 35] there are 52 types of 4-dimensional parallelotopes.
+2
+
+There are obvious obstructions to this plan. The subspace V must be chosen so that the lattice V ⊥∩Zn
+is sufficiently rich yet it contains no short nonzero vectors. Furthermore, the orthogonal projection Λ
+of Zn onto V is not Zm, so we must assume a stronger inductive hypothesis and also apply a suitable
+“correction” to Λ so as to be able to continue the induction. It turns out that there is tension between how
+large the packing radius of V ⊥∩Zn could be, the loss that we incur due to the aforementioned correction,
+and the total cost of iteratively applying the procedure that we sketched above. Upon balancing these
+constraints, we will see that the best choice for the dimension m of V is m = n exp(−Θ(
+�
+logn)). The rest
+of the ensuing text will present the details of the implementation of this strategy.
+2. PROOF OF THEOREM 1
+Below, for each n ∈ N the normed space ℓn
+2 = (Rn,∥·∥ℓn
+2 ) will denote the standard Euclidean space, i.e.,
+∀x = (x1,...,xn) ∈ Rn,
+∥x∥ℓn
+2
+def
+=
+�
+x2
+1 +···+ x2
+n.
+The standard scalar product of x, y ∈ Rn will be denoted 〈x, y〉
+def
+= x1y1+···+xnyn. The coordinate basis of
+Rn will be denoted e1,...,en, i.e., for each i ∈ {1,...,n} the ith entry of ei is 1 and the rest of the coordinates
+of ei vanish. We will denote the origin of Rn by 0 = (0,...,0). For 0 < s ⩽ n, the s-dimensional Hausdorff
+measure on Rn that is induced by the ℓn
+2 metric will be denoted by vols(·). In particular, if K ⊆ Rn is a
+convex body (compact and with nonempty interior), then the following identity holds (see, e.g., [27]):
+voln−1(∂K ) = lim
+δ→0+
+voln(K +δBn)−voln(K )
+δ
+.
+(2)
+If V is a subspace of Rn, then its orthogonal complement (with respect to the ℓn
+2 Euclidean structure)
+will be denoted V ⊥ and the orthogonal projection from Rn onto V will be denoted ProjV . When treating
+a subset Ω of V we will slightly abuse notation/terminology by letting ∂Ω be the boundary of Ω within V ,
+and similarly when we will discuss the interior of Ω we will mean its interior within V . This convention
+results in suitable interpretations of when K ⊆ V is a convex body or a parallelohedron (with respect to a
+lattice of V ). The variant of (2) for a convex body K ⊆ V becomes
+voldim(V )−1(∂K ) = lim
+δ→0+
+voldim(V )
+�
+K +δ(V ∩Bn)
+�
+−voldim(V )(K )
+δ
+.
+(3)
+Proof of Lemma 3. Since K ⊇ RBn, for every δ > 0 we have
+K +δBn ⊆ K + δ
+R K =
+�
+1+ δ
+R
+��
+R
+R +δK +
+δ
+R +δK
+�
+=
+�
+1+ δ
+R
+�
+K ,
+(4)
+where the last step of (4) uses the fact that K is convex. Consequently,
+voln−1(∂K )
+(2)
+= lim
+δ→0+
+voln(K +δBn)−voln(K )
+δ
+(4)
+⩽ lim
+δ→0+
+�
+1+ δ
+R
+�n −1
+δ
+voln(K ) = n
+R voln(K ).
+□
+The sequence {Q(n)}∞
+n=1 that we introduce in the following definition will play an important role in
+the ensuing reasoning:
+Notation 4. For each n ∈ N let Q(n) be the infimum over those Q ⩾ 0 such that for every lattice Λ ⊆ Zn of
+rank n there exists a Λ-parallelotope K ⊆ Rn that satisfies
+voln−1(∂K )
+voln(K )
+⩽Q.
+(5)
+As voln(K ) = 1 for any integer parallelotope K ⊆ Rn, Theorem 1 is a special case of the following result:
+Theorem 5. There exists a universal constant C ⩾ 1 such that Q(n) ≲ �neC�
+logn for every n ∈ N .
+The following key lemma is the inductive step in the ensuing proof of Theorem 5 by induction on n:
+3
+
+Lemma 6. Fix m,n,s ∈ N with s ⩽ m ⩽ n. Suppose that B ∈ Mm×n(Z) is an m-by-n matrix all of whose
+entries are integers such that B has rank m and any s of the columns of B are linearly independent. Then,
+Q(n) ⩽ 2(n −m)
+�s
++Q(m)∥B∥ℓn
+2 →ℓm
+2 ,
+where ∥·∥ℓn
+2 →ℓm
+2 denotes the operator norm from ℓn
+2 to ℓm
+2 .
+The fact that Theorem 5 treats any sublattice of Zn of full rank (recall howQ(n) is defined), even though
+in Theorem 1 we are interested only in Zn itself, provides a strengthening of the inductive hypothesis
+that makes it possible for our proof of Lemma 6 to go through. If Λ is an arbitrary full rank sublattice of
+Zn, then a Λ-parallelotope K ⊆ Rn need no longer satisfy voln(K ) = 1, so the inductive hypothesis must
+incorporate the value of voln(K ), which is the reason why we consider the quantity voln−1(∂K )/voln(K )
+in (5). Observe that this quantity is not scale-invariant, so it might seem somewhat unnatural to study it,
+but it is well-suited to the aforementioned induction thanks to the following simple lemma:
+Lemma 7. Fix m,n ∈ N and an m-dimensional subspace V of Rn. Let O ⊆ V ⊥ be an open subset of V ⊥ and
+let G ⊆ V be an open subset of V . Then, for Ω = O +G we have
+voln−1(∂Ω)
+voln(Ω)
+= voln−m−1(∂O)
+voln−m(O)
++ volm−1(∂G)
+volm(G)
+.
+(6)
+Furthermore, if T : Rm → V is a linear isomorphism and K ⊆ Rm is a convex body, then
+volm−1(∂T K )
+volm(T K )
+⩽ volm−1(∂K )
+volm(K )
+∥T −1∥(V,∥·∥ℓn
+2 )→ℓm
+2 ,
+(7)
+where ∥·∥(V,∥·∥ℓn
+2 )→ℓm
+2 is the operator norm from V , equipped with the norm inherited from ℓn
+2 , to ℓm
+2 .
+Proof. For (6), note that since O ⊥ G we have voln(Ω) = voln−m(O)volm(G), and ∂Ω = (∂O +G)∪(O +∂G)
+where voln−1((∂O +G)∩(O +∂G)) = 0, so voln−1(∂Ω) = voln−m−1(∂O)volm(G)+voln−m(O)volm−1(∂G).
+For (7), denote ρ = ∥T −1∥(V,∥·∥ℓn
+2 )→ℓm
+2 , so that T −1(V ∩Bn) ⊆ ρBm. Consequently,
+∀δ ∈ R,
+T K +δ(V ∩Bn) = T
+�
+K +δT −1(V ∩Bn)
+�
+⊆ T (K +δρBm).
+By combining this inclusion with (3), we see that
+volm−1(∂T K ) ⩽ lim
+δ→0+
+volm
+�
+T (K +δρBm)
+�
+−volm(T K )
+δ
+⩽ det(T ) lim
+δ→0+
+volm(K +δρBm)−volm(K )
+δ
+(2)
+= det(T )volm−1(∂K )ρ = volm(T K )
+volm(K ) volm−1(∂K )ρ.
+□
+Remark 8. We stated Lemma 7 with K being a convex body since that is all that we need herein. However,
+the proof does not rely on its convexity in an essential way; all that is needed is that K is a body in Rm whose
+boundary is sufficiently regular so that the identity (2) holds (with n replaced by m).
+Any matrix B as in Lemma 6 must have a row with at least n/m nonzero entries. Indeed, otherwise the
+total number of nonzero entries of B would be less than m(n/m) = n, so at least one of the n columns B
+would have to vanish, in contradiction to the assumed linear independence (as s ⩾ 1). Thus, there exists
+j ∈ {1,...,m} such that at least ⌈n/m⌉ of the entries of B∗e j ∈ Rn do not vanish. Those entries are integers,
+so ∥B∗e j∥ℓn
+2 ⩾
+�
+⌈n/m⌉. Hence, the quantity ∥B∥ℓn
+2 →ℓm
+2 = ∥B∗∥ℓm
+2 →ℓn
+2 in (6) cannot be less than
+�
+⌈n/m⌉.
+Question 9. Given m,n ∈ N and C > 1, what is the order of magnitude of the largest s = s(m,n,C) ∈ N for
+which there exists B ∈ Mm×n(Z) such that any s of the columns of B are linearly independent and
+∥B∥ℓn
+2 →ℓm
+2 ⩽C
+� n
+m .
+The following lemma is a step towards Question 9 that we will use in the implementation of Lemma 6:
+4
+
+Lemma 10. Suppose that m,n ∈ N satisfy 4 ⩽ m ⩽ n and n ⩾ (m logm)/4. There exist s ∈ N with s ≳ m2/n
+and B ∈ Mm×n(Z) of rank m such that any s of the columns of B are linearly independent and
+∥B∥ℓn
+2 →ℓm
+2 ≲
+� n
+m .
+Lemma 10 suffices for our purposes, but it is not sharp. We will actually prove below that in the setting
+of Lemma 10 for every 0 < ε ⩽ 1 there exist s ∈ N with s ≳ m1+ε/nε = m(m/n)ε ⩾ m2/n and B ∈ Mm×n(Z)
+of rank m such that any s of the columns of B are linearly independent and ∥B∥ℓn
+2 →ℓm
+2 ≲ε
+�
+n/m.
+While Question 9 arises naturally from Lemma 6 and it is interesting in its own right, fully answering
+Question 9 will not lead to removing the o(1) term in Theorem 1 altogether; the bottleneck in the ensuing
+reasoning that precludes obtaining such an answer to Question 2 (if true) is elsewhere.
+Proof of Theorem 5 assuming Lemma 6 and Lemma 10. We will proceed by induction on n. In prepara-
+tions for the base of the induction, we will first record the following estimate (which is sharp when the
+lattice is Zn). The Voronoi cell of a rank n sublattice Λ of Zn, namely the set
+K =
+�
+x ∈ Rn : ∀y ∈ Λ, ∥x∥ℓn
+2 ⩽ ∥x − y∥ℓn
+2
+�
+,
+is a Λ-parallelotope that satisfies K ⊇ 1
+2Bn. Indeed, if y ∈ Λ∖{0}, then ∥y∥ℓn
+2 ⩾ 1 since y ∈ Zn∖{0}. Hence,
+∀x ∈ 1
+2Bn,
+∥x − y∥ℓn
+2 ⩾ ∥y∥ℓn
+2 −∥x∥ℓn
+2 ⩾ ∥x∥ℓn
+2 .
+By Lemma 3, it follows that voln−1(∂K )/voln(K ) ⩽ 2n. This gives the (weak) a priori bound Q(n) ⩽ 2n.
+Fix n ∈ N and suppose that there exists m ∈ N satisfying 4 ⩽ m ⩽ n and n ⩾ (m logm)/4. By using
+Lemma 6 with the matrix B from Lemma 10 we see that there is a universal constant κ ⩾ 4 for which
+Q(n) ⩽ κ
+�
+n
+3
+2
+m +Q(m)
+� n
+m
+�
+.
+(8)
+We will prove by induction on n ∈ N the following upper bound on Q(n), thus proving Theorem 5:
+Q(n) ⩽ 4κ
+�
+ne
+�
+2(logn)log(2κ).
+(9)
+If n ⩽ 4κ2, then by the above discussion Q(n) ⩽ 2n ⩽ 4κ�n, so that (9) holds. If n > 4κ2, then define
+m
+def
+=
+�
+ne−�
+2(logn)log(2κ)�
+.
+(10)
+It is straightforward to verify that this choice of m satisfies 4 ⩽ m < n and n ⩾ (m logm)/4 (with room to
+spare). Therefore (8) holds. Using the induction hypothesis, it follows that
+Q(m)
+� n
+m ⩽ 4κ
+�
+ne
+�
+2(logm)log(2κ) (10)
+⩽ 4κ
+�
+ne
+�
+2
+�
+logn−�
+2(logn)log(2κ)
+�
+log(2κ)
+⩽ 4κ
+�
+ne
+��
+2logn−�
+log(2κ)
+��
+log(2κ) = 2
+�
+ne
+�
+2(logn)log(2κ),
+(11)
+where the penultimate step of (11) uses the inequality
+�
+a −b ⩽ �a − b/(2�a), which holds for every
+a,b ∈ R with a ⩾ b; in our setting a = logn and b =
+�
+2(logn)log(2κ) and a > b because we are now
+treating the case n > 4κ2. A substitution of (11) into (8), while using that m ⩾ 1
+2n exp
+�
+−
+�
+2(logn)log(2κ)
+�
+holds thanks to (10), gives (9), thus completing the proof of Theorem 5.
+□
+We will next prove Lemma 6, which is the key recursive step that underlies Theorem 1.
+Proof of Lemma 6. We will start with the following two elementary observations to facilitate the ensuing
+proof. Denote the span of the rows of B by V = B∗Rm ⊆ Rn and notice that dim(V ) = m as B is assumed
+to have rank m. Suppose that Λ is a lattice of rank n that is contained in Zn. Firstly, we claim that the rank
+of the lattice V ⊥ ∩Λ equals n −m. Indeed, we can write V ⊥ ∩Λ = C(Zn ∩C−1V ⊥) where C is an invertible
+matrix with integer entries, i.e., C ∈ Mn(Z) ∩GLn(Q), such that Λ = CZn. Furthermore, V ⊥ = Ker(B), so
+5
+
+the dimension over Q of Qn ∩V ⊥ equals n − m. As C−1 ∈ GLn(Q), it follows that C−1V ⊥ contains n − m
+linearly independent elements of Zn. Secondly, we claim that the orthogonal projection ProjV Λ of Λ
+onto V is a discrete subset of V , and hence is a lattice; its rank will then be dim(V ) = m because we
+are assuming that span(Λ) = Rn, so span(ProjV Λ) = ProjV (span(Λ)) = ProjV (Rn) = V . We need to check
+that for any {x1,x2,...} ⊆ Λ such that limi→∞ProjV xi = 0 there is i0 ∈ N such that ProjV xi = 0 whenever
+i ∈ {i0,i0 +1,...}. Indeed, as V ⊥ = Ker(B) we have Bx = BProjV x for every x ∈ Rn, so limi→∞Bxi = 0. But,
+Bxi ∈ Zm for every i ∈ N because B ∈ Mm×n(Z) and xi ∈ Λ ⊆ Zn. Consequently, there is i0 ∈ N such that
+Bxi = 0 for every i ∈ {i0,i0 +1,...}, i.e., xi ∈ Ker(B) = V ⊥ and hence ProjV xi = 0.
+Let K1 ⊆ V ⊥ be the Voronoi cell of V ��� ∩Λ, namely K1 = {x ∈ V ⊥ : ∀y ∈ V ⊥ ∩Λ,
+∥x∥ℓn
+2 ⩽ ∥x − y∥ℓn
+2 }. If
+y = (y1,..., yn) ∈ V ⊥ = Ker(B), then y1Be1 +··· + ynBen = 0. By the assumption on B, this implies that if
+also y ̸= 0, then |{i ∈ {1,...,n} : yi ̸= 0}| > s. Consequently, as the entries of elements of Λ are integers,
+∀y ∈ (V ⊥ ∩Λ)∖{0},
+∥y∥ℓn
+2 >
+�
+s.
+Hence, if x ∈
+�s
+2 (V ⊥ ∩Bn), then
+∀y ∈ (V ⊥ ∩Λ)∖{0},
+∥x − y∥ℓn
+2 ⩾ ∥y∥ℓn
+2 −∥x∥ℓn
+2 >
+�
+s −
+�s
+2 =
+�s
+2 ⩾ ∥x∥ℓn
+2 .
+This means that K1 ⊇
+�s
+2 (V ⊥ ∩Bn), and therefore by Lemma 3 we have
+voln−m−1(∂K1)
+voln−m(K1)
+⩽ n −m
+1
+2
+�s
+= 2(n −m)
+�s
+.
+(12)
+Next, fix i ∈ {1,...,m}. By the definition of V , the i’th row B∗ei of B belongs to V , so
+∀(x,i) ∈ Rn ×{1,...,m},
+〈x,B∗ei〉 = 〈ProjV x,B∗ei〉.
+(13)
+Since all of the entries of B are integers, it follows that
+∀(x,i) ∈ Zn ×{1,...,m},
+〈BProjV x,ei〉 = 〈ProjV x,B∗ei〉
+(13)
+= 〈x,B∗ei〉 ∈ Z.
+In other words, BProjV Zn ⊆ Zm, and hence the lattice BProjV Λ is a subset of Zm. Furthermore, B is
+injective on V because Ker(B) = V ⊥, so BProjV Zn is a rank m sublattice of Zm. By the definition of Q(m),
+it follows that there exists a BProjV Λ-parallelotope K 0
+2 ⊆ Rm such that
+volm−1(∂K 0
+2 )
+volm(K 0
+2 )
+⩽ Q(m).
+(14)
+Because V ⊥ = Ker(B) and the rank of B is m = dim(V ), the restriction B|V of B to V is an isomorphism
+between V and Rm. Letting T : Rm → V denote the inverse of B|V , define K2 = T K 0
+2. By combining (the
+second part of) Lemma 7 with (14), we see that
+volm−1(∂K2)
+volm(K2)
+⩽Q(m)∥B∥ℓn
+2 →ℓm
+2 .
+(15)
+Let K = K1 +K2 ⊆ Rn. By combining (the first part of) Lemma 7 with (12) and (15), we have
+voln−1(∂K )
+voln(K )
+⩽ 2(n −m)
+�s
++Q(m)∥B∥ℓn
+2 →ℓm
+2 .
+Hence, the proof of Lemma 6 will be complete if we check that K is a Λ-parallelotope. Our construction
+ensures by design that this is so, as K1 is a (V ⊥ ∩Λ)-parallelotope and K2 is a ProjV Λ-parallelotope; veri-
+fying this fact is merely an unravelling of the definitions, which we will next perform for completeness.
+Fix z ∈ Rn. As Rm = BProjV Λ+K 0
+2, there is x ∈ Λ with BProjV z ∈ BProjV x+K 0
+2. Apply T to this inclusion
+and use that TB|V is the identity mapping to get ProjV z ∈ ProjV x +K2. Next, V ⊥ = K1 +V ⊥ ∩Λ since K1
+is the Voronoi cell of V ⊥ ∩Λ, so there is y ∈ V ⊥ ∩Λ such that ProjV ⊥z −ProjV ⊥x ∈ y +K1. Consequently,
+z = ProjV ⊥z +ProjV z ∈ ProjV ⊥x + y +K1 +ProjV x +K2 = x + y +K ∈ Λ+K . Hence, Λ+K = Rn.
+6
+
+It remains to check that for every w ∈ Λ∖{0} the interior of K does not intersect w +K . Indeed, by the
+definition of K , if k belongs to the interior of K , then k = k1+k2, where k1 belongs to the interior of K1 and
+k2 belongs to the interior of K2. Since B is injective on K2 ⊆ V , it follows that Bk2 belongs to the interior
+of BK2 = K 0
+2 . If ProjV w ̸= 0, then BProjV w ∈ BProjV Λ∖ {0}, so because K 0
+2 is a BProjV Λ-parallelotope,
+Bk2 ∉ BProjV w + K 0
+2 . By applying T to is inclusion, we see that k2 ∉ ProjV w + K2, which implies that
+k ∉ w +K . On the other hand, if ProjV w = 0, then w ∈ (V ⊥ ∩Λ)∖{0}. Since K1 is a V ⊥ ∩Λ-parallelotope,
+it follows that k1 ∉ w +K1, so k ∉ w +K .
+□
+To complete the proof of Theorem 5, it remains to prove Lemma 10. For ease of later reference, we first
+record the following straightforward linear-algebraic fact:
+Observation 11. Fix m,n,s ∈ N with s ⩽ m ⩽ n. Suppose that there exists A ∈ Mm×n(Z) such that any s
+of the columns of A are linearly independent. Then, there also exists B ∈ Mm×n(Z) such that any s of the
+columns of B are linearly independent, B has rank m, and
+∥B∥ℓn
+2 →ℓm
+2 ⩽
+�
+1+∥A∥2
+ℓn
+2 →ℓm
+2 .
+(16)
+Proof. Let r ∈ {1,...,m} be the rank of A. By permuting the rows of A, we may assume that its first r rows,
+namely A∗e1,...,A∗er ∈ Rn are linearly independent. Also, since we can complete A∗e1,...,A∗er to a
+basis of Rn by adding n−r vectors from {e1,...,en} ⊆ Rn, by permuting the columns of A, we may assume
+that the vectors A∗e1,...,A∗er,er+1,...,em ∈ Rn are linearly independent. Let B ∈ Mm×n(Z) be the matrix
+whose rows are A∗e1,...,A∗er,er+1,...,em, so that B has rank m by design. Also,
+∀x ∈ Rn,
+∥Bx∥2
+ℓm
+2 =
+r�
+i=1
+(Ax)2
+i +
+m
+�
+j=r+1
+x2
+j ⩽
+�
+∥A∥2
+ℓn
+2 →ℓm
+2 +1
+�
+∥x∥2
+ℓn
+2 .
+Therefore (16) holds. It remains to check that any s of the columns of B are linearly independent. Indeed,
+fix S ⊆ {1,...,n} with |S| = s and {αj }j∈S ⊆ R such that �
+j∈S αjBi j = 0 for every i ∈ {1,...,m}. In particular,
+�
+j∈S αjAi j = 0 for every i ∈ {1,...,r}. If k ∈ {r +1,...,m}, then since the k’th row of A is in the span of the
+first r rows of A, there exist βk1,...,βkr ∈ R such that Ak j = �r
+i=1βkiAi j for every j ∈ {1,...,n}. Conse-
+quently, �
+j∈S αjAk j = �r
+i=1βki
+�
+j∈S αjAi j = 0. This shows that �
+j∈S αjAi j = 0 for every i ∈ {1,...,m}. By
+the assumed property of A, this implies that αj = 0 for every j ∈ S.
+□
+The following lemma is the main existential statement that underlies our justification of Lemma 10:
+Lemma 12. There exists a universal constant c > 0 with the following property. Let d,m,n ⩾ 3 be integers
+that satisfy d ⩽ m ⩽ n and n ⩾ (m logm)/d. Suppose also that s ∈ N satisfies
+s ⩽ c
+d
+�
+md
+n2
+�
+1
+d−2
+.
+(17)
+Then, there exists an m-by-n matrix A ∈ Mm×n({0,1}) with the following properties:
+• Any s of the columns of A are linearly independent over the field Z/(2Z);
+• Every column of A has at most d nonzero entries;
+• Every row of A has at most 5dn/m nonzero entries.
+The ensuing proof of Lemma 12 consists of probabilistic reasoning that is common in the literature
+on Low Density Parity Check (LDPC) codes; it essentially follows the seminal work [18]. While similar
+considerations appeared in many places, we could not locate a reference that states Lemma 12.3 A pecu-
+liarity of the present work is that, for the reason that we have seen in the above deduction of Theorem 5
+from Lemma 6 and Lemma 10, we need to choose a nonstandard dependence of m on n; recall (10).
+3The standard range of parameters that is discussed in the LDPC literature is, using the notation of Lemma 12, either when
+m ≍ n, or when s,d are fixed and the pertinent question becomes how large n can be as m → ∞; sharp bounds in the former
+case are due to [18] and sharp bounds in the latter case are due to [29, 32]. Investigations of these issues when the parameters
+have intermediate asymptotic behaviors appear in [15, 14, 2, 9, 21, 23].
+7
+
+In the course of the proof of Lemma 12 we will use the following probabilistic estimate:
+Lemma 13. Let {W (t) = (W (t,1),...,W (t,m))}∞
+t=0 be the standard random walk on the discrete hypercube
+{0,1}m, starting at the origin. Thus, W (0) = 0 and for each t ∈ N the random vector W (t) is obtained from
+the random vector W (t −1) by choosing an index i ∈ {1,...,m} uniformly at random and setting
+W (t) =
+�
+W (t −1,1),...,W (t −1,i −1),1−W (t −1,i),W (t −1,i +1),...,W (t −1,m)
+�
+.
+Then, Prob[W (t) = 0] ⩽ 2(t/m)t/2 for every t ∈ N.
+Proof. If t is odd, then Prob[W (t) = 0] = 0, so suppose from now that t is even. Let P ∈ M{0,1}m×{0,1}m(R)
+denote the transition matrix of the random walk W , i.e.,
+∀f : {0,1}m → R, ∀x ∈ {0,1}m,
+Pf (x) = 1
+m
+m
+�
+i=1
+f (x +ei mod 2).
+Then, Prob[W (t) = 0] = (Pt)00. By symmetry, all of the 2m diagonal entries of Pt are equal to each other,
+so (Pt)00 = Trace(Pt)/2m. For every S ⊆ {0,1}m, the Walsh function (x ∈ {0,1}m) �→ (−1)
+�
+i∈S xi is an eigen-
+vector of P whose eigenvalue equals 1−2|S|/m. Consequently,
+Prob[W (t) = 0] = 1
+2m Trace(Pt) = 1
+2m
+m
+�
+k=0
+�
+m
+k
+��
+1− 2k
+m
+�t
+.
+(18)
+Suppose that β1,...,βm are independent {0,1}-valued unbiased Bernoulli random variables, namely,
+Prob[βi = 0] = Prob[βi = 1] = 1/2 for any i ∈ {1,...,m}. By Hoeffding’s inequality (e.g., [37, Theorem 2.2.6]),
+∀u ⩾ 0,
+Prob
+�����
+m
+�
+i=1
+�
+βi − 1
+2
+����� ⩾ u
+�
+⩽ 2e− 2u2
+m .
+(19)
+Observing that the right hand side of (18) is equal to the expectation of
+�
+1− 2
+m
+�m
+i=1βi
+�t, we see that
+Prob[W (t) = 0]
+(18)
+=
+�
+− 2
+m
+�t
+E
+�� m
+�
+i=1
+�
+βi − 1
+2
+��t�
+=
+� 2
+m
+�t �∞
+0
+tut−1Prob
+�����
+m
+�
+i=1
+�
+βi − 1
+2
+����� ⩾ u
+�
+du
+(19)
+⩽ 2t
+� 2
+m
+�t �∞
+0
+ut−1e− 2u2
+m du = 2
+� 2
+m
+� t
+2 � t
+2
+�
+! ⩽ 2
+� 2
+m
+� t
+2 � t
+2
+� t
+2
+= 2
+� t
+m
+� t
+2
+.
+□
+With Lemma 13 at hand, we can now prove Lemma 12.
+Proof of Lemma 12. Consider the random matrix A ∈ Mm×n({0,1}) whose columns are independent iden-
+tically distributed copies W1(d),...,Wn(d) of W (d), where W (0) = 0,W (1),W (2),... is the standard ran-
+dom walk on {0,1}m as in Lemma 13. By design, this means that each column of A has at most d nonzero
+entries. Fixing (i, j) ∈ {1,...,m}×{1,...,n}, if Wj(d,i) = 1, then in at least one of the d steps of the random
+walk that generated Wj(d) the ith coordinate was changed. The probability of the latter event equals
+1−(1−1/m)d. Hence, Prob[Wj(d,i) = 1] ⩽ 1−(1−1/m)d ⩽ d/m and therefore for every fixed S ⊆ {1,...,n},
+the probability that Wj(d,i) = 1 for every j ∈ S is at most (d/m)|S|. Consequently, the probability that all
+of the rows of A have at most ℓ = ⌈4dn/m⌉ nonzero entries is at least
+1−m
+�
+n
+ℓ
+�� d
+m
+�ℓ
+⩾ 1−m
+�en
+ℓ
+�ℓ � d
+m
+�ℓ
+= 1−m
+�edn
+mℓ
+�ℓ
+⩾ 1−m
+�e
+4
+�4logm
+⩾ 1
+3,
+where the first step is an application of Stirling’s formula, the penultimate step uses ℓ ⩾ 4dn/m and the
+assumption n ⩾ (m logm)/d, and the final step holds because m ⩾ 3.
+It therefore suffices to prove that with probability greater than 2/3 the vectors {Wi (d)}i∈S ⊆ {0,1}m are
+linearly independent over Z/(2Z) for every ∅ ̸= S ⊆ {1,...,n} with |S| ⩽ s, where s ∈ N satisfies (17) and the
+universal constant c > 0 that appears in (17) will be specified later; see (23). So, it suffices to prove that
+with probability greater than 2/3 we have �
+i∈S Wi(d) ̸≡ 0 mod 2 for every ∅ ̸= S ⊆ {1,...,n} with |S| ⩽ s.
+8
+
+Hence, letting D denote the number of ∅ ̸= S ⊆ {1,...,n} with |S| ⩽ s that satisfy �
+i∈S Wi(d) ≡ 0 mod 2, it
+suffices to prove that 2/3 < Prob[D = 0] = 1−Prob[D ⩾ 1]. Using Markov’s inequality, it follows that the
+proof of Lemma 12 will be complete if we demonstrate that E[D] < 1/3.
+The expectation of D can be computed exactly. Indeed,
+E[D] = E
+�
+�
+S⊆{1,...,n}
+1⩽|S|⩽s
+1{
+�
+i∈S Wi(d)≡0 mod 2}
+�
+=
+s�
+r=1
+�
+n
+r
+�
+Prob[W (dr) = 0],
+(20)
+where we used the fact that �
+i∈S Wi(d) mod 2 ∈ {0,1}m has the same distribution as W (d|S|) for every
+∅ ̸= S ⊆ {1,...,n}. By substituting the conclusion of Lemma 13 into (20) we see that
+E[D] ⩽ 2
+s�
+r=1
+�
+n
+r
+��dr
+m
+� dr
+2
+⩽ 2
+s�
+r=1
+�ed
+d
+2 r
+d
+2 −1n
+m
+d
+2
+�r
+,
+(21)
+where in the last step we bounded the binomial coefficient using Stirling’s formula. For every r ∈ {1,...,s},
+ed
+d
+2 r
+d
+2 −1n
+m
+d
+2
+⩽ ed
+d
+2 s
+d
+2 −1n
+m
+d
+2
+(17)
+⩽ edc
+d
+2 −1 < 1
+7,
+(22)
+provided that
+c < inf
+d⩾3
+� 1
+7ed
+�
+2
+d−2
+∈ (0,1).
+(23)
+Therefore, when (23) holds we may substitute (22) into (21) to get that E[D] < 2�∞
+r=1
+1
+7r = 1
+3.
+□
+We can now prove Lemma 10, thus concluding the proof of Theorem 5.
+Proof of Lemma 10. We will prove the following stronger statement (Lemma 10 is its special case ε = 1).
+If 0 < ε ⩽ 2 and m,n ∈ N satisfy 2 + ⌊2/ε⌋ ⩽ m ⩽ n and n ⩾ (m logm)/(2 + ⌊2/ε⌋), then there exist s ∈ N
+with s ≳ εm1+ε/nε, and B ∈ Mm×n(Z) such that any s of the columns of B are linearly independent, the
+rows of B are linearly independent, and
+∥B∥ℓn
+2 →ℓm
+2 ≲ 1
+ε
+� n
+m .
+Indeed, apply Lemma 12 with d = 2 + ⌊2/ε⌋ ⩾ 3 (equivalently, d ⩾ 3 is the largest integer such that
+2/(d −2) ⩾ ε) to deduce that there exist an integer s with
+s ≍ 1
+d
+�
+md
+n2
+�
+1
+d−2
+= m
+d
+�m
+n
+�
+2
+d−2 ≍ εm
+�m
+n
+�ε
+= εm1+ε
+nε
+,
+and a matrix A ∈ Mm×n({0,1}) ⊆ Mm×n(Z) such that any s of the columns of A are linearly independent
+over Z/(2Z), every column of A has at most d nonzero entries, and every row of A has at most 5dn/m
+nonzero entries. If a set of vectors v1,...,vs ∈ {0,1}m is linearly independent over Z/(2Z), then it is also
+linearly independent over R (e.g., letting V ∈ Mm×s({0,1}) denote the matrix whose columns are v1,...,vs,
+the latter requirement is equivalent to the determinant of V∗V ∈ Ms({0,1}) being an odd integer, so in
+particular it does not vanish). Hence, any s of the columns of A are linearly independent over R. Also,
+∥A∥ℓn
+2 →ℓm
+2 ⩽
+�
+max
+i∈{1,...,m}
+n�
+j=1
+|Ai j|
+� 1
+2 �
+max
+j∈{1,...,n}
+m
+�
+i=1
+|Ai j|
+� 1
+2 ⩽
+�
+5dn
+m ·
+�
+d ≍ 1
+ε
+� n
+m ,
+where the first step is a standard bound which holds for any m-by-n real matrix (e.g. [20, Corollary 2.3.2]).
+Thus, A has all of the properties that we require from the matrix B in Lemma 10, except that we do not
+know that A has rank m, but Observation 11 remedies this (minor) issue.
+□
+We end by asking the following question:
+9
+
+Question 14. Fix n ∈ N. Does there exist an integer parallelotope K ⊆ Rn such that the (n−1)-dimensional
+area of the orthogonal projection Projθ⊥K of K along any direction θ ∈ Sn−1 is at most no(1)?
+An application of Cauchy’s surface area formula (see [27, Section 5.5]), as noted in, e.g., [31, Sec-
+tion 1.6], shows that a positive answer to Question 14 would imply Theorem 1. Correspondingly, a posi-
+tive answer to Question 14 with no(1) replaced by O(1) would imply a positive answer to Question 2.
+Apart from the intrinsic geometric interest of Question 14, if it had a positive answer, then we would
+deduce using [31] that there exists an integer parallelotope K ⊆ Rn such that the normed space X whose
+unit ball is K has certain desirable nonlinear properties, namely, we would obtain an improved random-
+ized clustering of X and an improved extension theorem for Lipschitz functions on subsets of X; we refer
+to [31] for the relevant formulations since including them here would result in a substantial digression.
+REFERENCES
+[1] A. D. Alexandrov, Convex polyhedra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005, Translated from
+the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, With comments and bibliography by V.
+A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov.
+[2] Noga Alon and Uriel Feige, On the power of two, three and four probes, Proceedings of the Twentieth Annual ACM-SIAM
+Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2009, pp. 346–354.
+[3] Noga Alon and Bo’az Klartag, Economical toric spines via Cheeger’s inequality, J. Topol. Anal. 1 (2009), 101–111.
+[4] Tomaso Aste and Denis Weaire, The pursuit of perfect packing, second ed., Taylor & Francis, New York, 2008.
+[5] Károly Bezdek, Sphere packings revisited, European J. Combin. 27 (2006), 864–883.
+[6] Mark Braverman and Dor Minzer, Optimal tiling of the Euclidean space using permutation-symmetric bodies, 36th Com-
+putational Complexity Conference, LIPIcs. Leibniz Int. Proc. Inform., vol. 200, Schloss Dagstuhl. Leibniz-Zent. Inform.,
+Wadern, 2021, pp. Art. No. 5, 48.
+[7] Jaigyoung Choe, On the existence and regularity of fundamental domains with least boundary area, J. Differential Geom.
+29 (1989), 623–663.
+[8] B. Delaunay, Sur la partition régulière de l’espace à 4 dimensions. I, II., Bull. Acad. Sci. URSS 2 (1929), 79–110 (French).
+[9] D. Dellamonica, Jr., P. Haxell, T. Łuczak, D. Mubayi, B. Nagle, Y. Person, V. Rödl, M. Schacht, and J. Verstraëte, On even-degree
+subgraphs of linear hypergraphs, Combin. Probab. Comput. 21 (2012), 113–127.
+[10] N. P. Dolbilin, Properties of faces of parallelohedra, Tr. Mat. Inst. Steklova 266 (2009), 112–126.
+[11] N. P. Dolbilin, Parallelohedra: a retrospective and new results, Trans. Moscow Math. Soc. (2012), 207–220.
+[12] Peter Engel, Geometric crystallography, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993,
+pp. 989–1041.
+[13] E. S. Fedorov, Naˇcala uˇceniya o figurah, Izdat. Akad. Nauk SSSR, Moscow, 1953.
+[14] Uriel Feige, Small linear dependencies for binary vectors of low weight, Building bridges, Bolyai Soc. Math. Stud., vol. 19,
+Springer, Berlin, 2008, pp. 283–307.
+[15] Uriel Feige, Jeong Han Kim, and Eran Ofek, Witnesses for non-satisfiability of dense random 3cnf formulas, 47th Annual
+IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Pro-
+ceedings, IEEE Computer Society, 2006, pp. 497–508.
+[16] Uriel Feige, Guy Kindler, and Ryan O’Donnell, Understanding parallel repetition requires understanding foams, 22nd An-
+nual IEEE Conference on Computational Complexity (CCC 2007), 13-16 June 2007, San Diego, California, USA, IEEE Com-
+puter Society, 2007, pp. 179–192.
+[17] László Fejes Tóth, Über das kürzeste Kurvennetz, das eine Kugeloberfläche in flächengleiche konvexe Teile zerlegt, Math.
+Naturwiss. Anz. Ungar. Akad. Wiss. 62 (1943), 349–354.
+[18] R. G. Gallager, Low-density parity-check codes, IRE Trans. IT-8 (1962), 21–28.
+[19] Apostolos Giannopoulos, Alexander Koldobsky, and Petros Valettas, Inequalities for the surface area of projections of convex
+bodies, Canad. J. Math. 70 (2018), 804–823.
+[20] Gene H. Golub and Charles F. Van Loan, Matrix computations, fourth ed., Johns Hopkins Studies in the Mathematical
+Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.
+[21] Venkatesan Guruswami, Pravesh K. Kothari, and Peter Manohar, Algorithms and certificates for Boolean CSP refutation:
+smoothed is no harder than random, STOC ’22—Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of
+Computing, ACM, New York, [2022] ©2022, pp. 678–689.
+[22] T. C. Hales, The honeycomb conjecture, Discrete Comput. Geom. 25 (2001), 1–22.
+[23] Jun-Ting Hsieh, Pravesh K. Kothari, and Sidhanth Mohanty, A simple and sharper proof of the hypergraph Moore bound,
+Preprint available at https://arxiv.org/abs/2207.10850, 2022.
+[24] Lord Kelvin, On homogeneous division of space., Lond. R. S. Proc. 55 (1894), 1–16 (English).
+10
+
+[25] Guy Kindler, Ryan O’Donnell, Anup Rao, and Avi Wigderson, Spherical cubes and rounding in high dimensions, 49th An-
+nual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, IEEE
+Computer Society, 2008, pp. 189–198.
+[26] Guy Kindler, Anup Rao, Ryan O’Donnell, and Avi Wigderson, Spherical cubes: optimal foams from computational hardness
+amplification, Commun. ACM 55 (2012), 90–97.
+[27] Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge
+University Press, Cambridge, 1997.
+[28] Zsolt Lángi, An isoperimetric problem for three-dimensional parallelohedra, Pacific J. Math. 316 (2022), 169–181.
+[29] Hanno Lefmann, Pavel Pudlák, and Petr Savický, On sparse parity check matrices, Des. Codes Cryptogr. 12 (1997), 107–130.
+[30] H. Minkowski, Allgemeine Lehrsätze über die convexen Polyeder., Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1897 (1897),
+198–219 (German).
+[31] Assaf
+Naor,
+Extension,
+separation
+and
+isomorphic
+reverse
+isoperimetry,
+Preprint
+available
+at
+https://arxiv.org/abs/2112.11523, 2021.
+[32] Assaf Naor and Jacques Verstraëte, Parity check matrices and product representations of squares, Combinatorica 28 (2008),
+163–185.
+[33] Ran Raz, A counterexample to strong parallel repetition, SIAM J. Comput. 40 (2011), 771–777.
+[34] C. A. Rogers, A note on coverings and packings, J. London Math. Soc. 25 (1950), 327–331.
+[35] M. I. Shtogrin, Regular Dirichlet-Vorono˘ı partitions for the second triclinic group, Izdat. “Nauka”, Moscow, 1973, Trudy Mat.
+Inst. Steklov. 123 (1973).
+[36] B. A. Venkov, On a class of Euclidean polyhedra, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (1954), 11–31.
+[37] Roman Vershynin, High-dimensional probability, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 47,
+Cambridge University Press, Cambridge, 2018, An introduction with applications in data science, With a foreword by Sara
+van de Geer.
+MATHEMATICS DEPARTMENT, PRINCETON UNIVERSITY, FINE HALL, WASHINGTON ROAD, PRINCETON, NJ 08544-1000, USA
+Email address: [email protected]
+DEPARTMENT OF COMPUTER SCIENCE, COURANT INSTITUTE OF MATHEMATICAL SCIENCES, NEW YORK UNIVERSITY, 251
+MERCER STREET, NEW YORK, NY 10012, USA
+Email address: [email protected]
+11
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